Elasto-Plasticity of Frame Structure Elements 2014

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Andreas chsner

Elasto-Plasticity

of FrameStructure

ElementsModeling and Simulation of Rodsand Beams


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Elasto-Plasticity of Frame Structure Elements


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Andreas chsner

Elasto-Plasticity of FrameStructure ElementsModeling and Simulation of Rods and Beams

1 3


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Andreas chsnerGriffith UniversitySouthportAustralia

and

The University of NewcastleCallaghanAustralia

ISBN 978-3-662-44224-1 ISBN 978-3-662-44225-8 (eBook)DOI 10.1007/978-3-662-44225-8

Library of Congress Control Number: 2014945136

Springer Heidelberg New York Dordrecht London

Springer-Verlag Berlin Heidelberg 2014This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar

methodology now known or hereafter developed. Exempted from this legal reservation are briefexcerpts in connection with reviews or scholarly analysis or material supplied specifically for thepurpose of being entered and executed on a computer system, for exclusive use by the purchaser of thework. Duplication of this publication or parts thereof is permitted only under the provisions ofthe Copyright Law of the Publishers location, in its current version, and permission for use mustalways be obtained from Springer. Permissions for use may be obtained through RightsLink at theCopyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exemptfrom the relevant protective laws and regulations and therefore free for general use.While the advice and information in this book are believed to be true and accurate at the date ofpublication, neither the authors nor the editors nor the publisher can accept any legal responsibility for

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Honest disagreement is often a good sign ofprogress

Mahatma Gandhi (18691948)


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Preface

The scientific literature offers many excellent textbooks on the elasto-plasticbehavior of structural members. However, many students who are entering thisresearch area find it difficult to master this topic based on classical textbooks. Tofacilitate the understanding of the theory, concepts, and involved algorithms, thistextbook offers for students and researchers who are new in this topic a simplerintroduction to elasto-plastic material behavior and its computational treatment.

In this intention, the focus of this textbook is on two simple one-dimensionalstructural members, i.e., the rod and the beam. A classical approach is presented

where first the governing differential equations are derived based on the basicequations of continuum mechanics. The corresponding analytical solutions servemany times to judge the accuracy of derived approximate solutions. The focus inregard to approximate solutions is on the finite element method. This method canbe regarded nowadays as the standard tool to solve engineering problems in theindustrial context. The treatment of one-dimensional members allows to keep themathematical notation and treatment relatively simple. Nevertheless, the deriva-tions are presented in such a way that a transfer to higher dimensions can beachieved. Thus, the considered one-dimensional elements are from an educational

point of view a good demonstrator to introduce basic concepts of continuummechanics and numerical strategies for approximate solutions. Pure elasticity istreated first and then the important case of nonlinear material behavior in theelasto-plastic range is considered.

Furthermore, rod and beam elements have structural analogies in higherdimensions. The procedures applied to the rod element can be transferred to three-dimensional solid elements. Knowing the theory of one-dimensional beam ele-ments allows the understanding of two-dimensional plate elements. Thus, a solidunderstanding of the basic equations and procedures for one-dimensional elementsallows an easier access to elements of the two- and three-dimensional space.

Chapter1illustrates a few technical applications, which can be modeled in asimplified approach based on one-dimensional elements. Chapter2introduces thebasic equations of plasticity theory. The yield condition, the flow rule, andthe hardening rule are introduced. After presenting the equations for the

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one-dimensional stress and strain state, the equations are generalized for atwo-component case. Looking at the two-component case, the notation stays rel-atively simple and the major differences between the one-dimensional case and ahigher dimensionality can easily be shown. Chapter 3 covers the analytical

description of rod or bar members. Based on the three basic equations, i.e., thekinematics relationship, the constitutive law, and the equilibrium equation, thepartial differential equation which describes the problem is derived. Analyticalsolutions in the elastic and elasto-plastic range for different loading and boundaryconditions are derived and discussed. Chapter4follows the approach of the pre-vious chapter and introduces beam elements. Three different types of theories aretreated, i.e., the EULERBERNOULLItheory, the TIMOSHENKOtheory and higher-orderbeam theories. Chapter5introduces the finite element theory for elastic problemsbased on rod, beam, and generalized beam elements. First, each element type is

considered separately, then in pure one-dimensional structures and finally the caseof plane frame structures which allows to arrange and combine different elements.Chapter6covers elasto-plastic finite element simulations, an important type ofnonlinear problems. The solution strategy is derived for pure one-dimensionalproblems. However, the derivations are presented in such a way that a transfer tohigher dimensions is easily possible. Chapter 7 is devoted to an alternativeapproach to derive approximate solutions. Based on one-dimensional problems,the finite difference method is introduced for elastic and elasto-plastic problems.Chapter8summarizes the basic equations for a three-dimensional continuum. The

derived partial differential equations serve to derive finite element procedure forsolid elements. The chapter concludes with an introduction to elasto-plastic finiteelement simulations.

In order to deepen the derived equations and theories, each technical chaptercollects at its end supplementary problems. In total over 120 of such additionalproblems are provided, and a short solution for each problem is included in thisbook. It should be noted that these short solutions contain major steps for thesolution of the problem and not only, for example, a numerical value for the finalresult. This should ensure that students are able to successfully master these

problems. I hope that students find this book a useful complement to many clas-sical textbooks. I look forward to receive their comments and suggestions.

Gold Coast, Australia, May 2014 Andreas chsner

viii Preface
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Acknowledgments

It is important to highlight the contribution of many undergraduate and post-graduate students, which helped to finalize the content of this book. Their ques-tions and comments during different lectures and their work in the scope of finalyear projects helped to compile this book. Furthermore, I would like to express mysincere appreciation to Springer-Verlag, especially to Dr. Christoph Baumann, forgiving me the opportunity to realize this book. A professional publishing companywith the right understanding was the prerequisite to complete this long-termproject. Corrections, comments, and suggestions by Prof. Graeme E. Murch and

Prof. Holm Altenbach are greatly appreciated. Finally, I like to thank my familyfor the understanding and patience during the preparation of this book.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Continuum Mechanics of Plasticity . . . . . . . . . . . . . . . . . . . . . . . . 52.1 General Comments and Observations. . . . . . . . . . . . . . . . . . . . 52.2 Yield Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Flow Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Hardening Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 Isotropic Hardening. . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.2 Kinematic Hardening. . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.3 Combined Hardening. . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Effective Stress and Effective Plastic Strain. . . . . . . . . . . . . . . 182.6 Elasto-Plastic Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Consideration of Unloading, Reversed Loading

and Cyclic Loading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.8 Consideration of Damage. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.8.1 Lemaitres Damage Model . . . . . . . . . . . . . . . . . . . . . . 252.8.2 Gursons Damage Model. . . . . . . . . . . . . . . . . . . . . . . 29

2.9 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Axial Loading of Rods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Constitutive Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3 Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4 Governing Differential Equation. . . . . . . . . . . . . . . . . . . . . . . 403.5 Analytical Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5.1 Elastic Range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.5.2 Elasto-Plastic Range. . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.6 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

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4 Bending of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Bernoulli Beam Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2.3 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . 644.2.4 Governing Differential Equation. . . . . . . . . . . . . . . . . . 674.2.5 Analytical Solutions in the Elastic Range. . . . . . . . . . . . 694.2.6 Analytical Solutions in the Elasto-Plastic Range. . . . . . . 81

4.3 Timoshenko Beam Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.3.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.3.3 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.3.4 Governing Differential Equation. . . . . . . . . . . . . . . . . . 1144.3.5 Analytical Solutions in the Elastic Range. . . . . . . . . . . . 116

4.4 Higher-Order Beam Theories . . . . . . . . . . . . . . . . . . . . . . . . . 1194.4.1 Overview on Different Concepts. . . . . . . . . . . . . . . . . . 1194.4.2 Analytical Solutions in the Elasto-Plastic Range. . . . . . . 131

4.5 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.5.1 Bernoulli Beam Problems . . . . . . . . . . . . . . . . . . . . . . 1394.5.2 Timoshenko Beam Problems . . . . . . . . . . . . . . . . . . . . 1464.5.3 Third Order Beam Problems. . . . . . . . . . . . . . . . . . . . . 148

5 Review of Linear-Elastic Finite Element Simulations. . . . . . . . . . . 1515.1 General Concept of the Weighted Residual Method. . . . . . . . . . 151

5.1.1 Methods Based on the Inner Product . . . . . . . . . . . . . . . 1525.1.2 Methods Based on the Weak Formulation. . . . . . . . . . . 1565.1.3 Procedure on Basis of the Inverse Formulation. . . . . . . . 1585.1.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.2 Rod Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665.2.1 Derivation of the Principal Finite Element Equation. . . . 166

5.2.2 Derivation of Shape Functions. . . . . . . . . . . . . . . . . . . 1795.2.3 Assembly of Elements and Considerationof Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 183

5.2.4 Post-computation: Determination of Strain,Stress and Further Quantities. . . . . . . . . . . . . . . . . . . . 192

5.2.5 Analogies to Other Field Problems. . . . . . . . . . . . . . . . 1945.2.6 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

5.3 Bernoulli Beam Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2045.3.1 Derivation of the Principal Finite Element Equation. . . . 2045.3.2 Derivation of Shape Functions. . . . . . . . . . . . . . . . . . . 2135.3.3 Assembly of Elements and Consideration

of Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 218

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5.3.4 Post-computation: Determination of Stress,Strain and Further Quantities . . . . . . . . . . . . . . . . . . . . 224

5.3.5 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2245.4 Timoshenko Beam Elements. . . . . . . . . . . . . . . . . . . . . . . . . . 236

5.4.1 Derivation of the Principal Finite Element Equation. . . . 2365.4.2 Linear Shape Functions for the Deflection

and Displacement Field. . . . . . . . . . . . . . . . . . . . . . . . 2415.4.3 Higher-Order Shape Functions for the Beam

with Shear Contribution . . . . . . . . . . . . . . . . . . . . . . . . 2525.4.4 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

5.5 Assembly of Elements to Plane Truss Structures . . . . . . . . . . . . 2675.5.1 Rotational Transformation in a Plane . . . . . . . . . . . . . . 2675.5.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

5.6 Assembly of Elements to Plane Frame Structures. . . . . . . . . . . 2735.6.1 Rotation of a Beam Element . . . . . . . . . . . . . . . . . . . . 2735.6.2 Generalized Beam Element . . . . . . . . . . . . . . . . . . . . . 2765.6.3 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

5.7 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2935.7.1 Weighted Residual Method Problems . . . . . . . . . . . . . . 2935.7.2 Rod Element Problems . . . . . . . . . . . . . . . . . . . . . . . . 2935.7.3 Bernoulli Beam Element Problems. . . . . . . . . . . . . . . . 3005.7.4 Timoshenko Beam Element Problems. . . . . . . . . . . . . . 307

5.7.5 Plane Truss Structure Problems . . . . . . . . . . . . . . . . . . 3075.7.6 Plane Frame Structure Problems. . . . . . . . . . . . . . . . . . 309

6 Elasto-Plastic Finite Element Simulations. . . . . . . . . . . . . . . . . . . 3136.1 Integration of the Constitutive Equations. . . . . . . . . . . . . . . . . 3136.2 Derivation of the Fully Implicit Backward-Euler Algorithm

for Isotropic Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3196.3 Derivation of the Fully Implicit Backward-Euler

Algorithm for Kinematic Hardening. . . . . . . . . . . . . . . . . . . . . 325

6.4 Derivation of the Fully Implicit Backward-EulerAlgorithm for Combined Hardening. . . . . . . . . . . . . . . . . . . . . 3276.5 Consideration of Damage. . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

6.5.1 Lemaitre Damage Model. . . . . . . . . . . . . . . . . . . . . . . 3306.5.2 Gurson Damage Model . . . . . . . . . . . . . . . . . . . . . . . . 337

6.6 Solved Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3406.7 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

7 Alternative Approach: The Finite Difference Method . . . . . . . . . . 3817.1 Idea and Derivation of the Method . . . . . . . . . . . . . . . . . . . . . 3817.2 Investigation of Rods in the Elastic Range. . . . . . . . . . . . . . . . 389

7.2.1 Constant Material and Geometry Parameters . . . . . . . . . 3897.2.2 Varying Material and Geometry Parameters . . . . . . . . . . 393

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7.3 Investigation of Bernoulli Beams in the Elastic Range . . . . . . . . 4017.3.1 Approximation of the Differential Equation. . . . . . . . . . 4017.3.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

7.4 Investigation of Timoshenko Beams in the Elastic Range. . . . . . 413

7.5 Consideration of Bernoulli Beams with PlasticMaterial Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417


8 Prelude to the General Three-Dimensional Case. . . . . . . . . . . . . . 4378.1 Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4378.2 Constitutive Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4398.3 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4408.4 Governing Differential Equation. . . . . . . . . . . . . . . . . . . . . . . 441

8.5 Derivation of the Principal Finite Element Equation . . . . . . . . . 4428.6 Hexahedron Solid Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 446

8.6.1 Evaluation of the Boundary Force Matrix . . . . . . . . . . . 4518.6.2 Evaluation of the Body Force Matrix . . . . . . . . . . . . . . 454

8.7 Derivation of the Fully Implicit Backward Euler Algorithmfor Isotropic Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456


Appendix A: Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

Appendix B: Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

Appendix C: Units and Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . 501

Answers to Supplementary Problems. . . . . . . . . . . . . . . . . . . . . . . . . 505

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

xiv Contents
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Symbols and Abbreviations

Latin Symbols (Capital Letters)

A Area, cross-sectional areaA Effective resisting area

AD Total area of microcracks and cavitiesA Matrix, cf. derivation of shape functionsB Matrix which contains derivatives of shape functions

Cijkl Fourth-order elasticity tensorC Elasticity matrixCelpl Elasto-plastic modulus matrixD Damage variableDpl Generalized plastic modulusDc Critical damage at crack initiation in pure tensionD Compliance matrixDpl Matrix of generalized plastic moduliE YOUNGs modulus

E Elastic modulus of damaged materialEelpl Elastic-plastic modulusEpl Plastic modulus (isotropic hardening)EA Axial tensile stiffnessEI Bending stiffnessF Force, yield conditionFellim Maximum elastic forceG Shear modulusGA Shear stiffness

H Kinematic hardening modulusI Second moment of area, abbreviation for an integral statementI Identity matrix (diagonal matrix), I d1 1 1. . .cJ Jacobian, cf. coordinate transformation

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K Global stiffness matrixK

e Elemental stiffness matrixKT Tangent stiffness matrixL Element length

M MomentMellim Maximum elastic bending moment

Mpllim

Limiting bending moment

Mpls;lim

Limiting bending moment under shear influence

N Normal force (internal), shape functionN Column matrix of shape functionsNi 3 93 Matrix of shape functions for node iP LEGENDREpolynomial, point

Q Shear force (internal), plastic potential function (plastic potential)_Q Heat transfer rate

Qpllim

Plastic shear force

R Equivalent nodal force, radius of curvature of a curve, stress ratioT Transformation matrixU PerimeterV VolumeVD Total volume of microcracks and cavitiesW Weight function

W

Fundamental solutionW Column matrix of weight functionsX Global Cartesian coordinateY Global Cartesian coordinate, damage energy release rateZ Global Cartesian coordinate

Latin Symbols (Small Letters)

a Basis coefficient, geometric dimension

a Column matrix of basis coefficientsb Coefficient, function, geometric dimensionb Column matrix of body forces acting per unit volumec Constant of integration, coefficient, geometric dimensiond Coefficientf Body force, scalar functionf Column matrix of loadsg Scalar function, standard gravityh Evolution function of hardening parameter, geometric dimension

h Evolution function of hardening parametersi Iteration index, node numberj Iteration index

xvi Symbols and Abbreviations


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k Auxiliary function, elastic embedding modulus, elastic foundationmodulus, layer number, stiffness, spring constant,thermal conductivity, yield stress

ks Shear correction factor, shear yield stress

kt Tensile yield stresskrt Uniaxial stress at rupturem Distributed moment, element number, massm Matrix functionn Node number, increment numbernj Components of the normal vectorn Normal vectorp Distributed load in x-directionq Distributed load in y-direction, internal variable (hardening)

q Column matrix of hardening variables_q Heat fluxr Damage evolution parameter, residualr Plastic flow direction, residual column matrixs Damage evolution parameters Column matrix of stress deviator componentst Time, traction forceti Components of the traction force vectortend Convergence value

t Column matrix of traction forcesu Displacementu0 Exact solutionu Column matrix of displacements, column matrix of nodal unknowns Auxiliary function Variable matrixw Weight for numerical integrationx Cartesian coordinatex Column matrix of Cartesian coordinates

y Cartesian coordinatez Cartesian coordinate

Greek Symbols (Capital Letters)

Boundary Factor Domain

Greek Symbols (Small Letters)

Back stress/kinematic hardening parameter, parameter, rotation angle Column matrix of kinematic hardening parameter

Symbols and Abbreviations xvii


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Parameter Shear strain (engineering definition), load per unit volume,

specific weight per unit volume, g DIRACdelta function, geometric dimension

Strain Effective strainel Elastic strainpl Plastic strain

plD

Damage plastic strain threshold in pure tension

pleff

Effective plastic strain (equivalent plastic strain)

plr Plastic strain at rupture

plV

Volumetric plastic strain

ij Second-order strain tensorc Elastic limit strain in compressiont Elastic limit strain in tension Column matrix of strain components Natural coordinate Natural coordinate_ Rate of energy generation per unit volume Curvature, isotropic hardening parameterellim Maximum elastic curvature

LAMSconstant, consistency parameter (cf. plasticity) LAMSconstant POISSONs ratio Natural coordinate, relative stress

;:

Rate of energy generation per unit length

Mass density Stress, normal stress Effective stress (damage mechanics)eff Effective stress (yield condition)

m Mean stress (hydrostatic stress)ij Second-order stress tensor Column matrix of stress components Coordinate where elasto-plastic interface reaches beam surface Shear stress Rotation (TIMOSHENKObeam) Basis function, rotation (BERNOULLIbeam) Column matrix of basis functions Basis function

Material integrity

xviii Symbols and Abbreviations


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Mathematical Symbols

9 Multiplication sign (used where essential). . . Matrix

d. . .c Diagonal matrix. . .T Transposeh. . .i MACAULAYs bracketh. . .;. . .i Inner productL f g Differential operatorL Matrix of differential operatorssgn. . . Signum (sign) functiono Partial derivative symbol (rounded d)IR Set of real numbers

DIRACdelta function1 Identity column matrix, 1 111000 T

L Diagonal scaling matrix, L d111000c

Indices, Superscripted

. . .e Element

. . .el Elastic

. . .init Initial

. . .pl Plastic

. . .trial Trial state (return mapping)

Indices, Subscripted

b Bending c Center, compression lim Limit

p Nodal value (point) s Shear t Tensile

Abbreviations

1D One-dimensional2D Two-dimensional3D Three-dimensional

a.u. Arbitrary unitBC Boundary conditionBEM Boundary element methodconst. Constant

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dim. DimensionDOF Degree(s) of freedomEBT EULERBERNOULLIbeam theory (elementary beam theory)FD Finite difference

FDM Finite difference methodFEM Finite element methodFGM Functionally graded materialFSDT First-order shear deformation theoryinc Incrementmax MaximumPDE Partial differential equationRVE Representative volume elementSI International system of units

SSDT Second-order shear deformation theorysym. SymmetricTBT TIMOSHENKObeam theoryTSDT Third-order shear deformation theoryWRM Weighted residual method

Some Standard Abbreviations

ca. About, approximately (from Latin circa)

cf. Compare (from Latin confer)ead. The same (woman) (from Latin eadem)e.g. For example (from Latin exempli gratia)et al. And others (from Latin et alii)et seq. And what follows (from Latin et sequens)etc. And others (from Latin et cetera)i.a. Among other things (from Latin inter alia)ibid. In the same place (the same), used in citations (from Latin ibidem)id. The same (man) (from Latin idem)

i.e. That is (from Latin id est)loc. cit. In the place cited (from Latin loco citato)N.N. Unknown name, used as a placeholder for unknown names (from Latin

nomen nescio)op. cit. In the work cited (from Latin opere citato)pp. Pagesq.e.d. Which had to be demonstrated (from Latin quod erat demonstrandum)viz. Namely, precisely (from Latin videlicet)vs. Against (from Latin versus)

xx Symbols and Abbreviations


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Chapter 1

Introduction

Abstract The first chapter classifies the content as well as the focus of this textbook.Based on common applications taken from structural engineering, the investigation

of rod and beam elements in spatial arrangements is motivated.

Frame structures are spatial arrangements or skeletons composed of beams and rods.The design principle of arranging one-dimensional members in a spatial frameworkcan be found in many areas of civil, mechanical and structural engineering. A majoridea of these structures is to optimize the use of a material such as steel, wood orconcrete and to arrange it in such a way to obtain maximum stiffness and strengthat a lower weight and costs than solid structures. Typical examples are some of theconstructions of Gustave Eiffel,1 such as the railway arch bridge Garabit Viaductwhich spans the river La Truyre in Southern France (Auvergne), see Fig. 1.1.

Some other typical examples from civil engineering are:

Eiffel Tower, Paris, France (construction period 18871889), Maria Pia Bridge (Douro Viaduct), Porto, Portugal (construction period 1876

1877), internal frame structure of the Statue of Liberty,2 New York, United States (con-

struction period 18751886).

In the same context, one may mention, for example, tower structures for power

transmission, cranes, and wind energy converters. Within this textbook, we distin-guish two different types of basic members of such spatial structures, i.e. rods andbeams. A rod is understood as a prismatic body where loads act only in the directionof the principal axis of the body. This member is sometimes called a bar and in someapplications a rotational deformation around the principal axis is also considered.

1 Alexandre Gustave EIFFEL (18321923), French civil engineer and architect.2 The Statue of Liberty was realized by the French sculptor Frdric Auguste BARTHOLDI (18341904). A bronze model of the statue stands in the Jardin du Luxembourg (Paris, France) and wasused by Bartholdi as part of the preparatory work for the New York statue. The second Statue ofLiberty in Paris is near the Grenelle Bridge on the le aux Cygnes.

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2 1 Introduction

Fig. 1.1 Railway arch bridge (Garabit Viaduct), France (construction period 18841884)

However, we are going to consider in the following only the elongation and compres-sion of rods. If rods are spatially arranged, a so-called truss structure is obtained. Thecommon characteristics and simplifications for truss structures are as follows [59]:

two- or three-dimensional framework of straight rod members, the ends are connected by frictionless hinged joints (not subjected to moments,

rotations are zero), the loads and reactions act only at the joints, the rods are only subjected to axial compressive or tensile forces.

The second basic member is a beam element which is understood as a prismaticbody where loads act only perpendicular to the principal axis of the body. Thismember can be generalized by superimposing a rod so that a member which can bendandelongateisobtained(generalizedbeam).Asamatterofcourse,torsionaroundtheprinciple axis could be also superimposed. The spatial arrangement of generalizedbeams is called a frame structure and has the following common characteristics andsimplifications [59]:

two- or three-dimensional framework of straight generalized beam members, the ends are connected by rigid (and/or hinged) connections (rigid connections

allow rotations), the members are subjected to axial forces, shear forces, and bending moments.

Naturally we can combine rods and beams as separated members in a spatialarrangement and we simply call such a structure a frame structure. As an example,


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1 Introduction 3

Fig. 1.2 Simplified two-dimensional model of the railway arch bridge (Garabit Viaduct in France,see Fig. 1.1) based on rods and beams

Table 1.1 Some steps in the early historical development of the finite element methodYear Author Comment Ref.

. . . . . . . . . . . .

1909 W. Ritz First discrete variational method. Formulation of theapproximate solution within thewholedomain

[89]

1915 B.G. Galerkin Orthogonality of the residuum and the weight function.Formulation of the approximate solution and the weightfunction within thewholedomain

[42]

1943 R. Courant Subdivision of the whole domain into triangular ele-ments for solving the torsion problem in a variational

approach. Consideration of a very small number of ele-ments

[27]

1956 M.J. Turner,R.W. Clough,H.C. Martin,L.J. Topp

First engineering formulation of the FEM based onthe principle of virtual work. Derivation of a triangu-lar plane stress element with six degrees of freedom.Deduction of a quatrilateral and a rectangular based tri-angular element

[108]

. . . . . . . . . . . .

1960 R.W. Clough This conference manuscript introduces for the first timethe expression finite element

[24]

. . . . . . . . . . . .

these basic elements can be used to create simple models of complex structuresas shown in Fig. 1.2where a two-dimensional model of the Garabit Viaduct (seeFig. 1.1)based on straight frame elements is schematically indicated.

The focus concerning approximation methods within this book is on the finiteelement method. Chapter 5introduces the finite element method for linear-elastic

material behavior while Chap. 6extends the procedures to the treatment of elasto-plastic material response. A few steps in the very early historical development of thiswidespread numerical tool are collected in Table 1.1,see [40, 91, 97, 119, 121].
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Chapter 2

Continuum Mechanics of Plasticity

Abstract This chapter introduces first the basic equations of plasticity theory. Theyield condition, the flow rule, and the hardening rule are introduced. After deriving

and presenting these equations for the one-dimensional stress and strain state, theequations are generalized for a two-component -stress state. Based on these basicequations, the concept of effective stress and strain as well as the elasto-plasticmodulus is introduced. The chapter finishes with an introduction to two differentdamage concepts, i.e. the Lemaitre and Gurson damage model, which are derivedfor the one-dimensional case.

2.1 General Comments and Observations

Let us consider in the following a uniaxial tensile test whose idealized specimenis schematically shown in Fig.2.1. The original dimensions of, for example, thecylindrical specimen are characterized by the cross-sectional area Aand length L.ThisspecimenisnowelongatedinauniversaltestingmachineanditslengthincreasestoL +L. In the caseofa real specimenmadeofa commonengineering material, thecross-sectional area would reduce to A A. This phenomenon could be describedbased on Poisssons ratio.1 However, if we assume an idealized uniaxial state, i.e. a

uniaxial stressandstrain state, the contraction is disregarded and the cross-sectionalarea is assumed to remain constant, see Fig.2.1b. During the tensile test, the force isnormally recorded by a load cell attached to the movable or fixed cross-head of themachine. If this force is divided by the (initial) cross-sectional area, the engineeringstress is obtained as:

= FA

. (2.1)

The deformation or elongation of the specimen can be measured, for example,

by an external extensometer which should be directly attached to the specimen.

1 Simon Denis Poisson(17811840), French mathematician, geometer, and physicist.

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6 2 Continuum Mechanics of Plasticity

(a) (b)

Fig. 2.1 Schematic representation ofaan unloaded andban idealized uniaxial tensile specimenloaded by a force For a displacement u

These devices are either realized as strain gauge or inductive extensometers.2 Anymeasurement based on the movement of the cross-head must be avoided since itdoes not guarantee an accurate determination of the specimens behavior. The defi-nition of strain is given in its simplest form as elongation over initial length and theengineering strain can be calculated as:

=

L

L. (2.2)

Relating the stress to its corresponding strain, the engineering stress-strain dia-gram can be plotted as schematically shown in Fig.2.2. In the pure elastic range, alinear line is observed and its slope is equal to Youngs3 modulus:

E=

or E= dd

. (2.3)

2 More modern devises are contactless laser or video based extensometers.3 ThomasYoung(17731829), English polymath.


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2.1 General Comments and Observations 7

(a) (b)

Fig. 2.2 Uniaxial stress-strain diagrams for different isotropic hardening laws:aarbitrary harden-

ing;blinear hardening and ideal plasticity

The last relation is also known as Hookes4 law and often written in the followingform for linear elastic behavior:

= E . (2.4)

As soon as the initial yield stress kinitt is reached, plastic material behavior occurs

and the slope of the stress-strain diagram changes.The characteristic feature of plastic material behavior is that a remaining strain

pl occurs after complete unloading, see Fig.2.2a. Only the elastic strainsel returnsto zero at complete unloading. An additive composition of the strains by their elasticand plastic parts

= el + pl (2.5)

is permitted at restrictions to small strains. The elastic strains el can hereby be

determined via Hookes law, wherebyin Eq. (2.3) has to be substituted by el

.Furthermore,no explicitcorrelation is given anymore for plastic materialbehaviorin general between stress and strain, since the strain state is also dependent on theloading history. Due to this, rate equations are necessary and need to be integratedthroughout the entire load history. Within the framework of the time-independentplasticity investigated here, the rate equations can be simplified to incremental rela-tions. From Eq. (2.5) the additive composition of the strain increments results in:

d = del + dpl . (2.6)

4 Robert Hooke(16351703), English natural philosopher, architect and polymath.


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The constitutive description of plastic material behavior includes

a yield condition, a flow rule and a hardening law.

In the following, the case of the monotonic loading5 is considered first, so thatisotropichardening is explained first in thecase of material hardening. This importantcase, for example, occurs in the experimental mechanics at the uniaxial tensile testwith monotonic loading. Furthermore, it is assumed that the yield stress is identicalin the tensile and compressive regime:kt= kc= k.

2.2 Yield Condition

The yield condition enables one to determine whether the relevant material suffersonly elastic or also plastic strains at a certain stress state at a point of the relevantbody. In the uniaxial tensile test, plastic flow begins when reaching the initial yieldstresskinit, see Fig.2.2.The yield condition in its general one-dimensional form canbe set as follows(IR IR IR):

F= F(, ) , (2.7)

where represents the inner variable of isotropic hardening. In the case of idealplasticity, see Fig.2.2b, the following is valid: F= F(). The values ofFhave thefollowing mechanical meaning, see Fig.2.3a:

F(, )


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2.2 Yield Condition 9

F(, )=0 plastic material behavior, (2.9)F(, ) >0 invalid. (2.10)

A further simplification results under the assumption that the yield condition canbe split into a pure stress fraction f(), the so-called yield criterion,6 and into anexperimental material parameterk(), the so-called flow stress:

F(, ) = f() k() . (2.11)

For a uniaxial tensile test (see Fig. 2.2) the yield condition can be noted in thefollowing form:

F(, ) = || k() 0 . (2.12)

If one considers the idealized case of the linear hardening (see Fig.2.2b), Eq. (2.12)can be written as (Fig.2.4)

F(, ) = || (kinit +Epl) 0 . (2.13)

The parameter Epl is the plastic modulus (see Fig.2.5), which becomes zero inthe case of ideal plasticity:

F(, )= |

|

kinit

0 . (2.14)

In the following, we are going to extend this concept to a stress state where inaddition to a normal stress ()a shear stress component ()is acting. This two-component stress state occurs, for example, in the case of general beam formulations,seeSects.4.3 and 4.4. Twoclassicalrepresentativesofyieldconditionsforsoliddense

(a) (b)

Fig. 2.4 Graphical representation of the yield condition according toa von Mises andb Trescain the two-component-space

6 If the unit of the yield criterion equals the stress, f()represents the equivalent stress or effectivestress. In the general three-dimensional case the following is valid under consideration of thesymmetry of the stress tensoreff: (IR6 IR+).
http://dx.doi.org/10.1007/978-3-662-44225-8_4http://dx.doi.org/10.1007/978-3-662-44225-8_4http://dx.doi.org/10.1007/978-3-662-44225-8_4http://dx.doi.org/10.1007/978-3-662-44225-8_4http://dx.doi.org/10.1007/978-3-662-44225-8_4


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Fig. 2.5 Flow curve for different isotropic hardening laws

materials (classical metals) will be introduced for this special two-component stress

state7 in the following [53]. The Tresca8 yield condition [107] is based on theassumption that yielding begins when the maximum shear stress reaches a certainvalue and can be written, for example, in its mathematical form based on the twoacting stress components as

F=

2 + 42 kt= 0 (Tresca) , (2.15)

where the relationship between shear (ks)and tensile yield stress (kt)is given byks

= kt2, see [120].

The von Mises9 yield condition [69] is based on the assumption that yieldingbegins when the elastic energy of distortion reaches a critical value [50]. The math-ematical formulation of this condition reads in the-space as

F=

2 + 32 kt= 0 (Von Mises) , (2.16)

where the relationship between shear and tensile yield stress is given by ks= kt3 ,see [20]. Both yield conditions can be written in a more general form as

F= F() = 0 , (2.17)

where = T is the column matrix of the stress components. The graphicalrepresentation of both yield conditions is given in Fig.2.4. As can be seen, centeredellipses are obtained where the major and minor axes of the ellipses are parallel tothe stress axes of the coordinate system. Comparing both shapes, it can be concludedthat the Trescayield condition gives a more conservative prediction of the yieldstress since the minor axis is smaller compared to the von Misescondition.

7 A two-dimensional stress state would comprise in its general form, for example, the three com-ponentsx,yand x y .8 Henri douard Tresca(18141885), French mechanical engineer.9 Richard Edler von Mises(18831953), Austrian scientist and mathematician.


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2.2 Yield Condition 11

Table 2.1 Differentformulations of the vonMisesand Trescayieldcondition in the two-component-space

von Mises Tresca

Based on tensile yield stress

2 + 32 kt= 0

2 + 42 kt= 02

+32

k2

t =0 2

+42

k2

t =0

Based on shear yield stress

3

3

2 + 32 ks= 0 1

2

2 + 42 ks= 0

1

3

2 + 32

k2s= 0 14 2 + 42 k2s= 0

It should be noted here that both conditions given in Eqs. (2.15) and(2.16) reduce

for a uniaxial stress state where only a normal stress is acting to the formulationpresented in Eq. (2.12). Finally, Table2.1summarizes different formulations of theconsidered yield conditions in the two-component-space.

2.3 Flow Rule

Theflowruleservesasamathematicaldescriptionoftheevolutionoftheinfinitesimal

increments of the plastic strain dpl in the course of the load history of the body. Inits most general one-dimensional form, the flow rule can be set up as follows [96]:

dpl = d r(, ) , (2.18)

whereupon the factor dis described as the consistency parameter (d 0)andr: (IR IR IR)as the function of the flow direction.10 One considers that solelyfor dpl = 0 then d= 0 results. Based on the stability postulate of Drucker11[33] the following flow rule can be derived12:

dpl = d F(, )

. (2.19)

Such a flow rule is referred to as the normal rule13 (see Fig.2.3a) or due to r=F(, )/as theassociatedflow rule.

10 In the general three-dimensional caserhereby defines the direction of the vector dpl, while thescalar factor defines the absolute value.11

Daniel Charles Drucker(19182001), US engineer.12 A formal alternative derivation of the associated flow rule can occur via the Lagrange multipliermethod as extreme value with side-conditions from the principle of maximum plastic work [12].13 In the general three-dimensional case the image vector of the plastic strain increment has to bepositioned upright and outside oriented to the yield surface, see Fig.2.3b.


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Experimental results, among other things from the area of granular materials [13]can however be approximated better if the stress gradient is substituted through adifferent function, the so-called plastic potential Q. The resulting flow rule is thenreferred to as thenon-associatedflow rule:

dpl = d Q(, )

. (2.20)

In quite complicated yield conditions often the case occurs that a more simple yieldcondition is used for Qin the first approximation, for which the gradient can easilybe determined.

The application of the associated flow rule (2.19) to the yield conditions accordingto Eqs. (2.12)(2.14) yields for all three types of yield conditions (meaning arbitrary

hardening, linear hardening and ideal plasticity):dpl = d sgn() , (2.21)

where sgn() represents the so-called sign function,14 which can adopt the followingvalues:

sgn() =

1 for 0. (2.22)

For the two-component -stress space, the associated flow rule (2.19) can bewritten as

dpl = d F()

, (2.23)

where dpl = dpl dplT is the column matrix of the plastic strain increments.Application of this definition to the yield conditions given in Eqs.(2.15)and (2.16)gives finally:

dpl =

dpl

dpl

= d

2 + 42

4

(Tresca) , (2.24)

dpl =

dpl

dpl

= d

2 + 32

3

(Von Mises) . (2.25)

These two equations can be generally expressed in the manner of Eq.(2.18) as:

d

pl

= dr(

, ) . (2.26)

14 Also signum function; from the Latin signum for sign.


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2.4 Hardening Rule 13

2.4 Hardening Rule

The hardening law allows the consideration of the influence of material hardeningon the yield condition and the flow rule.

2.4.1 Isotropic Hardening

In the case of isotropic hardening, the yield stress is expressed as being dependenton an inner variable:

k= k(). (2.27)

If the equivalent plastic strain15 is used for the hardening variable(= |pl|), thenone talks about strain hardening.

Another possibility is to describe the hardening being dependent on the specific16

plasticwork ( = wpl = dpl).Thenonetalksaboutworkhardening.IfEq.( 2.27)is combined with the flow rule according to (2.21), the evolution equation for theisotropic hardening variable results in:

d = d|pl| = d . (2.28)

Figure 2.5 shows the flow curve, meaning the graphical illustration of the yield stressbeing dependent on the inner variable for different hardening approaches.

The yield condition which was expressed in Eq.(2.17)for the case of ideal plas-ticity can now be expanded to the formulation

F= F(, q) = 0 , (2.29)

where the internal variable qconsiders the influence of the material hardening onthe yield condition. The evolution equation for this internal variable can be stated inits most general form based on Eq. (2.29) as

dq= d h(, q) , (2.30)

15 Theeffectiveplastic strainis in thegeneral three-dimensionalcase thefunctionpleff: (IR6 IR+).In the one-dimensional case, the following is valid: pleff=

plpl = |pl|. Attention: Finite

element programs optionally use themore general definitionfor theillustration in thepost processor,

this means pleff=

23

pli j

pli j , which considers the lateral contraction at uniaxial stress

problems in the plastic area via the factor 23 . However in pure one-dimensional problems withoutlateralcontraction,thisformulaleadstoanillustrationoftheeffectiveplasticstrain,whichisreduced

by the factor

23 0.816.

16 This is the volume-specific definition, meaning

wpl

= Nm

2mm=

kg ms2m2

mm=

kg m2

s2m3= J

m3.


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where the function hdefines the evolution of the hardening parameter. Assigningfor the internal variableq= (in the case thatequals the effective plastic strains,one talks about a strain space formulation) and considering the case of associatedplasticity, a more specific rule for the evolution of the internal variable is given as

d = d

Dpl1 F(, )

= d

k()

F(, )

= d 1Epl

F(, )

, (2.31)

where Dpl is the generalized plastic modulus. Considering the yield stresskas theinternal variable, one obtains a stress space formulation as F= F(, k)and thecorresponding evolution equation for the internal variable is given by:

dk= d Dpl F(, k)k

= d Epl F(, k)k

, (2.32)

where dkcan be written as Epld. Thus, one may alternatively formulate:

d = d F(, k)k

. (2.33)

Application of the instruction for the evolution of the internal variable according toEq. (2.31) or(2.33) to the Trescaor von Misesyield condition with kt= kt()(cf. Eqs.(2.15)and (2.16)) gives:

d = d . (2.34)

Thus, it turned out thath , i.e. the evolution equation for the hardening parameter inEq. (2.30), simplified toh=1. However, in the case of more complex yield condi-tions, the function hmay take a more complex form. This will be shown in Sect. 2.8.2

where the Gursons damage model is introduced. The graphical representation ofthe initial and subsequent yield surface for isotropic hardening in the two-component-space is shown in Fig.2.6.

2.4.2 Kinematic Hardening

In the case of a pure monotonic loading, i.e. pure tensile or pure compression, itis not possible to distinguish from the stress-strain diagram the cases of isotropic

or kinematic hardening. Let us look in the following on a uniaxial test with plasticdeformation and stress reversal as schematically shown in Fig.2.7. The test startswithout any initial stress or strain in the origin of the stress-strain diagram (point 0)and a tensile load is continuously increased. The first part of the path, i.e. as long as


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Fig. 2.6 Initial and subsequent yield surface for isotropic hardening in the two-component-space

(a) (b)

Fig. 2.7 Uniaxial kinematic hardening: a idealized stress-strain curve with Bauschingereffectandbkinematic hardening parameter as a function of the internal variable (linear hardening)

the stress is below the yield stress k, is in the pure elastic range and Hookes lawdescribes the stress-strain behavior. Reaching the yield stressk(point 1), the slopeofthediagramchangesandplasticdeformationoccurs.Withongoingincreasingload,

the plastic deformation and the plastic strain increases in this part of the diagram.Let us assume now that the load is reversed at point 2. The unloading is completelyelastic and compressive stress develops as soon as the load path passes the strain axis.The interesting question is now when thesubsequent plastic deformations starts in thecompressive regime. This plastic deformation occurs now in the case of kinematichardening at a stress level 3 which is lower than the initial yield stress kor thesubsequent stress2. This behavior is known as the Bauschinger17 effect [8] andrequires plastic pre-straining with subsequent load reversal.

The behavior shown in Fig.2.7a can be described based on the following yieldcondition

F= | ()| k= 0 , (2.35)

17 Johann Bauschinger(18341893), German mathematician and engineer.


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where the initial yield stress kis constant and the kinematic hardening parameter18

isafunctionofaninternalvariable.Figure2.7bshowsthecaseoflinearhardeningwhere a linear relationship between kinematic hardening parameter and internalvariable is obtained.

The simplest relation between the kinematic hardening parameter and the internalvariable was proposed in [68] as

=Hpl or d=Hdpl , (2.36)

where His a constant called the kinematic hardening modulus and the plastic strainis assigned as the internal variable. Thus, Eq. (2.36) describes the case of linear hard-ening. A more general formulation of Eq.(2.36)is known asPragers19 hardeningrule [80, 81]:

d=H(, i )dpl , (2.37)where the kinematic hardening modulus is now a scalar function which depends onthe state variables (, i ). One suggestion is to use the effective plastic strain

pleff

as internal variable [6]. A further extension is proposed in [60] where the hardeningmodulus is formulated as a tensor.

Another formulation was proposed by Ziegler20 [94, 118] as

d= d( ) , (2.38)

where the proportionality factor dcan be expressed as:

d = adpl , (2.39)

or in a more general way as a= a(, i ). The rule given in Eq. (2.38) is known inthe literature as Zieglers hardening rule. It should be noted here that the plasticstrain increments in Eqs. (2.39) and (2.37) can be calculated based on the flow rulesgiven in Sect.2.3.Thus, the kinematic hardening rules can be expressed in a more

general way as: d= dh(, ). (2.40)

For the two-component-/-space, the kinematic hardening rules according toPragerand Zieglercan be written as

d = H(,)dpl (Prager) , (2.41)d = d( ) (Ziegler) , (2.42)

18 An alternative expression for the kinematic hardening parameter is back-stress.19 William Prager(19031980), German engineer and applied mathematician.20 Hans Ziegler(19101985), Swiss scientist.


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or in components as

dd = H(,)

dpl

dpl (Prager) , (2.43)dd

= d

(Ziegler) . (2.44)

Thus, the generalization of the one-dimensional yield condition given in Eq. (2.35)can be written in the case ofTrescaand von Misesas:

F=

( )2 + 4( )2 kt= 0 (Tresca) , (2.45)F= ( )

2 + 3( )2 kt= 0 (von Mises) . (2.46)

The graphical representation of Eqs. (2.45) and(2.46)is schematically shownin Fig. (2.8) where it can be seen that the center of the subsequent yield surface isdescribed by the back-stress vector T = T.

2.4.3 Combined Hardening

The isotropic and kinematic hardening rules presented in Sects. 2.4.1and2.4.2can simply be joined together to obtain a combined hardening rule for the one-dimensional yield condition as:

F(, q) = | | k() , (2.47)

or for the special case of isotropic linear hardening as

F= | | (kinit +Eplpleff) , (2.48)

Fig. 2.8 Initial and subsequent yield surface for kinematic hardening in the two-component-space


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where the back-stress can be a function as indicated in Eqs. (2.36)(2.38). Theassociated flow rule is then obtained according to Eq.(2.19) as:

dpl

= dF

= d sgn( ) (2.49)and the isotropic and kinematic hardening (Prager) laws can be written as:

d = d|pl| = |d sgn( )| = d , (2.50)d=Hdpl = Hd sgn( ) . (2.51)

The last two equations can be combined and generally expressed as:

dq= dh(, q) , (2.52)

or dd

= d

1

Hsgn( )

. (2.53)

Forthetwo-component -/-space, Eqs. (2.45), (2.46) and(2.27)canbecombinedto obtain the yield conditions for combined hardening. In this case, the evolution

equations for the isotropic and kinematic hardening parameters can be generallyexpressed as:

dq= dh(, q) . (2.54)

2.5 Effective Stress and Effective Plastic Strain

The previous sections introduced a two-component stress state

= T. If amaterial is subjected to such a multi-axial stress state, it is difficult to directly com-pare these different stress components with the experimental uniaxial stress-strainas schematically shown in Fig.2.2.How to decide if the state is still in the elasticor already in the plastic range? To overcome this problem, one can define the so-called effective stress.21 This is a mathematical equation which calculates, based onthe acting stress components, a single scalar stress valueeffwhich can be directlycompared with the experimental stress value k. If a yield condition is defined, forexample, as F(, )= f()k(), then f()can be considered as the effec-tive stress definition. However, yield conditions may have different formulations

21 Sometimes called the equivalent stress.


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2.5 Effective Stress and Effective Plastic Strain 19

(see Table2.1) and we may indicate the yield condition in a more general form as

F(, ) = f() g(k) = 0 . (2.55)

Thus,theyieldconditionissplitintoafraction f() whichdependsonlyontheactingstress state and a fraction which involves the experimental stress-strain response ofthe material. Then, it follows that the function f()must be some constant ctimesthe effective stress to some power d[22]:

f() = c deff. (2.56)

Let us assume, for example, a yield condition of the form 2 + 32 k2t =0, seeTable2.1.Since the effective stress should reduce to the normal stress componentin the uniaxial tensile test (

=0

=0), we can write that

2 = c deff, (2.57)

and comparing coefficients givesc=1 andd=2. Thus, the effective stress in thisspecific case is given as:

eff= 1

2 + 32 1

2. (2.58)

For the definition of the effective plastic strain pl

eff

, different approaches can be foundin the literature [22]. We will use in the following a definition which is based on thevolume-specific plastic work which can be expressed either by the stress and plasticstrain components or the corresponding effective values as:

dwpl = Tdpl != effdpleff, (2.59)

or rearranged for the effective plastic strain increment as:

dpleff=

Tdpl

eff. (2.60)

Under the assumptionofanassociatedflow rule, the last equationcan be expressed as:

dpleff= d

TF

eff

h

. (2.61)

Inthefollowingletuslookatthe von Mises yieldconditionasgiveninEq.(2.16),i.e.

F=

2 + 32 eff

kt= 0 , (2.62)


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where the effective stress is easily identified. The derivative of the yield conditionwithrespecttothestresscomponentscanbeobtainedfromEq.(2.25)andtheeffectiveplastic strain increment can be written as:

dpleff= d

T 12+32

3

2 + 32

= d 1. (2.63)

2.6 Elasto-Plastic Modulus

The stiffness of a material changes during plastic deformation and the strain state

is dependent on the loading history. Therefore, Hookes law which is valid forthe linear-elastic material behavior according to Eq. (2.3)must be replaced by thefollowing infinitesimal incremental relation:

d= Eelpld , (2.64)

whereEelpl is the elasto-plastic modulus. The algebraic expression for this moduluscan be obtained in the following manner. The total differential of a yield conditionF= F(, q), see Eq.(2.47), is given by:

dF(, q) =

F

d +

F

q

Tdq= 0 . (2.65)

If Hookes law (2.3) and the flow rule (2.18) are introduced in the relation for theadditive composition of the elastic and plastic strain according to Eq.(2.6), oneobtains:

d =1

Ed + dr. (2.66)

Multiplication of Eq. (2.66) from the left-hand side with

F

Eand inserting in

Eq.(2.65) gives, under the consideration of the evolution equation of the hardeningvariables (2.52), the consistence parameter as:

d =

F

E

F Er Fq T

h

d . (2.67)


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2.6 Elasto-Plastic Modulus 21

This equation for the consistency parameter can be inserted into Eq. (2.66) andsolving for dd gives the elasto-plastic modulus as:

Eelpl = EEF Er

F

Er

Fq

Th

. (2.68)

Let us consider now the case of combined linear kinematic and isotropic hardening[see Eq. (2.48)] where the kinematic hardening modulusH(Prager) and the plasticmodulus Epl are constant. Furthermore, the flow rule is assumed to be associated.We assume in the following that the yield condition is a function of the followinginternal variables: F= F(, q)= F(, pl, pleff). The corresponding terms in theexpression for the elasto-plastic modulus are as follows:

F

= sgn

Hpl , (2.69)r= F

= sgn

Hpl , (2.70)F

q

= Eplsgn H

pl

H

, (2.71)

h=

1sgn

Hpl . (2.72)

Introducing these four expressions in Eq. (2.68) gives finally:

Eelpl = dd

= E(H+ Epl)

E+ (H+ Epl) . (2.73)

A slightly different derivation is obtained by considering the yield condition depend-

ing on the following internal variables: F= F(, q)= F(, , pleff). Then, thefollowing different expressions are obtained:

F

q

= Eplsgn Hpl

, (2.74)

h=

1Hsgn

Hpl

, (2.75)

which result again in Eq.(2.73). The different general definitions of the moduli usedin this derivation are summarized in Table2.2.


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Table 2.2 Comparison of the different definitions of the stress-strain characteristics (moduli) inthe case of the one-dimensional-space

Range Definition Graphical representation

Elastic E=d

del Figure2.2

Plastic Eelpl = dd

for > init Figure2.2b

Epl = dkd|pl| Figure2.5

For the two-component -/- space, the incremental relation between the

stresses and strains reads d= Celpldwhere Celpl is the elasto-plastic modulusmatrix. The yield condition can be generally stated as F= F(, q)and the totaldifferential of such a yield condition is given as:

dF(, q) =

F

Td +

F

q

Tdq= 0 . (2.76)

Following the same line of reasoning as in the one-dimensional case, the consistenceparameter is obtained as

d =

F

TC

F

TC r

Fq

Th

d , (2.77)

and finally the elasto-plastic modulus matrix as:

Celpl

= C CF (C r)

T

F

TC r

Fq

Th

, (2.78)

or under consideration of the dyadic product (see Sect. A.14.3):

Celpl = C

CF

(C r)

F

TC r

Fq

Th

. (2.79)

Let us consider now the case of combined linear kinematic and isotropic hardening(see Sect. 2.4.3) where the kinematic hardening modulusH(Prager) and the plasticmodulus Epl are constant. Furthermore, the flow rule is assumed to be associated.We assume in the following that the yield condition is a function of the following


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2.6 Elasto-Plastic Modulus 23

internal variables: F= F(, q)= F(, pleff, , ). The corresponding terms inthe expression for the elasto-plastic modulus matrix for the two-component-/-space are as follows:

C= E 00 G

, (2.80)

F

= 1

( )2 + 3 ( )2

3( )

, (2.81)

r= 1( )2 + 3 ( )2

3( )

, (2.82)

Fq =

Epl

( )2+3()2 3()

( )2+3()2

, (2.83)

h=

1H( )

( )2+3()2H3()

( )

2+3()2

. (2.84)

Introducing these five relationships in Eq. (2.78) gives finally the following spe-

cific expression for the elasto-plastic modulus matrix:

Celpl =

E 00 G

1

(E+H+ Epl)( )2 + (9G+ 9H+ 3Epl)( )2

E2( )2 3E G( )( )3E G( )( ) 9G2( )2

. (2.85)

The last equation can be simplified to the special case of a one-dimensional stressandandstrainstatebyassigning G

=0and

=0.ThenoneobtainsagainEq. (2.73).

At the end of this section, Table 2.3 compares the different equations and formula-tions of one-dimensional plasticity with the general two-dimensional representations(see for example [10, 96]).

2.7 Consideration of Unloading, Reversed Loading

and Cyclic Loading

The previous sections considered only monotonic loading either in the tensile orcompressive regimes. We will now briefly look at the cases where the loading direc-tion can change. Figure2.9a shows the case of loading in the elastic (01)and

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Elasto-Plasticity of Frame Structure Elements 2014

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