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CLARKSON UNIVERSITY Nonlocal Theory and Finite Element Modeling of Nano-Composites A Dissertation by Ali Alavinasab Department of Mechanical and Aeronautical Engineering for the degree of Doctor of Philosophy, Mechanical Engineering August 2009 Accepted by the Graduate School _____________ _________________________ Date Dean
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CLARKSON UNIVERSITY
Nonlocal Theory and Finite Element Modeling of Nano-Composites
A Dissertation
by
Ali Alavinasab
Department of Mechanical and Aeronautical Engineering
for the degree of
Doctor of Philosophy, Mechanical Engineering
August 2009
Accepted by the Graduate School
_____________ _________________________ Date Dean
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The undersigned have examined the dissertation entitled “Nonlocal Theory and Finite Element Modeling of Nano-Composites” presented by Ali Alavinasab, a candidate for the degree of Doctor of Philosophy (Mechanical Engineering), and hereby certify that it is worthy of acceptance.
__________ _______________________________
Date Prof. Ratneshwar Jha (Advisor)
_______________________________
Prof. Goodarz Ahmadi (Co-Advisor)
_______________________________ Prof. Weiqiang Ding
_______________________________ Prof. John Moosbrugger
_______________________________ Prof. Hayley Shen
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Abstract This research is concerned with fundamentals of modeling nano-composites. The study contains two major parts, namely, numerical modeling of nanocomposites and nonlocal theory based approach for predicting behavior of Carbon Nanotubes (CNTs). Computational modeling of glass (silica) fibers having micro-scale outer dimensions and nano-scale internal structures was performed to assess its mechanical behavior. Self-assembly technique was used to synthesize the individual fibers of approximately 5 µm in length with a hexagonal cross-section (2µm between two opposite sides) and honeycomb-like internal nano-structures. These fibers have several potential applications including synthesis of multifunctional composite materials. Numerical modeling of the individual fibers was performed using continuum mechanics based approach wherein linear elastic elements were utilized within a commercial finite element (FE) analysis software. A representative volume element approach was adopted for computational efficiency. Appropriate loads and boundary conditions were used to derive stress-strain relationship (stiffness matrix) which has six independent constants for the individual fiber. Force-displacement relationships under simulated nanoindentation were obtained for the actual fiber (with six independent constants) and under transversely isotropic approximation. The contact problem was solved for the transversely isotropic case, which indicated a much stiffer fiber compared to the FE predictions. This difference is likely due to the geometric nonlinearity considered in FE analysis yielding accurate results for large displacements.
The effective mechanical properties of randomly oriented nano-structured glass fiber composite are evaluated by using a continuum mechanics based FE model. The longitudinal and transverse properties of aligned fiber are calculated. Then the equivalent material properties for tilted fiber with different fiber orientations are obtained. Based on equivalent modulus of elasticity for different fiber orientations and using a distribution function for fiber orientation, the overall material properties of randomly oriented glass fibers are calculated. The finite element simulation results are compared with Halpin-Tsai and Mori-Tanaka results. A modified analytical modeling of CNT based on nonlocal theory is proposed. By considering the numerical finite element as an exact solution, a calibration between FEM results and analytical results has been performed. Using second order approximation in nonlocal theory provides more accurate results especially in nano scale. In nonlocal theory, stress is a function of the strains in the entire domain which is the first step for considering the effects of interactions between atoms in nano scale. The proposed analytical method yields the first moment and the total force from stress distribution equal to FEM results. The wave propagation in the nano-structured solids such as CNT composites using the second order approximation in nonlocal theory is studied. The nonlocal theoretical model for modeling CNT nano-sensors is derived when CNT is modeled as Euler-Bernoulli beam. Various boundary conditions and load conditions are considered for modeling of
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CNT beam. The first and second order approximations in nonlocal theory are considered and the nonlocal analytical model is applied to simply supported, cantilever, propped cantilever and clamped beams. The effects of small scale parameters on deflections and bending moment of CNT beam are obtained. The results illustrate that the deflection and bending moment of nonlocal beam depend on the small scale parameters and also on the boundary condition of the beam and the applied load. Finally, the small scale parameters in nonlocal theory are determined for CNT using the reported experimental results for transverse vibration of a nonlocal cantilever beam. The nonlocal length scale parameters are obtained by comparison of the analytical results with the experimental data. Using an optimization technique, the nonlocal small scale parameters and modulus of elasticity are estimated for a CNT for both the first and the second order approximations.
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Dedication
I dedicate this thesis to my dear wife, Azadeh, for all her love and encouragement. With the joy and
happiness she brings to me, my whole existence gains new meaning. I also dedicate this thesis to
my mother and father for their love and support over the years, and especially for their
encouragement that one can do anything that one wants to if you approach each task with enough
confidence and optimism. I also dedicate this thesis to my sister and my brother Maryam and Amir
who have always been supportive.
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Acknowledgement
This research project would not have been possible without the support of many people. I wish to
express my deepest gratitude to my supervisors, Prof. Jha and Prof. Ahmadi who were abundantly
helpful and offered invaluable assistance, support and guidance. My gratitude is also due to the
members of the dissertation committee, Prof. Moosbrugger, Prof. Shen and Prof. Ding, without
whose knowledge and assistance this study would not have been successful.
I would also like to convey thanks to the U.S. Army for providing the financial support under grant
number: W911NF-05-1-0339 for this research. In addition, I am very grateful to Mechanical and
Aeronautical Department at Clarkson University for their financial support and instructor position
to teach the fundamental engineering courses for one year.
I wish to express my love and gratitude to my beloved family members for their understanding and
endless love throughout the duration of my studies.
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Table of Contents CHAPTER 1. INTRODUCTION .................................................................................................... 1
1.1. OBJECTIVES .........................................................................................................................4 1.1.1. Nano-structured glass fiber composite .........................................................................4 1.1.2. Carbon Nanotube Composite ........................................................................................4
CHAPTER 2. COMPUTATIONAL MODELING OF NANO-STRUCTURED GLASS FIBERS6 2.1. INTRODUCTION ..................................................................................................................6 2.2. REPRESENTATIVE VOLUME ELAMENT MODELING .................................................8 2.3. STIFNESS MATRIX ............................................................................................................11 2.4. LOAD-DISPLACEMENT RELATIONSHIP ......................................................................13
2.4.1. Contact simulation for isotropic material ...................................................................14 2.4.2. Contact model for transversely isotropic material ......................................................16 2.4.3. Contact simulation for the glass fiber .........................................................................17
2.5. CONCLUSIONS ..................................................................................................................19 CHAPTER 3. COMPUTATIONAL MODELING OF NANO-STRUCTURED GLASS FIBER COMPOSITE
3.1. INTRODUCTION ................................................................................................................21 3.2. MODELING OF ALIGNED SHORT GLASS FIBER COMPOSITES ..............................23
3.2.1. Representative Volume Element Modeling ................................................................23 3.2.2. Numerical Modeling of the RVE ................................................................................25 3.2.2. 1 Longitudinal modulus of elasticity .............................................................................25 3.2.2. 2 In-plane Poisson’s ratio xyυ ........................................................................................27
3.2.3. Analytical modeling of the RVE ................................................................................29 3.2.4. Tilted glass fiber composites ......................................................................................31
3.3. MODULUS OF ELASTICITY ............................................................................................33 3.3.1. Numerical modeling of randomly oriented glass fiber composites ............................33 3.3.2. Analytical modeling of randomly oriented glass fiber composites ............................34
3.4. CONCLUSIONS ..................................................................................................................37 CHAPTER 4. NONLOCAL MODELING OF CARBON NANOCOMPOSITE ......................... 38
4.1. INTRODUCTION ................................................................................................................38 4.2. NONLOCAL CONTINUM THEORY .................................................................................43
4.2.1. First Order Approximation .........................................................................................45 4.2.2. Second Order Approximation .....................................................................................46
4.3. NUMERICAL EXAMPLE ...................................................................................................48 4.4. WAVE PROPAGATION .....................................................................................................53
4.4.1. First order approximation ...........................................................................................54 4.4.2. Second order approximation .......................................................................................54
4.5. CONCLUSIONS ..................................................................................................................56 CHAPTER 5. EFFECTS OF NONLOCAL SMALL SCALE PARAMETERS ON BEHAVIOR OF CARBON NANOTUBE BEAMS .............................................................................................. 57
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5.1. INTRODUCTION ................................................................................................................57 5.2. NONLOCAL EULER-BERNOULLI BEAM MODEL .......................................................58
5.2.1. Simply supported beam ..............................................................................................60 5.2.2. Clamped beam ............................................................................................................62 5.2.3. Cantilever beam ..........................................................................................................65 5.2.4. Propped cantilever beam .............................................................................................67
5.3. NUMERICAL RESULTS ....................................................................................................70 5.4. CONCLUSIONS ..................................................................................................................78
CHAPTER 6. FIRST- AND SECOND-ORDER NONLOCAL BEAM MODELS FOR CARBON NANOTUBE
6.1. INTRODUCTION ................................................................................................................80 6.2. NATURAL FREQUENCY OF NONLOCAL CNT BEAM ................................................81
6.2.1. First Order Approximation .........................................................................................82 6.2.2. Second Order Approximation .....................................................................................86
6.3. ESTIMATION OF NONLOCAL MODEL PARAMETERS ..............................................89 CHAPTER 7. FUTURE WORK ................................................................................................... 93 CHAPTER 8. REFRENCES ......................................................................................................... 94
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List of Tables Table (2-1) Engineering constants for the glass fiber ............................................................... 12
Table (6-1) Experimental data of Gao et al. (2000) ....................................................................... 89
Table (6-2) Nonlocal material constants for the CNT using the first order approximation ................ 90
Table (6-3) Comparison of the predicted natural frequencies with the experiment values using the first
order approximation of nonlocal Euler-Bernoulli beam .................................................................... 91
Table (6-4) Nonlocal material constants for the CNT using the second order approximation............ 91
Table (6-5) Comparison between the experimental natural frequencies and the nonlocal second
order approximation .................................................................................................................... 92
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List of Figures Figure (2-1) Glass fibers synthesized using self-assembly ....................................................... 7
Figure (2-2) Geometry of glass fibers ....................................................................................... 9
Figure (2-3) The RVE geometry ............................................................................................... 9
Figure (2-4) 3D finite element model of RVE .......................................................................... 10
Figure (2-5) RVE boundary conditions for evaluating E22 ....................................................... 11
Figure (2-6) In-plane displacement of RVE ............................................................................. 12
Figure (2-7) Shear displacement of the RVE ............................................................................ 12
Figure (2-8) Schematic of the nanoindentor tip placed on top of glass fiber ............................ 14
Figure (2-9) Geometry for FE modeling of nanoindentation assuming material isotropy ....... 15
Figure (2-10 )Hertz’s contact model (analytical) and FE results for nanoindentation of isotropic
material ............................................................................................................................ 15
Figure (2-11) Selected glass fiber geometry for contact modeling ............................................. 17
Figure (2-12) FE mesh for (a) selected geometry (b) full glass fiber ......................................... 18
Figure (2-13) Force-displacement diagram for actual and transversely isotropic ...................... 19
Figure (3-1) SEM micrographs of SBA-15 hexagonal tubular silica microcapsules used to retain
epoxy resin .......................................................................................................................... 22
Figure (3-2) The RVE model of the matrix with and aligned short fiber ................................. 24
Figure (3-3) Effective modulus of elasticity of aligned glass fiber composite with volume
fraction 1.0=fV .................................................................................................................. 25
Figure (3-4) Displacements of the RVE under longitudinal load, (a) Displacement of the RVE
in the z-direction, (b) Displacement of the RVE in the y-direction .................................... 26
Figure (3-5) Load and boundary conditions of the RVE for evaluating Poisson’s ratio xyυ .... 27
Figure (3-6) Strain and displacement of the RVE under above load condition ........................ 29
Figure (3-7) RVE partitions for modeling glass fiber composite ............................................. 30
Figure (3-8) Tilted short fiber composite .................................................................................. 31
Figure (3-9) Coordinate transformation for tilted short fiber composite (a) Original tilted RVE,
(b) transformed to aligned RVE .......................................................................................... 32
Figure (3-10) Modulus of elasticity of tilted glass fiber composite .......................................... 33
Figure (3-11) Effective modulus of elasticity of the randomly oriented glass fiber composite using
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the Mori-Tanaka and Halpin-Tsai methods ........................................................................ 36
Figure (4-1) Values of L/γ for CNT composites using Equation (4-23) ................................... 48
Figure (4-2) A representative volume element (RVE) for CNT composite ............................. 49
Figure (4-3) FE results for RVE of CNT composite, (a) Displacement of mid-plane
(b) longitudinal stress, and (c) longitudinal strain .............................................................. 50
Figure (4-4) Nonlocal stress distribution of CNT composite for the first three roots .............. 50
Figure (4-5) Cross section of CNT composite .......................................................................... 51
Figure (4-6) Stress distribution in CNT composite using nonlocal theory, average elasticity and
FEM for various values of matrix modulus ....................................................................... 52
Figure (4-7) The first moment of stress in CNT composite using FEM, average elasticity, and
nonlocal theory for various values of matrix modulus ...................................................... 53
Figure (4-8) Dispersion curve for the Born-von Karman lattice dynamic and nonlocal theory - 56
Figure (5-1) Deflection of nonlocal beam under uniform load distribution ............................... 72
Figure (5-2) Ratio of maximum deflection of nonlocal beam to maximum deflection in local
elasticity under uniform load distribution ................................................................................... 73
Figure (5-3) Ratio of maximum bending moment of nonlocal beam to maximum bending moment
in local elasticity under uniform load distribution ...................................................................... 73
Figure (5-4) Ratio of maximum positive bending moment in nonlocal beam to maximum positive
bending moment in local elasticity beam under uniform load distribution ................................ 74
Figure (5-5) Ratio of maximum nonlocal deflection to maximum deflection in local elasticity 75
Figure (5-6) Ratio of maximum deflection of nonlocal cantilever beam to local elasticity beam
under sinusoidal load condition .................................................................................................. 75
Figure (5-7) Ratio of maximum bending moment in nonlocal beam to local elasticity beam under
sinusoidal load condition ............................................................................................................ 76
Figure (5-8) Effects of nonlocal small scale parameters on the ratio of maximum positive bending
moment in nonlocal theory to local elasticity under sinusoidal load condition .......................... 77
Figure (5-9) Ratio of maximum bending moment in nonlocal beam to local elasticity beam under
approximated point load condition ............................................................................................. 77
Figure (5-10) Effects of nonlocal small scale parameters on ratio of maximum deflection in
nonlocal beams to local elasticity beams under approximated point load condition .................. 78
Figure (6-1) Effect of the small scale parameter on nonlocal beam first harmonic constant ..... 85
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Figure (6-2) Ratio of the first natural frequencies of a cantilever beam as predicted by the first order
approximation of nonlocal theory to that of local elasticity ....................................................... 86
Figure (6-3) Ratio of the first natural frequencies of a cantilever beam as predicted by the second
order approximation of nonlocal theory to that of local elasticity .............................................. 88
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1
CHAPTER 1. INTRODUCTION
The use of composite materials is rapidly increasing because of their advantageous
properties such as high specific strength/modulus, corrosion resistance and fatigue life. For
example, the Boeing 787 uses composite materials accounting for about 50% of structural weight
which leads to significantly increased fuel efficiency and reduced part count compared to the
similar sized airplanes. Composite materials are also being used in other industries such as
automotive and various sporting goods. Composite materials result from the integration of two or
more distinct components (fiber and matrix) such that superior physical and mechanical properties
are realized. In addition, some composite materials have other advantages, like electrical
conductivity and thermal properties, which make them suitable as multifunctional materials. The
development of multifunctional composite materials/structures is aimed at providing innovative
functionality to structures in addition to their load carrying capability (Gates, 2003).
Self-healing composites, because of their unique multifunctional properties, have received
increased attention of researchers during the last decade. The US Army is interested in a jacket for
its soldiers that have self-healing property along with light weight and flexibility. Self-healing
composites may be the solution to this Army requirement. In self-healing composites, fibers filled
with polymeric glue (such as epoxy) are incorporated into the matrix at the fabrication stage. Once
a fiber is ruptured due to damage to the structure, the healing polymer seals the crack tip and
thereby arrests crack propagation. Experimental work reported earlier has shown prevention of
micro-cracks and significant self-healing of composites using micro-scale spherical capsules of
polymeric material (White et al., 2001; Brown et al., 2002, 2004, 2005a, 2005b).
Nano-composites, that is, composites with at least one dimension of fibers at the nano scale,
have received a lot of interest lately because they have superior material properties compared to
micro-composites. Due to their large surface area in a given volume, nano materials yield
significant beneficial changes in material properties of composites. Since the material properties of
composites result from the physical and chemical interaction between surface areas of fiber and
matrix, nano-composites have a stronger bond compared to micro-composites. Toyota Central
Research Laboratory has reported significant thermal and mechanical property improvements by
using Nylon-6 nano-composites (Hussain et al., 2006). Nano-composites have been pursued in
recent years with a goal of creating multifunctional composite structures. In general, one of the
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most important (and challenging to achieve) multifunctional properties is self-healing. Nano-
structured silica fibers used in the present study are much smaller in size than micro-scale spherical
capsules used by other researchers (Alavinasab et al., 2008) and the encapsulated glue would have
longer shelf life. Therefore, the nano-structured glass fiber composite could improve the self-
healing properties of composite. In addition to other benefits, the chance of missing a self-healing
capsule is much less in a nano-composite compared to micro-composite. A computational model is
urgently needed to understand the behavior of these materials.
Much research is also being done on another specific nano-composite: Carbon Nanotubes
(CNT) composite. Iijima (Harris, 2002) discovered multiwall CNT and single wall CNT in 1991
and 1993, respectively and since then significant effort has been underway to understand the
behavior of CNT. When CNTs are incorporated into a matrix, outstanding mechanical properties of
CNT composites are achieved. The incorporation of CNT into a matrix increases not only the
strength but also the toughness of the composite. CNT reinforced composites possess unique
properties such as light weight, high strength, and high electrical conductivity (Qian et al., 2000).
Due to both electrical and mechanical properties of CNT, such composites have multi functional
behavior. The CNT composites have been used as high strength composite, energy storage devices,
sensors, and semi-conductor devices. A robust and practical (computationally efficient) theory for
modeling nano-structures (including both nano and macro scale features) is not yet available.
Several researchers have used continuum theory for modeling nano-composites (Bhushan and
Agrawal, 2002). Reich et al. (2004) have modeled CNT as a closed hollow cylindrical shell and
other researchers have considered classical elasticity for modeling CNT-composite (Liu and Chen,
2003; Chakraborty, 2006).
Previous studies of CNT composite have provided significant understanding of the
promising behavior of CNT composites. However, a robust analytical method which is well
matched to the atomic characteristics of CNT composite is not available in the literature. Some
researchers have used the classical elasticity theory for modeling of CNT composite, but the
internal atomic interactions were neglected. Therefore, classical elasticity could not consider the
effect of internal characteristic length. Several researchers have attempted to use molecular
dynamics (MD) for considering the interactions between atoms in CNT composite. Unfortunately,
the application of MD simulation for real engineering problems was unsuccessful. In MD
simulation, by increasing the length, the number of atoms in the simulation increases enormously
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and computational time becomes prohibitive. In addition, a statistical averaging method is needed
for comparison of the MD results with the experimental results.
This research is concerned with fundamentals of modeling CNT and nano-composites. The study
contains two major parts, namely, numerical modeling of nanocomposites and nonlocal theory
based approach for predicting behavior of CNTs. A comprehensive literature review is given at the
beginning of each following chapter. For the numerical studies, the finite element modeling of
nano-structure self-healing composite is presented in Chapters 2 and 3. The numerical modeling of
nano-structured glass fiber is done by using representative volume element and classical (local)
elasticity. The proposed computational method is applicable for modeling of nano-composite by
considering that the continuum theory is valid at the nano scale. In Chapter 4, analytical modeling
of carbon nanotube composite based on nonlocal elasticity theory is proposed. A novel approach for
analytical modeling of CNT composite is proposed in the present work using nonlocal theory with
both macro and nano scale features. The solution to nonlocal constitutive equation is obtained by
matching the dispersion curve of nonlocal theory with lattice dynamic simulation. Numerical
example results using nonlocal, FEM, and classical elasticity for CNT composites are presented in
Chapter 4. Unlike the classical (local) elasticity results, the first moment obtained from stress
distribution using the nonlocal theory is equal to the results obtained from the numerical FEM
calculation. Furthermore, wave propagation in nonlocal theory, and the equations of motion for
lattice dynamic simulation are presented. Phonon dispersion relationships between lattice dynamic
and nonlocal theory are presented. In Chapter 5, the nonlocal theoretical model for modeling CNT
nano-sensors is derived when CNT is modeled as Euler-Bernoulli beam. Various boundary
conditions and load conditions are considered for modeling of CNT beam. The first and second
order approximation in nonlocal theory are considered and the nonlocal analytical model is applied
to simply supported, cantilever, propped cantilever and clamped beams. The effects of small scale
parameters on deflections and bending moment of CNT beam are obtained. The results illustrate
that the deflection and bending moment of nonlocal beam depend on the small scale parameters and
also on the boundary condition of the beam and the applied load. In chapter 6, the small scale
parameters in nonlocal theory are determined for CNT using the reported experimental results for
transverse vibration of a nonlocal cantilever beam. The nonlocal length scale parameters are
obtained by comparison of the analytical results with the experimental data. Using an optimization
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technique, the nonlocal small scale parameters and modulus of elasticity are suggested for a CNT
for both the first and the second order approximations.
1.1. OBJECTIVES
1.1.1. Nano-structured glass fiber composite
The general goal is to provide a numerical method based on finite element analysis for better
understanding of mechanical behavior of nano-structured glass fiber composite. The numerical
modeling of glass fiber composite comprises FE modeling of nano-structured glass fiber and glass
fiber composite. The specific objectives are:
• To model nano-structured glass fibers using the finite element method.
• To evaluate the stiffness matrix for the nano-structured glass fibers.
• To simulate the response of the fibers under nano-indenter tip load.
• To model composite material with nano-structured glass fiber by using the FE method.
• To evaluate the overall effective modulus of elasticity using Weibull’s distribution
function for fiber orientation.
• To compare the FE results with analytical (Mori-Tanaka) method.
• To improve the accuracy of numerical modeling of glass fiber composite by considering
fiber geometry.
1.1.2. Carbon Nanotube Composite The general goal is to provide an accurate and efficient analytical modeling of CNT composite.
Modified nonlocal theory based model for accurate modeling of CNT composite is proposed. The
specific objectives are:
• To present a new approach for obtaining accurate stress distribution in nano-composite
in a computationally efficient manner using nonlocal continuum theory.
• To compute stress/strain by considering the effect of entire domain, which produces
more accurate results especially in nano-scale structures.
• To obtain a stress distribution in which the first moment of nonlocal continuum theory is
equal to that from FEM results, which is not the case for classical elasticity.
• To investigate the effects of the proposed method on wave propagation in structures.
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• To develop a theoretical model which can be used for modeling the CNT nano-sensors.
• To evaluate the effects of nonlocal small scale parameters on the behavior on CNT
beams such as deflection and bending moment.
• To obtain the nonlocal parameters such as nonlocal small scale parameters and modulus
of elasticity by comparing the experimental data with the nonlocal analytical equations.
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CHAPTER 2. COMPUTATIONAL MODELING OF NANO-STRUCTURED GLASS FIBERS
2.1. INTRODUCTION
The development of multifunctional composite materials/structures is aimed at providing
innovative functionality to structures in addition to their load carrying capability (Gates, 2003). The
additional functionalities under active research include embedded antennae, health monitoring,
vibration suppression, and self-healing materials. Nano-structured materials have been pursued in
recent years with a view to create multifunctional composite structures.
The self-assembly technique has been used to synthesize micro-scale silica fibers that have
nano-scale internal structures (Figure (2-1)) (Kievsky and Sokolov, 2005; Sokolov and Kievsky,
2005; Privman et al., 2007). A typical glass fiber is approximately 5 µm in length and its hexagonal
cross-section is 2 µm in width. The diameter of internal cylindrical pores is approximately 3 nm,
and 0.6-0.8 nm is the inter-pore wall thickness. Such fibers have several potential applications,
including synthesis of self-healing (multifunctional) composite structures. Fibers filled with
polymeric glue (such as epoxy) may be incorporated into composite materials at the fabrication
stage. Once a fiber is ruptured due to damage to the structure, the healing polymer would seal the
crack tip and thereby arrest crack propagation. Experimental work reported earlier has shown
prevention of micro-cracks and significant self-healing of composites using micro-scale spherical
capsules of polymeric material (Kessler et al., 2003). Nano-structured silica fibers used in the
present study are much smaller in size and the encapsulated glue would have longer shelf life. This
research focuses on computational modeling of the glass (silica) fibers to ascertain their mechanical
behavior by obtaining stiffness matrix and load-displacement relationship.
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Figure (2-1): Glass fibers synthesized using self-assembly (a) Large –area Scanning Electron Microscope image (bar size 22μm) (b) Zoomed images of fibers (bar size 5μm) (c) Schematic of
nanoporous arrangement within fibers (d) Transmission Electron Microscope image near fiber edge showing periodicity of about 3 nm (Privman et al., 2007)
Material modeling and characterization covering nano- to macro-scale pose significant
challenges due to both length and time scales involved. Several researchers have used the molecular
dynamics (MD) approach for modeling nanomaterials which is very expensive computationally
(Lin et al., 2004). Lin and Huang (2004) used MD to study stress-strain behavior of nano-sized
copper wires under uniaxial tension. Interpretations of their results indicate that a linear elastic
relationship may be applied at nano-scale. Continuum mechanics based approach provides much
more computationally efficient model for modeling nano-structures. The modeling of nano-beam
structures using continuum mechanics principles is considered by Bhushan and Agrawal (2002). A
comparison of numerical and experimental data indicates the applicability of the linear elastic
model for the stress-strain relationship in silicon nano-beams. Nikishkov et al., (2003) used
continuum based finite elements for modeling self-positioning micro- and nano-structures by using
finite elements with geometric nonlinearity, small strain, and large deformation.
This research evaluates mechanical properties of the glass fiber by using a continuum
mechanics based finite element (FE) model with nonlinear geometry and adaptive meshing. The
effects of relatively large displacements (in the nano-scale) are included by using nonlinear
geometry. Adaptive meshing reduces computational error by improving distribution and size of
elements in the FE analysis. The max/min error indicator is used to determine adaptive mesh in the
simulations. The stress-strain relationship (stiffness matrix) of the glass fiber is obtained by using a
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representative volume element (RVE) for computational efficiency. Force-displacement diagram
using a nanoindentor is generally used for modulus of elasticity and Poisson’s ratio of materials at
nano and micro-scale (Huang and Pelegri, 2003). A series of numerical simulations is performed to
obtain force-displacement relationship of the glass fiber under simulated nanoindentor probe loads.
Analytical contact problem is studied for isotropic and transversely isotropic materials and the
results are compared with FE analysis.
2.2. REPRESENTATIVE VOLUME ELEMENT MODELING
The main objective of current study is to analyze mechanical behavior of the glass fiber
under different loading conditions. The overall relationship between stress-strain is represented by
the stiffness matrix (constitutive relation). The representative stiffness matrix of a material with
complex geometry can be obtained by using RVE in FE analysis. The external geometry and
dimensions of the glass fiber are shown in Figure (2-2(a)). Figure (2-2(b)) indicates the details of
the arrangement of any three adjoining cells within the fiber considering an average wall thickness
of 0.7 nm (that is, the distance between the centers of two cylindrical cells is 3.7 nm) (Privman et
al., 2007). The filled area in Figure (2-2(b)) represents fiber material (silica) and the rest is void.
The elastic modulus of elasticity and Poisson ratio of silica are considered 73.1 GPa and 0.17
respectively. A typical single fiber consists of approximately 300,000 tubular cells. FE model of a
full fiber would lead to prohibitively high number of degree of freedom. Modeling a part of the
fiber cross-section due to symmetric boundary conditions reduces the number of elements;
however, it is still very large for regular finite element analysis. Therefore, a RVE having much
reduced number of elements is considered. The use of RVE to obtain mechanical properties of
composite materials is well established (Sun and Vaidya, 1996). Numerical calculation of the
effective properties of the glass fiber involves computation of stresses and strains for an RVE that
represents the nano-structure of the glass fiber. These stresses and strains are averaged over the
volume of the RVE. A commercial finite element software (ABAQUS) is used for numerical
analysis.
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Figure (2-2): Geometry of glass fibers (a) Overall dimensions, (b) Arrangement of internal cells.
As depicted in Figure (2-3), the RVE used in the analysis has a length of 345 nm and its
cross-section is 64 nm×74 nm which contains 20×20 cells. The RVE has the same aspect ratio as
the glass fiber. The structure of the glass fiber is similar to honeycomb, which is considered
transversely isotropic (Gibson and Ashby, 1998). Chung and Waas (2002) obtained very good
agreement between numerical analysis and experimental data for a honeycomb structure with
12×12 cells. Therefore, 20×20 cells for the RVE cross-section is considered adequate for the
present numerical analyses. Furthermore, by using the smaller dimensions, the side effects could
have some influences on modulus of elasticity of the RVE. The in-plane axes are numbered 1 and 2
and the longitudinal direction is along 3-axis.
Figure (2-3): (a) The RVE geometry, (b) Cross section of the RVE.
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The 3D FE model of RVE is shown in Figure (2-4) wherein 8-node isoparametric brick
elements based on reduced integration are used. The effect of large displacement/strain can be
accounted for by using geometric nonlinearity in the model. For increased accuracy of the
predictions, geometric nonlinearity is included in the model by using the full Newton technique and
linear load variation over steps; therefore, the effects of large displacements are considered in the
analysis.
Furthermore, adaptive mesh based on element energy is applied for reducing error in FE
analysis. Adaptive meshing maintains a high quality mesh and eliminates deformation induced
mesh distortion, especially for large deformations. Adaptive meshing also improves the quality of
the solutions while controlling the cost of analysis. The adaptive mesh has been applied to the FE
modeling of glass fiber by setting the indicator target to 5% near contact area (and 30% on the far
boundary) for minimal error near the contact area.
Figure (2-4): (a) 3D finite element model of RVE, (b) Enlarged view of upper right corner of RVE.
In addition to the 3D FE modeling of RVE, the in-plane properties are also obtained by
considering a 2D plane strain model. For the 2D model, 4-node bilinear plane strain quadrilateral
elements are utilized. The results show that the 2D and the 3D analyses compare very well.
11
2.3. STIFNESS MATRIX
In an orthotropic material, the constitutive relation between stress-strain at each point is
represented by nine material constants. The material constants of RVE are obtained by applying
appropriate boundary conditions and loads (Sun and Vaidya, 1996). As an example, for finding E22
(in-plane modulus of elasticity) a rigid plate is placed on the top of RVE (Figure 2-5). Symmetric
boundary conditions are applied to both lateral sides and the bottom is fixed. The boundary
conditions are defined by Equation (2-1) wherein wvu ,, , respectively, correspond to displacements
in x, y, and z directions.
zdyxwyxwxzyduzyu
zdxvzxv
in.Sym),,()0,,(in.Sym),,(),,0(
constant),,(0),0,(
3
1
22
======
=δ
(2-1)
where d1, d2, and d3, respectively, are 74nm, 64nm, and 345nm. A load applied to the rigid
plate generates stress and strain in the RVE, which are used to evaluate E22. Similar procedures
have been used for evaluating E11 and E33 with the boundary conditions given by the Equation (2-
2). For evaluating the Poisson’s ratios, the symmetric boundary conditions on the sides are removed
as shown in Figure (2-6) to compute the resulting strains.
Figure (2-5): RVE boundary conditions for evaluating E22.
33
11
constant),,(0)0,,(constant),,(
0),,0(
δ
δ
===
===
dyxwyxw
zyduzyu
(2-2)
12
Figure (2-6): In-plane displacement of RVE.
Figure (2-7): Shear displacement of the RVE.
The shear modulus G13 is computed using boundary conditions similar to those shown in
Figure (2-5) except that the symmetric boundary condition perpendicular to the 3-axis is removed.
Figure (2-7) shows the shear displacement of the RVE for G13. A similar procedure is applied for
finding G12 and G23.
After evaluating all nine material constants it was observed that some of the constants (such
as E11 and E22) have relatively small numerical differences. Therefore, their average value is
considered to be representative of the material property. The number of material constants then
reduces to six as presented in Table (2-1). The in-plane shear modulus of RVE, G12, is 2.44 GPa
which is much smaller than the value of 4.5 GPa obtained based on transversely isotropic
assumption. Since the RVE cross-section is similar to a honeycomb (which is considered as
13
transversely isotropic material), this difference needed further investigation. Sun and Vaidya (1996)
suggest that a shear deformation shape which satisfies periodicity and symmetry in the RVE yields
more accurate predictions than the parallelogram shape. The appropriate constraints are that shear
deformations at the corners must remain perpendicular and the deformation of two opposite edges
must be the same. The value of G12 based on equations in (Sun and Vaidya, 1996) is equal to 2.13
GPa, which is close to the value obtained for the RVE.
Table (2-1). Engineering constants for the glass fiber (Alavinasab et al., 2008).
11E
22E
33E 12υ 13υ
23υ
12G 13G
23G
13.74 31.65 0.52 0.07 2.44 7.18
The engineering constants are used to derive the stiffness matrix given by Equation (2-3)
following well established relationships described in (Daniel and Ishai, 1994). These engineering
constants (in GPa) are listed in Table (2-1).
The stress-strain relationship given by Equation (2-3) contains six independent constants for
the glass fiber; thus, in general the fiber is considered an orthotropic material.
437.2000000185.7000000185.7000000311.33144.5144.5000144.5643.19614.10000144.5614.10643.19
12
13
23
3
2
1
12
13
23
3
2
1
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
γγγεεε
τττσσσ
(2-3)
2.4. LOAD-DISPLACEMENT RELATIONSHIP
Nanoindentation is a common technique for investigating mechanical properties of materials
at nano- and micro-scale. The numerical model of the glass fiber is used to obtain a load–
displacement diagram, which can be used for direct correlation of numerical analysis with
experimental data. In this analysis, nanoindentor tip is modeled as a rigid sphere of 200 nm radius.
14
The tip exerts load on the fiber gradually and the fiber displacement is computed. The fiber is
considered fixed at the bottom with the load applied at the top as shown in Figure (2-8).
Figure (2-8): Schematic of the nanoindentor tip placed on top of glass fiber.
2.4.1. Contact simulation for isotropic material
Simulation of nanoindentation is performed essentially through contact modeling. Based on
Hertz’s theory for contact between a rigid body and a half plane, the contact force between two
isotropic bodies is given by Johnson (1985):
2/3δckF = (2-4)
where F is contact force and δ is the resulting indentation. The term ck is given as
REk cc 34
= (2-5)
where R is the tip radius. The contact modulus cE is calculated from
∑= ′
=2
1
11
i ic EE (2-6)
where the term iE′ is given by
21 i
ii
EEυ−
=′ (2-7)
In Equation (2-7) iE is the modulus of elasticity of material i. In this study, the indenter is much
stiffer than the sample and considered as an infinitely rigid body. Therefore the contact modulus is
taken equal to iE′ of the sample material.
15
Figure (2-9): Geometry for FE modeling of nanoindentation assuming material isotropy
An FE analysis of nanoindentation, assuming material isotropy, is performed initially to
compare the findings with Hertz’s model. As shown in Figure (2-9) the dimensions of the FE
model are taken greater than five times the probe radius; so that the boundaries have negligible
effect on the results. The modulus of elasticity and the Poisson’s ratio for the isotropic material are
assumed 13.8 GPa (similar to E11 for the glass fiber) and 0.33, respectively. The contact problem is
modeled in ABAQUS using normal behavior of hard contact interaction (that is, there is no friction
between bodies in contact). The sample material is modeled by 4-node tetrahedral solid elements.
The boundary conditions for the model are symmetric on the front and fixed at the bottom (Figure
(2-9)).
0 10 20 30 40 500
20
40
60
80
100
120
Displacement (nm)
P( μ
N)
AnalyticalFE
Figure (2-10): Hertz’s contact model (analytical) and FE results for nanoindentation of isotropic
material.
16
The FE predictions compare well with the analytical Hertz’s model as shown in Figure (2-10). The
largest difference between the two results is about 5%. Figure (2-10) verifies the accuracy
numerical FE model with analytical results.
2.4.2. Contact model for transversely isotropic material
As noted in section 2.3, the glass fiber has six independent engineering constants. In
addition to FE analysis, we have used transversely isotropic approximation for analytical prediction
of its contact behavior. The analytical indentation solution for transversely isotropic material is
given by Swanson (2005)
3/2
*43
⎟⎟⎠
⎞⎜⎜⎝
⎛=
TIERFδ (2-9)
where *TIE is the effective modulus for transversely isotropic materials. The effective modulus is
given as
⎟⎟⎠
⎞⎜⎜⎝
⎛=
31
* 2ααTIE (2-10)
where
2/1
2
2
1 1/
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−
=xy
xzzx EEν
να (2-11)
22 1
)1(12
1
xy
xyxzxz
xGE
ν
ννα
−
+−⎟⎟⎠
⎞⎜⎜⎝
⎛−+
= (2-12)
⎟⎟⎠
⎞⎜⎜⎝
⎛ −⎟⎠⎞
⎜⎝⎛ +
=xy
xy
Gνααα
12
2/121
3 (2-13)
Using Equation (2-9), the analytical solution of force-displacement of a sphere on a half
plane of transversely isotropic material can be evaluated.
17
2.4.3. Contact simulation for the glass fiber
In this section computer simulation of nanoindentation of the glass fiber is presented. Figure (2-11)
shows a segment of the glass fiber selected for contact simulations. The FE meshes for the selected
part and the full fiber are shown in Figure (2-12).
Figure (2-11): Selected glass fiber geometry for contact modeling.
Figure (2-12) shows the energy transfer through the glass fiber capsule under nanoindenter tip. On
the other hand, as shown in Figure (2-12(b)), most of the applied energy through nanoindenter tip
goes through the arc zone. By using the adaptive mesh method, the element sizes in the arc zone
was decreased significantly than the rest of glass fiber.
18
Figure (2-12): FE mesh for (a) selected geometry (b) full glass fiber.
As was noted before, between the two FE solutions, the only difference that exists in the
calculations pertains to material properties, i.e., the number of independent constants. The in-plane
shear modulus is considered an additional constant in the actual glass fiber, whereas it is inherently
derived in the transversely isotropic model. So, the equivalent model of glass fiber has six
independent constants. The behavior of glass fiber is approximated as a transversely isotropic
material with five independent constants. In Figure (2-13) the computed force-displacement
diagrams for the glass fiber from FE simulations for actual model and transversely isotropic
approximation with the analytical solution for the transversely isotropic approximation are
compared. It is observed that FE analysis results for the two cases are close to each other, whereas
the analytical prediction indicates a considerably stiffer material. The largest difference in force in
Figure (2-13) is 39%. The maximum difference between the FE simulation results for the actual
glass fiber and transversely isotropic approximation is about 12%, which results from the difference
in their in-plane shear moduli. The in-plane shear modulus of transversely isotropic fiber is about
two times the value for the actual glass fiber. Therefore, the results indicate stiffer behavior for
transversely isotropic fiber.
19
0 10 20 30 40 500
50
100
150
200
Displacement (nm)
P( μ
N)
FE Glass Fiber ( 6 independent const.)FE Glass Fiber ( 5 independent const.)Analytical Hertz Transversely Isotropic
Figure (2-13): Force-displacement diagram for actual and transversely isotropic.
The analytical solution differs from the FE solutions in that an infinite plane geometry is
assumed, whereas in the FE calculations hexagonal cross section geometry is used and the
corresponding edge effects are taken into consideration. This choice to neglect edge effects in the
analytical Hertz model stems from the lack of geometric parameters in the analytical solution. This
results in a discrepancy between the analytical and FE results for the displacement versus force
data. The Hertz contact model (and its extension) assumes small deflections whereas our FE
simulations consider geometric nonlinearity. This is important for the accuracy of the results for
large displacements, which is true for the transversely isotropic case. Also, Hertz’s contact model
as shown in Figure (2-9), considers the half-pane geometry whereas the geometry of glass fiber as
shown in Figure (2-11) is hexagonal as used for FE simulations.
2.5. CONCLUSIONS
A continuum mechanics based finite element modeling of nano-structured glass fibers
(Kievsky and Sokolov, 2005; Sokolov and Kievsky, 2005; Privman et al., 2007) is presented. The
stiffness matrix and force-displacement relation under simulated nanoindentation have been
obtained using ABAQUS. A representative volume element approach was used for its
computational efficiency. The constitutive relations for the fibers yield orthotropic property with six
independent constants. The load-displacement results for the actual glass fiber and its transversely
20
isotropic approximation have been obtained using nonlinear geometry and adaptive meshing in the
finite element simulations. The analytical contact model for the transversely isotropic case shows
significantly larger forces compared to FE results. It is believed that FE simulation results are
accurate since large displacements have been considered through geometric nonlinearity.
21
CHAPTER 3. COMPUTATIONAL MODELING OF NANO-STRUCTURED GLASS FIBER COMPOSITE
3.1. INTRODUCTION
The development of multifunctional composite materials promises to add novel
functionality to structures in addition to their load carrying capability. The self-healing composite
would make composites more reliable. The idea of self-healing is captured from biological systems.
Healing fracture in bones and ruptured blood vessels are examples of self-healing in nature.
Researchers have used glass fibers to create self-repairing concrete material (Dry, 1994; Dry and
McMillan, 1996; Li et al., 1998). Hollow glass fiber (~500 µm diameter) filled with superglue as
the healing material was mixed in concrete matrix. The reported bending test results showed the
restoration of flexural stiffness after fracture. Motuku et al. (1999) investigated a self-healing
material within a polymer composite. They used a reinforced polymer composite with woven S2-
glass fabric with embedded borosilicate glass micro-capillary pipettes of 1.15mm diameter.
Because the diameter of the hollow glass fibers used in (Motuku, 1999) was much larger than the
reinforcing fibers, initiation of failure occurred in the composites. Bleay et al. (2001) used smaller
diameter hollow fibers for both structural reinforcements and healing components. They used
hollow borosilicate glass capsules with a diameter of about 40-60µm. The experimental impact test
results showed a large number of healing component precipitated into the damage -zone, but the
strength after healing was much reduced. Instead of long hollow silica fibers, Zako et al. (1999)
used particles with average diameter of about 105µm. The experimental bending and tensile fatigue
tests showed stiffness recovery and self-healing properties.
The most promising self-healing results in the literature are reported by White et al. (2001)
and Brown et al. (2002, 2004, 2005a, 2005b). They used hollow spherical microcapsules filled with
dicyclopentadiene (DCPD). These microcapsules, having a diameter ranging from 180-460µm, are
embedded within a structural composite polymer matrix along with a reactant known as Grubbs
catalyst. When fracture occurs within the structure, the thin walled microcapsules are ruptured
releasing the DCPD monomer into the crack plane. As the DCPD monomer comes in contact with
the Grubbs catalyst, polymerization is triggered and the crack plane is sealed. By optimizing
microcapsule size, microcapsule concentration and catalyst concentration healing efficiency of
more than 90% is reported (Brown et al., 2002).
22
The proposed self-healing glass fiber composite uses a similar concept. The manufacturing
techniques for making these capsules called SBA-15 are presented in (Kievsky and Sokolov, 2005;
Yang et al., 1997, 1999a, 1999b). The dimension and geometry of SBA-15 are comprehensively
described in Chapter 2.
Figure (3-1): SEM micrographs of SBA-15 hexagonal tubular silica microcapsules used to retain
epoxy resin (presented by Kievsky et al., (2005)).
So far, researchers have mainly focused on synthesis and experimental verification of self-
healing composites. This research is focused on FEM modeling of self-healing composites as FEM
is used extensively for obtaining accurate model of micro-composites (Levy and Papazian, 1990;
Brinson and Knauss, 1992; Tucker and Liang, 1999; Kang and Gao, 2002). By using FEM,we can
obtain composite properties with variations in fiber and matrix properties and their volume
fractions. The numerical modeling of self-healing glass fiber composite can be performed similarly
as numerical modeling of carbon nanotube (CNT) composite. Numerical and analytical modeling
for evaluating properties of CNT composite is presented in (Chen and Liu, 2004). Chen and Liu
(2004) used continuum elastic theory for evaluating equivalent longitudinal and transverse material
properties. They considered cylindrical, square, and hexagonal representative volume element
(RVE) for numerical modeling and concluded that square RVE yields more accurate results. A
comprehensive literature review about modeling CNT composite is described in Chapter 4.
Analytical modeling of randomly oriented short fiber composites has been done using the
‘averaging method’. Tandon and Weng (1986) used micromechanics approach and average stresses
23
of the composite for evaluating modulus of elasticity of composite with randomly oriented fibers.
Tucker and Lian (1999) reviewed several analytical methods for modeling of aligned short fiber
composites such as dilute Eshelby, self-consistent, shear lag, Mori-Tanaka, and Halpin-Tsai
methods. They compared results with FEM results and found that Halpin-Tsai method leads to a
reasonable estimation, but Mori-Tanaka gives the best prediction for material properties of aligned
short fiber composites. Numerical modeling of composites using FEM method has been studied by
numerous researchers (Levy and Papazian, 1990; Brinson and Knauss, 1992; Tucker and Liang,
1999; Kang and Gao, 2002; Chen and Liu, 2004). Kang and Gao (2002) obtained elastic modulus of
randomly oriented aluminum short fiber composite using FEM. They used a square RVE for
evaluating modulus of elasticity with different fiber orientations and also considered interfacial
effects and fiber aspect ratio on the modulus of elasticity of aluminum composite.
In this research, the effective mechanical properties of randomly oriented nano-structured
glass fiber composite are evaluated by using continuum mechanics and FEM analysis. The stress-
strain relation (stiffness matrix) of the glass fibers was evaluated in Chapter 2, which was published
in Alavinasab et al., (2008). In this Chapter, a series of numerical simulations for the RVE are
performed and the effective modulus of elasticity and Poisson’s ratio of the composite are obtained.
Longitudinal and transverse properties of aligned fiber are calculated and then the equivalent
material properties for tilted fiber (with different fiber orientations) are obtained. Based on
equivalent modulus of elasticity for different fiber orientations and using a random distribution
function for fiber orientation, the overall material properties of randomly oriented glass fiber
composite are calculated. Finally, the finite element simulation results are compared with those
from Halpin-Tsai and Mori-Tanaka methods.
3.2. MODELING OF ALIGNED SHORT GLASS FIBER COMPOSITES
3.2.1. Representative Volume Element Modeling The main objective of this chapter is to evaluate the mechanical properties of composites
with embedded (self-healing) glass fibers. As mentioned in the introduction, the key for obtaining
material properties of randomly oriented glass fiber composite is evaluating the material properties
of unidirectional composite with the same constituents. The procedure for evaluating effective
mechanical properties of aligned short fiber composites is described by Sun and Vaidya (1996). The
authors compared different assumptions and configurations for modeling of a composite using an
24
RVE and proposed a more accurate model than the commonly practiced model at the time. Similar
approach has been applied for modeling the glass fiber composites (shown in Figure (3-2)).
Figure (3-2): The RVE model of the matrix with and aligned short fiber
A typical 3-D square RVE is shown in Figure (3-2) wherein L and S are longitudinal and
cross sectional lengths. The relationship between L and S can be found by assuming a volume
fraction of fibers in the composite. For finding the exact value for L and S, an additional equation is
needed. Lim (2003) proposed two schemes called Parallel-Series (PS) and Series-Parallel (SP)
scheme which shows the influence of aspect ratio of an RVE on the effective modulus of elasticity
of unidirectional short fiber composites. Both the PS and SP schemes are based on the rule of
mixtures in composite. The difference between PS and SP schemes is in dividing the RVE into two
parts. PS divides the RVE into two portions longitudinally, but SP divides the RVE transversely.
So, part one of the RVE (vertical for PS and horizontal for SP) is composed of matrix and fiber and
part two is composed entirely of matrix (shown in Figure (3-6)). By applying compatibility of
strain, equilibrium of stresses, and the rule of mixtures, the modulus of elasticity of the RVE can be
found. By considering volume fraction of glass fiber equal to 10%, the equivalent modulus of
elasticity of composite for different L and its corresponding S were obtained.
S
L
x
y
z
S
25
5 10 15 20 25 30 35 40 45 502.5
3
3.5
4
4.5
5
L
E (G
Pa)
22.533.544.555.562.5
3
3.5
4
4.5
5
S
E (G
Pa)
Eps
Esp
Eps
Esp
Figure (3-3): Effective modulus of elasticity of aligned glass fiber composite with volume fraction
1.0=fV
Figure (3-3) shows the variation of modulus of elasticity versus S and/or L for a volume
fraction of 0.1 for the aligned glass fibers composite. Eps and Esp are modulus of elasticity of
composite based on the PS and SP scheme, respectively. Figure (3-3) also shows upper and lower
limits of modulus of elasticity of glass fiber composite. The upper limit means the distance between
two glass fibers is equal to zero, or in other words, a continuous aligned glass fiber exists which
leads to the maximum modulus of elasticity. On the contrary, the lower limit indicates the
maximum matrix material between two fibers which yields the minimum modulus of elasticity.
Although the variation of modulus of elasticity is given based on L and S, the exact value of L and S
is not specified yet. In this research, dimensions of the RVE (L, S) are chosen in a way that the
aspect ratio of composite is equal to the aspect ratio of glass fiber. As shown in Figure (3-3) by a
dashed line, L and S are considered as 10.825 μm and 4 μm, respectively.
3.2.2. Numerical Modeling of the RVE
3.2.2. 1 Longitudinal modulus of elasticity The overall modulus of elasticity of a composite is obtained by considering a linear
relationship between average stress and strain in the RVE. A commercial finite element software
mμ
mμ
26
(ABAQUS) is used for the numerical analysis. By considering Figure (3-2), Equation (3-1) defines
the boundary conditions for evaluating the longitudinal modulus of elasticity of the RVE. The
quantities vu, and, w correspond to displacements in x, y, and z directions, respectively.
13
1
constant),,(constant)0,,(
δδ==
==dyxw
yxw (3-1)
where d3 and 1δ are 10.825 μm and 0.5 μm, respectively, and the applied strain is equal to 0.0924.
Figure (3-4) shows the displacement of the RVE under longitudinal strain or stress.
Figure (3-4): Displacements of the RVE under longitudinal load, (a) Displacement of the RVE in the z-direction, (b) Displacement of the RVE in the y-direction.
Now zE can be calculated based on FEM results by applying the following equation:
z
avgzE
εσ
= (3-2)
where avgσ represents average stress calculated from Equation (3-3):
F
Capsule
(a)
3 3
1
2
x
y
z
F (b)
27
∫= dxdyLyxAtot
avg )2/,,(1 σσ (3-3)
By using FEM results and Equations (3-2) and (3-3), the longitudinal modulus of elasticity zE is
obtained as 5.03 GPa.
3.2.2. 2 In-plane Poisson’s ratio xyυ
The in-plane Poisson’s ratio of the composite xyυ is obtained by considering the plane strain
assumption for the RVE (Levy and Papazian, 1990). Therefore, it is assumed that 0=zε and
)( yxzxz σσυσ += . The in-plane Poisson’s ratio xyυ is equal to:
x
yxy ε
ευ −= (3-4)
By applying the plane strain assumption, the stress-strain relationship is written as:
⎪⎭
⎪⎬⎫
⎩⎨⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
−−−=
⎪⎭
⎪⎬⎫
⎩⎨⎧
y
x
EEEE
EEEE
y
x
z
zx
xz
zx
x
xy
z
zx
x
xy
z
zx
x
σσ
εε
υυυ
υυυ
22
22
1
1
(3-5)
The corresponding applied load and boundary conditions for evaluating Poisson’s ratio is shown in
Figure (3-5).
Figure (3-5): Load and boundary conditions of the RVE for evaluating Poisson’s ratio xyυ
By using ∆x and ∆y as the average displacements change in the x and y directions
respectively, the applied load and boundary conditions can be written as:
Uniform Load
Sym. B.C.
Sym. B.C.
x
y
z
28
in.Sym),,()0,,(
2/constant),,(
2/constant),0,(
3
22
2
zdyxwyxw
zdxv
zxv
==
==
==
δ
δ
(3-6)
S/2y along 2
S/2 along 2
, ,0
y
x
±=Δ
=
±=Δ
=
==
Sy
xSx
pyx
ε
ε
σσ
where d2 , 2δ , and p are 4 μm, 1 μm and 1E5 nPa, respectively. The in-plane Poisson’s ratio is
calculated as:
⎟⎟⎠
⎞⎜⎜⎝
⎛+
Δ⎟⎟⎠
⎞⎜⎜⎝
⎛+
Δ−=
z
zx
z
zxxy EpS
yEpS
x 22 22 υυυ (3-7)
The in-plane Poisson’s ratio xyυ for the glass fiber composite is equal to 0.46. The FEM results for
evaluation of stress, strain and displacements are shown in Figure (3-6).
29
Figure (3-6): Strain and displacement of the RVE under above load condition, (a) Strain in the x-direction, (b) U1 displacement in the x-direction (c) Strain in the y-direction (d) U2 displacement in
the y-direction
3.2.3. Analytical modeling of the RVE The modulus of elasticity of the glass fiber composite can be evaluated analytically
following the method used by Chen and Liu (2004) for modeling CNT composites. They evaluated
the modulus of elasticity of CNT composites by modeling the CNT as a hollow cylinder and using
the rule of mixtures. For evaluating longitudinal and transverse moduli of aligned short glass fiber
composites, similar to the PS scheme, the RVE is divided into two portions. Part 1 is composed of
the fiber and matrix, whereas part 2 is entirely filled with the matrix. Figure (3-7) shows the
geometry of the RVE and its partitions.
30
Figure (3-7): RVE partitions for modeling glass fiber composite
3.2.3.1. Longitudinal modulus of elasticity of the RVE As shown in Figure (3-7), the sequential alignment of parts 1 and 2 is considered for
evaluating the longitudinal modulus of elasticity of the composite. Therefore, the analytical
equation for evaluating the modulus of elasticity of the glass fiber composite LE can be written as:
⎟⎠
⎞⎜⎝
⎛−+=
LL
ELL
EEe
ce
mL1111 (3-8)
In Equation (3-8), mE is the modulus of elasticity of the matrix, sL is the length of the glass
fiber and cLE is the effective longitudinal modulus of elasticity of part1, defined by the following
equation:
)1(33tot
fm
tot
fcL A
AE
AA
EE −+= (3-9)
where fA is the cross-sectional area of glass fiber and totA is the cross section area of the RVE. By
considering Young’s modulus of glass fiber equal to E33 (obtained in Chapter 2), and considering
Em equal to 3.5 (GPa), the longitudinal modulus of aligned short glass fiber composite (EL) is equal
to 4.96 (GPa).
3.2.3.2. Transverse modulus of elasticity of the RVE
Similarly, the transverse modulus of elasticity of the glass fiber composite can be obtained
by considering the parallel movement of parts 1 and 2 along the y-axis. The transverse modulus of
elasticity based on the rule of mixtures can be written as:
31
)1( 11 VEVEE mcTT −+= (3-10)
where V1 is the volume fraction of part 1 and cTE is the effective transverse modulus of part1 in the
y-direction, defined by the following equation:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−+=
tot
f
mtot
fcT V
VEV
VEE
11
221111 (3-11)
where 22E is the transverse modulus of elasticity of glass fiber obtained in Chapter 2, and 1fV and
totV are the volume fraction of part 1 of the glass fiber and the total volume of part 1, respectively.
By substituting the corresponding values into Equations (3-10) and (3-11), the transverse modulus
of elasticity of the short fiber composite TE is equal to 3.811 GPa.
3.2.4. Tilted glass fiber composites Although tilted short fiber composites are similar to aligned fiber composites, the
orientation angle between the fiber axis (1-axis) and load direction (x-axis), α, is different. Aligned
short fiber composites are a special case of tilted short fiber composites where α is zero. A
schematic of tilted fiber composite is shown in Figure (3-8).
Figure (3-8): Tilted short fiber composite.
For calculating the overall modulus of elasticity of the tilted glass fiber composite, the
coordinate system transformation was applied (Kang and Gao, 2002). Figure (3-9) illustrates the
coordinate system transformation for RVE.
y 2
1
x
32
Figure (3-9): Coordinate transformation for tilted short fiber composite
(a) Original tilted RVE, (b) transformed to aligned RVE
Figure (3-9(a)) shows the original tilted RVE composite and Figure (3-9(b)) shows the
transformation of the stresses in the composite by performing the transformation of the coordinate
system. The correlations between aligned and tilted short fiber as shown in Figure (3-9) can be
written as:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
αααααααααααααα
σ
τσσ
22
22
22
sincoscossincossincossin2cossin
cossin2sincos
00
T
Tc
LT
T
L
(3-12)
where α is the angle between the two coordinate systems (also, α is the angle between tilted fibers
and their corresponding aligned fibers). By applying the stresses as shown in Figure (3-9(b)), the
average strain in the RVE is calculated as:
∫
∫=
V
Vij
ijdV
dVε
ε (3-13)
where εij represents the strain tensor (i=1,2,3; j=1,2,3). By using the inverse transformation, the
average strain for the original tilted glass fiber shown in Figure (3-9(a)) can be obtained by the
following equation:
σcσc σL
(a) (b) σT
σT
σL
LTτ
LTτ
LTτ
33
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧−
LT
T
L
xy
y
x
Tγεε
γεε
21
1
21
(3-14)
Considering a linear relationship between stress and strain, the overall modulus of elasticity for
different fiber orientations is obtained.
3.3. MODULUS OF ELASTICITY
3.3.1. Numerical modeling of randomly oriented glass fiber composites The overall modulus of elasticity of randomly oriented glass fiber composites can be
obtained by considering the modulus of elasticity of the RVE for different fiber orientations and the
distribution function for fiber orientations. The overall mechanical properties of such composites
can be obtained by the following equation:
∫=2/
0 )()(πααα dfEErandom (3-15)
where E(α) is the modulus of elasticity of the tilted composite (with angle of α to horizontal) and
)(αf is the distribution function for fiber orientation. The variation of the modulus of elasticity of
the tilted glass fiber composite based on the different fiber orientations is given in Figure (3-10). As
shown in Figure (3-10), the maximum modulus of elasticity occurs when the glass fiber is aligned,
and the minimum when the fibers are oriented at 45 degrees.
Figure (3-10): Modulus of elasticity of tilted glass fiber composite.
34
Researchers usually determine the distribution function for fiber orientation by using
micrograph pictures of the composite. Kang et al. (2002) considered the Weibull distribution
function for a reinforced aluminum alloy composite. The Weibull distribution function is given by:
08.2)79.0/(08.1
79.06329.2)( ααα −⎟
⎠⎞
⎜⎝⎛= ef (3-16)
Both the uniform and Weibull distribution functions for evaluating the modulus of elasticity have
been used in this study. The uniform distribution function is considered as:
πα 2)( =f (3-17)
Considering uniform distribution of glass fibers, the effective modulus of elasticity of composite is
calculated as 4.01 GPa. The effective modulus of elasticity based on the Weibull distribution is
equal to 3.88 GPa.
3.3.2. Analytical modeling of randomly oriented glass fiber composites Analytical micromechanics theories have been successfully used for prediction of
mechanical properties of composites. They are typically an extension or modification of a single
inclusion or fiber in an infinite media. They predict mechanical properties of a composite based on
the volume fraction of fibers, the mechanical properties of matrix and fiber, and the fiber’s shape,
orientation, and distribution. The analytical methods can only predict the effective modulus of the
composite based on the continuum theory approach. Out of all analytical methods, Halpin-Tsai and
Mori-Tanaka methods give the best estimations for evaluating material properties of short fiber
composites (Tucker and Liang, 1999).
3.3.2.1. Halpin-Tsai method
Halpin and Tsai developed a semi-empirical method for modeling randomly oriented short
fiber composites. The model was developed first for aligned short fiber composites by using curve-
fitting parameters to match the experimental results. The Halpin-Tsai equations for short aligned
fiber composite can be written as (Lubin and Peters, 1998):
fL
fL
m
L
VV
EE
ηξη−
+=
11
(3-18)
35
( )( ) ξ
η+
−=
mf
mfL EE
EE 1 (3-19)
dl2
=ξ (3-20)
fT
fT
m
T
VV
EE
ηη
−
+=
121
(3-21)
( )( ) 2
1+
−=
mf
mfT EE
EEη (3-22)
whereξ is called the empirical factor which in general is a function of fiber geometry, fiber
distribution, and loading conditions. When ξ approaches infinity or zero, the Halpin-Tsai equations
become rule of mixture or inverse rule of mixture equations, respectively. EL and ET are the
longitudinal and transverse moduli of elasticity of short fiber composites. The modulus of elasticity
of random short fiber composites can be obtained experimentally by using the following equation:
TLrandom EEE85
83
+= (3-23)
3.3.2.2. Mori-Tanaka method
Mori-Tanaka is an analytical way to predict the elastic constants of randomly oriented short
fiber composite material. It has been found that the Mori-Tanaka method predicts the modulus of
elasticity close to the experimental results especially in low volume fraction composites. The basic
equations for the Mori-Tanaka method for a randomly oriented composite can be written as
(Bradshaw et al., 2003; Wang and Pyrz, 2004):
{ }( ) { }( ) 111011100
* −++= dildil AfIfACfCfC (3-24)
where the subscripts 0 and 1 denote matrix and fiber respectively and C* is the effective elastic
modulus tensor of a composite. The terms f and C are the volume fraction and moduli respectively
and N is the number of different types of fibers. I is the forth order identity tensor and {} represents
the average of quantity over all possible fiber orientations. dilA1 is the dilute strain concentration
obtained from following equation:
36
011 εε dilA= (3-25)
where 1ε , and 0ε are average inclusion strain and average matrix strain on the boundaries
respectively. The modulus of elasticity of the composite is obtained by calculating the shear and
bulk moduli. The relationship between the effective elastic modulus tensor of the composite and the
shear and bulk moduli is given as:
K2J2* μ+= KC (3-26)
where K and µ are bulk and shear moduli of the composite material respectively. J and K are two
tensors whose components are:
klijijkl δδ31J = (3-27)
)32(
21K klijjkiljlikijkl δδδδδδ −+= (3-28)
where δ is the Dirac delta function. As shown in Figure (3-11), the numerical results
obtained are compared with the Halpin-Tsai and Mori-Tanaka method results.
Figure (3-11): Effective modulus of elasticity of the randomly oriented glass fiber composite using the Mori-Tanaka and Halpin-Tsai methods
It should be noted that in the Mori-Tanaka and the Halpin-Tsai methods, the glass fiber is
considered isotropic; however, the material properties of glass fiber as described in Chapter 2 is
orthotropic. In addition, by considering Table (2-1), it can be concluded that the transverse modulus
37
of elasticity of the glass fiber is much smaller than the longitudinal modulus of elasticity. However,
in both Mori-Tanaka and Halpin-Tsai methods, material properties of the glass fiber are considered
isotropic and equal to the longitudinal material properties of the glass fiber. As shown in Figure (3-
11), the numerical FEM result is smaller than the analytical result. Therefore, it is observed that the
finite element simulation predicts a more accurate modulus of elasticity for the composite.
3.4. CONCLUSIONS
Numerical and analytical modeling for evaluating the overall modulus of elasticity of the
nano-structured glass fiber composite is presented. The FE simulation has been performed using
ABAQUS. A representative volume element approach for aligned and tilted fiber composites was
used for obtaining the overall modulus of elasticity of randomly oriented glass fiber composites.
The overall effective modulus of elasticity was computed using the uniform and Weibull’s
distribution functions. The numerical representation of the composite material is considered more
accurate because it accounts for the exact fiber geometry and the orthotropic material properties for
the glass fiber.
38
CHAPTER 4. NONLOCAL MODELING OF CARBON NANOCOMPOSITE
4.1. INTRODUCTION
Composite materials play a major role in advanced structures these days. By the discovery
of the Carbon Nano-Tube (CNT) by Iijima in 1991, a new window for making advanced
composites was opened. The task of modeling CNT composites is challenging and urgent since
industries are looking to exploit the benefits of such materials. A brief review of processing and
application of nano-composites is presented in (Hussain et al., 2006). CNT is a hollow cylinder of
graphite sheets typically with the size of nano meter in diameter and micro meter in length. The
Young’s modulus of CNT is between 1000-1500 GPa (Harris, 2002). When CNTs are incorporated
into a matrix, outstanding mechanical properties of CNT composites are achieved. The
incorporation of CNT into a matrix increases not only the strength but also the toughness of the
composite. CNT reinforced composites possess unique properties such as light weight, very high
strength, and electrical conductivity (Qian et al., 2000). Due to both electrical and mechanical
properties of CNT, such composites have multi functional behavior.
For more than two decades, researchers have been trying to find an accurate model for
mechanical properties of CNT. Effective material properties of composites, such as Young’s
modulus, are generally obtained using continuum theory and finite element method (FEM) (Liu and
Chen , 2003). The Molecular Dynamics (MD) modeling approach has the ability to yield accurate
results at the nano scale, but it has limitations in length and time scales and the associated
computational cost is very high (Yao et al., 2001). Due to these limitations, the majority of
researchers use the continuum modeling approach. Continuum models that capture the mechanics
of a microstructure are attractive due to the relative simplicity of solution and the availability of
existing methods. Some authors have used continuum approach for modeling individual CNTs with
spring, beam, solid, and shell elements as well as space frame structures. Reich et al. (2004)
modeled CNT based on the continuum theory using shell elements. They report that the results are
well matched with the experiments. Other researchers have combined MD simulations with FEM
modeling. Other researchers also have used shell structures (Chakraborty, 2006) and space frame
structures (Chunyu et al., 2005) for modeling CNT. Therefore, a number of authors have used a
modified continuum approach for modeling CNT by considering the strain energy of the continuum
model to be equal to the potential energy of the molecular structure of the solid. (The potential
energy of solid molecular structure is composed of the chemical bond energy and Van der Waals
39
bond energy.) However, the validity of continuum models based on classical elasticity is
questionable at nanoscales. Gao et al. (2000) measured the resonance frequency of a single CNT
using Transmission Electron Microscopy (TEM) and calculated the corresponding modulus of
elasticity of CNT using an elastic beam theory. The results show that the corresponding elastic
properties of CNTs, which are nano-scale structures, are size dependent.
Modeling of CNT considering the interaction of atoms and bonding between them by using
the Morse energy function has been reported in (Meo and Rossi, 2006). This approach indicates that
the behavior of CNT is dependent on the interaction of the whole atomic structure, and in that
regard it is nonlocal. The nonlocal continuum theory was developed to account for nonlocal stress-
strain relationships. In this approach, stress at any point in a structure is a function of strain in the
entire structure. Therefore, nonlocal theory is expected to yield more accurate results compared to
the classical elasticity, especially at very small scales such as those encountered in CNT
composites.
4.1.1 Benefits of nonlocal theory The continuum theory is considered as a local theory which is formulated based on two
major concepts. First of all, at every part of a body, no matter how small it is, all balance laws are
valid, and the state of the body at any point is influenced only by the state of the infinitesimal
neighborhood. The first concept neglects the effects of long-range load on the motion of the body
and the second concept neglects the long range inter atomic interactions. So, in the classical
elasticity theory, the internal scale is neglected. Therefore, when the internal characteristic length
and time scale (granular distance, lattice parameter, relation time) are large enough compared to
external length (the size of the area corresponding to the applied force, wave length, period, etc.),
the classical elastic theory fails. Several researchers have tried to apply MD for modeling, but until
now the application of MD has been unsuccessful for real engineering problems. In MD
simulations, the simulation time (cost) increases enormously if we increase the length and the
number of atoms. In addition, a statistical averaging theory is needed for postprocessing the MD
results to compare with experiments.
Nonlocal linear theory, which has both features of lattice parameter and classical elasticity,
could be considered a superior theory for modeling nano materials. The nonlocal continuum theory
was developed by Eringen (1972a) and Eringen and Edelen (1972) to account for nonlocal stress-
40
strain relationships. The departure from local elasticity theory to include couple stress effects was
originally initiated by Cosserat and Cosserat (1909). In 1960s new developments on couple stress
and micromorphic models were reported by Tupin (1962), Mindlin and Tiersten (1962), Minlin
(1964) and Eringen and Suhubi (1966), which formed the basis for nonlocal elasticity. In the
nonlocal theory, stress at any point in a structure is a function of strain in the entire structure.
Therefore, the nonlocal theory provides more accurate description of material behavior compared to
the classical (local) elasticity theory for nano-scale materials where long range forces compared to
the scale of the size of the sample are predicted as the small scale parameters. This is particularly
the case for the CNT structures.
Nonlocal (elastic) theory, which has features of both lattice parameters and classical
elasticity, has been considered for modeling nanomaterials. Edelen and Laws (1971) developed a
nonlocal theory by using the global laws of balance of momentum, moment of momentum, energy,
and local conservation of mass, and described the nonlocal form of entropy inequality. Eringen and
Edelen (1972) extended the theory of nonlocal elasticity and studied the formulation of nonlocal
constitutive equations. They obtained a set of constitutive equations for non-heat-conducting
nonlocal elastic solids. In addition, they developed a formulation for nonlocal polar elastic continua
(1972b). Nowinski (1984) studied the longitudinal wave propagation in an elastic circular bar and
determined that the velocity of short waves using the nonlocal theory is about 36% less compared
to the classical elasticity. Ahmadi and Farshad (1973) developed a nonlocal theory for vibration of
thin plates and also estimated the corresponding nonlocal material moduli. Lu et al. (2007)
presented the nonlocal theory for thick Kirchhoff and Mindlin plates, and solved bending and
vibration of rectangular plates. Eringen (1972a) studied the dispersion of plane waves in nonlocal
elastic solids. The reported results show a similarity between the nonlocal modulus of elasticity and
the inter-atomic potential in which the influence of distant atoms on the local stresses attenuates
rapidly. Ahmadi (1975) extended the nonlocal theory and derived the constitutive equations for
nonlocal viscoelastic materials.
Some of the properties of materials such as material hardness and electrical properties can be
explained by considering the static arrangement of the atoms. However, energy dispersion, sound
and wave propagation, thermal expansion and thermal conductivity of structures can be explained
by the atomic motion only known as lattice dynamics. Energy of the elastic /sound waves inside a
solid can be quantized in the form of phonons. In other words, phonons, similar to natural
41
frequencies and mode shapes in classical mechanics, represent the vibrational motion in which each
part of a lattice oscillates with the same frequency. Any arbitrary vibration of a lattice can be
considered as a superposition of the modes with corresponding frequencies. Therefore, these natural
frequencies and corresponding modes are important to study the dynamic behavior of materials. In
addition, phonon dispersion relation is the main key for studying lattice dynamics. The slope of the
dispersion curve represents the speed of sound in a material. The application of the nonlocal theory
can predict the behavior of physical phenomena at the nano scale. The dispersion curve obtained
from the nonlocal theory is similar to the one obtained from the lattice dynamic modeling. So, the
nonlocal theory can be considered as a bridge between the atomic interactions and the classic
elasticity. Furthermore, experimentalists have observed that the classical elasticity cannot capture
high frequency waves (short wavelength). This is due to the fact that classical elasticity does not
consider the long range inter-atomic forces. Eringen (1977) showed that the nonlocal theory results
are well matched with lattice dynamic phonon dispersion results. By matching the dispersion of the
nonlocal theory with lattice dynamics, the nonlocal modulus of elasticity in one dimension can be
written as a function of Young’s modulus of elasticity by the following equation
0
)()(ˆ 2
⎪⎩
⎪⎨⎧
>
<−=
ax
axxaaE
xE (4-1)
where E and E are nonlocal modulus of elasticity and Young’s modulus of elasticity, respectively
and a is the atomic distance. Eringen (1983, 1987) applied nonlocal theory and solved surface
wave, crack, and screw dislocation problems. Zhou (1998, 1999) used nonlocal theory for solving
crack problems without considering stress singularity at crack tip. Zeng et al. (2006) obtained
material constants of single crystal silicon and diamond using nonlocal micromorphic theory. They
found the material constant by matching the results with phonon dispersion relationship.
Wave propagation in CNT has also received a lot of attention from researchers. Xie et al.
(2007) studied the effects of small scale characteristics in nonlocal theory on dispersion
characteristics of waves in CNT. They considered a shell structure for modeling CNT and found
that the effects of small scale characteristics in wave propagation are more significant for CNT with
smaller inner radii. Wang and Varadan (2007) studied the wave propagation in CNT using nonlocal
elastic shell theory. They showed that the nonlocal modeling affects phonon dispersion relationship
especially for larger wavenumbers. Chakraborty (2007) studied wave propagation in laminated
42
composite layered media. He used spectral finite element method (SFEM) to model wave
propagation in nonlocal theory. He showed that for high frequency waves, nonlocal small scale
characteristics yield significant variation in results which can be seen as dispersive results.
Furthermore, the results obtained from classical elasticity compared to nonlocal theory give more
response (velocities).
This research presents an analytical approach for modeling of CNT composites using the
nonlocal theory. By having an exact solution, can be obtained from experiments, the internal
characteristic length in nonlocal theory can be achieved and the results can be extended for
modeling the entire composites. In addition, numerical FEM method in RVE with very fine meshes
is considered as accurate results in the RVE. In contrast, FEM uses classical elasticity theory which
its extension in entire structure cannot add any new feature to the classical elasticity. However, by
considering FEM in the RVE of CNT composite as an accurate model, the internal characteristic
length (atomic length) in nonlocal theory can be derived. As described in Chapter 5, the proposed
nonlocal method is capable to model the dispersion in high frequency waves.
In this chapter, a brief review of nonlocal theory is given and analytical modeling of carbon
nanotube composite based on nonlocal elasticity theory is proposed. A novel approach for modeling
stress and wave propagation in nanocomposites using the nonlocal theory with both macro and
nano scale features is proposed. The solution to nonlocal constitutive equation is obtained by
matching the dispersion curve of nonlocal theory with lattice dynamic simulation. The first and
second order approximations in nonlocal theory are described. It is shown that the second-order
approximation yields more accurate results when compared with the first-order approximation
which has been employed by various researchers so far. Numerical example results using nonlocal
theory, FEM, and classical elasticity (using constant stress) for a CNT composite are presented
herein. Unlike the classical elasticity results, the first moment obtained from stress distributions
using the nonlocal theory is equal to those obtained from numerical FEM calculations. Wave
propagation in elastic media is also studied using the second-order approximation in nonlocal
theory. The dispersion curves show that the second-order approximation yields values close to the
lattice dynamics results.
43
4.2. NONLOCAL CONTINUM THEORY
Basic equations of conservation using nonlocal theory can be written as:
Mass:
,0 ( ) 0k kV
d dv vdt t
ρρ ρ∂= ⇒ + =
∂∫ (4-2)
Momenta:
0)(, =−+dt
dvft k
kllk ρ (4-3)
lkkl tt = (4-4)
Energy:
0ˆ0 =−− hhηθ & (4-5)
Entropy (Clausius-Duhem):
bh ˆ ˆ0 θ≥ (4-6)
where ρ, ν , klt , kf , η , h, 0h , θ and b are the mass density, the velocity vector, the stress tensor,
the body force, the entropy density, the energy source density, a nonlocal energy term, the absolute
temperature and the nonlocal entropy source, respectively. Equation (4-2) represents the
conservation of mass and indicates that mass is invariant under motion. Equation (4-3), called the
principle of balance of linear momentum, considers changing rate of the momentum with respect to
time. On the other hand, variation of the rate of the momentum is equal to the resultant force acting
on the body. More details on the principle of nonlocal theory may be found in (Eringen, 2002). The
stress tensor, klt , in the nonlocal theory is given as
∫ ′′−′=υ
υσα dxxxxt klkl )()()( (4-7)
where x is the reference point in the body and )( xx −′α is the nonlocal kernel function, which
depends on the internal characteristic length. klσ is the macroscopic elastic stress tensor at any
point x′calculated from following equations:
44
klijklij C εσ = (4-8)
2/)( ,, kllkkl uu +=ε (4-9)
where klε , ijklC and u are the strain tensor, the elastic modulus component and the displacement,
respectively. The traction boundary condition in the nonlocal theory is calculated by using )(xtij .
The key for calculating nonlocal stress is the nonlocal kernel function )( xx −′α , which depends on
the internal characteristic length and has the dimensions of length-3. Therefore, the stresses in
nonlocal theory are influenced by the internal characteristic length, a, and the characteristic length
ratio, la / (l is an external characteristic length) (Eringen, 1983). The kernel function has the
following characteristics:
1)( =′′∫υ
υα dx (4-10)
xxatxx ′==−′ )( maxαα (4-11)
)()(lim 0 xxxxa −′=−′→ δα (4-12)
Equation (4-10) describes that the kernel function is normalized over the volume of the body. It can
be seen from Equation (4-12) that when the internal small scale, a, approaches zero, the nonlocal
theory reverts to classical elasticity. Eringen (1983, 2002) (presented α as a Green’s function of a
linear differential operator:
)()( xxxxL −′=−′ δα (4-13)
The nonlocal stress function, Equation (4-7), can be reduced to a simple differential equation using
the differential operator from Equation (4-13):
ijijLt σ= (4-14)
The approximation is obtained by matching a dispersion curve with the corresponding atomic
model and using the Fourier transform (as contained in Chapter 5 of this proposal describing wave
propagation). For small internal length scale, matching of the dispersion curves of plane waves with
those of lattice dynamics, Equation (4-14) can be written as (Eringen, 2002):
klklt σγε =−∇+∇− ...)1( 4422 (4-15)
45
where ε and γ are small parameters proportional to the internal length scale. Therefore, they may
be written in terms of a characteristic length scale (lattice constant) a as follows:
0 , ;0 , 0000 ≥=≥= γγγεεε aa (4-16)
The small parameters for nonlocal theory solution are derived using numerical finite element results
in the current work. Since modeling entire CNT composite is computationally expensive, a
Representative Volume Element (RVE) which represents the material properties of the CNT
composite is modeled. Alternatively, these small parameters could be obtained from either
experimental or MD stress distributions. Once these parameters are determined, the nonlocal theory
can be used for modeling the entire nanocomposite. The specification of the nonlocal small scale
parameters for CNT is still not fully resolved. Researchers have, generally, used the first order
approximation of nonlocal theory for modeling CNT (Wang et al., 2008; Duan et al., 2007; Zhang
et al., 2005; Peddieson et al., 2003; Wang et al., 2008; Reddy, 2007; Reddy and Pang, 2008; Kumar
et al., 2008; Eringen, 2002; Eringen, 1983; Zhang et al., 2006). Duan et al. (2007) proposed 0ε
ranges between 0 and 19 depending on single-walled carbon nanotube (SWCNT) aspect ratio and
boundary conditions. Zhang et al. (2005) calculated small scale parameter to be about 0.82. Zhang
et al. (2006) evaluated 0ε between 0.546 and 1.0043 for different chiral angles of SWCNTs. Wang
and Hu (2005) proposed 288.00 =ε . Eringen (1983) determined 39.00 =ε and 31.00 =ε lead to a
close match with the longitudinal and Rayleigh surface wave atomic dispersion curve respectively.
Wang and Wang (2007) proposed 288.00 =ε using the gradient method. Wang (2005) estimated
nm 1.20 <aε in SWCNT by investigation of wave propagation with wave frequency value greater
than 10 Hz. The above discussion shows that the value of 0ε (small scale parameter) is of the order
of 1. While there is very little information on value of 0 γ , which is a key parameter of the second
order approximation in the nonlocal theory, it is expected that its value also be of the order of unity.
4.2.1. First Order Approximation
Equation(4-15) can be approximated by considering only the first term (ε term) (Eringen,
2002), leading to:
46
klkltdyd
dxd σε =+− ))(1(
2
2
2
22 (4-17)
When a constant strain is imposed, the stress distribution obtained from finite element analysis
(FEM) shows a large variation of axial stresses in the y-direction. Thus, by neglecting the variation
in the x-direction, we have
11112
22 )1( σε =− t
dyd (4-18)
For a constant 11σ , the solution to Equation (4-18) is given as
)sinh()cosh(1111 εεσ yByAt ++= (4-19)
The constants A, B, and ε are obtained from boundary conditions; however, because of symmetry,
B=0. The appropriate boundary condition is obtained when the resultant force and moment derived
from Equation (4-19) are equal to those obtained for the RVE. However, no value of ε can be
found to satisfy the boundary conditions. Thus the expression given by Equation (4-19) is an
unacceptable solution and the governing equation given by Equation (4-18) is not appropriate for
this application.
4.2.2. Second Order Approximation In order to consider the second approximation of Equation (4-15), only the second term of Equation
(4-15) is retained. That is,
11114
44 )1( σγ =+ t
dyd (4-20)
The analytical solution to Equation (4-20) satisfying the symmetry condition is given by
)2
cos()2
cosh(1111 γγσ yyAt += (4-21)
As noted before, the applied boundary conditions are the equality of resultant force and moment
with those obtained for the RVE (through FEM). For the resultant force, the summation of forces
obtained from nonlocal theory should be equal to summation of forces in classical elasticity. The
resultant force in nonlocal theory is obtained from
47
dyyyAdydytLL
kl
L
kl ∫∫∫ +=000
)2
cos()2
cosh(γγ
σ (4-22)
where L is the cross-sectional dimension of RVE in the y-direction (Shown in Figure (4-5)). The
resultant force in average elasticity and the nonlocal theory model in Equation (4-22) should be
equal. Therefore, the remaining term in Equation (4-22) is equal to zero:
0)2
cos()2
cosh(0
=∫ dyyyAL
γγ (4-23)
By integrating Equation (4-23) we obtain
0)2
cosh()2
sin()2
sinh()2
cos( =+=γγγγ
LLLLF (4-24)
Solving Equation (4-24) we can find several values of γ in each length scale for CNT
composites. Roots of Equation (4-24) give the values of γL .
As shown in Figure (4-1), the first four roots of Equation (4-24) are 3.3446, 7.7751,
12.2179, and 16.6608, respectively. Furthermore, the values of function F around larger L/γ roots
vary steeply. Therefore, choosing the first root leads to more reasonable results which provides
smoother stress distribution curve. In addition to satisfying the equality of the resultant forces, the
first moment of forces needs to be balanced. First moment of forces in nonlocal theory is
calculated as:
dyyyyAdyyydytLL
kl
L
kl ∫∫∫ +=000
)2
cos()2
cosh(γγ
σ (4-25)
where the first term on right hand side of Equation (4-25) is the first moment of forces in average
elasticity. For a combination of γ and L from Figure (4-1), the variable A can be calculated as
Resultant force in nonlocal theory
Resultant force in elasticity
48
2
0
0
2
cosh cos2 2
L
kl kl
L
Lt ydyA
y y ydy
σ
γ γ
−
=⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
∫
∫ (4-26)
Using the FEM results for the RVE, we can evaluate the first moment of forces using the above
equations.
-6
-5
-4
-3
-2
-1
0
1
2
3 3.2 3.4 3.6 3.8 4 4.2
Fc
-60
-50
-40
-30
-20
-10
0
10
20
30
40
7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8F
-4000
-3500
-3000
-2500
-2000
-1500
-1000
-500
0
500
1000
12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13
F
-30000
-20000
-10000
0
10000
20000
30000
40000
16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 17
F
Figure (4-1). Values of L/γ for CNT composites using Equation (4-23).
4.3. NUMERICAL EXAMPLE
A rectangular RVE of CNT composite (Figure (4-2)) is considered for numerical modeling.
The RVE comprises matrix material in a rectangular shape and a hollow cylindrical CNT, similar to
the RVE used in (Liu and Chen, 2003). The geometry of RVE is as follows: length = 100 nm, inner
γL
γL
γL
γL
49
radius of CNT = 4.6 nm, and thickness of CNT = 0.4 nm. The volume fraction of CNT in the
composite is taken as 5%. The following material properties are used for CNT and matrix material:
CNT: ;3.0 ,nN/nm 1000 CNT
2 == νCNTE Matrix: ;3.0 ,nN/nm 200 ,100 ,20 ,5 m
2 == νmE
Figure (4-2). A representative volume element (RVE) for CNT composite
The 3D FE model of RVE shown in Figure (4-2) comprises 8-node isoparametric brick
elements based on reduced integration. A commercial finite element software (ABAQUS) is used
for numerical analysis. The maximum element size is 2 nm to obtain nano-scale resolution. The
CNT is modeled as hollow cylinder with thickness of 0.4 nm; the element size for CNT is 0.4 nm in
cross-section and 2 nm in length. Perfect bonding between CNT and matrix is considered. By fixing
one edge and applying uniform strain (equal to 5%) on opposite edge, the deformation of mid-
plane, stress, and strain in the RVE are obtained as shown in Figure (4-3).
(a)
50
(b) (c) Figure (4-3): FE results for RVE of CNT composite, (a) Displacement of mid-plane (b) longitudinal
stress, and (c) longitudinal strain
Figure (4-4): Nonlocal stress distribution of CNT composite for the first three roots
Different roots of Equation (4-24) present different curvatures for nonlocal stress
distributions. Figure (4-4) presents nonlocal stress distribution for the first three roots, which shows
that the higher roots cause more variation in stress distribution and also produce compressive
stresses (while a tensile force is applied). Therefore, the first root is considered for evaluating
nonlocal stress distributions.
51
As shown in Figure (4-3), the FEM results under constant strain shows the maximum stress
in CNT. The cross section of CNT composite which is used for numerical calculations is shown in
Figure (4-5).
Figure (4-5): Cross section of CNT composite
Figure (4-5) shows the cross-section of CNT composite and because of symmetry only one
quarter of CNT is considered for showing results. Figure (4-6) compares stress distributions in CNT
composite using nonlocal theory, average elasticity and FEM. The values of matrix modulus is
varied from Pa 5 GEm = to Pa 200 GEm = . FEM results correctly show that there is no stress
inside of CNT, maximum peak stress is obtained in CNT, and the stress is much less in the matrix
material. Classical elasticity gives a constant value for L = 0 to 10 (as expected) whereas the stress
from nonlocal theory varies similar to the variation of γ (lowest root) in Figure (4-4). Furthermore,
as shown in Figure (4-6(c)) and (4-6(d)), the difference in stress distribution between nonlocal
theory and average elasticity is larger for higher values of matrix modulus, Em. The total force
(integral of stress distribution) obtained using nonlocal theory, average elasticity, and FEM are
equal. The value of total force values for Em = 5, 20, 100, and 200 2 nN/nm (Figure 4-6 (a)-(d)) are
equal to 85, 100, 180, and 280 nN , respectively.
Sym.
Sym.
L
L
L=10
O
52
(a) (b)
(c) (d) Figure (4-6): Stress distribution in CNT composite using nonlocal theory, average elasticity and FEM for various values of matrix modulus (a) 2nN/nm 5=mE , (b) 2nN/nm 20=mE , (c) 2nN/nm 100=mE , (d) 2nN/nm 200=mE
The first moment (integral of stress times distance) computed from the stress distributions
(Figure (4-6)) show nearly identical values for FEM and nonlocal theory, but the average elasticity
results are different (Figure (4-7)). These results clearly indicate the superiority of nonlocal theory
approach over classical (local) elasticity. Nonlocal theory yields the same total force and first
moment as detailed finite element analysis, whereas classical elasticity fails to give the correct
value of first moment. Once the unknown constant in nonlocal theory solution is obtained from
FEM for an RVE, the computation for the composite is very efficient. Nonlocal theory is very
appropriate for nano-scale structures wherein the interaction between atoms (especially adjacent
atoms) has a big influence on behavior of the structure. By using nonlocal theory, stress is a
53
function of strain in the entire domain which is similar to the behavior of atoms in nano-scale
structures.
Figure (4-7): The first moment of stress in CNT composite using FEM, average elasticity, and nonlocal theory for various values of matrix modulus
(a) 2nN/nm 5=mE , (b) 2nN/nm 20=mE , (c) 2nN/nm 100=mE , (d) 2nN/nm 200=mE
4.4. WAVE PROPAGATION Wave propagation is widely used to study defects in structures and for evaluating material
properties such as Young’s modulus, shear modulus, etc. Since the speed of sound is unique in a
constant property material, the wave characteristics observably vary when passing through defects
or material changes. However, finite element analysis based on classical elasticity cannot capture
the dispersion of high frequency waves (Chakraborty, 2007). On the other hand, the lattice dynamic
model has been successfully implemented for modeling vibrations and heat transfer at the atomic
scale. However, using the lattice dynamic model for large scale structures is computationally
prohibitive. Nonlocal theory is capable of capturing wave dispersion curves in solid media with
micro- and nano-structures and it can be regarded as a good approximation for modeling nano-scale
(d) (c)
(a) (b)
54
effects in large scale structures. The approximation of the nonlocal equation of motion in an
isotropic material is given as (Eringen, 2002).
0...)1(.)2( 4422 =−∇+∇−−×∇×∇−∇∇+ uuu &&ργεμμλ (4-27)
where λ, μ are Lame parameters and u, ∇ and ρ are displacement, gradient and mass density,
respectively. The terms ε and γ are small parameters as discussed earlier and u&& denotes second
derivative of u with respect to time. Equation (4-27) is obtained by using Fourier transforms to
simplify the nonlocal equations and matching the dispersion curves of nonlocal theory and atomic
model.
4.4.1. First order approximation Using the first order approximation suggested by Eringen (1983,1987), the equation of motion for
wave propagation in a nonlocal elastic solid is written as
0)1(.)2( 22 =∇−−×∇×∇−∇∇+ uuu &&ρεμμλ (4-28)
Using the Helmholtz decomposition method similar to that used for the isotropic elastic solid,
harmonic wave solutions with the dispersion relations is given as (Eringen, 1983,1987)
)1(
122kkcL ε
ω
+=
(4-29)
)1(
122kkcT ε
ω
+=
(4-30)
Here ω and k are frequency and wave number whereas Lc and Tc are compression and shear wave
velocities, respectively.
4.4.2. Second order approximation It was shown in section 4.2.2 that the second order approximation of nonlocal theory leads to more
accurate results (compared to the first order approximation used by other researchers). The second
order approximation of Equation (4-27) is given as
0)1(.)2( 44 =∇+−×∇×∇−∇∇+ uuu &&ργμμλ (4-31)
55
The corresponding dispersion relations are obtained as
)1(
144kkcL γ
ω
+=
(4-32)
)1(
144kkcT γ
ω
+=
(4-33)
Equations (4-32) and (4-33) show the dispersion relation using the second order approximation.
The small parameters, 0ε and/or 0 γ , defined in Equation (4-16) are constant appropriate to each
material. Several efforts have been made for evaluating the small parameters for SWCNT. Zhang et
al. (2006) used the MD simulation for evaluating 0ε , and they found 043.1 546.0 0 ≤≤ ε for different
chiral angles of SWCNTs. Zhang et al. (2005) evaluated 82.00 ≈ε by comparing molecular
mechanics simulation with nonlocal buckling analysis of SWCNT. Wang and Hu (2005) proposed
288.00 =ε by using second-order strain gradient in elasticity theory and MD simulation. Eringen
(1983) determined that using the first order approximation of nonlocal theory, with 0.39aε =
where a is the lattice parameter, leads to a close match with the atomic model. As reported in
Eringen (1983, 1987), the maximum deviation between the first order approximation of nonlocal
theory and the Born-von Karman theory is less than 6%. Wang and Wang (2007) compared the
gradient method with nonlocal first order method ( 385.00 =ε ). The gradient method shows very
close agreement with Born-Karman only at smaller values of ka. The aforementioned small scale
parameters are obtained for SWCNT either by the comparison of nonlocal theory with lattice
dynamics or MD simulation. So far, no definitive study has been done for obtaining small scale
parameters in CNT composite. The nonlocal small scale parameter in the proposed method )( 0γ is
evaluated by matching the dispersion curve of the proposed nonlocal theory with lattice dynamic
simulation. For 0.39aε = and 0.35aγ = , Figure (4-8) shows the dispersion curves obtained using
first and second order approximations of nonlocal theory and the Born-von Karman atomic model.
The results show that the second order approximation gives slightly higher values compared to
atomic model whereas the first order approximation yields slightly lower values. The atomic model
values are close to the average of first order and second order approximation results.
56
Figure (4-8): Dispersion curve for the Born-von Karman lattice dynamic and nonlocal theory.
4.5. CONCLUSIONS
This study has presented a new approach for obtaining accurate stress distributions in
nanocomposites in a computationally efficient manner using nonlocal continuum theory. First- and
second-order approximations of the nonlocal continuum theory were considered. The first order
approximation (which has been used by several researchers so far) led to an unacceptable solution,
hence only the second-order approximation was retained for further investigations. Finite element
analysis of a representative volume element of CNT composite was used to evaluate unknown
constants in nonlocal theory solution. The major conclusion of this investigation is that the nonlocal
theory approach is superior to classical elasticity in the following ways:
1) The computed stress/strain considers the effect of the entire domain, which produces more
accurate results especially for nano-scale structures.
2) The first moment computed from nonlocal continuum theory based stress distributions is equal to
that from the FEM results which is not the case for classical elasticity.
3) The dispersion curve obtained from nonlocal theory is very close to the Born-von Karman
atomic model.
57
CHAPTER 5. EFFECTS OF NONLOCAL SMALL SCALE PARAMETERS ON BEHAVIOR OF CARBON NANOTUBE BEAMS
5.1. INTRODUCTION
Modeling Carbon nanotube (CNT) reinforced composites using nonlocal theory is described in
Chapter 4. Recently researchers have applied nonlocal theory for modeling of characteristic nano-
devices such as nano-sensors. Most of the researchers model nano-sensors as beams (cantilever
beams) and the formulation is obtained by combining Euler-Bernoulli or Timoshenko beam with
nonlocal theory (Lu et al., 2007; Peddieson et al., 2003; Wang et al., 2008). Comprehensive studies
on modeling CNT composite beams using nonlocal theory including analytical solutions for
bending, vibration and buckling of beams were reported by Reddy (2007) and Reddy and Pang
(2008). The nonlocal analytical model was applied to simply supported, cantilever, propped
cantilever, and clamped beams. Beam deflection, buckling load and natural frequency decreased in
all cases by using nonlocal theory (except increasing beam deflection in cantilever beams). Kumar
et al. (2008) also studied the buckling of CNTs using similar nonlocal one dimensional continua
with Euler-Bernoulli approach. Heireche et al. (2008) and Lu et al. (2007) considered the effects of
small scale characteristics on single wall CNT (SWCNT) based on nonlocal Euler-Bernoulli and
Timoshenko beams. They showed that the dynamic responses of CNT obtained from classical
elasticity are over estimated compared to nonlocal theory results. Wang and Varadan (2006) studied
vibration of SWCNT and double-walled CNT by using nonlocal elastic beam theory. They showed
that nonlocal results match well with the reported experimental results. Wang et al. (2006) used
nonlocal Timoshenko beam theory for elastic buckling analysis of micro and nano-tubes. They
found that the small scale characteristics reduce the buckling loads in nano-tubes.
In this chapter, a sensitivity analysis of the effects of nonlocal small scale parameters on
deflection and bending moment of CNT beam is studied. The first and second order nonlocal
approximation is considered where CNT is modeled as an Euler-Bernoulli nonlocal beam. Various
boundary conditions are considered for modeling of CNT beam such as simply supported,
cantilever, propped cantilever and clamped beam and three applied load conditions are considered
(uniform, sinusoidal and point loads). The effects of small scale parameters on deflections and
bending moment of CNT beam are obtained. The results indicate that the deflection and bending
58
moment of nonlocal beam depend not only the small scale parameters but also on the boundary
condition of the beam and the applied load.
5.2. NONLOCAL EULER-BERNOULLI BEAM MODEL
The equilibrium of forces and moments in the vertical direction of an infinitesimal element of a
beam structure is given as (Wang, 2005):
2
2
twA
xV
∂∂
=∂∂ ρ
(5-1)
Vx
M=
∂∂
(5-2)
where x is the axial coordinate, and V, M, ρ, A, w are resultant shear, bending moment, mass
density, area of cross section, and lateral deflection of beam, respectively. Considering definitions
of the moment in a beam structure, we have
∫= AdAzM σ
(5-3)
2
2
xwz
∂∂
−=ε (5-4)
where z is measured from the mid-plane along beam height. In the absence of axial forces, the
equation of motion based on Euler-Bernoulli theory is given as (Reddy, 2008):
22
4
22
2
02
2
txwm
twmq
xM
∂∂∂
−∂∂
=+∂∂
(5-5)
where q is the transverse force per unit length. The mass inertias m0 and m2 are defined as:
∫ ∫==A A
dAzmdAm 220 , ρρ
(5-6)
Substituting Equations (5-1) and (5-5) into the nonlocal constitutive relation Equation (4-15) leads
to (Reddy, 2008):
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
−∂∂
∂−
∂∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛−
∂∂∂
−∂∂
+∂∂
−= 2
2
42
6
222
4
04
22
4
22
2
02
2
2
xq
xtwm
txwmq
txwm
twm
xwEIM γε
(5-7)
where E is Young’s modulus and I is the moment of inertia. EI is the flexural stiffness of the beam.
By substituting Equation (5-7) into Equation (5-5), we have:
59
2 2 2 2 42
0 22 2 2 2 2 2
2 4 6 2 2 44
0 2 0 22 2 2 2 4 2 2 2 2
w w wEI m m qx x x t x t
w w q w wm m q m mx x t t x x t x t
ε
γ
⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂− + − −⎜ ⎟ ⎜ ⎟
∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂
− − − + = −⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠
(5-8)
The Euler-Bernoulli equation in local elasticity is obtained when the small parameters, ε and γ,
approach zero. The bending moment and deflection of a linear nonlocal beam with constant
material and geometric properties are obtained by setting nonlinear and time derivative terms to
zero.
02
242
2
2
2
2=+⎟⎟
⎠
⎞⎜⎜⎝
⎛+−− q
dxqdq
dxwdEI
dxd γε
(5-9)
where q is an arbitrary function. Integrating Equation (5-9) four times yields shear force, bending
moment, slope of the beam and beam deflection.
13
342
3
3)( cdq
dxqd
dxdq
dxwdEIQ
xE −−=+−−= ∫ ηηγε
(5-10)
212
242
2
2)()( cxcddq
dxqdxq
dxwdEIM
xE −−−=+−−= ∫ ∫ ξηηγε
ξ (5-11)
⎟⎟⎠
⎞⎜⎜⎝
⎛+++++−== ∫ ∫ ∫∫ 32
2
142
2)()(1 cxcxcdddq
dxdqdq
EIdxdw
xx
αξηηγηηεθξ η (5-12)
⎟⎟⎠
⎞⎜⎜⎝
⎛++++++−= ∫ ∫ ∫ ∫∫ ∫ 43
2
2
3
142
26)()()(1 cxcxcxcddddqxqddq
EIw
xxβαξηηγξηηε
ξ η αξ (5-13)
where c1, c2, c3, c4 are integration constants depending on the applied boundary conditions. The
most common boundary conditions are considered for nonlocal beam such as simply supported,
cantilever, propped cantilever and clamped beam. In addition, different loads, q(x), are considered
in this study:
0)( qxq = (5-14)
⎟⎠⎞
⎜⎝⎛=
Lxqxq πsin.)( 0
(5-15)
60
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞
⎜⎝⎛ −
=
0001.02
01.0)(20
Lx
qxq
π (5-16)
( )( )0001.001.0)( 20+−
=Lx
qxqπ (5-17)
Equations (5-14) and (5-15) imply uniformly distributed load and sinusoidal distributed load,
respectively. Equations (5-16) and (5-17) are approximations for point load (delta function) in the
middle and at the end of a beam. The approximations provide adequate accuracy for modeling CNT
because the maximum width of applied point load is two-order of magnitude less than the nano-
length scale which is considered as dimension of atoms or molecules. Equation (5-16) is used for
simply supported, clamped, propped cantilever beams whereas Equation (5-17) is used for
cantilever beam. Reddy and Pang (2008) have presented solutions of Equations ((5-10)-(5-13))
using uniformly distributed load and the first order approximation in nonlocal theory. In this study,
the effects of both nonlocal small scale parameters are considered and solutions are obtained for
different boundary and load conditions.
5.2.1. Simply supported beam
Boundary conditions for a nonlocal simply supported beam are:
2 22 4
2 20, ( ) 0 at 0d w d qw M EI q x x ,Ldx dx
ε γ= = − − + = = (5-18)
The constants of integration are obtained for different load conditions: a) Uniformly distributed load 0q
By applying Equation (5-18), the constants of integration for a nonlocal simply supported beam
under uniform distributed load condition are obtained as:
04
403
02
3201 ,241
21 ,0 ,
21 qcqLqLccLqc γε −=+==−= (5-19)
Thereore, the deflection and bending moment are obtained as:
[ ])12)((24
)( 2220 ε+−+−= xxLLxLxEI
qxw (5-20)
61
( )20
2)( xLx
qxM −= (5-21)
Equations (5-20) and (5-21) show that only the first small scale parameter contributes to the
displacement of nonlocal simply supported beam whereas none of the small scale parameters
contribute to the bending moment. The solution is the same as presented by Reddy and Pang
(2008). The maximum deflection and bending moment occur in the middle of the beam (x=L/2)
given by
[ ]222
0max 485
384ε+= L
EILq
w (5-22)
8
20
maxLq
M = (5-23) Equation (5-22) shows that the first nonlocal small scale parameter increases the deflection of the
beam.
b) Sinusoidal load
The constants of integration for simply support beam under sinusoidal load condition are:
( )0 , ,0 , 43
2220
320
1 =+
==−= cLLq
ccLq
cππε
π (5-24)
The deflection and bending moment are obtained as:
( )4442224
0 sin)( LLLx
EIq
xw ++⎟⎠⎞
⎜⎝⎛= πγπεπ
π (5-25)
⎟⎠⎞
⎜⎝⎛=
LxLq
xM ππ
sin)( 2
20
(5-26) Equations (5-25) and (5-26) show that both small scale parameters contribute to the displacement
whereas neither of the small scale parameters contribute to the bending moment. The maximum
deflection and bending moment at 2Lx = are:
( )4442224
0max LL
EIq
w ++= πγπεπ (5-27)
62
2
20
maxπ
LqM = (5-28) Equation (5-27) indicates that both nonlocal small scale parameters increase the deflection of
nonlocal beam.
c) Point load
The constants of integration for a nonlocal simply support beam using Equation (5-16) load
condition are:
0
1 0 2 4 2
2 2 4
3 0 5 2
3.533480.3183098862arctan(50 ) , 0, 2500L 1
3.97887 10 arctan(50 ) 7.9577 10
1.59155 10 arctan(50 ) 0.31831 arctan(50 )
qc L q c c
L L Lc q
L Lε
− −
−
= − = = −+
⎛ ⎞× − × += ⎜ ⎟⎜ ⎟× +⎝ ⎠
(5-29)
The corresponding bending moment for simply supported beam under the approximated point load
is given as:
⎟⎟⎠
⎞⎜⎜⎝
⎛
+++−
+−−++−=
)2500100001(00159.0)50100arctan(.159155.0
)50100arctan(.3183.0)50arctan(.159.0)12500(00159.0)(
2
2
0LxLLnLxL
LxxLLLLnqxM (5-30)
Equation (5-30) shows that neither of the small scale parameters contribute to the bending moment.
The deflection of nonlocal beam is found a function of both small scale parameters. It is shown in
Section 5.3, the effect of nonlocal small scale parameters increases the deflection of nonlocal beam.
5.2.2. Clamped beam
Boundary conditions for a nonlocal beam clamped at both ends are defined as:
,Lxdxdww 0at 0 ,0 ==== θ
(5-31) The constants of integration are obtained for different loads conditions:
a) Uniformly distributed load 0q The constants of integration are obtained as:
63
04
4322
0201 ,0 ,121 ,
21 qccLqcLqc γε −==⎟
⎠⎞
⎜⎝⎛ +=−= (5-32)
The deflection and bending moment are derived as:
( )EI
Lxxqxw24
)(22
0 −= (5-33)
( )2220 126612
)( LLxxq
xM ++−−= ε (5-34)
Equation (5-33) shows that none of the small scale parameters contributes to the displacement
whereas only the first nonlocal small scale parameter contributes to the bending moment. The
maximum deflection occurs at x=L/2 which is equal to:
EILqw
384
40
max = (5-35)
In a nonlocal clamped beam, the maximum positive bending moment occurs in the middle of the
beam and the maximum negative bending moment occurs at both ends of the beam:
( )
( ) LxLqM
LxLqM
,0 1212
2 24
24220
max
220max
=+−=
=−=⊕
ε
ε
(5-36)
The effect of the first order nonlocal parameter is to increase the maximum negative bending
moment and decrease the maximum positive bending moment for nonlocal beam subjected to a
uniformly distributed load.
b) Sinusoidal load
The constants of integration for clamped beam under sinusoidal load condition are:
( ) 0 , ,2 , 40
4
3442224
320
20
1 =−=++=−= cLqcLL
LqcLqc πγ
γππεππ (5-37)
The deflection and bending moment are given as:
64
( )24
224442220 sin
)(LEI
xxLLLxLLq
xwπ
ππππγπε ⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟
⎠⎞
⎜⎝⎛++
= (5-38)
⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛−−= 442224
230 22sin2)( πγπεππ
πL
LxL
LqxM (5-39)
Equations (5-38) and (5-39) show that both small scale parameters contribute to the displacement
and bending moment. The maximum deflection and positive bending moment at x=L/2 are:
( )( )4442224
0max 4
41 LL
EIq
w ++−−= πγπεππ (5-40)
( )( )44222423
0max 222 πγπεπ
π−−−=⊕ LL
Lq
M (5-41) and the maximum (negative) bending moment occurring at both ends of the beam is:
( ) LxLLL
qM ,0 2 44222423
0max =++−= πγπε
π (5-42) Equations (5-40) and (5-42) show that the nonlocal small scale parameters increase the deflection
and negative bending moment of the nonlocal beam. Equations (5-41) indicates that small scale
parameters decrease the positive bending moment in the middle of the beam.
c) Point load
The constants of integration for clamped beam using Equation (5-16) load condition are given as:
1250053348.3 ,
150001025.68.35334
58.706699577.7184.9947)6366.0102.32387.0
1.3183109788.383.596109736.4)(50arctan(12500
)106)50arctan(31831.0(
20
42460
3
2425
2326355
20
2
12
01
+−=
++×−=
⎟⎟⎠
⎞⎜⎜⎝
⎛
+−−+×++
+×++×
+=
×−−=
−
−
Lqc
LLLqc
LLL
LLLLLL
qc
LLqc
ε
εε (5-43)
It is found that the nonlocal clamped beam’s deflection under a point load is a function of both
small scale parametrs; however, the bending moment is only a function of the first small scale
65
parameter. The numerical results for the point load conditions are presented in section 5.3, however
the analytical results were not presented here because they are very large.
5.2.3. Cantilever beam
Boundary conditions for a nonlocal cantilever beam are written as:
LxVM
xdxdww
===
====
at 0 ,0
0at 0 ,0 θ
(5-44) The constants of integration are obtained based on the applied load conditions as follows.
a) Uniformly distributed load 0q The constants of integration for nonlocal cantilever beam under uniformly distributed load are
obtained as:
04
432
0201 ,0 ,21 , qccLqcLqc γ−===−= (5-45)
The deflection and bending moment are given as:
( )2222
0 124624
)( ε−−+= LxxLEIxqxw (5-46)
( )20
2)( Lx
qxM −−= (5-47)
Equation (5-46) shows that only the first small scale parameter contributes to displacement whereas
small scale parameters do not contribute to the bending moment. The maximum deflection occurs at
x=L, whereas the maximum bending moment occurs at x=0:
( )222
0max 4
8ε−= L
EILqw (5-48)
2
20
maxLq
M −= (5-49)
66
Equation (5-48) shows that the first small scale parameter decreases the deflection of nonlocal
cantilever beam under uniformly distributed load; however, the nonlocal small scale parameter has
no effect on the bending moment.
b) Sinusoidal load
The constants of integration for a cantilever beam under sinusoidal load condition are:
0 , , ,24
04
3
20
20
1 =−==−= cLqcLqcLqc πγ
ππ (5-50)
The deflection and bending moment are given as:
( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛−++−= LxxL
LxLxLL
LEIq
xw 3sin6666
)( 2234442224
0 ππππγπεπ (5-51)
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎠⎞
⎜⎝⎛+= L
LxLxLqxM πππ
πsin)( 2
0 (5-52)
Equation (5-51) shows that both small scale parameters contribute to the displacement of a
cantilever beam under sinusoidal load whereas no small scale parameter contributes to the bending
moment. The maximum deflection and bending moment are:
( ) LxLLEI
qw =−−−= 33)3(
34422224
30
max πγπεππ (5-53)
0 2
0max =−= x
LqM
π (5-54) Equation (5-53) indicates that both the first and the second nonlocal small scale parameters
decrease the deflection of the beam.
c) Point load
The constants of integration for cantilever beam using Equation (5-17) as an approximation of a
point load condition are:
67
110000L
53348.3 ,120000101
59.70669,)100arctan(3183.0)1 10000( 00159.0 ,)100arctan(3183.0
20
42480
3
02
0201
+−=
++×−=
++−=−=qc
LLLqc
LqLLLnqcqLc
(5-55)
The bending moment is obtained by using Equations (5-11), (5-17) and (5-55).
( ))1000020000100001(00159.0)100100arctan()(3183.0)( 220 LxLxLnLxLxqxM +−+++−−= (5-56)
Equation (5-56) shows that the small scale parameters do not affect the bending moment. In
addition, it is found that the deflection of nonlocal cantilever beam under a point load is a function
of both small scale parameters. The analytical deflection are not presented here because the
equations are very large, however the numerical results are presented later in this chapter.
5.2.4. Propped cantilever beam
Boundary conditions for a nonlocal propped cantilever beam at both ends of the beam are:
LxwM
xdxdww
===
====
at 0 ,0
0at 0 ,0 θ
(5-57) The constants of integration are obtained using different load conditions as follows.
a) Uniformly distributed load 0q The constants of integration of a nonlocal propped cantilever beam under a uniformly distributed
load are obtained as:
( ) ( ) 0
443
2202
220
1 ,0 ,128
,8
512qccL
qc
LLq
c γεε
−==−=+
−= (5-58)
The deflection and bending moment are given as:
( ))1223)(( 48
)( 222
0 ε+−−= xLLxLLEI
xqxw (5-59)
( ))124)((8
)( 220 ε+−−−= xLLxLL
qxM
(5-60)
68
Equations (5-59) and (5-60) show that the only first small scale parameter contribute to both
displacement and bending moment. The location of maximum deflection and positive bending
moment are given by Equations (5-61) and (5-62), respectively:
LLLLx
161296312331536)deflection(
422422
maxεεε ++−+
= (5-61)
LLx
8512)moment bending(
22
max+
=ε
(5-62) The location of maximum deflection and positive bending moment are a function of the first order
small scale parameter and the results obtained here are different from Reddy and Pang (2008).
Reddy and Pang (2008) did not consider the effects of small scale parameter, ε, on the location of
the maximum deflection and bending moment. On the other hand, they considered the location of
maximum bending moment and deflection as the same as the one in classical elasticity. In
Equations (5-61) and (5-62) when the first small scale parameter approaches zero, the location of
maximum deflection and positive bending moment approach those obtained in classical elasticity.
In addition, the maximum bending moment occurs at x=0. The maximum deflection and bending
moment at their corresponding locations are derived as:
( ) ( )4224
2424222224
0max
33312 1296 where
52112432241536 786432
LL
LLLLLEI
qw
++=
+++−−+=
εεκ
κκεεεκε (5-63)
( )42242
0max 816
1289 LL
LqM +−=⊕ εε (5-64)
( )220max 12
8ε+−= L
qM (5-65)
Equations (5-63), (5-64) and (5-65) show that the first small scale parameter affects the deflection
and bending moment of nonlocal beam.
b) Sinusoidal load
The constants of integration for propped cantilever beam under sinusoidal load condition are given
as:
69
( ) ( )
0 ,
,3
,333
40
4
3
44422232
02
4444222223
01
=−=
++=+++−=
cLq
c
LLL
qcLLL
Lq
c
πγ
γππεπ
γπππεπ
(5-66)
The deflection and bending moment are given as:
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−+−⎟
⎠⎞
⎜⎝⎛++= πππππγπε
π3223444222
340 32sin2
2)( xLxxL
LxLLL
LEIq
xw (5-67)
⎟⎟⎠
⎞⎜⎜⎝
⎛−−−+++⎟
⎠⎞
⎜⎝⎛= 5442324442225
330 333333sin)( LLLxLxLx
LxL
Lq
xM γππεγππεπππ (5-68)
Equations (5-67) and (5-68) show that both small scale parameters contribute to the displacement
and the bending moment. The location of the maximum deflection and the maximum positive
bending moment are obtained as:
Lx 5725362.0)deflection(max =
(5-69)
ππ
γππεπ ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ ++−
=42
444222
max
(3arccos
)moment bending(L
LLL
x (5-70)
The location of maximum negative bending moment is x=0. Equation (5-69) gives the location of
maximum deflection as identical to the classical elasticity. The maximum deflection, maximum
negative bending moment and maximum positive bending moment for propped cantilever nonlocal
beam are:
( )4442224
0max
4244.0LL
EIq
W ++= πγπεπ (5-71)
)(3 44422223
0max LL
LqM ++−= γππε
π (5-72)
444222
42284
220
max
where
3 arccos39
LL
LL
Lq
M
++=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛−−=⊕
γππεη
πηηηπ
π (5-73)
70
The nonlocal small scale parameters affect beam deflection and bending moment. Equation (5-71)
shows the small scale parameters increase the deflection of the nonlocal beam.
c) Point load
The constants of integration for propped cantilever beam using Equation (5-16) as an
approximation for point load are obtained as:
( )
( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−−
×
+−⎟
⎟⎠
⎞⎜⎜⎝
⎛×−×−
×−
+
++
⎟⎟⎠
⎞⎜⎜⎝
⎛ ×+−−−×−−×−
+=
−−−
−
LLLL
qLL
LL
LLnq
LLLL
L
Lqc
936.1178.149201006.1
1250010875.1105.1103
12500
)12500(
1077.4955.0676.065.477410968.58.2486107355.212500
)50arctan(
35
220369
13
22
20
2
522226246
220
1εεε
( )
( )( )
( )( )
1250053.3
,150001025.6
8.35334
936.1178.149201006.112500
10875.1105.110312500
)12500(
1077.4955.065.477410968.5358.025.8951046.712500
)50arctan(
20
42460
3
24522
046291322
20
522236355
22
02
+−=
++×−=
++×−+
−×−×+×+
++
⎟⎟⎠
⎞⎜⎜⎝
⎛ ×+++×+++×
+=
−−−
−
Lq
cLL
Lqc
LLL
qLL
L
LLnq
LLLLLL
L
Lqc εεε
(5-74)
It is shown in Section 5.3 that both small scale parameters contribute to deflection and bending
moment of nonlocal propped cantilever beam under a point load.
5.3. NUMERICAL RESULTS
Numerical results are presented for a carbon nanotube. The material properties of the carbon nanotube are the same as Reddy and Pang (2008).
4384
9
3
m1091.464
m,100.1 ,19.0 GPa, 1000
nm 142.0,kg/m 2300
−− ×==×===
==
dIdE
a
πν
ρ
Where, ρ is the density, a is the internal characteristic length which is the length of a C_C bond, E,
ν, d, I are Young’s modulus, Poisson ratio, diameter and moment inertia of CNT, respectively. The
maximum deflection of nonlocal beams is computed for simply supported, cantilever and propped
cantilever cases under uniform load distribution. Figure (5-1) shows the results using normalized
length. In Figure (5-1), x is the location of the displacement, ε=ε0a (nm) is the first nonlocal small
71
scale parameter and w is the deflection of the beam. As shown in Figure (5-1), under the uniformly
distributed load, the first nonlocal small scale parameter increases the deflection of simply
supported, clamped and propped cantilever beam whereas decreases the deflection of cantilever
beam.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8
9
Coordinate, x/L
w(x
) (nm
)
Local elasticityε0a 0.05 (nm)
ε0a 0.1 (nm)
ε0a 0.15 (nm)
ε0a 0.2 (nm)
0 0.2 0.4 0.6 0.8 10
10
20
30
40
50
60
Coordinate, x/L
w(x
) (nm
)
Local elasticityε0a 0.05 (nm)
ε0a 0.1 (nm)
ε0a 0.15 (nm)
ε0a 0.2 (nm)
a)
b)
72
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
3.5
Coordinate, x/L
w(x
) (nm
)
Local elasticityε0a 0.05 (nm)
ε0a 0.1 (nm)
ε0a 0.15 (nm)
ε0a 0.2 (nm)
Figure (5-1): Deflection of nonlocal beam under uniform load distribution
a) Simply supported b) Cantilever c) Propped cantilever
The maximum deflection and bending moment for nonlocal beam are compared with the local
elasticity results. The nonlocal results are obtained for different values of the small scale
parameters. Equations (5-22), (5-48) and (5-63) indicate that only the first small scale parameter
affects the deflection of the nonlocal beam. Figure (5-2) shows the ratio of maximum deflection of
nonlocal beam to that of local (classical) elasticity beam with the variation of small scale parameter.
c)
73
0 0.05 0.1 0.15 0.20.8
0.9
1
1.1
1.2
1.3
1.4
ε0a
w max
(non
loca
l)/ w
max
(loca
l)
Simply SupportedClampedCantileverPropped Cantilever
Figure (5-2): Ratio of maximum deflection of nonlocal beam to maximum deflection in local
elasticity under uniform load distribution
As shown in Figure (5-2), the first small scale parameter in simply supported and propped
cantilever beams increases the deflection of the nonlocal beam whereas it decreases the deflection
of cantilever beam. In addition, the nonlocal small scale parameter has no effect on the deflection of
clamped beam.
0 0.05 0.1 0.15 0.2
1
1.1
1.2
1.3
1.4
1.5
ε0a
Mm
ax(n
onlo
cal)/
Mm
ax(lo
cal)
Simply SupportedCantileverPropped CantileverClamped
Figure (5-3): Ratio of maximum bending moment of nonlocal beam to maximum bending moment
in local elasticity under uniform load distribution
(nm)
(nm)
74
As shown in Figure (5-3), the nonlocal small scale parameters have no effect on the bending
moment of simply supported and cantilever beams. The first small scale parameter increases the
maximum (negative) bending moment of propped cantilever and clamped beam significantly. As
shown in Figure (5-4), the first small scale parameter decreases the positive bending moment in
propped cantilever and clamped beams whereas it has no effect on the maximum positive bending
moment of simply supported beam. Figure (5-4) shows that the first small scale parameter has a
significant effect on the positive bending moment of nonlocal clamped and propped cantilever
beams.
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
1
ε0a
Mm
ax+(n
onlo
cal)
/Mm
ax+(lo
cal)
Simply SupportedClampedPropped Cantilever
Figure (5-4): Ratio of maximum positive bending moment in nonlocal beam to maximum positive
bending moment in local elasticity beam under uniform load distribution
In contrast to the uniformly distributed load condition, both first and second small scale parameters
affect the behavior of nonlocal beams under the sinusoidal load condition. By considering
Equations (5-27), (5-40) and (5-70), the ratio of maximum deflection of nonlocal beam to
maximum deflection of local elasticity beam for simply supported, clamped and propped cantilever
cases are exactly the same. Figure (5-5) shows the effects of nonlocal small scale parameters on the
behavior of the nonlocal beams.
(nm)
75
Figure (5-5): Ratio of maximum nonlocal deflection to maximum deflection in local elasticity
(identical results for simply supported, clamped and propped cantilever beams)
As shown in Figure (5-5), the first small scale parameter has greater influence on the deflection of
nonlocal beams compared to the second small scale parameter. In addition, both small scale
parameters increase the deflection of nonlocal beam under sinusoidal load condition.
Figure (5-6): Ratio of maximum deflection of nonlocal cantilever beam to local elasticity beam
under sinusoidal load condition
As shown in Figure (5-6), both small scale parameters decrease the deflection of nonlocal cantilever
beam under sinusoidal load condition wherein the second small scale parameter has lower effect
compared to the first small scale parameter. As presented in Equations (5-28) and (5-54), the
76
nonlocal small scale parameters do not have any effect on the bending moment of simply supported
and cantilever beam under sinusoidal load condition. The ratio of nonlocal maximum bending
moment to local elasticity beam for clamped and propped cantilever cases are exactly the same as
shown in Figure (5-7).
Figure (5-7): Ratio of maximum bending moment in nonlocal beam to local elasticity beam under
sinusoidal load condition (identical results for propped cantilever and clamped cases)
Figure (5-8): Effects of nonlocal small scale parameters on the ratio of maximum positive bending moment in nonlocal theory to local elasticity under sinusoidal load condition for
(a) Propped cantilever beam, and (b) Clamped beam
a) b)
77
The ratio of the maximum positive bending moment (at their corresponding locations) in clamped
and propped cantilever beams is shown in Figure (5-8). Figures (5-8)(a) and (5-8)(b) show that the
both small scale parameters reduce the maximum positive bending moment in the clamped and
propped cantilever beams under sinusoidal loads. Under the point load condition, the bending
moment of nonlocal clamped and propped clamped beam is only a function of the first small scale
parameter. The small scale parameters do not have any effect on the bending moment of nonlocal
simply supported and cantilever beams under approximated point load.
0 0.05 0.1 0.15 0.20.95
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
ε0a (nm)
Mm
ax(n
onlo
cal)/
Mm
ax(lo
cal)
Simply SupportedCantileverClampedPrepped Cantilever
Figure (5-9): Ratio of maximum bending moment in nonlocal beam to local elasticity beam under
approximated point load condition
As shown in Figure (5-9), the first nonlocal small scale parameter has a considerable effect on the
bending moment of clamped and propped cantilever beams under the approximated point load
condition whereas it does not affect simply supported and cantilever beams. In addition, both small
scale parameters affect the displacement of nonlocal beam under the point load condition.
78
Figure (5-10): Effects of nonlocal small scale parameters on ratio of maximum deflection in
nonlocal beams to local elasticity beams under approximated point load condition (a) Simply supported beam b) Clamped beam c) Cantilever beam d) Propped cantilever beam
As shown in Figure (5-10), the effects of nonlocal small scale parameters depend on the applied
boundary conditions. In nonlocal simply supported and cantilever beams, the second small scale
parameter has greater effect compared to the first small scale parameter. In contrast, the effect of
the first small scale parameter is much larger in nonlocal propped cantilever beams.
5.4. CONCLUSIONS
In this research, the effects of nonlocal small scale parameters are studied in CNT beams. The
equations for deflection and bending moment are obtained using nonlocal theory considering CNT
as Euler-Bernoulli beam. The static deformation of CNT beam is obtained through nonlocal theory
using both the first and second-order approximation in nonlocal theory. The atomistic length scale
parameters show considerable effects on the response of CNT beams. It is observed that the
a) b)
c) d)
79
response depends not only on scale parameters, but also on loads and applied boundary conditions.
The results are summarized as:
a) Under uniformly load condition:
it is shown that the first nonlocal small scale parameter increases the deflection of nonlocal
simply supported, clamped and propped cantilever beam whereas it decreases the deflection
of cantilever beam. In addition, the first small scale parameter increases the maximum
(negative) bending moment and decreases the maximum (positive) bending moment of
propped cantilever and clamped beams.
b) Under sinusoidal load condition:
Both the first and the second small scale parameters affect the behavior of nonlocal beams.
The nonlocal small scale parameters increase the deflection of simply supported, clamped
and propped cantilever beams whereas they decrease the deflection of cantilever beams.
The nonlocal small scale parameters do not have any effect on the bending moment of
simply supported and cantilever beams. However, they increase the maximum negative
bending moment and decrease the maximum positive bending moment in clamped and
propped cantilever beams.
c) Under point load condition:
Both nonlocal small scale parameters increase the deflection of the beams. The first small
scale parameter compared to the second small scale parameter has much larger effect on
deflection of propped cantilever beam whereas the second small scale parameter has a
larger effect on the deflection of simply supported and cantilever beams. The first nonlocal
small scale parameter increase the bending moment of nonlocal clamped and propped
cantilever beam whereas it does not affect on simply supported and cantilever beams.
80
CHAPTER 6. FIRST- AND SECOND-ORDER NONLOCAL BEAM MODELS FOR CARBON NANOTUBE
6.1. INTRODUCTION
The nonlocal small scale length effect becomes significant in modeling of nanostructures. However,
the identification of the nonlocal small scale parameters for CNT is not fully resolved. Researchers
so far used the first order approximation of nonlocal theory for modeling CNT. Wang et al. (2008)
estimated nonlocal stiffness of CNT using MD simulation. The small length scale parameters in
nonlocal theory, constant appropriate to each material, for the first order approximation in nonlocal
theory is obtained through comparison between nonlocal and MD simulation results. Similarly,
Duan et al. (2007) studied the effects of the small scale parameter, 0ε , on free vibration of single-
walled carbon nanotube (SWCNT). Their findings indicate that for matching MD simulation and
nonlocal theory results, the value of small scale parameter ranges between 0 and 19 depending on
SWCNT aspect ratio and boundary conditions. Zhang et al. (2005) calculated small scale parameter
to be about 0.82 and studied the effects of small scale parameter in free vibration of double-walled
carbon nanotube. They also compared molecular mechanics simulation with nonlocal buckling
analysis of SWCNT. In addition, for simply supported boundary condition, they found that the
effect of small scale parameters is related to the vibrational mode and the aspect ratio of CNT.
Zhang et al. (2006) used the MD simulation for evaluating 0ε , and they found 043.1 546.0 0 ≤≤ ε for
different chiral angles of SWCNTs. Wang and Hu (2005) proposed 288.00 =ε by using second-
order strain gradient in elasticity theory and MD simulation. Eringen (1983) determined that using
the first order approximation of nonlocal theory, with 39.00 =ε , leads to a close match with the
atomic model. As reported in Eringen (1983, 1987), the maximum deviation in dispersion curve
between the first order approximation of nonlocal theory and the Born-von Karman theory is less
than 6%. In addition, Eringen (1983) compared the dispersion curve of lattice dynamic modeling
with Rayleigh surface wave and obtained 31.00 =ε .Wang and Wang (2007) compared the gradient
method ( 288.00 =ε )(Wang and Hu, 2005) with nonlocal first order method ( 385.00 =ε ). The
gradient method shows very close agreement with Born-Karman only at smaller values of ka, where
k and a are the wave number and lattice parameter respectively. Wang (2005) estimated
nm 1.20 <aε in SWCNT by investigation of wave propagation with wave frequency value greater
81
than 10 Hz. Although, 0ε , key parameter in nonlocal theory, finding the rigorous estimation of
nonlocal small parameter for CNT under investigation.
Recent applications of nonlocal theory for modeling structural beam and plate elements in nano-
scale (representing nano-sensors) were reported in (Lu et al. 2007; Peddieson et al., 2003). Nano-
sensors are modeled as beams and the formulation is obtained using the nonlocal form of Euler-
Bernoulli or Timoshenko beam theory (Lu et al. 2007; Peddieson et al. 2003; Wang et al. 2008).
Comprehensive studies on modeling CNT composite beams using nonlocal theory including
analytical solutions for bending, vibration and buckling of beams were reported by Reddy (2007)
and Reddy and Pang (2008). The nonlocal analytical model was applied to simply supported,
cantilever, propped cantilever, and clamped beams. Beam deflection, buckling load and natural
frequency decreased in all cases by using nonlocal theory (except increasing beam deflection in
cantilever beams). Kumar et al. (2008) also studied the buckling of CNTs using similar nonlocal
one dimensional continua with Euler-Bernoulli approach.
In this chapter, analytical expressions for the deflection of a cantilever beam were derived using
the first and the second order approximation of nonlocal theory. The results were applied to carbon
nano-tube (CNT), which was modeled as an Euler-Bernoulli beam. The nonlocal material
parameters were estimated by comparing the nonlocal model predictions for the natural frequencies
with the experimental data. While the earlier works with the use of classical elasticity showed that
the estimated value of elastic modulus varies significantly with shape/aspect ratio of CNT. Here
the presented results showed that with the use of nonlocal theory with one additional parameter,
more universal material constants including the modulus of elasticity for CNT may be estimated.
Having universal material constants for nano-scale materials will provides a strong tool for
designing nanostructures.
6.2. NATURAL FREQUENCY OF NONLOCAL CNT BEAM
Using Equations (5-4), (5-5), (5-6), (5-7) into the nonlocal constitutive relation given by Equation
(4-2) leads to the following nonlocal Euler-Bernoulli model (Wang, 2005):
0)( 4
44
2
22
2
2
4
4=
∂
∂+
∂
∂−
∂
∂+
∂
∂
xw
xww
tA
xwEI γερ
(6-1)
Natural frequency of CNT, similar to local elastic beam (Meirovich, 1986), is obtained by using
Equation (6-1). By considering 2 EI Aκ ρ= , Equation (6-1) is written as:
82
0)( 4
44
2
22
2
2
4
42 =
∂
∂+
∂
∂−
∂
∂+
∂
∂
xw
xww
txw γεκ
(6-2)
Equation (6-2) is a partial differential equation of fourth order in space and second order in time.
Equation (6-2) is solved using separation of variables method, in which the deflection w is a
function of two independent functions:
( , ) ( ). ( )w x t X x T t= (6-3)
By separating variables X and T, Equation (6-2) is written as:
2
2
242 κω
κγε=−=
+′′− TT
)XXX(X
IV
IV &&
(6-4)
where 44 dxXdX IV = , and 22 dtTdT =&& , the space state and time space differential equations of
Equation (6-4) is rewritten as:
0422
2
=+′′−− )XXX(X IVIV γεκω
(6-5)
02 =+ TPT&& (6-6)
Equation (6-5) must satisfy the boundary conditions and the solution of Equation (6-6) is given as:
tsinBtcosAT ωω += (6-7)
where A, B, and ω are constants and ω is the angular frequency.
6.2.1. First Order Approximation Equation (4-2) presents the differential equation form of nonlocal theory involving infinite
series and higher order derivatives. The parameters ε and γ not only are small but also are
proportional to the internal length scale of the nonlocal media. For evaluating the transverse
vibration of a beam using nonlocal theory, Equation (6-5) can be approximated by considering only
the first term (ε term). This approximation of the nonlocal theory has been used by several
researchers (Meo and Rossi, 2006; Ahmadi and Farshad, (1973); Lu et al., 2007; Eringen, 1972a;
Ahmadi, 1975; Xiong et al., 2007; Wang et al., 2008; Duan et al., 2007; Zhang et al., 2005, 2006;
83
Wang and Hu, 2005; Peddieson et al., 2003; Wang et al., 2008; Reddy, 2007; Reddy and Pang,
2008; Kumar et al., 2008). The first order approximation leads to
0)( 22
2=′′−− XXX IV ε
κω
(6-8)
The solution of Equation (6-8) can be obtained by considering xeX λ= :
02
22
2
224 =−+
κωλ
κωελ (6-9)
The solution of Equation (6-9) is obtained by:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+±
−= 2
2422 4
2 κωε
κωε
κωλ
(6-10)
Because ε is a small parameter, the fourth power of ε in Equation (6-10) is neglected. Therefore,
Equation (6-10) is simplified to:
⎟⎟⎠
⎞⎜⎜⎝
⎛±
−±= 2
2
2
κωε
κωλ
(6-11)
By using perturbation around ε, the general solution of Equation (6-8) is written as:
)sinh()cosh()sin()cos()( 24231211 xCxCxCxCxX λλλλ +++= (6-12)
where
⎟⎟⎠
⎞⎜⎜⎝
⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛+=
κωε
κωλ
κωε
κωλ
41 ,
41
2
2
2
1 (6-13)
and C1, C2, C3, C4 are constants, which can be obtained depending on the applied boundary
conditions. For a cantilever beam, the deflection and slope at x=0 must be zero:
0)0( 31 =+= CCX (6-14)
0)0( 4221 =+=′ CCX λλ (6-15)
Bending moment and shear at the end of the beam (x=L) are equal to zero:
84
0)sinh()cosh()sin()cos()( 22 242
2 231
2 121
2 11 =++−−=′′ LCLCLCLCLX λλλλλλλλ (6-16)
0)cosh( )sinh( )cos( )sin( )( 2
3 242
3 231
3 121
3 11 =++−=′′′ LCLCLCLCLX λλλλλλλλ (6-17)
By substituting Equations (6-14), (6-15) into Equations (6-16), (6-17), we have:
0))cosh()cos(())sinh()sin((
0))sinh()sin(())cosh()cos((
212 21
3 122
3 21
3 11
21212
1222 21
2 11
=+−−
=+++
LLCLLC
LLCLLC
λλλλλλλλλ
λλλλλλλλλ
(6-18)
The only nontrivial solution of Equation (6-18) can be obtained when the following determinant is
equal to zero:
0))cosh()cos(()sinh()sin(
)sinh()sin()cosh()cos(
22 21
2 12
3 21
3 1
221122 21
2 1 =
+−−++
LLLLLLLLλλλλλλλλ
λλλλλλλλ
(6-19)
Substituting Equation (6-13) into Equation (6-19), we have:
0)sinh()sin(2
)cosh()cos(1 21
2
21 =−+ LLPLL λλκ
ελλ (6-20)
Assuming Lκωη = and a0εε = , Equation (6-20) can be restated as:
0))4
)(1(sinh())4
)(1(sin(2
)())4
)(1(cosh())4
)(1(cos(122
022
022
022
022
0 =−+−−++ηεηηεηηεηεηηεη LaLaLaLaLa
(6-21)
Equation (6-21) is the characteristic equation of a nonlocal cantilever beam. The solution of
Equation (6-21) must be obtained numerically which will provide a set of eigenvalues for the
system (Meirovich, 1986). The eigenvalues of Equation (6-21) are referred to as harmonic
constants. The first nonlocal harmonic constant which is a function of small scale parameter is
shown in Figure (6-1). The corresponding equation representing the relationship between the first
harmonic constant, η, and small scale parameter, ε0a, is obtained using polynomial curve fitting on
the data.
85
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.41.6
1.65
1.7
1.75
1.8
1.85
1.9
1.95
η
ε0a/L
η= 2.458*(ε0a/L)3 - 2.436*(ε0a/L)2 + 0.006707*(ε0a/L) + 1.875
Figure (6-1): Effect of the small scale parameter on nonlocal beam first harmonic constant
Figure (6-1) shows the effect of small scale parameter on the first harmonic constant of nonlocal
cantilever beam. When the small scale parameter approaches zero, the value of harmonic constant
is equal to the one obtained from classical elasticity. The natural frequency of nonlocal beam is
obtained using angular velocity in Equation (6-7):
2
2
1 2 ,2
Lff
πκηπω ==
(6-22)
where f1 is the first natural frequency of the cantilever beam as predicted by the first order
approximation to the nonlocal model. Substituting AEI ρκ =2
in Equation (6-22), we have:
AEI
Lf
ρπη
2
2
12
= (6-23)
For a tube with uniform structural and mass distribution, the natural frequency then is given as:
ρπη EDD
Lf
)(8
21
2
2
2
1+
= (6-24)
Here D is the tube outer diameter, D1 is the inner diameter, L is the length, ρ is the density, and E is
the elastic modulus.
86
Figure (6-2) shows the ratio between the harmonic constant using the first order approximation with
the local elasticity. It is seen that the natural frequency as predicted by the nonlocal model is
generally lower than that of the elastic beam. When the small scale parameter approaches zero, the
frequency ratio approaches one.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.75
0.8
0.85
0.9
0.95
1
η2 no
nloc
al/ η
2 elas
tic
ε0a/L
Figure (6-2): Ratio of the first natural frequencies of a cantilever beam as predicted by the first order approximation of nonlocal theory to that of local elasticity.
6.2.2. Second Order Approximation Most of the researchers have used the first order approximation (involving second order differential
equation) of nonlocal theory for modeling CNT (Meo and Rossi, 2006; Ahmadi and Farshad,
(1973); Lu et al., 2007; Eringen, 1972a; Ahmadi, 1975; Xiong et al., 2007; Wang et al., 2008; Duan
et al., 2007; Zhang et al., 2005, 2006; Wang and Hu, 2005; Peddieson et al., 2003; Wang et al.,
2008; Reddy, 2007; Reddy and Pang, 2008; Kumar et al., 2008) Lazar et al. (2006) modeled the
nonlocal elasticity of bi-Helmholtz type using the fourth order differential equation. They assumed
the nonlocal small parameters to be a function of wave velocity and evaluated the parameters by
matching the dispersion curves with the atomic model. They also showed that considering the
fourth order differential equation yields more accurate results. Alavinasab et al. (2008) used the
second order approximation of nonlocal theory for modeling a CNT composite where only the
second term in Equation (4-2) is retained. They showed that not only the dispersion curve of
nonlocal second order approximation is close to atomistic model, but also the resultant moment of
nonlocal stress distribution across a section has the correct trend of variation. Here, we used the
87
same second order approximation for the Euler-Bernoulli beam theory. Accordingly, Equation (6-5)
is written as:
0)( 42
2=+− IVIV XXX γ
κω
(6-25)
Similar to the first order approximation, the solution of Equation (6-25) can be obtained by
considering eigenfunction xeX λ= , which leads to the following dispersion equation,
0)1( 2
244
2
2=−−
κωλγ
κω
(6-26) The corresponding analytical solution to Equation (6-25) is given by:
)sinh()cosh()sin()cos()( 4321 xCxCxCxCxX λλλλ +++= (6-27)
where
422
24
γωκωλ−
= (6-28)
By applying boundary condition for the cantilever beam similar to Equations (6-14)-(6-19), the
nontrivial solution for second order approximation for the nonlocal beam is obtained when the
following characteristic function is satisfied:
0)cosh()cos(1 =+ LL λλ (6-29)
Equation (6-29) is identical to the one obtained for an elastic cantilever beam. The first harmonic
solution (eigenvalue) of Equation (6-29) is 875.1=Lλ . Using Equations (6-22), (6-23) and (6-29),
the fundamental natural frequency of the cantilever beam as predicted by the second order
approximation to nonlocal theory is given as:
42
2
2875.11
1.2875.1
⎟⎠⎞
⎜⎝⎛+
=
L
AEI
Lf
γρπ
(6-30)
88
Noting that a0γγ = , the natural frequency of CNT cantilever beam based on the second order
approximation to nonlocal theory, f2, can be written as a function of inner and outer diameter,
length, density, and nonlocal modulus of elasticity as:
40
21
2
2
2
2875.11
1.)(8875.1
⎟⎠⎞
⎜⎝⎛+
+=
La
EDDL
fγρπ
(6-31)
Equation (6-31) is similar to the natural frequency of an elastic beam, however involved the effect
of the internal scale parameter. The ratio of the first natural frequency as predicted by the second
order nonlocal theory to that of elastic beam is shown in Figure (6-3). It is seen that the natural
frequency as predicted by the nonlocal model is somewhat lower than that of the elastic beam. Also
when the small scale parameter approaches zero, the frequency ratio approaches one.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.975
0.98
0.985
0.99
0.995
1
1.005
η2 no
nloc
al/ η
2 elas
tic
γ0a/L
Figure (6-3): Ratio of the first natural frequencies of a cantilever beam as predicted by the second order approximation of nonlocal theory to that of local elasticity
By comparing Figure (6-2) and (6-3), it is shown that the first small scale parameters has more
effect on the reduction of the natural frequency of the cantilever beam when compared to the
second small scale parameter.
89
6.3. ESTIMATION OF NONLOCAL MODEL PARAMETERS
The natural frequencies of a cantilever CNT beam based on the first and the second order
approximations of nonlocal theory are given by Equations (6-24) and (6-31), respectively. In this
section, the elastic modulus and the nonlocal length scale parameters for CNT are estimated by
comparing the model predictions with the experimental data. Gao et al. (2000) experimentally
evaluated natural frequency of a single CNT cantilever beam excited by a sinusoidal time varying
excitation applied at its root. Their experimental data are reported in Table (6-1).
Table (6-1): Experimental data of Gao et al., (2000)
Outer diameter
Inner diameter Length Aspect
ratio Frequency Modulus of
Nanotube D (nm) D1 (nm) L (μm) Τ fnatural (MHz)
Elasticity (GPa)
1 33 18.8 5.5 212 0.658 32 2 39 19.4 5.7 195 0.644 26.5 3 39 13.8 5 189 0.791 26.3 4 45.8 16.7 5.3 170 0.908 31.5 5 50 27.1 4.6 119 1.42 32.1 6 64 27.8 5.7 124 0.968 23
In Table (6-1), aspect ratio is defined as the ratio of length to average diameter (average of inner
and outer diameter) of CNT. The modulus of elasticity is obtained from the classical elasticity
equations (Gao, 2000; Meirovich, 1986). It is show from Table (6-1) that that the estimated elastic
modulus of CNT shows large variation with geometry (aspect ratio). The disparities in modulus of
elasticity in the data range between 23 GPa to 32GPa. The significant range of variation of the
estimated elastic modulus further confirms that the simple elastic theory cannot properly describe
the vibration of the cantilever CNT, and the use of the higher level nonlocal theory is appropriate.
Table 1 also shows that the aspect ratio of the CNT studied varies from 119 to 212, so that a range
aspect ratios are covered.
Estimates of the model parameters from the data are obtained using an optimization technique.
Optimization objective functions for the first and the second order approximations to the nonlocal
model are defined as:
90
∑=
+−
n
i
EiDiDL
if1
21
2
2
2
.exp)()((
8)(
ρπη
(6-32)
∑= +
+−
n
i
La
EiDiDL
if1 40
21
2
2
2
.exp
)875.1(1
1.)()((8875.1)(
γρπ (6-33)
where in Equations (6-32) and (6-33), n is the total number of experimental samples. Optimization
constraints for first and second order approximations are
nmaEnmaE
nonlocal
nonlocal
1.20 ,380 1.20 ,380
0
0
≤≤≤≤≤≤≤≤
γε
(6-34)
The upper bond for nonlocal small scale parameters are chosen less than 2.1 nm similar to the one
proposed by Wang (2005). Parameter a is the internal characteristic length-scale. In CNT, the
length of a C–C bond is equal to 0.142 nm, which is chosen for the internal characteristic length-
scale. The nonlinear constraint minimization in MATLAB was used for the optimization algorithm.
The optimization results for the first order nonlocal model are shown in Table (6-2).
The nonlocal small scale parameter, ε0, is in the range of 0 to 19, which was suggested by Duan
(2007) based on MD simulations. Using the values of the material constants as given in Table (6-
2), the natural frequencies of the cantilever CNT are evaluated and are compared with the measured
values in Table (6-3). It is seen the predicted natural frequencies are in reasonable agreement with
the experimental data. The differences between the experimental natural frequencies and the
predicted results are also shown in Table (6-3). The maximum differences between the calculated
and the experimentally measured natural frequency are less than 13%. The average of the
differences in natural frequencies is about 6%.
Table (6-2): Nonlocal features for CNT using the first order approximation Case 1
nonlocalE (GPa) a0ε (nm) 0ε
33.91 0.52 3.66
91
Table (6-3): comparison between the experiment natural frequencies and the nonlocal first order approximation
Aspect Frequency Case1
Nanotube Ratio Fexp. (MHz)
fcal. (MHz) diff.%
1 212 0.658 0.647 1.7 2 195 0.644 0.685 6.6 3 189 0.791 0.892 12.79 4 170 0.908 0.908 0 5 119 1.42 1.3 8.41 6 124 0.968 1.057 9.26
The similar procedure is applied for the second order approximation of the nonlocal theory. The
corresponding values of the nonlocal material constants for the second order approximation are
given in Table (6-4). This table shows that the values of the elastic modulus for the first and second
order approximation are roughly the same.
Table (6-4): Nonlocal features for CNT using the second order approximation Case 1
nonlocalE (GPa) a0γ (nm) 0γ
33.7 0.32 2.25
The difference between the experimental results and the proposed nonlocal results is shown in
Table (6-5). Table (6-3) compares the natural frequencies as predicted by the second order model
with the experimental data. It is seen that the model predictions are in agreement with the
experimental data, and the average differences between the computed and the experimentally
measured natural frequency is about 6%. Comparing tables (6-3) and (6-5), it is seen that the
second order model has a slightly better predictions.
92
Table (6-5): Comparison between the experimental natural frequencies and the nonlocal second order approximation
Aspect Frequency Case1 Nanotube Ratio Eexp. (MHz) fcal. (MHz) diff.%
1 212 0.658 0.645 1.99 2 195 0.644 0.684 6.28 3 189 0.791 0.889 12.46 4 170 0.908 0.905 0.3 5 119 1.42 1.297 8.68 6 124 0.968 1.055 8.94
The proposed nonlocal parameters are obtained based on the reported experimental data which the
natural frequencies are in mega hertz.
4. Conclusions
In this study, the nonlocal material constants (nonlocal small scale parameters and modulus of
elasticity) were estimated for the CNT using the available experimental data. The CNT was
modeled as an Euler-Bernoulli beam. Analytical solutions for natural frequencies of the nonlocal
cantilever beam were derived using the first and the second order approximations. The nonlocal
parameters were estimated by comparing the nonlocal model predictions for the natural frequencies
with the experimental data. It was shown that the nonlocal model leads to roughly universal vales
of the material parameters that are applicable for a range of bean aspect ratios. The evaluated
material constants were used and the natural frequencies of the cantilever NCT of different length
were estimated. The predicted natural frequencies are in reasonable agreement with the
experimental data. In addition, it was shown that the predictions of the second order model were
slightly better than the first order model.
93
CHAPTER 7. FUTURE WORK One of the advantages of the nonlocal theory compared to classical elasticity is its ability to model
the high frequency waves. Therefore, the wave propagation of the proposed nonlocal theory for
modeling of the CNT composites needs to be expounded in future work.
The wave propagation using classical elasticity is widely used to obtain material constants such as
Young’s modulus, Shear modulus, Bulk and elastic constants. The reason is because of the speed of
sound within a material is a function of the properties of the material. In contrast, the application of
classical elasticity for modeling nano scale structures is still questionable. This thesis proposed the
nonlocal theory for modeling nanosturures specifically CNT composites which has the length scale
features. The following steps are suggested as an extension of the current research:
• Considering the wave propagation using the proposed method (in Chapter 4) for modeling
the surface waves such as Rayleigh and Lamb wave which has a potential in structural
health monitoring of nanostructures.
• Modeling of CNT beam described in Chapter 5 by Timoshenko beam theory for considering
the shear deformation.
• Applying the proposed constitutive law to a FEA software for modeling the complex
geometries.
• Implementing more experimental natural frequency of CNT for verification the proposed
nonlocal small scale parameters.
94
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