Jean-Marie Berthelot

Dynamics of Composite Materials and Structures

ISMANS Institute for Advanced Le Mans, France Materials and Mechanics

Jean-Marie Berthelot

Dynamics of Composite Materials

and Structures Jean-Marie Berthelot is an Emeritus Professor at the Institute for Advanced Materials and Mechanics (ISMANS), Le Mans, France. His current research is on the mechanical behaviour of composite materials and structures. He has published extensively in the area of composite materials and is the author of a textbook entitled Composite Materials, Mechanical Behavior and Structural Analysis published by Springer, New York, in 1999.

Jean-Marie Berthelot

Dynamics of Composite Materials and Structures

ISMANS Institute for Advanced Le Mans, France Materials and Mechanics

Preface

The objective of this textbook is to develop the fundamental concepts needed for the dynamic analysis of composite materials and composite structures. The book has been established for the undergraduate upper-levels and graduate level in Mechanical Engineering.

The basic elements needed to investigate the dynamic behaviour of laminate and sandwich materials and structures are developed in References 1 and 2 of this textbook. Chapter 1 summarizes these basic elements. First the chapter gives ele-ments on the constituents and the architecture of composite materials, and then introduces the mechanical behaviour of composite materials. Next the chapter considers the fundamental theories for modelling the mechanical behaviour of laminate and sandwich materials.

Chapter 2 treats the dynamics of systems with one degree of freedom. The topics considered are of general interest and practical usefulness for analyzing the dynamic behaviour of structures using mode superposition.

Chapter 3 develops the one-dimensional flexural vibrations of laminate or sandwich plates. This type of vibrations concerns the bending vibrations of beams and the vibrations of plates under cylindrical bending. The chapter considers the natural modes of beam vibrations and their properties. Beam vibrations with low damping are then analysed, introducing the modal coordinates. The effects of transverse shear are analysed in the case of cylindrical vibrations.

Chapter 4 analyses the flexural vibrations of rectangular laminate plates, with damping neglected. Analytical solutions for the free vibrations of plates are deri-ved in a few particular cases. Then, the chapter develops the Ritz method to obtain approximate solutions for the natural modes of rectangular plates with various conditions along the edges. The analysis of the dynamic behaviour of composite structure needs to evaluate the bending stiffnesses of composite materials. Chapter 4 shows how the stiffnesses can be estimated from the experimental analysis of rectangular plate vibrations. The problems and the difficulties of the evaluation are extensively discussed.

In fact, it is necessary to consider the damping of materials in the investigation of the dynamic behaviour of composite structures. Damping in composite mate-rials is considered in Chapter 5, first, as a function of the constituents. Then, bending vibrations of damped laminate beams are studied using viscous friction model and complex stiffness. Damping properties of orthotropic beams and plates are analyzed as a function of material orientation.

An extended experimental investigation of damping of composites is deve-loped in the case of unidirectional glass and Kevlar fibre composites in Chapter 6. A procedure for measuring laminate damping from the bending vibrations of beams is presented. Next, discussion is developed on the experimental results obtained. Temperature effect is also considered.

Inserting viscoelastic layers in composite laminates improves significantly the damped dynamic properties of the laminates. Chapter 7 develops an extended analysis in the case of rectangular plates which allows us to investigate the pro-cesses induced by interleaving viscoelastic layers.

vi Preface

The dynamic analysis of complex structures needs to use the finite element method. Chapter 8 provides the basic concepts of the finite element formulation applied to the dynamic analysis of composite structures. The formulation has been developed so as to make the chapter self-contained.

Finite element formulation derives the numerical equations of the dynamic behaviour of composite structures. Chapter 9 analyzes the numerical procedures used to solve these equations. Direct integration and mode superposition are presented. Lastly, the evaluation of the damping of composite structures is considered.

Chapter 10 develops an extensive analysis of the damping of sandwich mate-rials. First, modelling of sandwich materials and structures is implemented. Experimental investigation is next carried out in the case of sandwich materials constituted of a foam core and laminate skins.

The purpose of Chapter 11 is to report a general formulation of the different concepts introduced in the present textbook by applying these concepts to the analysis of the damping of different laminates and sandwich materials and to the analysis of the vibrations of a simple shape structure.

Le Mans, January 2010 Jean-Marie Berthelot

Contents

Preface v

Chapter 1 Basic Elements on Laminate and Sandwich Composite Materials 1

1.1 Constituents and Architecture of Composite Materials . . . . . . . . . . . . 1 1.1.1 Constituents of Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Laminate Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.3 Sandwich Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Elastic Behaviour of Composite Materials . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Unidirectional Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Orthotropic Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.3 Stress-Strain Relations for Off-Axis Layers . . . . . . . . . . . . . . . . . . . . . . 6 1.2.4 Plane Stress State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.4.1 Two-Dimensional Stress State . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 7 1.2.4.2 Elasticity Equations for Plane Stress . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.4.3 Elasticity Equations in Material Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.4.4 Off-Axis Reduced Stiffness Constants . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Basics of Laminate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 9 1.3.2 Displacement Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 Resultants and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.3.1 In-Plane Resultants . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 10 1.3.3.2 Transverse Shear Resultants . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.3.3 Resultant Moments . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11 1.3.4 Fundamental Equations for Plates in the case of First-Order Theory . . 12 1.4 Classical Laminate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.1 Assumptions of the Classical Theory of Laminates . . . . . . . . . . . . . . . . 15 1.4.2 Strain Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4.3 Stress Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.3.1 General Expression . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 17 1.4.3.2 Stress Field in the Case of the Classical Laminate Theory . . . . . . . . . . . . . . . . . . 18 1.4.4 Resultants and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.5 Constitutive Equation of a Laminate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.4.6 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4.7 Boundary Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.4.7.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 22 1.4.7.2 Simply Supported Edge . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 22 1.4.7.3 Clamped Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 22 1.4.7.4 Free Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 23 1.4.8 Energy Formulation of the Classical Laminate Theory . . . . . . . . . . . . . 24 1.4.8.1 Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 24 1.4.8.2 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 25

1.5 Laminate Theory Including the Transverse Shear Effects . . . . . . . . . 25 1.5.1 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.5.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.5.4 Introduction of Transverse Shear Coefficients . . . . . . . . . . . . . . . . . . . . 29

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1.6. Theory of Sandwich Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.6.2 Assumptions for the Sandwich Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.6.3 Displacement Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.6.4 Strain Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6.5 Stress Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.6.6 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.6.7 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Chapter 2 Dynamics of Systems with One Degree of Freedom 36

2.1 Equation of Motion of a System with One Degree of Freedom . . . . . 36 2.2 Undamped Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2.2 Free Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2.3 Forced Vibrations. Steady State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2.3.1 Case of a Harmonic Disturbing Force. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2.3.2 Case of a Harmonic Displacement of the Spring End . . . . . . . . . . . . . . . . . . . . 40 2.3 Vibrations with Viscous Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2.2 Equation of Motion with Viscous Damping . . . . . . . . . . . . . . . . . . . . . . 42 2.3.3 Free Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.3.3.1 Characteristic Equation . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 43 2.3.3.2 Case of Low Damping . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 43 2.3.3.3 Case of High Damping . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 46 2.3.3.4 Critical Damping . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 47 2.3.4 Vibrations in the case of Harmonic Disturbing Force . . . . . . . . . . . . . . 48 2.3.4.1 Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 48 2.3.4.2 Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 50 2.3.4.3 Effect of the Frequency of the Disturbing Force . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.3.4.4 Damping Modelling using Complex Stiffness .. . . . . . . . . . . . . . . . . . . . . . . . . 53 2.3.5 Vibrations in the case of Periodic Disturbing Force . . . . . . . . . . . . . . . . 55 2.3.6 Vibrations in the case of Arbitrary Disturbing Force . . . . . . . . . . . . . . . 56 2.4 Equivalent Viscous Damping Capacity . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.4.2 Energy Dissipated in the case of Viscous Damping . . . . . . . . . . . . . . . . 57 2.4.3 Loss Factor and Specific Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.4.4 Structural Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter 3 Beam Bending and Cylindrical Bending Vibrations of Undamped Laminate and Sandwich Materials 61

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Equation of Motion of Symmetric Laminate Beams . . . . . . . . . . . . . 62 3.3 Natural Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3.1 Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3.2 Properties of the Mode Shapes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4 Natural Modes of Beams with Different End Conditions . . . . . . . . . 66 3.4.1 Simply Supported Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4.2 Clamped Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4.3 Beam Clamped at One End and Simply Supported at the Other . . . . . 70 3.4.4 Beam Clamped at One End and Free at the Other . . . . . . . . . . . . . . . . 72

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3.4.5 Beam with Two Free Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.5 Normal Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.5.1 Motion Equation in Normal Coordinates . . . . . . . . . . . . . . . . . . . . . . . 75 3.5.2 Response to Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.5.2.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 76 3.5.2.2 Beam with Simply Supported Ends . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 77 3.5.2.3 Beam with Other End Conditions . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 78 3.5.3 Forced Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.5.3.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 79 3.5.3.2 Beam with Simply Supported Ends . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 79 3.6 Cylindrical Bending Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.6.2 Classical Laminate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.6.2.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 81 3.6.2.2 Plate Simply Supported . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 82 3.6.2.3 Plates with Other End Conditions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 83 3.6.3 Effect of Transverse Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.6.4 Cylindrical Vibrations of Sandwich Plates . . . . . . . . . . . . . . . . . . . . . . 86

Chapter 4 Flexural Vibrations of Undamped Rectangular Laminate Plates 88

4.1 Free Vibrations of Rectangular Orthotropic Plates Simply Supported . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.2 Vibrations of Orthotropic Plates with Various Conditions along the

Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 90 4.2.1 General Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.2.2 Rayleigh’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2.3 Two-Term Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2.4 Orthotropic Plates with Simply Supported or Clamped Edges . . . . . . 96 4.3 Vibrations of Symmetric Laminate Plates . . . . . . . . . . . . . . . . . . . . . . 100 4.3.1 General Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.3.2 Symmetric Plates with Clamped or Free Edges . . . . . . . . . . . . . . . . . . 101 4.4 Vibrations of Non-symmetric Laminate Plates . . . . . . . . . . . . . . . . . . 104 4.4.1 Plate Constituted of an Antisymmetric Cross-Ply Laminate . . . . . . . . 104 4.4.2 Plate Constituted of an Angle-Ply Laminate . . . . . . . . . . . . . . . . . . . . . 107 4.5 Evaluation of the Laminate Bending Stiffnesses by Analysis of Plate Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.5.2 Experimental Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.5.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 111 4.5.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 112 4.5.2.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 113 4.5.3 Introduction to the Experimental Modal Analysis of Orthotropic Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.5.3.1 Evaluation of the Natural Frequencies . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 114 4.5.3.2 Different Results . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 115 4.5.4 Experimental Results and Discussion in the case of Orthotropic Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 119 4.5.4.1 Values of the Natural Frequencies . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 119 4.5.4.2 Determination of the Stiffnesses by an Iterative Procedure . . . . . . . . . . . . . . . . 120 4.5.4.3 Evaluation of the Stiffnesses from Rayleigh’s Approximation . . . . . . . . . . . . . 124 4.5.4.4 Discussion of the Results and Conclusions .. . . . . . . . . . . . . . . . . . . . . . . . . . 128

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4.5.5 Case of Symmetric Laminates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.5.5.1 Materials and Results . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 130 4.5.2.2 Evaluation of the Bending Stiffnesses from the Natural Frequencies and Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 132

Chapter 5 Damping in Composite Materials 136

5.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.2 Damping in a Unidirectional Composite as a Function of the

Constituents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.3. Bending Vibrations of Damped Laminate Beams . . . . . . . . . . . . . . . 140 5.3.1 Damping Modelling using Viscous Friction . . . . . . . . . . . . . . . . . . . . . 140 5.3.2 Motion Equation in Normal Coordinates . . . . . . . . . . . . . . . . . . . . . . . 140 5.3.3 Forced Harmonic Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.3.4 Damping Modelling using Complex Stiffness . . . . . . . . . . . . . . . . . . . 142 5.3.5 Beam Response to a Concentrated Loading . . . . . . . . . . . . . . . . . . . . . 143 5.4 Evaluation of the Damping Properties of Orthotropic Beams as

Functions of Material Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.4.1 Energy Analysis of Beam Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 144 5.4.1.2 Adams-Bacon Approach . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 144 5.4.1.3 Ni-Adams Analysis . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 147 5.4.1.4 General Formulation of Damping . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 150 5.4.2 Complex Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.5 Evaluation of the Plate Damping as a Function of Material

Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.5.1 Orthotropic Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.5.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 151 5.5.1.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 155 5.5.2 Laminated Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Chapter 6 Experimental Investigation and Discussion on the Damping Properties of Laminates 158

6.1 Experimental Investigation in Literature . . . . . . . . . . . . . . . . . . . . . . 158 6.1.1 Experimental Processes for Evaluating Damping . . . . . . . . . . . . . . . . 158 6.1.2 Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.2 Damping Analysis of Unidirectional Glass and Kevlar Fibre Composites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.2.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.2.3 Experimental Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.2.4 Analysis of the Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.2.4.1 Determination of the Constitutive Damping Parameters . . . . . . . . . . . . . . . . . . 162 6.2.4.2 Plate Damping Measurement . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 163 6.2.5 Choice of the Frequency Range for the Experimental Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.2.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.2.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 165 6.2.6.2 Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 166 6.2.6.3 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 168

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6.3 Comparison of Experimental Results and Models for Unidirectional Beam Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.3.1 Models of Adams-Bacon and Ni-Addams . . . . . . . . . . . . . . . . . . . . . . 170 6.3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 170 6.3.1.2 Glass Fibre Composites . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 170 6.3.1.3 Kevlar Fibre Composites . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 170 6.3.2 Complex Stiffness Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.3.3 Using the Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.3.3.1 Damping Parameters . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 173 6.3.3.2 Influence of the Width of the Beams . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.3.3.3 Damping According to Modes of Beam Vibrations . . . . . . . . . . . . . . . . . . . . . 175 6.4 Damping of Laminated Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.5 Damping of Laminated Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.5.1 Damping Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.5.2 Plates with One Edge Clamped and the Other Edges Free . . . . . . . . . 184 6.5.3 Plates with Two Edges Clamped and the Other Edges Free . . . . . . . . 186 6.6 Longitudinal and Transverse Damping of Unidirectional Fibre Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.6.2 Longitudinal Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.6.3 Transverse Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.6.3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 190 6.6.3.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 192 6.7 Temperature Effect on the Damping Properties of Unidirectional Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.7.2 Materials and Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.7.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 6.7.3.1 Matrix properties . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 196 6.7.3.2 Composite Properties . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 197 6.7.3.3 Damping Evaluation Based on the Ritz Method . . . . . . . . . . . . . . . . . . . . . . 202

Chapter 7 Damping Analysis of Laminates with Interleaved Viscoelastic Layers 203

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 7.2 Damping Modelling of Orthotropic Laminates with Interleaved Viscoelastic Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 7.2.1 Laminate Configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 7.2.2 In-Plane Damping with Interleaved Viscoelastic Layers . . . . . . . . . . . 204 7.2.2.1 Case of a Single Interlaminar Viscoelastic Layer . . . . . . . . . . . . . . . . . . . . . . . 204 7.2.2.2 Case of Two Interlaminar Viscoelastic Layers . . . . . . . . . . . . . . . . . . . . . . . . . 207 7.2.3 Considering the Transverse Shear Effects in the Case of a Single

Interlaminar Viscoelastic Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 7.2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 208 7.2.3.2 Transverse Shear Stresses in Layers . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 208 7.2.3.3 Strain Energy Stored in xz-Transverse Shear . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.2.3.4 Strain Energy Stored in yz-Transverse Shear . . . . . . . . . . . . . . . . . . . . . . . . . . 215 7.2.3.5 Laminate Damping with a Single Interleaved Viscoelastic Layer Including the Transverse Shear Effects . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 215 7.2.4 Considering the Transverse Shear Effects in the case of Two Interleaved

Viscoelastic Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.2.4.1 Case of Symmetric Laminate . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 216 7.2.4.2 Transverse Shear Stresses in the (x, z) plane . . . . . . . . . . . . . . . . . . . . . . . . . . 217

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7.2.4.3 Transverse Shear Energies in the (x, z) plane . . . . . . . . . . . . . . . . . . . . . . . . . . 217 7.2.4.4 Transverse Shear Energies in the (y, z) plane . . . . . . . . . . . . . . . . . . . . . . . . . . 219 7.2.4.5 Laminate Damping with Two Interleaved Viscoelastic Layer Including the Transverse Shear Effects. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 219 7.2.5 Application to Angle-Ply Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 7.2.6 Laminates with External Viscoelastic Layers . . . . . . . . . . . . . . . . . . . . 221 7.2.7 Choice of the Basis Functions of the Ritz Method . . . . . . . . . . . . . . . 222 7.3 Experimental Investigation of Damping of Unidirectional Composites with Interleaved Viscoelastic Layers . . . . . . . . . . . . . . . . 223 7.3.1 Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.3.2 Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.3.3 Analysis of the Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 228 7.3.3.1 Dynamic Properties of the Viscoelastic Layers . . . . . . . . . . . . . . . . . . . . . . . . 228 7.3.3.2 Damping of the Glass Fibre Laminates with Interleaved Viscoelastic Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 229 7.4 Analysis of the Experimental Results Obtained in the case of Angle-Ply Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Chapter 8 Finite Element Method in the Dynamic Analysis of Composite Structures 242

8.1 Principle of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 8.2 Formulation of Structural Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 8.2.1 Isoparametric Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . 243 8.2.2 Example of a Four-Node Finite Element . . . . . . . . . . . . . . . . . . . . . . . . 244 8.2.2.1 Interpolation Functions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 244 8.2.2.2 Strain Formulation . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 245 8.3 Laminate Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 8.3.1 Displacement Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 8.3.2 In-Plane Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 8.3.3 Flexural Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 8.3.4 Transverse Shear Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 8.3.5 Stress Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 8.3.6 Energy Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 8.3.6.1 Strain Energy and Element Stiffness . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 252 8.3.6.2 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 253 8.3.6.3 Work of the External Loads . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 254 8.4 Finite Element Dynamic Equation of Laminate Structure . . . . . . . . 255

Chapter 9 Solution of Dynamic Equation in Finite Element Analysis 257

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.2 Direct Integration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.2.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.2.2 The Central Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 9.2.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

9.2.2.2 Characteristics of the Central Difference Method . . . . . . . . . . . . . . . . . . . . . 259

9.2.3 The Houbolt Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 9.2.4 The Wilson θ Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

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9.2.5 The Newmark Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 9.3 Mode Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 9.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 9.3.2 Dynamic Equation in Modal Coordinates . . . . . . . . . . . . . . . . . . . . . . . 267 9.3.2.1 Modal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

9.3.2.2 Motion Equation in Modal Coordinates . . . . . . . . . . . . . . . . . . . . . . . 268

9.3.3 Modal Analysis with Damping Neglected . . . . . . . . . . . . . . . . . . . . . . 268 9.3.4 Modal Analysis with Damping Included . . . . . . . . . . . . . . . . . . . . . . . . 269 9.4 Evaluation of Structure Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 9.4.1 Modal Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 9.4.2 Damping Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 9.5 Finite Element Nonlinear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

Chapter 10 Damping of Sandwich Materials and Structures 274

10.1 Modelling the Dampig of Sandwich Composite Materials and Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 10.1.1 Stress Field in Sandwich Composite Materials . . . . . . . . . . . . . . . . . . 274 10.1.2 In-Plane Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 10.1.3 Transverse Shear Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 10.1.2 Damping of a Sandwich Composite Structure . . . . . . . . . . . . . . . . . . . 282 10.2 Experimental Investigation of the Damping of Sandwich Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 10.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 10.2.2 Determination of the Constitutive Damping Parameters . . . . . . . . . . . 285 10.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 10.3.1 Determination of the Dynamic Characteristics of the Foams . . . . . . . 283 10.3.1.1 Test Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

10.3.1.2 Energies Stored in the Test Specimens and Procedure . . . . . . . . . . . . . . . . . 287

10.3.1.3 Dynamic Characteristics of the Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

10.3.2 Analysis of the Damping of Sandwich Materials . . . . . . . . . . . . . . . . . 291 10.3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

10.3.2.2 Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

10.3.2.3 Damping of the Sandwich Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

10.4 Characteristic Factors of the Damping of Sandwich Materials . . . . 294 10.4.1 Influence of the Shear Modulus of the Foam Core . . . . . . . . . . . . . . . . 294 10.4.2 Energies Dissipated in the Core and Skins . . . . . . . . . . . . . . . . . . . . . . 295 10.4.3 Effect of the Core Thickness on the Damping of Sandwich Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 10.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

Chapter 11 General Formulation of Damping of Composite Materials and Structures 301

11.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 11.2 Modelling Damping of Laminate Beams and Rectangular Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 11.3 Damping Modelling using Finite Element Analysis . . . . . . . . . . . . . 303 11.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 11.3.2 Stress Field in Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

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11.3.3 In-Plane Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 11.3.4 Transverse Shear Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 11.3.5 Damping of a Composite Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 11.3.6 Procedure for Evaluating the Damping of Composite Structure . . . . . 311 11.4 Investigation of the Damping of Composite Materials . . . . . . . . . . . 312 11.4.1 Determination of the Constitutive Damping Parameters . . . . . . . . . . . 312 11.4.2 Damping of the Glass Fibre Laminates . . . . . . . . . . . . . . . . . . . . . . . . . 313 11.4.3 Damping Comparison between Taffeta Laminates, Serge Laminates and Cross-Ply Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 11.4.4 Damping of the Unidirectional Glass Fibre Laminates with Interleaved Viscoelastic Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 11.4.5 Damping of the Sandwich Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 11.5 Dynamic Response of a Composite Structure . . . . . . . . . . . . . . . . . . 322 11.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

References 329

CHAPTER 1

Basic Elements on Laminate and Sandwich Composite Materials

1.1 CONSTITUENTS AND ARCHITECTURE OF COMPOSITE MATERIALS

1.1.1 Constituents of Composite Materials

A composite materials is constituted of two or more materials of different natures in such a way to obtain characteristics suited to a given applications. High mechanical performance composites are made of continuous fibre incorporated in a polymer matrix. The function of the matrix is to link the fibres together, to protect the fibres from the external environment and to transfer the external mechanical loading to the fibres.

Fibres confer upon composite materials their high mechanical characteristics as stiffness and strength. Fibres are elaborated with a diameter of a few microns (about 10 µm) and are gathered into strands, rovings or yarns which can be used to make surface tissues of various types. The principal fibres of composite mate-rials are glass fibres, carbon fibres and Kevlar fibres.

Finally, fibre composite materials lead to materials which have high mecha-nical properties associated to a low density.

The manufacturing processes used for moulding composite structures are based on the lamination of successive layers in the form of plates and shells. So, this fact justifies the analysis of composite materials in the form of plates of one or several layers. Next, shells can be modelled as a set of plates and their analysis deduced from the plate analysis. Furthermore, the theories of laminate and sandwich plates are the basic concepts which are used to elaborate finite elements for investigating the mechanical behaviour of laminate and sandwich structures.

1.1.2 Laminate Composite Materials

Laminates are made of successive layers (Figure 1.1) of fibre reinforcements in the forms of strands, rovings, cloths, etc., impregnated with matrix.

2 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

S

FIGURE 1.1. Constitution of a laminate.

Laminates with unidirectional strands or cloths constitute a basic laminate to which, in theory, every other type of laminate can be reduced. Several unidi-rectional layers can be stacked in a specified sequence of different orientations to obtain a laminate that will fit the mechanical properties required. The identi-fication of a laminate is obtained by reporting the successive layers with fibre orientation, from the lower face to the upper face. For example: [30/902/45/0/–45].

1.1.3 Sandwich Composites

The principle of sandwich construction consists (Figure 1.2) in coating to a core, made of light material or structure having good properties under com-pression, on both sides with two skins having good flexural properties. The objective is to obtain high flexural properties associated to lightness of the material. Generally, solid cores or hollow cores as honeycombs are used with laminated skins.

FIGURE 1.2. Sandwich material.

core

laminates

layers

laminate

1.2. Elastic Behaviour of Composite Materials 3

1.2 ELASTIC BEHAVIOUR OF COMPOSITE MATERIALS

1.2.1 Unidirectional Composite Materials

A unidirectional composite is constituted of parallel fibres arranged in a matrix (Figure 1.3a). An elementary cell can be considered, to a first approximation, as constituted of a fibre embedded in a matrix cylinder (Figure 1.3b). This cell has a revolution axis parallel to the fibres. This direction is called the longitudinal direction and denoted as axis 1 or L-axis. Every direction normal to the fibres is called a transverse direction and the composite is considered as transversely isotropic. The transverse plane will be described by the direction 2 and 3, also denoted by T and ,T ′ respectively. These directions are equivalent.

The elastic behaviour of a unidirectional composite is described by one of the two matrix forms:

( )

11 12 121 1

12 22 232 2

12 23 223 31

22 234 42

5 566

6 666

0 0 00 0 00 0 0

0 0 0 0 0

0 0 0 0 00 0 0 0 0

C C CC C CC C C

C C

CC

σ εσ εσ εσ εσ εσ ε

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥

= ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

, (1.1)

or

FIGURE 1.3. Unidirectional composite.

1, L

3, T ′

2, T

1, L

(a) (b)

4 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

( )

1 11 12 12 1

2 12 22 23 2

3 12 23 22 3

4 22 23 4

5 66 5

6 66 6

0 0 00 0 00 0 0

0 0 0 2 0 00 0 0 0 00 0 0 0 0

S S SS S SS S S

S SS

S

ε σε σε σε σε σε σ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

. (1.2)

These laws introduce the stiffness constants Cij and Sij. Relations (1.1) and (1.2) are written using the engineering notation for the stresses (Chapter 5 of Ref. 1) and for the strains (Chapter 6 of Ref. 1).

The stiffness and compliance matrices are inverses of each other and are cha-racterized by five independent constants: C11, C12, C22, C23, C66 or S11, S12, S22, S23, S66. These constants can be expressed (Chapter 9 of Ref. 1) as functions of five engineering constants, for example: EL and LTν , the Young’s modulus and the Poisson ratio measured in a longitudinal test; ET, the Young’s modulus measured in a transverse test; GLT and ,TTG ′ the shear moduli measured respectively in longitudinal and transverse shear tests.

1.2.2 Orthotropic Composite Materials

Laminates are constituted of layers of unidirectional fibre composites or of woven fabric composites. Usually the woven fabrics are made of unidirectional strands or rovings interlaced at 90°, one in the warp direction and the other in the weft (or fill) direction (Figure 1.4). These layers have three mutually orthogonal symmetry planes and so have the elastic behaviour of an orthotropic material. The

FIGURE 1.4. Layer of an orthotropic composite material.

3, T ′

2, T

1, L

fill direction

warp direction

1.2. Elastic Behaviour of Composite Materials 5

material directions (1, 2) will be respectively taken in the warp and fill directions and denoted as L and T directions (Figure 1.4). The direction 3 orthogonal to the plane (L, T) will be denoted .T ′

Hooke’s law of the elastic behaviour of an orthotropic layer is written in one of the matrix forms:

1 11 12 13 1

2 12 22 23 2

3 13 23 33 3

4 44 4

5 55 5

6 66 6

0 0 00 0 00 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

C C CC C CC C C

CC

C

σ εσ εσ εσ εσ εσ ε

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

, (1.3)

or

1 11 12 13 1

2 12 22 23 2

3 13 23 33 3

4 44 4

5 55 5

6 66 6

0 0 00 0 00 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

S S SS S SS S S

SS

S

ε σε σε σε σε σε σ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

, (1.4)

introducing the stiffness constants Cij and the compliance constants Sij. The elastic behaviour is characterized by nine independent coefficients. These constants can be expressed (Chapter 10 of Ref. 1) as functions of nine engineering constants: EL, LTν and ,LTν ′ the Young’s modulus and Poisson ratios measured in a longi-tudinal test; ET and ,TTν ′ the Young’s modulus and Poisson ratio measured in a transverse test in T direction; ,TE ′ the Young’s modulus measured in T ′ direction; GLT, LTG ′ and TTG ′ , the shear moduli measured respectively in planes (L, T), (L,

)T ′ and (T, ).T ′ For example the compliance matrix is:

1 0 0 0

1 0 0 0

1 0 0 0

10 0 0 0 0

10 0 0 0 0

10 0 0 0 0

LT LT

L L LLT TT

L T TLT TT

L T T

TT

LT

LT

E E E

E E E

E E ES

G

G

G

ν ν

ν ν

ν ν

′

′

′ ′

′

′

′

⎡ ⎤− −⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎢ ⎥− −⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

. (1.5)

The stiffness matrix is inverse of the compliance matrix.

6 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

1.2.3 Stress-Strain Relations for Off-Axis Layers

Laminates are constituted of different layers with different fibre or warp orient-tations. So, a unidirectional layer or a cloth reinforced layer will be characterized by its material directions (1, 2, 3) or (L, T, ),T ′ the plane (1, 2) or (L, T) being the plane of the layer and the direction 1 or T identified with the fibre direction or warp direction (Figure 1.5). The reference directions of laminate are ( )1 , 2 , 3′ ′ or (x, y, z), the fibre direction or the warp direction making an angle θ with the direction 1′ or x.

The stress-strain relations referred to the laminate directions (x, y, z) may be written in one of the two forms:

11 12 13 16

12 22 23 26

13 23 33 36

44 45

45 55

16 26 36 66

0 00 00 0

0 0 0 00 0 0 0

0 0

xx xx

yy yy

zz zz

yz yz

xz xz

xy xy

C C C CC C C CC C C C

C CC C

C C C C

σ εσ εσ εσ γσ γσ γ

′ ′ ′ ′⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥′ ′ ′ ′⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥′ ′ ′ ′⎢ ⎥

=⎢ ⎥ ⎢ ⎥⎢ ⎥′ ′⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥′ ′⎢ ⎥ ⎢ ⎥⎢ ⎥

′ ′ ′ ′⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

, (1.6)

or

11 12 13 16

12 22 23 26

13 23 33 36

44 45

45 55

16 26 36 66

0 00 00 0

0 0 0 00 0 0 0

0 0

xx xx

yy yy

zz zz

yz yz

xz xz

xy xy

S S S SS S S SS S S S

S SS S

S S S S

ε σε σε σγ σγ σγ σ

′ ′ ′ ′⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥′ ′ ′ ′⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥′ ′ ′ ′⎢ ⎥

=⎢ ⎥ ⎢ ⎥⎢ ⎥′ ′⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥′ ′⎢ ⎥ ⎢ ⎥⎢ ⎥

′ ′ ′ ′⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

. (1.7)

The expressions of ijC′ and ijS ′ are given in Chapter 11 of Ref. 1. These expressions show that the equations for a unidirectional layer or an orthotropic layer are identical for the terms ijC′ or ijS ′ with ij = 11, 12, 16, 22, 26, 66.

FIGURE 1.5. Material directions (1, 2, 3) of a layer and reference directions ( )1 , 2 , 3′ ′ or (x, y, z) of the laminate.

3, z

1

2

1 , x′

2 , x′

θ

1.2. Elastic Behaviour of Composite Materials 7

1.2.4 Plane Stress State

1.2.4.1 Two-dimensional Stress State

The elastic behaviour considered in the previous sections can be applied to solve any elasticity problem for a composite structure. In the case of a laminated structure the elasticity problem can be restricted to a two-dimensional stress state characterized by a stress tensor of the form:

0

( ) 00 0 0

xx xy

xy yyMσ σσ σ⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

′σ , (1.8)

at each point M of the material. The z direction is the principal direction with a zero eigen value.

1.2.4.2 Elasticity Equations for Plane Stress

It is shown (Chapter 11 of Ref. 1) that in the case of a plane stress:

0 if 1, 2, 6

0 if 3, 4, 5,i

i

i

i

σ

σ

′ ≠ =⎧⎪⎨

′ = =⎪⎩ and 0 if 1, 2, 3, 6

0 if 4, 5.i

i

i

i

ε

ε

′ ≠ =⎧⎪⎨

′ = =⎪⎩ (1.9)

The elasticity equations can be written in one of the forms:

1 11 12 16 1

2 12 22 26 2

6 16 26 66 6

S S SS S SS S S

ε σε σε σ

′ ′ ′ ′ ′⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥′ ′ ′ ′ ′=⎢ ⎥ ⎢ ⎥ ⎢ ⎥′ ′ ′ ′ ′⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

, (1.10)

with 3 13 1 23 2 36 6S S Sε σ σ σ′ ′ ′ ′ ′ ′ ′= + + , (1.11) or

1 11 12 16 1

2 12 22 26 2

6 16 26 66 6

Q Q QQ Q QQ Q Q

σ εσ εσ ε

′ ′ ′ ′ ′⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥′ ′ ′ ′ ′=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

′ ′ ′ ′ ′⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

, (1.12)

with

( )3 13 1 23 2 36 633

1 C C CC

ε ε ε ε′ ′ ′ ′ ′ ′ ′= − + +′

. (1.13)

The constants ijQ′ are the reduced stiffness constants which are expressed as func-tions of the stiffness constants by:

3 3

33, , 1, 2, 6,

.

i jij ij

ji ij

C CQ C i j

C

Q Q

′ ′′ ′= − =

′

′ ′=

(1.14)

8 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

The matrices ijS ′⎡ ⎤⎣ ⎦ and ijQ′⎡ ⎤⎣ ⎦ are inverses of each other.

1.2.4.3 Elasticity Equations in Material Directions

When referred to the material directions, a plane stress state is characterized by:

0 if 1, 2, 6

0 if 3, 4, 5,i

i

i

i

σ

σ

≠ =⎧⎪⎨

= =⎪⎩ and 0 if 1, 2, 3, 6

0 if 4, 5.i

i

i

i

ε

ε

≠ =⎧⎪⎨

= =⎪⎩ (1.15)

The elasticity equations can be written as:

1 11 12 1

2 12 22 2

6 66 6

00

0 0

S SS S

S

ε σε σε σ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

, (1.16)

with 3 13 1 23 2S Sε σ σ= + , (1.17)

or

1 11 12 1

2 12 22 2

6 66 6

00

0 0

Q QQ Q

Q

σ εσ εσ ε

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(1.18)

with

( )3 13 1 23 233

1 C CC

ε ε ε= − + . (1.19)

The matrices ijS⎡ ⎤⎣ ⎦ et ijQ⎡ ⎤⎣ ⎦ are inverses of each other.

The reduced stiffness constants Qij can be expressed as functions of the engineering moduli as:

112

22 112

12 22

66 .

,1 1

,1 1

,1

L L

TLT TL LTL

T T T

TLT TL LLTL

LT TLT

LT TL

LT

E EQ EE

E E EQ QE EE

EQ Q

Q G

ν ν ν

ν ν ν

ν νν ν

= =− −

= = =− −

= =−

=

(1.20)

1.3. Basics of Laminate Theory 9

TABLE 1.1. Reduced stiffness constants of a unidirectional or orthotropic layer, off its material directions.

( )

( ) ( )( ) ( )

( )

( )

4 4 2 211 11 22 12 66

2 2 4 412 11 22 66 12

3 316 11 12 66 12 22 66

4 4 2 222 11 22 12 66

326 11 12 66

cos sin 2 2 sin cos ,

4 sin cos sin cos ,

2 sin cos 2 sin cos ,

sin cos 2 2 sin cos ,

2 sin

Q Q Q Q Q

Q Q Q Q Q

Q Q Q Q Q Q Q

Q Q Q Q Q

Q Q Q Q

θ θ θ θ

θ θ θ θ

θ θ θ θ

θ θ θ θ

θ

′ = + + +

′ = + − + +

′ = − − + − +

′ = + + +

′ = − − ( )

( ) ( )

312 22 66

2 2 4 466 11 22 12 66 66

cos 2 sin cos ,

2 sin cos sin cos .

Q Q Q

Q Q Q Q Q Q

θ θ θ

θ θ θ θ

+ − +

′ = + − + + +⎡ ⎤⎣ ⎦

1.2.4.4 Off-Axis Reduced Stiffness Constants

The relations between the off-axis reduced stiffness constants ijQ′ and those

ijQ expressed in the material directions are identical for a unidirectional layer as well as for an orthotropic composite. They are transposed from the general relations in the three-dimensional case (Section 1.2.3). The expressions obtained are reported in Table 1.1.

1.3 BASICS OF LAMINATE THEORY

1.3.1 Introduction

A laminate consists (Figure 1.6) of n layers, numbered from the lower to the upper face. The middle plane is chosen as the reference plane (Oxy) and the axis Oz is directed in the direction of increasing layer number. Each layer k is referred to by the z coordinates of its lower face, 1kh − , and upper face, .kh The total laminate thickness will be denoted h.

The purpose of the laminate theory is to reduce the initial problem in three dimensions (x, y, z) of the mechanical behaviour of laminate structure to a less difficult analysis in two dimensions (x, y). Schematically, this problem is solved by integrating through the thickness of the laminate.

1.3.2 Displacement Field

The basic assumption of the general theory of plates lies in expressing the displacement at every point M, with coordinates (x, y, z), of a plate in the form of polynomials in z, usually limited to degree three and with coefficients dependent

10 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

FIGURE 1.6. Laminate element.

on (x, y). The displacement field is then written in the form:

2 3

2 3

2

( , , , ) ( , , 0, ) ( , , ) ( , , ) ( , , ),

( , , , ) ( , , 0, ) ( , , ) ( , , ) ( , , ),

( , , , ) ( , , 0, ) ( , , ) ( , , ).

x x x

y y y

z z

u x y z t u x y t z x y t z x y t z x y t

x y z t x y t z x y t z x y t z x y t

x y z t x y t z x y t z x y t

ϕ ψ φ

ϕ ψ φ

ϕ ψ

= + + +

= + + +

= + +

v v

w w

(1.21)

The simplest and widely used schemes reduce to a first-order theory of the form:

0

0

0

( , , , ) ( , , ) ( , , ),( , , , ) ( , , ) ( , , ),( , , , ) ( , , ),

x

y

u x y z t u x y t z x y tx y z t x y t z x y tx y z t x y t

ϕϕ

= += +

=

v v

w w

(1.22)

introducing the mid-plane displacements :

0

0

0

( , , ) ( , , 0, ),( , , ) ( , , 0, ),( , , ) ( , , 0, ).

u x y t u x y tx y t x y tx y t x y t

===

v vw w

(1.23)

The displacement field (1.22) includes the transverse shear effects. In this case, particles of the plate originally on a line that is normal to the non-deformed middle plane remain on a straight line during deformation, but this line is not necessary normal to the deformed middle plane.

1.3.3 Resultants and Moments

1.3.3.1 In-plane Resultants

The in-plane stress resultant matrix, denoted N(x, y), is defined by:

2

2( , ) ( ) d

h

kh

x y M z−

= ∫N σ , (1.24)

1 2

k

n

1kh −kh

2h 1h

0h

middle plane

layer number

z

1.3. Basics of Laminate Theory 11

FIGURE 1.7. In-plane resultants of the loads applied to a laminate element.

where ( )k Mσ is the in-plane stress matrix of elements , and ,xx yy xyσ σ σ in the layer k. Thus, the in-plane resultant matrix is written as:

11

( , ) dk

k

x xxn h

y yyhkxy xy k

Nx y N z

N

σσσ−=

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑∫N . (1.25)

Components Nx, Ny and Nxy are the in-plane resultants, per unit length, respec-tively, of the normal stresses in the directions x and y and of the shear stresses in the plane (x, y) They are illustrated in Figure 1.7.

1.3.3.2 Transverse Shear Resultants

The transverse shear resultants are defined in the same way by:

11

( , ) dk

k

n hx xz

hy yzk k

Qx y z

Q

σ

σ−=

⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑∫Q . (1.26)

As the in-plane resultants, the transverse shear resultants are loads per unit length of the cross-section of the laminate. They are illustrated in Figure 1.8.

1.3.3.3 Resultant Moments

The fundamental equations of laminates also introduce the moment of stresses applied to the element of the laminate considered. The resultant moments are defined as:

h

x

y

z

yN

xyN

xyN

xyN

xyN

yN

xN

xN

12 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

FIGURE 1.8. Transverse shear resultants.

1f

1

( , ) dk

k

x xxn h

y yyhk

xy xy k

M

x y M z z

M

σ

σ

σ−=

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑∫M , (1.27)

where Mx and My are the bending moments, and Mxy is the twisting moment. As previously, the resultant moments are moments per unit length. They are illustra-ted in Figure 1.9.

1.3.4 Fundamental Equations for Plates in the case of a First-Order Theory

The fundamental equations of plates are deduced from the fundamental

FIGURE 1.9. Resultant moments applied to a laminate element.

yQ

yQ

xQ

xQ

x y

z

h

x

y

z

yM xyM

yM xyM

xyM

xM

xyM xM

1.3. Basics of Laminate Theory 13

equations of deformable solids :

,

,

,

xx xy xz x x

yy yz xy y y

zz xz yz z z

f ax y z

f ay z x

f az x y

σ σ σ ρ

σ σ σ ρ

σ σ σ ρ

∂ ∂ ∂+ + + =

∂ ∂ ∂∂ ∂ ∂

+ + + =∂ ∂ ∂∂ ∂ ∂

+ + + =∂ ∂ ∂

(1.28)

where fx, fy and fz are the components of the body forces acting at the point M of the solid under consideration; ax, ay and az are the components of the acceleration vector at the point M and ρ is the material density at the point M.

In a first order theory the acceleration components are:

2 20

2 2

220

2 2

20

2

,

,

.

xx

yy

z

ua zt t

a zt t

at

ϕ

ϕ

∂ ∂= +

∂ ∂

∂∂= +

∂ ∂

∂=

∂

v

w

(1.29)

The integration of the first two equations (1.28) through the thickness of the laminate leads to the fundamental equations of a plate element for in-plane resul-tants. In the same way the third equation leads to the laminate equation for the transverse shear resultants. The fundamental equations for the moments are obtained by multiplying the first two equations (1.28) and integrating through the thickness.

Thus, we obtain the fundamental equations of laminates as: 2 2

01 2 2 2

xyx xx x x s

NN uF Rx y t t

ϕτ τ ρ∂∂ ∂ ∂

+ + + − = +∂ ∂ ∂ ∂

,

220

1 2 2 2y xy y

y y y sN N

F Ry x t t

ϕτ τ ρ

∂ ∂ ∂∂+ + + − = +

∂ ∂ ∂ ∂v ,

20

2yx

z sQQ q F

x y tρ

∂∂ ∂+ + + =

∂ ∂ ∂w , (1.30)

( )2 2

01 2 2 22

xyx xx x x x xy

MM h uP Q R Ix y t t

ϕτ τ∂∂ ∂ ∂

+ + + + − = +∂ ∂ ∂ ∂

,

( )22

01 2 2 22

y xy yy y y y xy

M M h P Q R Iy x t t

ϕτ τ

∂ ∂ ∂∂+ + + + − = +

∂ ∂ ∂ ∂v .

The components Fx, Fy and Fz are the resultants of the body forces and Px, Py are the moments:

14 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

( )

2

2, , , , d

h

x y z x y zh

F F F f f f z−

= ∫ , (1.31)

( )

2

2, , d

h

x y x yh

P P zf zf z−

= ∫ . (1.32)

The load q represents the pressure forces applied to each face of the plate:

( ) ( )( , ) 2 2zz zzq q x y h hσ σ= = − − . (1.33)

The fundamental equations of laminates also consider the case of possible shear stresses applied to the laminate faces:

( ) ( )( ) ( )

1 2

1 2

2 , 2 ,

2 , 2 .

x xz x xz

y yz y yz

h h

h h

τ σ τ σ

τ σ τ σ

= = −

= = − (1.34)

The quantity sρ is the weight per unit area of the laminate at the point (x, y):

2

2d

h

sh

zρ ρ−

= ∫ . (1.35)

Lastly, the previous fundamental equations introduce the rotational inertia terms:

( )

22

2, , d

h

xyh

R I z z zρ−

= ∫ . (1.36)

In the case of a plate made of n layers, the layer k having a material density ,kρ the quantities , and s xyR Iρ are:

( )

1

2

12 1 1

d dk

k

n nh h

s k k k kh hk k

z z h hρ ρ ρ ρ−

−− = =

= = = −∑ ∑∫ ∫ . (1.37)

( )2 21

1

12

n

k k kk

R h hρ −=

= −∑ , (1.38)

( )3 31

1

13

n

xy k k kk

I h hρ −=

= −∑ . (1.39)

When the effect of the transverse shear is neglected, an equation independent of the transverse shear resultants can be derived from the last three equations (1.30). In this case the fundamental equations of laminates are simplified as:

2 20

1 2 2 2xyx x

x x x sNN uF R

x y t tϕτ τ ρ

∂∂ ∂ ∂+ + + − = +

∂ ∂ ∂ ∂,

220

1 2 2 2y xy y

y y y sN N

F Ry x t t

ϕτ τ ρ

∂ ∂ ∂∂+ + + − = +

∂ ∂ ∂ ∂v ,

(1.40)

1.4. Classical Laminate Theory 15

2 22

2 2

32 3 3 30 0 0

2 2 2 2 2

2

.

y xyx

yxs xy

M MM qx yx y

uR It x t y t x t y t

ϕϕρ

∂ ∂∂+ + +

∂ ∂∂ ∂

⎛ ⎞⎛ ⎞ ∂∂ ∂ ∂ ∂⎜ ⎟= + + + +⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

w v

Generally the rotational inertia terms can be neglected and, in the absence of body forces and shear stresses applied on the laminate faces, the fundamental equations are simplified as:

20

2

20

2

20

2

,

,

,

0,

0.

xyxs

y xys

yxs

xyxx

y xyy

NN ux y t

N Ny x t

QQ qx y t

MM Qx y

M MQ

y x

ρ

ρ

ρ

∂∂ ∂+ =

∂ ∂ ∂

∂ ∂ ∂+ =

∂ ∂ ∂

∂∂ ∂+ + =

∂ ∂ ∂∂∂

+ − =∂ ∂

∂ ∂+ − =

∂ ∂

v

w (1.41)

These equations can also be written when the transverse shear effect is neglect-ted. We obtain:

20

2

20

2

2 22 20

2 2 2

,

,

2 .

xyxs

y xys

y xyxs

NN ux y t

N Ny x t

M MM qx yx y t

ρ

ρ

ρ

∂∂ ∂+ =

∂ ∂ ∂

∂ ∂ ∂+ =

∂ ∂ ∂

∂ ∂∂ ∂+ + + =

∂ ∂∂ ∂ ∂

v

w

(1.42)

1.4 CLASSICAL LAMINATE THEORY

1.4.1 Assumptions of the Classical Theory of Laminates

The classical laminate theory is investigated in Chapter 14 of Refs. 1 and 2. The general displacement field for a first-order theory is given by Equations (1.22). The strain field is deduced from these relations and is written as:

16 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

0

0

0

0

0

0 0

,

,

0,

2 ,

2 ,

2 .

xxx

yyy

zz

yz yz y

xz xz x

yxxy xy

u u zx x x

zy y y

z z

z y yuz x xu u zy x y x y x

ϕε

ϕε

ε

γ ε ϕ

γ ε ϕ

ϕϕγ ε

∂ ∂ ∂= = +∂ ∂ ∂

∂∂ ∂= = +∂ ∂ ∂

∂ ∂= = =∂ ∂

∂ ∂ ∂= = + = +

∂ ∂ ∂

∂ ∂ ∂= = + = +

∂ ∂ ∂∂⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂

= = + = + + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

v v

w w

v w w

w w

v v

(1.43)

This strain field is that of a first-order theory including the transverse shear effect. The classical laminate theory neglects the effect of the transverse shear. So, in

this model the transverse shear strains are zero, hence:

0, 0.xz yzγ γ= = (1.44)

This assumption implies from (1.43):

0

0

( , ) ,

( , ) .

x

y

x y

x y

ϕ

ϕ

∂= −

∂

∂= −

∂

wxwy

(1.45)

The displacement field is then, by (1.22), written as:

00

00

0

( , , , ) ( , , ) ( , , ),

( , , , ) ( , , ) ( , , ),

( , , , ) ( , , ).

u x y z t u x y t z x y tx

x y z t x y t z x y ty

x y z t x y t

∂= −

∂

∂= −

∂

=

w

wv v

w w

(1.46)

The deformation of the normal to the middle plane is then a straight line normal to the deformed middle plane.

1.4.2 Strain Field

Considering the general equation (1.43) of the stress field and taking account of Equation (1.44), the strain field is written as:

1.4. Classical Laminate Theory 17

20 0

2

20 0

2

20 0 0

,

,

0,

0, 0,

2 .

xx

yy

zz

yz xz

xy

u zx x

zy y

u zy x x y

ε

ε

ε

γ γ

γ

∂ ∂= −∂ ∂

∂ ∂= −∂ ∂

=

= =

⎛ ⎞∂ ∂ ∂= + −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

w

v w

v w

(1.47)

The strain matrix is reduced to the three non-zero components:

( )xx

yy

xy

M

ε

ε

γ

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎣ ⎦

ε . (1.48)

This strain field can be described as the superposition of in-plane strains (or midplane strains) m ( )Mε and flexural strains (bending and twisting strains)

f ( )Mε , as: m f( ) ( ) ( )M M M= +ε ε ε , (1.49) or

0

0

0

xx xx x

yy yy y

xy xy xy

z

ε ε κ

ε ε κ

γ γ κ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥

= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

, (1.50)

with :

0 0 00 0 0 0

2 2 20 0 0

2 2

( , , ), ( , , ), ,

( , , ), ( , , ), 2 ( , , ).

xx yy xy

x y xy

u ux y t x y tx y y x

x y t x y t x y tx yx y

ε ε γ

κ κ κ

∂ ∂ ∂ ∂= = = +

∂ ∂ ∂ ∂

∂ ∂ ∂= − = − = −

∂ ∂∂ ∂

v v

w w w (1.51)

The components 0 0 0, and xx yy xyε ε γ are the in-plane strains, and the components , and x y xyκ κ κ are the curvatures of the deformed laminate.

1.4.3 Stress Field

1.4.3.1 General expression

The stress field in the layer k of a laminate is deduced from the stress-strain relation (1.6). Hence:

18 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

11 12 13 16

12 22 23 26

13 23 33 36

44 45

45 55

16 26 36 66

0 00 00 0

0 0 0 00 0 0 0

0 0

xx xx

yy yy

zz zz

yz yz

xz xz

xy xyk k

C C C CC C C CC C C C

C CC C

C C C C

σ εσ εσ εσ γσ γσ γ

⎡ ⎤ ⎡ ⎤′ ′ ′ ′⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥′ ′ ′ ′⎢ ⎥ ⎢ ⎥⎢ ⎥

′ ′ ′ ′⎢ ⎥ ⎢ ⎥⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥′ ′⎢ ⎥ ⎢ ⎥⎢ ⎥′ ′⎢ ⎥ ⎢ ⎥⎢ ⎥

⎢ ⎥ ⎢ ⎥⎢ ⎥′ ′ ′ ′⎣ ⎦⎣ ⎦ ⎣ ⎦

, (1.52)

where the components ijC′ are the stiffness constants of the layer k. The elementary theory of plates makes the assumption that the normal stresses

zzσ are negligible within the volume of the plate. This assumption of a plane stress state is extended to the theory of laminates and leads to the strain field:

11 12 16

12 22 26

16 26 66

44 45

45 55

0 00 00 0

0 0 00 0 0

xx xx

yy yy

xy xy

yz yz

xz xzk k

Q Q QQ Q QQ Q Q

C CC C

σ εσ εσ γσ γσ γ

′ ′ ′⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥′ ′ ′⎢ ⎥ ⎢ ⎥ ⎢ ⎥

′ ′ ′⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥′ ′⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥′ ′⎣ ⎦ ⎣ ⎦ ⎣ ⎦

. (1.53)

The coefficients ijQ′ are the reduced stiffness constants of the layer k introduced

in Equation (1.14). They will be denoted by ijQ′ or kijQ . Equation (1.53) is the

general expression of the stress field when the transverse shear effect is considered.

1.4.3.2 Stress Field in the Case of the Classical Laminate Theory

In the case of the classical laminate theory, the transverse shear effect is neglected and the stress field (1.53) is simplified as:

011 12 16 11 12 16

012 22 26 12 22 26

016 26 66 16 26 66

xx xx x

yy yy y

xy xy xyk k k

Q Q Q Q Q Q

Q Q Q z Q Q Q

Q Q Q Q Q Q

σ ε κ

σ ε κ

σ γ κ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤′ ′ ′ ′ ′ ′⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

′ ′ ′ ′ ′ ′= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

′ ′ ′ ′ ′ ′⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

. (1.54)

1.4.4 Resultants and Moments

Expression (1.25) associated with Equation (1.54) leads to the expressions for the in-plane resultants. We obtain:

011 12 16 11 12 16

012 22 26 12 22 26

016 26 66 16 26 66

x xx x

y yy y

xy xy xy

N A A A B B B

N A A A B B B

N A A A B B B

ε κ

ε κ

γ κ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

= +⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦

, (1.55)

introducing the matrices:

1.4. Classical Laminate Theory 19

( )( )11

,n

ij k k ij kk

A h h Q−=

′= −∑ (1.56)

and

( )( )2 21

1

1 .2

n

ij k k ij kk

B h h Q−=

′= −∑ (1.57)

Equation (1.55) shows that the in-plane resultants are functions of the in-plane strains as well as the bending and twisting curvatures.

In the same way, the moments are obtained by introducing Equation (1.54) for the stresses into Equation (1.27). Hence:

011 12 16 11 12 16

012 22 26 12 22 26

016 26 66 16 26 66

x xx x

y yy y

xy xy xy

M B B B D D D

M B B B D D D

M B B B D D D

ε κ

ε κ

γ κ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

= +⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦

, (1.58)

on introducing the matrix:

( )( )3 31

1

1 .3

n

ij k k ij kk

D h h Q−=

′= −∑ (1.59)

The bending and twisting moments are therefore functions of the bending and twisting curvatures, but are also functions of the in-plane strains.

1.4.5 Constitutive Equation of a Laminate

The constitutive equation of a laminate expresses the resultants and the moments as functions of the in-plane strains and of the curvatures. This equation is deduced from Equations (1.55) and (1.58), which yields:

011 12 16 11 12 16

012 22 26 12 22 26

016 26 66 16 26 66

11 12 16 11 12 16

12 22 26 12 22 26

16 26 66 16 26 66

x xx

y yy

xy xy

x x

y y

xy xy

N A A A B B BN A A A B B BN A A A B B BM B B B D D DM B B B D D DM B B B D D D

ε

ε

γ

κ

κ

κ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥= ⎢⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥⎢⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦

⎥⎥⎥⎥⎥⎥⎥⎥

. (1.60)

The terms of the constitutive matrix are given by Expressions (1.56), (1.57) and (1.59). They can also be expressed by introducing the thickness ke and the coor-dinate kz of the midplane of layer k. We obtain:

20 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

( )1

,n

ij ij kkk

A Q e=

′=∑ (1.61)

( )1

,n

ij ij k kkk

B Q e z=

′=∑ (1.62)

( )3

2

1 12

nk

ij ij k kkk

eD Q e z=

⎛ ⎞′= +⎜ ⎟⎜ ⎟

⎝ ⎠∑ . (1.63)

The matrix ijA⎡ ⎤⎣ ⎦ is the stretching stiffness matrix, the matrix ijD⎡ ⎤⎣ ⎦ is the

bending-twisting stiffness matrix and the matrix ijB⎡ ⎤⎣ ⎦ is the coupling stiffness

matrix. The existence of the matrix ijB⎡ ⎤⎣ ⎦ leads to coupling effects between stretching and bending-twisting of laminates.

1.4.6 Governing Equations

The governing equations of the classical laminate theory are obtained by intro-ducing the constitutive equation (1.60) of laminates into Equations (1.30) or (1.40). Substituting, for example, Equation (1.60) into Equations (1.40), then taking account of (1.51), we obtain the three governing equations as:

( )2 2 2 2 2 2

0 0 0 0 0 011 16 66 16 12 66 262 2 2 22u u uA A A A A A A

x y x yx y x y∂ ∂ ∂ ∂ ∂ ∂

+ + + + + +∂ ∂ ∂ ∂∂ ∂ ∂ ∂

v v v

( )

3 3 3 30 0 0 0

11 16 12 66 263 2 2 33 2B B B B Bx x y x y y

∂ ∂ ∂ ∂− − − + −

∂ ∂ ∂ ∂ ∂ ∂w w w w (1.64)

2

02sut

ρ ∂=

∂,

( )2 2 2 2 2 2

0 0 0 0 0 016 12 66 26 66 26 222 2 2 22u u uA A A A A A A

x y x yx y x y∂ ∂ ∂ ∂ ∂ ∂

+ + + + + +∂ ∂ ∂ ∂∂ ∂ ∂ ∂

v v v

( )

3 3 3 30 0 0 0

16 12 66 26 223 2 2 32 3B B B B Bx x y x y y

∂ ∂ ∂ ∂− − + − −

∂ ∂ ∂ ∂ ∂ ∂w w w w (1.65)

2

02s

tρ ∂

=∂v ,

1.4. Classical Laminate Theory 21

( )4 4 4 4 4

0 0 0 0 011 16 12 66 26 224 3 2 2 3 44 2 2 4D D D D D D

x x y x y x y y∂ ∂ ∂ ∂ ∂

+ + + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂w w w w w

( )

3 3 3 3 30 0 0 0 0

11 16 12 66 26 163 2 2 3 33 2u u u uB B B B B Bx x y x y y x

∂ ∂ ∂ ∂ ∂− − − + − −

∂ ∂ ∂ ∂ ∂ ∂ ∂v

( )

3 3 30 0 0

12 66 26 222 2 32 3B B B Bx y x y y∂ ∂ ∂

− + − −∂ ∂ ∂ ∂ ∂v v v (1.66)

2

02sq

tρ ∂

= −∂w .

The preceding equations do not take into account the body forces, the possible shear stresses on the faces of the laminate and neglects the effects of inertia in rotation. The accounting for these factors leads to introduce additional terms. The governing equations, associated with the boundary conditions of the laminated structure (Section 1.4.7), allow us to find, in principle, the displacements

0 0 0( , , ), ( , , ) and ( , , ),u x y t x y t x y tv w which are solutions of the elasticity problem. Solving these equations is complex and can be done analytically only in some particular cases. General cases need to use a finite element analysis.

An important simplification appears in the case of symmetric laminates, for which the terms ijB are zero as well as the quantities R. The governing equations are then written in the form:

( )2 2 2 2 2 2

0 0 0 0 0 011 16 66 16 12 66 262 2 2 22u u uA A A A A A A

x y x yx y x y∂ ∂ ∂ ∂ ∂ ∂

+ + + + + +∂ ∂ ∂ ∂∂ ∂ ∂ ∂

v v v

2

02sut

ρ ∂=

∂, (1.67)

( )2 2 2 2 2 2

0 0 0 0 0 016 12 66 26 66 26 222 2 2 22u u uA A A A A A A

x y x yx y x y∂ ∂ ∂ ∂ ∂ ∂

+ + + + + +∂ ∂ ∂ ∂∂ ∂ ∂ ∂

v v v

2

02s

tρ ∂

=∂v , (1.68)

( )4 4 4 4 4

0 0 0 0 011 16 12 66 26 224 3 2 2 3 44 2 2 4D D D D D D

x x y x y x y y∂ ∂ ∂ ∂ ∂

+ + + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂w w w w w

2 4 4

0 0 02 2 2 2 2s xyq I

t x t y tρ

⎛ ⎞∂ ∂ ∂= − + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

w w w . (1.69)

In this case the in-plane behaviour 0 0( , )u v is decoupled from the flexural behaviour 0( )w . Equation (1.69) can be solved independently.

22 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

FIGURE 1.10. Plate boundary element.

1.4.7 Boundary conditions

1.4.7.1 Basics

A boundary element (Figure 1.10) is described at a point P(x, y, 0) of the boundary by its unit normal n and the tangential unit vector t in the middle plane. The deformed shape at point P of the laminate is characterized by the displacement of the point P expressed in the basis ( ), , n t k by its components

0 0 0( , , ), ( , , ), ( , , ),n tu x y t u x y t x y tw and by the orientation of the deformed

shape characterized by 0n

∂∂w . Vector k is the unit vector of z direction. The loads

applied at point P are characterized by the in-plane resultants , ,n ntN N the shear resultant ,nQ and the moments of bending nM and twisting ntM . The conditions prescribed at point P take on one of the quantities of each of the following pairs:

0

0 0 0, ; , ; , ; , .ntn n t t n nu N u N M Q

n t∂ ∂

+∂ ∂w Mw (1.70)

The quantity ntnQ

t∂

+∂M is known as the Kirchhoff’s boundary condition.

1.4.7.2 Simply Supported Edge

In the case of a simply supported edge (Figure 1.11), the boundary condition usually considered is: 0 0, 0, 0, 0.n n ntM N N= = = =w (1.71)

1.4.7.3 Clamped Edge

For a clamped boundary element (Figure 1.12), the condition usually retained

x

y

z

P t

n

nM

ntM

1.4. Classical Laminate Theory 23

FIGURE 1.11. Simply supported edge.

is the following one:

00 0 00, 0, 0, 0.n tu u

n∂

= = = =∂ww (1.72)

1.4.7.4 Free Edge

In the case of a free boundary, the resultants and moments are zero and lead to five conditions, whereas only four conditions are necessary. To remove this diffi-culty it is usual to introduce the Kirchhoff’s boundary condition. And so the boundary conditions at a free edge are written as:

0, 0, 0, 0.ntn nt n n

MN N M Qt

∂= = = + =

∂ (1.73)

FIGURE 1.12. Clamped edge.

n

t P

n

t P

24 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

1.4.8 Energy Formulation of the Classical Laminate Theory

1.4.8.1 Strain Energy

The strain energy of an elastic solid is expressed as:

( ) d1 d d d2 xx xx yy yy zz zz yz yz xz xz xy xyU x y zσ ε σ ε σ ε σ γ σ γ σ γ= + + + + +∫∫∫ ,

(1.74) where the integration is performed over the whole volume of the solid.

Considering the assumptions of the classical laminate theory, σzz = 0, γxz = 0, γyz = 0, and Relation (1.54) which expresses the stresses as functions of the strains, Expression (1.74) of the strain energy becomes:

(

)

2 2 2d 11 22 66 12

16 26

1 22

2 2 d d d .

k k k kxx yy xy xx yy

k kxx xy yy xy

U Q Q Q Q

Q Q x y z

ε ε γ ε ε

ε γ ε γ

= + + +

+ +

∫∫∫ (1.75)

This relation can be written as a function of the in-plane displacements 0 ,u 0 0 and ,v w by substituting into the preceding expression the strain-displacement

relations (1.50) and (1.51). Next, the expression of strain energy is obtained by integrating with respect to z through the thickness of the laminate (Chapter 16 of Refs. 1 and 2). The expression obtained for the strain energy introduces the stiffnesses constants , and ,ij ij ijA B D given by the relations (1.56), (1.57) and (1.59), respectively.

The general expression obtained for the strain energy may be simplified in the case of symmetric laminates for which the stretching/bending-twisting coupling terms ijB are zero. In this case the expression of the strain energy reduces to:

220 0 0 0

d 11 12 22

20 0 0 0 0 0

16 26 66

22 2 2 20 0 0 0

11 12 222 2 2 2

1 22

2 d d

1 22

u uU A A Ax x y y

u u uA A A x yx y y x y x

D D Dx x y y

⎡ ∂ ∂ ∂ ∂⎛ ⎞⎛ ⎞⎢= + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎢ ⎝ ⎠⎣⎤∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞⎛ ⎞ ⎛ ⎞⎥+ + + + +⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ⎥⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎦

⎛ ⎞ ⎛∂ ∂ ∂ ∂+ + +⎜ ⎟ ⎜⎜ ⎟ ⎜∂ ∂ ∂ ∂⎝ ⎠ ⎝

∫∫ v v

v v v

w w w w

2

22 2 2 20 0 0 0

16 26 662 2 4 4 d d .D D D x yx y x yx y

⎡ ⎞⎢ ⎟⎟⎢ ⎠⎣

⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ⎥+ + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ⎥∂ ∂⎝ ⎠ ⎝ ⎠ ⎦

∫∫w w w w

(1.76)

The strain energy appears as the sum of two terms: the first one contains only the

1.5. Laminate Theory Including the Transverse Shear Effect 25

in-plane displacements 0 0 and u v , the second one contains only the transverse displacement 0.w In the case of pure bending, the first term reduced to a constant C and the strain energy may be written as:

2 22 2 2 20 0 0 0

d 11 12 222 2 2 2

22 2 2 20 0 0 0

16 26 662 2

1 22

4 4 d d .

U D D Dx x y y

D D D x y Cx y x yx y

⎡ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎢= + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ⎥+ + + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ⎥∂ ∂⎝ ⎠ ⎝ ⎠ ⎦

∫∫ w w w w

w w w w(1.77)

1.4.8.2 Kinetic Energy

The kinetic energy of a structure is:

2 2 2

c1 d d d2

uE x y zt t t

ρ⎡ ⎤∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

∫∫∫ v w , (1.78)

where ρ is the density of the material at point (x, y, z). The integration is perfor-med over the whole volume of the structure.

In the case of the classical laminate theory the displacement field is expressed by (1.46). Substituting these expressions into expression (1.78), we obtain:

2 2 22 20 0 0 0 0

c1 d d d2

uE z z x y zt x t t y t t

ρ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞⎢ ⎥= − + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦

∫∫∫ w v w w .

(1.79)

Usually, the derivatives with respect to time of the laminate rotations can be neglected. In this case the expression of the kinetic energy is reduced to:

2 2 20 0 0

c s1 d d2

uE x yt t t

ρ⎡ ⎤∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

∫∫ v w , (1.80)

introducing the weight per unit area (1.35) of the laminate at point (x, y).

1.5 LAMINATE THEORY INCLUDING THE TRANSVERSE SHEAR EFFECT

1.5.1 Constitutive Equation

In the laminate theory including the transverse shear deformation, the displacement field is expressed in the general form (1.22) of a first-order theory. Then, a similar process as the one considered in the case of classical laminate

26 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

theory is implemented from the expressions (1.22) of the displacement field. This process leads to the constitutive equation of the laminate theory with transverse shear which may be written as:

11 12 16 11 12 16

12 22 26 12 22 26

16 26 66 16 26 66

11 12 16 11 12 16

12 22 26 12 22 26

16 26 66 16 26 66

44 45

45 55

0 00 00 00 00 00 0

0 0 0 0 0 00 0 0 0 0 0

x

y

xy

x

y

xy

y

x

N A A A B B BN A A A B B BN A A A B B BM B B B D D DM B B B D D DM B B B D D DQ F FQ F F

⎡ ⎤ ⎡⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢=⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥⎢ ⎥⎣ ⎦ ⎣

0

0

0

0

0

xx

yy

xy

x

y

xy

yz

xz

ε

ε

γκκκ

γ

γ

⎡ ⎤⎤ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎢ ⎥⎦ ⎢ ⎥⎣ ⎦

, (1.81)

with

0 0 00 0 0 0

0 00 0

, , ,

, , .

, .

xx yy xy

y yx xx y xy

yz y xz x

u ux y y x

x y y x

y x

ε ε γ

ϕ ϕϕ ϕκ κ κ

γ ϕ γ ϕ

∂ ∂ ∂ ∂= = = +∂ ∂ ∂ ∂

∂ ∂∂ ∂= = = +

∂ ∂ ∂ ∂∂ ∂

= + = +∂ ∂

v v

w w

(1.82)

The stiffness constants Aij, Bij and Dij have been introduced in the classical laminate theory and the new stiffness constants Fij are expressed as:

( )( ) ( )11 1

.n n

ij k k ij ij kk kk k

F h h C C e−= =

′ ′= − =∑ ∑ (1.83)

The transverse shear stiffness constants ijC′ are referred to the laminate directions and are given as functions of the stiffness constants ijC referred to the layer directions by:

( )

2 244 44 55

45 55 442 2

55 44 55

cos sin ,sin cos ,

sin cos ,

C C CC C C

C C C

θ θθ θ

θ θ

′ = +′ = −

′ = +

(1.84)

where C44 and C55 are the transverse shear moduli: 44 55, .TT LTC G C G′ ′= = (1.85)

1.5.2 Governing Equations

The governing equations of the theory of laminates that include the transverse shear effect are derived by introducing the constitutive equation (1.81) in the

1.5. Laminate Theory Including the Transverse Shear Effect 27

fundamental equations (1.30) of plates. We obtain:

( )2 2 2 2 2 2

0 0 0 0 0 011 16 66 16 12 66 262 2 2 22u u uA A A A A A A

x y x yx y x y∂ ∂ ∂ ∂ ∂ ∂

+ + + + + +∂ ∂ ∂ ∂∂ ∂ ∂ ∂

v v v

( )2 22 2 2

11 16 66 16 12 662 2 2 2 y yx x xB B B B B Bx y x yx y x

ϕ ϕϕ ϕ ϕ ∂ ∂∂ ∂ ∂+ + + + + +

∂ ∂ ∂ ∂∂ ∂ ∂

2 2 2

026 2 2 2 y x

suB R

y t tϕ ϕρ

∂ ∂ ∂+ = +

∂ ∂ ∂, (1.86)

( )2 2 2 2 2 2

0 0 0 0 0 016 12 66 26 66 26 222 2 2 22u u uA A A A A A A

x y x yx y x y∂ ∂ ∂ ∂ ∂ ∂

+ + + + + +∂ ∂ ∂ ∂∂ ∂ ∂ ∂

v v v

( )2 2 22 2 2

16 12 66 26 66 26 222 2 2 2 2y y yx x xB B B B B B Bx y x yx y x y

ϕ ϕ ϕϕ ϕ ϕ ∂ ∂ ∂∂ ∂ ∂+ + + + + + +

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

22

02 2

ys R

t tϕ

ρ∂∂

= +∂ ∂v , (1.87)

2 2 2

0 0 055 45 442 22y yx xF F F q

x y x x y yx yϕ ϕϕ ϕ⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂ ∂ ∂ ∂

+ + + + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

w w w

2

02s

tρ ∂

=∂

w , (1.88)

( )2 2 2 2 2 2

0 0 0 0 0 011 16 66 16 12 66 262 2 2 22u u uB B B B B B B

x y x yx y x y∂ ∂ ∂ ∂ ∂ ∂

+ + + + + +∂ ∂ ∂ ∂∂ ∂ ∂ ∂

v v v

( )2 22 2 2

11 16 66 16 12 662 2 2 2 y yx x xD D D D D Dx y x yx y x

ϕ ϕϕ ϕ ϕ ∂ ∂∂ ∂ ∂+ + + + + +

∂ ∂ ∂ ∂∂ ∂ ∂

20 0

26 55 452 yx yD F F

x yyϕ

ϕ ϕ∂ ⎛ ⎞∂ ∂⎛ ⎞+ − + − +⎜ ⎟⎜ ⎟∂ ∂∂ ⎝ ⎠ ⎝ ⎠

w w

2 2

02 2

xxy

uR It t

ϕ∂ ∂= +

∂ ∂, (1.89)

( )2 2 2 2 2 2

0 0 0 0 0 016 12 66 26 66 26 222 2 2 22u u uB B B B B B B

x y x yx y x y∂ ∂ ∂ ∂ ∂ ∂

+ + + + + +∂ ∂ ∂ ∂∂ ∂ ∂ ∂

v v v

( )2 22 2 2

16 12 66 26 66 262 2 2 2y yx x xD D D D D Dx y x yx y x

ϕ ϕϕ ϕ ϕ ∂ ∂∂ ∂ ∂+ + + + + +

∂ ∂ ∂ ∂∂ ∂ ∂

28 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

20 0

22 45 442 yx yD F F

x yyϕ

ϕ ϕ∂ ⎛ ⎞∂ ∂⎛ ⎞+ − + − +⎜ ⎟⎜ ⎟∂ ∂∂ ⎝ ⎠ ⎝ ⎠

w w

22

02 2

yxyR I

t tϕ∂∂

= +∂ ∂v . (1.90)

Equations (1.86) to (1.90) associated to the boundary conditions of the structure allow, in principle, to determine the five functions u0(x, y, t), v0(x, y, t), w0(x, y, t), ϕx(x, y, t) and ϕy(x, y, t), that are the solution of the elasticity problem.

In the case of symmetric laminates, Bij = 0 and R = 0, Equation (1.86) and (1.87) respectively reduce to Equations (1.67) and (1.68) of the classical theory. Equation (1.69) is unchanged, whereas Equations (1.89) and (1.90) simplify as:

( )2 22 2 2

11 16 66 16 12 662 2 22 y yx x xD D D D D Dx y x yx y x

ϕ ϕϕ ϕ ϕ ∂ ∂∂ ∂ ∂+ + + + +

∂ ∂ ∂ ∂∂ ∂ ∂

20 0

26 55 452 yx yD F F

x yyϕ

ϕ ϕ∂ ⎛ ⎞∂ ∂⎛ ⎞+ − + − +⎜ ⎟⎜ ⎟∂ ∂∂ ⎝ ⎠ ⎝ ⎠

w w

2

2x

xyItϕ∂

=∂

, (1.91)

( )2 22 2 2

16 12 66 26 66 262 2 2 2y yx x xD D D D D Dx y x yx y x

ϕ ϕϕ ϕ ϕ ∂ ∂∂ ∂ ∂+ + + + +

∂ ∂ ∂ ∂∂ ∂ ∂

20 0

22 45 442 yx yD F F

x yyϕ

ϕ ϕ∂ ⎛ ⎞∂ ∂⎛ ⎞+ − + − +⎜ ⎟⎜ ⎟∂ ∂∂ ⎝ ⎠ ⎝ ⎠

w w

2

2y

xyItϕ∂

=∂

. (1.92)

1.5.3 Boundary Conditions

The boundary conditions are imposed on one of the variables of each of the following pairs :

0 0 0, ; , ; , ; , ; , ;n n t t n n t t nu N u N M M Qϕ ϕ w where n and t are the normal and tangential directions at a point of the boundary element (Figure 1.10).

For a point of a free element, the boundary conditions are: 00, 0, 0, 0, 0.n t n tN N M ϕ= = = = =w (1.93)

For a simply supported element (Figure 1.11), the conditions are: 00, 0, 0, 0, 0.n t n tN N M ϕ= = = = =w (1.94)

1.6. Theory of Sandwich Plates 29

For a clamped element (Figure 1.12), the boundary conditions are:

0 0 00, 0, 0, 0, 0.n t n tu u ϕ ϕ= = = = =w (1.95)

1.5.4 Introduction of Transverse Shear Coefficients

An improvement of the theory of laminates including the transverse shear effect consists to introduce transverse shear coefficients. In this way, the part of Equation (1.81) related to the in-plane resultants and bending-twisting moments (Nx, Ny, Nxy, Mx, My, Mxy) is not modified. The part related to the transverse shear resultants is modified by replacing the coefficients Fij by new transverse shear stiffness constants Fij of the laminate as:

0

44 45045 55

y yz

x xz

Q H HQ H H

γ

γ

⎡ ⎤⎡ ⎤ ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎢ ⎥⎣ ⎦, (1.96)

with , , 4, 5.ij ij ijH k F i j= = (1.97)

The parameters kij are the shear correction factors. In the case of the initial theory (previous subsections):

1, , 4, 5ijk i j= = . (1.98)

Other values obtained in the case of isotropic homogeneous plate, and then applied to the case of orthotropic plates, are also used:

23 , , 4, 5ijk i j= = , (1.99)

56 , , 4, 5ijk i j= = . (1.100)

More generally the parameters kij can be evaluated by considering cylindrical bending around the directions x and y (Chapter 17 of Refs. 1 and 2).

1.6 THEORY OF SANDWICH PLATES

1.6.1 Introduction

A sandwich material (Subsection 1.1.3) is constituted of a material of low density (the core) with laminated face sheets (the skins) bonded to each of the core faces. The essential function of the core is to transfer, by transverse shear, the mechanical loading applied on one skin to the other.

In the general case, the skins are laminates of thickness h1 for the lower skin and of thickness h2 for the upper skin (Figure 1.13). The thickness of the core is h and the plane (x, y) of the coordinate system is the middle plane.

30 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

FIGURE 1.13. Notations for a sandwich plate.

1.6.2 Assumptions for the Sandwich Theory

The theory of sandwich plates considers the following basic assumptions: 1. The thickness of the core is much greater than that of the skins, 1 2,h h h .

2. The in-plane displacements uc and vc of the core are linear functions of the z coordinate.

3. The in-plane displacements u and v are uniform through the thickness of the skins.

4. The transverse displacement w is independent of the z coordinate: the strain εzz is neglected.

5. The core transmits only the transverse shear stresses σxz, σyz: the stresses σxx, σyy, σxy and σzz are neglected.

6. The transverse shear stresses σxz and σyz are neglected within the skins.

Lastly, the theory treats the elasticity problems of small deformations.

1.6.3 Displacement field

Assumption 2 implies a first-order model for the core displacements:

c 0

c 0

( , , , ) ( , , ) ( , , ),( , , , ) ( , , ) ( , , ),

x

y

u x y z t u x y t z x y tx y z t x y t z x y t

ϕϕ

= += +v v

(1.101)

with

0 c

0 c

( , , ) ( , , , 0),( , , ) ( , , , 0).

u x y t u x y tx y t x y t

==v v

(1.102)

2h

1h

2h

h

y

x

z

1.6. Theory of Sandwich Plates 31

The continuity of the displacements at the core-skin interfaces, associated with assumption 3, leads to the following expressions for the displacements within the skins:

— lower skin:

1 0

1 0

( , , , ) ( , , ) ( , , ),2

( , , , ) ( , , ) ( , , ),2

x

y

hu x y z t u x y t x y t

hx y z t x y t x y t

ϕ

ϕ

= −

= −v v (1.103)

— upper skin:

2 0

2 0

( , , , ) ( , , ) ( , , ),2

( , , , ) ( , , ) ( , , ).2

x

y

hu x y z t u x y t x y t

hx y z t x y t x y t

ϕ

ϕ

= +

= +v v (1.104)

Assumption 4 is written: 0( , , , ) ( , , ).x y z t x y t=w w (1.105)

The theory of sandwich plates is thus based on the determination of five functions of displacements and rotation, 0 0 0, , , and ,x yu ϕ ϕv w analogous to those introduced in the theory of laminates including the transverse shear effect (Section 1.5).

1.6.4 Strain field

The strain field is easily deduced from the displacement field. In the lower skin, the transverse shear strains are neglected and the strain field may be written in the form:

1 0

1 0

1 0 2

xx xx x

yy yy y

xy xy xy

hε ε κ

ε ε κ

γ γ κ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥

= −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

, (1.106)

with

0 0 00 0 0 0, , ,

, , .

xx yy xy

y yx xx y xy

u ux y y x

x y y x

ε ε γ

ϕ ϕϕ ϕκ κ κ

∂ ∂ ∂ ∂= = = +

∂ ∂ ∂ ∂∂ ∂∂ ∂

= = = +∂ ∂ ∂ ∂

v v

(1.107)

Similarly, the strain field is the upper skin is:

2 0

2 0

2 0 2

xx xx x

yy yy y

xy xy xy

hε ε κ

ε ε κ

γ γ κ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥

= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

. (1.108)

32 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

The strain field in the core can be expressed as the superposition of two strain fields :

— the in-plane strain field:

c 0

c 0

c 0

xx xx x

yy yy y

xy xy xy

z

ε ε κ

ε ε κ

γ γ κ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥

= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

, (1.109)

— the transverse shear strain field:

0

c

c0

yyz

xz x

y

x

ϕγ

γ ϕ

∂⎡ ⎤+⎢ ⎥⎡ ⎤ ∂⎢ ⎥=⎢ ⎥∂⎢ ⎥⎢ ⎥⎣ ⎦ +⎢ ⎥⎣ ⎦∂

w

w. (1.110)

1.6.5 Stress field

From the assumption 5, the core transmits only the transverse shear stresses:

c c c c 0xx yy xy zzσ σ σ σ= = = = , (1.111)

c c c c

44 45c c c c

45 55

yz yz

xz xz

C C

C C

σ γ

σ γ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤′ ′=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

′ ′⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦, (1.112)

where the transverse shear coefficients cijC′ are expressed as functions of the

transverse shear stiffnesses of the core as:

( )c c 2 c 2

44 44 55c c c

45 55 44c c 2 c 2

55 44 55

cos sin ,

sin cos ,

sin cos .

C C C

C C C

C C C

θ θ

θ θ

θ θ

′ = +

′ = −

′ = +

(1.113)

Assumption 6 implies that the transverse shear stresses are zero in all the layers k of the skins:

0k kxz yzσ σ= = . (1.114)

The other stresses are deduced from the strain field in the skins by the relation:

11 12 16

12 22 26

16 26 66

, 1, 2,

k ixx xxk iyy yyk ixy xyk

Q Q Q

Q Q Q i

Q Q Q

σ ε

σ ε

σ ε

⎡ ⎤ ⎡ ⎤ ⎡ ⎤′ ′ ′⎢ ⎥ ⎢ ⎥ ⎢ ⎥

′ ′ ′= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

′ ′ ′⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

(1.115)

for the layer k of the lower skin (i = 1) or upper skin (i = 2).

1.6. Theory of Sandwich Plates 33

1.6.6 Constitutive Equation

The in-plane resultants are obtained by the expression:

( )

2

1

2 2

2 2d d .

x xx xxh h h

y yy yyh h h

xy xy xy

NN z zN

σ σσ σσ σ

− +

− +

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

∫ ∫ (1.116)

The bending and twisting moments are:

( )

2

1

2 2

2 2d d ,

x xx xxh h h

y yy yyh h h

xy xy xy

MM z z z zM

σ σσ σσ σ

− +

− +

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

∫ ∫ (1.117)

and the shear resultants are expressed as:

2

2d

hy yz

hx xz

Qz

Qσσ−

⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦∫ . (1.118)

By substituting Expressions (1.12) and (1.115) into the preceding equations for the resultants and moments, we obtain the constitutive equation:

11 12 16 11 12 16

12 22 26 12 22 26

16 26 66 16 26 66

11 12 16 11 12 16

12 22 26 12 22 26

16 26 66 16 26 66

44 45

45 55

0 00 00 00 00 00 0

0 0 0 0 0 00 0 0 0 0 0

x

y

xy

x

y

xy

y

x

N A A A B B BN A A A B B BN A A A B B BM C C C D D DM C C C D D DM C C C D D DQ F FQ F F

⎡ ⎤ ⎡⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢=⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥⎢ ⎥⎣ ⎦ ⎣

0

0

0

0

0

xx

yy

xy

x

y

xy

yz

xz

ε

ε

γκκκ

γ

γ

⎡ ⎤⎤ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎢ ⎥⎦ ⎢ ⎥⎣ ⎦

, (1.119)

with

( )

( )

1 2

2 1

1 2

2 1

,

,2

,

,2

ij ij ij

ij ij ij

ij ij ij

ij ij ij

A A AhB A A

C C ChD C C

= +

= −

= +

= −

(1.120)

and

( )( )

( ) ( )

1 1

1 1

21

2 1 1

d dk

k

n nh h

ij ij ij ij kk k kh h hk k

A Q z Q z Q e−

−

− + = =

′ ′ ′= = =∑ ∑∫ ∫ , (1.121)

34 Chapter 1. Basic Elements on Laminate and Sandwich Composite Materials

( )( )

( ) ( )

1 1

1 1

21

2 1 1

d dk

k

n nh h

ij ij ij ij k kk k kh h hk k

C z Q z z Q z Q e z−

−

− + = =

′ ′ ′= = =∑ ∑∫ ∫ , (1.122)

( ) ( ) ( )

2 22

1

22

2 1 1

d dk

k

n nh h h

ij ij ij ij kk k kh hk k

A Q z Q z Q e−

+

= =

′ ′ ′= = =∑ ∑∫ ∫ , (1.123)

( ) ( ) ( )

2 22

1

22

2 1 1

d dk

k

n nh h h

ij ij ij ij k kk k kh hk k

C z Q z z Q z Q e z−

+

= =

′ ′ ′= = =∑ ∑∫ ∫ , (1.124)

cij ijF hC′= . (1.125)

where n1 and n2 are the numbers of layers respectively in the lower and upper skins.

The constitutive equation (1.119) has a form similar to Equation (1.81) obtain-ned in the case of the theory of laminates including the transverse shear effect. It differs from it by the terms Cij which induce an asymmetry in the stiffness matrix.

In the case of symmetric sandwich plates, the skins are identical. Hence:

1 2 1 2, .ij ij ij ijA A C C= = − (1.126) It results that:

2 22 , ,

0, 0.ij ij ij ij

ij ij

A A D hCB C

= =

= = (1.127)

In this case there is no coupling between stretching and bending. The constitutive equation takes a form identical to the constitutive equation of symmetric laminates with transverse shear.

1.6.7 Fundamental Equations

The governing equations of sandwich plates are obtained by introducing the constitutive equation (1.119) into the plate equations (1.30). The first three equations are identical to Equations (1.86) to (1.88), with Aij, Bij and Fij defined in (1.120) and (1.125). The last two equations are written:

( )2 2 2 2 2 2

0 0 0 0 0 011 16 66 16 12 66 262 2 2 22u u uC C C C C C C

x y x yx y x y∂ ∂ ∂ ∂ ∂ ∂

+ + + + + +∂ ∂ ∂ ∂∂ ∂ ∂ ∂

v v v

( )2 22 2 2

11 16 66 16 12 662 2 2 2 y yx x xD D D D D Dx y x yx y x

ϕ ϕϕ ϕ ϕ ∂ ∂∂ ∂ ∂+ + + + + +

∂ ∂ ∂ ∂∂ ∂ ∂

20 0

26 55 452 yx yD F F

x yyϕ

ϕ ϕ∂ ∂ ∂⎛ ⎞⎛ ⎞+ − + − +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠∂ ⎝ ⎠

w w

2 2

02 2

xxy

uR It t

ϕ∂ ∂= +

∂ ∂, (1.128)

1.6. Theory of Sandwich Plates 35

( )2 2 2 2 2 2

0 0 0 0 0 016 12 66 26 66 26 222 2 2 22u u uC C C C C C C

x y x yx y x y∂ ∂ ∂ ∂ ∂ ∂

+ + + + + +∂ ∂ ∂ ∂∂ ∂ ∂ ∂

v v v

( )2 22 2 2

16 12 66 26 66 262 2 2 2y yx x xD D D D D Dx y x yx y x

ϕ ϕϕ ϕ ϕ ∂ ∂∂ ∂ ∂+ + + + + +

∂ ∂ ∂ ∂∂ ∂ ∂

20 0

22 45 442 yx yD F F

x yyϕ

ϕ ϕ∂ ⎛ ⎞∂ ∂⎛ ⎞+ − + − +⎜ ⎟⎜ ⎟∂ ∂∂ ⎝ ⎠ ⎝ ⎠

w w

22

02 2

yxyR I

t tϕ∂∂

= +∂ ∂v . (1.129)

These equations differ from Equations (1.89) and (1.90) in the substitution of the coefficients Cij for the coefficients Bij.

The boundary conditions are identical to the conditions considered in Sub-section 1.5.3.

In the case of symmetric sandwich materials, the form of the governing equations is the same as that of symmetric laminates including the transverse shear effect.

CHAPTER 2

Dynamics of Systems with One Degree of Freedom

2.1 EQUATIONS OF MOTION OF A SYSTEM WITH ONE DEGREE OF FREEDOM

Mechanical vibrations are induced when an elastic system is disturbed from a position of stable equilibrium. In general, different modes of vibrations are gene-rated. The simplest configuration of a vibrating system is a system with one degree of freedom where the configuration is described by a single coordinate. The importance of the analysis of one degree of freedom system lies in the fact that the results which are established for this system constitute the basics for the analysis of the mechanical vibrations of complex structures.

As one degree of freedom, we consider the spring-mass system of Figure 2.1. A solid (S) of mass m is linked to a support (T) through an elastic spring cha-racterized by the stiffness constant k. Moreover, the solid (S) is connected with the support (T) such that horizontal displacement of the mass centre G of the solid is possible only along the axis (∆). The configuration of the system is determined completely by the coordinate x of the mass centre G from its equilibrium position.

Applying Newton law of motion, the motion equation of the solid (S) is written as: dmx k x X f= − + + , (2.1) where Xd is the component of the damping forces and f = f (t) is a disturbing force

FIGURE 2.1. Spring-mass system.

(T)

(R) (S) (∆)

OG x

y

2.2. Undamped Vibrations 37

acting externally on the solid (S). Equation of motion of any system with one degree of freedom can be expres-

sed in the form (2.1).

2.2 UNDAMPED VIBRATIONS

2.2.1 Equation of Motion

In the case where there is no damping, Xd = 0, the equation of motion (2.1) becomes:

mx k x f= − + . (2.2)

This equation can be written in the reduced form:

20x x qω+ = , (2.3)

introducing

20

km

ω = , (2.4)

and the reduced force:

1( ) ( )q q t f tm

= = . (2.5)

The quantity 0ω is the natural angular frequency of the system with one degree of freedom.

2.2.2 Free Vibrations

The free vibrations occur when the solid (S) is displaced from its equilibrium position and released. So, ( ) 0q t = , and the equation of motion is:

20 0x xω+ = . (2.6)

This equation is satisfied by taking 1 0cosx C tω= or 2 0sinx C tω= , where C1 and C2 are arbitrary constants. By addition of these solutions, we obtain the gene-ral solution of Equation (2.6) as:

1 0 2 0cos sinx C t C tω ω= + . (2.7)

The vibratory motion represented by this equation is a simple harmonic motion, where the constants C1 and C2 are deduced from the initial conditions. If at the initial instant ( 0)t = the solid (S) has a displacement x0 from its equili-brium position and if the solid is released with a velocity 0x , we have:

01 0 2

0, .xC x C

ω= = (2.8)

38 Chapter 2. Dynamics of Systems with one Degree of Freedom

FIGURE 2.2. Free vibrations as function of time.

Thus, the expression of the free vibrations of the solid (S) is:

00 0 0

0cos sinxx x t tω ω

ω= + . (2.9)

This expression can also be written in the form:

( )m 0cosx x tω ϕ= − , (2.10) with

2

2 0m 0

0

xx xω

⎛ ⎞= + ⎜ ⎟⎝ ⎠

, (2.11)

and

1 0

0 0tan x

xϕ

ω−= . (2.12)

The displacement x as a function of time t is reported in Figure 2.2. The maximum displacement xm is the amplitude of vibration and the angle ϕ is the phase difference. The interval of time T0 for which the motion repeats itself is the period of vibrations and is expressed as:

00

2T πω

= . (2.13)

The number f0 of cycles per unit of time is the natural frequency of the vibrations:

00

0

12

fT

ωπ

= = . (2.14)

2.2.3 Forced Vibrations. Steady State

2.2.3.1 Case of a Harmonic Disturbing Force

In numerous practical applications, the solid (S) is subjected to a periodic disturbing force or a periodic displacement is imposed to the spring support. The response of the system to these conditions is referred as forced vibrations.

0

-0.00

Time t

Dis

plac

emen

t x

x0 xm

T0

2.2. Undamped Vibrations 39

We consider the case where the solid (S) of the spring-mass system of Figure 2.1 is subjected to a periodic force of horizontal component m sinf tω . The term

m sinf tω is called a harmonic forcing function. Introducing this term into Equa-tion (2.3), we obtain:

20 m sinx x q tω ω+ = , (2.15)

with

m m1q fm

= . (2.16)

A particular solution of Equation (2.15) is:

3 sinx C tω= , (2.17)

where C3 is a constant which must satisfy Equation (2.15). We obtain:

m3 2 2

0

qCω ω

=−

. (2.18)

Thus, the particular solution is given by:

m2 20

sinqx tωω ω

=−

. (2.19)

The general solution of Equation (2.15) is obtained by adding this particular solution to the general solution (2.7) of the free vibrations. We obtain:

m1 0 2 0 2 2

0cos sin sinqx C t C t tω ω ω

ω ω= + +

−. (2.20)

The first two terms of this expression represent the free vibrations which were considered previously. These free vibrations are also called transient vibrations since in the practice these vibrations are rapidly damped by the damping forces (Section 2.3). The third term, depending on the disturbing force, represents the forced vibrations of the system, obtained in a steady state. These forced vibrations have the same period 2 /T π ω= as that of the disturbing force. They can be expressed as:

m2 2 20 0

1 sin1 /

qx tωω ω ω

=−

. (2.21)

The factor 2m 0/q ω is the displacement that the disturbing force qm would produce

if it were acting as a static force. The term ( )2 201/ 1 /ω ω− accounts for the dyna-

mical effect of the disturbing force. Its absolute value:

2 20

1( )1 /

K ωω ω

=−

, (2.22)

is usually called the magnification factor. It depends only of the frequency ratio 0/ω ω , ratio of the frequency of the disturbing force to the natural frequency of the

system. The variation of the magnification factor is plotted against the frequency ratio in Figure 2.3.

40 Chapter 2. Dynamics of Systems with one Degree of Freedom

FIGURE 2.3. Variation of the magnification factor as function of the frequency.

When the frequency of the disturbing force is small in comparison with the frequency of the free vibrations, the magnification factor is approximately equal to 1. The displacements are about the same as in the case of a static disturbing force.

When the frequency of the disturbing force approaches the natural frequency of the system, the magnification factor and thus the amplitude of the forced vibra-tions rapidly increases and becomes infinite when the force frequency exactly coincides with the natural frequency. The system is subjected to the resonance. In practice, there is a dissipation of energy due to damping and the amplitude of the vibrations is limited by the damping effects (Section 2.3).

When the frequency of the disturbing force increases beyond the natural frequency, the magnification factor decreases and approaches zero for high values of the frequency. The system may be considered as remaining stationary.

Considering the sign of the expression ( )2 201/ 1 /ω ω− , it is observed that for the

case where 0ω ω< this expression is positive. The displacement of the vibrating mass has the same sign as that of the disturbing force. The vibration is in phase with the excitation. In the case where 0ω ω> , the expression is negative and the displacement of the mass is in the direction opposite to that of the force. The vibration is out of phase.

2.2.3.2 Case of a Harmonic Displacement of the Spring Support

It is also possible to produce forced vibrations by imposing a displacement to the end support of the spring (Figure 2.4). In the case of a harmonic displacement, the support displacement is:

m sins sx x tω= , (2.23)

where xs is the displacement of the support from the equilibrium position. The displacement of the solid (S) referred to the support (T) is:

s rx x x= + , (2.24)

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

Frequency 0/ω ω

Mag

nific

atio

n fa

ctor

K

2.2. Undamped Vibrations 41

FIGURE 2.4. Displacement imposed to the spring end.

introducing the displacement xr referred to the end support of the spring. The component of the force exerted by the spring is:

rR k x= − , (2.25)

and the equation of motion (2.2) is modified as:

rmx k x= − . (2.26)

This equation leads to the motion equation of forced vibrations:

m sinsmx k x k x tω+ = − . (2.27)

This motion equation can be written in the reduced form (2.15), introducing:

2m m 0 ms s s

kq x xm

ω= = . (2.28)

The motion equation is reduced to the case of a disturbing force. In some applications, the end support of the spring is subjected to a harmonic

acceleration as: m sinsx a tω= . (2.29)

Considering relations (2.24) and (2.26), the equation of motion is written in the form:

s r rmx mx k x+ = − , (2.30) or

m sinr rmx kx ma tω+ = − . (2.31)

Whence, the reduced form of motion equation:

20 m sinr rx x q tω ω+ = , (2.32)

introducing m mq a= − . (2.33)

Again, the motion equation is reduced to the form (2.15) obtained in the case of a disturbing force.

y

(T)

(R) (S) (∆)

OG x

xs x

42 Chapter 2. Dynamics of Systems with one Degree of Freedom

However, it must be noted that Equation (2.32) is the motion equation expres-sed in the relative reference associated to the end support of the spring. The forced vibrations in this reference are transposed from Equation (2.21):

m2 20

20

1 sin1

rax tωω ω

ω

= −−

(2.34)

2.3 VIBRATIONS WITH VISCOUS DAMPING

2.3.1 Introduction

In previous discussions we did not consider the effects of dissipative forces. In practice, it is necessary to take into consideration the damping forces which may arise from several different sources, such as friction between dry sliding surfaces, friction between lubricated surfaces, air or fluid resistance, internal friction due to imperfect elasticity of materials, etc. Among all these processes of energy dissipation, the simplest case to deal with mathematically is the case where the damping force is proportional to the velocity. This damping process is called viscous damping. The damping processes of complex types are generally re-placed, for the purpose of the analyses, by an equivalent viscous damping. This equivalent damping is determined in such a way as to produce the same dissi-pation of energy by cycle as that produced by the actual damping processes (Section 2.4).

2.3.2 Equation of Motion with Viscous Damping

In the case of a viscous damping of the spring-mass system of Figure 2.1, the component Xd of the damping forces is proportional to the velocity. Thus:

dX cx= − . (2.35)

The coefficient c is the coefficient of viscous damping. The motion equation (2.1) becomes:

mx cx k x f+ + = . (2.36)

This equation can be rewritten in the reduced form:

202x x x qδ ω+ + = , (2.37)

introducing:

2cm

δ = . (2.38)

The parameter δ is the damping coefficient. Equation (2.37) is the general reduced form of the vibrations of a one degree of freedom system with viscous damping.

2.3 Vibrations with Viscous Damping 43

2.3.3 Free Vibrations

2.3.3.1 Characteristic Equation

The equation of free vibrations is derived from Equation (2.37) when f = 0. Hence:

202 0x x xδ ω+ + = . (2.39)

For solving this equation, a solution is assumed in the form:

rtx Ce= , (2.40)

where r is a parameter determined by reporting Expression (2.40) into Equation (2.39). Thus, we obtain the characteristic equation:

2 202 0r rδ ω+ + = . (2.41)

The solutions of this equation are:

1,2r δ ∆′= − ± , (2.42)

where ∆′ is the reduced discriminant of the characteristic equation:

2 20∆ δ ω′ = − . (2.43)

The final form of the solution of Equation (2.42) depends on the sign of ∆′ .

2.3.3.2 Case of Low Damping

In the case of low damping such as: 0δ ω< , (2.44)

the term ∆′ is negative and Equation (2.41) has two conjugated complex roots:

2

1,2 0 20

1r i δδ ωω

= − ± − . (2.45)

These two roots can be put in the form:

1,2 dr iδ ω= − ± , (2.46)

introducing the angular frequency:

2

0 20

1dδω ωω

= − . (2.47)

It is usual to introduce the viscous damping ratio ξ, defined as:

00

or δξ δ ξωω

= = . (2.48)

It results that:

20 1dω ω ξ= − , (2.49)

44 Chapter 2. Dynamics of Systems with one Degree of Freedom

and the two roots (2.46) are expressed as:

21,2 0 0 1r iξ ω ω ξ=− ± − . (2.50)

Finally, Equation (2.39) of the free vibrations can be written in the form:

20 02 0x x xξω ω+ + = . (2.51)

The two complex roots (2.46) are given by: 1 2, .d dr i r iδ ω δ ω= − + = − − (2.52)

or from (2.50):

( ) ( )2 21 0 2 01 , 1 r i r iω ξ ξ ω ξ ξ= − + − = − − − . (2.53)

Substituting these roots into Expression (2.40), we obtain two solutions of Equa-tion (2.39) or Equation (2.41). Any linear combination of these solutions is also a solution. For example:

( )1 211 1 cos ,

2r t r t t

dCx e e C e tδ ω−= + = (2.54)

( )1 212 2 sin .

2r t r t t

dCx e e C e t

iδ ω−= − = (2.55)

Adding these solutions, we obtain the general solution of Equation (2.39) in the form:

( )1 2cos sintd dx e C t C tδ ω ω−= + , (2.56)

where C1 and C2 are constants which are determined from the initial conditions. The factor te δ− in Solution (2.56) decreases with time, and the vibrations gene-

rated by the initial conditions are gradually damped out. The expression between brackets in Equation (2.56) is of the same form that the

one obtained in the case of vibrations without damping (Equation (2.7)). It repre- sents a harmonic function of angular frequency given by Equation (2.47) or (2.49). This frequency is the angular frequency of the damped vibrations. The variation

0/dω ω of this frequency referred to the natural frequency of the free undamped vi-brations is plotted in Figure 2.5 as function of the damping ratio 0/ξ δ ω= . From this figure, it is observed that the frequency of the damped vibrations is close to the frequency of undamped vibrations, even for notable value of the damping ratio. For 0.1ξ = , the damped frequency is 00.995dω ω= ; for 0.2ξ = , the fre-quency is equal to 00.98 ω and for 0.3ξ = , the damped frequency is still 00.95 .ω

Constants C1 and C2 in Expression (2.56) are deduced from the initial condi-tions at time 0t = : solid is displaced from its equilibrium position by a displace-ment x0 and the solid is released with a velocity 0x . Thus, we obtain:

0 01 0 2,

d

x xC x C δω+= = . (2.57)

Thus, the motion of damped free vibrations of a one degree of freedom system is:

0 00 cos sint

d dd

x xx e x t tδ δω ωω

− +⎛ ⎞= +⎜ ⎟⎝ ⎠

. (2.58)

2.3 Vibrations with Viscous Damping 45

FIGURE 2.5. Damped natural frequency variation as a function of damping.

This expression can be rewritten in the form:

( )m costdx x e tδ ω ϕ−= − , (2.59)

in which the maximum value is:

( )22 2 2 0 0

m 1 2 0 2d

x xx C C x δω+

= + = + , (2.60)

and the phase angle is given by:

1 1 0 02

1 0tan tan

d

x xCC x

δϕω

− − +⎛ ⎞= = ⎜ ⎟⎝ ⎠

. (2.61)

Equation (2.58) may be considered as representing a pseudo-harmonic motion, having an exponentially decreasing amplitude m

tx e δ− , a phase angle ϕ and a pseudo-period:

2d

dT π

ω= . (2.62)

The graph of the motion is plotted in Figure 2.6. The displacement-time curve is tangent to the envelopes m

tx e δ−± at the points m1, 1m′ , m2, 2m′ , etc., at instants of time separated by the time interval /2dT . Because the tangents at these points are not horizontal, the points of tangency do not coincide with the points of extreme displacements from the equilibrium position. If the damping ratio is low, the difference in these points may be neglected. For any damping, the time interval between two consecutive extreme positions is however equal to half the pseudo-period. Indeed, the velocity of the vibrating solid is:

( ) ( )m mcos sint td d dx x e t x e tδ δδ ω ϕ ω ω ϕ− −= − − − − . (2.63)

The velocity is equal to zero when:

( )tan dd

t δω ϕω

− = − , (2.64)

damping ξ 1 0 0

1

dfr

eque

ncy

ω

46 Chapter 2. Dynamics of Systems with one Degree of Freedom

FIGURE 2.6. Pseudo-harmonic motion.

which leads effectively to / /2d dt Tπ ω= = . The ratio between two successive amplitudes mix and m 1ix + is:

m

m 1dTi

i

x ex

δ

+= . (2.65)

The quantity l dTδ δ= is the logarithmic decrement and is given by:

m

m 1

2ln il d

i d

x Tx

πδδ δω+

= = = . (2.66)

This equation can be used for an experimental determination of the damping coef-ficient δ. However, a greater accuracy is obtained by measuring the extreme am-plitudes separated by n pseudo-cycles. In this case we have:

m

mdn Ti

i n

x ex

δ

+= , (2.67)

and the logarithmic decrement is obtained by:

m

m

1 ln il

i n

xn x

δ+

= . (2.68)

2.3.3.3 Case of High Damping

In the case of high damping such as:

0δ ω> , (2.69)

the term ∆′ is positive and Equation (2.41) has two roots r1 and r2 which are real and negative. The general equation of the motion equation (2.39) is:

0

0

m1

m2

m3 m4

1m′

2m′

3m′

/ dϕ ω

m3x m4x

m1xm2x

dT

mx

0x

0

Time t

Dis

plac

emen

t x

t

mx e δ−

2.3 Vibrations with Viscous Damping 47

FIGURE 21.7. Displacement as a function of time in the case of an aperiodic motion.

1 21 2

r t r tx C e C e= + . (2.70)

In this case the viscous damping is such as when the solid is displaced from its equilibrium position, it does not vibrate but creeps gradually back to that position. The motion is said aperiodic.

Constants C1 and C2 in Equation (2.70) are evaluated from the initial condi-tions which lead:

1 2 0 1 1 2 2 0, .C C x r C r C x+ = + = (2.71) We deduce:

0 2 0 1 0 01 2

1 2 1 2, ,x r x r x xC C

r r r r− −= =− −

(2.72)

and Equation (2.70) becomes:

1 20 2 0 1 0 0

1 2 1 2

r t r tx r x r x xx e er r r r

− −= +− −

. (2.73)

The motion depends on the values of 0 0, and x xδ . Figure 2.7 shows examples of displacement-time curves for a fixed value of the initial displacement x0 and seve-ral values of the initial velocity 0x (positive, zero or negative).

2.3.3.4 Critical Damping

The transition between the pseudo-harmonic motion and the aperiodic motion corresponds to a viscous damping crδ called critical damping given by:

cr 0δ ω= . (2.74)

In this particular case, Equation (2.41) has a repeated root:

1 2 0r r ω= = − , (2.75)

00

Time t

Dis

plac

emen

t x

x0 0 0x >

0 0x =

0 0x <

48 Chapter 2. Dynamics of Systems with one Degree of Freedom

and the solution of the motion equation is:

( ) 01 2

tx C C t e ω−= + . (2.76)

Taking the initial conditions into account, we obtain:

1 0 2 0 0 0, ,C x C x xω= = + (2.77)

and Solution (2.76) is written as:

( )[ ] 00 0 0 0

tx x x x t e ωω −= + + . (2.78)

The displacement-time curves are similar to the curves obtained in the case of aperiodic motion (Figure 2.7), but the solid comes back to the equilibrium posi-tion more rapidly.

2.3.4 Vibrations in the case of Harmonic Disturbing Force

2.3.4.1 Time Domain

As in Section 2.2.3 we consider the case where the solid (S) of the spring-mass system (Figure 2.1) is subjected to a harmonic force of horizontal component

m cosf tω . Under this condition, the motion equation (2.36) of the forced vibrations becomes:

m cosmx cx k x f tω+ + = . (2.79)

The reduced form (2.37) is written as :

20 m2 cosx x x q tδ ω ω+ + = , (2.80)

with

m m1q fm

= . (2.81)

Equation (2.80) constitutes the general form of the forced vibrations of a system with one degree of freedom in the case of harmonic disturbing force.

A particular solution of Equation (2.80) is: cos sinx A t B tω ω= + , (2.82)

where A and B are constants which are determined by substituting Expression (2.82) of this particular solution in the general equation of motion (2.80). We obtain:

( ) ( )2 2 2 20 m 02 cos 2 sin 0A B A q t B A B tω δω ω ω ω δω ω ω− + + − + − − + = . (2.83)

This equation is satisfied for all values of time t if :

2 2

0 m2 2

0

2 ,

2 0.

A B A q

B A B

ω δω ω

ω δω ω

− + + =

− − + = (2.84)

2.3 Vibrations with Viscous Damping 49

From which:

( )

( )

2 20

m22 2 2 20

m22 2 2 20

,4

2 .4

A q

B q

ω ω

ω ω δ ωδω

ω ω δ ω

−=− +

=− +

(2.85)

Next, the total solution of Equation (2.80) is obtained by adding the particular solution (2.82) to the general solution of Equation (2.80) with the second member equal to zero, thus to the general solution of Equation (2.39) of the free vibrations.

We consider hereafter the case of low damping for which the damping is lower than the critical damping. Thus, the solution of Equation (2.80) is given by:

( )1 2cos sin cos sintd dx e C t C t A t B tδ ω ω ω ω−= + + + . (2.86)

The first term represents the damped free vibrations, whereas the last two terms represent the damped forced vibrations. The free vibrations have the angular fre-quency dω as determined in Section 2.3.3, when the forced vibrations have the

angular frequency of the disturbing force. Due to the factor ,te δ− the free vibrations gradually decrease, then vanish, leaving only the steady forced vibra-tions. These vibrations are maintained as long as the disturbing force is applied. We study the forced vibrations hereafter.

In the case of steady-state, the harmonic response (2.82) may be written in the form:

( )m cosx x tω ϕ= − , (2.87) with

2 2 1m , tan .Bx A B

Aϕ −= + = (2.88)

Whence:

( )

m2

m 0m 2 2 22 2 2 2 2

02 00

24 1

qqx ω

ξωω ω δ ω ωωω

= =⎛ ⎞− + ⎛ ⎞− + ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

, (2.89)

and

1 1 02 2 20

20

22tan tan

1

ξωωδωϕ

ω ω ωω

− −= =− −

. (2.90)

When a static load fm is applied to the system, the static displacement xst is deduced from (2.79) as:

st m mk x f mq= = . (2.91)

50 Chapter 2. Dynamics of Systems with one Degree of Freedom

From which:

m mst 2

0

q fxkω

= = . (2.92)

Thus, considering Equations (2.89) and (2.92) the amplitude xm of the displace-ment may be written in the form:

m st( )x K xω= , (2.93)

in which K(ω) is the magnification factor expressed by:

2 22

2 00

1( )

1 2

K ωω ωξ

ωω

=⎛ ⎞ ⎛ ⎞− + ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

. (2.94)

So, the damped harmonic vibrations can be written as:

( )st ( ) cosx x K tω ω ϕ= − . (2.95)

2.3.4.2 Frequency Domain

The steady state of the harmonic forced vibrations can be studied in the fre-quency domain by representing the excitation ( )f t and the response ( )x t in com-

plex forms ( ) and ( ) ,i t i tF e X eω ωω ω respectively. The quantities ( )F ω and ( )X ω are the complex amplitudes associated to the excitation and response, respecti-vely. In the case of the harmonic forced vibrations considered previously, the complex amplitudes are:

m m( ) , ( ) .iF f X x e ϕω ω −= = (2.96)

Introducing these complex forms into the motion equation (2.80) leads to the complex equation of motion which may be written in the following forms:

( )2 20

12 ( ) ( )i X Fm

ω ω δω ω ω− + = , (2.97)

or

( )2 20 0

12 ( ) ( )i X Fm

ω ω ξω ω ω ω− + = . (2.98)

Thus, the response as a function of the excitation in complex form is expressed as:

1( ) ( ) ( )X H Fm

ω ω ω= , (2.99)

introducing the transfer function of the vibration system expressed by:

2 20 0

1( )2

Hi

ωω ω ξω ω

=− +

. (2.100)

2.3 Vibrations with Viscous Damping 51

When the frequency approaches zero, the transfer function ( )H ω approaches 201 ω and the function X(ω = 0) is identified with the static response xst introduced

in Equation (2.92). So, Expression (2.99) of the response may be rewritten as:

1( ) ( ) ( )rX H Fk

ω ω ω= , (2.101)

introducing the reduced transfer function:

2

2 00

1( )1 2

rHi

ωω ωξ

ωω

=− +

. (2.102)

So, the complex amplitude X(ω) is simply given by:

st( ) ( )rX H xω ω= . (2.103)

Next, the amplitude xm of the harmonic steady-state vibration is deduced from the previous expression by considering the modulus of X(ω ), which yields:

m st( )rx H xω= , (2.104) with

2 22

20 0

1( )21

rH ωξωω

ω ω

=⎛ ⎞ ⎛ ⎞− +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

.

The modulus of the function Hr(ω ) is identified with the magnification factor in-troduced in (2.94).

The phase ϕ is the opposite of the argument of the transfer function or of the reduced function Hr(ω ). Thus:

1 02 2

0

2 /arg ( ) tan1 /

rH ξω ωϕ ωω ω

−= − =−

, (2.105)

which is the result expressed in Equation (2.90).

2.3.4.3 Effect of the Frequency of the Disturbing Force

The amplitude xm of the harmonic forced vibrations, referred to the static displacement xst, is simply given either by the magnification factor (2.94) or by the modulus of the reduced transfer function:

m

st( ) ( )r

x H Kx

ω ω= = , (2.106)

Figure 2.8 shows the variation of the magnification factor as a function of the reduced frequency 0/ω ω for different values of damping. From these curves it is observed that when the angular frequency is small compared to the natural

52 Chapter 2. Dynamics of Systems with one Degree of Freedom

0.0 0.5 1.0 1.5 2.00

1

2

3

4

Frequency 0/ω ω

Mag

nific

atio

n fa

ctor

K

0ξ =

0.10

0.15

0.20

0.25

0.50

0.70 1.00

FIGURE 2.8. Variation of the reduced amplitude of harmonic vibrations as function of the frequency for different values of damping.

frequency, the value of the magnification factor is not greatly different from unity. Thus, the amplitude of vibrations is approximately the one which would be pro-duced by a static disturbing force.

When the angular frequency of the excitation is large compared to the natural frequency, the value of the magnification factor tends toward zero, regardless the value of damping. So, a high frequency disturbing force induces practically no forced vibrations of the system.

The curves of Figure 2.8 show that for low values of damping the magni-fication factor grows rapidly with the frequency, and its value near resonance is very sensitive to the values of damping. It is also observed that the maximum value occurs for a value of 0/ω ω less than unity. Setting the derivative of the ma-gnification factor with respect to 0/rω ω ω= equal to zero, we find that the maxi-mum occurs for a reduced frequency mrω defined by:

2mm

01 2r

ωω ξω

= = − . (2.107)

The maximum amplitude is then given by:

m m 21( ) ( )

2 1rK Hω ω

ξ ξ= =

−. (2.108)

For small damping ratios the maximum value of the magnification factor occurs very near to the undamped natural frequency and the maximum is approximately:

m m1( ) ( )

2rK Hω ωξ

= ≈ . (2.109)

2.3 Vibrations with Viscous Damping 53

For example, for 0.20ξ = , the maximum occurs for 00.96 ω and its value is 2.55. Then, when the damping increases, the value of angular frequency mω decreases and vanishes when 1ξ = .

In the case of low damping, the peak width of the magnification factor can be evaluated by considering the reduced frequencies rω for which the magnification factor is reduced by a factor 1/ 2 with respect to the maximum, corresponding to a reduction of –3 dB. We obtain:

( )2 22 2 2

1 1 12 2 11 4r r

ξ ξω ξ ω=

−− +, (2.110)

which leads to: ( )4 2 2 2 42 1 2 1 8 8 0r rω ξ ω ξ ξ− − + − + = . (2.111)

The solutions for this equation are:

( )( )

2 2 21

2 2 22

2 1 2 2 1 ,

2 1 2 2 1 .

r

r

ω ξ ξ ξ

ω ξ ξ ξ

= − + −

= − − − (2.112)

An approximate solution can be formulated in the case of low values of damping by expressing that 1rω and 2rω are not greatly different from the frequency mrω of the maximum. Whence:

2 21 2 1 2 1 2 m 1 2( )( ) 2 ( )r r r r r r r r rω ω ω ω ω ω ω ω ω− = + − ≈ − , (2.113)

or considering (2.107):

2 2 21 2 2 1 2r r rω ω ξ ∆ω− ≈ − , (2.114)

where r∆ω is the frequency band corresponding to –3 dB reduction of the magni-fication factor. Considering Equations (2.112), we obtain:

2

212

1 2r

ξ∆ω ξξ

−≈−

. (2.115)

In the case of low damping the bandwidth is simplified as: 2r∆ω ξ≈ . (2.116)

The frequency response of the damped system is also characterized by the phase angle ϕ expressed by Equations (2.90) and (2.105). Figure 2.9 shows the variation of the phase angle as a function of the frequency obtained for different values of damping.

2.3.4.4 Damping Modelling Using Complex Stiffness

Another way to take account of the energy dissipation in the case of forced harmonic vibrations consists in introducing a complex stiffness k∗ in the form:

54 Chapter 2. Dynamics of Systems with one Degree of Freedom

FIGURE 2.9. Variation of the phase angle as a function of the frequency, for different values of damping.

( )1k k iη∗ = + , (2.117)

where η is the structural damping coefficient. Then, the complex equation is deduced from the motion equation (2.2) without damping (c = 0) by introducing the complex forms (2.96) and substituting k∗ for k. We obtain:

( )2 2 20 0

1( ) ( )i X Fm

ω ω ηω ω ω− + = . (2.118)

This form is similar to Expressions (2.97) and (2.98), which leads to:

200

2 2ω ωη δ ξωω

= = . (2.119)

Substituting this expression into Equation (2.102), the reduced transfer function for the steady state response is written as:

2

20

1( )1

rHi

ωω ηω

=− +

, (2.120)

and the magnification factor becomes:

22

220

1( ) ( )

1

rK Hω ωω ηω

= =⎛ ⎞

− +⎜ ⎟⎝ ⎠

. (2.121)

0.0 0.5 1.0 1.5 2.0 2.5 3.00

20

40

60

80

100

120

140

160

180

0.1

0ξ =

0.2

0.5

1 2 4

Frequency 0/ω ω

Phas

e an

gle

φ (°

)

2.3 Vibrations with Viscous Damping 55

When the frequency is equal to the natural frequency of the vibrating system, the magnification factor is:

0 01( ) ( )rK Hω ωη

= = ,

and the amplitude of the forced vibrations is deduced from (2.106) as:

mm st

1 1fx xkη η

= = . (2.122)

At this frequency the phase angle is deduced from Equation (2.105), which yields:

tanϕ η= . (2.123)

Thus, coefficient η is also called the loss factor.

2.3.5 Vibrations in the case of Periodic Disturbing Force

A periodic disturbing force ( )f t of period T can be expressed in the form of Fourier series as:

( )01

( ) cos sinn nn

f t a a n t b n tω ω∞

=

= + +∑ , (2.124)

with 2 /Tω π= . The quantities a0, an and bn are obtained by:

00

1 ( ) dT

a f t tT

= ∫ , (2.125)

0

2 ( )cos dT

na f t n t tT

ω= ∫ , (2.126)

0

2 ( )sin dT

nb f t n t tT

ω= ∫ . (2.127)

Considering Equation (2.124), the motion equation leads to:

( )20 0 0

1

2 cos sinn nn

x x x q q n t p n tξω ω ω ω∞

=

+ + = + +∑ , (2.128)

with

00 , , .n n

n na a bq q pm m m

= = = (2.129)

The general solution of Equation (2.128) consists of the sum of the free vibra-tions and the forced vibrations. The free vibrations diminish and vanish with damping. The forced vibrations are obtained by superimposing the steady state forced vibrations produced by every terms of the second member of Equation (2.128). These vibrations can be obtained by applying the results obtained in the previous section (Subsection 2.3.4). In the practice, the coefficients of the terms of

56 Chapter 2. Dynamics of Systems with one Degree of Freedom

the series decrease when n increases. So, the analysis will be limited to a value N of n for which the terms of upper orders can be neglected. Considering the results established in Subsection 2.3.4, it can be concluded that forced vibrations with high amplitudes may occur when the period of one of the terms of series (2.124) coincides with the period of the natural vibrations of the system, i.e. if the period T of the disturbing force is equal to, or a multiple, of the damped period Td.

2.3.6 Vibrations in the case of Arbitrary Disturbing Force

The differential equation of motion for a damped one-degree system subjected to an arbitrary disturbing force is given by Equations (2.36) and (2.37). The arbi-trary reduced force ( )q t is represented in Figure 2.10.

At any instant t′ , we may consider (Figure 2.10) an impulse of height ( )q t q′ = and width dt′ . This impulse imparts to each unit of mass an instantaneous acceleration from the instant t′ given by:

dd

x x qt

= =′

, (2.130)

which leads to an increase in velocity from t′ given by: d dx q t′= , (2.131)

regardless of what other forces, such as the spring force, may be acting, and regardless of the displacement and velocity of solid (S) at the instant t′ . Then, the increment of displacement at instant t posterior to t′ , is deduced from Equation (2.58) by substituting the velocity increment (2.131) for the initial velocity 0x (with a zero initial displacement) and substituting the instant t t′− for instant t (in Equation (2.58), the disturbing force is exerted at instant 0t = , whereas the force

( )q t′ is applied at t t′= ). We obtain:

( ) dd sin ( )t td

d

q tx e t tδ ωω

′− − ′ ′= − . (2.132)

FIGURE 2.10. Arbitrary force as a function of time.

t′ dt′ Time t

Red

uced

forc

e q

(t) q

2.4 Equivalent Viscous Damping 57

Since each impulse ( ) dq t t′ ′ between 0t′ = and t t′ = produces an increment of displacement given by expression (2.132), the total displacement x(t) which results from the disturbing force is obtained by integration between 0 and t:

0( ) ( )sin ( )d

ttt

dd

ex t e q t t t tδ

δ ωω

−′ ′ ′ ′= −∫ . (2.133)

This form is referred as Duhamel’s integral. It includes both steady state and transient terms. The integral can be evaluated by an analytical method or a nume-rical process.

To take account of the effect of possible initial conditions of displacement x0 and velocity 0x , it is necessary to add to the results (2.133) the solution for the initial conditions considered in Equation (2.58). Thus, the total solution is:

0 00

0 0

1( ) cos sin ( )sin ( )dt

t td d d

d

x xx t e x t t e q t t t tδ δδω ω ωω ω

′− ⎡ ⎤+ ′ ′ ′= + + −⎢ ⎥⎣ ⎦∫ .

(2.134)

2.4 EQUIVALENT VISCOUS DAMPING

2.4.1 Introduction As reported in Section 2.3.1, the different types of damping may be replaced

by an equivalent viscous damping, leading to the linear differential equation (2.36) of the damped motion. Only structural damping will be considered here-after. The other damping processes can be analyzed in a similar way.

2.4.2 Energy Dissipated in the case of Viscous Damping

The work done per cycle by the disturbing force m( ) cosf t f tω= during the steady state response is:

0cos d

T

mW f x t tω= ∫ . (2.135)

The velocity x may be obtained by differentiating Equation (2.87) with respect to time. Whence:

m sin( )x x tω ω ϕ= − − . (2.136)

Substituting this expression into Equation (2.135), then integrating leads to the expression of work as:

m m sinW x fπ ϕ= . (2.137)

Similarly the energy U∆ dissipated per cycle by the viscous damping force cx is given by:

58 Chapter 2. Dynamics of Systems with one Degree of Freedom

0d

TU cxx t∆ = ∫ , (2.138)

which leads to:

2mU cxπ ω∆ = . (2.139)

For a harmonic steady state, the work done by the disturbing force is equal to the dissipated energy. From which the amplitude of the displacement is deduced as:

mm sinfx

cϕ

ω= . (2.140)

When the angular frequency is equal to the natural frequency ( 0ω ω= ), the phase angle ϕ is /2π and the displacement amplitude is:

mm 0

0( ) fx

cω

ω= . (2.141)

This result coincides with the result (2.109) obtained for low values of damping. The equivalent viscous damping constant will be obtained by equating Expres-

sion (2.139) of the energy dissipated by viscous damping to the energy dissipated by the actual damping process. This energy will be derived from Equation (2.130).

2.4.3 Loss Factor and Specific Damping Capacity

The maximum elastic energy stored per cycle in the case of the spring-mass system is:

2d

12 mU k x= . (2.142)

Thus, the energy dissipation is characterized by:

d

2U cU k

π ω∆ = , (2.143)

or considering Equations (2.38) and (2.119):

d

2UU

πη∆ = . (2.144)

In Equations (2.138) and (2.139) the dissipated energy is evaluated per cycle in the case of a harmonic disturbing force. The energy dissipated by viscous damping can be also evaluated during a pseudo-cycle of the damped vibrations as:

ddt T

TU cxx t

+∆ = ∫ . (2.145)

For low values of damping, it may be considered that the amplitude xa of

2.4 Equivalent Viscous Damping 59

vibrations during a pseudo-cycle is practically constant and equal to:

mt

ax x e δ−≈ . (2.146)

It results that the integration of Equation (2.145) leads to:

2a dU cxπ ω∆ ≈ . (2.147)

The maximum elastic energy stored during the pseudo-cycle is:

2d

12 aU kx= , (2.148)

and the energy dissipation per cycle is characterized by the coefficient:

d

UU

ψ ∆= , (2.149)

where U∆ and dU are expressed by Equations (2.147) and (2.148), respectively. We obtain:

d 0 0

2 2 4dl l

UU

ω δψ δ δ πω ω

∆= ≈ ≈ ≈ . (2.150)

Coefficient ψ is the specific damping capacity and the comparison of Equations (2.144) and (2.149) leads to the relation:

2ψ πη= . (2.151)

Finally, the relations between the different coefficients which characterize the viscous damping are reported in Table 2.1.

2.4.4 Structural Damping

The structural damping is associated to internal friction in materials that are not perfectly elastic. For these materials, the loading stress-strain curve for increasing TABLE 2.1. Parameters characterizing viscous damping of a one degree of freedom system.

δ ξ lδ η

damping coefficient δ δ 0ξω 2

dl

ω δπ

20

2ω η

ω

damping ratio ξ 0

δω

ξ 2 24l

l

δ

π δ+0

2ω ηω

logarithmic decrement lδ 2d

π δω

2

2

1

πξ

ξ−

lδ 20

d

ωπ ηωω

structural damping η ( 2η ψ π= ) 2

0

2ω δω

0

2 ω ξω

20

dl

ωω δπω

η

60 Chapter 2. Dynamics of Systems with one Degree of Freedom

FIGURE 2.11. Stress-strain curve for successive loading and unloading of a material

levels of stress and strain is different from the unloading curve. Figure 2.11 shows the hysteresis loop obtained in the case of one cycle of vibration. The expe-rimental results show that the energy dissipated per cycle is approximately pro-portional to the square of the strain amplitude. So, the work sU dissipated by structural damping may be written as:

2ms sU xα= , (2.152)

in which sα is a parameter which characterizes the structural damping of the material considered. Equating expressions (2.139) and (2.152) of the dissipated energies leads to the equivalent viscous damping constant:

eqsc α

πω= . (2.153)

This relation associated to relations (2.38) and (2.119) leads to the expression of the equivalent loss factor:

eqsk

αηπ

= . (2.154)

In fact, this relation introduces the stiffness k of the equivalent spring-mass system. In practice the loss factor or the specific damping capacity is evaluated in experimental investigation considering Equation (2.144) or (2.151), respectively, where dU is the strain energy stored per cycle and U∆ is the dissipated energy.

loading

unloading

Strain

Stre

ss

CHAPTER 3

Beam Bending and Cylindrical Bending Vibrations of Undamped Laminate and Sandwich Materials

3.1 INTRODUCTION

The theory of beams considers that the length a of a beam is much greater than its width b, a b (Figure 3.1). It results that the investigation of the beam ben-ding vibrations can be reduced to a one-dimension analysis. In this chapter we will consider the bending vibrations of beams made of symmetric laminates or sandwiches for which there is no stretching bending coupling. The importance of developing the analysis of beam behaviour is related to the use of beams as basic elements of structures and to the mechanical characterization of laminate and sandwich materials on test specimens in the form of beams.

Another type of one-dimension analysis is that of the investigation of cylin-drical bending which concerns plates that have a high length-to-width ratio. Cylindrical bending vibrations will be studied in the last section of this chapter.

FIGURE 3.1. Beam element.

a

b

h

y

z

x

2h

62 Chapter 3. Beam Bending and Cylindrical Bending Vibrations of Undamped Materials

3.2 EQUATION OF MOTION OF SYMMETRIC LAMINATE BEAMS

In the case of pure bending of a symmetric laminate beam, the constitutive equation (1.60) reduces to:

11 12 16

12 22 26

16 26 66

x x

y y

xy xy

M D D DM D D DM D D D

κκκ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

, (3.1)

where κx, κy and κxy are the curvatures of plate defined in Equations (1.51):

2 2 2

0 0 02 2( , , ), ( , , ), 2 ( , , ).x y xyx y t x y t x y t

x yx yκ κ κ∂ ∂ ∂= − = − = −

∂ ∂∂ ∂w w w (3.2)

Equation (3.1) can be written in the following inverted form:

1 1 111 12 16

1 1 112 22 26

1 1 116 26 66

x x

y y

xy xy

D D D MD D D M

MD D D

κκκ

− − −

− − −

− − −

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

, (3.3)

where the 1ijD− are the elements of the inverse matrix of [Dij]:

( ) ( )

( ) ( )

( ) ( )

1 2 111 22 66 26 12 16 26 12 66

1 1 216 12 26 16 22 22 11 66 16

1 1 226 12 16 26 11 66 11 22 12

1 1, ,

1 1, ,

1 1, ,

D D D D D D D D D

D D D D D D D D D

D D D D D D D D D

∆ ∆

∆ ∆

∆ ∆

− −

− −

− −

= − = −

= − = −

= − = −

(3.4)

and ∆ is the determinant of the matrix [Dij]:

2 2 211 22 66 12 16 26 11 26 22 16 66 122D D D D D D D D D D D D∆ = + − − − .

The beam theory makes the assumption that in the case of bending along the x-direction, the bending and twisting moments My and Mxy are zero:

0, 0.y xyM M= = (3.5) Relations (3.2) and (3.3) thus lead to:

2

10112x xD M

xκ −∂= − =

∂w . (3.6)

Lastly, the beam theory makes the additional assumption that the deflection is a function of x only:

3.3. Natural Modes 63

0 0 ( , )x t=w w . (3.7)

So, the mode shape of beam only depends on the coordinate x. In the framework of the beam theory, Equation (1.42) is written in the form:

2 2

02 2

xs

M qx t

ρ∂ ∂+ =∂ ∂

w . (3.8)

Equations (3.6) and (3.8) lead to:

2 4

0 02 1 4

11

1 ( , )s q x tt D x

ρ −∂ ∂+ =∂ ∂w w , (3.9)

where ( , )q x t is the pressure load applied to the beam. Finally, the differential equation of motion for an undamped laminate beam

may be written as:

2 4

0 02 4 ( , )s sk q x t

t xρ ∂ ∂+ =

∂ ∂w w , (3.10)

introducing the stiffness per unit area given by:

111

1sk

D−= . (3.11)

It has to be noted that the effective bending modulus Ex of the beam is exp-ressed as:

3 111

12xE

h D−= . (3.12)

3.3 NATURAL MODES

3.3.1 Mode Shapes

In the case of free vibrations, q = 0, and the motion equation (3.10) is:

2 4

0 02 4 0s sk

t xρ ∂ ∂+ =

∂ ∂w w . (3.13)

This equation of the free transverse vibrations may be rewritten in the form:

2 4

2 40 002 4 0,a

t xω∂ ∂+ =

∂ ∂w w (3.14)

introducing the angular frequency of the undamped beam:

3

0 2 2 1 211

1 1 1 1 .12s x

s ss

k E ha a D a

ω ρ ρρ −= = = (3.15)

64 Chapter 3. Beam Bending and Cylindrical Bending Vibrations of Undamped Materials

When the beam vibrates in its ith natural mode, the harmonic transverse dis-placement at a point of coordinate x is:

( ) ( )( )0 , cos sin ,i i ix t x A t B tω ω= +w w (3.16)

where ( )i xw is the mode shape of the natural mode and ωi is its angular fre-quency. Substitution of Equation (3.16) into Equation (3.14) results in:

4 2

4 4 20

d 1 0.d

i ii

X Xx a

ωω

− = (3.17)

The general solution for Equation (3.17) may be written in the form:

( ) 1 2 3 4sin cos sinh coshi i i i ix x x xx C C C Ca a a aλ λ λ λ= + + +w . (3.18)

Introducing this solution in Equation (3.17) leads to the evaluation of the natural frequency as: 2

0i iω λ ω= . (3.19)

The parameter iλ and the constants, C1, C2, C3 and C4, in Equation (3.18) are determined by considering the boundary conditions at the ends of the beam under consideration.

For example, at an end which is simply supported, the transverse displacement and the bending moment are equal to zero, and the boundary conditions are:

2

2d0, 0.d

ii

XXx

= = (3.20)

At a clamped end the transverse displacement and the slope are zero. In this case the conditions are:

d0, 0.di

iXX x= = (3.21)

At a free end the bending moment and the shear resultant both vanish. So, the conditions are:

2 3

2 3d d0, 0.d d

i iX Xx x

= = (3.22)

For the two ends of a beam, we will have four such end conditions. It results that the constants, C1, C2, C3 and C4, are determined to within an

arbitrary constant. So, we will express the shape mode as:

( ) ( )i i ix C X x=w , (3.23)

where the function ( )iX x is expressed in the form (3.18) introducing only three constants: one constant is taken equal to 1.

Then the normal modes can be superimposed to obtain the total response of the beam as:

( ) ( )( )01

, cos sini i ii

x t X x A t B tω ω∞

=

= +∑w . (3.24)

3.3. Natural Modes 65

3.3.2 Properties of the Mode Shapes

Equation (3.17) may be written for the mode i in the form:

4

4dd

ii i

X Xx

α= , (3.25)

where

2

4 20

1 ii

aωαω

= . (3.26)

For the mode j, Equation (3.25) is:

4

4d

dj

j jX

Xx

α= . (3.27)

Multiplying Equation (3.25) by jX and Equation (3.27) by iX and integrating over the length of the beam, we obtain:

4

40 0

d d dd

a ai

j ji iX X x X X xx

α=∫ ∫ , (3.28)

4

40 0

dd d

d

a ajjji i

XX x X X x

xα=∫ ∫ . (3.29)

Integration by parts of the left-hand sides of these two equations leads to:

23 2 2

3 2 2 20 00 0

d dd d d d ddd d d d

a a a aj ji i ij ji i

X XX X XX x X X xxx x x xα

⎡ ⎤ ⎡ ⎤− + =⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ∫ ∫ , (3.30)

3 2 2 2

3 2 2 20 00 0

d d dd d d ddd d d d

a a a aj j ji iji j i

X X XX XX x X X xxx x x xα

⎡ ⎤ ⎡ ⎤− + =⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ∫ ∫ . (3.31)

The end conditions (3.20)-(3.22) require that the integrated terms in the left-hand sides vanish. Therefore, subtraction of Equations (3.30) and (3.31) yields:

( )

0d 0

a

j ji iX X xα α− =∫ . (3.32)

When i and j are different, the values iα and jα are distinct, and to satisfy the preceding equation, we must have:

0d 0, .

a

jiX X x i j= ≠∫ (3.33)

Substituting this relation into Equation (3.30) leads to:

22

2 20

dd d 0, ,d d

a ji XX x i jx x

= ≠∫ (3.34)

66 Chapter 3. Beam Bending and Cylindrical Bending Vibrations of Undamped Materials

and from Equation (3.28) it is obtained:

4

40

d d 0, .d

ai

jX X x i jx

= ≠∫ (3.35)

Equations (3.33) to (3.35) constitute the orthogonality relationships for the mode shapes of the transverse vibrations of beams.

As constants, C1, C2, C3 and C4, in Equation (3.18) are determined to within an arbitrary constant, the integral in Equation (3.32), for the case ,i j= may be any constant ki:

2

0d

a

i iX x k=∫ . (3.36)

When the functions Xi are normalized by the constant ki, Equations (3.28) and (3.30) lead to:

24 2 2

4 2 4 20 0 0

d d 1d dd d

a ai i i

i i i iX XX x x k kx x a

ωαω

⎛ ⎞= = =⎜ ⎟

⎝ ⎠∫ ∫ . (3.37)

It is possible to set the normalization constant equal to unity ( 1ik = ), so that Equations (3.36) and (3.37) are simply:

2

0d 1

a

iX x =∫ , (3.38)

24 2 2

4 2 4 20 0 0

d d 1d dd d

a ai i i

iX XX x xx x a

ωω

⎛ ⎞= =⎜ ⎟

⎝ ⎠∫ ∫ . (3.39)

3.4 NATURAL MODES OF BEAMS WITH DIFFERENT END CONDITIONS

3.4.1 Simply Supported Beam

For a simply supported end the boundary conditions (1.71) are written: 0 0, 0,xM= =w (3.40)

where the moment Mx is expressed by (3.6). For the ith natural mode the trans-verse displacement is given by Equation (3.16) and the boundary conditions for the mode shape are:

― at the end 0x =

2

0 20

d0, 0,d

ii x

x

XXx=

== = (3.41)

― at the end x a=

2

2d0, 0.d

ii x a

x a

XXx=

== = (3.42)

3.4. Natural Modes of Beams with Different End Conditions 67

These conditions are deduced from the boundary conditions considered in Equa-tion (3.20).

From the first conditions in Equation (3.41), it is derived:

2 40, 0.C C= = (3.43)

From the conditions in Equation (3.42), it is obtained:

3 10, sin 0.iC C λ= = (3.44)

A non-zero solution for C1 is obtained when:

sin 0iλ = , (3.45)

which is the frequency equation for the case of a simply supported beam. The non-zero positive roots of this equation are:

, 1, 2, ... , .i i iλ π= = ∞ (3.46)

Considering Expressions (3.15) and (3.19), the natural angular frequencies of the beam are given by:

32 2 2 2

2 1 211

1 .12x

iss

E hi ia D aπ πω ρρ −= = (3.47)

The mode shapes of the beam for the various modes of bending vibrations are of the form:

1( ) sin , 1, 2, ... , .ixx C i iaπ= = ∞w (3.48)

Thus, the mode shapes are sine curves, the first three modes of which are shown in Figure 3.2. The value of the amplitude C1 of the vibrations depends on the initial deformation. FIGURE 3.2. Bending vibrations of a simply supported beam.

1 01 9.870i ω ω= =

2 02 39.478i ω ω= =

3 03 88.826i ω ω= =

3

0 21

12x

s

E ha

ωρ

=

68 Chapter 3. Beam Bending and Cylindrical Bending Vibrations of Undamped Materials

3.4.2 Clamped Beam

In the case of a beam clamped at the two ends, the boundary conditions (1.72) lead to:

― at the end 0x =

0 0

d0, 0,di

i x x

XX x= == = (3.49)

― at the end x a=

d0, 0.di

i x a x a

XX x= == = (3.50)

The first two conditions are satisfied if in the general solution (3.18) we take:

3 1 4 2, .C C C C= − = − (3.51)

Taking C2 = 1, the shape function can be written as:

( ) ( )cos cosh sin sinhi i i i i ix x x xX x a a a aλ λ γ λ λ= − − − . (3.52)

From the other two conditions at end x = a, it is derived the following equation:

( )( )

cos cosh sin sinh 0,sin sinh cos cosh 0.

i i i i i

i i i i i

λ λ γ λ λλ λ γ λ λ

− − − =

− − − − = (3.53)

A non-zero solution for the constant γi exists only when the determinant of Equa-tions (3.53) is equal to zero, which yields:

( )2 2 2cos cosh sin sinh 0i i i iλ λ λ λ− + − = . (3.54) Taking into account the equalities:

2 2 2 2cos sin 1, cosh sinh 1,i i i iλ λ λ λ+ = − = (3.55)

the previous expression reduces to: cos cosh 1i iλ λ = . (3.56) The first eight non-zero roots of this equation are reported in Table 3.1. Then the coefficients γi are derived from one of Equations (3.53). For example:

cos coshsin sinh

i ii

i i

λ λγ λ λ−=−

. (3.57)

The values of γi are reported in Table 3.1. It should be noted that an approximate solution can be given in the case where

λi is large enough. In fact in this case:

cosh 2i

ieλ

λ ≈ , (3.58)

3.4. Natural Modes of Beams with Different End Conditions 69

TABLE 3.1. Values of coefficients λi and γi of the function of a beam clamped at the two ends.

i λi γi

1

2

3

4

5

6

7

8

4.7300408

7.8532046

10.9956078

14.1371655

17.2787596

20.4203522

23.5619449

26.7035376

0.98250222

1.00077731

0.99996645

1.00000145

0.99999994

1.00000000

1.00000000

1.00000000

and Equation (3.56) may be written:

cos 2 0.ii e λλ −≈ ≈ (3.59)

Roots of this equation are:

( )2 1 .2i i πλ = + (3.60)

These approximate solutions are compared in Table 3.2 with the exact solutions of Equation (3.56).

The expression for the frequencies of the free bending vibrations is derived from Equations (3.15) and (3.19), which leads to:

2 2 3

2 1 211

1 .12i i x

iss

E ha D aλ λω ρρ −= = (3.61)

The fundamental frequency is:

3

1 222.373 .12

x

s

E ha

ω ρ= (3.62)

The beam mode shapes are of the form:

( ) ( ), 1, 2, ..., ,i i ix C X x i= = ∞w (3.63)

where the shape functions ( )iX x are given by Expression (3.52). The deformed shapes are shown in Figure 3.3 for the first three modes.

TABLE 3.2. Exact and approximate values of λi.

λ1 λ2 λ3 λ4 λ5

Solution of Equation (3.56) 4.730 7.853 10.996 14.137 17.279

Approximate solution (3.60) 4.712 7.854 10.996 14.137 17.279

70 Chapter 3. Beam Bending and Cylindrical Bending Vibrations of Undamped Materials

FIGURE 3.3. Bending vibrations of a clamped beam.

3.4.3 Beam Clamped at One End and Simply Supported at the Other

We consider the case of a beam clamped at the end x = 0 and simply supported at the end x = a. The boundary conditions are thus:

― at the end 0x =

0 0

d0, 0,d

ii x x

XXx=

== = (3.64)

― at the end x a=

2

2d0, 0.d

ii x a

x a

XXx=

== = (3.65)

The first two conditions are identical to conditions (3.49). It results that the shape functions ( )iX x are of the form (3.52). The other conditions (3.65) lead to the following equations:

( )( )

cos cosh sin sinh 0,cos cosh sin sinh 0.

i i i i i

i i i i i

λ λ γ λ λλ λ γ λ λ

− − − =

+ − + = (3.66)

A non-zero solution for the coefficient γi is obtained when:

cos cosh cos coshsin sinh sin sinh

i i i i

i i i i

λ λ λ λλ λ λ λ

− +=− +

, (3.67)

or tan tanhi iλ λ= . (3.68)

The first eight solutions of this equation are reported in Table 3.3. The coefficient γi is next determined by:

1 01 22.373i ω ω= =

2 02 61.673i ω ω= =

3 03 120.90i ω ω= =

3

0 21

12x

s

E ha

ωρ

=

3.4. Natural Modes of Beams with Different End Conditions 71

TABLE 3.3. Coefficients λi of the function of a simply supported-clamped beam.

i 1 2 3 4 5 6 7 8

λi 3.927 7.069 10.210 13.352 16.493 19.635 22.776 25.918

cos coshsin sinh

i ii

i i

λ λγλ λ

−=

−. (3.69)

The values of γi are almost exactly equal to 1. It should be noticed that for high enough values of iλ :

tanh 1iλ ≈ , (3.70) and Equation (3.68) is reduced to: tan 1iλ = . (3.71) Roots of this equation are: ( )0.25i iλ π= + . (3.72)

The calculation of these approximate solutions shows that they are practically identical to the exact values of the solutions of Equation (3.68).

The natural frequencies are expressed by Expression (3.61) where the values of iλ are given in Table 3.3. The fundamental frequency is:

3

1 215.421 .12

x

s

E ha

ω ρ= (3.73)

The deformed shapes are shown in Figure 3.4 for the first three modes. FIGURE 3.4. Bending vibrations of a beam clamped at one end and simply supported at the other.

1 01 15.421i ω ω= =

2 02 49.971i ω ω= =

3 03 104.24i ω ω= =

3

0 21

12x

s

E ha

ωρ

=

72 Chapter 3. Beam Bending and Cylindrical Bending Vibrations of Undamped Materials

3.4.4 Beam Clamped at One End and Free at the Other

In the case of a beam clamped at the end x = 0 and free at the other end x = a, the boundary conditions are:

― at the end 0x =

0 0

d0, 0,d

ii x x

XXx=

== = (3.74)

― at the end x a= , by (1.73) 0, 0.x xx a x aM Q= == = (3.75)

The moment is expressed by (3.6). The transverse shear resultant is obtained from the fourth equation for plates (1.41) as:

xx

MQx

∂=∂

. (3.76)

Thus, the boundary conditions (3.75) are:

2 3

2 3d d0, 0.d d

i i

x a x a

X Xx x= =

= = (3.77)

The first two conditions (3.74) are identical to the boundary conditions at the end 0x = of the beam considered in Sections 3.4.2 and 3.4.3. It results that the shape functions ( )iX x are again of the form (3.52). The conditions (3.77) at the end x a= lead to:

( )

( )cos cosh sin sinh 0,sin sinh cos cosh 0.

i i i i i

i i i i i

λ λ γ λ λλ λ γ λ λ

+ − + =

− + + = (3.78)

A non-zero solution for the coefficient γi is obtained when:

cos cosh sin sinhsin sinh cos cosh

i i i i

i i i i

λ λ λ λλ λ λ λ

+ −= −+ +

, (3.79)

or cos cosh 1i iλ λ = − . (3.80)

The coefficient γi is next determined by the expression:

cos coshsin sinh

i ii

i i

λ λγλ λ

+=

+. (3.81)

The first eight solutions of Equation (3.80) are reported in Table 3.4 with the corresponding values of γi. For high enough values of iλ , approximate values can be written in the form:

( )0.5i iλ π= − . (3.82)

These values are also reported in Table 3.4 and show that they are the same as the solutions (3.80) in practice for 3.i ≥

The natural frequencies are given by Expression (3.61) where the values of iλ are given in Table 3.4. The fundamental frequency is:

3.4. Natural Modes of Beams with Different End Conditions 73

TABLE 3.4. Coefficients of the clamped-free beam function.

i 1 2 3 4 5 6 7 8

λi 1.875 4.694 7.855 10.996 14.137 17.279 20.420 23.562

γi 0.734 1.018 0.999 1.000 1.000 1.000 1.000 1.000

(i – 0,5)π 1.571 4.712 7.854 10.996 14.137 17.279 20.420 23.562

3

1 23.516 .12

xs

E ha

ω ρ= (3.83)

The deformed shapes are shown in Figure 3.5 for the first three modes.

3.4.5 Beam with Two Free Ends

In the case of a beam with free ends, the boundary conditions are:

2 3 2 3

2 3 2 30 0

d d d d0, 0, 0, 0.d d d d

i i i i

x x x a x a

X X X Xx x x x= = = =

= = = = (3.84)

The first two conditions are satisfied if in the general solution (3.18) we have:

3 1 4 2, .C C C C= = (3.85)

Taking C2 = 1, the shape functions ( )iX x can be written as:

( ) ( )cos cosh sin sinhi i i i i ix x x xX x a a a aλ λ γ λ λ= + − + . (3.86)

FIGURE 3.5. Bending vibrations of a beam clamped at one end and free at the other.

1 01 3.516i ω ω= =

2 02 22.034i ω ω= =

3 03 61.701i ω ω= =

3

0 21

12x

s

E ha

ωρ

=

74 Chapter 3. Beam Bending and Cylindrical Bending Vibrations of Undamped Materials

The other two conditions at the free end x = a are satisfied if:

( )( )

cos cosh sin sinh 0,sin sinh cos cosh 0.

i i i i i

i i i i i

λ λ γ λ λλ λ γ λ λ

− + − − + =

+ − − + = (3.87)

A non-zero solution for the constant γi exists only when the determinant of Equa-tions (3.87) is equal to zero, which yields:

( ) ( )2 2 2cos cosh sin sinh 0i i i iλ λ λ λ− + − − − = . (3.88) or

cos cosh 1i iλ λ = . (3.89)

We obtain the same equation that the one (3.56) obtained in the case of a clamped-clamped beam. Then coefficients γi are derived from one of Equations (3.87) which gives expression (3.57).

Moreover, Equation (3.89) has a double root equal to zero which corresponds to the rigid-body motion of the beam which can be superimposed to the transverse vibrations of the beam. The rigid-body motions can be expressed in the form:

0 1 1 2 2( ) ( ) ( )x C X x C X x= +w , (3.90) with 1( ) 1X x = , (3.91)

( )2( ) 3 1 2 xX xa

= − . (3.92)

These functions correspond to the rigid modes of translation and rotation. They are normalized according to Equation (3.38). The two roots 1 20 and 0λ λ= = are associated with these functions. The other roots iλ of Equation (3.89) and the corresponding values of γi are identical to those found in the case of two clamped ends (Table 3.1). Finally, the values of iλ and γi for a free-free beam are reported in Table 3.5 for i varying from 1 to 9.

The first mode of the free vibrations is obtained for 3i = . Then the free vibra-tion frequencies are obtained with 4, 5, ...,i = and are the same as the fre-quencies obtained for the beam with clamped ends. In contrast, the deformed shapes are different (3.86). Figure 3.5 illustrates the first three mode shapes.

TABLE 3.5. Coefficients of the free-free beam functions.

i 1 2 3 4 5 6 7 8 9

λi 0 0 4.730 7.853 10.996 14.137 17.279 20.420 23.562

γi –0.9825 –1.0008 –1.000 –1.000 –1.000 –1.000 –1.000

3.5. Normal Mode Analysis 75

FIGURE 3.6. Bending vibrations of a beam with both ends free.

3.5 NORMAL MODE ANALYSIS

3.5.1 Motion Equation in Normal Coordinates

In this section we consider the normal mode analysis for determining the transverse response of a beam. In this analysis the transverse motion of the beam is expressed in terms of time functions ( )i tφ and normal displacement functions

( )iX x as:

( ) ( ) ( )01

, .i ii

x t t X xφ∞

=

= ∑w (3.93)

Substitution of this expression into the motion equation (3.10) leads to:

( )

4

41

d , d

is i i s i

i

XX k q x tx

ρ φ φ∞

=

⎛ ⎞+ =⎜ ⎟

⎝ ⎠∑ . (3.94)

Then, multiplying this expression by the normal function ( )jX x and integrating over the length of the beam, we obtain:

4

40 0 01

dd d dd

a a ai

j js i i s i ii

XX X x k X x qX xx

ρ φ φ∞

=

⎛ ⎞+ =⎜ ⎟

⎝ ⎠∑ ∫ ∫ ∫ . (3.95)

From the orthogonality and normalisation relations (3.33), (3.35) and (3.38), Equation (3.95) reduces to: ( )2 , 1,2,...,i i i ip t iφ ω φ+ = = ∞ , (3.96) where

1 01 22.373i ω ω= =

2 02 61.673i ω ω= =

3 03 120.90i ω ω= =

3

0 21

12x

s

E ha

ωρ

=

76 Chapter 3. Beam Bending and Cylindrical Bending Vibrations of Undamped Materials

( ) ( )

0, d ,

a

i ip t p x t X x= ∫ (3.97)

with

( ) ( )1, , .s

p x t q x tρ= (3.98)

Equation (3.96) is the motion equation for transverse vibrations expressed in normal coordinates, which is constituted of separate equations which allow to derive the time functions ( )i tφ for 1,2,...,i = ∞ .

3.5.2 Response to Initial Conditions

3.5.2.1 General Formulation

In this subsection we apply the normal mode analysis for determining the transverse response of a beam to initial conditions of transverse displacement and velocity. We assume that for instant t = 0, the initial transverse displacements and the initial velocities are:

00 1 0 2( , 0) ( ), ( , 0) ( , 0) ( ),x t f x x t x t f x

t∂= = = = = =∂ww w (3.99)

where 1( )f x and 2 ( )f x are given functions. The transverse motion of the beam is given by Expression (3.93) and it results

that the initial displacements and velocities (3.99) can be expressed in the forms:

0 0 11

( , 0) ( ),i ii

x t X f xφ∞

=

= = =∑w (3.100)

0 0 21

( , 0) ( ),i ii

x t X f xφ∞

=

= = =∑w (3.101)

where 0 0( , 0), ( , 0).i i i it tφ φ φ φ= = (3.102)

Multiplying Expressions (3.100) and (3.101) by jX and integrating over the length of the beam, it is obtained:

0 10 01

d ( ) da a

j ji ii

X X x f x X xφ∞

=

=∑ ∫ ∫ , (3.103)

0 20 01

d ( ) da a

j ji ii

X X x f x X xφ∞

=

=∑ ∫ ∫ . (3.104)

From the orthogonality and normalisation relations (3.33) and (3.37), Equations (3.103) and (3.104) lead to the following initial conditions expressed in normal coordinates:

3.5. Normal Mode Analysis 77

0 10

( ) d , 1, 2, ... , ,a

i if x X x iφ = = ∞∫ (3.105)

0 20

( ) d , 1, 2, ... , .a

i if x X x iφ = = ∞∫ (3.106)

The motion equation in normal coordinates for free vibrations of beam is deri-ved from (3.96) considering that ( ) 0ip t = . We obtain:

2 0, 1, 2,...,i i i iφ ω φ+ = = ∞ . (3.107)

Therefore, the free vibration responses of the normal modes are:

00( ) cos sin , 1,2,...,i

i i i ii

t t t iφφ φ ω ωω

= + = ∞ . (3.108)

Substitution of this expression into Equation (3.93) gives the combined response of all the modes:

( ) ( ) 00 0

1

, cos sin .ii i i i

ii

x t X x t tφφ ω ωω

∞

=

⎛ ⎞= +⎜ ⎟

⎝ ⎠∑w (3.109)

This equation constitutes the general form of the free transverse response of a beam to initial conditions.

3.5.2.2 Beam with Simply Supported Ends

We consider the transverse response to initial conditions of a beam with simply supported ends. The shape functions are derived from expression (3.48) where the constant C1 must be determined. Expression (3.48) verifies the orthogonality relations (3.33). To satisfy the normalization relations (3.38) and (3.39), it is necessary to use the constant 1 2C a= . Thus, the shape functions are:

2( ) sin , 1, 2, ... , .ixX x i ia aπ= = ∞ (3.110)

It results that the initial conditions derived from (3.105) and (3.106) are:

0 10

2 ( )sin d , 1, 2, ... ,a

ixf x i x i

a aφ π= = ∞∫ , (3.111)

0 20

2 ( )sin d , 1, 2, ... ,a

ixf x i x i

a aφ π= = ∞∫ . (3.112)

Substitution of these expressions into Expression (3.109) yields:

( ) ( )01

, cos sin sini i i ii

xx t A t B t ia

ω ω π∞

=

= +∑w , (3.113)

with

78 Chapter 3. Beam Bending and Cylindrical Bending Vibrations of Undamped Materials

10

2 ( )sin d , 1, 2, ... ,a

ixA f x i x i

a aπ= = ∞∫ , (3.114)

20

2 ( )sin d , 1, 2, ... ,a

ii

xB f x i x ia a

πω

= = ∞∫ . (3.115)

As an example, we consider the case of an impact for which an initial velocity v0 is given to the beam at point of coordinate x1. In this case we have:

1 2 0 1( ) 0, ( ) ( ),f x f x x xδ= = −v (3.116)

where 1( )x xδ − is the Dirac function localized at x1. Substituting these expres-sions into Equations (3.114) and (3.115), we obtain:

0 0 11

0

2 20, sin ( ) d sina

i ii i

xxA B i x x x ia a a a

π δ πω ω

= = − =∫v v . (3.117)

So, the beam response becomes:

( ) 0 10

1

2 1, sin sin sin iii

x xx t i i ta a a

π π ωω

∞

=

= ∑vw . (3.118)

If the impact is localized at the middle point of the span ( 1 2x a= ), the transverse displacement becomes:

( ) 0

0 1 31 3

55

2 1 1, sin sin sin 3 sin

1 sin 5 sin . . . ,

x xx t t ta a a

x ta

π ω π ωω ω

π ωω

⎛= −⎜⎝

⎞+ − ⎟⎠

vw (3.119)

where the angular frequencies are given by (3.47). So, the previous equation can be rewritten as:

( ) (

)0 0 1 332

5

122 1, sin sin sin 3 sin9

1 sin 5 sin . . . .25

s

x

x xx t a t ta aE h

x ta

ρ π ω π ωπ

π ω

= −

+ −

w v (3.120)

It results that only the symmetric modes of transverse vibrations are excited and the amplitude of the mode contributions decreases as 21 .i

3.5.2.3 Beam with Other End Conditions

The response of a beam to initial conditions involves the evaluation of the integrals (3.105) and (3.106). Direct integrations are possible in the case of a beam with simply supported ends, as considered in the previous Subsection 3.5.2.2. Beams with other conditions involve normal functions Xi with hyperbolic functions. In these cases the evaluation of the integrals (3.105) and (3.106) re-quires numerical integrations.

3.5. Normal Mode Analysis 79

3.5.3 Forced Response

3.5.3.1 General Formulation

In this section we consider the transverse response of a beam to a distributed or concentrated load. The motion equation expressed in normal coordinates of a beam submitted to the load ( , )q x t is given by Expression (3.96). The response

iφ of the ith mode of transverse vibrations is derived from the Duhamel integral (Section 2.3.6 of Chapter 2) to be:

0 0

1( ) ( ) ( , )sin ( )d da t

i i ii

t X x q x t t t t xφ ωω

′ ′ ′= −∫ ∫ . (3.121)

Substitution of this time function into Equation (3.93) gives the transverse vibration response as:

00 01

( )( , ) ( ) ( , ) sin ( )d da t

ii i

ii

X xx t X x q x t t t t xωω

∞

=

′ ′ ′= −∑ ∫ ∫w . (3.122)

If the load is concentrated at point x1 with an amplitude Q1(t), the load ( , )q x t is expressed as:

1 1( , ) ( ) ( )q x t Q t x xδ= − , (3.123)

where 1( )x xδ − is the Dirac function localized at point x1. From Equation (3.97) it results that:

1 1( ) ( ) ( )i ip t P t X x= , (3.124) with

1 11( ) ( )s

P t Q tρ

= . (3.125)

In this case the response (3.122) is simplified as:

0 1 101

1( , ) ( ) ( ) ( )sin ( )dt

i i iii

x t X x X x P t t t tωω

∞

=

′ ′ ′= −∑ ∫w . (3.126)

The response of a beam to a distributed or concentrated load involves the eva-luation of integrals (3.121) or (3.126). As in the case of a beam submitted to initial conditions, direct integrations are possible only in the case of a beam with simply supported ends.

3.5.3.2 Beam with Simply Supported Ends

In this case the angular frequencies and normalized functions are given by Expressions (3.47) and (3.110), respectively. Substitution of the normal functions (3.47) into the response (3.122) to a distributed load gives:

00 01

2 1( , ) sin sin ( , )sin ( )d da t

iii

x xx t i i q x t t t t xa a a

π π ωω

∞

=

′ ′ ′= −∑ ∫ ∫w . (3.127)

80 Chapter 3. Beam Bending and Cylindrical Bending Vibrations of Undamped Materials

Similarly, the response (3.126) for a concentrated load is:

10 1

01

2 1( , ) sin sin ( )sin ( )dt

iii

xxx t i i P t t t ta a a

π π ωω

∞

=

′ ′ ′= −∑ ∫w . (3.128)

As an example, we consider the case of a harmonic load 1( ) sinQ t Q tω= applied at point x1. The response given by Equation (3.128) is:

10

01

2 1( , ) sin sin sin sin ( )dt

is ii

Q xxx t i i t t t ta a a

π π ω ωρ ω

∞

=

′ ′ ′= −∑ ∫w . (3.129)

This expression leads to the transverse response: 23 2

10 4 3 4 2 2

1

2 12 1( , ) sin sin sin sin ( ),si i

s x i

Qa xx ax t i i t t Ka aE h i i a

ρ ωπ π ω ω ωπ ρ π

∞

=

⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠∑w

(3.130) introducing the magnification factor :

2

2

1( )1

i

i

K ωωω

=−

. (3.131)

The first part of Expression (3.130) represents the steady state forced vibrations of the beam, whereas the second part is the transient vibrations. These vibrations decrease rapidly and die out in the presence of damping. Next, only the steady- state response subsists.

3.6 CYLINDRICAL BENDING VIBRATIONS 3.6.1 Introduction

In the case of the beam bending the analyses developed in the previous sections were reduced to one-dimension analyses. Another type of one-dimension analysis concerns plates that have a high length-to-width ratio (Figure 3.7), so that the deformation of the plate can be considered to be independent of the coordinate along the length of the plate. Such behaviour is called cylindrical bending.

The plate is supported along the length of its edges x = 0 and x = a. If the transverse load is a function only of x and time, the deformation of the plate is cylindrical, that is:

0 0

0 0

0 0

( , , ) ( , ),( , , ) ( , ),( , , ) ( , ).

u x y t u x tx y t x tx y t x t

===

v vw w

(3.132)

Cylindrical bending vibrations will be considered in the present section in the case of plate constituted of laminate and sandwich materials.

3.6. Cylindrical Bending Vibrations 81

FIGURE 3.7. Plate with a great length.

3.6.2 Classical Laminate Theory

3.6.2.1 Equations

By substituting Expressions (3.132) into the fundamental equations (1.64) to (1.66) of the classical laminate theory, we obtain the one-dimension equations:

2 2 3 2

0 0 0 011 16 112 2 3 2s

u uA A Bx x x t

ρ∂ ∂ ∂ ∂+ − =

∂ ∂ ∂ ∂v w , (3.133)

2 2 3 2

0 0 0 016 66 162 2 3 2s

uA A Bx x x t

ρ∂ ∂ ∂ ∂+ − =

∂ ∂ ∂ ∂v w v , (3.134)

4 3 3 2

0 0 0 011 11 164 3 3 2s

uD B B qx x x t

ρ∂ ∂ ∂ ∂− − = −

∂ ∂ ∂ ∂w v w . (3.135)

For the free vibrations of the plates, no transverse load is applied. The vibra-tions are harmonic and the displacement field can be expressed in complex form as:

0 0

0 0

0 0

( , , ) ( , ) ,

( , , ) ( , ) ,

( , , ) ( , ) ,

i t

i t

i t

u x y t u x y e

x y t x y e

x y t x y e

ω

ω

ω

=

=

=

v v

w w

(3.136)

where ω is the angular frequency of the plate vibrations. Taking account of these expressions, the equations of free vibrations are deduced from Equations (3.133) to (3.135) as:

2 2 320 0 0

11 16 11 02 2 3d d d 0d d d

suA A B ux x x

ρ ω+ − + =v w , (3.137)

x

y

a

82 Chapter 3. Beam Bending and Cylindrical Bending Vibrations of Undamped Materials

2 2 320 0 0

16 66 16 02 2 3d d d 0d d d

suA A Bx x x

ρ ω+ − + =v w v , (3.138)

4 3 320 0 0

11 11 16 04 3 3d d d 0d d d

suD B B

x x xρ ω− − − =

w v w . (3.139)

3.6.2.2 Plate Simply Supported

In the case where the plate is simply supported along its edges x = 0 and x = a, the boundary conditions are satisfied by the displacements:

0

0

0

( ) cos ,

( ) cos ,

( ) sin .

m

m

m

xu x A maxx B maxx C ma

π

π

π

=

=

=

v

w

(3.140)

Substituting these expressions into Equations (3.137) to (3.139), we obtain:

2 2 2 2 3 32

11 16 112 2 3

2 2 2 2 3 32

16 66 162 2 3

3 3 3 3 4 42

11 16 113 3 4

0,

0,

0.

s m m m

m s m m

m m s m

m m mA A A B B Ca a a

m m mA A A B B Ca a a

m m mB A B B D Ca a a

π π πρ ω

π π πρ ω

π π π ρ ω

⎛ ⎞− − + =⎜ ⎟

⎝ ⎠⎛ ⎞

− + − + =⎜ ⎟⎝ ⎠

⎛ ⎞− − + − =⎜ ⎟

⎝ ⎠

(3.141)

As a function of the order of the magnitude of the vibration frequencies, it is pos-sible to neglect the term 2

sρ ω in the coefficients of Am and Bm in the two first equations. Equations (3.141) may then be written as:

11 16 11

16 66 16

3 3 3 3 4 42

11 16 113 3 4

0,

0,

0.

m m m

m m m

m m s m

mA A A B B Ca

mA A A B B Ca

m m mB A B B D Ca a a

π

π

π π π ρ ω

− − + =

− − + =

⎛ ⎞− − + − =⎜ ⎟

⎝ ⎠

(3.142)

Solving the first two equations leads to:

, ,m m m mm B m CA C B Ca A a Aπ π

= = (3.143)

introducing the parameters :

3.6. Cylindrical Bending Vibrations 83

211 66 16

66 11 16 16

11 16 16 11

,,.

A A A AB A B A BC A B A B

= −= −= −

(3.144)

Substituting Am and Bm into the last equation (3.142), it yields:

4 4

24 0s m

m D CAa

π ρ ω⎛ ⎞

− =⎜ ⎟⎝ ⎠

, (3.145) with 11 11 16D D A B B B C= − − . (3.146)

A non-zero solution of Equation (3.145) is obtained when the coefficient of Cm vanishes, which leads to the expression for the natural frequencies:

2 2

21

ms

m DAa

πωρ

= . (3.147)

The expressions of the natural frequencies can be written in the form:

1m m Hω ω′= − , (3.148) introducing the coefficient

11 16

11

B B B CHAD

+= . (3.149).

In Expression (3.148) mω′ is the bending vibration frequencies in the case where there exists no stretching/bending-twisting coupling (Bij = 0):

2 2

112m

s

m Daπω

ρ′ = . (3.150)

The stretching/bending-twisting coupling thus reduces the values of the natural vibration frequencies.

3.6.2.3 Plates with Other End Conditions

Other end conditions of plates can be easily investigated in the case of sym-metric laminates. In this case, Equations (3.137) to (3.139) become:

2 220 0

11 16 02 2d d 0d d

suA A ux x

ρ ω+ + =v , (3.151)

2 220 0

16 66 02 2d d 0d d

suA Ax x

ρ ω+ + =v v , (3.152)

4 320 0

11 11 04 3d d 0d d

suD B

x xρ ω− − =

w w . (3.153)

84 Chapter 3. Beam Bending and Cylindrical Bending Vibrations of Undamped Materials

The first two equations for the in-plane vibrations are decoupled from the third equation for the transverse vibrations. This last equation can be expressed as Equation (3.10) obtained for the transverse vibrations of beams introducing the stiffness:

11sk D= . (3.154)

So, the results obtained in the case of the transverse vibrations of symmetric laminated beams (Sections 3.2-3.5) can be applied to the cylindrical vibrations of symmetric plates.

3.6.3 Effect of Transverse Shear

In this section we consider the effect of transverse shear deformation on the cylindrical vibrations.

In the case of a laminate constituted of an arbitrary number of orthotropic layers, with cloth or unidirectional reinforcement, the material directions of which are parallel to the directions x and y of the plate, we have:

16 26 16 26 16 260, 0, 0.A A B B D D= = = = = = (3.155)

The laminate is orthotropic. The plate of great length in the y direction is consi-dered to be in a state of cylindrical deformation. That is: 0 0 0 0 0( , ), ( , ), 0, 0, ( , ).x x yu u x t x t x tϕ ϕ ϕ= = = = =v w w (3.156)

Equations (1.86) to (1.90), taking (1.97) into account, reduce to:

2 2 2 2

0 011 112 2 2 2

x xs

u uA B Rx x t t

ϕ ϕρ∂ ∂ ∂ ∂+ = +

∂ ∂ ∂ ∂, (3.157)

2 2

0 055 55 2 2

xsk F q

x x tϕ ρ

⎛ ⎞∂ ∂ ∂+ + =⎜ ⎟

∂⎝ ⎠∂ ∂w w , (3.158)

2 2 2 2

0 0 011 11 55 552 2 2 2

x xx xy

u uB D k F R Ixx x t t

ϕ ϕϕ∂ ∂ ∂ ∂ ∂⎛ ⎞+ − + = +⎜ ⎟⎝ ⎠∂∂ ∂ ∂ ∂

w . (3.159)

We now consider the case of an orthotropic laminate which is symmetric ( 0ijB = ). In this case, Equations (3.157) to (3.159) become:

0 02 2

0 055 55 2 2

2 20

11 55 552 2

0, 0,

,

.

xs

x xx xy

u

k F qx x t

D k F Ixx t

ϕ ρ

ϕ ϕϕ

= =

⎛ ⎞∂ ∂ ∂+ + =⎜ ⎟

∂⎝ ⎠∂ ∂

∂ ∂ ∂⎛ ⎞− + =⎜ ⎟⎝ ⎠∂∂ ∂

v

w w

w

(3.160)

In the case of simple supports, the boundary conditions are given by Equations (1.94) which yields:

0 0, 0, 0.x xN M= = =w (3.161)

The free vibrations are obtained for q = 0. Thus, the solutions for 0 and xϕ w

3.6. Cylindrical Bending Vibrations 85

which satisfy these boundary conditions and the free vibration equations are of the form:

0

cos ,

sin .

i tx m

i tm

xB e maxC e ma

ω

ω

ϕ π

π

=

=w (3.162)

Substituting these expressions into Equations (3.160) yields:

2 22

55 55 55 552

2 22

11 55 55 55 552

0,

0.

m s m

xy m m

m mk F B k F Ca a

m mD k F I B k F Caa

π π ρ ω

π πω

⎛ ⎞+ − =⎜ ⎟

⎝ ⎠⎛ ⎞

+ − + =⎜ ⎟⎝ ⎠

(3.163)

A non-zero solution is obtained when the determinant of these equations vanishes. Hence the expression for the natural frequencies is:

2 2 2 2

255 55 112 2

12m xy s s

s xy

m mI k F DI a a

π πω ρ ρ ∆ρ

⎡ ⎤⎛ ⎞= + + ±⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦, (3.164)

with

22 2 2 2 4 4

55 55 11 55 55 112 2 44xy s s s xym m mI k F D I k F D

a a aπ π π∆ = ρ ρ ρ

⎡ ⎤⎛ ⎞+ + −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦. (3.165)

In the case of a laminate with layers made from the same material but having different orientations and thicknesses, the density of each layer is identical. It results that:

3

0 0, ,12s xyhh Iρ ρ ρ= = (3.166)

where 0ρ is the density of the orthotropic material. The free vibration frequencies are given by:

2

2 2 2 2 2 255 55 112 3

0

612mha m k F m D

a hω π π ∆

ρ

⎡ ⎤⎛ ⎞′= + + ±⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦, (3.167)

with

22 4 4 4

2 2 2 2 255 55 11 55 55 11212 3

h m ha m k F m D k F Daπ∆ = π π

⎡ ⎤⎛ ⎞′ + + −⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦. (3.168)

If the rotatory inertia terms may be neglected (Ixy = 0), Equations (3.163) reduce to:

2 22

55 55 55 552

2 2

11 55 55 55 552

0,

0.

m s m

m m

m mk F B k F Ca a

m mD k F B k F Caa

π π ρ ω

π π

⎛ ⎞+ − =⎜ ⎟

⎝ ⎠⎛ ⎞

+ + =⎜ ⎟⎝ ⎠

(3.169)

The vibration frequencies may then be written in the form:

86 Chapter 3. Beam Bending and Cylindrical Bending Vibrations of Undamped Materials

2 21

1m m

m Sω ω

π′=

+, (3.170)

where S is the term that takes into account the transverse shear effect of the laminate:

112

55 55

DSa k F

= , (3.171)

and mω′ is the natural vibration frequencies with transverse shear neglected, given by Expression (3.150). Transverse shear deformation reduces the value of the vibration frequencies. The expression of the shear factor S can be rewritten by introducing an effective bending stiffness of the plate, 11,Q having the dimension of a modulus, and an effective shear modulus, 13,G expressed as:

11 5511 133

12 , .D FQ Ghh

= = (3.172)

So, the shear factor is written as:

2

11

55 13

112

Q hSk G a

⎛ ⎞= ⎜ ⎟⎝ ⎠

. (3.173)

The influence of the shear effect on the values of the vibration frequencies depends on the ratio 11 13Q G and the ratio a/h of the span length between the supports to the thickness of the laminate. The variation of the fundamental fre-quency (m = 1) as a function of the ratio a/h is plotted in Figure 3.8, in the case of a [0°/90°/90°/0°] laminate, of which the characteristics are:

230 GPa, 14 GPa, 5 GPa,4 GPa, 0.3.

L T LT

TT LT

E E GG ν′

= = == =

(3.174)

For large values of the ratio a/h, the results deduced from the analysis including the transverse shear effect tend to the ones obtained by the classical theory.

3.6.4 Cylindrical Vibrations of Sandwich Plates

As an example of sandwich plate vibrations, we consider the case of a plate constituted of:

― two identical orthotropic skins, the directions of which are parallel to the directions x and y of the plate:

16 26 16 26

0,0,

ij ijB CA A D D

= =

= = = = (3.175)

― a core, the principal directions 1 and 2 of which are parallel to the direc- tions x and y of the plate: 45 44 23, 55 13,0, F F hG F hG= = = (3.176) where G13 and G23 are the transverse shear moduli measured along directions 1 and 2.

3.6. Cylindrical Bending Vibrations 87

FIGURE 3.8. Influence of the transverse shear effect on the fundamental frequency of an orthotropic plate subjected to cylindrical bending.

The cylindrical deformation of the sandwich plate is described as: 0 0 0 00, ( , ), 0, 0, ( , ).x x yu x t x tϕ ϕ ϕ= = = = =v w w (3.177)

It results that the fundamental Equations (1.128) and (1.129) here reduce to:

2 2

011 132 2

x xx xyD hG I

xx tϕ ϕϕ∂ ∂ ∂⎛ ⎞− + =⎜ ⎟

⎝ ⎠∂∂ ∂w , (3.178)

2 2

0 013 2 2

xshG q

x x tϕ ρ

⎛ ⎞∂ ∂ ∂+ + =⎜ ⎟

∂⎝ ⎠∂ ∂w w . (3.179)

These equations have the same form as Equations (3.160). So, the results deduced for laminate plates can be transposed to sandwich plates.

In the case of simple supports, the results for the sandwich plates are deduced from the results (3.164) to (3.173) by changing 55 55 3into .k F hG In particular when the rotary inertia terms can be neglected, the natural frequencies are deduced from Equation (3.170) introducing the frequencies:

2 2

112m

s

m Daπω

ρ′ = , (3.180)

and considering the shear term:

112

13

DSa hG

= . (3.181)

0 5 10 15

length-to-thickness ratio a h

1

1 fu

ndam

enta

l fre

quen

cy ω

ω′

1.2

0.8

1.0

0.6

0.4

0.2

classical theory

transverse shear 55 1k =

155 3k =

CHAPTER 4

Flexural Vibrations of Undamped Rectangular Laminate Plates

4.1 FREE VIBRATIONS OF RECTANGULAR ORTHOTROPIC PLATES SIMPLY SUPPORTED

In the analysis of plate vibrations, the most complex problem is that of lami-nate made with an arbitrary stacking sequence which introduces coupling between stretching and bending-twisting. The first simplification of the analysis consists in the study of symmetric laminates for which there exists no coupling: Bij = 0. An additional simplification occurs when there exists no bending-twisting coupling: D16 = D26 = 0. The laminates are referred to as orthotropic laminates and the fundamental equations (1.67) to (1.69) may be written as:

( )

0 04 4 4 2

0 0 0 011 12 66 224 2 2 4 2

4 40 0

2 2 2 2

0, 0,

2 2

.

s

xy

u

D D D D qx x y y t

Ix t y t

ρ

= =

∂ ∂ ∂ ∂+ + + + +

∂ ∂ ∂ ∂ ∂

⎛ ⎞∂ ∂= +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

v

w w w w

w w

(4.1)

In the case where the rotary inertia terms can be neglected, Ixy = 0, the free vibration equation is deduced from the latter equation as:

( )4 4 4 2

0 0 0 011 12 66 224 2 2 4 22 2 0sD D D D

x x y y tρ∂ ∂ ∂ ∂

+ + + + =∂ ∂ ∂ ∂ ∂w w w w . (4.2)

The transverse displacement is expressed in the complex form:

0 0( , , ) ( , ) i tx y t x y e ω=w w , (4.3)

where ω is the angular frequency of the harmonic vibrations. Substituting expres-sion (4.3) in Equation (4.2) leads to:

( )4 4 4

20 0 011 12 66 22 04 2 2 42 2 0sD D D D

x x y yρ ω∂ ∂ ∂

+ + + − =∂ ∂ ∂ ∂w w w w . (4.4)

4.1. Free Vibrations of Rectangular Orthotropic Plates Simply Supported 89

In the case of a plate simply supported along its four edges, the boundary con-ditions are deduced from conditions (1.71):

― edges 0 and :x x a= = 0 0, 0,xM= =w (4.5) ― edges 0 and :y y b= = 0 0, 0,yM= =w (4.6)

From the constitutive equation (1.60), the conditions on the bending moments along the edges are:

― edges 0 and :x x a= =

2 2

0 011 122 2 0xM D D

x y∂ ∂

= − − =∂ ∂w w , (4.7)

― edges 0 and :y y b= =

2 2

0 012 222 2 0yM D D

x y∂ ∂

= − − =∂ ∂w w . (4.8)

It results that the boundary conditions are verified by the transverse displa-cement of the form:

0 ( , ) sin sinmnx yx y C m na b

π π=w . (4.9)

Substituting this expression into Equation (4.4) yields:

( )4 4 2 2 4 4 4

211 12 66 224 2 2 42 2 0s mn

m m n nD D D D Ca a b bπ π π ρ ω

⎡ ⎤+ + + − =⎢ ⎥

⎣ ⎦. (4.10)

A non-zero value of Cmn is obtained if the coefficient of Cmn vanishes, whence the expression for the natural frequencies of the transverse vibrations:

( )2

4 2 2 2 4 411 12 66 222

1 2 2mns

m D m n R D D n R Daπω

ρ⎡ ⎤= + + +⎣ ⎦ , (4.11)

where R is the length-to-width ratio of the plate (R = a/b). The deformed shape of the plate corresponding to the natural frequency ωmn is given by (4.9).

In the case of an isotropic plate, we have:

11 22 12 662D D D D D= = + = , (4.12)

and the expression for the vibration frequencies reduces to:

24 2 2 2 4 4

2 2mns

D m m n R n Raπω

ρ= + + . (4.13)

In the case of an orthotropic plate, the fundamental frequency corresponds to m = n = 1 and is given by:

( )2

2 411 11 12 66 222

1 2 2s

D R D D R Daπω

ρ⎡ ⎤= + + +⎣ ⎦ , (4.14)

and in the case of an isotropic plate we have :

90 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

( )

22

11 2 1s

D Raπω

ρ= + . (4.15)

The deformed shape of the fundamental mode is given for the two cases by:

0 11( , ) sin sinx yx y Ca b

π π=w . (4.16)

So as to evaluate the influence of anisotropy, we compare the behaviour of a square plate made from an orthotropic material with the characteristics:

11 22 12 66 2210 , 2 ,D D D D D= + = (4.17)

with the behaviour of a plate made of an isotropic material. In the case of the isotropic material, the natural vibration frequencies (4.13)

are:

22 2

2 , ,mn mn mns

Dk k m naπω

ρ= = + (4.18)

when for the isotropic material, the vibration frequencies are given by:

222

2 ,mn mns

Dkaπω

ρ= (4.19)

with

4 2 2 410 2 .mnk m m n n= + + (4.20)

The values of the frequencies and the corresponding modes of vibrations are reported in Table 4.1 for the isotropic plate and in Table 4.2 for the orthotropic plate. The results reported show that there is no privileged direction in the case of an isotropic plate. The vibration frequencies and the modes are the same for m = 1, n = 2 and m = 2, n = 1; for m = 1, n = 3 and m = 3, n = 1, etc. In contrast, in the case of the orthotropic plate, for example, the second mode corresponds to m = 1, n = 2 (with k12 = 5.83), whereas m = 2, n = 1 corresponds to the fourth mode mode (with k21 = 13.0), etc.

4.2 VIBRATIONS OF ORTHOTROPIC PLATES WITH VARIOUS CONDITIONS ALONG THE EDGES

4.2.1 General Expressions

In the preceding section, the exact solutions of Equation (4.2) were derived in the case of simply supported edges. In the case of other conditions, it is not pos-sible to solve Equation (4.2) directly. The determination of natural frequencies then requires using approximation methods. In this section we consider the Ritz method.

In the case of orthotropic laminates 16 26( 0)D D= = , the strain energy Ud is

4.1. Free Vibrations of Rectangular Orthotropic Plates Simply Supported 91

TABLE 4.1. Natural frequencies and vibration modes of a simply supported isotropic square plate.

2

2mn mns

Dkaπω

ρ=

1st mode 2nd mode 3rd mode 4th mode

m 1 1 2 2 1 3

n 1 2 1 2 3 1

kmn 2.0 5.0 5.0 8.0 10.0 10.0

Nodal lines

TABLE 4.2. Natural frequencies and vibration modes of a simply supported othotropic square plate.

222

2mn mns

Dkaπω

ρ=

1st mode 2nd mode 3rd mode 4th mode 5th mode 6th mode

m 1 1 1 2 2 1

n 1 2 3 1 2 4

kmn 3.61 5.83 10.44 13.0 14.42 17.26

Nodal lines

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y y

x x

y

x

y

x

92 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

deduced from Expression (1.77) as:

2 22 2 2 20 0 0 0

d 11 12 222 2 2 20 0

220

66

1 22

4 d d .

a b

x yU D D D

x x y y

D x y Cx y

= =

⎡ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎢= + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣⎤⎛ ⎞∂ ⎥+ +⎜ ⎟⎜ ⎟∂ ∂ ⎥⎝ ⎠ ⎦

∫ ∫ w w w w

w (4.21)

The maximum kinetic energy is obtained from Expression (1.80) on introducing the transverse displacement w0 in the form (4.3). We obtain:

2 2cmax 0

0 0

1 d d2

a b

sx y

E x yρ ω= =

= ∫ ∫ w . (4.22)

In the absence of transverse loads, the maximum energy function reduces to Ud max – Ec max with:

2 22 2 2 20 0 0 0

d max cmax 11 12 222 2 2 20 0

222 20

66 0

1 22

4 d d . (4.23)

a b

x y

s

U E D D Dx x y y

D x yx y

ρ ω

= =

⎡ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎢− = + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣⎤⎛ ⎞∂ ⎥+ −⎜ ⎟⎜ ⎟∂ ∂ ⎥⎝ ⎠ ⎦

∫ ∫ w w w w

w w

In the Ritz method the solution for the transverse displacement is expanded in the form of a double series as:

01 1

( , ) ( ) ( )M N

mn m nm n

x y A X x Y y= =

= ∑∑w , (4.24)

where the functions Xm(x) and Yn(y) have to form a functional basis and are chosen to satisfy the essential boundary conditions along the edges x = 0, x = a and y = 0, y = b. The coefficients Amn are next determined from the stationarity conditions which make extremum the energy function:

[ ]d max cmax1, 2, . . . , ,

0, 1, 2, . . . , ,mn

m MU E

n NA=∂

− ==∂

(4.25)

where d max cmaxU E− is the energy obtained by substituting Expression (4.24) for the transverse displacement into Expressions (4.22) and (4.23).

So, the calculation of the energy function requires to express the terms:

2 2 22 2 2 2 2

2 0 0 0 0 00 2 2 2 2, , , , .

x yx x y y

⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂⎜ ⎟ ⎜ ⎟ ⎜ ⎟

∂ ∂⎝ ⎠∂ ∂ ∂ ∂ ⎝ ⎠⎝ ⎠

w w w w ww

For example:

2 20

2 21 1

dd

M Nm

mn nm n

XA Yx x= =

∂=

∂ ∑∑w . (4.26)

4.2. Vibrations of Orthotropic Plates with Various Conditions along the Edges 93

Whence

22 2 20

2 2 21 1 1 1

d dd d

M N M Nm i

mn ij n jm n i j

X XA A Y Yx x x= = = =

⎛ ⎞∂=⎜ ⎟

⎝ ⎠∂ ∑∑∑∑w , (4.27)

and

22 2 20

2 2 21 1

1 d d2 d d

M Nm i

ij n jmn i j

X XA Y YA x x x= =

⎛ ⎞∂ ∂=⎜ ⎟

∂ ⎝ ⎠∂ ∑∑w . (4.28)

Integration of this term yields:

220

20 0

1 d d2

a b

x ymnx y

A x= =

⎛ ⎞∂ ∂=⎜ ⎟

∂ ⎝ ⎠∂∫ ∫ w

2 2

2 20 01 1

d d d dd d

M N a bm i

ij n ji j

X XA x Y Y yx x= =

∑∑ ∫ ∫ .

(4.29) In order to express these integrals, it is useful to introduce the reduced variables:

et x yua b

= =v . (4.30)

Expression (4.29) may then be written as:

220

20 0

1 d d2

a b

x ymnx y

A x= =

⎛ ⎞∂ ∂=⎜ ⎟

∂ ⎝ ⎠∂∫ ∫ w (4.31)

1 12 2

3 2 20 01 1

d d d dd d

M Nm i

ij n ji j

b X XA u Y Ya u u= =

∑∑ ∫ ∫ v

The integrals are then dimensionless. Proceeding in the same way for the other terms, we obtain:

( )

22 00 20 02 02 2011 122

1 111 11 2 00 22 4

66 22

1

4 ,

M Nd

mi nj mi nj mi njmn i j

mi nj mi nj ij

U D I J D I J I JA Ra

D I J R D I J R A= =

∂ ⎡= + +⎣∂

⎤+ +⎦

∑∑ (4.32)

where R is the length-to-width ratio and introducing the dimensionless integrals:

1

0

, 1, 2, . . . , ,d d d ,00, 02, 11, 20, 22.d d

p qpq m imi p q

m i MX XI upqu u

==

=∫ (4.33)

1

0

d , 1, 2, . . . , ,d d ,00, 02, 11, 20, 22.d d

srjrs n

nj r sY n j NYJ

rs=

==∫ v

v v (4.34)

The approximate expression for the maximum kinetic energy is obtained from the expression:

2 cmax0 01 1

1 d d2

M N a bjs m i n

x yi j

E X X x Y Y yρ ω= == =

= ∑∑ ∫ ∫ , (4.35)

94 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

which yields:

4 2 00 00cmax 1 1

1 M N

s mi njij i j

E a I JA R

ρ ω= =

∂=

∂ ∑∑ . (4.36)

Finally, the stationarity conditions (4.25) lead to the system of M N× homo-geneous equations:

( )

22 00 20 02 02 20 11 11 211 12 66

1 100 22 4 4 2 00 00

22

4

0,

for 1, 2, . . . , , 1, 2, . . . , .

M N

mi nj mi nj mi nj mi nji j

mi nj s mi nj ij

D I J D I J I J D I J R

D I J R a I J A

m M n N

ρ ω

= =

⎡ ⎤+ + + ⎦⎣

+ − =

= =

∑∑ (4.37)

This system of equations can also be written in a reduced form as:

( )

2200 2002 0220 1111 211 12 66

1 10022 4 4 2 0000

22

4

0,

for 1, 2, . . . , , 1, 2, . . . , ,

M N

minj minj minj minji j

minj s minj ij

D C D C C D C R

D C R a C A

m M n N

ρ ω

= =

⎡ ⎤+ + + ⎦⎣

+ − =

= =

∑∑ (4.38)

on writing the products of the integrals (4.33) and (4.34) in the form:

1 1

0 0

dd d dd dd d d d

sp q rjpqrs pq rs m i n

njminj mi p q r sYX X YC I J u

u u= = ∫ ∫ v

v v. (4.39)

Next, the system can be rewritten in the form of a dimensionless system as:

( )

2200 2002 0220 1111 212 66

1 10022 4 2 0000

22

4

0,

for 1, 2, . . . , , 1, 2, . . . , ,

M N

minj minj minj minji j

minj minj ij

C C C C R

C R C A

m M n N

α α

α Ω

= =

⎡ ⎤+ + + ⎦⎣

+ − =

= =

∑∑ (4.40)

by expressing the bending stiffness constants Dij as functions of D11:

12 12 11 66 66 11 22 22 11, , ,D D D D D Dα α α= = = (4.41)

and introducing the reduced frequency:

2

11

saDρΩ ω= . (4.42)

The system of Equations (4.37), (4.38) or (4.40) in Aij is a homogeneous system which can be solved as an eigenproblem. Eigenvalues are the natural frequencies and eigenvectors give coefficients Aij which determine the vibration modes.

4.2. Vibrations of Orthotropic Plates with Various Conditions along the Edges 95

4.2.2 Rayleigh’s Approximation

Rayleigh’s approximation consists of using for a given mode only the domi-nant term mn of the series (4.24):

( , ) ( ) ( )mn mn m nx y A X x Y y=w . (4.43) The vibration frequency of the mode mn is then obtained by equating the maxi-mum strain energy with the maximum kinetic energy associated with the maxi-mum transverse displacement wmn.

In this case, the maximum strain energy is given by:

( )2 22 00 20 02 11 11 2

d max 11 12 66

00 22 422

1 2 22

,

mn mm nn mm nn mm nn

mm nn

U A D I J D I J D I J R

D I J R ab

⎡= + +⎣

⎤+ ⎦

(4.44)

and the maximum kinetic energy is:

2 00 00 2

cmax12 s mm nn mnE ab I J Aρ ω= . (4.45)

The equality of these two expressions leads to:

20000mmnn

mnmmnn

BC

Ω = , (4.46)

where the coefficient Bmmnn is given by:

( )2200 2002 1111 2 0022 412 66 222 2mmnn mmnn mmnn mmnn mmnnB C C C R C Rα α α= + + + . (4.47)

In the case of transverse vibrations of orthotropic plates the difference between the value of the vibration frequencies obtained by Rayleigh’s approximation and the values deduced by Ritz method from an approximation with a large number of terms is small (less than a few percent) in the case of plates with edges clamped or simply supported. The difference increases when the geometric constraints impo-sed on the four edges decrease. Schematically, the change of a clamped or simply supported edge into a free edge increases the difference noticeably. The inter-section of two free edges produces the highest differences.

4.2.3 Two-term Approximation

In the case of a two-term approximation the transverse displacement is given, for example, by: 0 11 1 1 12 1 2( , ) ( ) ( ) ( ) ( )x y A X x Y y A X x Y y= +w , (4.48)

and the system (4.40) reduces to a system of two equations:

( ) ( )( ) ( )

2 0000 2 00001111 1111 11 1112 1112 12

2 0000 2 00001112 1121 11 1122 1122 12

0,

0,

B C A B C A

B C A B C A

Ω Ω

Ω Ω

− + − =

− + − = (4.49)

96 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

with

( )2200 2002 1111 2 0022 411 11 12 11 66 11 22 112 2 0, , 1, 2.ij ij ij ij ijB C C C R C R i jα α α= + − + = =

(4.50)

The vibration frequencies of modes 11 and 12 are obtained when the determinant of (4.49) vanishes. Whence:

2 0000 2 0000

1111 1111 1112 11122 0000 2 0000

1112 1121 1122 1122det 0

B C B C

B C B C

Ω Ω

Ω Ω

⎡ ⎤− −=⎢ ⎥

− −⎢ ⎥⎣ ⎦. (4.51)

The Rayleigh’s approximation of modes 11 et12 is obtained directly from the diagonal terms, that is:

2 21111 112211 120000 0000

1111 1122 and .B B

C CΩ Ω= = (4.52)

We recover the approximations deduced from (4.46).

4.2.4 Orthotropic Plates with Simply Supported or Clamped Edges

As an application we consider in this subsection the case of a rectangular ortho-tropic plate clamped or simply supported along its edges.

In the case of opposite edges being clamped it is possible to use, for the functions Xm(x) and Yn(y), the beam function (3.52):

— for clamped edges x = 0 and x = a:

( ) cos cosh sin sinhm m m m m mx x x xX xa a a a

λ λ γ λ λ⎛ ⎞= − − −⎜ ⎟⎝ ⎠

, (4.53)

— for clamped edges y = 0 and y = b:

( ) cos cosh sin sinhn n n n n ny y y yY yb b b b

λ λ γ λ λ⎛ ⎞= − − −⎜ ⎟⎝ ⎠

, (4.54)

where λm, λn, γm and γn are reported in Tables 3.1. According to approximation (3.60), we have: ( )1 4.730, 0.5 2, 3, . . ..i i iλ λ π= = + = (4.55)

In the case of simply supported opposite edges, the functions Xm(x) and Yn(y), are expressed as the sine functions introduced in Equation (3.48):

— for simply supported edges x = 0 and x = a:

( ) sinmxX x ma

π= , (4.56)

4.2. Vibrations of Orthotropic Plates with Various Conditions along the Edges 97

— for simply supported edges y = 0 and y = b:

( ) sinnyY y nb

π= . (4.57)

In the case where one edge is clamped and the other opposite edge is simply supported, the functions Xm(x) and Yn(y) can be expressed by the beam function (3.52):

— for clamped edge x = 0 and simply supported edge x = a:

( ) cos cosh sin sinhm m m m m mx x x xX xa a a a

λ λ γ λ λ⎛ ⎞= − − −⎜ ⎟⎝ ⎠

, (4.58)

— for clamped edge y = 0 and simply supported edge y = b:

( ) cos cosh sin sinhn n n n n ny y y yY yb b b b

λ λ γ λ λ⎛ ⎞= − − −⎜ ⎟⎝ ⎠

, (4.59)

where λm, λn, γm and γn are reported in Tables 3.3. The values of λm and λn are well approximated by Equation (3.72).

The natural frequencies and the corresponding vibration modes are next deter-mined by introducing functions (4.53)-(4.59) into the system of Equations (4.40). We have the relations:

0000 2002 0220 11111, .mnij mnij mnij mnijC C C C= = = (4.60)

So, the system of Equations (4.40) is written in the form:

( )

2200 1111 2 0022 4 212 66 22

1 1

2 2 0,

for 1, 2, . . . , , 1, 2, . . . , ,

M N

minj minj minj iji j

C C R C R A

m M n N

α α α Ω= =

⎡ ⎤+ + + − =⎣ ⎦

= =

∑∑ (4.61)

with 2200 22 00 22 1111 11 11 0022 00 22 22, , .minj mi nj mi minj mi nj minj mi nj njC I J I C I J C I J J= = = = = (4.62)

The values of these integrals must be evaluated using an analytical or nume-rical process. Some of these values are reported in Appendix B of Refs. 1 and 2.

We consider hereafter the case of the Rayleigh’s approximation (4.46). In this case the vibration frequency of the mode mn is given by:

( ) 2200 1111 2 0022 411

12 66 2221 2 2 .mn mmnn mmnn mmnn

s

D C C R C Ra

ω α α αρ

= + + + (4.63)

In the case of two opposite edges simply supported:

2200 4 4 1111 2 2 2 0022 4 4, , .mmnn mmnn mmnnC m C m n C nπ π π= = = (4.64) In the case of two clamped opposite edges, or of one edge clamped and the

other simply supported: 2200 4 0022 4, ,mmnn m mmnn nC Cλ λ= = (4.65)

98 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

and

1111 11 11mmnn mm nnC I J= . (4.66)

The evaluation of these integrals shows that:

( )11 1112.30, 2 2, 3, 4, . . ..ii ii i iI I iλ λ= ≈ − = (4.67)

in the case of two opposite edges clamped, and:

( )11 1 1, 2, 3, . . ..ii i iI iλ λ≈ − = (4.68)

in the case of one edge clamped and the other simply supported. Finally, Equation (4.63) associated with Equations (4.64) to (4.68) shows that

Rayleigh’s approximation of the vibration frequency of mode mn may be written in the following form:

( ) 4 2 4 4111 12 66 2 22 32

1 2 2 ,mns

D c R c R ca

ω α α αρ

= + + + (4.69)

where the values of coefficients c1, c2 and c3 are reported in Table 4.3 for each combination of clamped and simple supports along the plate edges.

In the case of an isotropic plate (4.12), the expression for the transverse vibra-tion frequencies may be written as:

4 2 4 41 2 32

1 2 .mns

D c R c R ca

ωρ

= + + (4.70)

For an isotropic plate with four clamped edges, the values reported in Table 4.3 lead to the following expression for the fundamental frequency:

11 236.1 .

s

Da

ωρ

= (4.71)

Using a 64-term series (M = N = 8), the solution of the system (4.61) leads to:

11 235.99 .

s

Da

ωρ

= (4.72)

So, the value deduced from one-term approximation is very close to the exact value for the fundamental frequency.

In the case of a square orthotropic plate, clamped along its four edges, with the stiffness constants: 11 22 12 66 2210 , 2 1.2 ,D D D D D= + = (4.73) the values of the natural frequencies obtained by the one-term approximation are compared in Table 4.4 with the values obtained by using a 64-term series. These results show that the values deduced from the one-term approximation are suffi-ciently precise for this type of edge conditions.

4.2. Vibrations of Orthotropic Plates with Various Conditions along the Edges 99

x

y

E

E E

E

x

y

E

E E

S

x

y

S

E E

S

x

y

S

S

S

S

x

y

SE

E

S

x

y

SE

S

S

TABLE 4.3. Coefficients introduced in the expression (4.69) for natural frequencies of orthotropic plates (clamped edges: E, simply supported edges: S).

Boundary conditions

m n 1c 3c 2c

1

1

2, 3, 4, . . .

2, 3, 4, . . .

1

2, 3, 4, . . .

1

2, 3, 4, . . .

4.730

4.730 ( 0.5)m π+ ( 0.5)m π+

4.730

( 0.5)n π+

4.730

( 0.5)n π+

212.3 151.3= 3 3( 2)12.3c c −

1 1( 2)12.3c c −

1 1 3 3( 2) ( 2)c c c c− −

1, 2, 3, . . . 1, 2, 3, . . . mπ nπ 2 2 4nm π

1, 2, 3, . . . 1, 2, 3, . . . ( 0.25)m π+ ( 0.25)n π+ 1 1 3 3( 2) ( 2)c c c c− −

1

2, 3, 4, . . .

1, 2, 3, . . .

1, 2, 3, . . . 4.730

( 0.5)m π+

nπ nπ

2 212.3n π 2 2

1 1( 2)n c cπ −

1

2, 3, 4, . . .

1, 2, 3, . . .

1, 2, 3, . . . 4.730 ( 0.5)m π+

( 0.25)n π+

( 0.25)n π+

3 3( 2)12.3c c −

1 1 3 3( 2) ( 2)c c c c− −

1, 2, 3, . . . 1, 2, 3, . . . ( 0.25)m π+ nπ 2 21 1( 2)n c cπ −

100 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

TABLE 4.4. Natural frequencies of flexural vibrations of an orthotropic square plate clamped along its four edges.

2

1mn mn

s

Dka

ωρ

=

kmn

m n Approximation (4.69) 64-Term Series

1 1 1 2 2 1 2 2

1 2 3 1 2 4 3 4

24.227 31.889 47.480 63.163 68.504 70.722 79.740 98.460

24.213 31.861 47.436 63.116 68.428 70.645 79.676 98.369

4.3 VIBRATIONS OF SYMMETRIC LAMINATE PLATES

4.3.1 General Expressions

The analysis of the free vibrations of symmetric plates can be implemented by the Ritz method as considered in the previous section. In the present case the strain energy is given by Expression (1.77). It results that the system (4.38) is modified by introducing the bending-twisting terms D16 and D26. Whence the system of M N× equations:

( )( ) ( )

2200 2002 0220 1111 2 0022 411 12 66 22

1 11210 2101 1012 0121 3 4 2 0000

16 26

4

2 2 0,

for 1, 2, . . .

M N

minj minj minj minj minji j

minj minj minj minj s minj ij

D C D C C D C R D C R

D C C R D C C R a C A

m

ρ ω

= =

⎡ ⎤+ + + +⎦⎣

+ + + + − =

=

∑∑

, , 1, 2, . . . , . (4.74)M n N=

As in the case of orthotropic plates, it is possible to derive the Rayleigh’s approximation (Subsection 4.2.2) that leads to expressions analogous to (4.46). However in the present case, the Rayleigh’s approximation for the natural fre-quencies differs notably from the value obtained with a large number of terms by the Ritz method. In effect, in the case of arbitrary symmetric laminates, the one-term approximation of the transverse displacement (4.43) does not describe the actual transverse displacement correctly enough.

4.3. Vibrations of Symmetric Laminate Plates 101

4.3.2 Symmetric Plates with Clamped or Free Edges

As an application of the preceding general formulation, we consider in this section the case of a symmetric rectangular plate with clamped or free edges.

The case of clamped opposite edges has already been considered in Section 4.2.4 (Expressions (4.53) to (4.55)).

For one edge clamped and the other opposite edge free, the transverse displa-cement is expressed by the beam function introduced in Section 3.4.4:

— clamped edge x = 0 and free edge x = a:

( ) cos cosh sin sinhm m m m m mx x x xX xa a a a

λ λ γ λ λ⎛ ⎞= − − −⎜ ⎟⎝ ⎠

, (4.75)

— clamped edge y = 0 and free edge y = b:

( ) cos cosh sin sinhn n n n n ny y y yY yb b b b

λ λ γ λ λ⎛ ⎞= − − −⎜ ⎟⎝ ⎠

, (4.76)

where λm, λn, γm and γn are reported in Table 3.4. In the case of free opposite edges, the transverse displacement is expressed by

the beam functions introduced in Equations (3.86) to (3.92): — free edges x = 0 and x = a:

1

2

( ) 1,

( ) 3 1 2 ,

( ) cos cosh sin sinh , 3,m m m m m m

X xxX xa

x x x xX x ma a a a

λ λ γ λ λ

=

⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞= + + + ≥⎜ ⎟⎝ ⎠

(4.77)

— free edges y = 0 and y = b:

1

2

( ) 1,

( ) 3 1 2 ,

( ) cos cosh sin sinh , 3.n n n n n n

Y yyY yb

y y y yY y nb b b b

λ λ γ λ λ

=

⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞= + + + ≥⎜ ⎟⎝ ⎠

(4.78)

The coefficients λm, λn, γm and γn are given in Table 3.5. It must be noted that if the beam functions (4.75) to (4.78) satisfy the boundary

conditions (3.84) exactly at the free ends of a beam, they satisfy the boundary conditions only approximately in the case of free edges of a plate. In fact, in the case of one free edge in the y direction, for example, the boundary conditions (1.73) are:

0, 0.xyx x

MM Q

y∂

= + =∂

(4.79)

102 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

The transverse shear resultant Qx is given by the fourth plate equation (1.41) and the boundary conditions become:

0, 2 0.xyxx

MMMx y

∂∂= + =

∂ ∂ (4.80)

The equations for the bending moment Mx and the twisting moment Mxy are deduced from the constitutive equation (1.60) for laminates. The boundary condi-tions thus are:

2 2 2

0 0 011 12 162 2 2 0D D D

x yx y∂ ∂ ∂

+ + =∂ ∂∂ ∂

w w w , (4.81)

( )3 3 3 3

0 0 0 011 16 12 66 263 2 2 34 4 2 0D D D D D

x x y x y y∂ ∂ ∂ ∂

+ + + + =∂ ∂ ∂ ∂ ∂ ∂w w w w . (4.82)

In the case of a free edge in the x direction, the boundary conditions are derived by interchanging the respective roles of x and y, and the indices 1 and 2.

The beam functions (4.75) and (4.77) in the x direction satisfy conditions (3.84) for the free ends of the beam, that is:

2 3

0 02 30, 0.

x x∂ ∂

= =∂ ∂w w (4.83)

It results that conditions (4.81) and (4.82) are only approximated by the beam functions. The approach by the Ritz method is then less precise in the case of free edges and a large number of terms of series (4.24) must be considered.

Considering the functions which correspond to the boundary conditions imposed on the four edges of the plates it is possible to evaluate the integrals

and pq rsnjmiI J , and to establish the corresponding system (4.74) of homogeneous

equations. This system can be solved as an eigenvalue and eigenvector problem, where the eigenvalues are the natural frequencies of the vibrations and the eigenvectors determine the vibration modes. Solving this problem can be carried out by the use of a general-purpose software package for scientific and engineering applications that integrates numerical analysis, matrix computation and graphics.

As a numerical application we consider the case of a square plate made of an orthotropic laminate with stiffness constants in the material directions given by:

0 0 0 0 0 022 11 12 11 66 110.25 , 0.075 , 0.125 . D D D D D D= = = (4.84)

We consider the case where the material directions are oriented at 30° to the plate axes. The bending stiffnesses with respect to the plate directions are then derived by applying to the constants (4.84) the transformation reported in Table 1.1 of Chapter 1. We obtain:

0 0 0

11 11 12 11 16 110 0 0

22 11 26 11 66 11

0.70 , 0.1875 , 0.2273 ,

0.325 , 0.0974 , 0.2375 .

D D D D D D

D D D D D D

= = =

= = = (4.85)

4.3. Vibrations of Symmetric Laminate Plates 103

TABLE 4.5. Natural frequencies of the first six flexural modes of a square plate constituted of a symmetric material, (C: clamped edge, F free edge).

011

2 for the mode ii

s

k D ia

ωρ

=

ki Boundary conditions Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6

CCCC FFFF CCFF CFCF CCCF CFFF

25.670 8.311 5.429 18.096 18.995 2.693

45.090 11.645 15.108 19.723 28.191 6.145

58.648 18.532 22.092 30.478 47.226 15.698

71.211 19.577 31.833 49.198 51.570 17.373

82.994 26.853 39.625 52.061 62.619 23.521

100.929 36.077 51.835 52.282 74.397 34.431

The values of the free vibration frequencies of the first six modes are reported in Table 4.5 for various combinations of clamped or free edges. The frequencies have been calculated by using a 64-term series for the transverse displacement. The shapes of the modes are shown in Figure 4.1 for the case of four clamped edges and Figure 4.2 for the case of two adjacent edges clamped with the other two free.

FIGURE 4.1. Free flexural modes of a square plate constituted of a symmetric material clamped along its four edges.

104 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

FIGURE 4.2. Free flexural modes of a square plate constituted of a symmetric material, two adjacent edges of which are clamped and the other two are free.

4.4 VIBRATIONS OF NON-SYMMETRIC LAMINATE PLATES

4.4.1 Plate Constituted of an Antisymmetric Cross-Ply Laminate Plate

We consider the case of a rectangular plate of length a and width b, constituted

of a [0°/90°]p cross-ply laminate. From Equations (1.56), (1.57) and (1.59), the laminate is characterized by:

16 26 12 16 26 66 16 26

22 11 22 11 22 11

0, 0, 0,, , .

A A B B B B D DA A B B D D

= = = = = = = == = − =

(4.86)

In the case of free vibrations the displacements can be expressed in the form:

0 0

0 0

0 0

( , , ) ( , ) ,

( , , ) ( , ) ,

( , , ) ( , ) ,

i t

i t

i t

u x y t u x y e

x y t x y e

x y t x y e

ω

ω

ω

=

=

=

v v

w w

(4.87)

where ω is the angular frequency of the harmonic vibrations of the plate. Introdu-cing these expressions into Equations (1.64) to (1.66) and neglecting the in-plane

4.4. Vibrations of Non-Symmetric Laminate Plates 105

inertia terms, the equations of free vibrations (q = 0) are written as:

( )

( )

( )

2 2 2 30 0 0 0

11 66 12 66 112 2 3

2 2 2 30 0 0 0

12 66 66 11 112 2 3

4 4 4 3 30 0 0 0

11 12 66 114 4 2 2 3

0,

0, (4.88)

2 2

u uA A A A Bx yx y x

uA A A A Bx y x y x

uD D D Bx y x y x

∂ ∂ ∂ ∂+ + + − =

∂ ∂∂ ∂ ∂

∂ ∂ ∂ ∂+ + + + =

∂ ∂ ∂ ∂ ∂

⎛ ⎞∂ ∂ ∂ ∂ ∂+ + + − −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

v w

v v w

w w w v 2003 0.sy

ρ ω⎛ ⎞

− =⎜ ⎟⎜ ⎟∂⎝ ⎠w

Equations (4.88) can be solved in the case of hinged edges, free in the in-plane normal direction. In this case the boundary conditions are:

— along edges 0x = and x a= :

2 20 0 0

0 11 11 122 2

20 0 0

0 11 12 11 2

0, 0,

0, 0,

x

x

uM B D Dx x y

uN A A Bx y x

∂ ∂ ∂= = − − =

∂ ∂ ∂

∂ ∂ ∂= = + − =

∂ ∂ ∂

w ww

v wv

(4.89)

— along edges 0y = and y b= :

2 20 0 0

0 11 12 112 2

20 0 0

0 12 11 11 2

0, 0,

0, 0.

y

y

M B D Dy x y

uu N A A Bx y x

∂ ∂ ∂= = − − − =

∂ ∂ ∂

∂ ∂ ∂= = + + =

∂ ∂ ∂

v w ww

v w (4.90)

These boundary conditions are satisfied by considering the displacements of the form:

0

0

0

cos sin ,

sin cos ,

sin sin .

mn

mn

mn

x yu A m na bx yB m na bx yC m na b

π π

π π

π π

=

=

=

v

w

(4.91)

Substituting these expressions into Equations (4.88), we obtain:

1 2 3

2 4 52

23 5 6 2

0,0,

0,

mn mn mn

mn mn mn

smn mn mn

a A a B a Ca A a B a C

aa A a B a Cρ ωπ

+ + =

+ + =

⎛ ⎞+ + − =⎜ ⎟

⎝ ⎠

(4.92)

with

106 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

( )

( ) ( )

2 2 21 11 66

2 12 66

33 11

2 2 24 66 11

3 35 11

24 4 4 2 2 2

6 11 12 662

,,

,

,

,

2 2 ,

.

a m A n R Aa mnR A A

a m Ba

a m A n R A

a n R Ba

a m n R D m n R D DaaRb

π

π

π

= += +

= −

= +

=

⎡ ⎤= + + +⎣ ⎦

=

(4.93)

A non-zero solution is obtained for the plate displacements when the determinant of the system (4.91) is zero. Whence the natural frequencies are derived as:

( ) ( )

( )

42 4 4 4 2 2 2

11 12 664

24 4 411

3 21

2 2

,

mns

m n R D m n R D Da

B m n R

πωρ

∆ ∆∆

⎡ ⎤= + + +⎣ ⎦

− +

(4.94)

on setting:

( )( ) ( )( )

( )

22 2 2 2 2 2 2 2 21 11 66 66 11 12 66

24 2 2 2 4 42 12 66 11 66

4 2 2 2 4 43 66 11 12 66

,

,

.

m A n R A m A n R A m n R A A

m A A m n R A n R A

m A m n R A n R A A

∆

∆

∆

= + + − +

= + + +

= + + +

(4.95)

When the stretching/bending-twisting coupling is neglected ( 11 0B = ), Equa-tion (4.94) is simplified as:

( ) ( )4

2 4 4 4 2 2 211 12 664 2 2 ,mn

sm n R D m n R D D

aπω

ρ⎡ ⎤= + + +⎣ ⎦ (4.96)

which is the expression (4.11) for the free vibrations of simply supported ortho-tropic plates for which D22 = D11.

In the case of orthotropic laminates, Equation (4.96) shows that the funda-mental frequency corresponds to 1m n= = . It is not the same in the case where a coupling exists. The m and n values corresponding to the fundamental frequency depend on the mechanical characteristics of the layers constituting the laminate.

We consider the case of an antisymmetric [0°/90°]p cross-ply laminate made of layers the engineering moduli of which are:

20 , 0.5 , 0.25.L T LT T LTE E G E ν= = = (4.97).

From expressions (1.56), (1.57) and (1.59), the stiffness constants are expressed as:

4.4. Vibrations of Non-Symmetric Laminate Plates 107

FIGURE 4.3. Variation of the fundamental frequency of a rectangular plate made of cross-ply laminates as a function of the aspect ratio of the plate.

11 11 12 12 66 661 1 , , ,2

T

L

EA Q h A Q h A Q hE

⎛ ⎞= + = =⎜ ⎟⎝ ⎠

211 11

1 1 ,8

T

L

EB Q hp E

⎛ ⎞= −⎜ ⎟⎝ ⎠

(4.98)

3 3 3

11 12 6611 12 66

1 1 , , ,2 12 12 12

T

L

E Q h Q h Q hD D DE

⎛ ⎞= + = =⎜ ⎟⎝ ⎠

where the reduced stiffness constants are expressed as functions of the engi-neering constants by Equations (1.20).

The variation of the fundamental frequency as a function of the length-to-width ratio (a/b) of the plate is reported in Figure 4.3 in the case of [0°/90°], [0°/90°]2 and [0°/90°]3 cross-ply laminates, and in the case of an orthotropic laminate ( 11 0B = ). The fundamental frequencies are obtained for all the cases for

1m n= = . The stretching/bending-twisting coupling reduces the values of the vibration frequencies, and the results of Figure 4.3 show that the values of the frequencies rapidly tend to the orthotropic solution (4.96) when the layer number increases.

4.4.2 Plate Constituted of an Angle-Ply Laminate

In this subsection we consider the case of a rectangular plate constituted of a [±θ ]n angle-ply laminate. The reduced stiffness constants of the +θ layers and −θ layers are related by the expressions:

0 0.5 1 1.5 length-to-width ratio a b

40

30

20

10

0

orthotropic laminate (B11 = 0)

[ ]0 / 90° °

2

2

113

fund

amen

tal f

requ

ency

s

Ta

Ehρ

ω

[ ]20 / 90° °

[ ]30 / 90° °

108 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

11 11 12 12 16 16

22 22 26 26 66 66

, , ,

, , ,

Q Q Q Q Q Q

Q Q Q Q Q Qθ θ θ θ θ θ

θ θ θ θ θ θ

− + − + − +

− + − + − +

′ ′ ′ ′ ′ ′= = = −

′ ′ ′ ′ ′ ′= = − = (4.99)

where the relations between the stiffness constants ijQ θ+′ and the constants ijQ referred to the layer directions are given in Table 1.1. From Expressions (1.56), (1.57) and (1.59), it results that:

16 26 11 12 22 66 16 260, 0, 0.A A B B B B D D= = = = = = = = (4.100)

Considering these relations, the introduction of Expressions (4.87) for the displacements into Equations (1.64) to (1.66), neglecting the in-plane inertia terms, leads to the equations of the free vibrations:

( )

( )

( )

2 2 2 3 30 0 0 0 0

11 66 12 66 16 262 2 2 3

2 2 2 3 30 0 0 0 0

12 66 66 22 16 262 2 3 2

4 4 40 0 0

11 12 66 224 2 2 4

3 0,

3 0,

2 2

u uA A A A B Bx yx y x y y

uA A A A B Bx y x y x x y

D D D Dx x y y

∂ ∂ ∂ ∂ ∂+ + + − − =

∂ ∂∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂+ + + − − =

∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂+ + +

∂ ∂ ∂ ∂

v w w

v v w w

w w w

3 3 3 320 0 0 0

16 26 02 3 3 2

3 3 0.su uB B

x y x y x yρ ω

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂− + − + − =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

v vw

(4.101)

Equations (4.101) can be solved in the case of hinged edges free in the in-plane tangential direction. In this case the boundary conditions are:

— edges 0x = and x a= :

2 20 0 0 0

0 16 11 122 2

2 20 0 0 0

0 66 16 262 2

0, 0,

0, 0,

x

xy

uM B D Dx y x y

uu N A B By x x y

∂ ∂ ∂ ∂⎛ ⎞= = + − − =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

∂ ∂ ∂ ∂⎛ ⎞= = + − − =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

v w ww

v w w (4.102)

— edges 0y = and y b= :

2 20 0 0 0

0 26 12 222 2

2 20 0 0 0

0 66 16 262 2

0, 0,

0, 0.

y

xy

uM B D Dy x x y

uN A B By x x y

∂ ∂ ∂ ∂⎛ ⎞= = + − − =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

∂ ∂ ∂ ∂⎛ ⎞= = + − − =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

v w ww

v w wv

(4.103)

These boundary conditions are verified by taking displacements in the form:

4.4. Vibrations of Non-Symmetric Laminate Plates 109

0

0

0

sin cos ,

cos sin ,

sin sin .

mn

mn

mn

x yu A m na bx yB m na bx yC m na b

π π

π π

π π

=

=

=

v

w

(4.104)

Substituting these expressions into Equations (4.101), we obtain:

1 2 3

2 4 52

23 5 6 2

0,0,

0.

mn mn mn

mn mn mn

smn mn mn

a A a B a Ca A a B a C

aa A a B a Cρ ωπ

′+ + =′ ′+ + =

⎛ ⎞′ ′ ′+ + − =⎜ ⎟

⎝ ⎠

(4.105)

This system has the same form as the system (4.92) with:

( )

( )

( )

2 2 23 16 26

2 2 24 66 22

2 2 25 16 26

24 2 2 2 4 4

6 11 12 66 222

3 ,

,

3 ,

2 2 .

a nR m B n R Ba

a m A n R A

a m m B n R Ba

a m D m n R D D n R Da

π

π

π

′ = − +

′ = +

′ = − +

⎡ ⎤′ = + + +⎣ ⎦

(4.106)

The expressions for the vibration frequencies may then be written in a form analo-gous to (4.94):

( )( ) ( )

42 4 2 2 2 4 4

11 12 66 224

2 2 2 2 2 216 26 2 16 26 3

1

2 2

1 3 3 ,

mns

m D m n R D D n R Da

m m B n R B nR m B n R B

πωρ

∆ ∆∆

⎡ ⎤= + + +⎣ ⎦

⎡ ⎤′ ′− + + +⎣ ⎦′

(4.107)

where

( )( ) ( )( )( )

( )( )( )( )

( )( )

22 2 2 2 2 2 2 2 21 11 66 66 22 12 66

2 2 2 2 2 22 11 66 16 26

2 2 2 2 212 66 16 26

2 2 2 2 2 23 66 22 16 26

2 2 2 2 212 66 16 26

,

3

3 ,

3

3 .

m A n R A m A n R A m n R A A

m A n R A m B n R B

n R A A m B n R B

m A n R A m B n R B

n R A A m B n R B

∆

∆

∆

′ = + + − +

′ = + +

− + +

′ = + +

− + +

(4.108)

110 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

FIGURE 4.4. Variation of the fundamental frequency of a rectangular plate made of angle-ply laminates as a function of the layer orientation.

When the stretching/bending-twisting coupling is neglected ( 16 26 0B B= = ), Expression (4.107) for the free vibration frequencies are reduced to Expression (4.11) obtained in the case of simply supported orthotropic plates. In the case where this coupling cannot be neglected, the m and n values corresponding to the fundamental mode depends on the mechanical characteristics of the laminate layers.

We consider the case of angle-ply laminates with the layer characteristics given in (4.97). In this case the fundamental frequency corresponds to 1m n= = and its variation as a function of layer angle is plotted in Figure 4.4 for a square plate constituted of [+θ /–θ ], [+θ /–θ ]2 and [+θ /–θ ]3. The variation is also plotted for an orthotropic laminate ( 16 26 0B B= = ). The results reported show that the values of the fundamental frequencies rapidly tend to the solution (4.11) of the ortho-tropic laminate when the number of layers increases.

4.5 EVALUATION OF THE LAMINATE BENDING STIFFNESSES BY ANALYSIS OF PLATE VIBRATIONS

4.5.1 Introduction

Different works [3-10] have been developed for evaluating the elastic para-meters of laminated composites from the vibrations of rectangular plates. In these works the elastic constants are deduced from the measurement of the natural fre-quencies of vibrations of a single plate. Works were also developed which take

0 5 10 15

angle ( )θ °

16

14

20

10

orthotropic laminate (B16 = B26 = 0)

[ ]θ±

20 2

113

fund

amen

tal f

requ

ency

s

Ta

Ehρ

ω

12

18

25 30 35 40 45

[ ]2θ±

[ ]3θ±

4.5. Evaluation of the Laminate Bending Stiffnesses by Analysis of Plate Vibrations 111

the measured mode shapes into consideration [11-14]. Deobald and Gibson [5] consider three different boundary conditions where the plate edges were clamped or free. In fact it has been considered that it is difficult to obtain perfect clamped conditions and the works developed in literature have been generally carried out in the case of free plates, where the four edges were free. The characteristic equations of free vibrations which model the flexural vibrations of rectangular plates have been derived using either the Ritz’s method, for example [5], or the finite element analysis, for example [8].

The characteristic equations of free vibrations relate the natural frequencies of vibrations and mode shapes to the bending stiffnesses and are obtained by solving an eigenvalue problem (previous sections). The inverse problem of deriving the bending stiffnesses from the natural frequencies is solved using an iterative proce-dure which minimizes an error function containing the deviations of natural fre-quencies between experiment and procedure. The experimental results obtained show that this inverse problem of deriving the bending stiffnesses is faced with difficulties associated with the inherent tendency of the equations of the inverse problem to be ill conditioned. This tendency was considered in [5].

So, the purpose of this section is to analyze the difficulties of evaluating the elastic constants of laminate materials from the flexural vibrations of plates and to show how some of these difficulties can be overcome. In such a way to have an extended analytical investigation of this problem, the Ritz method has been used to model the flexural vibrations of rectangular plates. This investigation was developed in [15, 16].

4.5.2 Experimental Features

4.5.2.1 Materials

The analysis developed hereafter is placed in an experimental context. So, the experimental work was performed in the case of glass fibre laminates. Unidi-rectional plates were made from epoxy resin with hardener and unidirectional glass fabric. The laminates were cured with pressure, and then post-cured in an oven. The engineering constants referred to the plate directions were measured in static tests. We obtained:

EL = 34.61 GPa, ET = 7.020 GPa, GLT = 3.190 GPa, νLT = 0.186, (4.109)

as mean values of 10 tests for each constant. Hence we deduce the values of reduced stiffnesses:

11 12 16

22 26 66

34.86 GPa, 1.315 GPa, 0,7.070 GPa, 0, 3.190 GPa,

Q Q QQ Q Q

= = == = =

(4.110)

The nominal thickness of the plates being equal to 2.1 mm, relation (1.59) allows then to evaluate the bending stiffnesses:

11 12 16

22 26 66

26.90 Nm, 1.015 Nm, 0,5.156 Nm, 0, 2.462 Nm.

D D DD D D

= = == = =

(4.111)

112 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

It has to be noted that these materials have an anisotropy ratio (EL /ET) of about 5, which is intermediate between those of balanced cross-plies (ratio equal to 1) and unidirectional carbon fibres (ratio about 10 to 20).

4.5.2.2 Boundary Conditions

Free boundary conditions are generally chosen [3-5, 10] for the experimental modal analysis of plates, considering that free conditions along edges can be more easily realized than clamped or simply-supported edges. The free boundary conditions are obtained by testing the plates on soft foam [5, 6] or cotton pads [3], or by hinging up the plates with thin threads [3, 4, 7, 10]. The stiffness of these supports would tend to increase the natural frequencies. Moreover the mode shapes are not easy to evaluate experimentally with actual free conditions. Deobald and Gibson [5] considered the case of plates with clamped edges, obtained by adhesively bonding steel shoulders to both surfaces of the plates. The shoulders were then clamped to an isolation table. In this case, it could be advanced that the actual clamped conditions along the edges would exhibit a degree of elasticity which would tend to lower the natural frequencies slightly.

In fact the tests we performed did not show any particular difficulties to implement a correct clamping along the edges of a plate. It results that we have considered more specially the case of plates the edges of which were either clamped (C) or free (F). The L direction of the materials coincides with the x-direction of the plate (figure 4.5), the edges are identified from 1 to 4 and the boundary conditions along the edges are noted in this order. Six arrangements will be considered hereafter: the CCCC, CFFF, FCFF, CCFF, CCCF and CFCC confi-gurations. The FFFF configuration was not investigated in this work on account of the different works in the literature, and finally considering the difficulty to control this type of boundary conditions in an industrial process.

Clamping the edges has been carried out by squeezing the plate directly in serrated jaws of a solid base (figure 4.6) where the experimental modal analysis was implemented. The edges were squeezed by applying gradually a pressure of about 500 kPa, until the natural frequencies of plates were stable. This equipment allows us to investigate rectangular plates the dimensions of which can be varied from 278 × 177 mm2 to 285 × 215 mm2 with any combination of clamped or free

FIGURE 4.5. Plate directions and identification of the edges.

a b

x y

z

1 2

3 4

4.5. Evaluation of the Laminate Bending Stiffnesses by Analysis of Plate Vibrations 113

FIGURE 4.6. Plate clamped along its four edges.

edges. The validation of this clamping process was established by the repetitive results obtained when plates were unset and set successively on the one hand, and by the good agreement between the experimental results and the results deduced from models.

4.5.2.3 Experiment

The impulse technique was chosen to perform modal analysis because of the ease of implementation and the quickness of the test. The equipment used is shown in Figure 4.7. An impulse hammer is used to induce the excitation of the flexural vibrations of the plate. A force transducer positioned on the hammer allows us to obtain the excitation signal as a function of the time. The width of the

FIGURE 4.7. Modal analysis equipment.

Impulse hammer

Signalconditionner

Laser vibrometer

Dynamic signal analyzer

Computer

Storage

Accelerometer

Signal conditionner

Signal conditionner

114 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

impulse and whence the frequency is controlled by the stiffness of the head of the hammer. Added mass can be placed at the back of the hammer to increase the energy dissipated by the impact and hence the height of the excitation signal. In such a way to simplify the tests, an accelerometer of low mass (equal to 0.7 g) was generally used in the tests. So as to justify the use of this accelerometer, verification tests were implemented by using a laser vibrometer (optical head and signal conditioner) which measured the velocity of the transverse displacement of a given point. It was observed that the use of low mass accelerometer can modify slightly the response at the level of the modal damping. But it was observed that this type of accelerometer had no measurable influence on the evaluation of the natural frequencies. Next, through amplifiers and conditioners, the excitation and the response signals were digitalized and processed by a dynamic analyzer of signals. This analyzer associated with a PC computer performs the acquisition of signals, controls the acquisition conditions (sensibility, frequency range, trigger conditions, etc.), and next performs the analysis of the signals acquired (Fourier transform, frequency response, mode shapes, etc.).Then, the signals and the associated processing can be saved for new post-processings.

4.5.3 Introduction to the Experimental Modal Analysis of Orthotropic Plates

4.5.3.1 Evaluation of the Natural Frequencies

The natural frequencies of rectangular plates can be evaluated using the Ritz method as considered in the previous sections. The natural frequencies are derived from Equation (4.40) for orthotropic laminates or (4.74) for arbitrary symmetric laminate plates. To solve the system of equations (4.40) or (4.74) it is necessary to choose the admissible functions Xm(x) and Yn(y) which satisfy the appropriate boundary conditions and then to calculate the integrals (4.33) and (4.34). As introduced by Young [17] and considered in the previous sections 4.2 and 4.3, the characteristic functions of the beam bending vibrations, introduced in Section 3.4 of Chapter 3, can be used as admissible functions. Polynomial functions [1, 2] satisfying the essential boundary conditions can also be considered. The dimen-sionless integrals (4.33) and (4.34) can be calculated by an analytical development or by a numerical process and then stored in computer. Next, the appropriate integral values can be read from computer memory to form system (4.40) or (4.74). Some of these integrals are tabulated in [1, 2, 15, 16]. It has to be noted that the beam functions satisfy orthogonality relations (Section 3.3.2) which make zero many of the integrals.

Calculation implemented to derive the natural frequencies from the system (4.40) or (4.74) when the bending stiffnesses are given shows a faster convergence of the values obtained with the characteristic beam functions when the number of terms used in series (4.24) increases. However the difference between the values obtained with beam functions and polynomial functions become low when the number of terms is great. This difference is smaller than

4.5. Evaluation of the Laminate Bending Stiffnesses by Analysis of Plate Vibrations 115

0.1% for the six first modes of the different configurations considered with a series of 100 terms (M = N = 10). At last the determination of the natural fre-quencies by finite element analysis shows that, for a series of 100 terms, the difference with the frequencies deduced from finite element is smaller than 0.3% for the beam functions and smaller than 0.2% for the polynomial functions. Finally the beam functions were chosen considering the interest of orthogonality relationship of these functions.

4.5.3.2 Different Results

The conditions of plate fabrication and realization of the boundary conditions lead to a value of the length-to-width ratio R of about 1.5. Table 4.6 reports for this value the contribution to the square of the natural frequencies (whence the contribution to the vibration energy) of each reduced stiffness as function of the boundary conditions and for the nine natural frequencies fij (i, j = 1, 2, 3), in the case of orthotropic laminates (4.109) to (4.111). Also table 4.6 reports the differ-rence between the Rayleigh’s approximation (4.2.2) and the Ritz’s evaluation with 100 terms (M = N = 10). The results reported show that this difference is generally small: difference smaller than 1.3% for 42 cases of the 54 cases reported in the table, 4 other cases showing a difference of 1.8% (hence 85% of the cases give a difference smaller than 1.8%). The other values calculated by the Rayleigh's approximation lead to differences ranging from 3 to 7.8%. The highest differences (7.6 and 7.8%) are observed in the case of the configurations with three free edges: CFFF configuration (frequency f12) and FCFF configuration (frequency f21). The other differences are smaller than 4.5%. The differences observed are the results of the capacity of the Rayleigh’s approximation to describe more or less accurately the actual mode shapes.

Next the results of table 4.6 show that the natural frequencies fi1 depend only on the reduced stiffness Q11 in the case of the CFFF configuration. In the same way, the natural frequencies f1j depend only on the reduced stiffness Q22 in the case of the FCFF configuration. These natural frequencies are associated with cylindrical bending modes along the directions 1 and 2 (Figures 4.8 and 4.9). Then it is possible to evaluate the reduced stiffnesses Q11 and Q22 from these two configurations. From relation (4.46) we obtain:

4 0000

2 21111 13 2200

1148 (CFFF)s ii

iii

a CQ fh C

ρπ= , (4.112)

00004112 2

22 13 4 002211

48 (FCFF)jjsi

jj

CaQ fh R Cρπ= . (4.113)

The differences between the values of fi1 and f1j frequencies calculated by the Rayleigh’s approximation and the values obtained by the 100-term Ritz’s estimation are smaller than 0.36%. Furthermore the determination of Q11 and Q22 from the different frequencies fi1 and f1j allows us to obtain an evaluation of the

116 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

TABLE 4.6. Contributions (expressed in percent) of the reduced stiffnesses to the squares of the natural frequencies of a rectangular orthotropic plate of length-to-width ratio equal to 1.5, constituted of material (4.109)-(4.111). Differences (in percent) between Rayleigh frequencies and 100-term Ritz frequencies.

f11 f12

Q11 Q22 Q12 Q66 Diff. Q11 Q22 Q12 Q66 Diff.

CCCC CFFF FCFF CCFC CCCF CCFF

42.97 100.0 0 2.163 89.62 28.77

44.12 0 100.0 89.91 2.273 29.54

2.206 0 0 0.313 0.321 0.291

10.700 0 8.2398.43241.40

0.1020.0800.0790.4870.4913.852

10.0721.200 0.30236.931.953

78.600 100.095.5536.7878.75

1.9360 0 0.1642.0490.306

9.391 78.80 0 4.313 24.24 19.60

0.147 7.786 0.364 0.458 0.556 0.569

f13 f21 Q11 Q22 Q12 Q66 Diff. Q11 Q22 Q12 Q66 Diff.

CCCC CFFF FCFF CCFC CCCF CCFF

2.994 1.731 0 0.080 9.466 0.294

89.78 71.99 100.0 97.55 73.91 92.84

1.236 0.251 0 0.094 1.813 0.159

5.99526.530 2.46114.817.029

0.1230.3690.0460.3200.3090.731

77.94100.00 35.8195.4378.26

10.530 21.6537.910.3192.046

1.9710 0 2.0480.1680.312

9.562 0 78.35 24.23 4.423 20.01

0.024 0.239 7.638 0.559 0.466 4.530

f22 f23 Q11 Q22 Q12 Q66 Diff. Q11 Q22 Q12 Q66 Diff.

CCCC CFFF FCFF CCFC CCCF CCFF

38.75 60.24 0 8.481 67.49 25.83

39.79 0 60.87 68.22 8.847 26.52

3.668 0 0 1.816 1.844 1.596

17.8039.7639.1321.4821.8246.05

0.0421.8221.7980.9690.9693.233

16.3020.670 2.64534.577.399

64.3221.8983.6581.7535.5259.57

3.3131.1820 1.2163.2611.579

16.08 56.26 16.35 14.38 26.65 31.46

0.185 3.851 0.593 0.946 0.386 1.196

f31 f32 Q11 Q22 Q12 Q66 Diff. Q11 Q22 Q12 Q66 Diff.

CCCC CFFF FCFF CCFC CCCF CCFF

89.46 100.0 71.42 73.21 97.49 92.65

3.147 0 1.811 9.885 0.085 0.309

1.264 0 0.256 1.844 0.096 0.163

6.1340 27.0315.062.5257.203

0.1770.0453.7790.3090.3271.295

63.4283.2821.3234.6081.3058.81

16.950 21.2335.502.7737.702

3.3550 1.1833.2621.2421.601

16.28 16.72 56.27 26.65 14.69 31.89

0.185 0.506 3.822 0.387 0.956 3.153

f33 FFFF plate

Q11 Q22 Q12 Q66 Diff. mode Q11 Q22 Q12 Q66

CCCC CFFF FCFF CCFC CCCF CCFF

37.15 50.31 6.539 14.71 57.11 29.29

38.15 6.793 51.06 57.99 15.27 30.08

4.221 1.267 1.252 2.978 3.012 2.753

20.4841.6441.1524.3324.6037.88

0.1101.7941.7850.4570.4821.191

1 2 3 4 5 6

2.51956.9536.3953.571.90019.40

2.47737.4258.391.91254.2219.84

0.134 5.621 5.174 0.297 0.286 1.429

94.87 0.007 0.051 44.22 43.59 59.33

4.5. Evaluation of the Laminate Bending Stiffnesses by Analysis of Plate Vibrations 117

FIGURE 4.8. Shapes of the cylindrical bending modes in the case of CFFF plates.

FIGURE 4.9. Shapes of the cylindrical bending modes in the case of FCFF plates.

variation of these stiffnesses as a function of the frequency. Also the results of Table 4.6 show that the natural frequency f22 depends only

on the stiffnesses Q11 and Q66 for the CFFF configuration and on the stiffnesses Q22 and Q66 for the FCFF configuration. The differences observed between Rayleigh’s approximation and Ritz evaluation are equal to 1.8%. For the mode 22, relation (4.46) leads to:

( )3

2 2200 2002 1111 2 0022 422 2222 11 2222 12 2222 66 2222 224 2 2

12 s

h C Q C Q C Q R C Q Ra

ωρ

⎡ ⎤= + + +⎣ ⎦ . (4.114)

In the case of two free opposite edges, the term 00222222C is zero. In addition we have

x z

y

x z

y

118 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

FIGURE 4.10. Shapes of mode 22 in CFFF and FCFF configurations

the following relations for the CFFF configuration:

2200 222222 22

1111 11 112222 22 22

11112222

(CFFF) (CF),

(CFFF) (CF) (FF),

(CFFF) 0.

C I

C I I

C

=

=

=

(4.115)

So Relation (4.114) associated with (4.115) gives:

3

2 22 11 11 222 22 11 22 22 662 4

1(CFFF) (CF) 4 (CF) (FF)48 s

hf I Q I I Q Raπ ρ

⎡ ⎤= +⎣ ⎦ . (4.116)

The FCFF configuration is obtained by transposing axes 1 and 2. Whence:

3 4

2 11 11 2222 22 22 66 22 222 4 2

1 1(FCFF) 4 (CF) (FF) (CF)48 s

h Rf I I Q I Qa Rπ ρ

⎡ ⎤= +⎢ ⎥⎣ ⎦. (4.117)

Figure 4.10 shows the shapes of modes 22 in the two configurations. Also the natural frequencies f12 and f32 for the CFFF configuration depend only

on the stiffnesses Q11 and Q66. In the same way the frequencies f21 and f23 for the FCFF configuration depend only on the stiffnesses Q22 and Q66. However for these frequencies either the difference between Rayleigh's approximation and Ritz’s evaluation is marked (about 8% for f12(CFFF) and f21(FCFF)) or the par-ticipation of the stiffness Q66 is rather low (about 16% for f23(FCFF) and f32(CFFF)).

Lastly, Table 4.6 shows very low contribution of the stiffness Q12 to the diffe-rent modes, and this result specifies clearly the tendency to have ill conditioned numerical equations for the evaluation of the bending stiffnesses from the natural frequencies of a single plate. In practice the best contributions are obtained in the case where the four edges of the plate are clamped: the contributions vary from 1.2% for the mode 13 to 4.2% for the mode 33. For comparison we report also in Table 4.6 the contributions of stiffnesses in the case of a plate free along its four edges. Again we observe a low contribution of the stiffness Q12 in this confi-guration.

x z

y

4.5. Evaluation of the Laminate Bending Stiffnesses by Analysis of Plate Vibrations 119

4.5.4 Experimental Results and Discussion in the case of Orthotropic Plates

4.5.4.1 Values of the Natural Frequencies

Before we evaluate the bending stiffnesses from the natural frequencies, it is needed to have an estimation of the consistency of the values measured for the frequencies. Table 4.7 compares the values of the natural frequencies of a plate made of the orthotropic material considered in Subsection 4.5.2.1 as determined by experimental modal analysis, Rayleigh’s approximation and 100-term Ritz’s evaluation. Different boundary conditions along the four edges have been consi-dered and the experimental analysis was implemented in the case of a plate of 278 mm length and 185 mm width. The comparison is limited to the first six modes. Rayleigh and Ritz frequencies were calculated using the values of stiffnesses TABLE 4.7. Natural frequencies (in Hz) measured and calculated for a rectangular plate. Differences (in percent) between the measured values and the calculated values: diff. 1 (Rayleigh’s approximation) and diff. 2 (100-term Ritz approximation).

Mode 1 Mode 2 Exp. Rayleigh Ritz Diff. 1 Diff. 2 Exp. Rayleigh Ritz Diff. 1 Diff. 2

CCCC CCFC CCCF CCFF CFFF FCFF

192.1 139.5 130.5 35.5 18.9 19.1

190.81 133.67 132.13 36.65 19.66 19.12

190.62133.02131.4835.29 19.64 19.11

–0.67–4.171.25 3.24 4.02 0.10

–0.77–4.690.75

–0.593.92 0.05

382.1 206.3 201.9 131.9 38.1 40.8

390.55 205.85 205.82 139.25 42.69 41.10

389.98 204.70 204.68 133.21 39.60 38.18

2.21 –0.22 1.94 5.57

12.05 0.74

2.06 –0.781.38 0.99 3.94

–6.42

Mode 3 Mode 4 Exp. Rayleigh Ritz Diff. 1 Diff. 2 Exp. Rayleigh Ritz Diff. 1 Diff. 2

CCCC CCFC CCCF CCFF CFFF FCFF

410.9 376.6 339.7 144.0 119.0 120.1

394.08 357.42 352.95 140.66 123.18 119.84

393.50355.79351.31139.87122.78119.40

–4.09–5.093.90

–2.323.51

–0.22

–4.23–5.523.42

–2.873.18 0.58

559.3 394.6 411.0 236.4 152.4 143.1

553.88 403.13 406.55 242.38 149.38 142.10

552.83 401.89 405.30 234.79 143.96 136.92

–0.97 2.16

–1.08 2.53

–1.98 –0.70

–1.161.85

–1.39–0.68–5.54–4.32

Mode 5 Mode 6

Exp. Rayleigh Ritz Diff. 1 Diff. 2 Exp. Rayleigh Ritz Diff. 1 Diff. 2

CCCC CCFC CCCF CCFF CFFF FCFF

699.1 435.4 421.4 341.5 155.3 156.2

714.64 422.99 419.69 358.34 158.71 153.60

713.37418.93415.66353.76155.87150.89

2.22 –2.85–0.414.93 2.20

–1.66

2.04 –3.78–1.363.59 0.37

–3.40

759.6 589.6 590.6 377.2 272.7 266.3

722.87 586.42 586.42 362.75 270.93 260.08

721.98 585.21 584.17 360.12 260.88 250.51

–4.84 –0.54 –0.71 –3.83 –0.65 –2.34

–4.95–0.74–1.09–4.53–4.33–5.93

120 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

(4.109) and (4.110) measured in static tests. The use of these values assumes that the stiffnesses are not depending on the frequency. The experimental values are the average of 10 tests by plate configuration, performed by setting and unsetting successively the different plates studied. The values reported in table 4.7 show that the differences observed are smaller than 5.6 percent, except for the mode 2 of the CFFF configuration. The analysis of the experimental results shows that the measured values are randomly distributed near the calculated frequencies. For the Rayleigh’s approximation the differences are in the range [–6%, 6%], 70 percent of the values being distributed in the interval [–3%, 3%].

Finally the differences observed appear to be associated with a usual scattering of the experimental results, which is not possible to reduce more in an industrial context. Previously we reported that setting and unsetting the plates do not modify perceptibly the measured values of the natural frequencies. Considering the differ-rent factors which can contribute to the scattering of the measured values, we may advance that the main factor is the one of the homogeneity of fabrication of the rectangular plates, which corresponds in our work to a usual industrial fabrication. Thus the results obtained validate the experimental process for measuring the natural frequencies as well as the use of clamping conditions.

4.5.4.2 Determination of the Stiffnesses by an Iterative Procedure

The inverse problem of deriving the bending stiffnesses from the natural fre-quencies needs the use of an iterative technique in the case of the evaluation of the flexural vibrations of rectangular plates by the Ritz’s method or by numerical methods. Starting from a set of initial values of the stiffnesses, these values are iteratively updated until the difference of natural frequencies between calculation and experiment is sufficiently small. In Ritz’s evaluation, Deobald and Gibson [5] introduce a “solution” matrix, when in numerical evaluation the iterative process is established by considering a “sensibility” matrix [4, 7, 10], introduced first by De Wilde [3]. The two procedures lead to similar developments and we consider the formulation of Deobald and Gibson hereafter.

For orthotropic plates, the system (4.40) with four indices m, n, i and j can be expressed in the following formulation with two indices k and l as:

( )2

1

0M N

kl kl lk

a b AΩ×

=

− =∑ , (4.118)

where the coefficients akl and bkl are given by the expressions:

( )2200 2002 2002 1111 2 0022 4

12 66 22

0000

4 ,

,

kl minj minj minj minj minj

kl minj

a C C C C R C R

b C

α α α⎡ ⎤= + + + +⎣ ⎦

= (4.119)

and the indices k and l are deduced from the indices m, n, i and j by the relations:

( ) ( )1 , 1 ,

for , 1, 2, . . . , , , 1, 2, . . . , .k m N n l i N j

m i M n j N= − + = − +

= = (4.120)

4.5. Evaluation of the Laminate Bending Stiffnesses by Analysis of Plate Vibrations 121

Then, the system (4.118) can be expressed in the matrix form:

( )2 0Ω− =a b A , (4.121)

where a and b are the square matrices of respective elements akl and bkl and A is the column matrix of the eigenvectors which give the mode shapes of the vibra-tions. Equation (4.121) can be solved as an eigenvalue and eigenvector problem.

Let 1 2 3 4, , and e e e ef f f f be the experimental values of four natural frequencies evaluated by modal analysis. The four corresponding eigenvalues of the system (4.121) are deduced from Expression (4.42) and can be expressed as:

( )2

11, 1, 2, 3, 4,

ee ii i

DλΩ = = (4.122)

with

( )222e ei i sf aλ π ρ= . (4.123)

The iterative process is initiated by solving the eigenvalue problem with four initial estimates of the bending stiffnesses D11, D22, D12 and D66. In this way the system (4.121) is rewritten in the form:

( ) 111

0M N

kl kl lk

D a b Aλ×

=

− =∑ , (4.124)

with

( )

222 sf aλ π ρ= . (4.125) The system (4.124) is simplified in the case where the assumed functions present orthogonality relationships:

00

00

1 if ,,

0 if ,

1 if ,,

0 if .

mi mi mi

nj nj nj

m iI

m i

n jI

n j

δ δ

δ δ

=⎧= = ⎨ ≠⎩

=⎧= = ⎨ ≠⎩

(4.126)

These relations are verified by the beam functions. In this case the system (4.124) is written as:

( ) 111

1 if ,0,

0 if .

M N

kl kl l klk

k lD a A

k lλ δ δ

×

=

=⎧− = = ⎨ ≠⎩

∑ (4.127)

If the initial estimates of the bending stiffnesses are reasonable, it is possible to identify, among the solutions of system (4.124) or (4.127), the four approximate eigenpairs (1) (2) (3) (4)

1 2 3 4, , and ,A A A Aλ λ λ λ corresponding to the

experimental eigenvalues 1 2 3 4, , and e e e eλ λ λ λ , respectively.

122 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

Next, the approximate eigenvectors are used to derive the matrix formulation which relates the four experimental eigenvalues to the four bending stiffnesses. For each eigenpair, ( ) and i

i Aλ , the system, for example (4.127), leads to the P equations (P = M × Ν ):

( )( )

( )

( ) ( ) ( ) ( )11 11 12 11 2 1

( ) ( ) ( ) ( )11 21 22 21 2 2

( ) ( ) ( ) ( )11 1 21 2

. . . ,

. . . ,.

. . . .

.

.

i i i iP iP

i i i iP iP

i i i iP P PP iP P

D a A a A a A A

D a A a A a A A

D a A a A a A A

λ

λ

λ

+ + + =

+ + + =

+ + + =

(4.128)

From these equivalent equations, it is practical to select the equation which cor-responds to the maximum value ( )i

qA in the second member. Hence we obtain:

( )( ) ( ) ( )( )11 1 21 2

. . . , 1, 2, 3, 4.i i iii q q q qP PA D a A a A a A iλ = + + + = (4.129)

Introducing expression (4.119) of akl, Equation (4.128) can be expressed in the following form:

( )( )

( )

( )( )1 1 11 2 1 22 3 1 12 4 1 661

( )1 2 11 2 2 22 3 2 12 4 2 662

( )1 11 2 22 3 12 4 66 . . . ,

iii q q q q q

iq q q q

iqP qP qP qPP

A A C D C D C D C D

A C D C D C D C D

A C D C D C D C D

λ = + + +

+ + + +

+ + + + +

(4.130)

with

( )

2200 0022 41 2

2002 0220 2 1111 23 4

, ,

, 4 ,

kl minj kl minj

kl minj minj kl minj

C C C C R

C C C R C C R

= =

= + = (4.131)

where subscripts k, l are defined as functions of subscripts m, n, i and j by Relation (4.120). Thus the relation between the eigenvalues and the bending stiffnesses can be written in the following matrix form:

11 12 13 14 111

21 22 23 24 222

31 32 33 34 123

41 42 43 44 664

H H H H DH H H H DH H H H DH H H H D

λλλλ

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ =⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

, (4.132)

where the coefficients Hij are expressed as:

( )( )

1

1 , , 1, 2, 3, 4.P

iij jpqPi

q p

H A C i jA =

= =∑ (4.133)

The matrix ijH⎡ ⎤⎣ ⎦ is the sensibility matrix.

4.5. Evaluation of the Laminate Bending Stiffnesses by Analysis of Plate Vibrations 123

The solution of system (4.132) for the stiffnesses 11 12 22 66( , , , )D D D D is searched by using an iterative process. This process is initiated with a set of the stiffnesses and approximate eigenvectors are obtained by solving system (4.118) At each step of the iterative process new estimates it it it it

11 12 22 66( , , , )D D D D of the bending stiffnesses are obtained by solving (4.132) with the use of eigenvalues from the experimental natural frequencies as:

it111 11 12 13 14it

21 22 23 24 222it31 32 33 34 123it41 42 43 44664

e

e

e

e

DH H H HH H H H DH H H H DH H H H D

λ

λ

λ

λ

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦

, (4.134)

where the coefficients Hij are calculated with the eigenvectors of the preceding iteration. The cycle is then repeated by solving again the system (4.118) with the new estimates of the bending stiffnesses upgrading the accuracy.

In fact and whatever the boundary conditions along the edges of the plates may be, the results obtained show that the iterative procedures depend strongly on the conditioning of the sensibility matrix, on accuracy of the experimental natural frequencies and on the initial estimates for the bending stiffnesses. The first feature is the inherent tendency for the sensibility matrix to be ill conditioned. This problem results from the fact that one or more of the stiffnesses does not contribute much to all four natural frequencies considered. The results of table 4.6 show clearly that the stiffness D12 is usually the cause of ill conditioned matrices when they occurred. It results that the convergence of iterative procedures is inconsistent. Thus mode combinations may produce sensibility matrix nearly singular, then the values of bending stiffnesses may diverge or be completely unreasonable.

The other major problem is that the experimental natural frequencies do not closely match the predicted values. We reported in Subsection 4.5.4.1 that the scattering observed on the experimental values of frequencies was associated with the inherent inhomogeneity of the materials. The experimental results obtained shown that the experimental values of the natural frequencies match the exact values within about 5 percent. In fact numerical simulations show that a slight error in the natural frequencies greatly magnifies the error in the bending stiffnesses, also depending on the initial estimates of stiffnesses. With the exact values of the natural frequencies (derived from system (4.118) and using the actual bending stiffnesses), it is observed that the iterative procedures converge and lead to the actual bending stiffnesses, even if the initial estimates are greatly different (about 50%). Next when the errors between the values used for the natural frequencies and the exact ones are increased slightly, it is observed that it is necessary to introduce initial estimates of the bending stiffnesses which are nearer and nearer to the actual stiffnesses. For industrial conditions, the iterative processes diverge or give values of the bending stiffnesses which are notably

124 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

distant from the true values, when the initial estimates are not accurate enough. It results that iterative procedures cannot be used in industrial applications.

4.5.4.3 Evaluation of the Stiffnesses from Rayleigh’s Approximation

4.5.4.3.1 Introduction

From the elements developed in the preceding subsection it becomes apparent that for industrial conditions it is not possible to have confidence in the results for the bending stiffnesses derived from the experimental natural frequencies of plate vibrations by an iterative procedure. The object of this present subsection is to consider procedures of evaluation which allow us to overcome the difficulties associated with the iterative processes.

4.5.4.3.2 Plate Clamped on its Four Edges

In Rayleigh’s approximation, the natural frequencies are given by relation (4.46). In the case of clamped opposite edges we have:

1111 2002mmnn mmnnC C= , (4.135)

and approximation (4.46) gives:

( )

32 2200 2002 2 0022 4

11 12 66 224 0000 2 2 12mn mmnn mmnn mmnn

s mmnn

h C Q C Q Q R C Q Ra C

ωρ

⎡ ⎤= + + +⎣ ⎦ .

(4.136) This relation shows that the reduced stiffnesses Q12 and Q66 are not separate. It results that measuring three natural frequencies allows us to obtain the stiffnesses Q11, Q22 and Q12 + 2Q66. These stiffnesses are solutions of the linear system:

( )4

21 11 2 22 3 12 663

12 2si i i i

a a Q a Q a Q Qhρ ω = + + + , (4.137)

introducing the coefficients:

2200 0022 22004 2

1 2 30000 0000 0000

1 2 3

, , 2 ,

, , ,

mmnn mmnn mmnni i i

mmnn mmnn mmnn

C C Ca a R a RC C C

i i i i

= = =

=

(4.138)

where i1, i2 and i3 are three given modes, determined by the respective values (mn)1, (mn)2, and (mn)3 of mn.

The results obtained are reported in Table 4.8 for different combinations of modes. The values obtained for the stiffnesses Q11 and Q22 are slightly scattered. The mean value of Q11 is equal to 36.12 GPa with values ranging from –3.4 to 4.5 percent of the mean value. The mean value of Q22 is equal to 7.249 GPa with values ranging from –4.6 to 4.5 percent of the mean value. By contrast the values of Q12 + 2 Q66 are largely scattered from 3.593 GPa to 8.190 GPa.

4.5. Evaluation of the Laminate Bending Stiffnesses by Analysis of Plate Vibrations 125

TABLE 4.8. Reduced stiffnesses deduced from the natural frequencies measured in the case of a CCCC plate.

f1 f2 f3 i1, i2, i3 (mn)1,(mn)2,(mn)3 Q11 Q22 Q12 + 2Q66

(Hz) (GPa)

192.1 192.1 192.1 382.1 382.1 192.1 559.3 382.1

382.1 382.1 382.1 410.9 410.9 382.1 699.1 559.3

410.9 559.3 699.1 559.3 699.1 759.6 759.6 759.6

1, 2, 3 1, 2, 4 1, 2, 5 2, 3, 4 2, 3, 5 1, 2, 6 4, 5, 6 2, 4, 6

11, 21, 12 11, 21, 22 11, 21, 31 21, 12, 22 21, 12, 31 11, 21, 13 22, 31, 13 21, 22, 13

37.74 34.88 36.86 35.15 36.90 36.31 35.98 35.13

7.576 6.917 7.374 7.151 7.438 7.246 7.155 7.136

3.593 8.190 4.998 7.579 4.882 5.897 7.266 7.616

Mean value (GPa) Lower deviation (%) Upper deviation (%)

36.12 –3.4 4.5

7.249 –4.6 4.5

6.253 –42.5 31.0

4.5.4.3.3 Plate with Three Clamped Edges and Free along the Other Edge

In the case of three clamped edges (CCCF or CCFC configuration), the reduced stiffnesses Q12 and Q66 are separate in expression (4.46) of Rayleigh’s appro-ximation. It results that the reduced stiffnesses Q11, Q22, Q12 and Q66 can be deduced from measurement of four different natural frequencies. The stiffnesses are solutions of the linear system:

4

21 11 2 22 3 12 4 663

12 si i i i i

a a Q a Q a Q a Qhρ ω = + + + , (4.139)

introducing the coefficients:

2200 00224

1 20000 0000

2200 11112 2

3 40000 0000

1 2 3 4

, ,

2 , 4 ,

, , , ,

mmnn mmnni i

mmnn mmnn

mmnn mmnni i

mmnn mmnn

C Ca a RC C

C Ca R a RC C

i i i i i

= =

= =

=

(4.140)

where i1, i2, i3 and i4 are four given modes determined by the values of mn. The results obtained are reported in Table 4.9 for different combinations of

modes. Again the values obtained for the stiffnesses Q11 and Q22 are slightly scat-tered and even less than previously: mean value equal to 35.41 GPa for Q11 with values ranging from –1.1 to 1.9%, and mean value equal to 7.182 GPa for Q22 with values ranging from –0.8 to 0.4%. The values of the stiffness Q66 are clearly more scattered with a mean value of 2.516 GPa and values ranging from −7 to 11.2%. Lastly the values obtained for the coupling stiffness Q12 show a large scattering: values ranging from 1.752 GPa to 3.833 GPa, very similar to the

126 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

TABLE 4.9. Reduced stiffnesses deduced from the natural frequencies measured in the case of a CCFC plate.

f1 f2 f3 f4 i1, i2, i3, i4 mn1, mn2, mn3, mn4 Q11 Q22 Q12 Q66

(Hz) (GPa)

139.5 206.3 376.6 139.5 206.3 139.5 139.5 139.5 139.5

206.3 376.6 394.6 206.3 376.6 376.6 376.6 206.3 206.3

376.6 394.6 435.4 376.6 394.6 394.6 394.6 394.6 394.6

394.6 435.4 589.6 435.4 589.6 589.6 435.4 435.4 589.6

1, 2, 3, 4 2, 3, 4, 5 3, 4, 5, 6 1, 2, 3, 5 2, 3, 4, 6 1, 3, 4, 6 1, 3, 4, 5 1, 2, 4, 5 1, 2, 4, 6

11, 21, 12, 31 21, 12, 31, 22 12, 31, 22, 32 11, 21, 12, 22 21, 12, 31, 32 11, 12, 31, 32 11, 12, 31, 22 11, 21, 31, 22 11, 21, 31, 32

35.00 35.60 35.36 36.09 35.33 35.36 35.34 35.33 35.28

7.208 7.154 7.205 7.212 7.178 7.209 7.209 7.124 7.136

3.833 1.752 3.382 3.509 2.673 3.487 3.509 2.417 2.613

2.402 2.797 2.390 2.331 2.622 2.356 2.359 2.717 2.673

Mean value (GPa) Lower deviation (%) Upper deviation (%)

35.41 –1.1 1.9

7.182 –0.8 0.4

3.019 –42.0 16.2

2.516 –7.0 11.2

scattering observed previously for Q12 + 2Q66 when they are deduced from the vibrations of plates with four clamped edges.

4.5.4.3.4 Plate with Two Consecutive Clamped Edges and Free along the Other Edges

In the case where two consecutive edges are clamped and the others free, the stiffnesses Q11 and Q22 are separate again in Rayleigh’s approximation. As previously the reduced stiffnesses are solutions of the linear system (4.139), the expressions (4.140) of the coefficients aij being modified according to the new boundary conditions.

The results obtained are reported in Table 4.10 for different combinations of modes. These results are very similar to the results deduced from the preceding tests of the plates with three clamped edges. The values of Q11 and Q22 are slightly scattered: mean value equal to 34.60 GPa for Q11 with values ranging from –1.4 to 0.7%, and mean value of 7.069 GPa for Q22 ranging from –0.9 to 0.8%. As previously the values obtained for Q12 are largely scattered ranging from 0.308 to 3.6 GPa. The values obtained for Q66 lead to a mean value of 2.984 GPa with values ranging from –14.3 to 91%. This high range can be associated with the anomalous high value of 5.693 GPa obtained in the third case of Table 4.10. This high deviation can be connected with the instability introduced by the stiffness Q12. However this process is not induced systematically, as it can be observed in the sixth case of Table 4.9.

4.5. Evaluation of the Laminate Bending Stiffnesses by Analysis of Plate Vibrations 127

TABLE 4.10. Reduced stiffnesses deduced from the natural frequencies measured in the case of a CCFF plate.

f1 f2 f3 f4 i1, i2, i3, i4 mn1,mn2,mn3,mn4 Q11 Q22 Q12 Q66

(Hz) (GPa)

35.5 35.5 35.5 35.5 35.5 35.5 35.5 131.9

131.9 131.9 131.9 131.9 131.9 144.0 144.0 144.0

144.0 144.0 144.0 236.4 341.5 341.5 236.4 236.4

236.4 341.5 377.2 377.2 377.2 377.2 341.5 341.5

1, 2, 3, 4 1, 2, 3, 5 1, 2, 3, 6 1, 2, 4, 6 1, 2, 5, 6 1, 3, 5, 6 1, 3, 4, 5 2, 3, 4, 5

11, 21, 12, 22 11, 21, 12, 31 11, 21, 12, 13 11, 21, 22, 13 11, 21, 31, 13 11, 12, 31, 13 11, 12, 22, 31 21, 12, 22, 31

34.8434.6634.1034.7334.6734.4534.7234.67

7.129 7.100 7.007 7.046 7.042 7.006 7.124 7.101

3.600 2.895 0.652 3.497 3.239 0.308 3.575 3.267

2.556 2.588 5.692 2.593 2.602 2.677 2.564 2.599

Mean value (GPa) Lower deviation (%) Upper deviation (%)

34.60–1.4 0.7

7.069 –0.9 0.8

2.629 –88.3 36.9

2.984 –14.3

91

4.5.4.3.5 Plate with Four Different Configurations Another possibility for determining the bending stiffnesses consists in consi-

dering the natural frequencies of a given mode for four different configurations of the boundary conditions along the edges of a plate, for example CCCC, CCFC, CCFF and CFFF configurations. The reduced stiffnesses are solutions of a linear system similar to system (4.139). In the present case, the subscripts i are related to the four configurations considered.

Tables 4.11 and 4.12 report the results obtained when different modes are considered: Table 4.11 when a mode of given shape is considered and Table 4.12 when modes are considered according to increasing values of the natural fre-quencies. Again we observe general features which are similar to the ones noticed when the bending stiffnesses are deduced from the natural frequencies of a plate with given boundary conditions considered in the previous subsections.

TABLE 4.11. Reduced stiffnesses deduced from the natural frequencies of a mode of given shape (mn) for four different boundary conditions of a plate.

Mode Frequencies (Hz) Stiffnesses (GPa)

mn CCCC CCFC CCFF CFFF Q11 Q22 Q12 Q66

11 21 12 22

192.1 382.1 410.9 559.3

139.5 206.3 376.6 435.4

35.5 131.9 144.0 236.4

18.9 119.0 38.1 155.3

34.96 35.29 35.62 37.14

8.592 9.275 7.974 8.277

2.264 2.326 1.938 – 0.01

2.816 2.548 2.348 2.533

Mean value (GPa) Lower deviation (%) Upper deviation (%)

35.75 –2.2 3.9

8.529 –6.5 8.7

1.630 –101 42.7

2.561 –8.3 10

128 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

TABLE 4.12. Reduced stiffnesses deduced from the natural frequencies of a mode of given order (1, 2, 3 or 4) for four different boundary conditions of a plate.

Mode Frequencies (Hz) mn Q11 Q22 Q12 Q66

CCCC CCFC CCFF CFFF CCCC CCFC CCFF CFFF (GPa)

1 192.1 139.5 35.5 18.9 11 11 11 11 34.961 8.592 2.664 2.816

2 382.1 206.3 131.9 38.1 21 21 21 12 35.098 9.108 2.604 2.635

3 410.9 376.6 144.0 119.0 12 12 12 21 35.290 8.654 2.649 2.594

4 559.3 394.6 236.4 152.4 22 31 22 13 35.750 8.600 2.286 2.887

Mean value (GPa) Lower deviation (%) Upper deviation (%)

35.274 –0.9 1.3

8.738 –1.7 4.2

2.551 –10.4 4.4

2.733 –5.1 5.6

4.5.4.4 Discussion of the Results and Conclusions

4.5.4.4.1 Discussion

The results reported in the previous subsections show clearly that the use of Rayleigh’s approximation improves greatly the determination of the bending stiffnesses, since the problem of the divergence of the iterative processes is overcome. However the results obtained show the tendency for the evaluation of the bending stiffnesses from the natural frequencies of the flexural vibrations of a plate to be an ill conditioned problem.

The results show a low difference between the values deduced from the natural frequencies of vibrations and the value obtained in static tests for the longitudinal stiffness Q11, and this for the different configurations of the boundary conditions. A similar observation can be done for the values deduced for the transverse stiffness Q22 from the frequencies of four different modes of a plate in a given configuration.

We have reported the high scattering of the experimental values obtained for the coupling stiffness Q12 from the natural frequencies, whatever the configu-ration of the plate may be. Lastly, the evaluation of the shear stiffness Q66 from the natural frequencies under-estimates systematically the value measured in static tests from 6.5 to 21 percent. This fact can be related with the instability of the evaluation of Q12.

An extensive analysis of the sensibility of the bending stiffnesses to the variation of the natural frequencies of orthotropic plates has been implemented [15, 16]. The results obtained clearly corroborate the general features deduced from the previous experimental analysis. The variations of the reduced stiffnesses Q11 and Q22 are always of the same order as the variations of the natural fre-quencies of the plates. For all the configurations of plates, the variations obtained for the stiffness Q12 show a high instability with regard to low variations of the natural frequencies. It results that the stiffness Q12 cannot be evaluated from the

4.5. Evaluation of the Laminate Bending Stiffnesses by Analysis of Plate Vibrations 129

measurement of the natural frequencies of the flexural vibrations of plates. Lastly the variations of the stiffness Q66 are depending on the configuration and modes. This fact is related to the contribution of the stiffness Q66 to the different modes.

4.5.4.4.2 Conclusions

The analysis developed and the results reported in the preceding subsections show how Rayleigh’s approximation allows us to evaluate the bending stiffnesses Q11 and Q22 from the natural frequencies measured in a single flexural test of a plate with one of the configurations CCCC, CCCF or CCFF. The errors observed in the evaluations of Q11 and Q22 are of the same order as the natural frequencies errors. The process under-estimates the shear stiffness Q66 of about 20% and does not allow to evaluate the coupling stiffness Q12. At last the procedure assumes that the bending stiffnesses are independent of the frequency.

Another way for evaluating the bending stiffnesses is suggested by the results reported which consists in evaluating first the bending stiffnesses Q11 and Q22 separately from the cylindical bending modes of the CFFF configuration (Relation (4.112)) and from the cylindrical bending modes of the CFFF configuration (relation (4.113)), respectively. Moreover this procedure allows us to estimate the variations of the stiffnesses as functions of the frequency. Next the shear stiffness Q66 can be derived from the frequency of the mode 22 in the FCFF or CFFF configuration (relations (4.116) and (4.117)). Table 4.13 shows the values obtained by this procedure for the stiffnesses Q11 and Q22. These results show that the values of Q11 ( 34.96 GPa) and Q22 (7.053 GPa) obtained for low frequencies are very close to the values derived from the static tests (Q11 = 34.86 GPa and Q22 = 7.070 GPa). Furthermore the values reported show an increase (about 10% in the range 0 to 1 kHz) of the stiffnesses Q11 and Q22 with the frequency. Next Table 4.14 gives the values of the stiffness Q66 deduced from the frequencies of the modes 22 of the CFFF and FCFF configurations, considering the values of Q11 and Q22 evaluated previously for the corresponding frequencies of modes 22 (about 155 Hz). The values obtained (3.40 and 3.44 GPa) are slightly higher (of about 7 percent) than the value measured in static tests.

Finally the results reported in this subsection show that the present determination of stiffnesses Q11, Q22 and Q66 from the cylindrical bending modes and from mode 22 of CFFF and FCFF configurations appears as the procedure the best suited for deriving the bending stiffnesses from the flexural vibrations of rectangular plates. Furthermore this procedure allows us to evaluate the variation of the stiffnesses as a function of the frequency.

Moreover, the increase of the stiffnesses with the frequency (about 8 percent for Q11 and 6 percent for Q22 in the frequency domain studied) pointed out by this procedure shows the limits of the evaluation of the bending stiffnesses from the natural frequencies of a single plate in a given configuration, evaluation in which the stiffnesses are assumed to be independent of the frequency.

130 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

TABLE 4.13. Evaluation of the reduced stiffnesses Q11 and Q22 from the frequencies of the cylindrical bending modes of CFFF and FCFF plates.

Mode 11 21 31 41

Frequency CFFF (Hz) 18.9 119.0 342.0 675.9

Q11 (GPa) 34.96 35.29 37.18 37.81

Mode 11 12 13 14

Frequency FCFF (Hz) 19.1 120.5 340.9 677.2

Q22 (GPa) 7.053 7.148 7.297 7.498

TABLE 4.14. Evaluation of the reduced shear stiffness Q66 from the frequencies of the modes 22 of CFFF and FCFF plates.

Configuration CFFF FCFF

Frequency 22 (Hz) 155.3 156.2

Q66 (GPa) 3.40 3.39

4.5.5 Case of Symmetric Laminates

4.5.5.1 Materials and Results

Two types of plates have been investigated: plates made of the orthotropic material considered in Subsection 4.5.2.1 and oriented at 30° with respect to the length of the plates and plates made of unidirectional glass fibre-epoxy layers arranged with the stacking sequence [30/02/30] of nominal thickness equal to 2.5 mm. Table 4.15 reports the reduced stiffnesses deduced from static tests and the bending stiffnesses obtained by relation (1.59).

The experimental investigation of the flexural vibrations of rectangular plates were performed by the procedure reported in Subsection 4.5.2.3. An extended TABLE 4.15. Stiffnesses of the 30° plates and [30/02/30] plates measured in static tests. 0° plates

Q11 (GPa) Q22 (GPa) Q12 (GPa) Q16 (GPa) Q26 (GPa) Q66 (GPa) h (mm) 22.93 9.04 6.29 8.89 3.14 8.17 2.1

D11 (N m) D22 (N m) D12 (N m) D16 (N m) D26 (N m) D66 (N m) 17.70 6.98 4.85 6.86 2.43 6.30

[30/02/30] plates Q11 (GPa) Q22 (GPa) Q12 (GPa) Q16 (GPa) Q26 (GPa) Q66 (GPa) H (mm)

31.80 11.45 7.38 10.13 3.58 9.82 2.5 D11 (N m) D22 (N m) D12 (N m) D16 (N m) D26 (N m) D66 (N m)

24.42 8.79 5.67 7.78 2.75 7.54

4.5. Evaluation of the Laminate Bending Stiffnesses by Analysis of Plate Vibrations 131

TABLE 4.16. Values of the natural frequencies (in Hz) measured and calculated by 100-term Ritz evaluation in the case of [30/02/30] plates, and differences (in percent).

Mode 1 Mode 2 Measure Ritz Difference Measure Ritz Difference

CCCC CCFC CCFF CFFF

247.3 185.2 51.0 16.4

251.09 186.47 51.31 16.10

1.53 0.69 0.61 –1.83

435.9 285.4 150.2 53.3

448.70 284.23 156.66 52.28

2.94 –0.41 4.30 –1.91

Mode 3 Mode 4 Measure Ritz Difference Measure Ritz Difference

CCCC CCFC CCFF CFFF

559.5 479.2 202.7 107.6

260.78 477.24 199.26 107.32

0.23 –0.41 –1.70 –0.26

695.5 497.5 300.7 175.9

727.39 499.13 311.58 183.14

4.59 0.33 3.62 4.12

Mode 5 Mode 6 Measure Ritz Difference Measure Ritz Difference

CCCC CCFC CCFF CFFF

788.2 610.1 386.6 204.1

789.34 613.22 391.10 197.80

0.14 0.51 1.16 –3.09

1026.9 762.0 493.9 317.0

1028.8 757.73 503.35 318.87

0.19 –0.56 1.91 0.59

experimental analysis [15, 16] was implemented in the case of plates with CCCC, CCFF and CFFF configurations. Table 4.16 compares the experimental values of the natural frequencies of [30/02/30] plates with the values obtained by the Ritz’s method with 100 terms (M = N = 10). The results reported show that the experi-mental values match the Ritz values within 4.5 percent. The scattering of the experimental natural frequencies is very similar to the one observed for the ortho-tropic laminates (Section 4.5.4)

Also we investigated the contribution of the bending stiffnesses to the energy of plate vibrations. Table 4.17 shows an example of the results obtained in the case of [30/02/30] plate in the CCFC configuration and for an aspect ratio of the

TABLE 4.17. Contributions (expressed in percent) of the stiffnesses to the strain energy of [30/02/30] plate in CCFC configuration and for a length-to-width ratio equal to 1.5.

R = 1.5 Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7

D11 2.00 16.84 31.08 2.37 9.25 29.34 36.20

D22 78.58 30.78 16.43 85.64 52.54 14.76 23.11

D12 0.11 6.49 6.59 0.17 5.89 7.03 6.67

D16 1.57 7.68 13.37 1.45 0.74 17.60 1.67

D26 6.02 7.72 7.95 3.35 5.41 7.79 6.13

D66 11.72 30.49 24.58 7.02 26.16 23.48 26.11

132 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

plate equal to 1.5. The results obtained for the different configurations and values of the aspect ratio are similar. The results show that the bending stiffnesses D12, D16 and D26 have the lowest contributions, although these contributions are clearly higher that the contribution of D12 in the case of orthotropic plates. As in the case of orthotropic plates the bending stiffnesses are not separate in the CCCC confi-guration.

4.5.5.2 Evaluation of the Bending Stiffnesses from the Natural Frequencies and Mode Shapes

In the case of a plate constituted of a symmetric laminate the material direc-tions of which are different from the plate edges, it is no more possible to model the flexural vibrations properly by Rayleigh’s approximation, because the domi-nant term (4.43) does not fit really the actual mode shape. Thus it is necessary to consider the Ritz method using a great number of terms in series (4.24). In this case the homogeneous system of the vibrations is given by Equation (4.74). This system can be expressed in the form (4.118) where the coeffients akl and bkl are given by the expressions:

( ) ( )( )

2200 2002 2002 1111 2 1210 210112 66 16

1012 0121 3 0022 426 22

0000

4 2

2 , (4.141)

,

kl minj minj minj minj minj minj

minj minj minj

kl minj

a C C C C R C C R

C C R C R

b C

α α α

α α

⎡ ⎤= + + + + +⎣ ⎦

+ + +

=

Hence the relation between the natural frequencies and the bending stiffnesses of a symmetric plate is extended from relation (4.132), which gives:

1 11 12 13 14 15 16 11

2 21 22 23 24 25 26 22

3 31 32 33 34 35 36 12

4 41 42 43 44 45 46 16

5 51 52 53 54 55 56 26

6 61 62 63 64 65 66 66

H H H H H H DH H H H H H DH H H H H H DH H H H H H DH H H H H H DH H H H H H D

λλλλλλ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

. (4.142)

The eigenvalues iλ (i = 1 to 6) are related to the natural frequencies by Relation (4.125) and the expressions of the coefficients Hij are extended from (4.133) as:

( )

( )1

1 , , 1 to 6,

, 1 to .

Pi

ij jpqPiq p

H A C i jA

q p P M N=

= =

= = ×

∑ (4.143)

The vector ( ) ( ) ( )( ) ( )1 2, , ... , , ... , , ... , i i ii i

p q PA A A A A is the vector of coefficients

Amn (reduced to one subscript), which determine the shape of the mode i.

4.5. Evaluation of the Laminate Bending Stiffnesses by Analysis of Plate Vibrations 133

185.2 Hz

286.0 Hz

480.7 Hz

497.5 Hz

610.1 Hz

762.0 Hz

FIGURE 4.11. Mode shapes and frequencies determined experimentally in the case of a [30/02/30] plate clamped along three edges.

FIGURE 4.12. Mode shapes reconstituted from the eigenvectors deduced from the experimental mode shapes of Figure 4.11.

( )iqA is the coefficient which has the maximum value. Lastly the terms Cjpq are

obtained by extension of Relations (4.131) as:

( ) ( )( )

2200 0022 41 2

2002 0220 2 1210 21013 4

1012 0121 3 1111 45 6

, ,

, 2 ,

2 , 4 ,

kl minj kl minj

kl minj minj kl minj minj

kl minj minj kl minj

C C C C R

C C C R C C C R

C C C R C C R

= =

= + = +

= + =

(4.144)

where the subscripts k and l are given by Relation (4.120).

x z y

x z y

134 Chapter 4. Flexural Vibrations of Undamped Rectangular Laminate Plates

When the coefficients Hij are unknown, it is necessary to use an iterative pro-cedure (Subsection 4.5.4.2) for deriving the bending stiffnesses from the natural frequencies. In the case of orthotropic laminates we have pointed out the tendency of this problem to be ill conditioned. In the case of symmetric laminates, this feature is strongly emphasized, and in practice the iterative procedure does not converge when any set of six experimental natural frequencies is used. It results that the only way to solve equation (4.142) is to determine the coefficients Hij. These coefficients can be obtained by evaluating the shapes of the six modes into consideration. Figure 4.11 shows the results obtained for the shapes of the first six modes in the case of a [30/02/30] plate in CCFC configuration where three edges are clamped. These shapes are deduced from measurement performed along a mesh of 25 points (5 points along the length and 5 points along the width of plate) away from the clamped edges. Next the eigenvectors are derived by applying Relation (4.24) to the points of the mesh. Figure 4.12 shows the mode shapes reconstituted from these eigenvectors using a mesh of 28 × 18 points. The number of points required depends on the complexity of the mode shapes. This number has to be minimized considering the heaviness for deriving mode shapes expe-rimentally. The experimental investigation we carried out shows that the mesh of 5 × 5 points is sufficient for the modes considered in Figures 4.11 and 4.12. The eigenvector being obtained for each mode i, the coefficients Hij are calculated by relation (4.142). Lastly the bending stiffnesses are deduced from Relation (4.141) using the experimental natural frequencies.

The values of the reduced stiffnesses obtained by applying this procedure to the experimental results are reported in Tables 4.18 and 4.19 in the case of 30° and [30/02/30] plates, respectively. The results, compared to the values derived from static tests, show a good evaluation of the stiffnesses Q11, Q22 and Q66. The values obtained for Q12, Q16 and Q26 show a low instability of the method for evaluating these stiffnesses, instability which is clearly less marked than for the evaluation of Q12 in the case of orthotropic laminates. This low instability is associated with the contributions of the stiffnesses Q12, Q16 and Q26 which are rather low (lower than 18%), but however clearly higher than the participation of Q12 in the case of orthotropic plates.

TABLE 4.18. Values of the reduced stiffnesses derived from the experimental modal analysis of orthotropic plates oriented at 30°.

Q11 Q22 Q12 Q16 Q26 Q66

Modes considered (GPa)

1, 2, 3, 4, 5, 6 23.20 8.936 6.016 7.893 2.753 8.019

Static deviation (%) 1.15 –1.16 –4.36 –11.2 –12.4 –1.79

1, 2, 3, 4, 6, 7 23.26 9.141 5.660 10.11 3.548 8.475

Static deviation (%) 1.40 1.11 –10.0 13.7 12.9 3.80

Static tests 22.93 9.041 6.290 8.888 3.143 8.165

4.5. Evaluation of the Laminate Bending Stiffnesses by Analysis of Plate Vibrations 135

TABLE 16. Values of the reduced stiffnesses derived from the experimental modal analysis of [30/02/30] plates.

Q11 Q22 Q12 Q16 Q26 Q66

Modes considered (GPa)

1, 2, 3, 4, 5, 6 24.07 8.712 4.982 8.881 3.165 8.335

Static deviation (%) –1.45 –0.93 –12.1 14.2 15.1 10.5

1, 2, 3, 4, 6, 7 24.86 8.890 5.101 9.099 3.162 8.146

Static deviation (%) 1.77 0.97 –11.1 14.5 13.0 7.40

Static tests 24.42 8.794 5.668 7.777 2.750 7.543

CHAPTER 5

Damping in Composite Materials

5.1 GENERAL COMMENTS

Damping is an important parameter for vibration control, fatigue endurance, impact resistance, etc. Although the damping of composite materials is not very high, it is significantly higher than that measured for most usual metallic mate-rials. At the constituent level, the energy dissipation in fibre reinforced com-posites is induced by different mechanisms such as the viscoelastic nature of the matrix, the damping at the fibre-matrix interface, the damping due to defects or damage, etc. At the laminate level, damping is strongly depending on the layer constituent properties as well as layers orientations, interlaminar effects and stac-king sequence.

Among all the sources of energy dissipation, the case of viscous damping is the simplest to deal with mathematically (Chapter 2). For this reason damping forces of a complicated nature are generally replaced by equivalent viscous damping (Section 2.4 of Chapter 2).

Viscoelastic materials combine the capacity of an elastic type material to store energy with the capacity to dissipate energy. So, the use of an energy approach for evaluating the material or structure damping is widely considered. In this energy approach, the dissipated energy is related to the strain energy stored by intro-ducing a damping parameter.

The various forms that viscoelastic stress-strain relations can take have been considered by Gross [18]. Further aspects have been treated by Christensen [19] and Pipkin [20]. A form of viscoelastic stress-strain relations is that involving the complex moduli of materials. In the case of harmonic oscillations, the stress-strain relation can be expressed in a complex form, introducing a complex stiffness matrix which is a function of the oscillation frequency. Thus, in the case where the strain is a harmonic function of time, the strain field is expressed in the complex form as: ( ) 0 ,i tt e ω=ε ε (5.1)

where ω is the frequency of harmonic oscillations. Then, the stress field can be expressed as:

5.2. Damping in a Unidirectional Composite as Function of the Constituents 137

0*( ) ( ) ,i tt e ωω= Cσ ε (5.2)

introducing the complex stiffness matrix *( ).ωC The previous form can also be inverted considering the compliance matrix *( )ωS defined as the inverse matrix of *( ).ωC

Thus, it results that static elastic solutions can be converted to steady state har-monic viscoelastic solutions simply by replacing the elastic moduli by the corres-ponding complex viscoelastic moduli, and reinterpreting the elastic fields as com-plex harmonic viscoelastic field variables. This correspondence principle consi-dered first in the case of homogeneous materials was extended then to hetero-geneous materials.

The elastic-viscoelastic correspondence principle was developed by Hashin [21, 22] in the case of composite materials. The author shows that the effective complex moduli of viscoelastic composites can be determined on the basis of analytical expressions for effective elastic moduli of composites. The method was applied to particulate composites [21] and fibre composites [22]. Furthermore, Sun et al [23] and Crane and Gillespie [24] applied the correspondence principle to the laminate relations derived from the classical laminate theory.

5.2 DAMPING IN A UNIDIRECTIONAL COMPOSITE AS A FUNCTION OF THE CONSTITUENTS

The elastic behaviour of a unidirectional orthotropic material is characterized by the engineering constants , , and ,L T LT LTE E Gν measured in the material directions ( ), , ,L T T ′ also noted ( )1, 2, 3 . In the same way, the damping proper-ties can be described by four damping coefficients. In practice, damping associa-ted to the Poisson ratio is neglected, and the evaluations of the damping coefficients associated to the longitudinal and transverse Young’s moduli and to the shear modulus are generally based on an energy approach.

The use of the energy approach to evaluate the damping properties of a structure was introduced by Ungar and Kerwin [25], considering that the struc-tural damping η can be described as a function of the constitutive elements of the structure and of the energy stored in these elements:

1

1

n

i ii

n

ii

U

U

η

η =

=

=∑

∑. (5.3)

Applying this relation to a fibre composite leads to express the damping ηc of the composite as a function of the fibre damping ηf and matrix damping ηm according

138 Chapter 5. Damping in Composite Materials

to the expression:

f f m mc

c

U UU

η ηη += , (5.4)

where f m c, and U U U are the elastic energies stored in fibres, matrix and com-posite material, respectively. Expression (5.4) is general, but in practice the appli-cation is restricted to simple fibre-matrix arrangements and loading conditions for which the elastic energies stored can be derived easily.

Applying Expression (5.4) to the case of a unidirectional fibre composite loa-ded in the fibre direction leads to the expression for the longitudinal damping as:

( )f mf f m f1L

L L

E EV VE E

η η η= + − , (5.5)

where fV is the fibre volume fraction, fE and mE are the Young’s moduli of fibres and matrix, respectively, and LE is the Young’s modulus of the unidi-rectional composite. This modulus is well evaluated by the law of mixtures and Expression (5.5) can be written as:

( ) ( )

f ff m

m ff f f

f m

1

1 1L

f

V VE EV V V VE E

η η η−= +

+ − − +. (5.6)

In the case where the damping of fibres can be neglected, Expression (5.6) is simply reduced to:

( )

fm

ff f

m

1

1L

VEV VE

η η−≈

− +. (5.7)

If now the unidirectional fibre composite is loaded in the transverse direction, Expression (5.4) leads to the transverse damping which can be expressed as:

( )f f m ff m

1T TT

E EV VE E

η η η= + − , (5.8)

introducing the transverse Young’s modulus of composite. This modulus can be expressed by an inverse law of mixtures, but a better evaluation can be obtained [1, 2] using expression:

22

1 1 22 2

TLT

L TT L

E

K G Eν

′

=+ +

, (5.9)

where LK is the lateral compression modulus of the unidirectional composite and TTG ′ the transverse shear modulus. These coefficients are obtained from the

expressions established by Hashin [26, 27] and Hill [28]:

5.2. Damping in a Unidirectional Composite as Function of the Constituents 139

( )

fm

f1 4

f m f m m m3 3

1 1LVK K V

k k G G k G

= +−

+− + − +

, (5.10)

and by Christensen and Lo [29, 30]:

( ) ( )f

mm m m

ff m m m

1 2 12

TTVG G G K G V

G G K G

′⎡ ⎤= +⎢ ⎥+

+ −⎢ ⎥− +⎣ ⎦

. (5.11)

The bulk moduli m f( , ),k k the shear moduli m f( , )G G and the lateral compres-sion moduli m f( , )K K of the matrix and fibres are expressed as functions of the Young’s moduli and Poisson ratios of the matrix and fibres by:

( ) ( )

, , , m, f.3 1 2 2 1 3

i i ii i i i

i i

E E Gk G K k iν ν

= = = + =− +

(5.12)

The Poisson ratio LTν in the Expression (5.9) can be evaluated by the law of mixtures.

Lastly, in the case of a longitudinal shear loading, the expression of composite damping is similar to Expression (5.8) obtained in the case of a transverse loading:

( )f f m ff m

1LT LTLT

G GV VG G

η η η= + − , (5.13)

where the longitudinal shear modulus can be evaluated [27, 28] by:

( ) ( )( ) ( )

f f m f

f f m f

1 11 1LT m

G V G VG G

G V G V+ + −

=− + +

. (5.14)

Limited experimental results are reported in literature on the processes of com-posite damping at the scale of fibres, matrix and fibre-matrix interface. Adams et al. [31] found that the longitudinal damping of unidirectional carbon-fibre com-posites and glass-fibre composites fell rapidly with increasing the fibre volume fraction. Both composites have essentially the same damping for a given volume fraction. It was found by Adams [32] that expression (5.7) considerably under-estimates the experimental values of the longitudinal damping. Several factors were thought to contribute to the discrepancy: fibre misalignment, imperfections in the materials (matrix cracks and fibre-matrix debonding), effect of fibre-matrix interface. Fibre interaction and fibre-matrix interphase were considered in [32-35], in the case of discontinuous fibres. Authors estimate the strain energies stored in fibres and matrix using a finite element analysis. Then, composite damping was derived from expression (5.4). More recently, Yim [36], and Yim and Gillepsie [37] have considered the evaluation of the damping parameters in the case of unidirectional carbon-fibre epoxy composites. According to the results obtained by Adams [31], Yim [36] introduced a curve fitting parameter α in Relation (5.7)

140 Chapter 5. Damping in Composite Materials

and expressed the longitudinal damping as:

( )

fm

ff f

m

1

1L

V

EV VE

αη η−=

⎛ ⎞− + ⎜ ⎟

⎝ ⎠

. (5.15)

In fact, the curve fitting parameter α is obtained by Yim considering the only fibre fraction equal to 0.65. In the same way, parameters were introduced in Expression (5.8) for the transverse damping and in Expression (5.13) for the longitudinal shear damping.

5.3 BENDING VIBRATIONS OF DAMPED LAMINATE BEAMS

5.3.1 Damping Modelling using Viscous Friction

In the case of a viscous damping, the damping force is proportional to the velo-city. Thus, the differential equation of motion for a damped beam is deduced from Equation (3.13) and is written as:

2 4

0 0 02 4 ( , ),s s sc k q x t

tt xρ ∂ ∂ ∂

+ + =∂∂ ∂

w w w (5.16)

introducing the coefficient of viscous damping cs by unit area. Then, Equation (5.16) can be rewritten in the following form:

2 4

2 40 0 002 4 ( , ),s

s

c a p x ttt x

ωρ

∂ ∂ ∂+ + =

∂∂ ∂

w w w (5.17)

introducing the angular frequency (3.15) of the undamped beam and where the reduced load ( , )p x t is defined in Equation (3.98).

5.3.2 Motion Equation in Normal Coordinates

As in the case of undamped beam (Section 3.5.1 of Chapter 3), the motion equation (5.17) can be transformed in an equation in normal coordinates by introducing the transverse displacement expressed by Equation (3.93). We obtain:

22 , 1, 2, ... , i i i i i i ip iφ ξ ω φ ω φ+ + = = ∞ , (5.18)

introducing the modal damping coefficient ξi, related to the coefficient of viscous damping by:

2 .si i

s

c ξ ωρ

= (5.19)

Each of Equations (5.18) is uncoupled from all the others, and the response φi(t)

5.3. Damping Vibrations of Damped Laminate Beams 141

of each mode i can be determined in the same manner as for one-degree system with viscous damping (Section 2.3. of Chapter 2). Then, the analysis can be per-formed by the same processing as that applied in Section 3.5 of Chapter 3 to the case of undamped beams.

5.3.3 Forced Harmonic Vibrations

In the case of a beam of length a, submitted to a harmonic load:

m( , ) ( ) cosq x t q x tω= , (5.20)

the component of the reduced load for the mode i is given by:

m( ) ( ) cos ,i ip t p x tω= (5.21) with

m m

0

1 .a

i is

p q X dxρ

= ∫ (5.22)

Equation (5.18) of motion in normal coordinates becomes:

2m2 cos , 1, 2,..., .i i i i i i ip t iφ ξ ω φ ω φ ω+ + = = ∞ (5.23)

Considering the results obtained in the case of a system with one degree of freedom (Section 2.3), the steady-state response for mode i is given by;

( )m2( ) ( ) cos sin ,i

i i i ii

pt K a t b tφ ω ω ωω

= + (5.24)

with

2

21 , 2 ,i i iii

a bω ωξωω

= − = (5.25)

2 22

2

1( ) .

1 2

i

iii

K ωω ωξ

ωω

=⎛ ⎞ ⎛ ⎞− +⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

(5.26)

Then, the transverse displacement is deduced from (3.93), which gives:

( )m0 2

1( , ) ( ) ( ) cos sin .i

i i i ii i

px t K X x a t b tω ω ωω

∞

== +∑w (5.27)

According to the results obtained in Section 2.3.4.2, the equation of the har-monic motion can be expressed in the frequency domain in the complex form:

( ) ( ) ( ) 1,2,...,i i iH P iΦ ω ω ω= = ∞ (5.28)

where Φi(ω) and Pi(ω) are the complex amplitudes associated to the time functions φi(t) and pi(t), respectively, and introducing the transfer function:

142 Chapter 5. Damping in Composite Materials

21( ) ( ),i rii

H Hω ωω

= (5.29)

with

2

2

1( ) .1 2

ri

iii

Hi

ωω ωξ

ωω

=⎛ ⎞

− +⎜ ⎟⎝ ⎠

(5.30)

Hri is the reduced transfer function. The time response φi(t) is then deduced from Equation (5.28) and expressed in the form (5.27) with

( ) ( ) ,i riK Hω ω= (5.31) and

( ) ( )Re , Im .i ri i ria H b Hω ω= =⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ (5.32)

5.3.4 Damping Modelling using Complex Stiffness

As considered in the case of one degree system (Subsection 2.3.4.4), the energy dissipation in the case of harmonic vibrations can be accounted for by introducing the complex stiffness per unit area:

( )* 1 ,s sk k iη= + (5.33)

where η is the structural damping coefficient or the loss factor introduced in (2.117). It results that motion equation (5.16) can be transposed in complex form using the procedure considered in Section 3.5.1. This procedure leads to:

4

24

0 0

4

40

1d d

d ( ),

a ai

s i i j s i j

ai

s i j s j

i

d XX X x i k X xdx

d Xk X x Pdx

ρ ω Φ η Φ

Φ ρ ω

∞

=

⎛− +⎜

⎝

⎞+ =⎟

⎠

∑ ∫ ∫

∫ (5.34)

introducing the complex amplitudes Φi(ω), Xi(ω), Xj(ω), and Pj(ω) of φi(t), xi(t), xj(t) and pj(t), respectively. Considering the orthogonality and normality relations, Equation (5.34) can be rewritten as:

( )2 2 2 ( ) ( ), 1, 2,..., ,i i i i ii P iω ω ω η Φ ω ω⎡ ⎤− + = = ∞⎣ ⎦ (5.35)

introducing the loss factor ηi of each mode. Equations (5.35) constitute the motion equation in normal coordinates. These equations are uncoupled. They can be written in form (5.28) with:

2

2

1( ) .1

rii

i

Hi

ωω ηω

=⎛ ⎞

− +⎜ ⎟⎝ ⎠

(5.36)

5.3. Damping Vibrations of Damped Laminate Beams 143

Finally, the transverse displacement can be expressed in form (5.27), with

2

21 , ,i i ii

a bω ηω

= − = (5.37)

and

22

22

1( ) .

1

i

ii

K ωω ηω

=⎛ ⎞

− +⎜ ⎟⎜ ⎟⎝ ⎠

(5.38)

5.3.5 Beam Response to a Concentrated Loading

In the case of a load concentrated at point x = x1 of a beam, the exerted loading can be written as: 1 1 1( , ) ( , ) ( ) ( ),q x t q x t x x q tδ= = − (5.39)

where 1( )x xδ − is the Dirac function localized at point x1. According to Equation (3.97), the modal component of the reduced load is:

1 1

0( ) ( ) ( ) ( ) d ,

a

i ip t p t X x x x xδ= −∫ (5.40)

which yields: 1 1( ) ( ) ( ),i ip t X x p t= (5.41) with

1 11( ) ( ).

sp t q t

ρ= (5.42)

In the case of an impact, the reduced load can be expressed as:

1 1( ) ( ),p t p tδ= (5.43)

where p1 is constant and δ (t) is the impulse Dirac function localized at time t = 0. This function can be expanded in Fourier transform as:

( ) d .i tt e ωδ ω+∞

−∞= ∫ (5.44).

Thus, the impact loading generates the whole frequency domain, and for every frequency the motion equation in normal coordinates is written in form (5.23) with: m 1 1( ).i ip p X x= (5.45)

Equation (5.24) can also be written in form (5.28) where the transfer function is expressed by (5.30) in the case of the damping modelling using viscous friction or

144 Chapter 5. Damping in Composite Materials

by (5.36) in the case of modelling using complex stiffness. Thus, it results that the transverse displacement can be written as:

0 1 1 21

1( , ) ( ) ( ) ( ) ( cos sin ),i i i i iii

x t p X x X x K a t b tω ω ωω

∞

=

= +∑w (5.46)

where ai, bi and Ki are given by (5.25) and (5.26) in the case of the damping mo-delling using viscous friction and by (5.37) and (5.38) in the case of the damping modelling using complex stiffness.

5.4 EVALUATION OF DAMPING PROPERTIES OF ORTHOTROPIC BEAMS AS FUNCTIONS

OF MATERIAL ORIENTATION

5.4.1 Energy Analysis of Beam Damping

5.4.1.1 Introduction

The prediction for damping properties of orthotropic beams as a function of material orientation was developed by Adams and Bacon [38] and Ni and Adams [39]. These analyses also consider cross-ply laminates and angle ply laminates, as well as more general types of symmetric laminates. The damping concept of Adams and Bacon was also applied by Adams and Maheri [40] to the invest-tigation of angle ply laminates made of unidirectional glass fibre or carbon layers. More recently the analysis of Adams and Bacon was applied by Yim [36] and Yim and Jang [41] to different types of laminates, then extended by Yim and Gillespie [37] including the transverse shear effect in the case of 0° and 90° unidirectional laminates.

5.4.1.2 Adams-Bacon Approach

For an orthotropic material the strain-stress relationship in material axes (L, T, T') = (1, 2, 3) is given (1.15) by:

1 11 12 1

2 12 22 2

6 66 6

00 ,

0 0

S SS S

S

ε σε σε σ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(5.47)

where the components Sij are the compliance constants related [1, 2] to the engi-neering moduli EL, ET, GLT and νLT by the following expressions:

11 12 22 661 1 1, , , .LT

L L T LTS S S S

E E E Gν

= = − = = (5.48)

5.4. Evaluation of Damping Properties of Orthotropic Beams 145

Adams and Bacon [38] consider that the strain energy stored in a volume element δV can be separated into three components associated respectively to the stresses σ1, σ 2 and σ 6 expressed in the material axes as:

11 22 66 ,U U U Uδ δ δ δ= + + (5.49) with

( )11 1 1 1 11 1 12 21 1 ,2 2

U V S S Vδ σ ε δ σ σ σ δ= = + (5.50)

( )22 2 2 2 12 1 22 21 1 ,2 2

U V S S Vδ σ ε δ σ σ σ δ= = + (5.51)

266 6 6 6 66

1 1 .2 2

U V S Vδ σ ε δ σ δ= = (5.52)

Thus, Adams and Bacon consider that the energy δU11 is the strain energy stored in tension-compression in the longitudinal direction, δU22 is the strain energy stored in tension-compression in the transverse direction and δU66 is the strain energy stored in in-plane shear. Then, the strain energy dissipation in the longi-tudinal direction is written as:

( )11 11 11,U Uδ ψ δ∆ = (5.53)

introducing the longitudinal specific damping capacity ψ11 measured in the case of traction-compression tests of 0° materials and assuming the damping is inde-pendent of the applied stress σ1. Expressions (5.50) and (5.53) yield:

( ) ( )11 11 1 11 1 12 21 .2

U S S Vδ ψ σ σ σ δ∆ = + (5.54)

Similarly, the strain energy dissipation in the transverse direction is expressed as:

( ) ( )22 22 2 12 1 22 21 ,2

U S S Vδ ψ σ σ σ δ∆ = + (5.55)

introducing the transverse specific damping capacity ψ22. And the strain energy dissipation in shear deformation is given by:

( ) 266 66 6 66

1 ,2

U S Vδ ψ σ δ∆ = (5.56)

introducing the in-plane shear damping specific capacity ψ66. Hence, the total energy dissipated in the element can be written as:

( ) ( ) ( ) ( )11 22 66 .U U U Uδ δ δ δ∆ = ∆ + ∆ + ∆ (5.57)

This expression can be extended to the whole volume of the laminate to derive the total energy dissipation:

( )

,V

U Uδ∆ = ∆∫ (5.58)

and the specific damping capacity of the laminate is then:

146 Chapter 5. Damping in Composite Materials

,UU

ψ ∆= (5.59)

with

V

U Uδ= ∫ (5.60)

The stresses 1 2 6, and ,σ σ σ expressed in the material directions are related (Chapter 5 of Refs. 1 and 2) to the stresses , and ,xx yy xyσ σ σ in the beam directions by the relation:

2 21

2 22

2 26

cos sin 2sin cos

sin cos 2sin cos

sin cos sin cos cos sin

xx

yy

xy

θ θ θ θσ σσ θ θ θ θ σσ σθ θ θ θ θ θ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥= −⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ − −⎢ ⎥ ⎣ ⎦⎣ ⎦

, (5.61)

where θ is the orientation of the orthotropic material with respect to the beam directions.

In the case of free flexure of beam along the x direction, the stresses σyy and σxy are zero, and the stresses in the material directions are:

21

22

6

cos ,

sin ,sin cos .

xx

xx

xx

σ σ θ

σ σ θσ σ θ θ

=

== −

(5.62)

The energy dissipated in an element of unit volume is given by:

( )

( )

2 2 2 211 11 12

2 2 2 2 222 12 22 66 66

1 cos sin cos2

cos sin sin cos sin .

xxU S S

S S S

σ ψ θ θ θ

ψ θ θ θ ψ θ θ

⎡∆ = +⎣

⎤+ + + ⎦

(5.63)

The strain energy stored in the element is:

211

1 1 ,2 2xx xx xxU Sσ ε σ ′= = (5.64)

where

4 4 2 211

1 1 1 1cos sin 2 cos sin ,LT

x L T LT LS

E E E G Eνθ θ θ θ

⎛ ⎞′ = = + + −⎜ ⎟

⎝ ⎠ (5.65)

introducing the Young's modulus measured in the x direction (Chapter 11 of Refs. 1 and 2). Thus, Relations (5.63) to (5.65) lead to the expression of the specific damping capacity in the x direction:

( )4 4 2 26611 2211 22cos sin cos sin .LT

x xL T LT L

EE E G E

ψψ ψ νψ θ θ ψ ψ θ θ⎧ ⎫⎡ ⎤⎪ ⎪= + + − +⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

(5.66)

5.4. Evaluation of Damping Properties of Orthotropic Beams 147

5.4.1.3 Ni-Adams Analysis

In this section the analysis of Ni-Adams [39] is developed in the particular case of the bending of a beam constituted of an orthotropic or unidirectional material. The beam of length a and width b is caused to vibrate along its length (the x direction). In the analysis, only the principal bending moment Mx is applied along the x direction, the other moments being zero: My = Mxy = 0, according to the assumptions of the classical laminate theory. Thus curvatures are expressed from Equation (3.3) as:

111

112

116

,

,

.

x x

y x

xy x

D M

D M

D M

κ

κ

κ

−

−

−

=

=

=

(5.67)

where the 1ijD− coefficients are the flexural compliance matrix components, derived

as the elements of the inverse matrix [Dij] expressed in the beam axes. The curva-ture κx is due to bending along the x direction, the curvature κy is due to the Poisson coupling and the curvature κxy results from the bending-twisting cou-pling. In the case of beam bending, the strain field (1.50) is reduced to:

,,.

xxx

yyy

xyxy

zzz

ε κε κ

γ κ

==

=

(5.68)

The stresses in the material, referred to the plate directions, are deduced from Equation (1.53):

11 12 16

12 22 26

16 26 66

.xx x

xy y

yy xy

Q Q Qz Q Q Q

Q Q Q

σ εσ ε

σ γ

⎡ ⎤ ⎡ ⎤′ ′ ′⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥′ ′ ′=⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥′ ′ ′⎢ ⎥⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(5.69)

The reduced stiffness ijQ′ are referred to the plate axes x and y, and are expressed as functions of the material orientation by the expression reported in Table 1.1. Considering Equations (5.67) to (5.69) leads to:

( )( )( )

1 1 111 11 12 12 16 16

1 1 112 11 22 12 26 16

1 1 116 11 26 12 66 16

,

,

.

xx x

yy x

xy x

z Q D Q D Q D M

z Q D Q D Q D M

z Q D Q D Q D M

σ

σ

σ

− − −

− − −

− − −

′ ′ ′= + +

′ ′ ′= + +

′ ′ ′= + +

(5.70)

Then, the stresses expressed in the material directions are deduced from Equation (5.61).

As previously, Ni and Adams consider that, in the case of free bending beam, the stresses σyy and σxy can be neglected. Thus, the stresses in material directions

148 Chapter 5. Damping in Composite Materials

are given by:

( )( )

( )

1 1 1 21 11 11 12 12 16 16

1 1 1 22 11 11 12 12 16 16

1 1 16 11 11 12 12 16 16

cos ,

sin ,

sin cos .

x

x

x

z Q D Q D Q D M

z Q D Q D Q D M

z Q D Q D Q D M

σ θ

σ θ

σ θ θ

− − −

− − −

− − −

′ ′ ′= + +

′ ′ ′= + +

′ ′ ′= − + +

(5.71)

The strains in the material directions can be expressed as functions of the strains in the beam directions considering the strain transformations (Chapter 6 of Refs. 1 and 2):

2 21

2 22

2 26

cos sin sin cos

sin cos sin cos ,

2sin cos 2sin cos cos sin

xx

yy

xy

θ θ θ θε εε θ θ θ θ εγ γθ θ θ θ θ θ

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ = −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ − − ⎣ ⎦⎢ ⎥⎣ ⎦

(5.72)

Considering that yyε is much smaller than xxε and ,xyγ the strain yyε can be ne-glected, and the strains in the material directions are given by:

( )( )

( )( )

1 2 11 11 16

1 2 12 11 16

1 1 2 26 11 16

cos sin cos ,

sin sin cos ,

2 sin cos cos sin .

x

x

x

z D D M

z D D M

z D D M

ε θ θ θ

ε θ θ θ

ε θ θ θ θ

− −

− −

− −

= +

= −

⎡ ⎤= − − −⎢ ⎥⎣ ⎦

(5.73)

As in the Adams-Bacon approach, the energy dissipation is separated into three components associated with the stress components σ1, σ 2 and σ 6 expressed in the material directions. Thus, the energy dissipation can be expressed as:

11 22 66,U U U U∆ = ∆ + ∆ + ∆ (5.74) with

/ 2

11 11 1 10 0

12 d d ,2

a h

x zU b x zψ σ ε

= =∆ = ∫ ∫ (5.75)

/ 2

22 22 2 20 0

12 d d ,2

a h

x zU b x zψ σ ε

= =∆ = ∫ ∫ (5.76)

/ 2

66 66 6 60 0

12 d d .2

a h

x zU b x zψ σ ε

= =∆ = ∫ ∫ (5.77)

These expressions lead to:

( )( )

1 1 1 211 11 11 11 12 12 16 16

1 2 1 211 16

0

1 cos2

cos sin cos d ,a

x

U I Q D Q D Q D

D D M x

ψ θ

θ θ θ

− − −

− −

′ ′ ′∆ = + +

× + ∫ (5.78)

5.4. Evaluation of Damping Properties of Orthotropic Beams 149

( )( )

1 1 1 222 22 11 11 12 12 16 16

1 2 1 211 16

0

1 sin2

sin sin cos d ,a

x

U I Q D Q D Q D

D D M x

ψ θ

θ θ θ

− − −

− −

′ ′ ′∆ = + +

× − ∫ (5.79)

( )

( )

1 1 166 66 11 11 12 12 16 16

1 1 2 2 211 16

0

1 sin cos2

2 sin cos cos sin d ,a

x

U I Q D Q D Q D

D D M x

ψ θ θ

θ θ θ θ

− − −

− −

′ ′ ′∆ = + +

⎡ ⎤× − −⎣ ⎦ ∫ (5.80)

introducing the quadratic moment I of the cross-section of the beam with respect to the (x, y) plane:

3

,12

b hI = (5.81)

where h is the beam thickness. The total strain energy of the beam can be expressed [1, 2] as:

( )

/ 2

0 / 2

1 d d .2

a b

x x y y xy xyx y b

U M M M x yκ κ κ= =−

= + +∫ ∫ (5.82)

The moments My and Mxy are neglected and the total strain energy can be ex-pressed as:

1 211

0d .

2

a

xx

bU D M x−

== ∫ (5.83)

Then, the specific damping capacity fxψ for the beam bending along the x-direction is given by:

11 22 66 .fxU U U

Uψ ∆ + ∆ + ∆= (5.84)

In the case of a beam constituted of the same orthotropic or unidirectional material, the stiffness constants Dij of the beam are related to the reduced stiffness constants ijQ′ of the material by the expression:

3

,12ij ijhD Q′= (5.85)

and the compliance components 1ijD− are given by:

1 13

12 ,ij ijD Qh

− −= (5.86)

where 1ijQ− are the components of the inverse matrix [ ] 1

ijQ − of the reduced stiffness matrix [ ].ijQ′

150 Chapter 5. Damping in Composite Materials

5.4.1.4 General Formulation of Damping

Expressions obtained by the analysis of Adams-Bacon (5.66), then by the analysis of Ni-Adams (5.84) show that the specific damping capacity evaluated in the direction θ can be expressed in the general form:

( ) ( ) ( ) ( )11 11 22 22 66 66 .a a aψ θ ψ θ ψ θ ψ θ= + + (5.87)

Functions aij(θ ) differ according to the analysis which is considered. In the case of Adams-Bacon approach, functions aij(θ ) are expressed as:

( ) ( )

( ) ( )

( )

2 2 211 11 12

11

2 2 222 12 22

11

2 266 66

11

1 cos sin cos ,

1 cos sin sin ,

1 sin cos .

a S SS

a S SS

a SS

θ θ θ θ

θ θ θ θ

θ θ θ

= +′

= +′

=′

(5.88)

In the case of the analysis of Ni-Adams, functions aij(θ ) are given by:

( ) ( )( )

( ) ( )( )

( ) ( )

1 1 111 11 11 12 12 16 161

111 2 1 2

11 16

1 1 122 11 11 12 12 16 161

111 2 1 2

11 16

1 1 166 11 11 12 12 16 161

111

11 1

1

cos sin cos cos ,1

sin sin cos sin ,1

2 sin cos

a Q Q Q Q Q QQ

Q Q

a Q Q Q Q Q QQ

Q Q

a Q Q Q Q Q QQ

Q Q

θ

θ θ θ θ

θ

θ θ θ θ

θ

θ θ

− − −−

− −

− − −−

− −

− − −−

−

′ ′ ′= + +

× +

′ ′ ′= + +

× −

′ ′ ′= + +

× − ( )1 2 26 cos sin sin cos .θ θ θ θ−⎡ ⎤−⎣ ⎦

(5.89)

5.4.2 Complex Moduli

The correspondence principle (Section 5.1) can be applied to the effective bending modulus of a beam expressed by relation (3.12). In complex form this bending modulus is expressed as:

1311

**

12 ,fxEh D −= (5.90)

where 111*D − is expressed as function of the complex moduli of the laminated

material. Relation (5.90) allows us to evaluate the loss factor fxEη associated to

the bending modulus as:

( )* 1 .fxfx fx EE E iη= + (5.91)

5.5. Evaluation of the Plate Damping as Function of Material Directions 151

This complex modulus has been also considered by Yim and Jang [41]. Previous relations correspond to the case of the free flexure of laminate beam

where Mx is the only applied moment, curvatures being expressed by (5.66). Adams and Bacon [38] also consider the case of a pure flexure for which the twisting would be constrained to zero κxy = 0. Considering the curvature-moment relations [1, 2] this pure flexure would be obtained when the twisting moment would be equal to:

1

16

66* ,xy x

DM MD

−

= − (5.92)

and the curvature-moment relations yield:

( )21

16111 1

66.x x

DD M

Dκ

−−

−

⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

(5.93)

This expression is substituted for expression (5.67) of κx obtained in the case of free flexure and expression (5.90) of the effective bending modulus becomes:

( )21

161 1

11 66

1 .

1

fpx fxE ED

D D

−

− −

=

−

(5.94)

In fact, the scheme of pure flexure is theoretic, since there exists a bending-twisting coupling for off-axis materials. Moreover .fpx fxE E≈ However this scheme was considered by Yim and Jang [41] and applied to the damping of beam flexure introducing the complex bending modulus:

( )21

161 1

11 66

* **

* *

1 .

1

fpx fxE ED

D D

−

− −

=

−

(5.95)

5.5 EVALUATION OF THE PLATE DAMPING AS A FUNCTION OF MATERIAL DIRECTIONS

5.5.1 Orthotropic Plates

5.5.1.1 Formulation

The energy approach considered in the previous section for the damping of beams can also be applied for evaluating the damping properties of plates. The energy approach is based on the evaluation of the strain energy, which can be derived by finite element analysis in the case of a complex structure or by using

152 Chapter 5. Damping in Composite Materials

the Ritz method in the case of the analysis of rectangular plates. This analysis has been developed in [42, 43] and is considered hereafter.

In the Ritz method (Section 4.2), the transverse displacement is expressed (4.24) in the form of a double series of the coordinates x and y, where the coeffi-cients Amn are determined by considering the stationarity conditions (4.25) of the total potential energy.

The strain energy Ud can be expressed as a function of the strain energies related to the material directions as:

d 1 2 6 ,U U U U= + + (5.96) with

1 1 1

2 2 2

6 6 6

1 d d d ,2

1 d d d ,2

1 d d d ,2

U x y z

U x y z

U x y z

σ ε

σ ε

σ ε

=

=

=

∫∫∫∫∫∫∫∫∫

(5.97)

where the triple integrations are extended over the volume of the plate. Considering the case of a plate constituted of a single layer of unidirectional or

orthotropic material, the strains ε1, ε2 and ε6 are related to the strains εxx, εyy and γxy in the beam directions according to the strain transformations. The strain transformations are obtained inverting Expression (5.72):

2 21

2 22

2 2 6

cos sin sin cos

sin cos sin cos ,

2sin cos 2sin cos cos sin

xx

yy

xy

θ θ θ θε εε θ θ θ θ εγ γθ θ θ θ θ θ

⎡ ⎤−⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦− −⎣ ⎦ ⎢ ⎥⎣ ⎦

(5.98)

Next the stresses σ1, σ2 and σ6 can be evaluated considering the elasticity relations of plates:

1 11 1 12 2

2 12 1 22 2

6 66 6

,,

.

Q QQ QQ

σ ε εσ ε εσ ε

= += +=

(5.99)

It results that the strain energy U1, stored in tension-compression in the fibre direction, can be written as: 1 11 12,U U U= + (5.100) with

2 11 11 1

12 12 1 2

1 d d d ,2

1 d d d .2

U Q x y z

U Q x y z

ε

ε ε

=

=

∫∫∫∫∫∫

(5.101)

Expression (5.100) separates the energy U11 stored in the fibre direction and the

5.5. Evaluation of the Plate Damping as Function of Material Directions 153

coupling energy U12 induced by the Poisson’s effect. They are given by

(

)

2 4 2 4 2 2 2 11 11

2 2 3

3

1 cos sin sin cos2

2 sin cos 2 sin cos

2 sin cos d d d ,

xx yy xy

xx yy xx xy

yy xy

U Q

x y z

ε θ ε θ γ θ θ

ε ε θ θ ε γ θ θ

ε γ θ θ

= + +

+ +

+

∫∫∫ (5.102)

( ) ( )

( )

2 2 2 2 2 2 2 2 2 12 12

4 4 2 2

2 2

1 sin cos sin cos sin cos2

sin cos sin cos sin cos

cos sin sin cos d d d . (5.103)

xx yy xy

xx yy xx xy

yy xy

U Q

x y z

ε θ θ ε θ θ γ θ θ

ε ε θ θ ε γ θ θ θ θ

ε γ θ θ θ θ

⎡= + −⎣

+ + + −

⎤+ − ⎦

∫∫∫

In the case of bending vibrations of plates, the strains are deduced from Equations (5.68), which leads to the relations with the transverse displacement:

20

2

20

2

20

,

,

2 .

xx

yy

xy

zx

zy

zx y

ε

ε

γ

∂= −

∂

∂= −

∂

∂= −

∂ ∂

w

w

w

(5.104)

Then, the strain energies U11 and U12 are expressed as functions of the transverse displacement introducing expressions (5.104) in Equations (5.102) and (5.103), respectively. Next, considering the Ritz method, the transverse displacement is introduced in the form (4.24) and the expressions of the energies are integrated over the plate volume. Calculation leads to a formulation similar to the one deve-loped in Section 4.2 of Chapter 4. Considering this formulation leads to the follo-wing correspondences:

( )

( )( )

2 2200 2 0022 4 2 1111 2

2002 0220 2

1210 2101

1012 0121 3

, , 4 ,12 ,2

2 ,

2 ,

xx minj yy minj xy minj

xx yy minj minj

xx xy minj minj

yy xy minj minj

C C R C R

C C R

C C R

C C R

ε ε γ

ε ε

ε γ

ε γ

→ → →

→ +

→ +

→ +

(5.105)

where the coefficients pqrsminjC are given by Equation (4.39).

It results that the strain energies U11 and U12 can be written in the form:

( )11 11 1121 1 1 1

1 ,2

M N M N

mn ijm n i j

U A A D fR a

θ= = = =

= ∑∑∑∑ (5.106)

154 Chapter 5. Damping in Composite Materials

with ( ) ( )2200 4 0022 4 4 1111 2002 2 2 2

11

2101 3 0121 3 3

cos sin 2 2 sin cos

4 sin cos 4 sin cos ,

minj minj minj minj

minj minj

f C C R C C R

C R C R

θ θ θ θ θ

θ θ θ θ

= + + +

+ +(5.107)

3

11 11 ,12hD Q= (5.108)

and

( )12 12 1221 1 1 1

1 ,2

M N M N

mn ijm n i j

U A A D fR a

θ= = = =

= ∑∑∑∑ (5.109)

with ( ) ( ) ( )

( )( )

2200 0022 4 1111 2 2 2 2002 2 4 412

2101 0121 3 2 2

4 sin cos cos sin

2 sin cos sin cos , (5.110)

minj minj minj minj

minj minj

f C C R C R C R

C R C R

θ θ θ θ θ

θ θ θ θ

= + − + +

+ − −

3

12 12 .12hD Q= (5.111)

These expressions introduced the length-to-width ratio of the plate (R = a/b). In the same way, the energy U2 stored in tension-compression in the direction

transverse to the fibre direction is obtained as:

2 21 22,U U U= + (5.112) with 21 12,U U= (5.113) and

( )22 22 2221 1 1 1

1 ,2

M N M N

mn ijm n i j

U A A D fR a

θ= = = =

= ∑∑∑∑ (5.114)

with

( ) ( )2200 4 0022 4 4 1111 2002 2 2 2

22

2101 3 0121 3 3

sin cos 2 2 sin cos

4 sin cos 4 sin cos , (5.115)

minj minj minj minj

minj minj

f C C R C C R

C R C R

θ θ θ θ θ

θ θ θ θ

= + + +

− −

3

22 22 .12hD Q= (5.116)

Lastly, the strain energy U66 stored in in-plane shear can be written as:

( )66 66 6621 1 1 1

1 ,2

M N M N

mn ijm n i j

U A A D fR a

θ= = = =

= ∑∑∑∑ (5.117)

with

( ) ( )

( ) ( )( )

2200 0022 4 2002 2 2 266

21111 2 2 2 0121 3 2101 2 2

4 2 sin cos (5.118)

4 cos sin 8 cos sin sin cos ,

minj minj minj

minj minj minj

f C C R C R

C R C R C R

θ θ θ

θ θ θ θ θ θ

= + −

+ − + − −

5.5. Evaluation of the Plate Damping as Function of Material Directions 155

3

66 66 .12hD Q= (5.119)

Then, the energy dissipated by damping in the material is written in the form

11 11 12 12 22 22 66 662 ,U U U U Uψ ψ ψ ψ∆ = + + + (5.120)

introducing the damping coefficients ψ11, ψ12, ψ22 and ψ66 associated to the strain energies, respectively. The strain energy U12 is generally negative, due to the coupling between ε1 and ε2, and the corresponding dissipated energy must be taken positive. In fact, this energy can be neglected with regard to the other energies. Next, the damping ψx in the x direction of the plate along its length is evaluated by the relation

.xU

Uψ ∆= (5.121)

5.5.1.2 Procedure

In the Ritz method, the functions Xm(x) and Yn(y) introduced in expression (4.24) of the transverse displacement can be chosen [1, 2] as polynomials or as beam functions which give the characteristics shapes of the natural vibrations of beams (Section 3.4). The beam functions satisfy orthogonality relations which make zero many of the integrals (4.33) and (4.34).

Functions Xm(x) and Yn(y) depend on the boundary conditions imposed along the plate edges (Sections 4.2 and 4.3). Integrals (4.33) and (4.34) can be next calculated by an analytical development or by a numerical process and stored. Then, the values of the integrals allow us to establish the system (4.40) of homo-geneous equations for the undamped flexural vibrations of the plates. This system can be put in the form (4.118) or (4.121) with two indices and this system is solved as an eigenvalue and eigenvector problem where the eigenvectors determine the vibration modes, whence the coefficients Amn for the transverse displacement (4.24) corresponding to the different modes. Next, the different strain energies are derived, for a given mode, by reporting the values of the coef-ficients Amn in the energy expressions (5.106), (5.109), (5.114) and (5.117). Hence the laminate damping is derived from Relation (5.121).

5.5.2 Laminated Plates

The Ritz method used in the previous section for analyzing the damping pro-perties of orthotropic plates can be also applied to arbitrary laminated plates [43]. In the present section we consider the case of a laminated plate constituted of n orthotropic layers (Figure 1.6 of Chapter 1). Each layer is referred to by the z coordinates of its lower face 1kh − and upper face .kh Layer can also be charac-terized by introducing the thickness ek and the z coordinate zk of the middle plane of the layer. Layer orientation is defined by the angle kθ of layer axes with the

156 Chapter 5. Damping in Composite Materials

directions (x, y) of the plate. For a laminate, the strain energy relation (5.96) con-sidered for a single orthotropic layer can be written in the axes of each layer as:

d 1 2 6k k k kU U U U= + + , (5.122)

and the total energy of laminate is given by:

( )d 1 2 61

nk k k

k

U U U U=

= + +∑ . (5.123)

In the case of the vibrations of a rectangular plate of length a and width b, the strain energies are expressed by:

11 1 1

0 0d d d ,

k

k

a b hk

x y z hU x y zσ ε

−= = == ∫ ∫ ∫ (5.124)

12 2 2

0 0d d d ,

k

k

a b hk

x y z hU x y zσ ε

−= = == ∫ ∫ ∫ (5.125)

16 6 6

0 0d d d .

k

k

a b hk

x y z hU x y zσ γ

−= = == ∫ ∫ ∫ (5.126)

As in the previous subsection, the strain energy can be written in the form:

d1

,n

kpq

k pq

U U=

= ∑∑ (5.127)

with

10 0

1 d d d ,2

11, 12, 22, 66.

k

k

a b hk k k kpq pq p q

x y z hU Q x y z

pq

ε ε−= = =

=

=

∫ ∫ ∫ (5.128)

By considering the Ritz method, the transposition of the results obtained previously in the case of a single layer leads to:

( )

1

22

1 1 1 1

1 d2

k

k

hM N M Nk kpq mn ij pq k pq

hm n i j

U A A f Q z zRa

θ−= = = =

= ∑∑∑∑ ∫ . (5.129)

Hence:

21 1 1 1

1 ( ),2

M N M Nk k kpq mn ij pq pq

m n i j

U A A D fRa

θ= = = =

= ∑∑∑∑ (5.130)

with

( )3

3 3 21

1 .3 12

k k kkpq k k pq k k pq

eD h h Q e z Q−⎛ ⎞

= − = +⎜ ⎟⎜ ⎟⎝ ⎠

(5.131)

Then, the total energy dissipated by damping in the laminated plate is expressed as:

5.5. Evaluation of the Plate Damping as Function of Material Directions 157

( )11 11 12 12 22 22 66 661

2 ,n

k k k k k k k k

k

U U U U Uψ ψ ψ ψ=

∆ = + + +∑ (5.132)

introducing the specific damping coefficient kpqψ of each layer. Next, the damping

xψ in the x direction of the plate along its length is evaluated by relation:

xU

Uψ ∆= , (5.133)

where the dissipated energy is given by relation (5.132) and the total strain energy by relation (5.123).

The functions ( )kpqf θ of each layer are simply derived from the functions

( )pqf θ expressed previously in the case of a single layer of orthotropic material as: ( ) ( )k k

pq pq kf fθ θ θ= + , (5.134)

where functions ( )pqf θ are given by (5.107), (5.110), (5.115) and (5.118).

5.5.3 Conclusion

The process for evaluating the laminate damping from the dissipated energy has been implemented by using the Ritz method. This procedure can also be carried out using a vibration analysis by the finite element method (Chapters 8 and 9). In this case it is necessary to have access to the strain and stress fields for each vibration mode. Next, the energies and the loss damping are obtained in the same way as for the Ritz method by considering the stored energies and the dissipated energies.

The interest of the Ritz method lies in the fact that the process can be easily implemented with usual tools. However, the method is restricted to the analysis of rectangular plates. In contrast, the finite element analysis can be applied to the case of a laminated structure of complex shape.

CHAPTER 6

Experimental Investigation and Discussion on the Damping Properties of Laminates

6.1 EXPERIMENTAL INVESTIGATION

IN LITERATURE

6.1.1 Experimental Processes for Evaluating Damping

The first equipment for measuring the damping of composite materials was developed by Adams et al. [31] and Adams and Bacon [44]. Flexural damping of beams was evaluated over a frequency range of 100 to 800 Hz. Symmetric free-free flexural modes of vibrations were excited by a coil/electromagnet drive trans-ducer, the coil being fixed to the midpoint of the beam. A coil/magnet transducer was also used for measuring the central amplitude of the beam. The signal of the input transducer was tuned to the fundamental natural frequency of the beam, and damping was evaluated from the current input into the driving coil and the voltage induced in the pick-up coil. Forced flexural vibration technique was also used by Gibson et al [45, 46] to test E-glass/epoxy specimens having different fibre aspect ratios.

The impulse method was initially developed by Suarez et al [47], Suarez and Gibson [48] and Crane and Gillespie [49]. The test specimen to characterize was supported as a flat cantilever beam in a clamping block. Impulsive excitation was applied using an electromagnetic hammer and the transverse displacement of the beam was measured versus time by means of non-contact eddy current probe positioned near the tip of the beam. Then, Fourier transform was performed to obtain the frequency response function. Curve fitting to Fourier transform was used by Suarez et al [47] to yield the complex modulus, when damping was evaluated by Crane and Gillespie [49] from the loss factor determined by the half-power bandwidth method.

The evaluation of damping from the logarithmic decrement of the free vibra-tions of cantilever beams was considered by Hadi and Ashton [50] using the

6.1. Experimental Investigation in Literature 159

experimental process developed by Wray et al [51]. In this process, the specimen was clamped at one end and an initial displacement was achieved by striking the free end with a controlled striker mechanism. Next, displacement of the free end versus time was detected using a capacitance transducer and damping was de-duced from the logarithmic decrement obtained from the decaying voltage-time signal of the transducer.

The evaluation of damping from flexural vibrations of plates was considered by Sol et al [8] and De Visscher [52] using a procedure which estimates the complex bending stiffness of plates. This procedure was used by De Santis et al [53] for characterizing the dynamic behaviour of shape memory alloy fibre rein-forced epoxide. However the fundamental concepts of the procedure have not been developed extensively in the papers of the authors. Schematically, the proce-dure is implemented in two steps. In the first step, the real parts of the plate stiffnesses are determined from vibration data obtained in the case of flexural vibration of a free rectangular plate suspended vertically by two thin threads. In a second step, the stiffness loss factors are determined from measurements of the free vibration responses after cutoff of an acoustic excitation: i) at the first natural frequency of two free-free beam samples oriented in the longitudinal and transverse directions of the orthotropic materials under consideration, and ii) at the first three natural frequencies of a free plate. The logarithmic decrements are deduced from these responses and the stiffness loss factors of the materials are derived from these experimental decrements according to relations [8, 52] esta-blished by the authors, considering the energy dissipation of the beams and plates.

6.1.2 Experimental Results

The effect of aerodynamic damping in the case of flexural beam vibrations was considered by Baker et al [54], Adams and Bacon [44] and Crane and Gillespie [49]. Air damping is due to both air viscosity and inertia effects, and increases with the increase in the amplitude of vibrations. Adams and Bacon found that the effect of air damping was significant for specimens of low damping as usual materials when vibration amplitude is increased. The results obtained by Crane and Gillespie shown that no significant variation of loss factor was obtained for low amplitude in the case of composite materials, air damping staying within the experimental accuracy of the testing procedure. So it can be considered that the measured loss factor is from material damping for low amplitude of composite beam vibrations.

The first significant experimental results obtained on the evaluation of the damping of unidirectional composite materials were derived by Adams and Bacon [38] and Ni and Adams [39]. The specimens were made from pre-impregnated unidirectional fibres: E-glass fibres in DX 210 epoxy matrix and surface treated graphite fibres (HMS) in the same epoxy matrix. The longitudinal and transverse properties were determined from free-free flexural modes of beam vibrations, when the shear properties were derived from torsional tests. Next, the effect of fibre orientation was investigated. The results obtained by Adams and Bacon [38]

160 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

and Ni and Adams [39] for the damping are very similar. The main difference is that the damping maximum has moved to about 25° fibre orientation for carbon epoxy composites and 60° fibre orientation (with a maximum little marked) for glass epoxy composites [39], compared to 35° and no maximum, respectively, according to Adams and Bacon [38].

Glass and carbon fibre reinforced epoxy matrix were also investigated by Adams and Maheri [40]. All the tests were carried out in air and at room tempe-rature, and the damping was evaluated as function of fibre orientation and beam aspect ratio. In the case of unidirectional materials, the experimental results show that damping is maximum for about 75° fibre orientation, again with a maximum little marked.

Damping properties of unidirectional glass fibre epoxy composites were studied by Hadi and Ashton [50] with various fibre orientations (0°, 30°, 45°, 60° and 90°) and for each fibre orientation at three fibre volume fractions (0.35, 0.45 and 0.60). The experimental results show that damping increases as fibre volume decreases at each fibre orientation and damping maximum is observed for 30° orientation.

6.2 DAMPING ANALYSIS OF UNIDIRECTIONAL GLASS AND KEVLAR FIBRE COMPOSITES

6.2.1 Introduction

An extended experimental investigation of damping properties of glass and Kevlar fibre composites has been developed in [42, 43]. In this investigation the damping properties of unidirectional materials are deduced from the experimental analysis of bending vibrations of beams. We report in this section the essential results.

6.2.2 Materials

Laminates were prepared by hand lay-up process from epoxy resin with hardener and unidirectional fabrics of weights 300 gm–2 for E-glass fibres and 400 gm–2 for Kevlar fibres. Plates of 450 mm length and 300 mm width were cured at room temperature with pressure using vacuum moulding process, and then post-cured in an oven. Plates had a nominal thickness of 2.5 mm with a volume frac-tion of fibres equal to 0.40.

The engineering constants of E-glass laminates referred to the fibre direction were measured in static tests:

29.9 GPa, 5.85 GPa, 2.45 GPa, 0.24,L T LT LTE E G ν= = = = (6.1)

as mean values of 10 tests for each constant. Then the values of the reduced

6.2. Damping Analysis of Unidirectional Glass and Kevlar Fibre Composites 161

stiffnesses are derived as:

11 12 16

22 26 66

30.24 GPa, 1.42 GPa, 0,5.92 GPa, 0, 2.45 GPa.

Q Q QQ Q Q

= = == = =

(6.2)

Similarly, the engineering constants of Kevlar laminates measured in static tests were obtained:

50.70 GPa, 4.50 GPa, 2.10 GPa, 0.29,L T LT LTE E G ν= = = = (6.3)

which leads to the values of the reduced stiffnesses:

11 12 16

22 26 66

51.08 GPa, 1.31 GPa, 0,4.53 GPa, 0, 2.10 GPa.

Q Q QQ Q Q

= = == = =

(6.4)

Beam specimens were cut from the laminates and damping properties were measured for different orientations of fibres: 0°, 15°, 30°, 45°, 60°, 75° and 90°.

6.2.3 Experimental Equipment

The equipment used for damping measurement is shown in Figure 6.1. The test specimen is supported horizontally as a cantilever beam in a clamping block. An impulse hammer is used to induce the excitation of the flexural vibrations of the beam. A force transducer positioned on the hammer allows us to obtain the excitation signal as a function of the time. The width of the impulse and hence the

FIGURE 6.1. Experimental equipment for damping analysis.

Signal conditioner

Dynamic analyser

Laser vibrometer Impact hammer

Beam Clamping block

162 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

frequency domain is controlled by the stiffness of the head of the hammer. The beam response is detected by using a laser vibrometer which measures the velocity of the transverse displacement of a point near the free end of the beam. Next, the excitation and the response signals are digitalized and processed by a dynamic analyzer of signals. This analyzer associated with a PC computer per-forms the acquisition of signals, controls the acquisition conditions (sensibility, frequency range, trigger conditions, etc.), and next performs the analysis of the signals acquired (Fourier transform, frequency response, mode shapes, etc.). Then, the signals and the associated processings can be saved for post-processings. The system allows the simultaneous acquisition of two signals with a maximum sampling frequency of 50 kHz with a resolution of 13 bits for each channel.

6.2.4 Analysis of the Experimental Results

6.2.4.1 Determination of the Constitutive Damping Parameters

Impulse excitation of the flexural vibrations of beam was induced (Figure 6.2) at point x1 near the clamping block and the beam response was detected at point x near the free end of the beam. Figure 6.3 gives an example of the Fourier trans-form of the beam response to an impulse input. This response shows peaks which correspond to the natural frequencies of the bending vibrations of the beam. Expe-rimental analysis was performed on beams of different lengths 160, 180 and 200 mm so as to have a variation of the values of the peak frequencies.

The transverse response to an impact loading is given by Expression (5.46). In fact, the laser vibrometer measures the velocity of the transverse displacement and the beam response detected by the vibrometer is proportional to:

00 1 1 2

1

( ) ( ) ( ) ( sin cos ).i i i i iii

p X x X x K a t b tt

ω ω ω ωω

∞

=

∂= = − +∂ ∑ww (6.5)

The Fourier transform gives the complex amplitude as function of the frequency expressed by:

2 20 1 1 2

1

( ) ( ) ( ) ( ).i i i i iii

p X x X x a b Kωω ωω

∞

=

= +∑W (6.6)

FIGURE 6.2. Impact and measuring points on the cantilever beams.

x

x l

x1 Measuring Impact point

6.2. Damping Analysis of Unidirectional Glass and Kevlar Fibre Composites 163

Frequency (Hz)0 100 200 300 400 500 600 700

Am

plitu

de (d

B)

0

10

20

30

40

50

60

70

FIGURE 6.3. Typical frequency response to an impulse of a unidirectional glass compo-site beam.

So, the experimental analysis was implemented by fitting the experimental res-ponses with relation (6.6), considering either the viscous friction model (Equations (5.25) and (5.26)) or the complex stiffness model (Equations (5.37) and (5.38)). This fitting was obtained by a least square method using the optimisation toolbox of Matlab, which allows us to derive the values of the natural frequencies fi, and the modal damping coefficient ξi (case of damping using viscous friction model-ling) or the loss factor ηi (case of damping using the complex stiffness model). This method can be applied for notable damping of materials.

According to Relations (3.15) and (3.19) of Chapter 3, each natural frequency of the undamped beam is related to the stiffness by unit area by relation:

4

2 244 .i s

is

kfaκπ

ρ= (6.7)

This relation allows us to evaluate the stiffness ks for each natural frequency of beams in the case of low damping.

6.2.4.2 Plate Damping Measurement Rectangular plates with two adjacent edges clamped with the other two free

and plates with one edge clamped and the others free were tested to determine the damping characteristics for the first modes of flexural vibrations. As in the case of beams, the excitation of vibrations was induced by the impulse hammer and the plate response was detected by using the laser vibrometer. The damping para-meters were derived from the Fourier transform of the plate response. Vibration excitation and response detection were carried out at different points of the plates so as to generate and detect all the modes.

164 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

6.2.5 Choice of the Frequency Range for the Experimental Investigation

Performances of the acquisition system are associated to the maximum sam-pling frequency and to the maximum number of samples which can be stored. We noted (Section 6.2.3) that the maximum sampling frequency is equal to 50 kHz with a resolution of 13 bits. The domain of the experimental analysis extending until frequencies of 1000 to 1500 Hz, the analysis is not limited for the high frequencies. However, the sample number which can be acquired during the acquisition of a time response with calculation in real time of the Fourier trans-form is limited to 8192. A sample number which can be extended up to 1.5 ¥ 106 is possible in the case of the storage of the time signal only. Next, the Fourier transform can be calculated off line.

Before the signal acquisition, the operator chooses the value of the number ns of samples which will be stored and the value f2 of the frequency range. This value determines the sampling frequency used by the system: fs = 2f2 and the frequency resolution rf is determined by the number nf of points for which the Fourier transform is calculated:

2 .ff

frn

= (6.8)

In the case of low damping of the structure under consideration, the 3 dB bandwidth near the modal frequency fi of the structure is given by:

,ii

ff

η∆ = (6.9)

where ηi is the loss factor of mode i. If np is the number of points within the band-width needed for accurate evaluation of the modal response, the resolution neces-sary near the frequency fi is:

1 .fi i ip p

fr fn n

η∆= = (6.10)

According to Equation (6.8), the maximum resolution of the analyzer is

2max

max,f

f

frn

= (6.11)

where nf max is the maximum number of points for which the Fourier transform can be calculated.

For obtaining the needed resolution (6.10), it is necessary that rf max ≤ rfi, which yields: 1,if f≥ (6.12) with

1 2max

1 .p

f i

nf f

n η= (6.13)

6.2. Damping Analysis of Unidirectional Glass and Kevlar Fibre Composites 165

The frequency bandwidth which can be analyzed is then limited to the interval [f1, f2].

The frequency f1 must be strictly lower than the frequency f2. Thus, it results that the loss factors which can be evaluated for a given number nf max of samples is such as:

max

> .pi

f

nn

η (6.14)

For a sample number equal to 8192 and considering that 10 samples within the 3 dB bandwidth is necessary, only loss factors higher than 1.22 ¥ 10–3 can be mea-sured, when loss factors of 2 ¥ 10–6 can be measured in the case of a sample number equal to 500 ¥ 103. Moreover, for a loss factor equal to 2 ¥ 10–3, relation (5.146) leads to f1 = 0.61f2 when the Fourier transform is calculated with 8192 samples and f1 = 0.01f2 when it is calculated with 500 ¥ 103

samples. Tables 6.1 and 6.2 show the bandwidths which can be analyzed in the case of a loss factor equal to 2 ¥ 10–3 and for the two sample numbers 8192 and 500 ¥ 103. The values reported in these tables show the need to perform damping measurements by recording first time signals using a high sample number, then calculating Fourier transform off line.

6.2.6 Experimental Results

6.2.6.1 Introduction

This subsection reports the experimental results deduced from the analysis of unidirectional glass fibre and Kevlar fibre beam. As reported previously, the TABLE 6.1. Frequency bandwidth as function of the frequency range, in the case of a sample number equal to 8192.

f2 (Hz) f1 (Hz) - f2 (Hz)

33 99

297 891 1782

20 - 33 60.2 - 99

18.2 - 297 543.5 - 891 1087 - 1782

TABLE 6.2. Frequency bandwidth as function of the frequency range, in the case of a sample number equal to 500 × 103.

f2 (Hz) f1 (Hz) - f2 (Hz)

33 297 1782

0.33 - 33 3 - 297

17.8 - 1782

166 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

Frequency (Hz)

0 100 200 300 400 500

Ben

ding

mod

ulus

Ef x

(GPa

)

5

10

15

20

25

30

35θ = 0 °

θ = 15 °

θ = 30 °

θ = 45 °

θ = 60 °

θ = 90 °θ = 75 °

experimental investigation of damping was performed on beams of different lengths: 160, 180 and 200 mm so as to have a variation of the values of the peak frequencies. Beams had a nominal width of 20 mm and a nominal thickness of 2.4 mm.

6.2.6.2 Stiffness

Measurement of the natural frequencies of beams leads to the estimation of the bending modulus Efx of beams according to Relation (6.7) associated with Rela-tions (3.12). The experimental results obtained for the variation of the modulus with the frequency and for the different orientations of fibres are given in Figure 6.4 for glass fibre composites and in Figure 6.5 for Kevlar fibre composites. For a given fibre orientation, it is observed a light increase (about 5 to 8 %) of the bending modulus with the frequency within the domain under consideration.

The variation of the bending modulus with the fibre orientation is then given in Figures 6.6 and 6.7 for glass fibre composites and Kevlar fibre composites, respectively, for a frequency of 100 Hz. These experimental results are compared with the analytical results deduced from Relation (3.12). A good agreement is observed.

FIGURE 6.4. Experimental results obtained for the bending modulus Efx as a function of the frequency for different fibre orientations, in the case of glass fibre composites.

6.2. Damping Analysis of Unidirectional Glass and Kevlar Fibre Composites 167

FIGURE 6.5. Experimental results obtained for the bending modulus Efx as a function of the frequency for different fibre orientations, in the case of Kevlar fibre composites. FIGURE 6.6. Variation of the bending modulus Efx as a function of fibre orientation for a frequency equal to 100 Hz, in the case of glass fibre composites.

Frequency (Hz)

0 200 400 600 800 1000 1200

Ben

ding

mod

ulus

Ef x

(GPa

)

0

10

20

30

40

50

60

θ = 0 °

θ = 15 °

θ = 30 °

θ = 45 °

θ = 60 °θ = 75 °

θ = 90 °

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Ben

ding

mod

ulus

Ef x

(GPa

)

0

5

10

15

20

25

30

35

Experimental resultsAnalytical curve

168 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

FIGURE 6.7. Variation of the bending modulus Efx as a function of fibre orientation for a frequency equal to 100 Hz, in the case of Kevlar fibre composites.

6.2.6.3 Damping

Fitting the experimental responses of beams with the analytical responses (Subsection 6.2.4.1) leads to the evaluation of the modal damping coefficient ξi or the loss factor ηi, associated to each mode i.

Figures 6.8 and 6.9 show the experimental results obtained in the case of glass fibre composites and Kevlar fibre composites. The results are reported for the first three bending modes and for the different lengths of the beams. The experimental results show that damping is maximum at a fibre orientation of about 60° for the glass fibre composites, when a maximum for about 30° fibre orientation is obser-ved in the case of the Kevlar composites.

For a given fibre orientation, it is observed that damping increases when the frequency is increased. The values of the damping increase when the frequency is increased from 50 Hz to 600 Hz are reported in Table 6.3 for the glass fibre composites and the Kevlar fibre composites. The table shows that the damping increase is fairly the same (from 21 to 27 %) for the different fibre orientations in

TABLE 6.3. Damping increase (%) in the frequency range [50, 600 Hz].

Fibre orientation (°) 0 15 30 45 60 75 90

Glass fibre composites 21 24 26 23 26 23 27

Kevlar fibre composites 5.4 10.2 16.5 17 18.3 18 11.6

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Ben

ding

mod

ulus

Ef x

(GPa

)

0

10

20

30

40

50

60

Analytical curve Experimental results

6.2. Damping Analysis of Unidirectional Glass and Kevlar Fibre Composites 169

FIGURE 6.8. Experimental results obtained for the damping as a function of the frequency for different fibre orientations, in the case of glass fibre composites.

FIGURE 6.9. Experimental results obtained for the damping as a function of the frequency for different fibre orientations, in the case of Kevlar fibre composites.

Frequency (Hz)0 200 400 600 800

Loss

fact

or η

i (%

)

0.3

0.5

0.7

0.9

1.1

1.3

1.5

1.7

θ = 0 °

θ = 15 °

θ = 30 °

θ = 45 °θ = 75 °θ = 90 °

θ = 60 °

Frequency (Hz)0 200 400 600 800 1000 1200

Loss

fact

or η

i (%

)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

θ = 60 °θ = 75 °

θ = 45 °

θ = 90 °

θ = 30 °

θ = 15 °

θ = 0 °

170 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

the case of the glass fibre composites, when the increase depends on the fibre orientation in the case of the Kevlar fibre composites with values varying from about 5 to 18 %.

6.3 COMPARISON OF EXPERIMENTAL RESULTS AND MODELS FOR UNIDIRECTIONAL BEAM

DAMPING

6.3.1 Models of Adams-Bacon and Ni-Adams 6.3.1.1 Introduction

The models are based on an energy analysis and lead to the evaluation (5.87) of the specific damping coefficient ψ measured in the direction θ as a function of the damping coefficients ψ11 in the 0° direction, ψ22 in the 90° direction and ψ66 the damping coefficient associated to in-plane shear. It is usual to consider the results obtained for the loss factor η related to ψ by the relation ψ = 2πη. Thus, the formulation (5.87) is simply transposed by considering the loss factors η11, η12 and η66. The values of these coefficients can be derived from the experimental results by considering the results obtained for fibre orientations of 0° and 90°, and for an intermediate orientation of 45°, for example. The analytical curve giving the damping η (θ) as a function of the fibre orientation is then derived using Equa-tion (5.87).

6.3.1.2 Glass Fibre Composites The results deduced from the Adams-Bacon and Ni-Adams models are com-

pared with the experimental results at frequency 50 Hz in Figure 6.10. The curve derived from the Adams-Bacon model is obtained with 11 22 660.40%, 1.24%, 1.48%.η η η= = = (6.15)

The one deduced from the Ni-Adams model is obtained with 11 22 660.40%, 1.24%, 1.72%.η η η= = = (6.16)

In Figure 6.10, it is observed a rather good agreement between the results deduced from the two models and the experimental results. However, the values of the shear loss factor deduced from the two models are fairly different.

6.3.1.3 Kevlar Fibre Composites

The same analysis was then applied to Kevlar fibre composites. The results deduced from the two models are compared with the experimental results in Figure 6.11. Fitting the results deduced from the two models with the experi-mental results was obtained with 11 22 661.60%, 2.90%, 3.14%,η η η= = = (6.17)

in the case of the Adams-Bacon model, and with

6.3. Comparison of Experimental Results and Models for Beam Damping 171

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (%

)

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

Experimental resultsAdams-Bacon analysis Ni-Adams analysis Complex stiffness modulus

Fibre orientation θ (°)0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (%

)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Experimental resultsAdams-Bacon analysis Ni-Adams analysis Complex stiffness model

FIGURE 6.10. Comparison between the experimental damping results and the results derived from Adams-Bacon, Ni-Adams and complex stiffness models, in the case of glass fibre composites.

FIGURE 6.11. Comparison between the experimental damping results and the results derived from Adams-Bacon, Ni-Adams and complex stiffness models, in the case of Kevlar fibre composites.

172 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

11 22 661.60%, 2.90%, 3.53%,η η η= = = (6.18)

in the case of the Ni-Adams model. In Figure 6.11, it is observed that, if the two models describe the overall varia-

tion of damping as function of the orientation of fibres measured in experiments, there is a notable deviation for fibre orientations from about 10° to 35°.

6.3.2 Complex Stiffness Model

The damping evaluation using the complex modulus of the beams was consi-dered in Subsection 5.4.2. The damping is evaluated by Relation (5.91), where the complex bending modulus is expressed (5.90) as a function of the element 1

11*D − of

the complex inverse matrix of *ijD⎡ ⎤⎣ ⎦ . According to the elastic-viscoelastic corres-

pondence principle, the complex bending-twisting matrix *ijD⎡ ⎤⎣ ⎦ is obtained as:

3

** ,12ij ijhD Q⎡ ⎤⎡ ⎤ ′=⎣ ⎦ ⎣ ⎦ (6.19)

where the complex reduced stiffnesses *ijQ′ are converted from the relations of

Table 1.1 giving the reduced stiffnesses ijQ′ with reference to the fibre orientation

as functions of the reduced stiffnesses ijQ referred to the material directions. Thus, the complex reduced stiffnesses in the material directions are expressed as:

11 12

22 66

* * ** *

* *2 2* ** *

** * *

*2**

, ,1 1

, ,1

L TL L

T TLT LT

L L

TLT

TLT

L

E EQ QE EE E

EQ Q GEE

ν

ν ν

ν

= =− −

= =−

(6.20)

introducing the engineering moduli in the complex form:

( ) ( )

( ) ( )* *

* *

1 , 1 ,

1 , 1 .

L L L T T T

LT LT LT LT LT LT

E E i E E i

G G i i ν

η η

η ην ν= + = +

= + = + (6.21)

When the fibre orientation is equal to 0°, the effective bending modulus can be identified with the longitudinal modulus EL of the material. Hence, the longitudinal loss factor can be identified with the damping η0° measured for the 0° fibre orientation. In the same way, the transverse loss factor can be identified with the loss factor η90° measured for the 90° fibre orientation.

6.3. Comparison of Experimental Results and Models for Beam Damping 173

Fibre orientation θ (°)0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (%

)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

f = 50 Hz f = 300 Hzf = 600 Hzf = 50 Hzf = 300 Hzf = 600 Hz

Experimental results

Ritz analysis

The results obtained by the complex stiffness model are reported in Figures 6.10 and 6.11 in the case of glass fibre composites and Kevlar fibre composites, respectively. The results were obtained by considering that the damping asso-ciated to the Poisson's ratio is zero and fitting the shear loss factor ηLT so that the complex stiffness model gives the value of the loss factor measured for the 45° fibre orientation. The comparison between the results obtained shows that the experimental results are not well described by the complex stiffness model for fibre orientations ranging from about 10° to 45°. Besides, the use of the complex modulus approach leads to problems associated with the consistency of the complex moduli [55].

6.3.3 Using the Ritz Method

6.3.3.1 Damping Parameters

The analysis using the Ritz method (Section 5.5) was applied to the expe-rimental results obtained for the bending of beams. The beams were considered in the form of plates with one edge clamped and with the others free. Damping was evaluated by the Ritz method (5.121) considering the beam functions introduced previously in Section 4.2 and 4.3 of Chapter 4. Thus, the present evaluation of the beam damping takes account of the effect of the beam width. The results deduced from the Ritz method are reported in Figures 6.12 and 6.13 in the case of glass fibre composites and Kevlar fibre composites, respectively. A good agreement is obtained with the experimental results. The values of the loss factors considered

FIGURE 6.12. Comparison of the experimental results and the results deduced from the Ritz’s method for damping as a function of fibre orientation, in the case of glass fibre composites.

174 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

FIGURE 6.13. Comparison of the experimental results and the results deduced from the Ritz’s method for damping as a function of fibre orientation, in the case of Kevlar fibre composites.

for modelling are reported in Tables 6.4 and 6.5 for the frequencies 50, 300 and 600 Hz. These results show that the shear damping evaluated by using the Ritz method is fairly higher that the values of the shear loss factor deduced from the Adams-Bacon analysis or from the Ni-Adams analysis which do not consider the width of the beam.

TABLE 6.4. Loss factors derived from the Ritz method in the case of unidirectional glass fibre laminates.

f (Hz) η11 (%) η12 η22 (%) η66 (%)

50 0.35 0 1.30 1.80

300 0.40 0 1.50 2.00

600 0.45 0 1.65 2.22

TABLE 6.5. Loss factors derived from the Ritz method in the case of unidirectional Kevlar fibre laminates.

f (Hz) η11 (%) η12 η22 (%) η66 (%)

50 1.50 0 2.50 3.80

300 1.63 0 2.60 4.10

600 1.69 0 2.78 4.50

Fibre orientation θ (°)0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (%

)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

f = 50 Hz f = 300 Hzf = 600 Hzf = 50 Hzf = 300 Hzf = 600 Hz

6.3. Comparison of Experimental Results and Models for Beam Damping 175

6.3.3.2 Influence of the Width of the Beams

The influence of the beam width can be analyzed by the Ritz method. Figures 6.14 show the results obtained for the loss factor of the first mode of beams with a nominal length of 200 mm and for different length-to-width ratio of the beam: 100, 20, 10, 7 and 5, in the case of glass fibre composites (Figure 6.14a) and Kevlar fibre composites (Figure 6.14b). These figures show that the results reach a limit for high values of the length-to-width ratio of the beams. Furthermore, the results deduced from the Ni-Adams analysis, considering the values of damping derived from the experimental results using the Ritz method in the case of beams with a length to width ratio equal to 10 (Tables 6.4 and 6.5), are compared in Figures 6.15a (glass fibre composites) and 6.15b (Kevlar fibre composites) with the results derived from the Ritz method in the case of a length-to-width ratio of the beams equal to 100. The results are rather similar. This shows that the Ni-Adams analysis can be applied to the evaluation of damping properties of beams with high values of the length-to-width ratio. In fact, in order to minimize the edge effects especially for off-axis materials it is difficult to implement an expe-rimental analysis with a high value of the length-to-width ratio of the beams. A ratio about 10 which leads to a beam width of 20 mm for a length of 200 mm appears to be a good compromise. In this case it is necessary to analyse the experimental results with a modelling which takes the width of the beams into account.

6.3.3.3 Damping According to Modes of Beam Vibrations

The Ni-Adams analysis is established using the beam theory which considers the case of bending along the x axis of beams and assumes that the transverse displacement of beams is a function of the x coordinate only:

w0 = w0(x). (6.22)

According to this theory, only the bending modes of beams are described and the damping of unidirectional beams will all the more high as the beam deformation will induce bending in the direction transverse to fibres and in-plane shearing for intermediate orientations of fibres. The Ni-Adams analysis does not take account of the effects of beam twisting which can induce notable twisting deformation of beams for which the transverse displacement is not anymore independent of the y coordinate.

Figures 6.16 show the variations of beam damping deduced from the Ritz method for the first four modes of unidirectional beams in the case of beam length equal to 180 mm and a length-to-width ratio equal to 10: glass fibre beams (Figure 6.16a) and Kevlar fibre beams (Figure 6.16b). For the damping evaluation of beams we have considered that the loss factors of the materials depend on the frequency according to the results obtained in Subsection 6.3.3.1. The natural frequencies and modes of the beams were first derived using the Ritz method. Next, the damping evaluation of laminated beams was derived according to the modelling developed in Section 5.5 of Chapter 5 and considering that the damping

176 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

Fibre orientation θ (°)0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (%

)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Experimental resultsR = 100R = 20 R = 10 R = 7 R = 5

(a) (b)

FIGURE 6.14. Unidirectional beam damping obtained with different values of the length-to-width ratio R : a) in the case of glass fibre composites and b) in the case of Kevlar fibre composites.

Fibre orientation θ (°)0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (%

)

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

Experimental results R = 100R = 20 R = 10 R = 7R = 5

6.3. Comparison of Experimental Results and Models for Beam Damping 177

(a) (b)

FIGURE 6.15. Comparison of the results deduced from the Ni-Adams analysis and the Ritz method in the case of a length-to-width ratio of 100: a) glass fibre beams and b) Kevlar fibre beams.

Fibre orientation θ (°)0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (%

)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Experimental resultsNi-Adams analysisRitz analysis

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (%

)

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

Experimental resultsNi-Adams analysis Ritz analysis

178 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

(a)

(b)

FIGURE 6.16. Variation of the damping of unidirectional beams of length equal to 180 mm, derived from the Ritz method for the first four modes: a) glass fibre beams and b) Kevlar fibre beams.

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Ritz analysis

Experimentalbending modes

mode 1mode 2mode 3mode 4mode 1mode 2mode 3

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

1.4

1.8

2.2

2.6

3.0

3.4

3.8

4.2

mode 1mode 2mode 3mode 4mode 1mode 2mode 3

Ritz analysis

Experimentalbending modes

6.4. Damping of Laminated Beams 179

factors η11, η22 and η66 increased linearly in the frequency range [50, 600 Hz] according to the values reported in Tables 6.4 and 6.5. The results for the first two modes are similar, differing by the increase of the damping with the frequency.

In the case of the third mode (Figure 6.16), it is observed a high beam damping for fibre orientations of 0° and 10° with a value which is fairly near of the shear damping. The shapes of the modes 1 to 4 for a fibre orientation of 0° are given in Figure 6.17. The results show that the shapes of modes 1, 2 and 4 satisfy the assumption (6.22), whereas an important twisting of the beam is observed for mode 3 inducing a notable in-plane shear deformation. Finally, the beam damping results from the respective contributions of the energies induced in bending along the x direction of the beam, bending along the transverse y direction and beam twisting. These energies are taken into account by the damping analysis based on the Ritz method.

Figure 6.18 reports the mode shapes deduced in the case of 30° fibre orientation showing the participation of the different deformation modes. In this case, it is observed that beam twisting of the mode 4 is associated to a lower damping of the beam. These results show that beam twisting induces an increase of damping for fibre orientations near the material directions: 0° direction for mode 3 and 90° direction for mode 4 (Figures 6.16 and 6.17), resulting from the increase of in-plane shear deformation of materials. In contrast, the beam twisting results in a decrease of damping for intermediate orientations (mode 4, Figures 6.16 and 6.18) associated to the decrease of in-plane shear deformation. Similar results are observed in the case of unidirectional Kevlar composites.

The variations of beam damping deduced from the Ritz method are compared in Figure 6.16 with the experimental results obtained for the first three bending modes of beams. These bending modes were obtained by exciting the beams by an impulse applied on the beam axis so as to induce vibration modes without beam twisting. The damping evaluation by the Ritz method agrees fairly well with the experimental results when only the bending modes of the beams are considered.

6.4 DAMPING OF LAMINATED BEAMS

Laminated beams with three different stacking sequences were analyzed: [0/90/0/90]s cross-ply laminates, [0/90/45/−45]s laminates and [θ/−θ/θ/−θ]s angle- ply laminates with θ varying from 0° to 90°. The laminates were prepared from 8 plies of the unidirectional materials studied in the previous section. The nominal thickness of the laminates was 2.4 mm and the analysis was implemented in the case of beams 200 mm long and 20 mm width.

Figures 6.19 and 6.20 show the results obtained for the damping in the case of glass fibre laminates and Kevlar fibre laminates, respectively. Figures report the results deduced for the damping by the Ritz method for the first four modes and the experimental damping measured for the first mode. The evaluation of laminate damping by the Ritz method takes account of the variation of the loss factors η11, η22 and η66 with frequency (Tables 6.4 and 6.5). For the cross-ply laminates (Figures 6.19a and 6.20a) and [0/90/45/−45]s laminates (Figures 6.19b and 6.20b),

180 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

FIGURE 6.17. Free flexural modes of a unidirectional glass fibre beam for 0° fibre orientation. FIGURE 6.18. Free flexural modes of a unidirectional glass fibre beam for 30° fibre orientation.

the material damping is derived as function of laminate orientation. For the [θ/−θ/θ/−θ]s angle ply laminates (figures 6.19c and 6.20c), damping is reported as function of the ply orientation θ. The damping deduced from the Ritz method was evaluated by applying the results of Section 5.5.2 to the different laminates.

mode 1 mode 2

mode 3 mode 4

mode 1 mode 2

mode 3 mode 4

6.4. Damping of Laminated Beams 181

FIGURE 6.19. Damping variation as a function of laminate orientation for beams of differ-rent glass fibre laminates: a) [0/90/0/90]s cross-ply laminates, b) [0/90/45/−45]s laminates and c) [θ/−θ/θ/−θ]s angle ply laminates.

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2mode 1mode 2mode 3mode 4mode 1

Ritz analysis

Experimental results

(c)

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

0.4

0.6

0.8

1.0

1.2

1.4

1.6mode 1mode 2mode 3mode 4mode 1

Ritz analysis

Experimental results

(b)

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

mode 1mode 2mode 3mode 4mode 1

Ritz analysis

Experimental results

(a)

182 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

FIGURE 6.20. Damping variation as a function of laminate orientation for beams of differ-rent Kevlar fibre laminates: a) [0/90/0/90]s cross-ply laminates, b) [0/90/45/−45]s laminates and c) [θ/−θ/θ/−θ]s angle ply laminates.

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

1.5

2.0

2.5

3.0

3.5

4.0

4.5mode 1mode 2mode 3mode 4mode 1

Ritz analysis

Experimental results

(c)

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2mode 1mode 2mode 3mode 4mode 1

Ritz analysis

Experimental results

(b)

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

1.5

2.0

2.5

3.0

3.5

4.0

4.5

mode 1mode 2mode 3mode 4mode 1

Ritz analysis

Experimental results

(a)

6.5. Damping of Laminated Plates 183

The in-plane behaviour of the [0/90/0/90]s cross-ply laminates is the same in the 0° and 90° directions, when the external 0° layers of the stacking sequence leads to a slight increase of the bending properties in the 0° direction. Thus, com-pared to the damping of unidirectional composites (Figures 6.12 and 6.13), the stacking sequence [0/90/0/90]s leads to a more symmetric variation of damping as function of the orientation with damping characteristics which are slightly higher in the 90° direction. Near 45° orientations damping of the [0/90/0/90]s laminates is clearly reduced (about 1.2 % for glass fibre laminates and 2.6 % for Kevlar fibre laminates, for the first two modes) compared to the damping of the uni-directional laminates (about 1.4 % and 3.0 %, respectively). This reduction results from the in-plane shear deformation which is constrained by the [0/90] stacking sequence. For the third mode it is observed a high damping for directions near 0° and 90° associated to the effects of beam twisting as in the case of the unidi-rectional laminates. For the fourth mode the beam twisting leads to a decrease of the beam damping. The use of the [90/0/90/0]s stacking sequence would lead to a damping behaviour where the 0° and 90° directions would be inverted.

For [0/90/45/−45]s laminates (Figures 6.19b and 6.20b), the damping beha-viour is practically symmetric as a function of the fibre orientation with an in-plane shear constrain effect which is more important than in the case of cross-ply laminates, leading to a reduction of the damping near 45° orientation, for modes 1 and 2: loss factor of about 0.98 % for glass fibre laminates and 2.2 % for Kevlar fibre laminates in the case of mode 1.

In the case of the [θ/−θ/θ/−θ]s angle ply laminates and for the first three modes (Figures 6.19c and 6.20c), the damping for ply angles higher than 60° is practically the same as damping observed for the unidirectional beams with fibre orientation equal to θ. For lower values of ply angle, it is observed a reduction of laminate damping comparatively to the unidirectional composites, associated to the in-plane constrain effect induced by the [θ/−θ/θ/−θ]s sequence. For mode 4 the damping reduction of angle ply laminates is observed for all the ply orientations, except for orientations near 0° and 90° where angle ply laminates are similar to unidirectional laminates.

6.5 DAMPING OF LAMINATED PLATES

6.5.1 Damping investigation

The damping of rectangular laminated plates with different edge conditions can be evaluated using the Ritz method. The results obtained in the case of glass fibre plates and in the case of Kevlar fibre plates are very similar and differ by the levels of the damping of plate vibrations. Figures 6.21 and 6.22 show the results derived for the damping by the Ritz analysis for the first four modes in the case of rectangular plates of glass fibre laminates with two edge conditions: one clamped edge and the other edges free (Figure 6.21) and two adjacent edges clamped and the other two free (Figure 6.22). The plates were clamped along the width and

184 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

FIGURE 6.21. Damping variation as a function of laminate orientation for rectangular plates with one edge clamped and the other edges free, in the case of different glass fibre laminates: a) unidirectional laminates, b) [0/90/0/90]s cross-ply laminates, c) [0/90/45/−45]s laminates and d) [θ/−θ/θ/−θ]s angle ply laminates.

the investigation was performed on plates 200 mm wide and 300 mm long with a nominal thickness of 2.4 mm. The experimental results obtained are also reported for the first mode and show a good agreement with the results derived from the Ritz analysis.

6.5.2. Plates with One Edge Clamped and the Other Edges Free

The mode shapes of the unidirectional plates with one clamped edge and the other edges free are rather similar for the different orientations of fibres. Figure 6.23 shows an example obtained for the shapes of the first four modes of uni-directional plates in the case of 15° fibre orientation.

Modes 1 and 3 correspond to plate bending in the x and y directions of plates. So the damping variation of plates corresponding to these modes (Figure 6.21a) are comparable to the damping variation observed in the case of the first and second modes of the unidirectional beams considered in Subsection 6.3.3 (Figure 6.16a). The difference between the results is induced by the effects of the length-

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8mode 1mode 2mode 3mode 4mode 1

Ritz analysis

Experimental results

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

mode 1mode 2mode 3mode 4mode 1

Ritz analysis

Experimental results

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

mode 1mode 2mode 3mode 4mode 1

Ritz analysis

Experimental results

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

mode 1mode 2mode 3mode 4mode 1

Ritz analysis

Experimental results

(a)

(c) (d)

(b)

6.5. Damping of Laminated Plates 185

FIGURE 6.22. Damping variation as a function of laminate orientation for rectangular plates with one edge clamped and the other edges free, in the case of different Kevlar fibre laminates: a) unidirectional laminates, b) [0/90/0/90]s cross-ply laminates, c) [0/90/45/−45]s laminates and d) [θ/−θ/θ/−θ]s angle ply laminates.

FIGURE 6.23 Flexural mode shapes of unidirectional plate with one edge clamped and the others free for 45° fibre orientation.

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

0.4

0.6

0.8

1.0

1.2

1.4mode 1mode 2mode 3mode 4mode 1

Ritz analysis

Experimental results

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

0.4

0.6

0.8

1.0

1.2

1.4

mode 1mode 2mode 3mode 4mode 1

Ritz analysis

Experimental results

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

0.4

0.6

0.8

1.0

1.2

1.4

mode 1mode 2mode 3mode 4mode 1

Ritz analysis

Experimental results

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (

%)

0.4

0.6

0.8

1.0

1.2

1.4

mode 1mode 2mode 3mode 4mode 1

Ritz analysis

Experimental results

(b) (a)

(c) (d)

mode 1 mode 2

mode 3 mode 4

186 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

to-width ratio: 10 in the case of the beams and 1.5 in the case of the plates. The influence of the in-plane shear is lower for fibre orientations near 45° in the case of the plates.

For modes 2 and 4, it is observed (Figure 6.23) an important twisting of plates which leads to similar effects as the ones observed in the case of unidirectional beams: plate twisting induces an increase of damping for fibre orientations near the material directions and a decrease of damping for intermediate orientations (Figure 6.21a).

For [0/90/0/90]s and [0/90/45/−45]s plates the shape modes are similar to the ones observed in the case of unidirectional plates (Figure 6.23) and it is observed (Figures 6.21b and 6.21c) a more symmetric variation of damping as function of the fibre orientation. Lastly, damping variation of [θ/−θ/θ/−θ]s angle ply plates (Figure 6.21d) is practically the same as the variation obtained in the case of unidirectional plates (Figure 6.21a).

6.5.3 Plates with Two Edges Clamped and the Other Edges Free

Figures 6.24 give examples of the mode shapes of unidirectional plates with two adjacent edges clamped for three fibre orientations: 0°, 30° and 60°. The mode shapes combine bending vibrations along the two free edges and the figures show an evolution of the mode shapes with fibre orientation.

For mode 1 it is observed (Figure 6.22a) a rather high damping induced by the plate twisting for fibre orientation near 0° (Figure 6.24a) and next the damping decreases regularly when the fibre orientation increases. The mode shape of the mode 2 is rather similar for the different fibre orientations resulting in a low varia-tion of damping. For mode 3 plate damping is nearly constant for fibre orient-tations from 0° to 40° and next plate damping is clearly increased up to 90° fibre orientation. Lastly, the shapes of mode 4 (Figures 6.24) show an important twisting of the plates for 0° and 30° (Figures 6.24a and 6.24b), inducing a high damping of the plates for fibre orientations from 0° to 40°. Then, the damping decreases according to a mode shape with low plate twisting (Figure 6.24b).

The shape modes of [0/90/0/90]s and [0/90/45/−45]s plates with two adjacent edges clamped are very similar to the ones obtained in the case of the unidi-rectional plates. As it was observed previously for beams and plates, Figures 6.22b and 6.22c show a more symmetric variation of damping with fibre orient-tation. Also, the damping variation of angle ply laminates (Figure 6.22d) is rather similar to the variation observed in the case of unidirectional plates.

6.5. Damping of Laminated Plates 187

FIGURE 6.24. Flexural mode shapes of unidirectional plate with two adjacent edges clamped and the others free for three fibre orientation: a) 0° orientation, b) 30° orientation and c) 60° orientation.

mode 1 mode 2

mode 3 mode 4

mode 1 mode 2

mode 3 mode 4

mode 1 mode 2

mode 3 mode 4

(a)

(b)

(c)

188 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

6.6 LONGITUDINAL AND TRANSVERSE DAMPING OF UNIDIRECTIONAL FIBRE COMPOSITES

6.6.1 Introduction

Longitudinal and transverse damping of unidirectional composites as functions of the constituent properties has been considered in Section 5.2 of Chapter 5. This section develops an analysis of the longitudinal and transverse damping based on the experimental results obtained by Adams, Bacon and Ni [38, 39] in the case of glass fibre and carbon fibre composites and the experimental results reported in the previous sections in the case of glass fibre and Kevlar fibre composites. The experimental results obtained for the engineering constants and damping pro-perties of matrices, fibres and composites are reported in Tables 6.6. and 6.7.

TABLE 6.6. Unidirectional glass and carbon fibre composites investigated by Adams, Bacon and Ni [38, 39].

Materials Vf EL

(GPa) ET

(GPa) GLT

(GPa) νLT

ηL

(%) ηT

(%) ηLT

(%)

Matrix DX 210 − 3.21 3.21 1.20 0.338 1.04 1.04 1.06

Carbon fibres − 345 15 − − − − −

Glass fibres − 73 73 − 0.22 − − −

Carbon fibre/DX 210 composites 0.50 172.7 7.20 3.76 0.29 0.072 0.67 1.12

Glass fibre/DX 210 composites 0.50 37.78 10.90 4.91 0.32 0.138 0.804 1.10

TABLE 6.7. Unidirectional glass and Kevlar fibre composites investigated in the previous sections.

Materials Vf EL

(GPa) ET

(GPa) GLT

(GPa) νLT

η11,ηL

(%) η22,ηT

(%) η66,ηLT

(%)

Matrix SR 1500 − 2.80 2.80 1.08 0.33 1.57 1.57 −

Glass fibres − 73 73 30 0.22 − − −

Kevlar fibres − 135 − 12 0.37 − − −

Glass fibre/SR 1500 composites 0.39 29.9 5.85 2.45 0.26 0.35 1.30 1.80

Kevlar fibre/SR 1500 composites 0.37 50.7 4.50 2.10 0.33 1.50 2.50 3.80

6.6. Longitudinal and Transverse Damping of Unidirectional Fibre Composites 189

6.6.2 Longitudinal Damping

In the case of unidirectional composites with polymer matrix and glass or carbon fibres, the participation of fibres to longitudinal composite damping is generally neglected in the literature, expressing the composite damping by expres-sion (5.7). This assumption results from a negligible value of the damping mea-sured on glass and carbon in bulk form. However, the experimental results obtained on the damping properties (Tables 6.6 and 6.7) show that expression (5.7) considerably underestimates the experimental values of the longitudinal damping. The effect of the fibre-matrix interface was considered by Vantomme [56] introducing a three-phase model: the matrix, the fibre and the interphase between fibres and matrix. The results obtained show that, in the case of a low stiffness interphase, the introduction of the interphase material does not affect the longitudinal damping of composite appreciably. Very few papers on the damping of Kevlar-fibre composites are reported in the literature [57-59]. However, in contrast to carbon or glass fibre composites, it is considered that Kevlar fibres contribute to the longitudinal damping of composites.

In fact, the experimental results obtained for all the types of unidirectional composites (Tables 6.6 and 6.7, for examples) lead to consider that the results obtained for the longitudinal damping can be described simply by introducing a damping parameter fLη , associated to the longitudinal motion of the fibres in a viscoelastic matrix and no more associated to the effective damping of the bulk material. In this way, expression (5.5) can be simply rewritten as

( )f mf f m f1L L

L L

E EV VE E

η η η= + − . (6.23)

The value of the longitudinal fLη cannot be measured. This value can only be estimated from the experimental results obtained from the longitudinal damping of composites, considering Expression (6.23). Table 6.8 reports the results obtained in the case of the composites considered in Tables 6.6 and 6.7. In Table

TABLE 6.8. Longitudinal damping of fibres and respective contributions, deduced from expression (6.23), in the case of composites materials considered in Tables 6.6 and 6.7.

Materials η11,ηL

(%) ηfL (%)

Matrix contribution (%)

Fibre contribution (%)

Glass fibre/DX 210 composites 0.138 0.12 26 74

Carbon fibre/DX 210 composites 0.072 0.062 13.5 86.5

Glass fibre/SR 1500 composites 0.35 0.17 26 74

Kevlar fibre/SR 1500 composites 1.50 1.65 3.4 96.6

190 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

6.8, also are reported the respective contributions of matrix and fibres to the effective damping of composites. The results obtained show that the participation of the matrix to the damping is rather low for all the composites considered, that is in contrast with considerations in literature which neglect the fibre contribution. In the case of Kevlar fibre composites, composite damping is induced essentially by the longitudinal damping of fibres.

Further to the preceding results, it is interesting to consider how the damping of unidirectional composites is changed as a function of fibre fraction. The influence of the volume fraction of fibres was studied experimentally by Adams et al. [31] in the case of glass-fibre and carbon-fibre composites with a polyester matrix. Figure 6.25 shows the experimental results for the longitudinal Young’s modulus deduced from bending tests. These experimental values are less than that predic-ted by the law of mixtures for the longitudinal tensile modulus. The authors attribute this fact to the lower longitudinal compressive modulus than the tensile modulus of the composites. The experimental results of Figure 6.25 can be described using a law of mixtures with lower value of the fibre modulus than the actual one: fE = 61 GPa for the glass-fibre composites and fE = 345 GPa for the carbon-fibre composites. The variation of the specific damping capacity Lψ ( 2L Lψ πη= ) with fibre volume fraction is reported in Figure 6.26 for glass and carbon fibre composites. The experimental values seem to lead to a damping somewhat independent of the fibre type. In Figure 6.26, damping evaluated by Equation (6.25) is reported by considering a specific damping capacity of fibres:

fLψ = 1.2 % ( fLη = 0.19 %) for the glass-fibre composites and fLψ = 1.6 % ( fLη = 0.25 %) for the carbon-fibre composites. These results lead to a good evaluation of the variation of composite damping with fibre volume fraction for glass-fibre composites when they are significantly distant from the experimental results for carbon-fibre composites. For high longitudinal modulus of fibres, Expression (6.23) leads to a composite damping which results essentially from fibre damping since low values of the fibre fraction. Effects of the fibre-matrix interaction, depending on the fibre fraction, could be introduced in Expression (6.23) and would lead to a better description of damping of carbon-fibre composites.

6.6.3 Transverse Damping

6.6.3.1 Formulation

The description of the transverse damping by expression (5.8), which is based on the evaluation of the energies dissipated in matrix and fibres, leads to a contri-bution of fibre damping which is negligible. Moreover for glass-fibre composites, expression (5.8) gives a composite damping which is clearly higher than the experimental results and this expression cannot be considered to evaluate the transverse composite damping.

A new model for evaluating the transverse damping has been developed in [60]. This model introduces local damping coefficients of the composite consti-tuents and considers that the transverse damping coefficient ηfT of fibres is

6.6. Longitudinal and Transverse Damping of Unidirectional Fibre Composites 191

FIGURE 6.25. Variation of the longitudinal Young’s modulus with fibre volume fraction: a) in the case of unidirectional glass-fibre composites and b) in the case of unidirectional carbon-fibre composites, from Ni and Adams [31].

Fibre volume fraction Vf

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Long

itudi

nal m

odul

us E

L (G

Pa)

0

10

20

30

40

50

experiment

Law of mixtures Ef = 61 GPa

(a)

Fibre volume fraction Vf

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Long

itudi

nal m

odul

us E

L (G

Pa)

0

40

80

120

160

200

240

experiment

Law of mixtures Ef = 345 GPa

(b)

192 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

FIGURE 6.26. Variation of the longitudinal specific damping capacity with fibre volume fraction for unidirectional glass-fibre and carbon-fibre composites [31].

associated to the transverse motion of fibres in the viscoelastic matrix. From this model [60], the transverse specific damping capacity is expressed as:

m fT T Tψ ψ ψ= + , (6.24) with

( ) ( ) ( )3 2 2mm m f f f f2

m f f

9 31 1 14 2

T T TT f

E E E EV V V V VE E E

ψ ψ ⎡ ⎤= − + − + −⎢ ⎥⎣ ⎦

, (6.25)

( ) ( )23 2 ff f f f f f2f m m

3 31 12 4

TT TT T f

E EE EV V V V VE E E

ψ ψ ⎡ ⎤= + − + −⎢ ⎥⎣ ⎦

. (6.26)

where mTψ and fTψ are the transverse contributions of matrix and fibre, respec-tively. These contributions are expressed as functions of the specific damping capacity of matrix mψ and the transverse damping capacity of fibre fTψ by Equations (6.25) and (6.26), respectively.

6.6.3.2 Application

1. Glass fibre composites

The application of expressions (6.25) and (6.26) to the case of the unidirec-tional glass fibre composites considered in Table 6.7 leads to:

m m f f0.501 , 6.20 .T T Tψ ψ ψ ψ= = (6.27)

Considering the matrix damping of Table 6.7: m m 2 1.57 %η ψ π= = , the

Fibre volume fraction Vf

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Long

itudi

nal d

ampi

ng ψ

L (%

)

0

1

2

3

4

5

6

7

8

9

glass-fibre composites

carbon-fibre composites experiment

experimentmodelling

modelling

6.6. Longitudinal and Transverse Damping of Unidirectional Fibre Composites 193

transverse damping of the composites is given by:

f0.782 6.20T Tη η= + . (6.28)

The transverse damping measured on the composites (Table 6.7) is 1.30 %Tη = . Hence, the preceding relation shows that matrix takes part in the transverse damping for about 60 %. Moreover, the relation leads to an evaluation of transverse damping induced by fibres equal to:

f f0.083 %, 0.52 %T Tη ψ= = . (6.29)

This damping is about 2 times lower than the fibre damping evaluated in the longitudinal direction (Table 6.8).

2. Kevlar fibre composites

Although the transverse behaviour of composites with Kevlar fibres is not well understood, we applied the preceding analysis to Kevlar composites (Table 6.7). The application of Expressions (6.25) and (6.26) leads to: m m f f0.413 , 8.43 .T T Tψ ψ ψ ψ= = (6.30)

So, the transverse damping is expressed as: f0.649 8.43T Tη η= + . (6.31)

The damping measured in the case of Kevlar composites is 2.50 %Tη = (Table 6.7). It results that matrix takes part in the transverse damping for about 26 %. Then, relation (6.26) leads to the evaluation of the transverse damping induced by fibres f f0.22 %, 1.38 %T Tη ψ= = . (6.32)

This damping is about 7.5 times lower than the fibre damping evaluated in fibre direction (Table 6.8).

3 Variation of transverse damping with fibre volume fraction

The variation of the transverse damping with fibre volume fraction was approached by Ni and Adams [61] in the case of the unidirectional glass-fibre composites. However, the analysis is limited since only two fibre volume fractions are considered: Vf = 0.50 and Vf = 0.78 and the results considered are not very consistent. So, an experimental analysis have been implemented in the case of the glass fibre composites considered previously, for three fibre volume fractions: Vf = 0.20, Vf = 0.39 and Vf = 0.59. The experimental results obtained for the transverse modulus and the transverse damping are reported in Figures 6.27 and 6.28, respectively.

The transverse modulus as a function of the fibre volume fraction can be evaluated considering Expression (5.9). The results obtained are reported in Figure 6.27 which shows a good fitting with the experimental results.

Next, the transverse damping capacity of composites was evaluated using expression (6.24). A good fitting was obtained by taking a transverse damping of fibres:

194 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

Fibre volume fraction Vf

0.0 0.2 0.4 0.6

Tran

sver

se d

ampi

ng ψ

T (%

)

0

2

4

6

8

10

12

Experimental results

Modelling

Fibre volume fraction Vf

0.0 0.2 0.4 0.6

Tran

sver

se m

odul

us E

T (G

Pa)

0

10

Experimental resultsModelling

f f0.49 %, 0.078 %T Tψ η= = . (6.33)

The results obtained are reported in Figure 6.28. The micromechanics model used to evaluate the transverse composite damping

is based on a transverse damping of fibres which is considered as constant with the fibre fraction. A more extended experimental analysis as a function of fibre volume fraction would be necessary to derive complementary elements on this hypothesis.

FIGURE 6.27. Variation of the transverse modulus with fibre volume fraction for unidirectional glass-fibre composites.

FIGURE 6.28. Variation of the transverse specific damping capacity with fibre volume fraction for unidirectional glass-fibre composites.

6.7. Temperature Effect on the Damping Properties of Unidirectional Composites 195

6.7 TEMPERATURE EFFECT ON THE DAMPING PROPERTIES OF UNIDIRECTIONAL COMPOSITES

6.7.1 Introduction

Few papers have been considered in literature about the temperature effect. Maheri et al. [62] compared the effect of the temperature on the stiffness and the damping of two composites with thermoplastic matrices and two composites with thermosetting matrices reinforced by carbon fibres. The properties were derived from the flexural vibrations of free-free beams, for temperatures going from –200 °C to 300 °C, depending on the glass transition temperature of the materials. Only fibre orientations equal to 0° and 90° were considered. The experimental results obtained show that the dynamic properties are appreciably kept near to the temperature of glass transition, where damping increases sharply in a low interval of temperature.

Works on the effect of temperature, were also achieved recently by Benchek-chou et al. [63], and Gibson [64], then by Zhang et al. [65]. The results obtained confirm the experimental analysis of Maheri et al. [62].

Melo and Radford [66] studied the evaluation of time and temperature on the viscoelastic properties of unidirectional reinforced laminates and cross-ply lami-nates with carbon fibres.

Recently, works have been implemented by Sefrani and Berthelot [67] to study the temperature effect on damping and bending modulus, for temperatures higher than room temperature, of glass fibre composites with low glass transition tempe-rature of the matrix. Some of the results are reported in this section.

6.7.2 Materials and Experiment

The experimental study was achieved in the case of glass fibre composites con-sidered in Sections 6.2 and 6.3.

To analyze the temperature effect, tests were carried out over a range of tempe-rature from room temperature using a regulated oven. The beam specimen was placed in a clamping block inside the oven and the excitation of the flexural vibrations of the specimen beam was induced by using a light carbon rod fixed to the output of a vibration electromagnetic exciter. The beam response was detected across a glass window by using a laser vibrometer and the analysis was imple-mented by the procedure developed in Section 6.2.4.

Figure 6.29 reports the frequency response of the specimen beams obtained for three different temperatures in the case of 30° fibre orientation. These responses show peaks which correspond to the natural frequencies of the flexural vibrations of the beams. For temperature (25 °C) near room temperature the material damping is low and the frequency peaks are clearly separated. In this case the damping can be deduced from the half-power bandwidth. The frequency bandwidth increases when the test temperature is increased. For high temperature the damping is important and the frequency response of a given peak has an

196 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

Frequency (HZ)

0 50 100 150 200

Am

plitu

de (d

B)

-200

-180

-160

-140

-120

-100

-80

-60

25 °C

50 °C

70 °C

Temperature (°C)

20 30 40 50 60 70 80 90 100

Stru

ctur

al d

ampi

ng η

(%

)

0

10

20

30

40

50

60

70

80

Temperature (°C)

20 30 40 50 60 70 80 90 100

Ben

ding

mod

ulus

Efx

(G

Pa)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

FIGURE 6.29. Temperature influence on the frequency responses of unidirectional glass fibre composites, for 30° fibre orientation. influence on the nearest peaks. So, for the analysis of the frequency response it is necessary to analyze the beam response with the procedure considered in Section 6.2.4.

6.7.3. Experimental results

6.7.3.1. Matrix Properties Figure 6.30 reports the experimental results deduced from the flexural vibra-

tions of matrix beams, as function of the temperature. A damping peak is obser-ved for the glass transition temperature around 80°C, when the stiffness of the matrix decreases in the range of the temperatures considered.

FIGURE 6.30. Variations of the bending modulus and damping of the matrix with temperature.

6.7. Temperature Effect on the Damping Properties of Unidirectional Composites 197

6.7.3.2 Composite Properties

The experimental results obtained for the variations in the frequency of the bending modulus and damping of the glass fibre composites are given for differ-rent temperatures in Figures 6.31 to 6.33, for three fibre orientations: 0° (Figure 6.31), 45° (Figure 6.32) and 90° (Figure 6.33). For a given fibre orientation and a given temperature, it is observed an increase of the bending modulus and an increase of damping with frequency within the domain under consideration.

The variation of the bending modulus with temperature is then given in Figures 6.34 to 6.36, for a frequency of 100 Hz and for the three fibre directions. From the present results it is observed a significant shift in the value of the temperature of the damping peak with respect to fibre orientation: 100°C for 0° fibre orientation, 90°C for 45° orientation and 80°C for 90° orientation. So it seems that the value of glass transition temperature of composites depends on fibre orientation. This phenomenon was observed by Maheri et al. [62] in the case of PEEK composites. Furthermore, in polymers shear damping is greater than damping in tension-compression and polymer damping depends on the fractions of tension-com-pression energy and shear energy. So in a composite, the structural configuration that generates the maximum fraction of total strain energy due to shear will produce the greatest damping. It results that the shift in the damping peak with changing fibre orientation may be due to the change in strain energy distribution.

The E-glass fibres keep their mechanical characteristics up to temperatures of the order of 200°C. For 0° fibre direction the bending modulus deduced from the bending vibrations of beams is identical to the longitudinal modulus of the unidirectional materials. This modulus can be evaluated using the law of mixtures. The results obtained are reported in figure 6.34 until 90°C considering the variation of the matrix modulus with temperature reported in figure 6.30. In fact the participation of the matrix to the longitudinal modulus is very low. The results obtained in the range of temperatures considered are very similar to the experimental results. Then, when the temperature is increased, it is observed a sharp decrease of the bending modulus around 90-100°C up to temperatures about 120°C. In this range of temperatures the law of mixture is no more verified, indicating that the matrix transfers only partially the mechanical loading to the fibres. Next, for temperatures higher than 120°C, it is observed a stabilisation of the bending modulus.

For 90° fibre direction the bending modulus of beams gives the transverse modulus of the unidirectional materials. This modulus can be evaluated using the procedure considered in [1, 2]. The results obtained are reported in figure 6.36. The results are similar to the experimental results in the range of temperatures considered.

The variation of the bending modulus of the beams with fibre orientation can be derived from Expression (3.12) and relations reported in Table 1.1 of Chapter 1. The results obtained for a frequency of 100 Hz are reported in Figure 6.36 and show a good estimation of the experimental results.

198 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

FIGURE 6.31. Experimental results obtained a) for the bending modulus and b) for the damping, as functions of the frequency and for different temperatures, in the case of glass fibre composites with 0° fibre orientation.

Frequency (Hz)

0 100 200 300 400 500

Ben

ding

mod

ulus

Efx

(GPa

)

0

5

10

15

20

25

30

35

25 °C 50 °C

70 °C 90 °C

100 °C

120 °C

130 °C

150 °C

110 °C

Frequency (Hz)

0 100 200 300 400 500

Loss

fact

or η

(%)

0

2

4

6

8

10

12

14

25 °C50 °C

70 °C

90 °C

100 °C

110 °C

120 °C

130 °C

150 °C

(a)

(b)

6.7. Temperature Effect on the Damping Properties of Unidirectional Composites 199

Frequency (Hz)

0 100 200 300

Ben

ding

mod

ulus

Efx

(GPa

)

0

2

4

6

8

10

12

14

16

50 °C

70 °C

90 °C100 °C

25 °C

Frequency (Hz)

0 100 200 300

Loss

fact

or η

(%)

0

10

20

30

40

50

60

25 °C50 °C

70 °C

90 °C

100 °C

FIGURE 6.32. Experimental results obtained a) for the bending modulus and b) for the damping, as functions of the frequency and for different temperatures, in the case of glass fibre composites with 45° fibre orientation.

(a)

(b)

200 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

0 100 200 300 4000

1

2

3

4

5

6

7

8

25 °C B

endi

ng m

odul

us E

fx (

GPa

)

Frequency (Hz)

50 °C

70 °C

90 °C 100 °C

Frequency (Hz)

0 100 200 300 400

Loss

fact

or η

(%)

0

10

20

30

40

50

60

70

25 °C50 °C

70 °C

90 °C

100 °C

FIGURE 6.33. Experimental results obtained a) for the bending modulus and b) for the damping, as functions of the frequency and for different temperatures, in the case of glass fibre composites with 90° fibre orientation.

(a)

(b)

6.7. Temperature Effect on the Damping Properties of Unidirectional Composites 201

Temperature (°C)

20 40 60 80 100 120 140 160

Ben

ding

mod

ulus

Efx

(GPa

)

0

5

10

15

20

25

30

35

ExperienceLaw of mixtures

Temperature (°C)

20 40 60 80 100 120 140 160

Stru

ctur

al d

ampi

ng η

(%)

0

1

2

3

4

5

6

7

8

Temperature (°C)

20 40 60 80 100 120

Ben

ding

mod

ulus

Efx

(GPa

)

0

2

4

6

8

10

12

14

16

Temperature (°C)

20 40 60 80 100 120

Stru

ctur

al d

ampi

ng η

(%)

0

10

20

30

40

50

60

Temperature (°C)

20 40 60 80 100 120

Ben

ding

mod

ulus

Efx

(GPa

)

0

1

2

3

4

5

6ExperienceEvaluation [1, 2]

Temperature (°C)

20 40 60 80 100 120

Stru

ctur

al d

ampi

ng η

(%)

0

10

20

30

40

50

FIGURE 6.34. Variations of the stiffness and damping of the glass fibre composites as functions of the temperature, for a frequency of 100 Hz and for 0° fibre orientation.

FIGURE 6.35. Variations of the stiffness and damping of the glass fibre composites as functions of the temperature, for a frequency of 100 Hz and for 45° fibre orientation.

FIGURE 6.36. Variations of the stiffness and damping of the glass fibre composites as functions of the temperature, for a frequency of 100 Hz and for 90° fibre orientation.

202 Chapter 6. Experimental Investigation and Discussion on Damping Properties of Laminates

FIGURE 6.37. Damping evaluation as a function of fibre orientation deduced from model-ling using the Ritz method, for a frequency of 100 Hz, and different temperatures.

6.7.3.3 Damping Evaluation Based on the Ritz Method

The Ritz method was applied to the experimental results obtained for the bending of beams at different temperatures. The beams were considered in the form of plates with one edge clamped and the others free. Thus, the present eva-luation of the beam damping takes account of the effect of the beam width. Figure 6.37 gives the results obtained for different temperatures and shows that the expe-rimental results are fairly well described using the Ritz method. The values used for the loss factors 11 22,η η and 66η are reported in Table 6.9 as functions of temperature.

TABLE 6.9. Loss factors 11 22,η η and 66η of glass fibres laminates as functions of tempe-rature.

Temperature (°C) 25 50 60 70 90 100

( )11 %η 0.35 0.60 1.50 2.00 4.50 6.30

( )22 %η 1.30 5.00 20.5 35.0 45.0 23.0

( )66 %η 1.80 6.50 25.0 45.0 60.0 30.0

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Stru

ctur

al d

ampi

ng η

(%)

0

10

20

30

40

5090 °C

70 °C

100 °C60 °C

50 °C

25 °C

experiencemodellingexperiencemodelling

25 °C

50 °C

60 °C experiencemodellingexperiencemodellingexperiencemodellingexperiencemodelling

70 °C

90 °C

100 °C

CHAPTER 7

Damping Analysis of Laminates with Interleaved Viscoelastic Layers

7.1 INTRODUCTION

Constrained damping layers in isotropic metallic materials have been investi-gated in literature and the results obtained show that the layers provide significant higher damping than the initial materials. In the same way, inserting viscoelastic layers in laminates improves significantly the damped dynamic properties of the laminates. Moreover, the interlaminar damping concept is highly compatible with the fabrication processes of laminated structures.

Limited analytical and experimental papers on the analysis of composite damping with viscoelastic layers have been reported in literature [68-73]. Sara-vanos and Pereira [68] develop a discrete-layer laminate theory for analysing the damping of composite laminates with interlaminar damping layers. Experimen-tally measured and predicted dynamic responses of graphite epoxy plates with co-cured damping layers are compared to illustrate the accuracy of the theory. Liao et al. [69] analyse the vibration-damping behaviour of unidirectional and symmetric angle-ply laminates as well as their interleaved counterparts with a layer of PEAA (polyethylene-co-acrylic acid) at the mid-plane. The introduction of the PEAA layer significantly improves the damping capability of laminates. The experi-mental results are compared with the results obtained by extending to laminate materials the evaluation of damping performances derived by Liao and Hsu [70] in the case of conventional constrained-layer configuration: two isotropic outer layers and a thin viscoelastic interlayer. Shen [71] proposes a hybrid damping design which consists of a viscoelastic layer sandwiched between piezoelectric constraining cover sheets. The active damping component produces significant and adjustable damping, when the passive component increases gain. A first order shear deformation theory is used by Cupial and Niziol [72] to evaluate the natural frequencies and loss factors of a rectangular three-layered plate with a viscoelastic core layer and laminated faces. Simplified forms are discussed in the case of symmetric plate and for orthotropic faces. Comparison is made between the present shear deformation theory and simplified models. More recently the

204 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

damping behaviour of a 0° laminated sandwich composite beam inserted with a viscoelastic layer was investigated by Yim et al. [73]. It is shown that the Ni-Adams theory [39] for evaluating the damping of laminate beams can be extended to evaluate the damping characteristics of laminated sandwich composite beams. Results show the capability of laminated sandwich composites with embedded viscoelastic layer to significantly enhance laminate damping.

A finite element for predicting modal damping of thick composite and sandwich beams was developed by Plagianakos and Saravanos [74]. Previous linear layerwise formulations [75, 76] provided the basis for developing a discrete-layer higher order theory satisfying compatibility in interlaminar shear stress and modal damping was calculated by modal strain energy dissipation method. The effect of ply orientation of composite beams with interply visco-elastic damping layers was investigated. Experimental investigation of modal damping illustrated the accuracy of the developed formulation.

The purpose of this chapter is to extend the analysis of laminate damping deve-loped in Section 5.5 of Chapter 5 to the case of the damping analysis of laminated plates with viscoelastic layers. Modelling was developed in [77] and experimental investigation was implemented in [78].

7.2 DAMPING MODELLING OF ORTHOTROPIC LAMINATES WITH INTERLEAVED

VISCOELASTIC LAYERS

7.2.1 Laminate Configurations

Two types of laminates with viscoelastic layers were considered: laminates with a single viscoelastic layer of thickness e0 interleaved in the middle plane of laminates (Figure 7.1) and laminates with two viscoelastic layers of thickness e0 interleaved away from the middle plane (Figure 7.2). The layers of the initial laminates are constituted of unidirectional or orthotropic materials with material directions making an angle θ with the x direction oriented along the length of plates under consideration. The total thickness of the unidirectional or orthotropic layers is e and the interlaminar layers are assumed to have an isotropic behaviour. This section develops an analysis of damping of rectangular plates.

7.2.2 In-plane Damping with Interleaved Viscoelastic Layers

7.2.2.1 Case of a Single Interlaminar Viscoelastic Layer

The laminate is constituted of a unidirectional or orthotropic material of thick-ness e in which a single viscoelastic layer of thickness e0 is interleaved (Figure 7.1). Material directions make an angle θ with the plate directions.

7.2. Damping Modelling of Orthotropic Laminates with Interleaved Viscoelastic Layers 205

FIGURE 7.1. Laminate with a single viscoelastic layer.

FIGURE 7.2. Laminate with two interleaved viscoelastic layers.

According to the results established in Section 5.5 of Chapter 5, the total strain energy stored in the laminate with the viscoelastic layers can be expressed as

11 12 22 662U U U U U= + + + , (7.1) with ort v , 11, 12, 22, 66.pq pq pqU U U pq= + = (7.2)

The energies U11 and U22 are the strain energies stored in tension-compression in the material directions, U12 is the coupling energy induced by the Poisson’s effect,

viscoelastic layer 2

3

1 orthotropic layer

orthotropic layer 2h 2e

2e

0e

viscoelastic layer 2

3

1 orthotropic layer

orthotropic layer 2 2 2

ed α=

0e

0e

1 1 2ed α=

middle plane

0 02eh e⎛ ⎞= − +⎜ ⎟

⎝ ⎠

1 1 02eh eα⎛ ⎞= − +⎜ ⎟

⎝ ⎠

2 1 2eh α= −

3 2 2eh α=

4 2 02eh eα= +

5 02eh e= +

viscoelastic layer 4

5 orthotropic layer

z

206 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

and U66 is the strain energy stored in in-plane shear. Each energy is separated as the strain energy ort

pqU stored in the orthotropic layers and the strain energy vpqU

stored in the viscoelastic layer. Applying the results obtained in Section 5.5, the energy stored in the orthotropic layers 1 and 3 (Figure 7.1) can be written as:

( )3 33

ort 1 3 0 02

1 1 1 1

1 1 ,122

11, 12, 22, 66.

M N M N

pq pq pq mn ij pq pqm n i j

e e eU U U A A f Qe eRa

pq

θ= = = =

⎡ ⎤⎛ ⎞ ⎛ ⎞= + = + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

=

∑∑∑∑

(7.3)

where a is the length and R the length-to-width ratio of the plate, 1pqU and 3

pqU are the strain energies stored in layers 1 and 3, Qpq are the reduced stiffness constants of the materials and fpq(θ) are functions of the material directions introduced in Section 5.5.

In the same way, the strain energy stored in the viscoelastic layer is given by:

( )

3v 2 v v 0

v21 1 1 1

1 ,122

11, 12, 22, 66,

M N M N

pq pq mn ij pq pqm n i j

eU U A A f QRa

pq

θ= = = =

= =

=

∑∑∑∑ (7.4)

where the reduced stiffness constants vpqQ are expressed as:

( )

2 2

v2 2

01 1

0 ,1 1

0 02 1

pq

E E

E EQ

E

νν ν

νν ν

ν

⎡ ⎤⎢ ⎥− −⎢ ⎥⎢ ⎥⎡ ⎤ = ⎢ ⎥⎣ ⎦ − −⎢ ⎥⎢ ⎥⎢ ⎥+⎣ ⎦

(7.5)

by introducing the Young’s modulus E and the Poisson ratio ν of the viscoelastic layer. This layer being considered as isotropic, the results are independent of the direction. Thus the function ( )v

vpqf θ can be deduced by considering an orientation equal to zero, which leads:

( ) ( )

( ) ( )

v 2200 v 2002 211 v 12 v

v 0022 4 v 1111 222 v 66 v

, ,

, 4 ,

minj minj

minj minj

f C f C R

f C R f C R

θ θ

θ θ

= =

= = (7.6)

In the case of a viscoelastic layer with a low thickness, Expression (7.3) shows that the in-plane strain energy stored in the interleaved laminate is practically the same as the strain energy stored in the material without the viscoelastic layer.

7.2. Damping Modelling of Orthotropic Laminates with Interleaved Viscoelastic Layers 207

The energy dissipated by viscous damping is then expressed by:

( )ort ort v v , 11, 12, 22, 66,pq pq pq pqpq

U U U pqψ ψ∆ = + =∑ (7.7)

introducing the specific damping coefficients ortpqψ of the orthotropic material

considered and the coefficients vpqψ of the viscoelastic layer. The damping

coefficients ortpqψ are the in-plane damping coefficients of the orthotropic layer

considered in Chapters 5 and 6. They will be simply noted pqψ . The damping of the viscoelastic layer is related essentially to the Young’s modulus and it can be written:

v v v v11 22 66 v 12, 0.ψ ψ ψ ψ ψ≈ ≈ ≈ = (7.8)

Next, the damping of laminated plate with a single viscoelastic layer is evaluated by:

UU

ψ ∆= . (7.9)

7.2.2.2 Case of Two Interlaminar Viscoelastic Layers

This subsection considers the case of a unidirectional or orthotropic material of thickness e in which two viscoelastic layers of thicknesses e0 are interleaved in the initial material (Figure 7.2). So as to obtain a general analysis, the viscoelastic layers are considered to be interleaved at distances d1 and d2 from the middle plane, respectively. These distances will be expressed as:

1 1 2 2, .2 2e ed dα α= = (7.10)

As previously, the strain energy stored in the laminate with the two interleaved viscoelastic layers can be expressed by Relations (7.1) and (7.2) where ort

pqU is the

strain energy stored in the orthotropic layers and vpqU is the strain energy stored

in the two viscoelastic layers. The strain energy stored in the orthotropic layers 1, 3 and 5 (Figure 7.2) can be expressed as:

( )ort 1 3 5 ort

21 1 1 1

1 ,2

11, 12, 22, 66,

M N M N

pq pq pq pq mn ij pq pqm n i j

U U U U A A f DRa

pq

θ= = = =

= + + =

=

∑∑∑∑ (7.11)

with

3 3 3 3

ort 3 30 0 01 2 1 22 1 2 2 2 .

24pq pqe e e eD Qe e e

α α α α⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + + − + − +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ (7.12)

208 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

The strain energy stored in the two viscoelastic layers 2 and 4 (Figure 7.2) can be written as:

( )v v vv2

1 1 1 1

1 , 11, 12, 22, 66,2

M N M N

pq mn ij pq pqm n i j

U A A f D pqRa

θ= = = =

= =∑∑∑∑ (7.13)

with

( )3 3 3

3v v0 01 2 1 22 2 ,

24pq pqe e eD Qe e

α α α α⎡ ⎤⎛ ⎞ ⎛ ⎞= + + + − +⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

(7.14)

where the functions ( )vvpqf θ are given by Expressions (7.6). In the particular

case of two viscoelastic layers which are interleaved at the same distance from the middle plane:

1 2 2ed d α= = , (7.15)

and the bending stiffness induced by the viscoelastic layers are simply written as:

3 3

v 3 v0212pq pq

e eD Qe

α α⎡ ⎤⎛ ⎞= + −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

. (7.16)

7.2.3 Considering the Transverse Shear Effects in the Case of a Single Interlaminar Viscoelastic Layer

7.2.3.1 Introduction

Applying the previous results obtained for the in-plane damping shows that the analysis does not describe the experimental results obtained for damping in the case where one or two viscoelastic layers are interleaved. Indeed the in-plane energy stored in the viscoelastic layers is too low. This observation shows that the energy dissipation is induced by another process, which leads to consider the transverse shear effects induced in the viscoelastic layers.

The classical laminate theory which is considered in the previous analyses does not take account of the transverse shear effects induced in laminates. However, the classical laminate theory allows us to evaluate the transverse shear stresses in layers and the analysis developed in the following subsections is based on this concept.

7.2.3.2 Transverse Shear Stresses in Layers

We consider again the case of a laminate constituted of a unidirectional or orthotropic layer of thickness e in which a viscoelastic layer of thickness e0 is interleaved (Figure 7.1). The total thickness of the laminate is:

0h e e= + . (7.17)

7.2. Damping Modelling of Orthotropic Laminates with Interleaved Viscoelastic Layers 209

The in-plane stresses in the orthotropic layers are given by the relations:

ort11 12 16

12 22 26

16 26 66

,xx xx

yy yy

xy xy

Q Q QQ Q QQ Q Q

σ εσ εσ γ

′ ′ ′⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥′ ′ ′=⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥′ ′ ′⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

(7.18)

where the elements ijQ′ are the reduced stiffness constants of the materials expres-sed in the (x, y) directions of the plate, which are deduced from the reduced stiffness constants Qij in the material directions, according to the relations reported in Table 1.1 of Chapter 1. In the same way, the in-plane stresses in the viscoelastic layer are written as:

v vv11 12v v12 22

v66

0

0 ,

0 0

xx xx

yy yy

xy xy

Q Q

Q Q

Q

σ εσ εσ γ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

(7.19)

where the reduced stiffness constants are given by Relation (7.5). The in-plane strains are expressed as functions of the transverse displacement

by the following relations:

2 2 2

0 0 02 2, , 2 .xx yy xyz z z

x yx yε ε γ∂ ∂ ∂

= − = − = −∂ ∂∂ ∂

w w w (7.20)

Thus, the in-plane stresses in the orthotropic layers are written as:

2 2 2

ort 0 0 011 12 162 2 2 ,xx z Q Q Q

x yx yσ

⎛ ⎞∂ ∂ ∂′ ′ ′= − + +⎜ ⎟∂ ∂∂ ∂⎝ ⎠

w w w (7.21)

2 2 2

ort 0 0 012 22 262 2 2 ,yy z Q Q Q

x yx yσ

⎛ ⎞∂ ∂ ∂′ ′ ′= − + +⎜ ⎟∂ ∂∂ ∂⎝ ⎠

w w w (7.22)

2 2 2

ort 0 0 016 26 662 2 2 ,xy z Q Q Q

x yx yσ

⎛ ⎞∂ ∂ ∂′ ′ ′= − + +⎜ ⎟∂ ∂∂ ∂⎝ ⎠

w w w (7.23)

and the in-plane stresses in the viscoelastic layer are:

2 2

v v v0 011 122 2 ,xx z Q Q

x yσ

⎛ ⎞∂ ∂= − +⎜ ⎟

∂ ∂⎝ ⎠

w w (7.24)

2 2

v v v0 012 222 2 ,yy z Q Q

x yσ

⎛ ⎞∂ ∂= − +⎜ ⎟

∂ ∂⎝ ⎠

w w (7.25)

2

v v 0662 .xy z Q

x yσ ∂

= −∂ ∂w (7.26)

210 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

The classical laminate theory neglects the transverse shear effects. However, the transverse shear stresses in the laminate layers can be derived from the fundamental equations of motion which can be evaluated, neglecting the inertia terms, by using the fundamental equations:

0, ort, v,ii ixyxx xz i

x y zσσ σ∂∂ ∂

+ + = =∂ ∂ ∂

(7.27)

0, ort, v.i i ixy yy yz ix y z

σ σ σ∂ ∂ ∂+ + = =

∂ ∂ ∂ (7.28)

1. xz-shear

The first equation (7.27) leads to:

ort, v,ii ixyxz xx i

z x yσσ σ ∂∂ ∂

= − − =∂ ∂ ∂

(7.29)

which yields for the unidirectional or orthotropic layers:

( )ort

ort , ,xzxzA x y z

zσ∂

=∂

(7.30)

with

( )3 3 3 3

ort 0 0 0 011 12 66 16 263 2 2 32 3 .xzA Q Q Q Q Q

x x y x y y∂ ∂ ∂ ∂′ ′ ′ ′ ′= + + + +∂ ∂ ∂ ∂ ∂ ∂w w w w (7.31)

Integrating Relation (7.30), the transverse shear stress in the unidirectional or orthotropic layers is written as:

( )ort ort 2ort

1 , .2xz xzA x y z Cσ = + (7.32)

Similarly, the transverse shear in the viscoelastic layer is given by:

( )v v 2v

1 , .2xz xzA x y z Cσ = + (7.33)

with

( )3 3

v v v v0 011 12 663 22 .xzA Q Q Q

x x y∂ ∂

= + +∂ ∂ ∂w w (7.34)

The constants ortC and vC in each layer are determined by considering the continuity of the transverse shear stress at the interfaces between the viscoelastic layer and the orthotropic layers and that the transverse shear stress vanishes on the lower and upper faces of the laminate:

v ort0 02 2xz xze eσ σ⎛ ⎞ ⎛ ⎞± = ±⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠, (7.35)

7.2. Damping Modelling of Orthotropic Laminates with Interleaved Viscoelastic Layers 211

ort 02xzhσ ⎛ ⎞± =⎜ ⎟

⎝ ⎠. (7.36)

These conditions lead to:

( )2

ort ort 21 , ,2 4xz xz

hA x y zσ⎛ ⎞

= −⎜ ⎟⎝ ⎠

(7.37)

( ) ( )( )2

v v 2 ort 2 200

1 1, , .2 4 8xz xz xz

eA x y z A x y h eσ⎛ ⎞

= − − −⎜ ⎟⎝ ⎠

(7.38)

2. yz-shear Equation (7.28) leads to:

, ort, v.i i iyz yy xy iz y x

σ σ σ∂ ∂ ∂= − − =

∂ ∂ ∂ (7.39)

Compared to the xz-shear, the x coordinate is changed for y and the y coordinate is changed for x. Similarly, the subscript 1 of the reduced stiffness constants is changed for 2 and the subscript 2 for 1.

7.2.3.3 Strain Energy Stored in xz-Transverse Shear

The strain energy stored in xz-transverse shear by volume unit can be evaluated by the relation:

21 , ort, v,

2

ii xzxz i

xzu i

Gσ

= = (7.40)

in which ixzG is the xz-transverse shear modulus of the layers.

1. Unidirectional or orthotropic layers According to Relations (7.37) and (7.40), the transverse shear strain energy is

expressed as:

( )22

ort ort2 21 1 , ,2 4 4xz xz

xz

hu A x y zG

⎛ ⎞= −⎜ ⎟

⎝ ⎠ (7.41)

where xzG is the xz-transverse shear modulus in the θ direction, expressed as a function of the transverse moduli LTG and TTG ′ in the material directions:

2 21 1 1cos sin .xz LT TTG G G

θ θ′

= + (7.42)

The laminates considered are symmetric and the total strain energy stored in the unidirectional or orthotropic layers is given by:

0

2ort ort

0 02

2 d d d .ha b

xz xzex y zU u x y z

= = == ∫ ∫ ∫ (7.43)

212 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

Combining Relations (7.41) and (7.43) leads to the expression of the total strain energy which can be written in the form:

( )

5 5 3ort ort 2

0 0

1 151 , d d ,240 4 10 3 2

a bh h h

xz xzx yxz

h r r rU A x y x yG = =

⎡ ⎤⎛ ⎞= − − +⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦ ∫ ∫ (7.44)

where the ratio

0h

erh

= (7.45)

has been introduced and the coefficient ( )ort 2 , xzA x y is expressed as:

( ) ( )

( )

( )

2 2 23 3 32ort 2 2 20 0 0

11 12 66 163 2 2

23 3 3 3 32 0 0 0 0 0

26 11 12 66 11 163 3 2 3 2

3 30 0

11 26 12 66 163 3

, 2 9

2 2 6

2 6 2

xzA x y Q Q Q Qx x y x y

Q Q Q Q Q Qy x x y x x y

Q Q Q Q Qx y

⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂′ ′ ′ ′= + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞∂ ∂ ∂ ∂ ∂′ ′ ′ ′ ′ ′+ + + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

∂ ∂ ∂′ ′ ′ ′ ′+ + +∂ ∂

w w w

w w w w w

w w

( )

3 30 02 2

3 3 3 30 0 0 0

12 66 26 16 262 3 2 32 2 6 . (7.46)

x y x y

Q Q Q Q Qx y y x y y

∂∂ ∂ ∂ ∂

∂ ∂ ∂ ∂′ ′ ′ ′ ′+ + +∂ ∂ ∂ ∂ ∂ ∂

w w

w w w w

Finally, considering Relations (7.44) and (7.46), the total strain energy stored in the unidirectional or orthotropic layers can be written as:

( ) ( )5 5 3

ort ort4

1 1 1 1

1 151 ,240 4 10 3 2

M N M Nh h h

xz mn ij xzxz m n i j

h r r rU A A FG R a

θ θ= = = =

⎡ ⎤⎛ ⎞= − − +⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦∑∑∑∑

(7.47) where the function ( )ort

xzF θ is expressed as:

( ) ( )

( )

( ) ( )

2ort 2 3300 1122 4 2 2211 2 2 0033 611 12 66 16 26

3102 2 3201 3003 311 12 66 11 16 11 26

1221 3 1012 66 16 12 66 26

2 9

2 2 6 2

6 2 2 2

xz minj minj minj minj

minj minj minj

minj minj

F Q C Q Q C R Q C R Q C R

Q Q Q C R Q Q C R Q Q C R

Q Q Q C R Q Q Q C

θ ′ ′ ′ ′ ′= + + + +

′ ′ ′ ′ ′ ′ ′+ + + +

′ ′ ′ ′ ′ ′+ + + +

23 5

2013 416 266 .minj

R

Q Q C R′ ′+

(7.48)

The coefficients pqrsminjC have been introduced in Section 5.5 of Chapter 5 using

the Ritz method, for p, q, r, s = 0, 1 and 2, to evaluate the integrals of the form:

2 2

0 0

0 0d d

a b

p q r sx yx y

x y x y= =

∂ ∂∂ ∂ ∂ ∂∫ ∫ w w . (7.49)

7.2. Damping Modelling of Orthotropic Laminates with Interleaved Viscoelastic Layers 213

The formulation can be extended to the present analysis for p, q, r, s equal to 3. Thus, the coefficients pqrs

minjC are expressed as:

,pqrs pq rsnjminj miC I J= (7.50)

introducing the dimensionless integrals:

1

0

, 1, 2 . . . ,d d d , , 0 ,1, 2, 3,d d

p qpq m imi p q

m i MX XI up qu u

==

=∫ (7.51)

1

0

d , 1, 2 . . . ,d d ,, 0, 1, 2, 3,d d

srjrs n

nj r sY n j NYJ v

r sv v=

==∫ (7.52)

where X(x) and Y(y) are the functions introduced in Expression (4.24) of Chapter 4 which gives the transverse displacement at point (x, y) of the laminated plate. The integrals pq

miI and rsnjJ are calculated using the reduced coordinates u = x/a

and v = y/b, where a and b are the length and the width of the plate, respectively.

2. Viscoelastic layer

The transverse shear strain energy stored by volume unit in the viscoelastic layer is given by:

2

v 1 ,2

xzxzu

Gσ

= (7.53)

where G is the shear modulus of the viscoelastic layer:

( )

v66 .

2 1EG Q

ν= =

+ (7.54)

Introducing Expression (7.38) of the transverse shear stress, the transverse shear strain energy can be rewritten as:

( ) ( )( )22

v v 2 ort 2 200

1 1 1, , .2 4 4 4xz xz xz

eu A x y z A x y h eG

⎡ ⎤⎛ ⎞= − − −⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦ (7.55)

The total strain energy stored in the viscoelastic layer is then obtained by the relation:

02v v

0 0 02 d d d

ea b

xz xzx y z

U u x y z= = =

= ∫ ∫ ∫ , (7.56)

which leads to :

214 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

( )

( ) ( )

( ) ( ) ( )

5v v 20

0 0

522 ort 2

0 0

52 3 v ort

0 0

1 , d d4 60

1 , d d32

1 , , d d .24

a b

xz xzx y

a b

h h xzx y

a b

h h xz xzx y

eU A x y x yG

hr r A x y x y

hr r A x y A x y x y

= =

= =

= =

⎡= ⎢

⎣

+ −

⎤+ − ⎥

⎦

∫ ∫∫ ∫∫ ∫

(7.57)

The coefficient ( )ort 2 , xzA x y is expressed by (7.46) and the two other coefficients are given by

( ) ( )

( )

2 23 32v 2v2 v v0 012 6611 3 2

3 3v v 0 0

12 12 66 3 2

, 2

2 2 ,

xz

v

A x y Q Q Qx x y

Q Q Qx x y

⎛ ⎞ ⎛ ⎞∂ ∂= + +⎜ ⎟ ⎜ ⎟

∂ ∂ ∂⎝ ⎠ ⎝ ⎠

∂ ∂+ +

∂ ∂ ∂

w w

w w (7.58)

and

( ) ( )

( )

( )

v ort

23 3 3v v0 0 011 11 11 12 663 3 2

3 3 3 3v v0 0 0 011 16 11 263 2 3 3

3 3v v 0 012 66 11 122 3

, ,

2

3 (7.59)

2

xz xzA x y A x y

Q Q Q Q Qx x x y

Q Q Q Qx x y x y

Q Q Q Qx y x

=

⎛ ⎞∂ ∂ ∂′ ′ ′+ +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

∂ ∂ ∂ ∂′ ′+ + +∂ ∂ ∂ ∂ ∂

∂ ∂′+ + +∂ ∂ ∂

w w w

w w w w

w w ( )( )

( ) ( )

23v v 0

66 12 66 2

3 3 3 3v v v v0 0 0 012 66 16 12 66 262 2 2 3

2 2

3 2 2 .

Q Q Qx y

Q Q Q Q Q Qx y x y x y y

⎛ ⎞∂′ ′+ + ⎜ ⎟∂ ∂⎝ ⎠

∂ ∂ ∂ ∂′ ′+ + + +∂ ∂ ∂ ∂ ∂ ∂ ∂

w

w w w w

As considered previously, the integrals in Expression (7.57) can be evaluated considering the Ritz method, and the expression of the total shear strain energy stored in the viscoelastic layer can be expressed in the form:

( ) ( ) ( ) ( )

( ) ( )

5 2v 5 v 2 v1 24

1 1 1 1

32 v

3

1 160 324

1 ,24

M N M Nh h

xz mn ij xz h xzm n i j

hh xz

r rU h A A F r FG R a

rr F

θ θ θ

θ

= = = =

⎡= + −⎢

⎣

⎤+ − ⎥⎦

∑∑∑∑ (7.60)

where the functions vxziF are given by:

( ) ( ) ( )2v2 v v v v vv 3300 1122 4 3102 21 11 12 66 11 12 662 2 2 ,xz minj minj minjF Q C Q Q C R Q Q Q C Rθ = + + + + (7.61)

7.2. Damping Modelling of Orthotropic Laminates with Interleaved Viscoelastic Layers 215

( ) ( )v ort2 ,xz xzF Fθ θ= (7.62)

( ) ( ) ( )( ) ( )

( ) ( )

v v v vv 3300 3102 23 11 12 66 1111 11 12 66

v v v v3201 3003 3 1122 416 26 12 6611 11 12 66

v v v v1221 3 1023 516 2612 66 12 66

2 2

3 2 2

3 2 2 .

xz minj minj

minj minj minj

minj minj

F Q Q C Q Q Q Q Q Q C R

Q Q C R Q Q C R Q Q Q Q C R

Q Q Q C R Q Q Q C R

θ ⎡ ⎤′ ′ ′ ′= + + + +⎣ ⎦

′ ′ ′ ′+ + + + +

′ ′+ + + + (7.63)

7.2.3.4 Strain Energy Stored in yz-Transverse Shear

Compared to the xz-transverse shear (previous subsection), the x coordinate is changed for y coordinate, the y coordinate is changed for x coordinate and the directions 1 and 2 are interchanged.

7.2.3.5 Laminate Damping with a Single Interleaved Viscoelastic Layer Including the Transverse Shear Effects

The total strain energy stored in the laminate with a viscoelastic layer can be written as: ort v ort ort v v

p p .xz yz xz yzU U U U U U U= + + + + + (7.64)

The strain energy ortpU is the in-plane strain energy stored in the orthotropic

layers deduced from expression (7.3). This energy can be written as:

ort ort ort ort ortp 11 12 22 662 .U U U U U= + + + (7.65)

The in-plane strain energy vpU stored in the viscoelastic layer is deduced from

Expression (7.4). The strain energies stored in transverse shear have been evalua-ted in the preceding subsections.

The specific damping coefficient ψ(θ) of the laminate can thus be evaluated by the relation: p p s s

v ort vort( )ψ θ ψ ψ ψ ψ= + + + , (7.66) where

( )p ort ort ort ort11 11 12 12 22 22 66 66ort

1 2U U U UU

ψ ψ ψ ψ ψ= + + + , (7.67)

p

p vv v

UU

ψ ψ= , (7.68)

( )s ort ort ort ortort

1 ,xz xz yz yzU UU

ψ ψ ψ= + (7.69)

v v

sv v .xz yzU U

Uψ ψ

+= (7.70)

216 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

These expressions introduce the specific damping coefficients ortxzψ and ort

yzψ characterising the transverse shear energy dissipated in the unidirectional or orthotropic layers. For unidirectional materials these coefficients can be assi-milated with the in-plane shear coefficient:

ort ort66xz yzψ ψ ψ= = . (7.71)

7.2.4 Considering the Transverse Shear Effects in the Case of Two Interlaminar Viscoelastic Layer

7.2.4.1 Case of a Symmetric Laminate

We consider the case of a unidirectional or orthotropic material in which two viscoelastic layers are interleaved at a distance d from the middle plane (Figure 7.3). The thicknesses of the viscoelastic layers are e0 and the total thickness of the orthotropic layers is e. Moreover, it will be noted:

( )00 1 0, 2 , 2 ,

2 2 2ee e e ed h e e d d e r

eα α α⎛ ⎞= = + = + = + = +⎜ ⎟

⎝ ⎠ (7.72)

introducing the thickness ratio:

02e

ere

= . (7.73)

FIGURE 7.3. Laminate with two viscoelastic layers interleaved at the same distance from the middle plane.

z

viscoelastic layer 2

3

1 orthotropic layer

orthotropic layer2ed α=

0e

0e 2ed α=

middle plane

viscoelastic layer 4

5 orthotropic layer

1d

1d 2h

2h

7.2. Damping Modelling of Orthotropic Laminates with Interleaved Viscoelastic Layers 217

Thus, the laminate is symmetric and constituted for the upper part by a unidi-rectional or orthotropic layer for 0 z d≤ ≤ , a viscoelastic layer for 1d z d≤ ≤ and a unidirectional or orthotropic layer for 1 2d z h≤ ≤ .

7.2.4.2. Transverse Shear Stresses in the (x, z) Plane

The transverse shear stresses in the (x, z) plane are expressed again by Expres-sions (7.32) and (7.33) in the unidirectional or orthotropic layers and the visco-elastic layers, respectively. These stresses have to satisfy the vanishing conditions on the laminate faces and the continuity conditions at the interfaces between the viscoelastic layers and the orthotropic layers:

( ) ( )

( ) ( )

ort

ort v

v ort1 1

0,2

,

.

xz

xz xz

xz xz

h

d d

d d

σ

σ σ

σ σ

⎛ ⎞± =⎜ ⎟⎝ ⎠

± = ±

± = ±

(7.74)

Applied to the upper half of the laminate, these conditions lead to:

— upper unidirectional or orthotropic layer, 1 2d z h≤ ≤ :

( )2

ort ort 21 , ,2 4xz xz

hA x y zσ⎛ ⎞

= −⎜ ⎟⎝ ⎠

(7.75)

— viscoelastic layer, 1d z d≤ ≤ :

( ) ( ) ( )2

v v 2 2 ort 21 1

1 1, , ,2 2 4xz xz xz

hA x y z d A x y dσ⎛ ⎞

= − − −⎜ ⎟⎝ ⎠

(7.76)

— lower unidirectional or orthotropic layer, 0 z d≤ ≤ :

( ) ( )[ ]2

ort ort 2 v0

1 1, 1 2 1 .2 4 2 2

exz xz e xz

e rA x y z r A eeσ α α⎧ ⎫⎪ ⎪ ⎛ ⎞= − + − − +⎨ ⎬ ⎜ ⎟

⎝ ⎠⎪ ⎪⎩ ⎭ (7.77)

7.2.4.3. Transverse Shear Energies in the (x, z) Plane

1. Energy stored in the upper and lower orthotropic layers

The total strain energy stored in the orthotropic layers by transverse shear effects in the (x, z) plane is given by:

1

2ort ort1

0 02 d d d ,

ha b

xz xzx y z d

U u x y z= = =

= ∫ ∫ ∫ (7.78)

where ortxzu is the strain energy stored in the volume unit of the layers:

218 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

ort 2

ort ,2

xzxz

xzu

Gσ

= (7.79)

the transverse shear stress being expressed by Relation (7.75). Thus, the strain energy stored in the upper and lower layers is given by:

5 5 3

ort ort 21 1 11

0 0

1 1 60 d d ,240 5 6 16

a bd d d

xz xzx yxz

h r r rU A x yG = =

⎡ ⎤⎛ ⎞= − − +⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦ ∫ ∫ (7.80)

in which the following ratio is introduced:

1 01

1 .2 1

ed

e

d d e rrh h r

α+ += = =

+ (7.81)

Using the Ritz method, the strain energy (7.80) is written as:

( )5 5 3

ort ort1 1 11 4

1 1 1 1

1 1 60 ,240 5 6 16

M N M Nd d d

xz mn ij xzxz m n i j

h r r rU A A FG R a

θ= = = =

⎡ ⎤⎛ ⎞= − − +⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦∑∑∑∑ (7.82)

where the function ( )ortxzF θ was introduced in (7.48).

2. Energy stored in the viscoelastic layers

The total strain energy stored in the viscoelastic layers by transverse shear effects in the (x, z) plane is expressed as:

1v v

0 02 d d d ,

a b d

xz xzx y z d

U u x y z= = =

= ∫ ∫ ∫ (7.83)

in which vxzu is the strain energy stored in the volume unit of the viscoelastic

layers:

v2

v ,2

xzxzu

Gσ

= (7.84)

the transverse shear stress being expressed by Relation (7.76). Hence, the strain energy (7.83) is given by:

( )

( ) ( ) ( )

( ) ( ) ( )

5 v 5 5 3 v 2

1 1 1 1 0 0

2 2 3 v ort11 1 1

0 0

22 ort 21 1 1

0 0

1 8 15 21 , 4 15 8 3

1 31 4 1 1 , , 3 2 3

1 1 4 1 , 16

a b

xz d xzx y

a b

d d xz xzx y

a b

d d xzx y

hU r r r r A x y dx dyG

rr r r A x y A x y dx dy

r r r A x y dx dy

= =

= =

= =

⎧ ⎡ ⎤⎛ ⎞= − − +⎨ ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎩⎡ ⎤⎛ ⎞

+ − − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎫+ − − ⎬

∫ ∫∫ ∫

∫ ∫ ,⎭

(7.85) where

7.2. Damping Modelling of Orthotropic Laminates with Interleaved Viscoelastic Layers 219

11

.e

drd r

αα

= =+

(7.86)

Considering the Ritz method, the strain energy (7.85) can be expressed as:

( )( ) ( )

( ) ( ) ( )

v 5 5 5 3 v1 1 1 1 14

1 1 1 1

32 3 v11 1 1 3

22 ort1 1 1

1 8 15 2115 8 34

1 31 4 13 2 3

1 1 , (7.87)

M N M N

xz mn ij d xzm n i j

d d xz

d d xz

U h A A r r r r FGR a

rr r r F

r r r F

θ

θ

θ

= = = =

⎡ ⎤⎛ ⎞= − − +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞+ − − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

+ − −

∑∑∑∑

where the functions ( )v1xzF θ and ( )v

3xzF θ have been introduced in (7.61) and (7.63), respectively.

3. Energy stored in the middle unidirectional or orthotropic layer

The evaluation of the total strain energy stored in the middle orthotropic layer by transverse shear effects in the (x, z) plane leads to:

( )( )

( )( ) ( )

( )( )[ ] ( ) ( )

5ort 5

2 4 21 1 1 1

ort4

224 v v

3 1

1 10 11 1 1160 34

15 1 1

3 1 1 1 .24 2 8 2

M N M N

xz mn ij exz m n i j

e xz

e e e ee xz xz

U e A A rG R a

r F

r r r rr F F

α αα

α θα

α α α θ α α θ

= = = =

⎧⎪ ⎡= − + −⎨ ⎢⎣⎪⎩

⎤+ + − ⎥⎦⎫⎪⎛ ⎞ ⎛ ⎞+ + + − − + + ⎬⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎪⎭

∑∑∑∑

(7.88)

7.2.4.4. Transverse Shear Energies in the (y, z) Plane

The transverse shear energies stored by shear effects in the (y, z) plane are transposed from Expressions (7.78) to (7.88) by changing the functions xzF for the functions yzF deduced from Relations (7.61) and (7.63) by introducing the correspondence of Subsection 7.2.3.4.

7.2.4.5. Laminate Damping with Two Interleaved Viscoelastic Layers Including the Transverse Shear Effects

The total strain energy stored in the laminate with two interleaved viscoelastic layers can be written as:

ort v ort ort v v ort ortp p 1 1 2 2.xz yz xz yz xz yzU U U U U U U U U= + + + + + + + (7.89)

220 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

The strain energy ortpU is the in-plane strain energy stored in the orthotropic

layers expressed in Subsection 7.2.2.2. This energy can be written in the form (7.65) introducing the energies:

ort 1 3 5 , 11, 12, 22, 66,pq pq pq pqU U U U pq= + + = (7.90)

given by Relation (7.11). The in-plane strain energy vpU stored in the viscoelastic

layers is given by Relation (7.13). The strain energies stored in transverse shear have been evaluated in the preceding subsections.

The specific damping coefficient of the laminate can thus be evaluated by Expression (7.66) introducing the coefficients:

( )p ort ort ort ort11 11 12 12 22 22 66 66ort

1 2U U U UU

ψ ψ ψ ψ ψ= + + + (7.91)

p

p vv v

UU

ψ ψ= , (7.92)

( ) ( )s ort ort ort ort ort ortort 1 2 1 2

1 ,xz xz xz yz yz yzU U U UU

ψ ψ ψ⎡ ⎤= + + +⎣ ⎦ (7.93)

v v

sv v .xz yzU U

Uψ ψ

+= (7.94)

As previously in the case of a single interleaved viscoelastic layer, the coefficients ortxzψ and ort

yzψ can be assimilated with the in-plane shear damping coefficient of materials (Equation (7.71)).

7.2.5 Application to Angle-Ply Laminates

The previous modelling considered in the case of unidirectional or orthotropic laminates can also be applied to evaluate the damping properties of [ ]θ± angle-ply laminates with interleaved viscoelastic layers. The reduced stiffness constants of θ and –θ layers are related by the expressions:

11 11 12 12 16 16

22 22 26 26 66 66

, , ,, , ,

Q Q Q Q Q QQ Q Q Q Q Q

θ θ θ θ θ θ

θ θ θ θ θ θ

− + − + − +

− + − + − +

′ ′ ′ ′ ′ ′= = = −′ ′ ′ ′ ′ ′= = − =

(7.95)

where ijQ θ+′ are the reduced stiffness constants of θ layers expressed in the beam directions (Table 1.1 of Chapter 1). Compared to the orthotropic laminates, only the stiffness constants 16Q′ and 26Q′ are changed. Thus, the in-plane stresses in the –θ layers are deduced from Expressions (7.21) to (7.25) by changing 16Q′ and

26Q′ by 16Q′− and 26Q′− , respectively. Then, the transverse shear stresses in the laminate layers are derived from Equations (7.27) and (7.28). The integration of the stresses through the thickness of the laminate layers introduces constants

7.2. Damping Modelling of Orthotropic Laminates with Interleaved Viscoelastic Layers 221

which are determined by considering the vanishing conditions of the transverse shear stresses on the laminate faces and the continuity conditions at the interfaces between the viscoelastic layers and the orthotropic layers as well as between θ layers and –θ layers. Next the evaluation of the different strain energies and the laminate damping are derived following the procedure previously implemented in the case of orthotropic layers. The in-plane energies and damping are expressed following the results reported in Section 5.5.2 of Chapter 5. The final results for the damping depend on the type of angle-ply laminates.

More generally, the modelling can be applied to an arbitrary laminate.

7.2.6 Laminates with External Viscoelastic Layers

Increase of the damping capacity of composite laminates can be achieved by the application of damping layers to the surfaces of the laminates after fabrication (Figure 7.4). In this case the laminate damping can be derived by considering the in-plane energies as evaluated in Subsection 7.2.2. The damping of laminates with two external viscoelastic layers of thicknesses 0 2e is given by expressions (7.9) and the strain energies are modified according the following relations:

— strain energy stored in the orthotropic layers:

( )

3ort

21 1 1 1

1 ,122

11, 12, 22, 66,

M N M N

pq mn ij pq pqm n i j

eU A A f QRa

pq

θ= = = =

=

=

∑∑∑∑ (7.96)

FIGURE 7.4. Laminate with two external viscoelastic layers.

viscoelastic layer

orthotropic layer e

0 2e

0 2e

middle plane

viscoelastic layer

z

222 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

— strain energy stored in the viscoelastic layers:

( )

3 3v v v 0

v21 1 1 1

1 1 1 ,122

11, 12, 22, 66.

M N M N

pq mn ij pq pqm n i j

e eU A A f QeRa

pq

θ= = = =

⎡ ⎤⎛ ⎞= + −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

=

∑∑∑∑ (7.97)

The functions pqf are expressed in Section 5.5 of Chapter 5 and the functions vpqf are given by Expressions (7.6).

7.2.7 Choice of the Basis Functions of the Ritz Method

The solution to the governing equations of a problem of structural mechanics has to satisfy the boundary conditions which are separated in two classes: the essential and natural boundary conditions. The essential boundary conditions, also called geometric boundary conditions, correspond to prescribed displacements and rotations. The natural boundary conditions are also called the force boundary conditions and correspond to prescribed forces and moments.

The effect of the natural boundary conditions is included as a potential in the expression of the total potential energy (the functional) of the system under consi-deration. Hence the natural boundary conditions are implicitly contained in the functional, whereas the essential boundary conditions are stated separately. From this important consideration, it results that, in the Ritz method, the trial functions only need to satisfy the essential boundary conditions and not the natural boun-dary conditions. The reason for this relaxed requirement on the trial functions is that the stationarity conditions of the total potential energy minimizes the violation of the internal equilibrium requirements induced by the approximation (4.24) considered for the transverse displacement, and thus minimizes the violation of the natural boundary conditions. Actually, it can be expected that in most cases the approximate solution will be more accurate if the trial functions also satisfy the natural boundary conditions. However, it is observed that it is more effective in some cases to use rather a larger number of functions that only satisfy the essential boundary conditions.

The analysis of the damping which includes the transverse shear effects of the viscoelastic layers, considered in this section, was applied to the experimental results reported in the next section. First, the beam functions (Section 3.4 of Chapter 3), which satisfy the essential and natural boundary conditions, were used as trial functions of the transverse displacement. Considering the results deduced from this analysis leads to questioning on the conditioning of these trial functions. A more extended analysis has shown that the evaluation of the integrals of these beam functions was confronted to singularity problems when some of the integrals Ii3 (i = 0, 1, 2, 3) were calculated. So, this fact leads us to consider polynomials which are a simple and convenient way to construct the trial functions. Thus, we introduced polynomials of the form:

— along the length of beams or plates in the x direction (clamped end-free end or clamped edge-free edge):

7.3. Experimental Investigation of Unidirectional Laminates with Viscoelastic Layers 223

1( ) , , 1,ii

xX u u u ia

+= = ≥ (7.98)

— along the width of beams or plates in the y direction (free edges):

( )11 , , 1.

2

j

jyY jb

−⎛ ⎞= − = ≥⎜ ⎟⎝ ⎠

v v v (7.99)

These functions satisfy the essential boundary conditions of beams or plates. They do not satisfy the natural boundary conditions.

The analysis of damping including the transverse shear effects and considering the previous polynomials shows however a tendency to be ill conditioned when the number of functions used for the transverse displacement is increased. A complete study would need to implement an extensive analysis of the condi-tioning problem. The results reported hereafter in the next sections were obtained by considering 16 terms (M = N = 4) in the transverse displacement series (Equation (4.24)).

7.3 EXPERIMENTAL INVESTIGATION OF DAMPING OF UNIDIRECTIONAL LAMINATES WITH INTERLEAVED VISCOELASTIC LAYERS

7.3.1 Materials

The materials investigated are the unidirectional glass fibre composites, consi-dered in Section 6.2 of Chapter 6, in which a single or two viscoelatic layers were interleaved. The volume fraction of fibres is equal to 0.40 and the nominal thickness e of the unidirectional layers is 2.4 mm. The engineering constants and the modal loss factors η (related to the specific damping coefficients ψ by relation:

2ψ πη= ) referred to the material directions were evaluated in Section 6.2 and are reported in Table 7.1. The viscoelastic layers are constituted of Neoprene based layers of nominal thickness e0 = 0.2 mm. Three types of laminates have been investigated: a laminate with a single viscoelastic layer of thickness e0 interleaved in the middle plane (Figure 7.1), a laminate with a single viscoelastic layer of thickness 2e0 in the middle plane and a laminate with two viscoelastic layers of

TABLE 7.1. Properties of the glass fibre composites without viscoelastic layers.

EL (GPa)

ET (GPa)

GLT (GPa)

νLT

η11, ηL (%)

η22, ηT (%)

η66, ηLT (%)

29.9 5.85 2.45 0.24 0.40 1.50 2.00

224 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

thickness e0 interleaved at the distance e/2 from the middle plane (Figue 7.2). Plates were hand laid up and cured at room temperature with a pressure of 70 kPa using vacuum moulding process.

In the case of a single interleaved viscoelastic layer, the nominal thickness of the laminates is 2.6 mm with a weight of 3.7 kgm–2. Interleaving two viscoelastic layers leads to a laminate thickness of 2.8 mm with a weight of 3.85 kgm–2. Alu-minium spacers of the same thicknesses as the viscoelastic layers were added in the root section between the unidirectional layers. Beam specimens were next cut from the plates and the damping properties were measured for different orient-tations of the fibres.

7.3.2 Experimental Results

The material damping was deduced from impulse tests using the experimental procedure described in Section 6.2 of Chapter 6. The experimental evaluation of damping was performed on beams 20 mm wide of different lengths: 160, 180 and 200 mm, so as to have a variation of the values of the natural frequencies of the beams. Only the first two modes were considered. Figures 7.5, 7.6 and 7.7 report the experimental results obtained for the beam damping as function of the fibre orientation for the three beam lengths (Figures 7.5a, 7.6a and 7.7a) and the beam damping as function of the frequency for the different fibre orientations (Figures 7.5b, 7.6b and 7.7b). The results are given in the case of laminates with a single viscoelastic layer of thickness e0 = 0.2 mm (Figure 7.5), laminates with a single viscoelastic layer of thickness e0 = 0.4 mm (Figure 7.6) and laminates with two viscoelastic layers of thicknesses e0 = 0.2 mm (Figure 7.7). The experimental results obtained for the unidirectional materials without viscoelastic layers are also reported (Figures 7.5a, 7.6a and 7.7a). The experimental results show that the damping of laminates increases significantly upon interleaving a single or two viscoelastic layers. The fibre orientation dependence of damping appears some-what similar to that of the laminates without viscoelastic layers, but with a damping maximum which is moved from 60° fibre orientation to 30° fibre orient-tation when viscoelastic layers are interleaved. Moreover in contrast to the non-interleaved laminates, damping increases significantly with frequency depending on the vibration mode. In the case of a single viscoelastic layer interleaved in the middle plane, the laminate damping is increased all the more since the viscoelastic layer is thick. The damping of laminate with two interleaved viscoelastic layers of thicknesses e0 is lower (about 1.6 time) than the one measured in the case of laminate with a single viscoelastic layer of thickness 2e0, when the damping is fairly similar to the damping of laminate with a single layer of thickness e0. This results from the fact that the energy is essentially dissipated by transverse shear of the viscoelastic layers and the associated energy is maximum in the middle plane of laminates.

7.3. Experimental Investigation of Unidirectional Laminates with Viscoelastic Layers 225

FIGURE 7.5. Experimental results obtained in the case of glass fibre composites with a single viscoelastic layer of thickness 0.2 mm interleaved in the middle plane and for three lengths of the test specimens: a) laminate damping as function of the fibre orientation and b) laminate damping as function of the frequency.

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%

)

0

1

2

3

4

5

61st mode2nd mode1st mode2nd mode1st mode2nd modewithout viscoelastic layer

l = 160 mm

l = 180 mm

l = 200 mm

Frequency (Hz)0 100 200 300 400 500 600

Loss

fact

orη

(%)

1

2

3

4

5

6

θ = 0°

2nd mode

θ = 15°θ = 30°

θ = 45°

θ = 60°

θ = 75°

θ = 90°

θ = 0°

θ = 15°

θ = 30°

θ = 45°

θ = 60°θ = 75°

θ = 90°

1st mode

(a)

(b)

226 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%

)

0

2

4

6

8

101st mode2nd mode1st mode2nd mode1st mode2nd modewithout viscoelastic layer

l = 160 mm

l = 180 mm

l = 200 mm

Frequency (Hz)0 100 200 300 400 500 600

Loss

fact

orη

(%)

1

2

3

4

5

6

7

8

9

θ = 0°

2nd mode

θ = 15°θ = 30°

θ = 45°

θ = 60°

θ = 75°

θ = 90°θ = 0°

θ = 15°

θ = 30°

θ = 45°

θ = 60°

θ = 75°

θ = 90°

1st mode

(a)

(b)

FIGURE 7.6. Experimental results obtained in the case of glass fibre composites with a single viscoelastic layer of thickness 0.4 mm interleaved in the middle plane and for three lengths of the test specimens: a) laminate damping as function of the fibre orientation and b) laminate damping as function of the frequency.

7.3. Experimental Investigation of Unidirectional Laminates with Viscoelastic Layers 227

FIGURE 7.7. Experimental results obtained in the case of glass fibre composites with two viscoelastic layer of thickness 0.2 mm interleaved away from the middle plane and for three lengths of the test specimens: a) laminate damping as function of the fibre orientation and b) laminate damping as function of the frequency.

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%

)

0

1

2

3

4

5

61st mode2nd mode1st mode2nd mode1st mode2nd modewithout viscoelastic layer

l = 160 mm

l = 180 mm

l = 200 mm

Frequency (Hz)0 100 200 300 400 500 600

Loss

fact

orη

(%)

0

1

2

3

4

5

6

θ = 0°

2nd mode

θ = 15°θ = 30°

θ = 45°

θ = 60°

θ = 75°θ = 90°

θ = 0°

θ = 15°

θ = 30°

θ = 45°

θ = 60°θ = 75°

θ = 90°

1st mode

(a)

(b)

228 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

Frequency (Hz)

10 100 1000

You

ng's

mod

ulus

(M

Pa)

35

40

60

50

50 500

70

80

7.3.3 Analysis of the Experimental Results

7.3.3.1 Dynamic Properties of the Viscoelastic Layers

In the case of laminates with interleaved viscoelastic layers, the laminate damping is evaluated by the modelling developed in Section 7.2. This evaluation needs to obtain the values of the Young’s modulus and the loss factors of the viscoelastic layers. These characteristics depend on the frequency and are gene-rally derived according to the standard ASTM E 756 [79]. Following this standard, the damping characteristics of the viscoelastic material were evaluated from the flexural vibrations of a clamped-free beam 10 mm wide and constituted of two aluminium beams with a layer of the viscoelastic material interleaved between the aluminium beams. An aluminium spacer was added in the root section between the two aluminium beams of the test specimens. The roots were machined as part of the aluminium beams to obtain a root section 40 mm long and 10 mm high and then the root section was closely clamped in a rigid fixture. The free length and the thicknesses of the aluminium beams were selected so as to measure the damping characteristics on the frequency range [50, 600 Hz] consi-dered in the case of the experimental analysis of interleaved laminates (Subsection 7.3.2). Thus, the beam dimensions used were a free length varying from 200 to 300 mm, a thickness of the viscoelastic layer of 0.2 mm and thicknesses of aluminium beams of 1 mm. The Young’s modulus of the viscoelastic layer was deduced from the natural frequencies of the test specimens and the loss factor was evaluated by applying the results of the modelling considered in Section 7.2 to the case of the aluminium-viscoelastic layer laminates.

Figures 7.8 and 7.9 report the experimental results obtained, using logarithmic scales for the Young’s modulus and for the frequency. In the frequency range

FIGURE 7.8. Frequency dependence of the Young’s modulus of the viscoelastic layers.

7.3. Experimental Investigation of Unidirectional Laminates with Viscoelastic Layers 229

Frequency (Hz)

10 100 1000

Loss

fact

or η

(%

)

20

22

24

26

28

30

32

34

FIGURE 7.9. Frequency dependence of the loss factor of the viscoelastic layers.

studied, it is observed linear variations for the logarithm of the Young’s modulus and for the loss factor of the viscoelastic material. The results of Figures 7.8 and 7.9 lead to: v vlog 0.106log 1.52, (MPa),E f E= + (7.100)

for the variation of the Young’s modulus of the viscoelastic layer with the frequency, and: v v39.4 5.56log , (%)fη η= − , (7.101)

for the loss factor of the viscoelastic material.

7.3.3.2 Damping of the Glass Fibre Laminates with Interleaved Viscoelastic Layers

The loss factor of the glass fibre laminates with interleaved viscoelastic layers was derived from the modelling (Section 7.2) and the results obtained are com-pared with the experimental results in Figures 7.10, 7.11 and 7.12, for the first two modes of the test specimens:

― in the case of a single interleaved viscoelastic layer 0.2 mm thick, for the different free lengths of the test specimens l = 160 mm (Figure 7.10a), l = 180 mm (Figure 7.10b) and l = 200 mm (Figure 7.10c);

― in the case of a single interleaved viscoelastic layer 0.4 mm thick, for the different free lengths of the test specimens l = 160 mm (Figure 7.11a), l = 180 mm (Figure 7.11b) and l = 200 mm (Figure 7.11c);

― in the case of two interleaved viscoelastic layers 0.2 mm thick, for the different free lengths of the test specimens l = 160 mm (Figure 7.12a), l = 180 mm (Figure 7.12b) and l = 200 mm (Figure 7.12c).

230 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

FIGURE 7.10. Comparison between the experimental results and the results deduced from the modelling, in the case of a single viscoelastic layer 0.2 mm thick, for test specimen lengths of : a) l = 160 mm, b) l = 180 mm, c) l = 200 mm.

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%)

0

1

2

3

4

5

1st mode2nd mode1st mode2nd modewithout viscoelastic layer

modelling

experiment

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%)

0

1

2

3

4

5

1st mode2nd mode1st mode2nd modewithout viscoelastic layer

modelling

experiment

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%)

0

1

2

3

4

5

6

1st mode2nd mode1st mode2nd modewithout viscoelastic layer

modelling

experiment

(a)

(b)

(c)

7.3. Experimental Investigation of Unidirectional Laminates with Viscoelastic Layers 231

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%)

0

1

2

3

4

5

6

7

1st mode2nd mode1st mode2nd modewithout viscoelastic layer

modelling

experiment

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%)

0

1

2

3

4

5

6

7

8

1st mode2nd mode1st mode2nd modewithout viscoelastic layer

modelling

experiment

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%)

0

2

4

6

8

10

1st mode2nd mode1st mode2nd modewithout viscoelastic layer

modelling

experiment

(a)

(b)

(c)

FIGURE 7.11. Comparison between the experimental results and the results deduced from the modelling, in the case of a single viscoelastic layer 0.4 mm thick, for test specimen lengths of : a) l = 160 mm, b) l = 180 mm, c) l = 200 mm.

232 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

FIGURE 7.12. Comparison between the experimental results and the results deduced from the modelling, in the case of two viscoelastic layers 0.2 mm thick, for test specimen lengths of : a) l = 160 mm, b) l = 180 mm, c) l = 200 mm.

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%)

0

1

2

3

4

5

1st mode2nd mode1st mode2nd modewithout viscoelastic layer

modelling

experiment

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%)

0

1

2

3

4

5

1st mode2nd mode1st mode2nd modewithout viscoelastic layer

modelling

experiment

(b)

(c)

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%)

0

1

2

3

4

5

6

1st mode2nd mode1st mode2nd modewithout viscoelastic layer

modelling

experiment

(a)

7.4. Analysis of the Experimental Results Obtained in the Case of Angle-Ply Laminates 233

It is observed that the results deduced from the modelling describe fairly well the experimental damping variation obtained as a function of the fibre orientation. Furthermore the modelling results corroborate that damping of laminates with two viscoelastic layers of thicknesses e0 introduced at the quarters of the thickness of laminates (Figure 7.12) is equal to the damping of laminates with a single visco-elastic layer of thickness e0 interleaved in the middle plane (Figure 7.10).

7.4 ANALYSIS OF THE EXPERIMENTAL RESULTS OBTAINED IN THE CASE OF ANGLE-PLY

LAMINATES

The experimental analysis of the effects of a viscoelastic layer was also carried out by Liao et al. [69] in the case of carbon fibre composites. The results are compared in this section with the results deduced from the modelling.

The laminates studied by Liao et al. are angle-ply laminates stacked into [±θ]3s lay-ups, the layer orientations varying from 0° to 90° in 15° intervals. The nomi-nal thickness of the laminates was 1.50 mm with a density equal to 1560 kg/m3. The properties of the unidirectional layers measured by the authors are reported in Table 7.2. The longitudinal Young’s modulus EL, the longitudinal loss factor ηL, the transverse Young’s modulus ET and the transverse loss factor ηT were deduced from flexural vibration tests of the unidirectional layers with fibre orientations of 0° and 90°, respectively. The longitudinal shear modulus GLT and the loss factor ηLT were derived from the low frequency (1 Hz) torsional response of a unidi-rectional 0° laminate.

The laminates with a viscoelastic layer were fabricated interleaving a layer 0.10 mm in thickness of PEAA (copolymer of ethylene and acrylic acid). The properties of this material were studied by Liao et al. [70], establishing the master curves of the shear modulus and loss factor as functions of the frequency.

The damping properties of the interleaved laminates were derived from flexural vibration testing of cantilever beams. The effective beam length was 180 mm with a width equal to 10 mm. The experimental results obtained for the loss factor of the laminates are shown in figure 7.13. The results obtained for the first three modes are reported for the specimens with ply angle of 0° and 15°, when results from all resonance modes below 1600 Hz are reported except those of the first mode for the other orientations.

Figure 7.14 compares the experimental results obtained by Liao et al. in the case of angle-ply laminates without viscoelastic layers with the results deduced from the modelling considered in Section 5.5.2 of Chapter 5. The best fitting

TABLE 7.2. Properties measured on the unidirectional layers of carbon fibre composites (Liao et al. (69)).

EL (GPa)

ET (GPa)

GLT (GPa)

νLT

η11, ηL (%)

η22, ηT (%)

η66, ηLT (%)

110 8.6 6.0 0.28 0.14 0.66 0.80

234 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

FIGURE 7.13. Loss factors measured by Liao et al. (69) as function of the layer direction θ : a) in the case of angle-ply [±θ]3s carbon fibre laminates, b) with a single interleaved viscoelastic layer.

between modelling and experimental results is obtained with

11 22 660.18 %, 0.52%, 0.85%.L T LTη η η η η η= = = = = = (7.102)

These values are fairly similar to the ones obtained by Liao et al. (Table 7.2). It is observed that the damping of carbon fibre laminates is notably lower than damping of glass fibre laminates.

Frequency (Hz)

0 200 400 600 800 1000 1200 1400 1600

Loss

fact

or

η (%

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0°15°

30°

45°

60° 75°

90°

(a)

Frequency (Hz)

0 200 400 600 800 1000 1200 1400 1600

Loss

fact

or

η (%

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0°15°

30°

45°

60°75°

90°

(b)

7.4. Analysis of the Experimental Results Obtained in the Case of Angle-Ply Laminates 235

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1st mode 2nd mode 1st mode 2nd mode

modelling

experiment

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%

)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1st mode2nd mode1st mode2nd modewithout viscoelastic layer

modelling

experiment

The modelling considered previously to evaluate the damping of unidirectional laminates with interleaved viscoelastic layers can be extended to the case of angle-ply laminates (Section 7.2.5). The experimental results and the results derived from modelling are compared in Figure 7.15. Modelling results were established considering the frequency dependences of the Young’s modulus and loss factor of the viscoelastic layers obtained by Liao et al. [70]. The good agree-ment between the results of Figure 7.15 confirms the ability for the modelling to evaluate the damping properties of interleaved angle-ply laminates.

FIGURE 7.14. Comparison of the results deduced from the Ritz’s method with the experi-mental results obtained by Liao et al., without viscoelastic layer.

FIGURE 7.15. Comparison of the results deduced from the Ritz’s method with the experi-mental results obtained by Liao et al., with an interleaved viscoelastic layer.

236 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

Fibre orientation (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%

)

0

2

4

6

8

10

12

e0 = 0

e0 = 0.1 mm

e0 = 0.4 mm

e0 = 0.2 mm

e0 = 0.6 mm

e0 = 0.8 mm

e0 = 1 mm

7.5 DISCUSSION

First, this section presents some results deduced from the modelling in the case of cantilever beams considered previously in Section 7.3 when a viscoelastic layer is interleaved in the middle plane. The free length of the beams is 200 mm and the width 20 mm. The characteristics of the unidirectional layers are kept constant: total thickness of 2.4 mm, mechanical and damping properties defined in Table 7.1, and the effects of the properties of the viscoelastic layer are investigated.

The variations of the first modal damping are plotted in Figure 7.16 as functions of the fibre orientation for different values of the thickness e0 of the viscoelastic layer. Figure 7.17 reports the variations of the modal damping and the natural frequency of the beams with the ratio of the thickness of the viscoelastic layer to the total thickness e of the unidirectional layers, using a logarithmic scale for the thickness ratio. For all fibre orientations the modal damping increases with the thickness of the viscoelastic layer, but the highest increase is observed for fibre orientations about 40°. This results from the fact that laminates induce higher interlaminar shear stresses in the interleaved layer for these fibre orientations. Also, it is observed a higher increase of damping for fibre orientations near 0° than for fibre orientations near 90°, which results from higher stiffness of the unidirectional layers for near 0° orientations. Moreover, it is observed that the natural frequency is not changed much for low thicknesses of the viscoelastic layer. Thus, it seems that an optimum viscoelastic layer thickness exists such that viscoelastic layers with less or equal thickness induce significant damping with negligible or low effects on the structure properties.

FIGURE 7.16. Damping of unidirectional laminates with a viscoelastic layer interleaved in middle plane as function of fibre orientation, for different values of the thickness of the viscoelastic layer.

7.5. Discussion 237

FIGURE 7.17. Effect of viscoelastic layer thickness on damping and natural frequency of laminates for fibre orientations equal to 0°, 40° and 90°.

Figure 7.18 presents the variation of the modal damping for the first two modes in the case of laminates with a viscoelastic layer of thickness 0.2 mm, for a Young’s modulus of the layer Ev = 50 MPa and for four values of the layer damping: ηv = 0.10, 0.20, 0.30 and 0.40. In the same way, Figure 7.19 shows the results obtained in the case of a viscoelastic layer with a given damping: ηv = 0.30 and for different values of the Young’s modulus Ev = 30, 75, 100 and 150 MPa.

Col 1 vs Col 2 Col 1 vs Col 3

Col 1 vs Col 4

Thickness ratio e0/e

0.001 0.01 0.1

Nat

ural

freq

uenc

y (

Hz)

20

40

60

80

0.5

θ = 0°

θ = 40°

θ = 90°

Thickness ratio e0/e

0.001 0.01 0.1

Loss

fact

or η

(%

)

0

2

4

6

8

10

12

0.5

θ = 0°

θ = 40°

θ = 90°

238 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%)

0

1

2

3

4

5

6

7

8

1st mode 2nd mode

ηv =0.10

ηv =0.40

ηv =0.20ηv =0.30

ηv =0.10

ηv =0.20

ηv =0.30

ηv =0.40

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%

)

0

2

4

6

81st mode 2nd mode

Ev = 30 MPa

Ev =75 MPa

Ev = 100 MPa

Ev = 150 MPa

Ev = 150 MPa

Ev =75 MPaEv = 100 MPa

Ev = 30 MPa

The results obtained show that the damping of the interleaved laminate increases when the damping of the viscoelastic layer increases or when the Young’s modulus of the layer decreases. Thus the ratio ηv/Ev allows us to characterise the resulting laminate damping for a given thickness of the interleaved viscoelastic layer.

FIGURE 7.18. Laminate damping as function of the fibre orientation in the case of a viscoelastic layer interleaved in middle plane for a Young’s modulus of the layer equal to 50 MPa and for different values of the damping layer.

FIGURE 7.19. Laminate damping as function of the fibre orientation in the case of a viscoelastic layer interleaved in middle plane for a loss factor of the layer equal to 0.30 and for different values of the Young’s modulus.

7.5. Discussion 239

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%

)

0.0

0.5

1.0

1.5

2.0

2.5

1st mode 2nd mode

e0 /2 = 0.2 mm

e0 /2 = 0.6 mm

e0 /2 = 0.4 mm

e0 /2 = 0.8 mm

e0 /2 = 1 mm

Increase of the damping capacity of composite laminates also can be achieved by the application of external damping layers to the surfaces of the laminates after fabrication. The damping of laminates with two external viscoelastic layers of thicknesses 0 2e was derived in Section 7.2.6. Figure 7.20 shows the results deduced from the modelling in the case of viscoelastic layers with a Young’s modulus v 50 MPaE = and a loss factor of the layers v 0.30η = . The results are reported for the first two modes for different thicknesses of the viscoelastic layers:

0 2 0.2, 0.4, 0.6, 0.8 and 1mme = . The damping of laminates is similar for the two modes and for a given thickness of the viscoelastic layers the laminate damping is notably lower than when viscoelastic layers are interleaved in the middle plane of laminates. So, to obtain high damping in the case of external viscoelastic layers it is necessary to use layers with high thicknesses. When the thicknesses of the viscoelastic layers are increased, it has to be noted that the actual laminate behaviour is not well described by the classical laminate theory used in the damping modelling. Furthermore, in the case of external viscoelastic layers, the energy is dissipated in the viscoelastic layers by the in-plane behaviour of layers. It results that the resulting damping of laminates increases with the Young’s modulus of layers for a given layer damping. Figure 7.21 reports the results deduced from the modelling in the case of external viscoelastic layers of thickness 0 2 0.8 mme = , loss factor v 0.30η = and for three values of the Young’s modulus: Ev = 50, 100 and 150 MPa. FIGURE 7.20. Laminate damping as function of the fibre orientation in the case of external viscoelastic layers for a Young’s modulus of the layers equal to 50 MPa and different thicknesses of the layers.

240 Chapter 7. Damping Analysis of Laminates with Interleaved Viscoelastic Layers

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%

)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1st mode 2nd mode

Ev = 50 MPa

Ev = 150 MPa

Ev = 100 MPa

FIGURE 7.21. Laminate damping as function of the fibre orientation in the case of external viscoelastic layers for a thickness of the layers equal to 0.8 mm and different values o theYoung’s modulus of the layers.

7.6 CONCLUSIONS

This chapter investigates the damping properties of unidirectional laminates in which a single or two viscoelastic layers are interleaved. Modelling was first developed and experimental investigation was implemented in the case of glass fibre laminates with a single viscoelastic layer interleaved in the middle plane or with two viscoelastic layers interleaved on both sides from the middle plane. The damping characteristics of laminates were deduced from the flexural vibrations of beams as function of fibre orientation. Interleaving viscoelastic layers increases significantly the damping of laminates and the fibre orientation dependence of damping appears somewhat similar to that of the initial laminates, but with a damping maximum moved from 60° fibre orientation to about 30° fibre orientation when viscoelastic layers are interleaved. In contrast to the non-interleaved laminates, damping increases consistently with frequency in the entire fibre orientation range depending on the vibration mode.

The increase of laminates damping when viscoelastic layers are interleaved is associated essentially to the transverse shear energy dissipated in the viscoelastic layers. The experimental results have been analysed considering the modelling developed in Section 7.2. The analysis introduces the variation of the Young’s modulus and damping of the viscoelastic layers with the frequency. A good agreement is obtained between modelling and experimental investigation. This agreement was corroborated by applying the modelling to the experimental results obtained by Liao et al. in the case of angle-ply laminates.

7.6. Conclusions 241

Next, the chapter studies the effects of the thickness, the Young’s modulus and the damping of a viscoelastic layer interleaved in the middle plane of laminates. For a given thickness of the layer, the damping of laminates is increased when the loss factor of the viscoelastic layer increases or when its Young’s modulus decreases. Thus, the ratio of the loss factor to the Young’s modulus of the viscoelastic layer characterises the laminate damping for a given thickness of the viscoelastic layer interleaved in the middle plane.

Lastly, the chapter considers the damping of laminates with external visco-elastic layers. The results obtained show that the laminate damping is notably lower, compared to the case of laminates with interleaved layers.

CHAPTER 8

Finite Element Method in the Dynamic Analysis of Composite Structures

8.1 PRINCIPLE OF THE METHOD

The basic concept of the finite element analysis is based on the concept that a structure can be approximated by replacing the structure with an assemblage of discrete elements. Since these elements can be put together in a variety of ways, they can be used to represent complex shapes. So, the first step of finite element analysis is to represent the structure as a mesh of discrete elements (Figure 8.1). Next, the field of displacements within each element is expressed as function of the displacements at the nodes of the element using interpolation functions. Then, the equations of elasticity allow to express as functions of the nodal displacements the strain energy, the kinetic energy and the work of external loads acting on the structure. Expressing the equilibrium of the structure or applying the principle of virtual displacements leads, in the case of a displacement formulation, to an equa-tion system of the nodal displacements. Thus, the nodal displacements become the new unknowns and the finite element discretisation procedure reduces the initial continuum problem to one of a finite number of unknowns.

FIGURE 8.1. Meshing of a structure.

element node

8.2. Formulation of Structural Elements 243

8.2 FORMULATION OF STRUCTURAL ELEMENTS

8.2.1 Isoparametric Finite Element Formulation

The basic procedure in the finite element formulation is to express the element coordinates in the form of interpolations in terms of the nodal coordinates. For a general three-dimensional element, the coordinate interpolations are:

1

1

1

( , , ) ,

( , , ) ,

( , , ) ,

q

i iiq

i iiq

i ii

x h x y z x

y h x y z y

z h x y z z

=

=

=

=

=

=

∑

∑

∑

(8.1)

where x, y and z are the local coordinates at any point of the element, and xi, yi and zi, i = 1, 2, …, q, are the coordinates of the q element nodes. Functions hi are the interpolation functions, also called the shape functions. The fundamental property of the interpolation functions is that its values is unity at node i and is zero at all other nodes.

The first step to obtain the interpolation functions is to relate the global coor-dinates of the element to a natural coordinate system which has variables r, s and t that each vary from –1 to 1. Variables r, s and t are functions of coordinates x, y and z. An example is given in Figure 8.2 for a two dimensional element of 9 nodes.

FIGURE 8.2. Two-dimensional element with four to nine nodes.

node 1

2

3

4

5

6

7

8 9 r

r = 0

r = –1

s

r = 1

s = –1

s = 0

s = 1 y

x

244 Chapter 8. Finite Element Method in the Dynamic Analysis of Composite Structures

8.2.2 Example of a Four-Node Finite Element

8.2.2.1 Interpolation Functions

As an example we consider the case of the four-node two-dimensional element of Figure 8.3. The coordinate interpolations can be expressed as:

1 1 2 2 3 3 4 4

1 1 2 2 3 3 4 4

,,

x h x h x h x h xy h y h y h y h y= + + += + + +

(8.2)

introducing the interpolation functions:

( )( ) ( )( )

( )( ) ( )( )

1 2

3 4

1 11 1 , 1 1 ,4 41 11 1 , 1 1 .4 4

h r s h r s

h r s h r s

= + + = − +

= − − = + − (8.3)

In the isoparametric formulation the element in-plane displacements are inter-polated in the same way as the node coordinates. So, we have for the in-plane displacements:

0 0 1 1 2 2 3 3 4 4

0 0 1 1 2 2 3 3 4 4

( , ) ,( , ) ,

u x y u h u h u h u h ux y h h h h

= = + + += = + + +v v v v v v

(8.4)

where 0u and 0v are the in-plane displacements at any point of the element, and iu and iv , i = 1 to 4, are the corresponding displacements at the nodes of the

element.

FIGURE 8.3. Four-node two-dimensional element.

r

s

4 ( r = 1, s = –1)

y, v

x, u

1 ( r = 1, s = 1) ( r = –1, s = 1) 2

( r = –1, s = –1) 3

x4

local coordinates

y4

8.2. Formulation of Structural Elements 245

8.2.2.2 Strain Formulation

The in-plane strains of a two-dimensional element are given (Equation (1.51) of Chapter 1) by:

0

0m

0

( , )xx

yy

xy

x y

ε

ε

γ

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

ε , (8.5)

where

0 0 00 0 0 0, , .xx yy xyu ux y y x

ε ε γ∂ ∂ ∂ ∂= = = +

∂ ∂ ∂ ∂v v (8.6)

To evaluate the displacement derivatives, we have to use the following relations:

,

.

r sx r x s x

r sy r y s y

∂ ∂ ∂ ∂ ∂= +∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂= +∂ ∂ ∂ ∂ ∂

(8.7)

These relations need to calculate ,r x∂ ∂ ,s x∂ ∂ r y∂ ∂ and ,s y∂ ∂ which means that the explicit inverse relationships giving r and s as functions of x and y, res-pectively, would need to be evaluated. These inverse relationships are in general difficult to derive explicitly, and the required derivatives are evaluated in the following way. The derivatives are expressed with respect to the natural coor-dinates r and s as:

,

,

yxr x r y r

yxs x s y s

∂∂ ∂ ∂ ∂= +∂ ∂ ∂ ∂ ∂

∂∂ ∂ ∂ ∂= +∂ ∂ ∂ ∂ ∂

(8.8)

which can be expressed in matrix notation as:

yxxr r r

yxys s s

∂∂ ⎡ ⎤∂ ∂⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ ∂∂ ∂ ∂= ⎢ ⎥⎢ ⎥⎢ ⎥ ∂∂ ∂∂ ⎢ ⎥⎢ ⎥⎢ ⎥∂⎢ ⎥∂⎣ ⎦ ∂ ∂⎣ ⎦ ⎣ ⎦

. (8.9)

This matrix relation can be rewritten in a reduced form as:

,= Jr x∂ ∂∂ ∂

(8.10)

where J is the Jacobian matrix relating the natural coordinate derivatives to the local ones:

246 Chapter 8. Finite Element Method in the Dynamic Analysis of Composite Structures

yxr r

yxs s

∂∂⎡ ⎤⎢ ⎥∂ ∂= ⎢ ⎥∂∂⎢ ⎥∂ ∂⎣ ⎦

J . (8.11)

The Jacobian matrix can be easily deduced from Equation (8.2). Next the local coordinate derivatives are obtained as:

1 ,−= Jx r∂ ∂∂ ∂

(8.12)

which requires that the inverse matrix of J exists. This inverse matrix exists provided that there is a one-to-one correspondence between the natural and the local coordinates of the element.

Thus, the Jacobian matrix (8.11) is evaluated considering Equations (8.2) and (8.3). We obtain:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 1 1 11 1 1 1 ,4 4 4 41 1 1 11 1 1 1 ,4 4 4 41 1 1 11 1 1 1 ,4 4 4 41 1 1 11 1 1 1 .4 4 4 4

x s x s x s x s xrx r x r x r x r xsy s y s y s y s yry r y r y r y r ys

∂ = + − + − − + −∂∂ = + + − − − − +∂∂ = + − + − − + −∂∂ = + + − − − − +∂

(8.13)

Therefore, we can form the Jacobian matrix J for any values of r and s, by using these expressions. Then, the local derivatives are expressed as:

1x r

y s

−

∂⎡ ⎤ ∂⎡ ⎤⎢ ⎥ ⎢ ⎥∂ ∂=⎢ ⎥ ⎢ ⎥∂ ∂⎢ ⎥ ⎢ ⎥∂⎢ ⎥ ∂⎣ ⎦⎣ ⎦

J . (8.14)

The strains in the element are evaluated considering Equations (8.13). We have:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

01 2 3 4

01 2 3 4

01 2 3 4

01 2 3 4

1 1 1 11 1 1 1 ,4 4 4 41 1 1 11 1 1 1 ,4 4 4 41 1 1 11 1 1 1 ,4 4 4 41 1 1 11 1 1 1 .4 4 4 4

u s u s u s u s ur

u r u r u r u r us

s s s sr

r r r rs

∂ = + − + − − + −∂∂ = + + − − − − +∂∂ = + − + − − + −∂∂ = + + − − − − +∂

v v v v v

v v v v v

(8.15)

Combining Equations (8.14) and (8.15) we obtain:

8.3 Laminate Element 247

( ) ( )

( ) ( )

0

1

0

1 0 1 0 1 0 1 014 1 0 1 0 1 0 1 0 e

us s s sx

u r r r ry

−

∂⎡ ⎤⎢ ⎥ ⎡ + − + − − − ⎤∂

=⎢ ⎥ ⎢ ⎥∂ + − − − − +⎣ ⎦⎢ ⎥∂⎢ ⎥⎣ ⎦

J u , (8.16)

and

( ) ( )

( ) ( )

0

1

0

0 1 0 1 0 1 0 114 0 1 0 1 0 1 0 1 e

s s s sxr r r r

y

−

∂⎡ ⎤⎢ ⎥ ⎡ + − + − − − ⎤∂

=⎢ ⎥ ⎢ ⎥∂ + − − − − +⎣ ⎦⎢ ⎥∂⎢ ⎥⎣ ⎦

J u

v

v, (8.17)

where eu is the vector listing the element nodal point displacements expressed as:

[ ]t1 1 2 2 3 3 4 4e u u u u=u v v v v . (8.18)

8.3 LAMINATE ELEMENT

8.3.1 Displacement Field

We consider the formulation of a plate element that is based on the theory of plates with transverse shear deformation included (Section 1.5 of Chapter 1). This theory uses the assumption that particles of the plate originally on a line that is normal to the non-deformed middle plane remain on a straight line during defor-mation, this line being not necessary normal to the deformed middle surface. With this assumption, the displacement components of a point of coordinates (x, y, z) are given by Equation (1.22):

0

0

0

( , , , ) ( , , ) ( , , ),( , , , ) ( , , ) ( , , ),( , , , ) ( , , ),

x

y

u x y z t u x y t z x y tx y z t x y t z x y tx y z t x y t

ϕϕ

= += +

=

v v

w w

(8.19)

where the displacements 0 ( , , ),u x y t 0( , , ) and x y tv 0 ( , , )x y tw are the in-plane displacements at point (x, y, 0) of the middle plane. Functions and x yϕ ϕ can be related to the rotations (Figure 8.4) and x yβ β during deformation of the normal lines to the non-deformed middle plane in the (x, z) and (y, z) planes, respectively, or to the rotations and x yθ θ about the axes x and y, respectively:

, .x y y xθ β θ β= = − (8.20)

Functions and x yϕ ϕ are given by:

( , ) ( , ) ( , ),( , ) ( , ) ( , ).

x x y

y y x

x y x y x yx y x y x y

ϕ β θ

ϕ β θ

= − =

= − = − (8.21)

248 Chapter 8. Finite Element Method in the Dynamic Analysis of Composite Structures

FIGURE 8.4. Flexural deformation in analysis of plate including shear deformation.

Therefore the displacement equation (8.19) can be rewritten as:

0

0

0

( , , , ) ( , , ) ( , , ),( , , , ) ( , , ) ( , , ),( , , , ) ( , , ),

x

y

u x y z t u x y t z x y tx y z t x y t z x y tx y z t x y t

ββ

= −= −

=

v v

w w

(8.22)

or

0

0

0

( , , , ) ( , , ) ( , , ),( , , , ) ( , , ) ( , , ),( , , , ) ( , , ).

y

x

u x y z t u x y t z x y tx y z t x y t z x y tx y z t x y t

θ

θ

= +

= −=

v vw w

(8.23)

It results that the laminate behaviour with transverse shear included is charac-terised by 5 degrees of freedom: 0 0 0, , , and ,x yu θ θv w which are the gene-ralised displacements. In finite formulation these degrees of freedom are expres-sed in the form of interpolations in terms of the degrees of freedom at each node. In the case of a four-node element (Figure 8.3) the displacements are expressed as:

0 1 1 2 2 3 3 4 4

0 1 1 2 2 3 3 4 4

0 1 1 2 2 3 3 4 4

( , , ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ),( , , ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ),( , , ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , )

u x y t h x y u t h x y u t h x y u t h x y u tx y t h x y t h x y t h x y t h x y tx y t h x y t h x y t h x y t h x y

= + + += + + += + + +

v v v v vw w w w w

1 2 3 4 1 2 3 41 2 3 4 1 2 3 4

( ),

( , , ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ),

( , , ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ),x x x x x

y y y y y

t

x y t h x y t h x y t h x y t h x y t

x y t h x y t h x y t h x y t h x y t

θ θ θ θ θ

θ θ θ θ θ

= + + +

= + + +

(8.24)

where 1 4 1 1 4 4, , ... , , ... , , , ..., ,y yu uθ θv v are the generalised displacements at

z

x, u y, v

xβ yβ

xθ yθ

8.3 Laminate Element 249

nodes 1, 2, 3 and 4, respectively, and where h1, h2, h3 and h4 are the interpolation functions.

It results that the displacement field in the element is expressed as:

1

1

10 1 2 3 4 1

0 1 2 3 4 10 1 2 3 4

1 2 3 4

1 2 3 4 44

4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

..0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

x

y

x

y

x

y

u

u h h h hh h h h

h h h hh h h h

h h h h

θ

θ

θθ

θ

θ

⎡⎢⎢⎢⎢⎡ ⎤ ⎡ ⎤⎢⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥=⎢⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥⎢⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎢⎢

⎣

vw

vw

w

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎢ ⎥⎢ ⎥

⎦

.

(8.25) This relation can be rewritten for the finite element e in the matrix form:

( , , ) ( , ) ( ), ( , ) element ,e ex y t x y t x y e= ∈U N u (8.26)

introducing the matrix of displacements and rotations:

t 0 0 0( , , ) x yx y t u θ θ= ⎡ ⎤⎣ ⎦U v w , (8.27)

the interpolation matrix of element e:

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

e

h h h hh h h h

h h h hh h h h

h h h h

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

N , (8.28)

and the matrix of nodal displacements and rotations of element e:

t 1 1 4 41 1 1 4...e x y x yu θ θ θ θ⎡ ⎤= ⎣ ⎦u v w w . (8.29)

8.3.2 In-Plane Behaviour

In-plane strains are given by Equations (1.82) of Chapter 1:

00

0 0m

00 0

xx

yy

xy

ux

yuy x

εεγ

∂⎡ ⎤⎢ ⎥⎡ ⎤ ∂⎢ ⎥⎢ ⎥ ∂= = ⎢ ⎥⎢ ⎥ ∂⎢ ⎥⎢ ⎥∂ ∂⎣ ⎦ ⎢ ⎥+⎢ ⎥∂ ∂⎣ ⎦

v

v

ε . (8.30)

250 Chapter 8. Finite Element Method in the Dynamic Analysis of Composite Structures

Considering Expression (8.25), the in-plane strains can be expressed as:

m ( , ) ( , ) ( )ex y x y tε= B uε , (8.31) introducing the matrix:

31 2 4

31 2 4

3 31 1 2 2 4 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

hh h hx x x x

hh h hy y y y

h hh h h h h hy x y x y x y x

ε

∂∂ ∂ ∂⎡ ⎤⎢ ⎥∂ ∂ ∂ ∂⎢ ⎥∂∂ ∂ ∂⎢ ⎥=

∂ ∂ ∂ ∂⎢ ⎥⎢ ⎥∂ ∂∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦

B . (8.32)

The matrix εB introduces the local derivatives of the interpolation functions. According to the results considered in Section 8.2.2, these derivatives are next expressed as functions of the natural coordinates of the finite element.

8.3.3 Flexural Behaviour

The flexural strains are deduced from Relations (1.82), considering Relations (8.21). Hence:

f

f ff

( , , ) ( , , )xx

xy

xy

x y t z x y t

ε

ε

γ

⎡ ⎤⎢ ⎥⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦

ε κ , (8.33)

where the curvature matrix is given by:

( , , )

y

x

y x

x

x y ty

y x

θ

θ

θ θ

∂⎡ ⎤⎢ ⎥∂⎢ ⎥

∂⎢ ⎥= −⎢ ⎥∂⎢ ⎥∂ ∂⎢ ⎥−⎢ ⎥∂ ∂⎣ ⎦

κ . (8.34)

Considering Expression (8.25), the curvature matrix is expressed as:

( , , ) ( , ) ( )ex y t x y tκ= B uκ , (8.35) introducing the matrix:

31 2 4

31 2 4

3 31 1 2 2 4 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

hh h hx x x x

hh h hy y y y

h hh h h h h hx y x y x y x y

κ

∂∂ ∂ ∂⎡ ⎤⎢ ⎥∂ ∂ ∂ ∂⎢ ⎥∂∂ ∂ ∂⎢ ⎥= − − − −

∂ ∂ ∂ ∂⎢ ⎥⎢ ⎥∂ ∂∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥− − − −

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦

B .

(8.36)

8.3 Laminate Element 251

8.3.4 Transverse Shear Behaviour

The transverse shear strains are given by Equations (1.82):

0

s0

xyz

xzy

y

x

θγγ

θ

∂⎡ ⎤−⎢ ⎥⎡ ⎤ ∂= = ⎢ ⎥⎢ ⎥ ∂⎣ ⎦ ⎢ ⎥+⎢ ⎥∂⎣ ⎦

w

wγ . (8.37)

So, in the first order theory including the transverse shear effects, the trans-verse shear strains are constant though the laminate.

Introducing interpolation (8.25), the transverse shear strains can be expressed as:

s ( , ) ( , ) ( )ex y x y tγ= B uγ , (8.38) introducing the matrix:

31 2 41 2 3 4

31 2 41 2 3 4

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

hh h hh h h hy y y y

hh h hh h h hx x x x

γ

∂∂ ∂ ∂⎡ ⎤− − − −⎢ ⎥∂ ∂ ∂ ∂= ⎢ ⎥

∂∂ ∂ ∂⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦

B .

(8.39)

8.3.5 Stress Formulation

The in-plane stresses in the layer k of laminate are given (1.54) by:

011 12 16 11 12 16

012 22 26 12 22 26

016 26 66 16 26 66

xx xx x

yy yy y

xy xy xyk k k

Q Q Q Q Q Q

Q Q Q z Q Q Q

Q Q Q Q Q Q

σ ε κ

σ ε κ

σ γ κ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤′ ′ ′ ′ ′ ′⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

′ ′ ′ ′ ′ ′= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

′ ′ ′ ′ ′ ′⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

, (8.40)

where the constants ijQ′ are the reduced stiffnesses in the plate axes expressed in Table 1.1 of Chapter 1. Relation (8.40) may be expressed in the matrix form as:

m m( , , , ) ( , , ) ( , , )kk kx y z t x y t z x y t′ ′= +Q Qσ ε κ , (8.41)

where mkσ is the in-plane stress matrix of layer k and introducing the reduced

stiffness matrix k′Q of layer k. Substituting Relations (8.31) and (8.35) into Equation (8.41), we obtain:

[ ] m ( , , , ) ( , ) ( , ) ( )kk k ex y z t x y z x y tε κ′ ′= +Q Q uσ Β Β . (8.42)

The transverse shear stresses in the layer k are written (1.52) as:

44 45

45 55

yz yz

xz xzk k

C CC C

σ γσ γ

′ ′⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥′ ′⎣ ⎦ ⎣ ⎦ ⎣ ⎦

, (8.43)

252 Chapter 8. Finite Element Method in the Dynamic Analysis of Composite Structures

where the constants ijC′ are the transverse shear stiffnesses of layer k. This relation is expressed in the matrix form as:

s s( , , , ) ( , , )kkx y z t x y t′= Cσ γ , (8.44)

where skσ is the transverse shear stress of layer k and k′C is the transverse shear

matrix of layer k. Introducing Relation (8.38) into Equation (8.44), we obtain:

s ( , , , ) ( , ) ( )kk ex y z t x y tγ′= C uσ Β . (8.45)

8.3.6 Energy Formulation

8.3.6.1 Strain Energy and Element Stiffness

The strain energy stored in the element e of the laminate is given by:

d1 d d d2

1 d d d ,2

xxe

xx yy xy yye

xy k

yzyz xz

e xz k

U x y z

x y z

σε ε γ σ

σ

σγ γ

σ

⎡ ⎤⎢ ⎥= ⎡ ⎤⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤+ ⎡ ⎤ ⎢ ⎥⎣ ⎦

⎣ ⎦

∫∫∫

∫∫∫ (8.46)

where the integration is extended over the whole volume of the element e and where ( ), , xx yy xyε ε γ is the strain field resulting from the in-plane behaviour and flexural behaviour of the laminate element:

0

0

0

xxxx x

yy yy y

xy xyxy

z

εε κε ε κγ κγ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥= +⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

. (8.47)

Introducing the different expressions of strains and stresses into Expression (8.46) of the strain energy, then integrating through the laminate thickness, the strain energy stored in the element e may be expressed as:

t

d12

ee e eU = u K u , (8.48)

where the matrix eK is the stiffness matrix of the element expressed as:

t t

t t

t

d d d d

d d d d

d d .

e e

e e

e

e e eS S

e eS S

eS

x y x y

x y x y

x y

ε ε ε κ

κ ε κ κ

γ γ

= +

+ +

+

∫∫ ∫∫∫∫ ∫∫∫∫

K B A B B B B

B B B B D B

B F B

(8.49)

8.3 Laminate Element 253

This expression introduces the in-plane stiffness matrix ,eA the bending stiffness matrix ,eD the in-plane/bending coupling stiffness matrix ,eB and the transverse shear stiffness matrix ,eF for the element e of the laminate. These matrices are expressed in Equations (1.56) to (1.63), and (1.83).

To account for the actual non uniformity of the transverse shear stresses, shear coefficients may be introduced in the expression of the transverse shear stiffness matrix (Section 1.5.4 of Chapter 1).

The total strain energy stored in the laminate structure is obtained by sum-mation on the elements of the structure as:

td

12

U = U K U , (8.50)

where U is the matrix of displacements and rotations at all the nodes of the structure and K is the stiffness matrix of the structure given by:

elementse= ∑K K , (8.51)

where the summation includes all the elements. To perform the summation, each element matrix Ke is written as a matrix of the same order as the stiffness matrix K, where all the entries are zero, except those which correspond to the different degrees of freedom of the element

8.3.6.2 Kinetic Energy

Neglecting the rotatory inertia terms, the kinetic energy of a finite element of the structure is given (1.80) by:

( )

2 2 2c 0 0 0

1 d d2 e

e es

SE u x yρ= + +∫∫ v w , (8.52)

where esρ is the mass per unit area of the element e.

The in-plane displacements u0, v0 and w0 are derived from Equation (8.25), which leads to:

0 ( , , ) ( , ) ( )eu ex y t x y t=u H u , (8.53)

introducing the matrix of in-plane displacements in the element e:

[ ]t0 0 0 0u=u v w , (8.54)

and the interpolation matrix of the element:

1 2 3 4

1 2 3 4

1 2 3 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

eu

h h h hh h h h

h h h h

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

H . (8.55)

It results that the kinetic energy of the element e may be expressed as:

t

c12

ee e eE = u M u , (8.56)

254 Chapter 8. Finite Element Method in the Dynamic Analysis of Composite Structures

where the matrix eu is the vector of the nodal point velocities of the element e and introducing the mass matrix of the element:

t d de

e e ee s u u

Sx yρ= ∫∫M H H . (8.57)

The kinetic energy of the structure is obtained by summation on the elements of the structure, which leads to:

tc

12

E = U M U , (8.58)

where U is the vector of the velocities of all the nodal points of the structure. The mass matrix of the structure can be written as:

elements

e= ∑M M , (8.59)

where the matrix M is obtained by assemblage of the mass matrices of the elements by the same procedure as the stiffness matrix.

8.3.6.3 Work of the External Loads

In the case of transverse loads acting on the lower and upper faces of the laminate element, the expression of the work of the external loads is given by:

R 0 d de

ee

SW q x y= ∫∫ w , (8.60)

where qe is the transverse external loading per unit area of the element e. The transverse displacement w0 is deduced from Equation (8.25), which leads

to: 0 ( , , ) ( , ) ( )e

ex y t x y t=w H uw , (8.61)

where the matrix eHw is the interpolation matrix of the element associated to the transverse displacement:

[ ]1 2 3 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0e h h h h=Hw . (8.62)

It results that the work of the transverse loading may be expressed as:

t

Re

e eW = u R , (8.63)

where eR is the load matrix of the element e expressed as:

t( ) d de

ee e e

St q x y= = ∫∫R R Hw . (8.64)

The total work of the transverse loading is obtained by summation on the elements of the structure. Whence:

8.4 Finite Element Dynamic Equation of Laminate Structure 255

tfW = U R , (8.65)

where R = R(t) is the external load vector obtained by assemblage of the load matrices of all the elements.

elements

e= ∑R R . (8.66)

8.4 FINITE ELEMENT DYNAMIC EQUATION OF LAMINATE STRUCTURE

Considering the dynamic equilibrium of the laminate structure or that the total potential energy must be stationary, we deduce the dynamic equation of laminate structure as:

+ =MU KU R , (8.67)

where U is the matrix which lists the nodal point accelerations of the structure. The matrices K, M and R are expressed in (8.51), (8.59) and (8.66), respectively.

In vibration analysis the structure damping is usually taken into account by introducing viscous damping where the damping forces depend on velocity. In this case the energy dissipated by the structure is expressed in a quadratic form as:

t12aU = U C U , (8.68)

where C is the damping matrix of the structure. The dynamic equation of the laminate structure is thus modified as:

+ =MU CU + KU R . (8.69)

The formulation (8.68) is introduced due to the fact that this formulation is simple to deal with mathematically. But in practice, it is not possible to determine for general finite element assemblages the damping parameters. For this reason, the damping matrix C generally is constructed using the mass and stiffness matrices of the complete element assemblage, considering the results obtained on the structure damping which are deduced from experimental investigation or modelling (Section 9.4).

As an example, we have considered in the previous sections the case of a four-node plate element associated to a first order theory including the transverse shear effect. This formulation leads to the formulation of the matrix stiffness K expressed by Equations (8.49) and (8.51). In practice this element is somewhat too stiff for modelling the behaviour of thin laminates. In this case, the behaviour of the element can be improved by neglecting the transverse shear strain energy in the finite element formulation and imposing the Kirchhoff assumption that the transverse shear deformations are zero at the nodes of the elements.

Furthermore, isoparametric elements based on the formulation considered

256 Chapter 8. Finite Element Method in the Dynamic Analysis of Composite Structures

previously are more effective when they are used as high-order elements and specifically the 9-node quadrilateral elements. Also, the formulation of plate elements can be obtained as a special case of a formulation of a general three-dimensional shell element [80].

CHAPTER 9

Solution of Dynamic Equation in Finite Element Analysis

9.1 INTRODUCTION

In Section 8.4 of Chapter 8 the dynamic equation of a system of finite elements was derived in the form: + =MU CU + KU R , (9.1)

where M, C and K are the mass, damping and stiffness matrices; R is the external load vector; and U, U and U are the displacement, velocity and acceleration vec-tors of the finite element assemblage. In practice it is difficult, if not impossible, to determine the element damping properties for general finite element assem-blages, in particular because the damping properties are dependent on frequency. For this reason, the damping matrix C is in general not assembled from damping matrices of the elements, but is constructed using the mass and stiffness matrices of the complete element assemblage considering the experimental results obtained on the structure damping or considering damping modelling.

Equation (9.1) is a system of linear differential equations of second order the solution of which can be obtained, in principle, by standard procedures used for solving differential equations with constant coefficients. However, these standard procedures can become very expensive when the matrices M, C and K are large. So, in finite element analysis only a few effective methods are used. The proce-dures are divided into two types: direct integration and mode superposition. The choice for one method or the other is determined by their numerical effectiveness.

9.2 DIRECT INTEGRATION METHODS

9.2.1 Principle

In direct integration, the differential equations in Equation (9.1) are integrated using a numerical step-by-step procedure.

258 Chapter 9. Solution of Dynamic Equation in Finite Element Analysis

The displacement, velocity and acceleration vectors at time t = 0 are known and denoted by U0, 0U and 0U , respectively. The solution to Equation (9.1) is searched from time t = 0 to time T. In direct integration the time interval T is divided into n equal time intervals ∆t ( t T n∆ = ). Then, the integration method calculates approximate solutions at times 0, ∆t, 2∆t, … , t, t + ∆t, … ,T. The gene-ral algorithm of the method is obtained by considering that the solution is known at time t and the solution at time t + ∆t is required next.

The dynamic equation (9.1) can be considered as a system of ordinary differ-rential equation with constant coefficients. It results that any usual convenient differrence formulation can be used to approximate the acceleration, velocities and displacements (see for example [81, 82]). So, a large number of formulations could be used. However, formulations should have to lead to efficient algorithms, and it results that only few schemes are considered.

9.2.2 The Central Difference Method

9.2.2.1 Formulation

One procedure that can be efficient to obtain the solution of the dynamic equation (9.1) is the central difference method. This method is derived from the Euler’s method and Euler-Cauchy’s method [81, 82].

In the central difference method the velocity is expressed as:

( )12t t t t tt +∆ −∆= −∆

U U U . (9.2)

The velocity tU at the central time t is expressed as function of the displacements at t t+ ∆ and t t− ∆ , respectively. Next, the acceleration is expressed by the same expression used on the interval 1 1

2 2, t t t t⎡ ⎤− ∆ + ∆⎣ ⎦ . Thus we have:

( )1 12 2

1t t t t tt + ∆ − ∆= −

∆U U U , (9.3)

with

( )12

1t t tt t t +∆+ ∆ = −

∆U U U , (9.4)

( )12

1t t tt t t +∆− ∆ = −

∆U U U . (9.5)

Finally, Equations (9.3) to (9.5) lead to:

( )21 2t t t t t tt

+∆ −∆= −∆

U U U + U . (9.6)

The error in Expressions (9.3) and (9.5) are of second order. The displacement for time t t+ ∆ is obtained by introducing the expressions for and t tU U in the dynamic equation at time t:

9.2. Direct Integration Methods 259

t t t t+ =MU CU + KU R . (9.7)

Substituting Equations (9.2) and (9.6) into Equation (9.7), we obtain:

2 2 21 1 2 1 1

2 2t t t t t tt tt t t+∆ −∆

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ = − − − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∆ ∆∆ ∆ ∆⎝ ⎠ ⎝ ⎠ ⎝ ⎠M C U R K M U M C U . (9.8)

This equation can be solved for the displacement t t+∆U at time t t+ ∆ as function of the displacements at times t and t t−∆ .

The solution for t t+∆U is based on the dynamic equation (9.7) expressed at time t. For this reason the integration procedure is called as explicit integration method. Such an integration scheme does not require a factorisation of the stiff-ness matrix in the step-by-step procedure.

Another characteristic of Equation (9.8) is that the calculation of t t+∆U invol-ves the displacements and t t t+∆U U at times t and t t− ∆ , respectively. There-fore, a special starting procedure must be used at time t t= ∆ . At this time, Equa-tion (9.8) leads to:

0 02 2 21 1 2 1 1

2 2t tt tt t t∆ −∆

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ = − − − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∆ ∆∆ ∆ ∆⎝ ⎠ ⎝ ⎠ ⎝ ⎠M C U R K M U M C U . (9.9)

This expression needs to have the displacement t−∆U at time t−∆ . This display-cement can be deduced from Equations (9.2) and (9.6), which leads to:

( )01

2 t tt ∆ −∆= −∆

U U U , (9.10)

( )0 021 2t tt

∆ −∆= −∆

U U U + U . (9.11)

These two equations allow us to derive t−∆U as:

2

0 0 0 2ttt−∆

∆= − ∆U U U + U . (9.12)

It must be noted that with 0 0and U U known at time 0t = , the acceleration 0U can be calculated using the dynamic equation (9.1) at time 0t = .

Thus, the time integration scheme as it might be implemented in a computer is reported in Table 9.1.

9.2.2.2 Characteristics of the Central Difference Method

In the central difference method it can be shown that it is not necessary to calculate the stiffness and mass matrices of the complete element assemblage of the structure under consideration. The solution for the displacements can be ob-tained by considering only the element level and so relatively little storage is required. The effectiveness of the method is even increased if the stiffness and mass matrices are the same for the elements of the structure. In this case it is necessary to calculate and store only the matrices corresponding to one element.

260 Chapter 9. Solution of Dynamic Equation in Finite Element Analysis

TABLE 9.1. Step-by-step procedure using the central difference method.

A. Initial Calculations 1. Form the mass matrix M, the damping matrix C and the stiffness matrix

K.

2. Initialize 0 0 0, and U U U . 3. Select the time step t∆ and calculate the constants:

0 1 2 0 32 2

1 1 1, , 2 , .2

a a a a at at

= = = =∆∆

4. Form the effective mass matrix:

0 1a a= +M M C .

B. At each step of the procedure 1. Calculate the effective load at time t:

( ) ( )2 0 1t t t t ta a a −∆= − − − −R R K M U M C U .

2. Solve for the displacement at time t t+ ∆ the equation:

t t t+∆ =MU R .

3. Evaluate the velocities and accelerations at time t:

( )1t t t t ta +∆ −∆= −U U U ,

( )0 2t t t t t ta +∆ −∆= −U U U + U .

A second important characteristic of the central difference method is that this scheme requires that the time step t∆ must be smaller than a critical value crt∆ which depends on the mass and stiffness properties of the complete assemblage of the structure. It is shown that we must have:

crnTt tπ

∆ < ∆ = , (9.13)

where Tn is the smallest period of the finite element assemblage having n degrees of freedom. Schematically, this period is a function of the dimensions of the smallest elements in the mesh of the structure.

Integration schemes that require the use of a time step smaller than a critical time step, such as the central difference method are said to be conditionally stable. If a time step larger than the critical one is used, the integration is unstable resul-ting in an increase of the errors along the step integration procedure.

9.2. Direct Integration Methods 261

9.2.3 The Houbolt Method

As for the central difference method, the Houbolt scheme expresses the velocity and acceleration in terms of the displacements. The finite difference expansions used in the Houblot integration is:

( )21 11 18 9 2

6t t t t t t t t tt+∆ +∆ −∆ − ∆= − + −∆

U U U U U , (9.14)

( )221 2 5 4t t t t t t t t tt

+∆ +∆ −∆ − ∆= − −∆

U U U + U U , (9.15)

which are difference expressions with errors of second order. The solution for displacement at time t t+ ∆ is obtained by substituting the

expressions for and t t t t+∆ +∆U U in the dynamic equation at time t t+ ∆ :

t t t t t t t t+∆ +∆ +∆ +∆+ =MU CU + KU R . (9.16) We obtain:

2 2

22 2

2 11 5 36

4 3 1 1 .2 3

t t t t t

t t t t

t tt t

t tt t

+∆ +∆

−∆ − ∆

⎛ ⎞ ⎛ ⎞+ = + +⎜ ⎟ ⎜ ⎟∆ ∆∆ ∆⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞+ − + + +⎜ ⎟ ⎜ ⎟∆ ∆∆ ∆⎝ ⎠ ⎝ ⎠

M C + K U R M C U

M C U M C U (9.17)

A basic difference with the central difference scheme is the appearance of the stiffness matrix K as a factor to the displacement t t+∆U . The terms t t+∆KU appears because the dynamic equation is considered at time t t+ ∆ (9.16) and not at time t as in the central difference scheme (9.7). For this reason it results that the Houtbolt method is an implicit integration scheme.

Furthermore there is not critical time step limit for the time step t∆ . The Houlbot scheme is unconditionally stable and t∆ can in general be taken much larger than the limit given in (9.13) for the central difference method.

Expression (9.17) shows that the calculation of t t+∆U requires the knowledge of 2, and t t t t t−∆ − ∆U U U . It results that a starting procedure must be used to cal-culate 2 and t t∆ ∆U U .

The step-by-step procedure using the Houbolt method is reported in Table 9.2.

9.2.4 The Wilson θ method

The Wilson θ method is an extension of the linear acceleration scheme, for which the acceleration variation is assumed to have a linear variation from time t to time t t+ ∆ . In the Wilson θ method, the acceleration is assumed to be linear from time t to time t tθ+ ∆ , with 1.θ ≥ In fact, it is shown that for unconditional stability of the method it is necessary to take 1.37,θ ≤ and usually θ is taken as

1.40.θ = Then, if τ denotes the increase in time, with 0 tτ θ≤ ≤ ∆ , the acceleration at

262 Chapter 9. Solution of Dynamic Equation in Finite Element Analysis

TABLE 9.2. Step-by-step scheme using the Houbolt method.

A. Initial Calculations 1. Form the mass matrix M, the damping matrix C and the stiffness matrix

K. 2. Initialize 0 0 0, and U U U . 3. Select the time step t∆ and calculate the constants:

0 1 2 32 2

3 0 34 0 5 6 7

2 11 5 3, , , ,62

2 , , , .2 2 9

a a a at tt t

a a aa a a a a

= = = =∆ ∆∆ ∆

= − = − = =

4. Use a starting procedure to obtain 2 and t t∆ ∆U U . 5. Form the effective stiffness matrix:

0 1a a= + +K K M C .

B. At each step of the procedure 1. Calculate the effective load at time t t+ ∆ :

( )

( )

2 4 6 2

3 5 7 2 .t t t t t t t t

t t t t t

a a a

a a a+∆ −∆ − ∆

−∆ − ∆

= + + +

+ + +

R R M U U U

C U U U.

2. Solve for the displacement at time t t+ ∆ the equation:

t t t t+∆ +∆=KU R .

3. Evaluate the velocities and accelerations at time t:

1 3 5 7 2t t t t t t t t ta a a a+∆ +∆ −∆ − ∆= − − −U U U U U , 0 2 4 6 2t t t t t t t t ta a a a+∆ +∆ −∆ − ∆= − − −U U U U U .

time t τ+ is given by:

( )t t t t t tτ θτθ+ + ∆= + −∆

U U U U . (9.18)

Integrating this expression, we obtain:

( )2

2t t t t t t tτ θττθ+ + ∆= + + −∆

U U U U U , (9.19)

and next:

( )2

212 6t t t t t t t tτ θ

ττ τθ+ + ∆= + + −∆

U U + U U U U . (9.20)

9.2. Direct Integration Methods 263

From these expressions, we obtain the velocity and the displacement at time t tθ+ ∆ as:

( )2t t t t t t

tθ θ

θ+ ∆ + ∆

∆= + +U U U U , (9.21)

( )2 2

26t t t t t t t

ttθ θθθ+ ∆ + ∆∆= + ∆ + +U U U U U . (9.22)

These last two expressions allow to obtain and t t t tθ θ+ ∆ + ∆U U in terms of displa-cement t tθ+ ∆U :

( )2 26 6 2t t t t t t ttt

θ θ θθ+ ∆ + ∆= − − −

∆∆U U U U U , (9.23)

( )3 22t t t t t t t

ttθ θ

θθ+ ∆ + ∆

∆= − − −∆

U U U U U . (9.24)

These expressions constitute the difference expansion of the Wilson θ method. The equations for the displacements, velocities and accelerations at time t t+ ∆

are obtained by considering the dynamic equation (9.1) at time t tθ+ ∆ :

t t t t t t t tθ θ θ θ+ ∆ + ∆ + ∆ + ∆+ =MU CU + KU R . (9.25)

In this expression appears the external load vector at time t tθ+ ∆ . As the acce-leration, this load vector is obtained by assuming a linear variation of the load:

( )t t t t t tθ θ θ+ ∆ + ∆= −R R + R R . (9.26)

Substituting Equations (9.23) and (9.24) into Equation (9.25), we obtain an equation from which the displacement t tθ+ ∆U can be solved:

( )

( ) ( )2 26 3 3

6 2 2 .2

t t t t t t t

t t

t ttt

t

θ θθ θθ

θθ

+ ∆ +∆⎛ ⎞+ = + − +⎜ ⎟∆ ∆∆⎝ ⎠

∆+ + + +∆

M C + K U R R R CU

M C U M C U (9.27)

Solving this equation, the displacement t tθ+ ∆U is then substituted into Equation (9.23) to obtain the acceleration t tθ+ ∆U at time t tθ+ ∆ . Next, this acceleration is introduced in Equations (9.18), (9.19) and (9.20), and the equations obtained are evaluated at time tτ = ∆ to derive , and ,t t t t t t+∆ +∆ +∆U U U at time t t+ ∆ . The algorithm for the integration is reported in Table 9.3.

The Wilson θ scheme is an implicit integration method, because the stiffness matrix K is a coefficient matrix to the unknown displacement vector. No starting procedure is necessary, since the displacements, velocities and accelerations at time t t+ ∆ are expressed in terms of displacements, velocities and accelerations at time t only.

264 Chapter 9. Solution of Dynamic Equation in Finite Element Analysis

TABLE 9.3. Step-by-step procedure using the Wilson θ method.

A. Initial Calculations 1. Form the mass matrix M, the damping matrix C and the stiffness matrix

K.

2. Initialize 0 0 0, and U U U . 3. Select the time step t∆ and calculate the constants (usually θ = 1.40):

00 1 2 1 3 42 2

22

5 6 7 8

6 3, , 2 , , ,2

3, 1 , , .2 2 6

ata a a a a att

a t ta a a a

θθ θθ

θ

∆= = = = =∆∆

∆ ∆= − = − = =

4. Form the effective stiffness matrix:

0 1a a= + +K K M C .

B. At each step of the procedure 1. Calculate the effective load at time t t+ ∆ :

( ) ( ) ( ) 0 2 1 32 2 .t t t t t t t t t t ta a a aθ +∆= + + + + + +R R R - R + M U U U C U U U

2. Solve for the displacement at time t tθ+ ∆ the equation:

t t t tθ θ+ ∆ + ∆=KU R .

3. Evaluate the accelerations, velocities and displacements at time t t+ ∆ :

( )4 5 6t t t t t t ta a aθ+∆ + ∆= − + +U U U U U ,

( )7t t t t t ta+∆ +∆= + +U U U U ,

( )8 2t t t t t t tt a+∆ +∆= + ∆ + +U U U U U .

9.2.5 The Newmark Method

As the Wilson method, the Newmark integration can be considered as an extension of the linear acceleration scheme. Expressions (9.21) and (9.22) of the Wilson method are modified as:

( )1t t t t t t tθ δ δ+ ∆ +∆⎡ ⎤= + − ∆⎣ ⎦U U U + U , (9.28)

( ) 212t t t t t t tt tα α+∆ +∆

⎡ ⎤= + ∆ + − + ∆⎢ ⎥⎣ ⎦U U U U U . (9.29)

For 1 12 6 and ,δ α= = these expressions correspond to the linear acceleration

9.2. Direct Integration Methods 265

scheme which is also obtained with 1θ = in the Wilson method. For 1 1

2 4 and ,δ α= = Expressions (9.28) of the velocity vector and (9.30) of the displacement vector are reduced to:

( )2t t t t t tt

+∆ +∆∆= +U U U + U , (9.30)

( )2

4t t t t t t ttt+∆ +∆

∆= + ∆ + +U U U U U . (9.31)

These expressions correspond to the integration method called the constant-average-acceleration method which is an unconditionally stable scheme.

In the general case of Expressions (9.28) and (9.29), it is shown that the sche-me is unconditionally stable if:

( )21 1 1, .2 4 2

δ α δ≥ ≥ + (9.32)

From Equation (9.29) we deduce the acceleration vector at time t t+ ∆ as:

( ) ( )21 1 1 1

2t t t t t t ttt α αα+∆ +∆ − − + −

∆∆U = U U U U . (9.33)

Then, substituting this result into Equation (9.30) leads to the velocity vector at time t t+ ∆ :

( ) ( ) ( )1 12t t t t t t tt

δ δ δα α α+∆ +∆ − + − + −∆

U = U U U U . (9.34)

Expressions (9.33) and (9.34) are similar in forms to Expressions (9.23) and (9.24) obtained in the case of the Wilson method.

Substituting Equations (9.33) and (9.34) into the dynamic equation (9.16) at time t t+ ∆ , we obtain an equation for which the displacement t t+∆U can be solved:

( ) ( ) ( )

2 21 1

1 11 1 1 .2

t t t t t

t t

t tt t

t

δ δα αα α

δ δα α α α

+∆ +∆⎛ ⎞ ⎛ ⎞+ = + +⎜ ⎟ ⎜ ⎟∆ ∆∆ ∆⎝ ⎠ ⎝ ⎠

⎡ ⎤ ⎡ ⎤+ − − − − − −⎢ ⎥ ⎢ ⎥∆⎣ ⎦ ⎣ ⎦

M C + K U R M C U

M C U M C U (9.35)

Solving this equation, the displacement vector t tθ+ ∆U is then introduced in Equa-tions (9.33) and (9.34) to derive the acceleration vector t t+∆U and the velocity vector t t+∆U .

The algorithm using the Newmark integration is reported in Table 9.4. Accor-ding to Equations (9.23) and (9.24), (9.33) and (9.34), the algorithm is similar to the algorithm for Wilson method (Table 9.3). So, the two integration methods can be easily implemented in a single computer program.

266 Chapter 9. Solution of Dynamic Equation in Finite Element Analysis

TABLE 9.4. Step-by-step procedure using the Newmark integration method.

A. Initial Calculations 1. Form the mass matrix M, the damping matrix C and the stiffness matrix

K.

2. Initialize 0 0 0, and U U U . 3. Select the time step t∆ , the parameters α and δ, and calculate the

constants:

( )21 1 1, ,2 4 2

δ α δ≥ ≥ +

( ) ( )

0 1 2 32

4 5 6 7

1 1 1, , , 1,2

1, 2 , 1 , .2

a a a at tt

ta a a t a t

δα α αα

δ δ δ δα α

= = = = −∆ ∆∆

∆= − = − = − ∆ = ∆

4. Form the effective stiffness matrix:

0 1a a= + +K K M C .

B. At each step of the procedure 1. Calculate the effective load at time t t+ ∆ :

( ) ( ) 0 2 3 1 4 5 .t t t t t t t t ta a a a a a+∆= + + + + + +R R M U U U C U U U

2. Solve for the displacement at time t t+ ∆ the equation:

t t t t+∆ +∆=KU R .

3. Calculate the accelerations and velocities at time t t+ ∆ :

( )0 2 3t t t t t t ta a a+∆ +∆= − − −U U U U U .

6 7t t t t t ta a+∆ +∆= + +U U U U ,

9.3 MODE SUPERPOSITION

9.3.1 Introduction

The number of operations in the direct integration methods is directly propor-tional to the number of time steps used in the integration. Therefore, direct integration can be effective in the case of a response with relatively short dura-tion. In contrast, if the integration must be carried out for many time steps, it may be more effective to transform the dynamic equation (9.1) into a form in which

9.3. Mode Superposition 267

the step-by-step solution is less expensive. In this way, we consider in the present section the case where the response is derived by using mode superposition.

9.3.2 Dynamic Equation in Modal Coordinates

9.3.2.1 Modal Coordinates

The free vibration equations with damping neglected are given in the matrix form as:

0+ =MU KU . (9.36)

When the structure vibrates, the displacement is harmonic and can be expressed in the following form:

( ) cos sinA t B tω ω= +U X , (9.37)

where X is a vector of order n, ω is the angular frequency of the harmonic vibrations, and A and B are depending on the initial conditions at time 0t = .

Substituting Equation (9.37) into (9.36), we obtain:

2ω=KX MX , (9.38)

which is the equation of the generalised eigenproblem. Solving this equation yields to the n eigensolutions: ( )

21 1, ,ω X ( )

22 2, ,ω X … , ( )

2 , ,n nω X where 1,ω

2 ,ω … , ,nω are the eigenvalues and 1,X 2 ,X … , ,nX are the eigenvectors. The eigenvectors can be M-orthonormalised, that is:

t 1, if , 0, if ,i j

i ji j=⎧

= ⎨ ≠⎩X M X (9.39)

and the eigenvalues can be ranged as increasing values:

2 2 21 2 . . . 0 nω ω ω≤ ≤ ≤ ≤ . (9.40)

The eigenvalue iω is the natural frequency of vibration of the mode i and the vector iX is the corresponding mode shape vector. Equation (9.37) is satisfied using any of the n displacement solutions:

( ) cos sin , 1, 2, ... , .i i i iA t B t i nω ω= + =U X (9.41)

This process generalises in a three-dimensional analysis the modal concept intro-duced in the one-dimensional analysis developed in the case of beam vibrations (Section 3.3 of Chapter 3).

Introducing the n n× modal matrix Λ the columns of which are the eigen-vectors iX and a diagonal matrix 2Ω with the eigenvalues on its diagonal:

268 Chapter 9. Solution of Dynamic Equation in Finite Element Analysis

[ ]

21

22

21 2

2

. . .0 0

. . .0 0. . . , . . . .. . . .. . . .

. . .0 0

n

n

ω

ω

ω

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= =⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

X X XΛ Ω , (9.42)

the n solutions to Equation (9.38) can be expressed as:

2=K MΛ ΛΩ . (9.43) Since the eigenvectors are M-orthonormal, we have:

t 2 t, ,=K MΛ Λ Ω Λ Λ = Ι (9.44) where I is the unit matrix.

9.3.2.2 Motion Equation in Modal Coordinates

When external loads are imposed, the dynamic equation is given by Equation (9.1). A solution for this motion equation can be expressed in terms of time functions ( )tφ and modal displacement iX as:

( ) ( )t t=U Λ φ . (9.45)

Substitution of this expression into Equation (9.1), then considering the ortho-normality properties (9.44) leads to:

t 2 t( ) ( ) ( ) ( )t t t t+ + =C Rφ Λ Λ φ Ω φ Λ . (9.46)

This equation is the motion equation expressed in modal coordinates. This form extends in three-dimension the motion equation (5.18) obtained in the analysis of beam vibrations.

The initial conditions on ( )tφ are derived from Equation (9.45) and consi-dering the M-orthonormality properties of Λ . So the initial conditions at time

0t = are:

t t0 0 0 0, ,= =M U M Uφ Λ φ Λ (9.47)

where 0U is the displacement matrix and 0U the velocity matrix at time 0t = .

9.3.3 Modal Analysis with Damping Neglected

If the damping effects are neglected, Equation (9.46) reduces to:

2 t( ) ( ) ( )t t t+ = Rφ Ω φ Λ . (9.48)

This equation leads to n equations of the form:

2( ) ( ) ( ), 1, 2, ... , ,i i i it t r t i nφ ω φ+ = = (9.49) where

t( ) ( )i ir t t= X R . (9.50)

9.3. Mode Superposition 269

The initial conditions of the structure are derived from Equation (9.47) as:

t t0 0 0 0, .i i i iφ φ= =X M U X M U (9.51)

In fact, Equation (9.49) is the motion equation of a single degree of freedom system. The form is the same as the motion equation (3.96) obtained in the case of undamped beam vibrations. So, the results of the Section 3.5 of Chapter 3 can be transposed to the present analysis.

In the general case, the solution to each equation in (9.49) can be obtained using a numerical integration or using the Duhamel integral according to Equation (2.134) of Chapter 2 as:

0

1( ) cos sin ( )sin ( ) dt

i i i i i i ii

t A t B t r t t t tφ ω ω ωω

+ ′ ′ ′= + −∫ , (9.52)

where and i iA B are derived from the initial conditions (9.51). Then, the complete response of the structure is obtained by superposition of the

response in each mode using Equation (9.45). We obtain:

1

( ) ( )n

i ii

t tφ=

=∑U X . (9.53)

It results that the response analysis by mode superposition requires to obtain, first the solution for the eigenvalues and eigenvectors of the eigenproblem (9.38), then the solution of the decoupled motion equations in (9.49) for each mode, and, finally, the superposition of the responses as expressed by (9.53).

9.3.4 Modal Analysis with Damping Included

The general form with damping included of the dynamic equation of a finite element structure is given by Equation (9.46) in the basis of the eigenvectors Xi. When the damping effects are neglected (previous section), Equation (9.46) leads to motion equations (9.49) which are decoupled and which can be solved indivi-dually. In the case where damping cannot be neglected, the same property of decoupled motion equations would be searched for in such a way to implement the same procedure whether damping are neglected or included.

Indeed, the damping matrix C cannot be constructed from assemblage of the element damping matrices, such as the mass and stiffness matrices of the element assemblage. So, the damping properties are deduced, in general, from the overall energy dissipation of the structure under consideration. In this way, the mode superposition analysis is particularly effective in the case where the structure damping can be expressed in the form:

2 , , 1, 2, ... , ,ti j i i ij i j nω ξ δ= =X CX (9.54)

where iξ is the modal damping coefficient and ijδ is the Kronecker notation ( 1ijδ = for i j= and 0ijδ = for i j≠ ). Therefore, Equation (9.54) assumes that

270 Chapter 9. Solution of Dynamic Equation in Finite Element Analysis

the eigenvectors are also C-orthogonal. In this case the dynamic equation (9.46) leads to n decoupled equations of the form:

2( ) 2 ( ) ( ) ( ), 1, 2, ... , ,i i i i i i it t t r t i nφ ω ξ φ ω φ+ + = = (9.55)

where ( )ir t and the initial conditions on ( )i tφ have already expressed in (9.50) and (9.51), respectively.

In the general case, the solution to each equation in (9.55) can be obtained using a numerical integration or using the Duhamel integral according to Equation (2.134) as:

( )

( )

0

1( ) cos sin ( ) sin ( ) d ,

1,2, ... , , (9.56)

i i i it

t t ti i i i i i di

dit e A t B t r t e t t t

i n

ξ ω ξ ωφ ω ω ωω

′− − −+ ′ ′ ′= + −

=

∫

with the angular frequency:

21di i iω ω ξ= − . (9.57)

The constants and i iA B are evaluated using the initial conditions (9.51). The complete response is then obtained by superposition of the response in

each mode using Equation (9.53). Equation (9.55) is the motion equation of a single degree of freedom system

with damping. The form is also the same as the motion equation (5.18) of Chapter 5 obtained in the case of damped beam vibrations. The results obtained in Section 5.3 of Chapter 5 can be transposed to the present analysis.

Finally, considering Equation (9.54) leads to the same procedure for analysis with damping neglected or damping included. The mode superposition analysis yields decoupled motion equations for which the damping matrix C is not calcu-lated, but only the stiffness and mass matrices K and M.

9.4 EVALUATION OF THE STRUCTURE DAMPING

9.4.1 Modal Damping

The damping in composite materials has been extensively considered in Chapters 5 and 6. In the case of orthotropic layers, the damping properties are characterised by the damping coefficients 11 22 66, and η η η evaluated in the material directions of the layer. These damping coefficients can be associated to the reduced stiffnesses 11 22 66, and Q Q Q of the layer, respectively. In the energy analysis of damping developed in Sections 5.4 and 5.5 of Chapter 5, the damping coefficients 11 22 66, and η η η are related to the energy dissipations in traction-compression in the longitudinal direction of the material, in traction-compression in the transverse direction and in in-plane shear, respectively. Energy is also dissi-pated by transverse shear effects and damping can be characterised by the damping coefficients 44 55and η η associated to transverse shear stiffnesses C44 and C55.

9.4. Evaluation of the Structure Damping 271

However in the case of laminate materials, the transverse shear strain energy is much lower than the in-plane energies and the transverse shear damping can be neglected.

Considering the elastic-viscoelastic correspondence principle (Section 5.1 of Chapter 5), the complex stiffness matrix e

∗K of the finite element e may be evaluated by introducing the complex reduced stiffnesses ijQ ∗′ and complex trans-verse shear moduli ijC ∗′ into Expression (8.49). Stiffnesses ijQ ∗′ are derived from the relations reported in Table 1.1 of Chapter 1 by introducing the complex stiff-nesses ijQ∗ in the material directions in the forms:

( ) ( ) ( )11 11 11 22 22 11 66 66 111 , 1 , 1 .Q Q i Q Q i Q Q iη η η∗ ∗ ∗= + = + = + (9.58)

In a similar way, the transverse shear stiffnesses ijC ∗′ are deduced from Relations (1.84) by introducing the complex transverse moduli ijC∗ in the forms:

( ) ( )44 44 44 55 55 551 , 1 .C C i C C iη η∗ ∗= + = + (9.59)

So the complex matrix of the element is obtained in the form:

e e ei∗ ′= +K K K . (9.60)

Next, the complex stiffness matrix ∗K is obtained by assemblage on the elements according to Expression (8.51). Finally the complex stiffness matrix is expressed as: i∗ ′= +K K K , (9.61) where the real part K is obtained by assemblage of the real parts of the element matrices:

elementse= ∑K K , (9.62)

and the imaginary part ′K is obtained by assemblage of the imaginary parts:

elements

e′ ′= ∑K K . (9.63)

For a structure with low damping, the eigenvalues and the eigenvectors of the damped structure are rather similar to the eigenvalues and the eigenvectors of the undamped structures. They are solutions of Equation (9.38). If Xi is the eigen-vector of mode i deduced from this equation, the energy dissipated by the damping processes is:

a12

i ti iU ′= X K X . (9.64)

The strain energy stored in the structure for the mode i is expressed as:

d12

i ti iU = X K X . (9.65)

272 Chapter 9. Solution of Dynamic Equation in Finite Element Analysis

Next, the ratio of the dissipated energy to the strain energy allows us to derive the modal damping coefficient as:

a

d

i ti i

i i ti i

UU

η′

= = X K XX K X

. (9.66)

9.4.2 Damping Matrix

As considered in Section 9.3.4, the damping effects can easily be taken into account in mode superposition analysis by introducing Equation (9.54). However, in some cases it will be more effective to use direct step-by-step integration, the modal damping coefficient iξ being evaluated for each mode i: i = 1, 2, …, n. Mode superposition analysis can be very effective if only some vibration modes are excited by the loading. In this case it is necessary to derive the damping matrix C explicitly, which yields the damping ratio iξ when C is substituted in Equation (9.54). A practical way is to assume a Rayleigh damping which is of the form:

α β= +C M K , (9.67)

where and α β are constants to be determined from given damping coefficients and p qξ ξ corresponding to two different frequencies of structure vibrations pω

and qω , respectively. Substituting the Rayleigh relation (9.67) into Equation (9.54), we obtain:

( )

t 2i i i iα β ω ξ+ =X M K X , (9.68)

and considering (9.38), this relation becomes:

2 2i i iα βω ω ξ+ = . (9.69)

Applying this relation for ( , p pω ξ ) and ( , q qω ξ ), we obtain:

2

2

2 ,

2 .

p p p

q q q

α ω β ω ξ

α ω β ω ξ

+ =

+ = (9.70)

The solution is:

22 2 2 24 4 , 4 .q q p p q q p p

p p pq p q p

ω ξ ω ξ ω ξ ω ξα ω ξ ω β

ω ω ω ω

− −= − =

− − (9.71)

With the damping evaluated by the Rayleigh damping matrix (9.67), the dam-ping ratio iξ for any value of iω is derived from Equation (9.69) as:

12i i

i

αξ βωω⎛ ⎞= +⎜ ⎟⎝ ⎠

. (9.72)

In actual analysis, the damping coefficients are known for the different modes: i = 1, 2, …, n. In this case two average values ( , p pω ξ ) and ( , q qω ξ ) will be used to

9.5 Finite Element Nonlinear Analysis 273

evaluate and α β . The average values will be chosen in such a way they fit the actual damping variation at the best.

It is possible to obtain a more complicated damping matrix when more two damping coefficients are used to establish C. However, the damping matrix which is obtained is not banded and the cost of analysis is increased significantly. So, in most practical analyses using direct integration, Rayleigh damping evaluation is considered.

9.5 FINITE ELEMENT NONLINEAR ANALYSIS

In the finite element formulation developed in Chapter 8 and in the present chapter, it was assumed that the displacements of the finite element assemblage are infinitesimally small and that the material has a linear elastic behaviour. In addition, it was also assumed that the boundary conditions remain unchanged during the application of the external loads on the finite element assemblage. With these assumptions, the dynamic finite element equations (8.69) or (9.1) corres-pond to a linear analysis of the structural problem under consideration. So, the displacement response U is a linear function of the applied load vector R: if the external loads are αR instead of R, where α is a constant, the corresponding displacements are .αU If this is not the case, the analysis is not linear.

Different nonlinear processes can be considered. Non linear structure beha-viour can be induced by material nonlinear behaviour for which the stress-strain relation is nonlinear. In the linear analysis, the displacements and strains are infinitesimally small. When material is subjected to large displacements and large strains, the stress-strain relation is also usually nonlinear. Another type of non linear analysis is the analysis of problems for which the boundary conditions change during the motion of the structure. This process arises in particular in the case of contact problems.

The solution of the non linear dynamic response of a finite element system can be obtained using the procedures already discussed: direct integration method or mode superposition. The most common explicit integration is probably the central difference method (Section 9.2.2). The shortcoming in the use of this method lies in the time step restriction and a proper choice of the time step must be used.

All the implicit time integration schemes considered in Section 9.2 for linear analysis can also be used in nonlinear dynamic analysis. A common technique is the trapezoidal scheme, which is a Newmark method with 1 1

2 4 and δ α= = (Sec-tion 8.2.5).

The basic principles of mode superposition are also applicable to nonlinear analysis of dynamic problems. In this case the vibration mode shapes and natural frequencies change. The calculation of the vibration mode shapes and frequencies at time t when these quantities have been calculated at a previous time should be achieved economically as functions of these quantities. Furthermore, the mode superposition analysis of nonlinear dynamic response is generally only effective when the solution can be obtained without updating the stiffness matrix too frequently.

CHAPTER 10

Damping of Sandwich Materials and Structures

The purpose of this chapter is to develop modelling of the damping of sandwich materials and structures based on a finite element analysis. The results derived from this modelling are then applied to the experimental evaluation of the dam-ping parameters of sandwich materials from the flexural vibrations of beam spe-cimens. Parametric studies which are essential to the design of structures are next considered.

10.1 MODELLING THE DAMPING OF SANDWICH COMPOSITE MATERIALS AND STRUCTURES

10.1.1 Stress Field in Sandwich Composite Material

A sandwich material (Chapter 1) is made from a material of low density (the core) with face sheets (the skins) of high stiffness bonded to each other of the surfaces of the core. The essential function of the core is to transfer, by transverse shear, the mechanical loads developed on one skin to the other.

In the general case, the skins are laminates of thickness hs1 for the lower skin and of thickness hs2 for the upper skin (Figure 10.1). The thickness hc of the core is usually much greater than that of the skins. The coordinate system is chosen in such a way that the (x, y) plane is the middle plane of the core.

The theory of sandwich plates considers (Section 1.6 of Chapter 1) the follo-wing fundamental assumptions on the displacement field:

i) The in-plane displacements uc and vc in the core are linear functions of the transverse z coordinate.

ii) The in-plane displacements u and v are uniform through the thickness of the skins.

iii) The transverse displacement w is independent of the z coordinate. Thus, the strain zzε is neglected in the core and the skins.

10.1. Modelling the Damping of Sandwich Materials and Structures 275

FIGURE 10.1. Sandwich Plate

The strain field and the stress field are deduced from the previous displacement field and then from the strain-stress relations. It results that the transverse shear stresses c c and xz yzσ σ are constants through the thickness of the core. Further-more, the in-plane stresses c c c, and xx yy xyσ σ σ are linear functions of the z coor-dinate through the thickness of the core. These in-plane stresses are usually neglected.

From the displacement field, it is also deduced that the in-plane stresses ,xxkσ and yyk xykσ σ are constant in each layer k of the skins (lower or upper skin), as

well as the transverse shear stresses and .xzk yzkσ σ These transverse shear stresses are usually neglected. Indeed, the stresses in each layer can be obtained consi-dering a linear variation as a function of the z coordinate.

Thus, when finite elements based on the theory of sandwich plates are used, finite element analysis gives (Figure 10.2) the values of the in-plane stresses ,xxσ

, ,yy xyσ σ in each layer k of the skins of each finite element e of the structure under consideration:

, , ,xxk yyk xykσ σ σ (10.1)

and the values of the stresses ,xxσ , , , ,yy xy xz yzσ σ σ σ on the lower face (l) and upper face (u) of the core for each finite element:

lc lc lc lc lc

uc uc uc uc uc

, , , , ,

, , , , ,xx yy xy xz yz

xx yy xy xz yz

σ σ σ σ σ

σ σ σ σ σ (10.2)

with uc lc c uc lc c, .xz xz xz xz yz yzσ σ σ σ σ σ= = = = (10.3)

It results that the in-plane stresses in the sandwich core can be expressed for the element e as: c c c( , ) ( , ), , , ,e

p p pa x y z b x y p xx yy xyσ = + = (10.4)

c 2h

s1h

s2h

ch

y

x

z

core

skin

276 Chapter 10. Damping of Sandwich Materials and Structures

FIGURE 10.2. Stresses derived from finite element analysis in the layers of the skins and in the core.

with

( )

uc lcc

c

c uc lc

,

1 .2

p pp

p p p

ah

b

σ σ

σ σ

−=

= + (10.5)

The transverse shear stresses in the layers of skins can be expressed consi-dering a linear variation through the thicknesses of skins, and considering that the shear stresses vanish on the upper and lower faces of sandwich material and are equal to the transverse shear stresses in the sandwich core. Thus, we have:

1 1 1( , ) ( , ), , ,r r rx y z x y r xz yzσ α β= + = (10.6) with

c c1 1

s1 s1, 1 ,

2r

r rh

h hσα β ⎛ ⎞= = +⎜ ⎟

⎝ ⎠ (10.7)

for the lower skin, and:

2 2 2( , ) ( , ), , ,r r rx y z x y r xz yzσ α β= + = (10.8) with

c c2 2

s2 s2, 1 ,

2r

r rh

h hσα β ⎛ ⎞= − = +⎜ ⎟

⎝ ⎠ (10.9)

for the upper skin of the sandwich material.

z

, , xxk yyk xykσ σ σ

lc lc lc lc lc, , , ,xx yy xy xz yzσ σ σ σ σ

element e

uc uc uc uc uc, , , ,xx yy xy xz yzσ σ σ σ σ

k1kh −

kh c 2h−

c 2h

0

s2h

s1h

10.1. Modelling the Damping of Sandwich Materials and Structures 277

10.1.2 In-Plane Strain Energy

We consider the case where the layers of the skins and the core of the sandwich material are constituted of orthotropic materials.

The in-plane energy deU stored in a given finite element e can be expressed as a

function of the in-plane strain energies related to the material directions as:

d 1 2 6 ,e e e eU U U U= + + (10.10) with

1 1 1

2 2 2

6 6 6

1 d d d ,2

1 d d d ,2

1 d d d ,2

e

e

e

e

e

e

U x y z

U x y z

U x y z

σ ε

σ ε

σ ε

=

=

=

∫∫∫∫∫∫∫∫∫

(10.11)

where the integrations are extended over the volume of the finite element e. The in-plane strains 1 2 6, and ε ε ε related to the material directions of the core

or layer k of the skins are expressed as functions of stresses 1 2 6, and σ σ σ in the material directions according to the strain-stress relations as:

1 11 1 12 2

2 12 1 22 2

6 66 6

,,

,

S SS SS

ε σ σε σ σε σ

= += +=

(10.12)

where the components ijS are the compliance constants of the material of core or layer k related to the engineering moduli , , and L T LT LTE E G ν by the following expressions:

11 22 12 661 1 1, , , .LT

L T L LTS S S S

E E E Gν= = = − = (10.13)

It results that Expression (10.11) of the strain energy 1eU , stored in tension-

compression in the longitudinal direction can be written in the form:

1 11 12e e eU U U= + , (10.14)

with

211 11 1

12 12 1 2

1 d d d ,2

1 d d d .2

e

e

e

e

U S x y z

U S x y z

σ

σ σ

=

=

∫∫∫∫∫∫

(10.15)

In each layer k or in the core, stresses 1 2 6, and σ σ σ , related to the material directions, can be expressed as functions of the in-plane stresses , andxx yyσ σ

278 Chapter 10. Damping of Sandwich Materials and Structures

,xyσ related to the finite element directions (x, y, z), according to the stress trans-formation relation:

2 21

2 22

2 26

cos sin 2sin cos

sin cos 2sin cos

sin cos sin cos cos sin

xx

yy

xy

θ θ θ θσ σσ θ θ θ θ σσ σθ θ θ θ θ θ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥= −⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ − −⎢ ⎥ ⎣ ⎦⎣ ⎦

, (10.16)

where θ is the orientation of material in layer k or in the core. Whence:

( )

2 4 2 411 11

2 2 2

3 3

1 cos sin2

2 2 sin cos

4 sin cos 4 sin cos d d d ,

exx yy

e

xy xx yy

xx xy yy xy

U S

x y z

σ θ σ θ

σ σ σ θ θ

σ σ θ θ σ σ θ θ

⎡= +⎣

+ +

⎤+ + ⎦

∫∫∫ (10.17)

( )( )

( )( )

2 2 2 2 212 12

4 4

2 2

1 4 sin cos2

sin cos

2 sin cos sin cos d d d .

exx yy xy

e

xx yy

xx xy yy xy

U S

x y z

σ σ σ θ θ

σ σ θ θ

σ σ σ σ θ θ θ θ

⎡= + −⎣

+ +

⎤+ − − ⎦

∫∫∫ (10.18)

In the same way, the strain energy 2U , stored in tension-compression in the transverse direction can be written in the form:

2 22 12e e eU U U= + , (10.19)

with

( )

2 4 2 422 22

2 2 2

3 3

1 sin cos2

2 2 sin cos

4 sin cos 4 sin cos d d d ,

exx yy

e

xy xx yy

xx xy yy xy

U S

x y z

σ θ σ θ

σ σ σ θ θ

σ σ θ θ σ σ θ θ

⎡= +⎣

+ +

⎤− − ⎦

∫∫∫ (10.20)

Lastly, the in-plane shear strain energy is written as:

( )

( )( )

2 2 2 26 66 66

22 2 2

2 2

1 2 sin cos2

cos sin (10.21)

2 cos sin sin cos d d d

e exx yy xx yy

e

xy

yy xy xx xy

U U S

x y z

σ σ σ σ θ θ

σ θ θ

σ σ σ σ θ θ θ θ

⎡= = + −⎣

+ −

⎤+ − − ⎦

∫∫∫

.

The in-plane strain energies stored in the finite element e can be expressed as:

10.1. Modelling the Damping of Sandwich Materials and Structures 279

11 11c 11 22 22c 221 1

12 12c 12 66 66c 661 1

, ,

, ,

n ne e e e e e

k kk k

n ne e e e e e

k kk k

U U U U U U

U U U U U U

= =

= =

= + = +

= + = +

∑ ∑

∑ ∑ (10.22)

where cepqU and ( 11, 22, 12, 66)e

pqkU pq = are respectively the in-plane strain energies stored in the core and layer k of the skins of the element e and n is the total number of layers in the lower and upper skins.

It results from Expressions (10.17) to (10.21) that the in-plane strain energies stored in layer k of the skins or in the core of the element e can be expressed as:

( )

4 411 11

2 2

3 3

cos sin

2 2 sin cos

4 sin cos 4 sin cos ,

e e el l xxxxl l yyyyl l

e exyxyl xxyyl l l

e exxxyl l l yyxyl l l

U S U U

U U

U U

θ θ

θ θ

θ θ θ θ

⎡= +⎣

+ +

⎤+ + ⎦

(10.23)

( )( )

( )( )

2 212 12

4 4

2 2

4 sin cos

sin cos

2 sin cos sin cos ,

e e e el l xxxxl yyyyl xyxyl l l

exxyyl l l

e exxxyl yyxyl l l l l

U S U U U

U

U U

θ θ

θ θ

θ θ θ θ

⎡= + −⎣

+ +

⎤+ − − ⎦

(10.24)

( )

4 422 22

2 2

3 3

sin cos

2 2 sin cos

4 sin cos 4 sin cos ,

e e el l xxxxl l yyyyl l

e exyxyl xxyyl l l

e exxxyl l l yyxyl l l

U S U U

U U

U U

θ θ

θ θ

θ θ θ θ

⎡= +⎣

+ +

⎤− − ⎦

(10.25)

( )( )

( )( )

2 266 66

22 2

2 2

2 sin cos

cos sin

2 sin cos sin cos ,

e e e el l xxxxl yyyyl xxyyl l l

exyxyl l l

e exxxyl yyxyl l l l l

U S U U U

U

U U

θ θ

θ θ

θ θ θ θ

⎡= + −⎣

+ −

⎤+ − − ⎦

(10.26)

These expressions introduce the orientation kθ of layer k of the skins ( l k= ) or

the orientation cθ of the core ( cl = ). The terms epqlU ( , , , p q xx yy xy= ) are

derived by considering Expressions (10.1) and (10.4) for the in-plane stresses in the core and in layer k of the skins. We obtain:

— for the layer k of the skins ( l k= ):

1 , , , , ,2

e epql pq k pk qk e kU U S e p q xx yy xyσ σ= = = (10.27)

where Se is the area of the finite element e and ek is the thickness of the layer k;

280 Chapter 10. Damping of Sandwich Materials and Structures

— for the core ( cl = ):

2c

c c c c c c1 , , , , ,2 12

e epql pq p q p q e

hU U a a b b S h p q xx yy xy⎛ ⎞

= = + =⎜ ⎟⎝ ⎠

(10.28)

where the coefficients c c c c, , and ,p q p qa a b b are given by Expressions (10.5). Next, the in-plane strain energy stored in element e is given by Expressions

(10.22) and the total in-plane strain energies stored in the finite element assem-blage is then obtained by summation on the elements as:

11 11 12 12elements elements

22 22 66 66elements elements

, ,

, .

e e

e e

U U U U

U U U U

= =

= =

∑ ∑

∑ ∑ (10.29)

10.1.3 Transverse Shear Strain Energy

The transverse shear strain energy for a given element e can be expressed in the material directions as: s 44 55

e e eU U U= + , (10.30) with

44 4 4

55 5 5

1 d d d ,2

1 d d d ,2

e

e

e

e

U x y z

U x y z

σ γ

σ γ

=

=

∫∫∫∫∫∫

(10.31)

where the integration is extended over the volume of the finite element e. 4 4 and σ γ are respectively the transverse shear stress and strain in plane ( , T T ′ )

of material in the layer k of the skins or in the core of sandwich material. 5 5 and σ γ are the transverse shear stress and strain in plane ( , L T ′ ) of the

materials. The transverse shear strains and stresses are related by:

4 4 5 5, ,TT LTG Gσ γ σ γ′ ′= = (10.32)

where and TT LTG G′ ′ are the transverse shear moduli in planes ( , T T ′ ) and ( , L T ′ ), respectively. It results that the transverse shear strain energies (10.31) can be written as:

24

44

25

55

1 d d d ,2

1 d d d .2

e

TTe

e

LTe

U x y zG

U x y zG

σ

σ

′

′

=

=

∫∫∫∫∫∫

(10.33)

10.1. Modelling the Damping of Sandwich Materials and Structures 281

In each layer k of the skins or in the core, stresses 4 5and ,σ σ related to material directions of the layer or the core, can be expressed as functions of the transverse shear stresses and yz xzσ σ in the finite element directions (x, y, z) according to the stress transformations:

4

5

cos sinsin cos

yz

xz

σ θ θ σσ θ θ σ

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦. (10.34)

So, the transverse shear strain energies (10.33) are expressed as:

( )

2 2 2 2

441 1 cos sin 2 sin cos d d d ,2

eyz xz yz xz

TTeU x y z

Gσ θ σ θ σ σ θ θ

′= + −∫∫∫

(10.35)

( )

2 2 2 2

551 1 sin cos 2 sin cos d d d .2

eyz xz yz xz

LTeU x y z

Gσ θ σ θ σ σ θ θ

′= + +∫∫∫

(10.36) The transverse shear energies can be expressed as:

44 44c 44 55 55c 551 1

, ,n n

e e e e e ek k

k k

U U U U U U= =

= + = +∑ ∑ (10.37)

where c and ( 44, 55)e ers rskU U rs = are respectively the transverse shear strain

energies stored in the core and layer k of the skins of the element.

It results from Expressions (10.35) and (10.36) that the transverse shear strain energies stored in layer k of the skins or in the core of the element e can be expressed as:

( )2 244

1 cos sin 2 sin cos ,e e e el yzyzl l xzxzl l yzxzl l l

TTU U U U

Gθ θ θ θ

′= + − (10.38)

( )2 255

1 sin cos 2 sin cos .e e e el yzyzl l xzxzl l yzxzl l l

LTU U U U

Gθ θ θ θ

′= + + (10.39)

As previously, these expressions introduce the orientation kθ of layer k of the

skins ( l k= ) or the orientation cθ of the core ( cl = ). The terms erslU

( , , r s yz xz= ) are derived by considering Expressions (10.2), (10.6) and (10.8) of the transverse shear stresses in the core and in the layer k of the skins. We obtain:

— for the core ( cl = ):

c c c c1 , , , ,2

e ersl rs r s eU U S h r s yz xzσ σ= = = (10.40)

282 Chapter 10. Damping of Sandwich Materials and Structures

— for the layer k of the skins ( l k= ):

( ) ( )3 3 2 2

1 11

1 ,2 3 2

, , , 1 for the lower skin and 2 for the upper skin, (10.41)

e ersl rsk

k k k kri si ri si si ri ri si k k e

U U

h h h h h h S

r s yz xz i i

α α α β α β β β− −−

= =

⎡ ⎤− −+ + + −⎢ ⎥⎣ ⎦

= = =

where the coefficients , , and ,ri si ri siα α β β are given by Expression (10.7) for the lower skin ( 1i = ) and Expression (10.9) for the lower skin ( 2i = ).

Next, the transverse shear strain energy stored in element e is given by Expressions (10.37) and the total transverse strain energies stored in the finite element assemblage is then obtained by summation on the elements as:

44 44 55 55elements elements

, .e eU U U U= =∑ ∑ (10.42)

10.1.4 Damping of a Sandwich Composite Structure

The damping of the finite element assemblage can be evaluated by extending the energy formulation approach considered in Section 5.5.

The total strain energy stored in the sandwich structure is given by:

d 11 22 12 66 44 552U U U U U U U= + + + + + , (10.43)

where the in-plane strain energies 11 22 12 66, , 2 and U U U U are expressed by Equations (10.29), and the transverse shear strain energies 44 55 and U U are given by Expressions (10.42).

Then, the energy dissipated by damping in the layer k of the skins or in the core of the sandwich material of the element e is derived from the strain energy stored in layer k or in the core as:

11 11 22 22 12 12 66 66

44 44 55 55

2

,

e e e e e e e e el l l l l l l l l

e e e el l l l

U U U U U

U U

ψ ψ ψ ψ

ψ ψ

∆ = + + +

+ + (10.44)

introducing the specific damping coefficients of the layer k of the skins or of the core. These coefficients are referred to the material directions ( , , L T T ′ ) of the

layer k of the skins ( l k= ) or of the core ( cl = ): 11 22 and e el lψ ψ are the damping

coefficients in traction-compression in the L direction and T direction of the layer k of the skins or of the core, respectively; 12

elψ is the in-plane coupling coeffi-

cient; 66e

lψ is the in-plane shear coefficient; 44 55 and e el lψ ψ are the transverse

shear damping coefficients in planes ( , T T ′ ) and ( , L T ′ ), respectively. The damping energy dissipated in the element e is next obtained by summation

on the core and on the layers of the skins of element e as:

10.1. Modelling the Damping of Sandwich Materials and Structures 283

c1

,n

e e ek

k

U U U=

∆ = ∆ + ∆∑ (10.45)

and the total energy U∆ dissipated in the finite element assemblage is then obtained by summation on the elements:

elements

.eU U∆ = ∆∑ (10.46)

Finally, the damping of the finite element assemblage is characterised by the damping coefficient ψ of the assemblage derived from relation:

d

UU

ψ ∆= . (10.47)

The in-plane coupling energy 12eU is much lower than the other in-plane ener-

gies and can be neglected. A general procedure was implemented to evaluate the damping of a sandwich

composite structure using finite element analysis. This procedure is based on the previous formulation and can be applied to any structure for which the damping characteristics are different according to the core and the layers of the skins and according to the elements of the assemblage.

This procedure is applied hereafter to the analysis of the damping of sandwich materials. The application to a sandwich composite structure is considered in Chapter 11.

10.2 EXPERIMENTAL INVESTIGATION OF THE DAMPING OF SANDWICH MATERIALS

10.2.1 Materials

Experimental investigation of sandwich materials is developed in [83]. We report hereafter the elements of this investigation.

The sandwich materials investigated were constructed with glass fibre lami-nates as skins and with PVC closed-cell foams as core.

The glass fibre laminates of the skins are cross-ply laminates constituted of unidirectional layers of E-glass fibres in an epoxy matrix, arranged in the sequence [0°/90°/90°/0°]. The unidirectional layers were fabricated with unidirec-tional fabrics of weight 300 gm–2 with glass fibres aligned in a single direction. The engineering constants of the unidirectional layers referred to the material directions ( , ,L T T ′ ) or (1, 2, 3) were measured in static tests as mean values of 10 tests for each constant. The values obtained are reported in Table 10.1. Experimental damping analysis of the unidirectional glass fibre layers is invest-tigated in Chapter 6 and the values of the loss factors derived from this analysis are reported in Table 10.2.

284 Chapter 10. Damping of Sandwich Materials and Structures

TABLE 10.1. Engineering constants of the unidirectional glass fibre layers derived from static tests.

EL (GPa) ET (GPa) νLT GLT (GPa)

29.9 7.50 0.24 2.25

TABLE 10.2. Loss factors in the material directions of the unidirectional glass fibre layers, derived from laminate beam vibrations.

f (Hz) η11 (%) η12 η22 (%) η66 (%)

50 0.35 0 1.30 1.80

300 0.40 0 1.50 2.00

600 0.45 0 1.65 2.22

The PVC closed-cell foams were supplied in panels of thickness of 15 mm. Three foams were considered in the experimental investigation differing in their densities: 60 kg m–3, 80 kg m–3 and 200 kg m–3. Mechanical characteristics of the foams were measured in static tensile tests for the Young’s modulus and the Poisson’s ratio, and in static shear tests for the shear modulus. The experimental results obtained show that the foams are fairly isotropic and the modulus values derived are reported in Table 10.3.

Sandwich materials were constructed with these foams and with cross-ply glass-fibre laminates prepared by hand lay-up process, which leads to a nominal thickness of 1.2 mm for the sandwich skins.

TABLE 10.3. Mechanical characteristics of the foams derived from static tests.

Density of the foam

(kg m–3)

Young’s modulus (MPa) Poisson’s ratio Shear modulus

(MPa)

60 59 0.42 22

80 83 0.43 30

200 240 0.45 80

10.2. Experimental Investigation of the Damping of Sandwich Materials 285

10.2.2 Determination of the Constitutive Damping Parameters

The damping characteristics of the materials were obtained by subjecting beams to flexural vibrations. The equipment used is the same as the one used for the experimental investigation of laminates (Figure 6.1). The test specimen is supported horizontally as a cantilever beam in a clamping block. An impulse hammer is used to induce the excitation of the flexural vibrations of the beam near the clamping box (Figure 10.3) and the beam response is detected near the free end of the beam by using a laser vibrometer. Next, the excitation and the response signals are digitalized and processed by a dynamic analyzer of signals. This analyzer associated with a PC computer performs the acquisition of signals, controls the acquisition conditions and next performs the analysis of the signals acquired (Fourier transform, frequency response, mode shapes, etc.).

The flexural beam responses were identified in the frequency domain using MATLAB Toolbox, and the identification procedure allows us to obtain the values of the natural frequencies fi and the modal loss factors ηi, related to the specific damping coefficient by the relation ψi = 2πηi.

10.3 RESULTS AND DISCUSSION

10.3.1 Determination of the Dynamic Characteristics of the Foams

10.3.1.1 Test Specimens

The experimental investigation of the dynamic properties of the foams has been implemented according to the experimental process considered in the standard ASTM E 756 [79], which was introduced in Subsection 7.3.3.1 of Chapter 7. Properties of the foams were evaluated from the flexural vibrations of a clamped-free beam 40 mm wide and constituted of two aluminium beams with foam material interleaved between the two aluminium beams (Figure 10.4). An aluminium spacer was added in the root section which was closely clamped in a

Figure 10.3. Impact and measuring points.

Clamping box Impact point Measuring point

x

286 Chapter 10. Damping of Sandwich Materials and Structures

FIGURE 10.4. Aluminium-foam beam for determining the dynamic characteristics of the foams.

rigid fixture. The free length and the thicknesses of the aluminium beams were selected so as to measure the foam characteristics on the frequency range [50, 1,000 Hz]. Thus, the beam dimensions used were a free length equal to 300, 350, 400 and 450 mm, with thicknesses of the aluminium beams of 4 mm.

The experimental results deduced for the damping of the aluminium-foam beams are reported in Figure 10.5 for the three densities of the foam core. In fact, these results do not represent the damping of the foam core. They have to be corrected by considering the damping induced by the aluminium in the damping of the aluminium-foam beams. This correction is considered hereafter. FIGURE 10.5. Experimental beam damping derived from the bending of aluminium-foam beams, for the three densities of the foam.

Clamping block

Foam core

Aluminium core Aluminium or laminate skins

Frequency ( Hz )0 200 400 600 800 1000 1200

Loss

fact

or η

ι ( %

)

0

1

2

3

4

5

6

60 kg/m3

80 kg/m3

200 kg/m3

10.3. Results and Discussion 287

FIGURE 10.6. Energies stored in the foam core of aluminium-foam beams for the first mode of the bending vibrations.

10.3.1.2 Energies Stored in the Test Specimens and Procedure

Figure 10.6 shows a typical example of the values of the different strain ener-gies stored in the foam core of the aluminium-foam beams for the first mode of the bending vibrations. These results were obtained using a finite element analysis in the case of a foam of density 60 kg m–3 and for a length of the beam equal to 300 mm. The results show that the longitudinal transverse shear energy is clearly higher than the other energies.

Figures 10.7 to 10.9 show the evolution of the energies stored in the aluminium beams and in the foam core as functions of the frequency for the first three modes. The results were obtained using a finite element analysis in the case of foams of density 60 kg m–3 (Figure 10.7), 80 kg m–3 (Figure 10.8) and 200 kg m–3 (Figure 10.9). The results obtained show that the energy stored in the foam cores is higher than the energy stored in the aluminium beams, except for the first mode of the beams with a foam core of 200 kg m–3.

Damping of aluminium was deduced from the flexural vibrations of aluminium beams. The results obtained show a fairly linear variation of the loss factor from 0.22 % to 0.32 % in the frequency domain of investigation [50, 1,000 Hz].

From the preceding results, it follows that it would be possible to consider only the longitudinal transverse shear energy stored in the foam cores. In fact, the shear modulus and the damping of the foams were deduced by fitting the experimental frequency responses of the aluminium-foam beams with the results deduced from the modelling considered in Section 10.1 introducing the damping of aluminium.

Frequency ( Hz )0 20 40 60 80 100 120 140 160 180 200

Ener

gy (

J )

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40Uxz

Uxy x 105

Uyy x 104

Uxx x 102

Uyz x 105

288 Chapter 10. Damping of Sandwich Materials and Structures

FIGURE 10.7. Energies stored in the aluminium beams and in the foam core as functions of the frequency for the first three modes, in the case of foam density of 60 kg m–3.

FIGURE 10.8. Energies stored in the aluminium beams and in the foam core as functions of the frequency for the first three modes, in the case of foam density of 80 kg m–3.

Frequency ( Hz )0 200 400 600 800 1000

Ener

gy

0.0

0.2

0.4

0.6

0.8

1.0

mode 1 mode 2 mode 3

Ucore/Utotal

Uskins/Utotal

Frequency ( Hz )0 200 400 600 800 1000

Ener

gy

0.0

0.2

0.4

0.6

0.8

1.0

mode 1 mode 2 mode 3

Uskins/Utotal

Ucore/Utotal

10.3. Results and Discussion 289

Frequency ( Hz )0 200 400 600 800 1000 1200

Ener

gy

0.0

0.2

0.4

0.6

0.8

1.0

mode 1

mode 2 mode 3

Ucore/Utotal

Uskins/Utotal

FIGURE 10.9. Energies stored in the aluminium beams and in the foam core as functions of the frequency for the first three modes, in the case of foam density of 200 kg m–3.

Furthermore, it has been noted (Section 10.2.1) that the foams were isotropic. So, in the investigation of the dynamic properties it was considered that the Young’s modulus and the shear modulus of foams was related by the usual relation of isotropic material, with a Poisson ratio independent of the frequency equal to the values determined in static tests (Table 10.3).

10.3.1.3 Dynamic Characteristics of the Foams

Figures 10.10 and 10.11 report the results obtained for the shear modulus (Fi-gure 10.10) and the loss factors (Figure 10.11) as functions of the frequency for the three densities of the foams: 60, 80 and 200 kg m–3. These results are deduced from the experimental investigation and finite element analysis according to the procedure described previously. In the frequency range studied, it is observed (Figure 10.10) a fairly linear variation of the shear modulus as a function of the logarithm of the frequency. The shear modulus increases clearly with the foam density: the shear modulus is multiplied by about 4 when the foam density is in-creased from 60 to 200 kg m–3.

So, the variation of the shear modulus of the foams as function of the fre-quency can be expressed in the linear form:

c clog , (MPa), (Hz),G A B f G f= + (10.48)

where the values of the parameters A and B depend on the density of the foams. The values are reported in Table 10.4.

290 Chapter 10. Damping of Sandwich Materials and Structures

FIGURE 10.10. Shear modulus of foams as function of frequency deduced from the experimental investigation and the finite element analysis. FIGURE 10.11. Damping of foams as function of frequency deduced from the experi-mental investigation and the finite element analysis.

Frequency ( Hz )10 100 1000 10000

Shea

r mod

ulus

( M

Pa )

20

40

60

80

100

120

140

60 kg/m3

80 kg/m3

200 kg/m3

Frequency ( Hz )0 150 300 450 600 750 900 1050 1200

Loss

fact

or η

i ( %

)

0

1

2

3

4

5

6

7

60 kg/m3

80 kg/m3

200 kg/m3

10.3. Results and Discussion 291

TABLE 10.4. Values of the parameters of the frequency variation of the shear modulus of the foams according to the foam density.

Foam density (kg/m3) A B

60 6.58 3.70

80 5.28 5.56

200 4.17 17.8

In contrast, the results of Figure 10.11 show that the foam damping is not much depending on the foam density. The foam damping decreases with the frequency according to a general property of the viscoelastic materials. Furthermore, compa-ring the results of Figure 10.11 with the results of Figure 10.5 shows that the actual damping of the foams is somewhat different from the damping of the aluminium-foam beams. So the results of Figure 10.5 need to be corrected using the finite element procedure to derive the damping of the foams.

10.3.2 Analysis of the Damping of Sandwich Materials

10.3.2.1 Introduction

Investigation of damping was implemented in the case of the sandwich mate-rials considered in Subsection 10.2.1. The experimental investigation was carried out in the case of flexural vibrations of clamped-free beams with the same condi-tions as the ones considered in the analysis of the foam properties (Subsection 10.3.1). An aluminium spacer was added in the root section which was closely clamped in a rigid fixture. The free length of the beams was equal to 300, 350, 400 and 450 mm, and the beam width was equal to 40 mm.

The experimental results obtained are compared with the results deduced from the modelling developed in Section 10.1. This analysis takes account of the varia-tions of the shear modulus and damping of foam core as functions of the fre-quency (Figures 10.10 and 10.11), as well as the variation of the damping of the unidirectional layers of the skins (Table 10.2).

10.3.2.2 Mode shapes

The shapes of the modes 1 to 6 deduced from finite element analysis are repor-ted in Figure 10.12 for the free flexural modes of the sandwich beams, in the case of a beam length of 350 mm and a density of 60 kg m–3 of the foam core. Beam twisting is observed for the modes 2 and 4. Because of the position of the measu-ring point (Figure 10.3), these modes are not really observed in the experimental investigation. So, the analysis implemented hereafter considers only the four bending modes 1, 3, 5 and 6.

292 Chapter 10. Damping of Sandwich Materials and Structures

FIGURE 10.12. Free flexural modes of sandwich beams, for a length of 350 mm of the beams and a density of 60 kg m–3 of the foam core.

10.3.2.3 Damping of the Sandwich Materials

Figure 10.13 compares the results deduced from the experimental investigation and from modelling for the sandwich materials, in the case of core with foams of 60, 80 and 200 kg m−3. The comparison shows a good agreement between the experimental results and the results deduced from modelling. This agreement

mode 1 mode 2

mode 3 mode 4

mode 5 mode 6

10.3. Results and Discussion 293

FIGURE 10.13. Comparison between the results deduced from experiment and modelling for the damping of sandwich materials, in the case of core with foams of 60, 80 and 200 kg m−3.

underlines that the modelling considered is well suited to describe the damping of sandwich materials constituted of a foam core and laminated skins.

Furthermore, it is observed that damping as a function of the frequency is discontinuous from one mode to the other, due to a change of the distribution of the strain energies in the beam volume according to the vibration mode. Damping of sandwich materials increases when the density of the core foam decreases, so when the shear modulus of the foam decreases, for a given mode. Also, for a given mode and foam density, damping of sandwich materials decreases when the frequency increases, that is associated to the increase of the shear modulus of the foam as function of the frequency (Figure 10.10) as well as the decrease of the foam damping with the frequency (Figure 10.11).

The evolution of the damping of the sandwich materials from one mode to the other depends in fact on different factors as the distribution of the strain energy between the skins and the foam core, the variation of the shear modulus of the foam as a function of the frequency (Figure 10.10), the variation of the foam damping (Figure 10.11) and the variation of the layer damping with the frequency (Table 10.2). The influence of these different factors is considered in the following section.

Frequency ( Hz )0 400 800 1200 1600 2000 2400

Loss

fact

or η

i ( %

)

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

mode 1mode 2

mode 3 mode 4

60 kg/m3

80 kg/m3

200 kg/m3Experiment

Modelling

294 Chapter 10. Damping of Sandwich Materials and Structures

10.4 CHARACTERISTIC FACTORS OF THE DAMPING OF SANDWICH MATERIALS

10.4.1 Influence of the Shear Modulus of the Foam Core

For a given damping of the foam core, one of the principal factors which governs the damping of sandwich materials is the shear modulus of the foam core. Figure 10.14 shows the results derived from a finite element analysis for the damping of sandwich materials as a function of the value of the shear modulus of the foam core, in the case of the two first modes of sandwich beams of lengths 300, 350, 400 and 450 mm. The damping considered in the analysis for the foam core is deduced from the results reported in Figure 10.11. Indeed, the foam dam-ping is fairly similar when the shear modulus of the foam is increased. The results of Figure 10.14 show clearly that an increase of the shear modulus of the foam core yields a significant decrease of the damping of sandwich materials: when the shear modulus is multiplied by 8, the sandwich damping is divided by a factor about 4 for the first mode and about 3 for the second mode.

For a given mode and shear modulus of the foam core, damping of sandwich materials decreases when the frequency increases, then when the length of the

FIGURE 10.14. Influence of the shear modulus of the foam core on the damping of sandwich materials, for the first two modes of sandwich beams.

Frequency ( Hz )0 100 200 300 400 500 600

Loss

fact

or η

i (

% )

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2mode 1

G = 10 MPa

mode 2

G = 20 MPa

G = 30 MPa

G = 80 MPa

10.4. Characteristic Factors of the Damping of Sandwich Materials 295

sandwich beams is increased. This result can be associated to a decrease of the part of the transverse shear strain energy stored in the foam core.

The variation of the damping from one mode to the other is associated to the distribution of the strain energies in the skins and the core. Elements on this aspect are considered in the following subsections.

10.4.2 Energies Dissipated in the Core and Skins

Figures 10.15, 10.16 and 10.17 report the energies dissipated in the core and in the skins for the first four modes of the bending vibrations of the sandwich beams of lengths 300, 350, 400 and 450 mm, in the case of foam cores of density 60, 80 and 200 kg m–3, respectively. These results were derived from finite element mo-delling.

For the densities of the foam core equal to 60 and 80 kg m–3, the energy dis-sipated in the foam core is clearly higher than the energy dissipated in the skins. Also, it is observed an increase of the energy dissipated in the foam core and a decrease of the energy dissipated in the skins when the frequency increases as well as the mode number increases. This behaviour can be associated to the domi-nant effect of the transverse shear in the foam core, which results from the low values of the shear modulus of the foam core and which increases with the mode because of the mode shape.

FIGURE 10.15. Energies dissipated in the core and in the skins for the first four modes of the bending vibrations of the sandwich beams of lengths 300, 350, 400 and 450 mm, in the case of a foam core of density 60 kg m–3.

Frequency ( Hz )0 200 400 600 800 1000 1200 1400 1600 1800

Dis

sipa

ted

ener

gy

0.0

0.2

0.4

0.6

0.8

1.0

mode 1

∆Uskins/∆Utotal

mode 2 mode 3 mode 4

∆Ucore/∆Utotal

296 Chapter 10. Damping of Sandwich Materials and Structures

FIGURE 10.16. Energies dissipated in the core and in the skins for the first four modes of the bending vibrations of the sandwich beams of lengths 300, 350, 400 and 450 mm, in the case of a foam core of density 80 kg m–3. FIGURE 10.17. Energies dissipated in the core and in the skins for the first four modes of the bending vibrations of the sandwich beams of lengths 300, 350, 400 and 450 mm, in the case of a foam core of density 200 kg m–3.

Frequency ( Hz )0 200 400 600 800 1000 1200 1400 1600 1800

Dis

sipa

ted

ener

gies

0.0

0.2

0.4

0.6

0.8

1.0

mode 1

∆Ucore/∆Utotal

mode 2 mode 3 mode 4

∆Uskins/∆Utotal

Frequency ( Hz )

0 500 1000 1500 2000 2500

Dis

sipa

ted

ener

gies

0.2

0.3

0.4

0.5

0.6

0.7

0.8

mode 1

∆Ucore/∆Utotal

mode 2 mode 3 mode 4

∆Ucore/∆Utotal

∆Uskins/∆Utotal

∆Uskins/∆Utotal

10.4. Characteristic Factors of the Damping of Sandwich Materials 297

Thickness ratio hcore/hskins

0 5 10 15 20 25 30

Dis

sipa

ted

ener

gies

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

mode 1

∆Ucore/∆Utotal

∆Uskins/∆Utotal

mode 2

mode 3

mode 2

mode 3

In contrast, for the density of 200 kg m–3 of the foam core, the energy dissipated in the foam core increases with the frequency and the mode number, when the energy dissipated in the skins decreases. Moreover, the energy dissipated in the foam core is smaller for the modes 1 and 2 and higher for the mode 4. For this foam density the value of the shear modulus is high, and the results observed can be associated with two different elements: the distribution of the energy in the core and skins according to the modes, and the increase of the shear modulus as a function of the frequency. For the modes 1 and 2, the transverse deformation of the foam core is less pronounced than for the mode 4, which yields, with a high value of the shear modulus of the foam core, to a low energy dissipated in the foam core.

The preceding effect according to the mode shape is also underlined when we consider the evolution of the energies dissipated as functions of the ratio of the core thickness to the skin thickness (Figure 10.18). In this figure, the dissipated energies derived from finite element analysis are reported for the first three modes of the bending vibrations of sandwich beams of 350 mm long and a foam density of 60 kg m–3. Skins of the beams are [0/90]s cross-ply laminates. It is observed that the energy dissipated in the core increases with the thickness of the foam core. For the first mode, the energy dissipated in the skins is higher than the energy dissi-pated in the foam core for low values of core thickness: values of core thickness smaller than about four times the skin thicknesses. Also, the energy dissipated in the foam core increases when the mode number increases, which shows clearly FIGURE 10.18. Energies dissipated in the core and in the skins for the first third modes of the bending vibrations of sandwich beams of lengths 350 mm as functions of the ratio of core and skins thicknesses, in the case of a foam core of density 60 kg m–3 and (0/90)s skins.

298 Chapter 10. Damping of Sandwich Materials and Structures

that the part of the transverse shear energy dissipated in the foam core increases with the mode number. This effect is induced by the shapes of the vibration mode.

10.4.3 Effect of the Core Thickness on the Damping of Sandwich Materials

Figure 10.19 shows the evolution of the damping of the sandwich materials as a function of the thickness of the foam core, for a foam density of 60 kg m–3. It is observed two different evolutions of the loss factors as function of the frequency according to the foam thickness: for thicknesses of the foam core of 3.5, 5 and 7.5 mm, the loss factor increases as functions of the frequency (Figure 10.19a), when the loss factor decreases for thicknesses of the core higher or equal to 10 mm (Figure 10.19b). These two different behaviours can be associated to the evolution of the distribution of the energies dissipated in the core and skins as functions of the core thickness. For low values of the core thickness, the effect of the damping of the skins is dominant and the increase of the damping of the skin layers with frequency (Table 10.2) induces an increase of the resultant damping of the sandwich materials. For high values of the skin thicknesses, the effect of the damping of foam core is dominant and the increase of the shear modulus of the foam with the frequency (Figure 10.10) as well as the decrease of the foam damping (Figure 10.11) lead to a decrease of the damping of the sandwich mate-rials as function of the frequency.

10.5 CONCLUSIONS

An evaluation of the damping of sandwich materials constituted of a foam core and laminated skins was presented based on the theory of sandwich plates and on a finite element analysis of the vibrations of a composite structure. The analysis derived the strain energies stored in the material directions of the foam core and in the material directions of the layers of the skins. Next, the energy dissipated by damping in the structure can be obtained as a function of the strain energies and the damping coefficients associated to the different energies stored in the material directions of the core and of the layers of the skins.

Damping characteristics of laminates were evaluated experimentally using beam specimens subjected to an impulse input. The flexural beam responses were identified in the frequency domain, and the identification procedure allows us to obtain the values of the natural frequencies and the modal loss factors.

Dynamic properties of the foams of different densities were first measured using free-clamped beams constituted of two aluminium beams with foam material interleaved between the two aluminium beams. The experimental invest-tigation shows a significant increase of the shear modulus of the foams associated to a decrease of the loss factors as functions of the frequency, depending on the

10.5. Conclusions 299

FIGURE 10.19. Evaluation of the damping of the sandwich materials for various thicknesses of the foam core, in the case of (0/90)s cross-ply laminate skins and a foam density of 60 kg m–3: (a) thickness core smaller than 7.5 mm and (b) thickness core higher than 10 mm.

Frequency ( Hz )0 100 200 300 400 500 600 700

Loss

fact

or η

i ( %

)

0.9

1.0

1.1

1.2

1.3

1.410 mm15 mm20 mm25 mm30 mm

Core thickness

mode 1mode 2

(b)

Frequency0 50 100 150 200 250 300 350

Loss

fact

or η

i ( %

)

0.6

0.7

0.8

0.9

1.0

1.1

1.2

mode 1

mode 2

3.5 mm 5 mm7.5 mm

Core thickness

(a)

300 Chapter 10. Damping of Sandwich Materials and Structures

foam density. These variations as functions of frequency have to be considered in the analysis of the damping of the sandwich materials.

Next, modelling was applied to the analysis of the dynamic properties of sandwich materials constituted of the foams as core and of glass [0/90]s cross-ply laminates as skins. The comparison between the results deduced from the experi-mental investigation and the results derived from the modelling shows that the modelling describes fairly well the damping properties of the foam sandwich materials.

Furthermore, modelling allows us to underline the characteristic factors which govern the damping of sandwich materials. The principal parameters are the shear modulus of the foam core and the ratio of the core thickness to the thickness of the skins. Also, modelling shows clearly the effect of the distribution of the energies dissipated in the foam core and in the skins on the damping of the sandwich mate-rials. Damping is more dependent of the energy dissipated in the skins for the first mode when damping depends essentially on the energy dissipated in the foam core for the other modes. This effect results from the shape of the modes which induce a higher transverse shear deformation of the core for modes of high order.

CHAPTER 11

General Formulation of Damping of Composite Materials and Structures

The purpose of this chapter is to report a general formulation of the different concepts introduced in the present textbook by applying these concepts to the analysis of the damping of different laminates and sandwich materials and to the analysis of the damping and vibrations of a simple shape structure.

11.1 MATERIALS

The materials considered in the analysis are laminate materials, sandwich mate-rials and laminates with interleaved viscoelastic layers.

The laminate materials are constituted of E-glass fibres in an epoxy matrix and were fabricated with different layers: unidirectional layers studied in Section 6.2 of Chapter 6, unidirectional woven fabric layers, taffeta layers and serge weave layers. The reinforcement for unidirectional layers was unidirectional fabrics of weight 300 g m–2. In the unidirectional woven fabric, glass fibres are aligned in the warp direction and are held together by fine weft glass threads, so that the fabric is mostly unidirectional. The weight of fabrics was 200 gm–2. The weights of taffeta fabrics and serge fabrics were 200 g m–2 and 300 g m–2, respectively.

Laminate materials were prepared by hand lay-up process from epoxy resin with hardener and glass fabrics. Plates of different dimensions were cured at room temperature with pressure using vacuum moulding process, and then post-cured in an oven. The plates were fabricated with 8 or 12 layers according to the weight of reinforcement in such a way to obtain the same plate thickness (nominal value of 2.4 mm) with the same reinforcement volume fraction (nominal value of 0.41). The engineering constants of the laminates referred to the material directions ( , ,L T T ′ ) or (1, 2, 3) were measured in static tests as mean values of 10 tests for each constant. The values obtained are reported in Table 11.1. Then, the values of the reduced stiffnesses were derived and are reported in Table 11.2.

The sandwich materials considered are the sandwich materials studied in Chapter 10. They were constructed with the previous unidirectional laminates as skins and with PVC closed-cell foams supplied in panels of thickness of 15 mm.

Chapter 11. General Formulation of the Damping of Composite Materials and Structures 302

TABLE 11.1. Engineering constants of laminates.

Laminate EL (GPa) ET (GPa) νLT GLT (GPa)

Unidirectional layer 29.9 7.50 0.24 2.25

Unidirectional woven layer 28 11.0 0.24 3.8

Taffeta layer 14 13.5 0.25 2.05

Serge layer 16 15.4 0.24 2.10

TABLE 11.2. Reduced stiffnesses of laminates.

Laminate Q11 (GPa) Q12 (GPa) Q22 (GPa) Q66 (GPa)

Unidirectional layer 30.3 1.83 7.61 2.25

Unidirectional woven layer 28.6 2.70 11.3 3.8

Taffeta layer 14.9 3.45 14.4 2.05

Serge layer 16.9 3.91 16.3 2.10

16 260, 0.Q Q= =

TABLE 11.3. Mechanical characteristics of the foams.

Density (kg m–3) Young’s modulus (MPa) Poisson’s ratio Shear modulus

(MPa) 60 59 0.42 22

80 83 0.43 30

200 240 0.45 80

Three foams were considered differing in their densities: 60 kg m–3, 80 kg m–3 and 200 kg m–3. Mechanical characteristics of the foams were measured in static ten-sile tests for the Young’s modulus and the Poisson’s ratio, and in static shear tests for the shear modulus. The values derived are reported in Table 11.3.

In addition the effect of interleaving viscoelastic layers in laminates (Chapter 7) was also considered. The characteristics of the viscoelastic layers and inter-leaved laminates are reported in Section 7.3.

11.2 MODELLING DAMPING OF LAMINATE BEAMS AND RECTANGULAR PLATES

Modelling of the damping of beams and rectangular plates made of unidirec-tional or orthotropic laminates has been considered in Chapter 5 and experimental investigation of damping of laminates has been developed extensively in Chapter

11.3. Damping Modelling using Finite Element Analysis 303

6. The results obtained show that the material damping can be evaluated by an energy approach based on the evaluation of the strain energy stored in the material directions and where the dissipated energy is characterized by damping coeffi-cients. In the case of beams or rectangular plates the strain energy can be derived by using the Ritz method.

11.3 DAMPING MODELLING USING FINITE ELEMENT ANALYSIS

11.3.1 Introduction

The Ritz method is restricted to the analysis of rectangular plates and to the case where the materials are unidirectional or orthotropic materials. In the case of other types of materials or complex shape structures, it is necessary to use a finite element analysis to analyze the dynamic behaviour.

Principle of finite element analysis of a dynamic problem of a structure with damping included has been considered in Chapters 8 and 9. The energy approach for the evaluation of damping which has been developed in Chapter 5 by consi-dering the Ritz method can be extended to any type of materials and to a complex shape structure by using finite element analysis. The formulation was applied in Chapter 10 to the case of sandwich materials using finite element analysis asso-ciated to sandwich plate theory. A general formulation which can be applied to different composite materials (laminates, laminates with viscoelastic layers, sandwich materials) and complex shape structures is developed in the present section. This formulation is based on the first-order laminate theory including the transverse shear effects.

11.3.2 Stress Field in Composite Materials

Laminates, laminates with interleaved viscoelastic layers and sandwich mate-rials are constituted of layers with different orientations and properties. The middle plane is chosen as the reference plane (Oxy). Each layer k is referred to (Figure 11.1) by the z coordinate of its lower face ( 1kh − ) and upper face ( kh ).

FIGURE 11.1. Stresses evaluated by finite element analysis in the layers of a finite element.

z

1kh −

kh

k ke

u u u u u, , , , xx k yy k xy k xz k yz kσ σ σ σ σ

l l l l l, , , , xx k yy k xy k xz k yz kσ σ σ σ σ

Chapter 11. General Formulation of the Damping of Composite Materials and Structures 304

The laminate theory taking into account the transverse shear effects is based on a first-order theory (Section 1.5 of Chapter 1) which expresses the displacement field as a linear function of z coordinate through the thickness of laminate element. Furthermore, the laminate theory assumes that the stress zzσ can be neglected through the thickness of laminate. The strain field and the stress field are deduced from the previous displacement field and then from the strain-stress relations.

When finite element based on the laminate theory with transverse shear effects included is used, finite element analysis gives the values of stresses

, , , , ,xx yy xy yz xzσ σ σ σ σ on the lower face (l) and upper face (u) of each layer k of each finite element e of the structure (Figure 11.1):

l l l l l

u u u u u

, , , , ,

, , , , .xx k yy k xy k yz k xz k

xx k yy k xy k yz k xz k

σ σ σ σ σ

σ σ σ σ σ (11.1)

The first-order theory leads to in-plane stresses in layer k which are linear func-tions of z coordinate of the forms:

( , ) ( , ), , , ,pk pk pka x y z b x y p xx yy xyσ = + = (11.2) with , , .e e e

pk xxk yyk xykσ σ σ σ= (11.3)

Coefficients and pk pka b in each element e can be deduced from the stresses cal-culated by the finite element analysis on the lower and upper faces of each layer k. Note that in-plane stresses are discontinuous at the layer interfaces. We obtain:

( )

u l

u u l

,

,

p k p kpk

k

kpk p k p k p k

k

ae

hbe

σ σ

σ σ σ

−=

= − − (11.4)

with

u u u u

l l l l

, , ,, , ,

p k xx k yy k xy k

p k xx k yy k xy k

σ σ σ σ

σ σ σ σ

=

= (11.5)

and where ke is the thickness of layer k. From the first-order theory it results that the transverse shear stresses are uni-

form through the layer thickness and discontinuous between. However a better estimate can be obtained considering the governing equations of the mechanics of materials:

0,

0.

xykxxk xzk

yyk xyk yzk

x y z

y x z

σσ σ

σ σ σ

∂∂ ∂+ + =∂ ∂ ∂

∂ ∂ ∂+ + =

∂ ∂ ∂

(11.6)

These expressions allow us to derive the transverse shear stresses and xzk yzkσ σ

11.3. Damping Modelling using Finite Element Analysis 305

as functions of the in-plane stresses , and .xxk yyk xykσ σ σ Considering Expres-sion (11.2), Equations (11.6) show that the transverse shear stresses are quadratic functions of the z coordinate. Moreover the transverse shear stresses are conti-nuous at the layer interfaces and are zero on the two outer faces of the laminate.

Finite element analysis gives the values of the transverse shear stresses ( l ,yz kσ

l u u, , xz k yz k xz kσ σ σ ) on the lower and upper faces of each layer of each element e. So, the transverse shear stresses can be expressed as:

2( , ) ( , ), , ,rk rk rkx y z x y r yz xzσ α β= + = (11.7)

where the coefficients are deduced from the values of the shear stresses on the lower and upper faces. Whence:

( )2l u

u1

, , , .r k r krk rk r k rk k

k k kh r yz xz

h h eσ σα β σ α

−

−= − = + =+

(11.8)

11.3.3 In-Plane Strain Energy

We consider the case where the layers of the composite material are constituted of orthotropic materials.

The in-plane energy deU stored in a given finite element e can be expressed as a

function of the in-plane strain energies related to the material directions as:

d 1 2 6 ,e e e eU U U U= + + (11.9) with

1 1 1

2 2 2

6 6 6

1 d d d ,2

1 d d d ,2

1 d d d ,2

e

e

e

e

e

e

U x y z

U x y z

U x y z

σ ε

σ ε

σ ε

=

=

=

∫∫∫∫∫∫∫∫∫

(11.10)

where the integrations are extended over the volume of the finite element e. The in-plane strains 1 2 6, and ε ε ε related to the material directions of the

layer are expressed as functions of stresses 1 2 6, and σ σ σ in the material direc-tions according to the strain-stress relations as:

1 11 1 12 2

2 12 1 22 2

6 66 6

,,

,

S SS SS

ε σ σε σ σε σ

= += +=

(11.11)

where the components ijS are the compliance constants of the layer related to the

Chapter 11. General Formulation of the Damping of Composite Materials and Structures 306

engineering moduli , , and L T LT LTE E G ν by the following expressions:

11 22 12 661 1 1, , , .LT

L T L LTS S S S

E E E Gν= = = − = (11.12)

It results that Expression (11.10) of the strain energy 1eU , stored in tension-

compression in the longitudinal direction can be written in the form:

1 11 12e e eU U U= + , (11.13)

with

211 11 1

12 12 1 2

1 d d d ,2

1 d d d .2

e

e

e

e

U S x y z

U S x y z

σ

σ σ

=

=

∫∫∫∫∫∫

(11.14)

In each layer, stresses 1 2 6, and σ σ σ , related to the material directions, can be expressed as functions of the in-plane stresses , and ,xx yy xyσ σ σ related to the finite element directions (x, y, z), according to the stress transformation relation:

2 21

2 22

2 26

cos sin 2sin cos

sin cos 2sin cos

sin cos sin cos cos sin

xx

yy

xy

θ θ θ θσ σσ θ θ θ θ σσ σθ θ θ θ θ θ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥= −⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ − −⎢ ⎥ ⎣ ⎦⎣ ⎦

, (11.15)

where θ is the orientation of the material in the layer. Whence:

( )

2 4 2 411 11

2 2 2

3 3

1 cos sin2

2 2 sin cos

4 sin cos 4 sin cos d d d ,

exx yy

e

xy xx yy

xx xy yy xy

U S

x y z

σ θ σ θ

σ σ σ θ θ

σ σ θ θ σ σ θ θ

⎡= +⎣

+ +

⎤+ + ⎦

∫∫∫ (11.16)

( )( )

( )( )

2 2 2 2 212 12

4 4

2 2

1 4 sin cos2

sin cos

2 sin cos sin cos d d d .

exx yy xy

e

xx yy

xx xy yy xy

U S

x y z

σ σ σ θ θ

σ σ θ θ

σ σ σ σ θ θ θ θ

⎡= + −⎣

+ +

⎤+ − − ⎦

∫∫∫ (11.17)

In the same way, the strain energy 2U , stored in tension-compression in the transverse direction can be written in the form:

2 22 12e e eU U U= + , (11.18)

with

11.3. Damping Modelling using Finite Element Analysis 307

( )

2 4 2 422 22

2 2 2

3 3

1 sin cos2

2 2 sin cos

4 sin cos 4 sin cos d d d ,

exx yy

e

xy xx yy

xx xy yy xy

U S

x y z

σ θ σ θ

σ σ σ θ θ

σ σ θ θ σ σ θ θ

⎡= +⎣

+ +

⎤− − ⎦

∫∫∫ (11.19)

Lastly, the in-plane shear strain energy is written as:

( )( )

( )( )

2 2 2 26 66 66

22 2 2

2 2

1 2 sin cos2

cos sin (11.20)

2 cos sin sin cos d d d

e exx yy xx yy

e

xy

yy xy xx xy

U U S

x y z

σ σ σ σ θ θ

σ θ θ

σ σ σ σ θ θ θ θ

⎡= = + −⎣

+ −

⎤+ − − ⎦

∫∫∫

.

The in-plane strain energies stored in the finite element e can be expressed as:

11 11 22 221 1

12 12 66 661 1

, ,

, ,

n ne e e e

k kk k

n ne e e e

k kk k

U U U U

U U U U

= =

= =

= =

= =

∑ ∑

∑ ∑ (11.21)

where ( 11, 22, 12, 66)eijkU ij = are the in-plane strain energies stored in the

layer k of the element e and n is the total number of layers in the lower and upper skins.

It results from Expressions (11.16) to (11.20) that the in-plane strain energies stored in layer k of the element e can be expressed as:

( )

4 411 11

2 2

3 3

cos sin

2 2 sin cos

4 sin cos 4 sin cos ,

e e ek k xxxxk k yyyyk k

e exyxyk xxyyk k k

e exx xyk k k yyxyk k k

U S U U

U U

U U

θ θ

θ θ

θ θ θ θ

⎡= +⎣

+ +

⎤+ + ⎦

(11.22)

( )( )

( )( )

2 212 12

4 4

2 2

4 sin cos

sin cos

2 sin cos sin cos ,

e e e ek k xxxxk yyyyk xyxyk k k

exxyyk k k

e exxxyk yyxyk k k k k

U S U U U

U

U U

θ θ

θ θ

θ θ θ θ

⎡= + −⎣

+ +

⎤+ − − ⎦

(11.23)

( )

4 422 22

2 2

3 3

sin cos

2 2 sin cos

4 sin cos 4 sin cos ,

e e ek k xxxxk k yyyyk k

e exyxyk xxyyk k k

e exx xyk k k yyxyk k k

U S U U

U U

U U

θ θ

θ θ

θ θ θ θ

⎡= +⎣

+ +

⎤− − ⎦

(11.24)

Chapter 11. General Formulation of the Damping of Composite Materials and Structures 308

( )( )

( )( )

2 266 66

22 2

2 2

2 sin cos

cos sin

2 sin cos sin cos .

e e e ek k xxxxk yyyyk xxyyk k k

exyxyk k k

e exxxyk yyxyk k k k k

U S U U U

U

U U

θ θ

θ θ

θ θ θ θ

⎡= + −⎣

+ −

⎤+ − − ⎦

(11.25)

Expressions (11.22) to (11.25) introduce the orientation kθ of layer k.

The energy terms epqkU ( , , , p q xx yy xy= ) introduced in Expressions (11.22)

to (11.25) can be expressed in the form:

, , , , ,2

e eepqk pqk

SU I p q xx yy xy= = (11.26)

where Se is the area of the finite element e and introducing the integral:

1

d , , , , .k

k

heppk pk qk

hI z p q xx yy xyσ σ

−

= =∫ (11.27)

Considering Expression (11.2) of the in-plane stresses, this integral can be expressed as:

( ) ( )( )

3 3 2 21 1

1 1 ,3 2

, , , .

epqk pk qk k k pk qk qk pk k k pk qk kI a a h h a b a b h h b b e

p q xx yy xy

− −= − + + − +

= (11.28)

Coefficients , , and ,pk qk pk pka a b bq are derived from Expressions (11.4). Next, the in-plane strain energies stored in element e are given by Expressions

(11.21) and the total in-plane strain energies stored in the finite element assemblage is then obtained by summation on the elements as:

11 11 12 12elements elements

22 22 66 66elements elements

, ,

, .

e e

e e

U U U U

U U U U

= =

= =

∑ ∑

∑ ∑ (11.29)

11.3.4 Transverse Shear Strain Energy

The transverse shear strain energy for a given element e can be expressed in the material directions as: s 44 55

e e eU U U= + , (11.30) with

44 4 4

55 5 5

1 d d d ,2

1 d d d ,2

e

e

e

e

U x y z

U x y z

σ γ

σ γ

=

=

∫∫∫∫∫∫

(11.31)

11.3. Damping Modelling using Finite Element Analysis 309

where the integration is extended over the volume of the finite element e. 4 4 and σ γ are respectively the transverse shear stress and strain in plane ( , T T ′ )

of material in layer k. 5 5 and σ γ are the transverse shear stress and strain in plane ( , L T ′ ).

The transverse shear strains and stresses are related by:

4 4 5 5, ,TT LTG Gσ γ σ γ′ ′= = (11.32)

where and TT LTG G′ ′ are the transverse shear moduli in planes ( , T T ′ ) and ( , L T ′ ), respectively. It results that the transverse shear strain energies (11.31) can be written as:

24

44

25

55

1 d d d ,2

1 d d d .2

e

TTe

e

LTe

U x y zG

U x y zG

σ

σ

′

′

=

=

∫∫∫∫∫∫

(11.33)

In each layer k, stresses 4 5 and ,σ σ relative to the material directions of the layer, can be expressed as functions of the transverse shear stresses and yz xzσ σ in the finite element directions (x, y, z) according to the stress transformations:

4

5

cos sinsin cos

yz

xz

σ θ θ σσ θ θ σ

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦. (11.34)

So, the transverse strain energies (11.33) are expressed as:

( )

2 2 2 2

441 1 cos sin 2 sin cos d d d ,2

eyz xz yz xz

TTeU x y z

Gσ θ σ θ σ σ θ θ

′= + −∫∫∫

(11.35)

( )

2 2 2 2

551 1 sin cos 2 sin cos d d d .2

eyz xz yz xz

LTeU x y z

Gσ θ σ θ σ σ θ θ

′= + +∫∫∫

(11.36)

The transverse shear energies can be expressed as:

44 44 55 551 1

, ,n n

e e e ek k

k k

U U U U= =

= =∑ ∑ (11.37)

where ( 44, 55)emmkU mm = are the transverse shear strain energies stored in

layer k of the element. Introducing the terms:

1 1

1

2 2d , d ,2 2

d ,2

k k

k k

k

k

h he ee eyzyzk yzk xzxz k xzk

h h

he eyzxzk yzk xzk

h

S SU z U z

SU z

σ σ

σ σ

− −

−

= =

=

∫ ∫∫

(11.38)

Chapter 11. General Formulation of the Damping of Composite Materials and Structures 310

the transverse shear energies stored in layer k are expressed as:

( )2 244

1 cos sin 2 sin cos ,e e e ek yzyzk k xzxzk k yzxzk k k

TTU U U U

Gθ θ θ θ

′= + − (11.39)

( )2 255

1 sin cos 2 sin cos .e e e ek yzyzk k xzxzk k yzxzk k k

LTU U U U

Gθ θ θ θ

′= + + (11.40)

The energy terms , and e e eyzyzk xzxzk yzxzkU U U are derived by introducing Ex-

pression (11.7) of the transverse shear stresses. It results that the energy terms can be written as:

, , , ,2

e eersk rsk

SU I r s yz xz= = (11.41)

with

( ) ( )( ) ( )5 5 3 31 1 1

1 1 ,5 3

, , . (11.42)

ersk rk sk k k rk sk sk rk k k rk rk k kI h h h h h h

r s yz xz

α α α β α β β β− − −= − + + − + −

=

Constants , , and rk rk sk skα β α β are deduced from Equations (11.8). Next, the transverse shear strain energy stored in element e is given by Expres-

sions (11.37) and the total transverse shear strain energies stored in the finite element assemblage is obtained by summation on the elements as:

44 44 55 55elements elements

, .e eU U U U= =∑ ∑ (11.43)

11.3.5 Damping of a Composite Structure

The damping of the finite element assemblage can be evaluated by extending to the finite element analysis the energy formulation approach considered in the case of the Ritz method (Chapter 5).

The total strain energy stored in the laminated structure is given by:

d 11 22 12 66 44 552U U U U U U U= + + + + + , (11.44)

where the in-plane strain energies 11 22 12 66, , 2 and U U U U are expressed by Equations (11.29), and the transverse shear strain energies 44 55 and U U are given by Equations (11.43).

Then, the energy dissipated by damping in the layer k of the element e is derived from the strain energy stored in layer as:

11 11 22 22 12 12 66 66

44 44 55 55

2

,

e e e e e e e e ek k k k k k k k k

e e e ek k k k

U U U U U

U U

ψ ψ ψ ψ

ψ ψ

∆ = + + +

+ + (11.45)

introducing the specific damping coefficients epqkψ of the layer. These coeffi-

cients are related to the material directions ( , , L T T ′ ) of the layer: 11 andekψ

11.3. Damping Modelling using Finite Element Analysis 311

22e

kψ are the damping coefficients in traction-compression in the L direction and

T direction of layer, respectively; 12e

kψ is the in-plane coupling coefficient; 66e

kψ

is the in-plane shear coefficient; 44 55 and e ek kψ ψ are the transverse shear damping

coefficients in planes ( , T T ′ ) and ( , L T ′ ), respectively. The damping energy dissipated in the element e is next obtained by summation

on the layers of element as:

1

,n

e ek

k

U U=

∆ = ∆∑ (11.46)

and the total energy U∆ dissipated in the finite element assemblage is then ob-tained by summation on the elements:

elements

.eU U∆ = ∆∑ (11.47)

Finally, the damping of the finite element assemblage is characterised by the damping coefficient ψ of the assemblage derived from relation:

d

UU

ψ ∆= . (11.48)

It has to be noted that the in-plane coupling energy 12eU is much lower than the

other in-plane energies and this energy can be neglected.

11.3.6 Procedure for Evaluating the Damping of Composite Structure

A general procedure was implemented to evaluate the damping of a structure using finite element analysis. This procedure is based on the previous formulation and can be applied to any structure for which the damping characteristics are dif-ferent according to the layers and the elements of the assemblage. In the procedure, the finite element analysis is used first to establish the eigen-equation of the structure vibrations. The equation is solved to obtain the natural frequencies and the corresponding mode shapes. Next, the stresses on the lower and upper faces of each layer are read in each element of the finite element assemblage for each vibration mode. The different energies are calculated according to the formu-lation developed in the previous sections and the damping iψ for each mode i is evaluated according to Equation (11.48).

The formulation considered is based on the laminate theory including the transverse shear effects. Results that we derived from the application of this for-mulation have shown that this formulation can be applied to all the materials considered in Section 11.1: laminate materials, laminate materials with inter-leaved viscoelastic layers and sandwich materials.

In the case of laminate materials, the results deduced from the formulation

Chapter 11. General Formulation of the Damping of Composite Materials and Structures 312

show that the transverse shear strain energies can be neglected with regard to the in-plane strain energies. So, the damping is induced by the in-plane behaviour of laminate layers.

In the case of sandwich materials, the results derived show that the behaviour of materials obtained by considering finite elements based on the laminate theory including the transverse shear effects is the same as the behaviour obtained by considering finite elements based on the sandwich theory (Chapter 10). Moreover, the results obtained show that the transverse shear strain energies are much higher than the in-plane strain energies. Damping in sandwich materials is induced by the transverse shear behaviour of sandwich core for large thicknesses of the core.

A similar behaviour as in the case of sandwich materials is observed in the case of laminate materials with interleaved viscoelastic layers. Damping is induced by the transverse behaviour of the viscoelastic layers.

Finally, the results deduced from the previous finite element damping formu-lation is general and can be applied to laminate materials, sandwich materials and laminate materials with interleaved viscoelastic layers.

11.4 INVESTIGATION OF THE DAMPING OF COMPOSITE MATERIALS

11.4.1 Determination of the Constitutive Damping Parameters

The damping characteristics of the materials can be obtained by subjecting beams to flexural vibrations. The equipment is shown in Figure 6.1. The test spe-cimen is supported horizontally as a cantilever beam in a clamping block. An impulse hammer is used to induce the excitation of the flexural vibrations of the beam and the beam response is detected by using a laser vibrometer. Next, the excitation and the response signals are digitalized and processed by a dynamic analyzer of signals. This analyzer associated with a PC computer performs the acquisition of signals, controls the acquisition conditions and next performs the analysis of the signals acquired (Fourier transform, frequency response, mode shapes, etc.).

In the case of laminate materials, the damping characteristics of the beams are deduced from the Fourier transform of the beam response to an impulse input by fitting this experimental response with the analytical response of the beam which was derived in Subsection 6.2.4.1 of Chapter 6 using the Ritz method. This fitting is obtained by a least square method which allows us to obtain the values of the natural frequencies fi and the modal loss factors ηi, related to the specific damping coefficient by the relation ψi = 2πηi.

In the case of laminate materials with interleaved viscoelastic layers and in the case of sandwich materials, it is not possible to derive the beam response using the Ritz method. So, this response is obtained by using a finite element analysis and fitting the results with the experimental response allows us to obtain the modal loss factors.

11.4. Investigation of the Damping of Composite Materials 313

11.4.2 Damping of the Glass Fibre Laminates The experimental evaluation of damping was performed on beams of different

lengths: 160, 180 and 200 mm so as to have a variation of the values of the peak frequencies. Beams had a nominal width of 20 mm and a nominal thickness of 2.4 mm.

Figures 11.2 to 11.5 show the experimental results obtained for damping in the case of unidirectional glass fibre composites (Figure 11.2), unidirectional glass cloth composites (Figure 11.3), glass taffeta composites (Figure 11.4) and glass serge composites (Figure 11.5). The results are reported for the first three bending modes and for the different lengths of the beams as functions of the frequency, considering different orientations of glass fibres. For a given fibre orientation, it is observed that damping increases when the frequency is increased. The values of the damping increase when the frequency is increased from 50 Hz to 600 Hz are reported in Table 11.4 for the different glass fibre composites considered. The table shows that the damping increase is fairly the same for the different fibre orientations in the case of a given laminate. The increase is fairly higher in the case of unidirectional laminates (21 to 26 %) and unidirectional cloth laminates (18 to 22 %) than in the case of taffeta laminates (15 to 20 %) and serge laminates (17 to 21 %).

The variations of the loss factor with fibre or cloth orientation are given in Figures 11.6 to 11.8 for the three frequencies 50, 300 and 600 Hz. In the case of unidirectional glass fibre laminates and unidirectional glass cloth laminates, the transverse damping is higher than the longitudinal damping, and the damping is maximum at a fibre orientation of about 60° for the glass fibre composites. In the case of taffeta and serge laminates, the damping variation is symmetric with a damping maximum for the orientation of 45°.

TABLE 11.4. Damping increase (%) of the different laminates, in the frequency range

[50, 600 Hz].

Fibre orientation (°) 0 15 30 45 60 75 90

Unidirectional glass fibre composites 21 24 26 23 26 23 27

Unidirectional glass cloth composites 19 20 21 19 22 20 18

Glass taffeta composites 15 17 20 19 20 18 15

Glass serge composites 18 19 21 17 21 19 18

Chapter 11. General Formulation of the Damping of Composite Materials and Structures 314

FIGURE 11.2. Experimental results obtained for the damping as functions of the frequency for different fibre orientations, in the case of unidirectional glass fibre composites. FIGURE 11.3. Experimental results obtained for the damping as functions of the frequency for different orientations, in the case of unidirectional glass cloth composites.

Frequency (Hz)0 200 400 600 800

Loss

fact

or η

i (%

)

0.3

0.5

0.7

0.9

1.1

1.3

1.5

1.7

θ = 0 °

θ = 15 °

θ = 30 °

θ = 45 °θ = 75 °θ = 90 °

θ = 60 °

Frequency (Hz)0 200 400 600 800 1000 1200

Loss

fact

or η

i (%

)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

θ = 0°

θ = 15°

θ = 90°θ = 75°θ = 45°θ = 60°

θ = 30°

11.4. Investigation of the Damping of Composite Materials 315

FIGURE 11.4. Experimental results obtained for the damping as functions of the frequency for different orientations, in the case of glass taffeta composites. FIGURE 11.5. Experimental results obtained for the damping as functions of the frequency for different orientations, in the case of glass serge composites.

Frequency (Hz)0 200 400 600 800 1000 1200

Loss

fact

or η

i (%

)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

θ = 0°, θ = 90°

θ = 15°, θ = 75°

θ = 30°, θ = 60°θ = 45°

Frequency (Hz)0 200 400 600 800 1000 1200

Loss

fact

or η

i (%

)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

θ = 0°, θ = 90°

θ = 15°, θ = 75°

θ = 30°, θ = 60°

θ = 45°

Chapter 11. General Formulation of the Damping of Composite Materials and Structures 316

FIGURE 11.6. Variation of the loss factor as a function of fibre orientation in the case of unidirectional glass fibre composites. Comparison between experimental results and modelling.

FIGURE 11.7. Variation of the loss factor as a function of orientation in the case of unidirectional cloth fibre composites. Comparison between experimental results and modelling.

Frequency (Hz)0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (%

)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

f = 50 Hz f = 300 Hzf = 600 Hzf = 50 Hzf = 300 Hzf = 600 Hz

Modelling

Experimental results

Fibre orientation θ (°)0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (%

)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

f = 50 Hz f = 300 Hzf = 600 Hzf = 50 Hzf = 300 Hzf = 600 Hz

Experimental results

Modelling

11.4. Investigation of the Damping of Composite Materials 317

FIGURE 11.8. Variation of the loss factor as a function of orientation in the case of glass taffeta or serge composites. Comparison between experimental results and modelling.

The analysis using the Ritz method (Section 5.5 of Chapter 5) or the analysis using the finite element method (Section 11.3 of this chapter) can be applied to the experimental results obtained for the bending of beams. The beams were con-sidered in the form of plates with one edge clamped and with the others free. The results obtained are identical by using either the Ritz analysis or the finite element analysis, and these results are reported in Figures 11.6 to 11.8. A good agreement is obtained with the experimental results. The values of the loss factors in the material directions considered for modelling the damping results are reported in Tables 11.5 to 11.8 for the frequencies 50, 300 and 600 Hz. The loss factors are fairly identical for the taffeta laminates and serge laminates. The increase of

TABLE 11.5. Loss factors derived from modelling in the case of unidirectional glass fibre laminates.

f (Hz) η11 (%) η12 η22 (%) η66 (%)

50 0.35 0 1.30 1.80

300 0.40 0 1.50 2.00

600 0.45 0 1.65 2.22

Fibre orientation (°)0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

i (%

)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

f = 50 Hz f = 300 Hzf = 600 Hzf = 50 Hzf = 300 Hzf = 600 Hz

Modelling

Experimental results

Chapter 11. General Formulation of the Damping of Composite Materials and Structures 318

TABLE 11.6. Loss factors derived from modelling in the case of unidirectional glass cloth laminates.

f (Hz) η11 (%) η12 η22 (%) η66 (%)

50 0.50 0 1.24 1.61

300 0.55 0 1.40 1.85

600 0.58 0 1.43 2.11

TABLE 11.7. Loss factors derived from modelling in the case of glass taffeta laminates.

f (Hz) η11 (%) η12 η22 (%) η66 (%)

50 0.66 0 0.65 1.50

300 0.83 0 0.84 1.75

600 0.89 0 0.9 1.80

TABLE 11.8. Loss factors derived from modelling in the case of glass serge laminates.

f (Hz) η11 (%) η12 η22 (%) η66 (%)

50 0.67 0 0.67 1.53

300 0.83 0 0.83 1.78

600 0.89 0 0.89 1.83

fibres in the 90° direction from the unidirectional laminates to taffeta and serge laminates leads to an increase of the loss factor 11η in the 0° direction, a decrease of the loss factor 22η in the 90° direction and a decrease of the shear loss factor

66η .

11.4.3 Damping Comparison between Taffeta Laminates, Serge Laminates and Cross-Ply Laminates

Figure 11.9 compares the results obtained for damping in the case of taffeta laminates, serge laminates and cross-ply laminates, for 0° orientation of the lami-nates. Two cross-ply laminates are considered: [(0/90)2]s and [02/902]s. Damping of [(0/90)2]s laminates is slightly higher than that of [02/902]s laminates. This fact results from the damping of the 90° layers which are more external in the [(0/90)2]s laminates.

11.4. Investigation of the Damping of Composite Materials 319

FIGURE 11.9. Comparison between damping of taffeta and serge laminates and damping of cross-ply laminates, for 0° orientation of the laminates.

In Figure 11.9, it is observed that the damping of taffeta and serge laminates is clearly higher than the damping of cross-ply laminates. This increase of damping may be associated with the energy which is dissipated by friction between the warp fibres and weft fibres, in the case of the taffeta and serge laminates.

11.4.4 Damping of the Unidirectional Glass Fibre Laminates with Interleaved Viscoelastic Layers

As in the case of unidirectional composites (Subsection 11.4.1), the experi-mental evaluation of damping was performed on beams 20 mm wide of different lengths: 160, 180 and 200 mm. An extended experimental investigation is deve-loped in Chapter 7 (Subsection 7.3) in the case of glass laminates with a single viscoelastic layers e0 of thickness 0.2 mm interleaved in the middle plane, lami-nates with a single layer of thickness 2e0 and laminates with two viscoelastic layers of thicknesses e0 interleaved on both sides from the middle plane.

The Young’s modulus and the loss factor of the viscoelastic layers were ob-tained (Chapter 7) following the standard ASTM E 756 [79]. These properties were evaluated from the flexural vibrations of clamped-free beams 10 mm wide and constituted of two aluminium beams with a layer of the viscoelastic material interleaved between the two aluminium beams of the test specimens. The free lengths and the thicknesses of the aluminium beams were selected so as to measure the dynamic properties on the frequency range [50, 600 Hz]. The Young’s modulus of the viscoelastic material was deduced from the natural

Frequency (Hz)0 200 400 600 800 1000 1200

Loss

fact

or η

i (%

)

0.4

0.6

0.8

1.0

1.2

[(0/90)2]s

[02/902]s

TaffetaSerge

Chapter 11. General Formulation of the Damping of Composite Materials and Structures 320

frequencies of the test specimens and the loss factor was evaluated by applying either the modelling considered in Section 7.2 of Chapter 7 or the modelling developed in Section 11.3 of this chapter. Figures 7.8 and 7.9 report the expe-rimental results obtained, using logarithmic scales for the Young’s modulus and for the frequency. In the frequency range studied, it is observed linear variations for the logarithm of the Young’s modulus and for the loss factor of the visco-elastic material as functions of the frequency.

Next, the loss factor of the glass fibre laminates with interleaved viscoelastic layers can be derived from the modelling considered in Section 11.3. As an example, the results obtained are compared with the experimental results in Figure 11.10, for the first two modes of the test specimens, in the case of a single interleaved viscoelastic layer of 0.2 mm thick, for the different free lengths of the test specimens: 160, 180 and 200 mm. It is observed that the results deduced from the modelling describe fairly well the experimental damping variation obtained as a function of the fibre orientation.

11.4.5 Damping of the Sandwich Materials

An extended analysis of the damping of the sandwich materials was developed in Chapter 10.

As in the case of the viscoelastic layers (previous subsection), the experimental investigation of the dynamic properties of the PVC foams were evaluated from the flexural vibrations of a clamped-free beam 40 mm wide and constituted of two aluminium beams with foam material interleaved between the two aluminium beams. An aluminium spacer was added in the root section which was closely clamped in a rigid fixture. The free length and the thicknesses of the aluminium beams were selected so as to measure the foam characteristics on the frequency range [50, 1,000 Hz]. Thus, the beam dimensions used were a free length equal to 300, 350, 400 and 450 mm, and thicknesses of the aluminium beams of 4 mm.

Investigation of damping was implemented in the case of sandwich materials constituted of glass [0/90]s laminates as skins and PVC foams as core. The experimental investigation was carried out in the case of flexural vibration of clamped-free beams with the same conditions as the ones considered in the previous subsection. An aluminium spacer was added in the root section which was closely clamped in a rigid fixture. The free length of the beams was equal to 300, 350, 400 and 450 mm, and the beam width was equal to 40 mm. The experimental loss factors were derived from the frequency responses for the first four modes. The results are reported in Figure 10.13 for the three foam densities under consideration.

Next, the damping of sandwich materials can be evaluated by considering modelling. The results obtained show that the evaluation of the damping is identical when either the modelling implemented in Section 10.1 based on the sandwich theory or the modelling developed in Section 11.3 is used. It has to be

11.4. Investigation of the Damping of Composite Materials 321

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%)

0

1

2

3

4

5

6

1st mode2nd mode1st mode2nd modewithout viscoelastic layer

modelling

experiment

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%)

0

1

2

3

4

5

1st mode2nd mode1st mode2nd modewithout viscoelastic layer

modelling

experiment

Fibre orientation θ (°)

0 10 20 30 40 50 60 70 80 90

Loss

fact

or η

(%)

0

1

2

3

4

5

1st mode2nd mode1st mode2nd modewithout viscoelastic layer

modelling

experiment

(a)

(b)

(c) FIGURE 11.10. Comparison between the experimental results and the results deduced from the modelling, in the case of a single viscoelastic layer 0.2 mm thick, for test specimen lengths of: (a) l = 160 mm, (b) l = 180 mm, (c) l = 200 mm.

Chapter 11. General Formulation of the Damping of Composite Materials and Structures 322

noted that modelling must take into account the variations of the properties of skins and core as functions of the frequency. The results derived from modelling are compared to the experimental results in Figure 10.13.

11.5 DYNAMIC RESPONSE OF A COMPOSITE STRUCTURE

As an application, modelling developed in Section 11.3 and the results deduced from the investigation developed in Section 11.4 for the damping of the different materials were applied to the analysis of the simple shape structure of Figure 11.11. Three types of materials were used for the structure: glass serge laminate of thickness of 5 mm; glass serge laminate with interleaved viscoelastic layer 0.2 mm thick; sandwich material with PVC foam 15 mm thick and density of 60 kg

m–3, and glass [0/90]s skins 1.2 mm thick. The damping properties of these different materials were considered in the previous section. The different mate-rials of the structure were chosen in such a way to have the same stiffness of the structure. The structure was clamped in a clamping block of dimensions 150 mm × 150 mm. An impulse hammer was used to induce the excitation of the vibrations of the structure. The response of the structure was detected by using a

Figure 11.11. Simple shape structure.

Clamping block

Measuring point

Impact point

560 mm

80 mm

150 mm

150 mm

11.5. Dynamic Response of a Structure 323

laser vibrometer. Different impact points and measuring points were considered to induce and to detect all the vibration modes of the structure.

Figure 11.12 shows the shapes of the first six modes deduced from finite element analysis in the case where the structure is constituted of serge laminate or serge laminate with interleaved viscoelastic layer. In the case of sandwich material, modes 1 and 2 are inverted. Mode 1 is a twisting mode, mode 2 a longi-tudinal bending mode and mode 3 a transverse bending mode. The other modes combine these different effects.

The loss factors of the modes were evaluated by applying the modelling developed in Section 11.3 to the structure considered. The results obtained for the damping are reported in Tables 11.9 to 11.11, for the three different materials. The modal loss factors were also deduced from experimental investigation where the responses of the structure were identified in the frequency domain using MATLAB Toolbox. The results are compared in Tables 11.9 to 11.11 for the first ten modes. Also, tables report the frequencies of the free natural modes of the structure deduced from experiment and finite element analysis. A good agreement is observed between the results derived from modelling and the experimental results. Interleaving viscoelastic layer does not change significantly the frequency of the modes. Compared to the damping of the structure constituted of serge laminate, the damping of the first two modes is increased by a factor of about 5 when the structure is constituted of the sandwich material. For the other modes, the damping is increased by a factor of 1.5 to 2. In the case of the structure constituted of the serge laminate with interleaved viscoelastic layer, the damping of mode 2 (a twisting mode) is lower than the structure with sandwich material. The damping of the other modes is greatly increased, by a factor 6 to 12 with respect to the structure constituted of the serge laminate.

Next, the modal responses of the structure were derived by finite element analysis using a mode superposition method (Chapter 9). This analysis considers the modal loss factors obtained previously and the analysis was nonlinear so as to take into account the variation of the moduli of the materials with the frequency. Figure 11.13 compares the frequency responses of the structure constituted of the different materials derived from the finite element analysis with the frequency responses obtained by the experimental investigation. For these responses the impact point and the measuring point considered are reported in Figure 11.11. The modal responses derived from finite element analysis were adjusted so as to have the amplitude response equal to zero for the frequency equal to zero. Next the experimental responses were fitted so as to have the same amplitude of the finite element analysis response and the experimental response for the first peak. The responses are reported with the same scale for the response amplitude.

Due to the mode shapes and the positions of the impact and measuring points (Figure 11.11), the vibration modes 1, 5 and 6 are not detected in the case where the structure is constituted of serge laminate or serge laminate with interleaved viscoelastic layer. Modes 2 and 3 combine to yield two resonance peaks at frequencies of 153 Hz and 215 Hz, and an anti-resonance peak at 180 Hz. Mode 4 leads to a resonance peak at 341 Hz, and modes 7 and 8 to a resonance peak about 475 Hz.

Chapter 11. General Formulation of the Damping of Composite Materials and Structures 324

FIGURE 11.12. Examples of the shapes of the vibration modes of the structure constituted of serge laminate or serge laminate with interleaved viscoelastic layer.

mode 1 mode 2

mode 3 mode 4

mode 5 mode 6

11.5. Dynamic Response of a Structure 325

TABLE 11.9. Comparison of the modal loss factors deduced from modelling and the ones obtained by experimental investigation, in the case of the structure constituted of the serge laminate.

Modelling Experiment

Mode frequency Loss factor Mode frequency Loss factor

mode 1 107 0.95 109 0.98

mode 2 153 0.94 155 0.96

mode 3 215 0.82 216 0.84

mode 4 341 0.97 341 1.05

mode 5 345 0.99 348 1.01

mode 6 457 1.03 456 1.05

mode 7 471 0.96 473 0.99

mode 8 475 0.99 476 1.15

mode 9 538 1.01 540 1.05

mode 10 556 1.05 559 1.07

TABLE 11.10. Comparison of the modal loss factors deduced from modelling and the ones obtained by experimental investigation, in the case of the structure constituted of the serge laminate with interleaved viscoelastic layer.

Modelling Experiment

Mode frequency Loss factor Mode frequency Loss factor

mode 1 108 5.74 110 5.80

mode 2 153 2.80 155 2.88

mode 3 216 9.74 215 9.65

mode 4 343 14.3 345 14.4

mode 5 348 11.3 351 11.1

mode 6 456 12.1 455 12.4

mode 7 471 7.39 474 7.52

mode 8 477 9.75 480 9.65

mode 9 540 9.41 543 9.45

mode 10 562 10.4 565 10.7

Chapter 11. General Formulation of the Damping of Composite Materials and Structures 326

TABLE 11.11. Comparison of the modal loss factors deduced from modelling and the ones obtained by experimental investigation, in the case of the structure constituted of the sandwich material.

Modelling Experiment

Mode frequency Loss factor Mode frequency Loss factor

mode 1 109 4.25 110 4.30

mode 2 148 4.99 150 4.80

mode 3 423 1.89 425 1.95

mode 4 545 1.69 543 1.82

mode 5 588 1.64 590 1.70

mode 6 630 1.62 633 1.68

mode 7 767 1.56 769 1.70

mode 8 830 1.54 834 1.60

mode 9 853 1.53 855 1.63

mode 10 955 1.50 958 1.58

In the case of the structure constituted of the sandwich material, the vibration modes 2, 3 and 5 are not detected. Modes 1, 4, 6 and 7 yield resonance peaks of 109, 545, 588 and 767 Hz, respectively.

The amplitudes of the peaks are slightly decreased in the case of the structure constituted of the sandwich material. However, the higher damping is obtained in the case of the structure constituted of the serge laminate with interleaved viscoelastic layer. A significantly higher damping could be obtained using a thicker viscoelastic layer.

In fact, the purpose of this section was to show that the modelling considered, associated to the experimental characterisation of the dynamic properties of the constituents, was well suited to the analysis of the damped response of a structure constituted of different composite materials. The agreement between the expe-rimental dynamic responses and the responses deduced from the modelling cor-roborates this ability.

11.6 CONCLUSIONS

Modelling of the damping properties of composite materials was developed considering the first-order laminate theory including the effects of the transverse shear. Finite element analysis allows us to derive the different strain energies stored in the material directions of the constituents of composite materials, and next, the energy dissipated by damping in the materials and the composite structure can be obtained as a function of the strain energies and the damping coefficients associated to the different energies stored in the material directions.

11.5. Dynamic Response of a Structure 327

FIGURE 11.13. Frequency responses of the structure constituted of three different materials: (a) glass serge laminate, (b) glass serge laminate with interleaved viscoelastic layer and (c) sandwich material.

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Chapter 11. General Formulation of the Damping of Composite Materials and Structures 328

FIGURE 11.13 (continued). Frequency responses of the structure constituted of three different materials: (a) glass serge laminate, (b) glass serge laminate with interleaved viscoelastic layer and (c) sandwich material.

Modelling so considered can be applied to structures made of laminates, laminates with interleaved viscoelastic layers, as well as sandwich materials.

Damping characteristics of laminates were evaluated experimentally using beam specimens subjected to an impulse input. Loss factors were then derived by fitting the experimental Fourier responses with the analytical motion responses expressed in modal coordinates.

The damping characteristics of the composite materials and of the constituents can be deduced by applying modelling to the flexural vibrations of free-clamped beams. So it can be obtained: the loss factors in the material directions of the dif-ferrent layers of laminated materials, the damping characteristics of the visco-elastic layers, as well as the ones of the foam cores. The analysis has to be imple-mented as a function of the frequency because of the variations with the frequency of the moduli and of the damping properties of the constituents.

Next, modelling can be applied to evaluate the damping properties of struc-tures constituted of laminates, laminates with interleaved viscoelastic layers or sandwich materials. Then, the dynamic responses of structures can be derived by using a nonlinear mode superposition method. The application to a simple shape structure showed that the procedure developed is well suited to the description of the experimental results obtained.

Frequency ( Hz )0 100 200 300 400 500 600 700 800 900 1000

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Jean-Marie Berthelot

Dynamics of Composite Materials and Structures Jean-Marie Berthelot is an Emeritus Professor at the Institute for Advanced Materials and Mechanics (ISMANS), Le Mans, France. His current research is on the mechanical behaviour of composite materials and structures. He has published extensively in the area of composite materials and is the author of a textbook entitled Composite Materials, Mechanical Behavior and Structural Analysis published by Springer, New York, in 1999.