Nonlocal Theory and Finite Element Modeling of Nano-Composites

Transcription

1 CARKSON 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 9 Accepted by the Graduate School Date Dean

2 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 ii

3 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 nanoscale 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 (µm 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 iii

4 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. iv

5 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. v

6 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: W9NF-5--9 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. vi

7 Table of Contents CHAPTER. INTRODUCTION..... OBJECTIVES Nano-structured glass fiber composite Carbon Nanotube Composite... CHAPTER. COMPUTATIONA MODEING OF NANO-STRUCTURED GASS FIBERS6.. INTRODUCTION REPRESENTATIVE VOUME EAMENT MODEING STIFNESS MATRIX..... OAD-DISPACEMENT REATIONSHIP Contact simulation for isotropic material Contact model for transversely isotropic material Contact simulation for the glass fiber CONCUSIONS...9 CHAPTER. COMPUTATIONA MODEING OF NANO-STRUCTURED GASS FIBER COMPOSITE.. INTRODUCTION..... MODEING OF AIGNED SHORT GASS FIBER COMPOSITES Representative Volume Element Modeling Numerical Modeling of the RVE ongitudinal modulus of elasticity In-plane Poisson s ratioυ xy Analytical modeling of the RVE Tilted glass fiber composites..... MODUUS OF EASTICITY Numerical modeling of randomly oriented glass fiber composites Analytical modeling of randomly oriented glass fiber composites..... CONCUSIONS...7 CHAPTER. NONOCA MODEING OF CARBON NANOCOMPOSITE INTRODUCTION NONOCA CONTINUM THEORY First Order Approximation Second Order Approximation NUMERICA EXAMPE WAVE PROPAGATION First order approximation Second order approximation CONCUSIONS...56 CHAPTER 5. EFFECTS OF NONOCA SMA SCAE PARAMETERS ON BEHAVIOR OF CARBON NANOTUBE BEAMS vii

8 5.. INTRODUCTION NONOCA EUER-BERNOUI BEAM MODE Simply supported beam Clamped beam Cantilever beam Propped cantilever beam NUMERICA RESUTS CONCUSIONS...78 CHAPTER 6. FIRST- AND SECOND-ORDER NONOCA BEAM MODES FOR CARBON NANOTUBE 6.. INTRODUCTION NATURA FREQUENCY OF NONOCA CNT BEAM First Order Approximation Second Order Approximation ESTIMATION OF NONOCA MODE PARAMETERS...89 CHAPTER 7. FUTURE WORK... 9 CHAPTER 8. REFRENCES... 9 viii

9 ist of Tables Table (-) Engineering constants for the glass fiber... Table (6-) Experimental data of Gao et al. () Table (6-) Nonlocal material constants for the CNT using the first order approximation... 9 Table (6-) Comparison of the predicted natural frequencies with the experiment values using the first order approximation of nonlocal Euler-Bernoulli beam... 9 Table (6-) Nonlocal material constants for the CNT using the second order approximation... 9 Table (6-5) Comparison between the experimental natural frequencies and the nonlocal second order approximation... 9 ix

10 ist of Figures Figure (-) Glass fibers synthesized using self-assembly... 7 Figure (-) Geometry of glass fibers... 9 Figure (-) The RVE geometry... 9 Figure (-) D finite element model of RVE... Figure (-5) RVE boundary conditions for evaluating E... Figure (-6) In-plane displacement of RVE... Figure (-7) Shear displacement of the RVE... Figure (-8) Schematic of the nanoindentor tip placed on top of glass fiber... Figure (-9) Geometry for FE modeling of nanoindentation assuming material isotropy... 5 Figure (- )Hertz s contact model (analytical) and FE results for nanoindentation of isotropic material... 5 Figure (-) Selected glass fiber geometry for contact modeling... 7 Figure (-) FE mesh for (a) selected geometry (b) full glass fiber... 8 Figure (-) Force-displacement diagram for actual and transversely isotropic... 9 Figure (-) SEM micrographs of SBA-5 hexagonal tubular silica microcapsules used to retain epoxy resin... Figure (-) The RVE model of the matrix with and aligned short fiber... Figure (-) Effective modulus of elasticity of aligned glass fiber composite with volume fraction V = f Figure (-) 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... 6 Figure (-5) oad and boundary conditions of the RVE for evaluating Poisson s ratioυ xy... 7 Figure (-6) Strain and displacement of the RVE under above load condition... 9 Figure (-7) RVE partitions for modeling glass fiber composite... Figure (-8) Tilted short fiber composite... Figure (-9) Coordinate transformation for tilted short fiber composite (a) Original tilted RVE, (b) transformed to aligned RVE... Figure (-) Modulus of elasticity of tilted glass fiber composite... Figure (-) Effective modulus of elasticity of the randomly oriented glass fiber composite using x

11 the Mori-Tanaka and Halpin-Tsai methods... 6 Figure (-) Values of /γ for CNT composites using Equation (-)... 8 Figure (-) A representative volume element (RVE) for CNT composite... 9 Figure (-) FE results for RVE of CNT composite, (a) Displacement of mid-plane (b) longitudinal stress, and (c) longitudinal strain... 5 Figure (-) Nonlocal stress distribution of CNT composite for the first three roots... 5 Figure (-5) Cross section of CNT composite... 5 Figure (-6) Stress distribution in CNT composite using nonlocal theory, average elasticity and FEM for various values of matrix modulus... 5 Figure (-7) The first moment of stress in CNT composite using FEM, average elasticity, and nonlocal theory for various values of matrix modulus... 5 Figure (-8) Dispersion curve for the Born-von Karman lattice dynamic and nonlocal theory - 56 Figure (5-) Deflection of nonlocal beam under uniform load distribution... 7 Figure (5-) Ratio of maximum deflection of nonlocal beam to maximum deflection in local elasticity under uniform load distribution... 7 Figure (5-) Ratio of maximum bending moment of nonlocal beam to maximum bending moment in local elasticity under uniform load distribution... 7 Figure (5-) Ratio of maximum positive bending moment in nonlocal beam to maximum positive bending moment in local elasticity beam under uniform load distribution... 7 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 Figure (5-7) Ratio of maximum bending moment in nonlocal beam to local elasticity beam under sinusoidal load condition 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 Figure (5-9) Ratio of maximum bending moment in nonlocal beam to local elasticity beam under approximated point load condition Figure (5-) Effects of nonlocal small scale parameters on ratio of maximum deflection in nonlocal beams to local elasticity beams under approximated point load condition Figure (6-) Effect of the small scale parameter on nonlocal beam first harmonic constant xi

12 Figure (6-) 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 Figure (6-) 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 xii

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14 CHAPTER. 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 5% 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, ). 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., ; Brown et al.,,, 5a, 5b). 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 aboratory has reported significant thermal and mechanical property improvements by using Nylon-6 nano-composites (Hussain et al., 6). Nano-composites have been pursued in recent years with a goal of creating multifunctional composite structures. In general, one of the

15 most important (and challenging to achieve) multifunctional properties is self-healing. Nanostructured silica fibers used in the present study are much smaller in size than micro-scale spherical capsules used by other researchers (Alavinasab et al., 8) and the encapsulated glue would have longer shelf life. Therefore, the nano-structured glass fiber composite could improve the selfhealing 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, ) discovered multiwall CNT and single wall CNT in 99 and 99, 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., ). 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, ). Reich et al. () have modeled CNT as a closed hollow cylindrical shell and other researchers have considered classical elasticity for modeling CNT-composite (iu and Chen, ; Chakraborty, 6). 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

16 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 and. 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, 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. 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

17 technique, the nonlocal small scale parameters and modulus of elasticity are suggested for a CNT for both the first and the second order approximations... OBJECTIVES... 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.... 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.

18 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. 5

19 CHAPTER. COMPUTATIONA MODEING OF NANO-STRUCTURED GASS FIBERS.. INTRODUCTION The development of multifunctional composite materials/structures is aimed at providing innovative functionality to structures in addition to their load carrying capability (Gates, ). 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 (-)) (Kievsky and Sokolov, 5; Sokolov and Kievsky, 5; Privman et al., 7). A typical glass fiber is approximately 5 µm in length and its hexagonal cross-section is µm in width. The diameter of internal cylindrical pores is approximately nm, and.6-.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., ). 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. 6

20 Figure (-): Glass fibers synthesized using self-assembly (a) arge area Scanning Electron Microscope image (bar size μm) (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 nm (Privman et al., 7) 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 (in et al., ). in and Huang () 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 (). 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., () 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 7

21 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, ). 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... REPRESENTATIVE VOUME EEMENT MODEING 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 (-(a)). Figure (-(b)) indicates the details of the arrangement of any three adjoining cells within the fiber considering an average wall thickness of.7 nm (that is, the distance between the centers of two cylindrical cells is.7 nm) (Privman et al., 7). The filled area in Figure (-(b)) represents fiber material (silica) and the rest is void. The elastic modulus of elasticity and Poisson ratio of silica are considered 7. GPa and.7 respectively. A typical single fiber consists of approximately, 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, 996). 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. 8

22 Figure (-): Geometry of glass fibers (a) Overall dimensions, (b) Arrangement of internal cells. As depicted in Figure (-), the RVE used in the analysis has a length of 5 nm and its cross-section is 6 nm 7 nm which contains 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, 998). Chung and Waas () obtained very good agreement between numerical analysis and experimental data for a honeycomb structure with cells. Therefore, 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 and and the longitudinal direction is along -axis. Figure (-): (a) The RVE geometry, (b) Cross section of the RVE. 9

23 The D FE model of RVE is shown in Figure (-) 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 % on the far boundary) for minimal error near the contact area. Figure (-): (a) D finite element model of RVE, (b) Enlarged view of upper right corner of RVE. In addition to the D FE modeling of RVE, the in-plane properties are also obtained by considering a D plane strain model. For the D model, -node bilinear plane strain quadrilateral elements are utilized. The results show that the D and the D analyses compare very well.

24 .. 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, 996). As an example, for finding E (in-plane modulus of elasticity) a rigid plate is placed on the top of RVE (Figure -5). Symmetric boundary conditions are applied to both lateral sides and the bottom is fixed. The boundary conditions are defined by Equation (-) wherein in x, y, and z directions. u, v, w, respectively, correspond to displacements v( x,, z) = v( x, d, z) = constant = δ u(, y, z) = u( d, y, z) = Sym. w( x, y,) = w( x, y, d ) = Sym. in in x z (-) where d, d, and d, respectively, are 7nm, 6nm, and 5nm. A load applied to the rigid plate generates stress and strain in the RVE, which are used to evaluate E. Similar procedures have been used for evaluating E and E with the boundary conditions given by the Equation (- ). For evaluating the Poisson s ratios, the symmetric boundary conditions on the sides are removed as shown in Figure (-6) to compute the resulting strains. Figure (-5): RVE boundary conditions for evaluating E. u(, y, z) = u( d, y, z) = constant = δ w( x, y,) = w( x, y, d ) = constant = δ (-)

25 Figure (-6): In-plane displacement of RVE. Figure (-7): Shear displacement of the RVE. The shear modulus G is computed using boundary conditions similar to those shown in Figure (-5) except that the symmetric boundary condition perpendicular to the -axis is removed. Figure (-7) shows the shear displacement of the RVE for G. A similar procedure is applied for finding G and G. After evaluating all nine material constants it was observed that some of the constants (such as E and E ) 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 (-). The in-plane shear modulus of RVE, G, is. GPa which is much smaller than the value of.5 GPa obtained based on transversely isotropic assumption. Since the RVE cross-section is similar to a honeycomb (which is considered as

26 transversely isotropic material), this difference needed further investigation. Sun and Vaidya (996) 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 G based on equations in (Sun and Vaidya, 996) is equal to. GPa, which is close to the value obtained for the RVE. Table (-). Engineering constants for the glass fiber (Alavinasab et al., 8). E E υ υ G G E υ G The engineering constants are used to derive the stiffness matrix given by Equation (-) following well established relationships described in (Daniel and Ishai, 99). These engineering constants (in GPa) are listed in Table (-). The stress-strain relationship given by Equation (-) contains six independent constants for the glass fiber; thus, in general the fiber is considered an orthotropic material. σ 9.6 σ.6 σ 5. = τ τ τ ε ε ε γ γ.7 γ (-).. OAD-DISPACEMENT REATIONSHIP 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 nm radius.

27 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 (-8). Figure (-8): Schematic of the nanoindentor tip placed on top of glass fiber.... 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 (985): / F = k c δ (-) where F is contact force and δ is the resulting indentation. The term k c is given as kc = Ec R (-5) where R is the tip radius. The contact modulus E c is calculated from E c E = i= i where the term E i is given by (-6) E E = (-7) i i υi In Equation (-7) E i 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 E i of the sample material.

28 Figure (-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 (-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.8 GPa (similar to E for the glass fiber) and., 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 -node tetrahedral solid elements. The boundary conditions for the model are symmetric on the front and fixed at the bottom (Figure (-9)). Analytical FE 8 P(μN) 6 5 Displacement (nm) Figure (-): Hertz s contact model (analytical) and FE results for nanoindentation of isotropic material. 5

29 The FE predictions compare well with the analytical Hertz s model as shown in Figure (-). The largest difference between the two results is about 5%. Figure (-) verifies the accuracy numerical FE model with analytical results.... Contact model for transversely isotropic material As noted in section., 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 (5) / F δ = * (-9) RE TI * where E TI is the effective modulus for transversely isotropic materials. The effective modulus is given as * E TI = (-) αα where / / E = x E z ν xz α (-) ν xy Ex + ν xz ( + ν xy ) G xz α = (-) ν xy / α + α ν xy α = (-) Gxy Using Equation (-9), the analytical solution of force-displacement of a sphere on a half plane of transversely isotropic material can be evaluated. 6

30 ... Contact simulation for the glass fiber In this section computer simulation of nanoindentation of the glass fiber is presented. Figure (-) 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 (-). Figure (-): Selected glass fiber geometry for contact modeling. Figure (-) shows the energy transfer through the glass fiber capsule under nanoindenter tip. On the other hand, as shown in Figure (-(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. 7

31 Figure (-): 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 (-) 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 (-) is 9%. The maximum difference between the FE simulation results for the actual glass fiber and transversely isotropic approximation is about %, 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. 8

32 5 FE Glass Fiber ( 6 independent const.) FE Glass Fiber ( 5 independent const.) Analytical Hertz Transversely Isotropic P(μN) 5 5 Displacement (nm) Figure (-): 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 (-9), considers the half-pane geometry whereas the geometry of glass fiber as shown in Figure (-) is hexagonal as used for FE simulations..5. CONCUSIONS A continuum mechanics based finite element modeling of nano-structured glass fibers (Kievsky and Sokolov, 5; Sokolov and Kievsky, 5; Privman et al., 7) 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 9

33 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.

34 CHAPTER. COMPUTATIONA MODEING OF NANO-STRUCTURED GASS FIBER COMPOSITE.. 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, 99; Dry and McMillan, 996; i et al., 998). Hollow glass fiber (~5 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. (999) investigated a self-healing material within a polymer composite. They used a reinforced polymer composite with woven S- glass fabric with embedded borosilicate glass micro-capillary pipettes of.5mm diameter. Because the diameter of the hollow glass fibers used in (Motuku, 999) was much larger than the reinforcing fibers, initiation of failure occurred in the composites. Bleay et al. () used smaller diameter hollow fibers for both structural reinforcements and healing components. They used hollow borosilicate glass capsules with a diameter of about -6µm. 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. (999) used particles with average diameter of about 5µm. 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. () and Brown et al. (,, 5a, 5b). They used hollow spherical microcapsules filled with dicyclopentadiene (DCPD). These microcapsules, having a diameter ranging from 8-6µm, 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 9% is reported (Brown et al., ).

35 The proposed self-healing glass fiber composite uses a similar concept. The manufacturing techniques for making these capsules called SBA-5 are presented in (Kievsky and Sokolov, 5; Yang et al., 997, 999a, 999b). The dimension and geometry of SBA-5 are comprehensively described in Chapter. Figure (-): SEM micrographs of SBA-5 hexagonal tubular silica microcapsules used to retain epoxy resin (presented by Kievsky et al., (5)). So far, researchers have mainly focused on synthesis and experimental verification of selfhealing composites. This research is focused on FEM modeling of self-healing composites as FEM is used extensively for obtaining accurate model of micro-composites (evy and Papazian, 99; Brinson and Knauss, 99; Tucker and iang, 999; Kang and Gao, ). 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 iu, ). Chen and iu () 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. Analytical modeling of randomly oriented short fiber composites has been done using the averaging method. Tandon and Weng (986) used micromechanics approach and average stresses

36 of the composite for evaluating modulus of elasticity of composite with randomly oriented fibers. Tucker and ian (999) 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 (evy and Papazian, 99; Brinson and Knauss, 99; Tucker and iang, 999; Kang and Gao, ; Chen and iu, ). Kang and Gao () 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 stressstrain relation (stiffness matrix) of the glass fibers was evaluated in Chapter, which was published in Alavinasab et al., (8). 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. ongitudinal 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... MODEING OF AIGNED SHORT GASS FIBER COMPOSITES... 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 (996). The authors compared different assumptions and configurations for modeling of a composite using an

37 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 (-)). y S z x S Figure (-): The RVE model of the matrix with and aligned short fiber A typical -D square RVE is shown in Figure (-) wherein and S are longitudinal and cross sectional lengths. The relationship between and S can be found by assuming a volume fraction of fibers in the composite. For finding the exact value for and S, an additional equation is needed. im () 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 (-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 %, the equivalent modulus of elasticity of composite for different and its corresponding S were obtained.

38 E (GPa) E ps E sp E (GPa) μm E ps E sp S μm.5.5 Figure (-): Effective modulus of elasticity of aligned glass fiber composite with volume fraction V f =. Figure (-) shows the variation of modulus of elasticity versus S and/or for a volume fraction of. for the aligned glass fibers composite. E ps and E sp are modulus of elasticity of composite based on the PS and SP scheme, respectively. Figure (-) 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 and S, the exact value of and S is not specified yet. In this research, dimensions of the RVE (, 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 (-) by a dashed line, and S are considered as.85 μm and μm, respectively.... Numerical Modeling of the RVE... ongitudinal 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 5

39 (ABAQUS) is used for the numerical analysis. By considering Figure (-), Equation (-) defines the boundary conditions for evaluating the longitudinal modulus of elasticity of the RVE. The quantities u, v and, w correspond to displacements in x, y, and z directions, respectively. w( x, y,) = constant = δ w( x, y, d ) = constant = δ (-) where d and δ are.85 μm and.5 μm, respectively, and the applied strain is equal to.9. Figure (-) shows the displacement of the RVE under longitudinal strain or stress. Capsule F y z x (a) F (b) Figure (-): 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 E can be calculated based on FEM results by applying the following equation: z E z σ avg = (-) ε z where σ avg represents average stress calculated from Equation (-): 6

40 σ = σ ( x, y, / dxdy (-) avg ) A tot By using FEM results and Equations (-) and (-), the longitudinal modulus of elasticity obtained as 5. GPa. E z is... In-plane Poisson s ratioυ xy The in-plane Poisson s ratio of the composite υ xy is obtained by considering the plane strain assumption for the RVE (evy and Papazian, 99). Therefore, it is assumed thatε = and σ z = υ zx ( σ x + σ y ). The in-plane Poisson s ratio υ xy is equal to: z υ xy ε y = (-) ε x By applying the plane strain assumption, the stress-strain relationship is written as: ε x = ε y Ex υ E xy x υ E zx z zx υ E z υ E E x xy x υ E υ E z zx z zx σ x σ y (-5) The corresponding applied load and boundary conditions for evaluating Poisson s ratio is shown in Figure (-5). Uniform oad y z x Sym. B.C. Sym. B.C. Figure (-5): oad and boundary conditions of the RVE for evaluating Poisson s ratioυ xy 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: 7

41 v( x,, z) = constant= δ / v( x, d, z) = constant= δ / (-6) w( x, y,) = w( x, y, d ) = Sym. in z σ =, x Δx ε x = S Δy ε y = S σ = p, y along x = ± S/ along y = ± S/ where d, δ, and p are μm, μm and E5 npa, respectively. The in-plane Poisson s ratio is calculated as: Δx υ zx Δy υ zx υ = + + xy (-7) ps Ez ps Ez The in-plane Poisson s ratio υ xy for the glass fiber composite is equal to.6. The FEM results for evaluation of stress, strain and displacements are shown in Figure (-6). 8

42 Figure (-6): Strain and displacement of the RVE under above load condition, (a) Strain in the x- direction, (b) U displacement in the x-direction (c) Strain in the y-direction (d) U displacement in the y-direction... 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 iu () 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 is composed of the fiber and matrix, whereas part is entirely filled with the matrix. Figure (-7) shows the geometry of the RVE and its partitions. 9

43 Figure (-7): RVE partitions for modeling glass fiber composite... ongitudinal modulus of elasticity of the RVE As shown in Figure (-7), the sequential alignment of parts and 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 E can be written as: E = e + e E c E m (-8) In Equation (-8), E m is the modulus of elasticity of the matrix, s is the length of the glass c fiber and E is the effective longitudinal modulus of elasticity of part, defined by the following equation: E c A f A f = E + Em ( ) (-9) A A tot tot where A f is the cross-sectional area of glass fiber and A tot is the cross section area of the RVE. By considering Young s modulus of glass fiber equal to E (obtained in Chapter ), and considering E m equal to.5 (GPa), the longitudinal modulus of aligned short glass fiber composite (E ) is equal to.96 (GPa).... 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 and along the y-axis. The transverse modulus of elasticity based on the rule of mixtures can be written as:

44 E T c T = E V + E ) (-) m ( V c where V is the volume fraction of part and E T is the effective transverse modulus of part in the y-direction, defined by the following equation: E c T = E V V f tot + E m V V f tot (-) where E is the transverse modulus of elasticity of glass fiber obtained in Chapter, and V f and V tot are the volume fraction of part of the glass fiber and the total volume of part, respectively. By substituting the corresponding values into Equations (-) and (-), the transverse modulus of elasticity of the short fiber composite ET is equal to.8 GPa.... Tilted glass fiber composites Although tilted short fiber composites are similar to aligned fiber composites, the orientation angle between the fiber axis (-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 (-8). y x Figure (-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, ). Figure (-9) illustrates the coordinate system transformation for RVE.

45 σ T τ T σ c σ c σ σ τ T (a) τ T σ T (b) Figure (-9): Coordinate transformation for tilted short fiber composite (a) Original tilted RVE, (b) transformed to aligned RVE Figure (-9(a)) shows the original tilted RVE composite and Figure (-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 (-9) can be written as: σ σ c σ T = T τ T cos α T = sin α sinα cosα sin cos α α sinα cosα sinα cosα sinα cosα cos α sin α (-) 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 (-9(b)), the average strain in the RVE is calculated as: εijdv ε ij = V dv (-) V where ε ij represents the strain tensor (i=,,; j=,,). By using the inverse transformation, the average strain for the original tilted glass fiber shown in Figure (-9(a)) can be obtained by the following equation:

46 ε x ε y γ xy = T ε ε T γ T (-) Considering a linear relationship between stress and strain, the overall modulus of elasticity for different fiber orientations is obtained... MODUUS OF EASTICITY... 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: π / E random = E( α) f ( α) dα (-5) where E(α) is the modulus of elasticity of the tilted composite (with angle of α to horizontal) and f (α) 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 (-). As shown in Figure (-), the maximum modulus of elasticity occurs when the glass fiber is aligned, and the minimum when the fibers are oriented at 5 degrees. Figure (-): Modulus of elasticity of tilted glass fiber composite.

47 Researchers usually determine the distribution function for fiber orientation by using micrograph pictures of the composite. Kang et al. () considered the Weibull distribution function for a reinforced aluminum alloy composite. The Weibull distribution function is given by:.8 α.8 ( α /.79) f ( α) =.69 e (-6).79 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: f ( α) = (-7) π Considering uniform distribution of glass fibers, the effective modulus of elasticity of composite is calculated as. GPa. The effective modulus of elasticity based on the Weibull distribution is equal to.88 GPa.... 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 iang, 999).... 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 curvefitting parameters to match the experimental results. The Halpin-Tsai equations for short aligned fiber composite can be written as (ubin and Peters, 998): E E m + ξη V = η V f f (-8)

48 η = ( E f Em ) ( E E ) + ξ f m (-9) l ξ = (-) d E E T m + η TV = η V T ( E f Em ) ( E E ) + f f (-) η = (-) T f m 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. E and E T 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: E random 5 = E + ET (-) 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., ; Wang and Pyrz, ): dil dil ( f C + f { C A })( f I + f { A }) * = C (-) where the subscripts and 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. dil A is the dilute strain concentration obtained from following equation: 5

49 dil ε = A ε (-5) where ε, and ε 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: * C = KJ + μk (-6) where K and µ are bulk and shear moduli of the composite material respectively. J and K are two tensors whose components are: J ijkl = δ ijδ kl (-7) K ijkl = ( δ ikδ jl + δ ilδ jk δ ijδ kl ) (-8) where δ is the Dirac delta function. As shown in Figure (-), the numerical results obtained are compared with the Halpin-Tsai and Mori-Tanaka method results. Figure (-): 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 is orthotropic. In addition, by considering Table (-), it can be concluded that the transverse modulus 6

50 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 (- ), 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... CONCUSIONS 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. 7

51 CHAPTER. NONOCA MODEING OF CARBON NANOCOMPOSITE.. INTRODUCTION Composite materials play a major role in advanced structures these days. By the discovery of the Carbon Nano-Tube (CNT) by Iijima in 99, 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., 6). 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 -5 GPa (Harris, ). 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., ). 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) (iu and Chen, ). 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., ). 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. () 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, 6) and space frame structures (Chunyu et al., 5) 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 8

52 bond energy.) However, the validity of continuum models based on classical elasticity is questionable at nanoscales. Gao et al. () 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, 6). 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 stressstrain 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... 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 (97a) and Eringen and Edelen (97) to account for nonlocal stress- 9

53 strain relationships. The departure from local elasticity theory to include couple stress effects was originally initiated by Cosserat and Cosserat (99). In 96s new developments on couple stress and micromorphic models were reported by Tupin (96), Mindlin and Tiersten (96), Minlin (96) and Eringen and Suhubi (966), 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 aws (97) 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 (97) 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 (97b). Nowinski (98) studied the longitudinal wave propagation in an elastic circular bar and determined that the velocity of short waves using the nonlocal theory is about 6% less compared to the classical elasticity. Ahmadi and Farshad (97) developed a nonlocal theory for vibration of thin plates and also estimated the corresponding nonlocal material moduli. u et al. (7) presented the nonlocal theory for thick Kirchhoff and Mindlin plates, and solved bending and vibration of rectangular plates. Eringen (97a) 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 (975) 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

54 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 (977) 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 ˆ E ( a x ) x < a E ( x ) = a (-) x > a where Ê and E are nonlocal modulus of elasticity and Young s modulus of elasticity, respectively and a is the atomic distance. Eringen (98, 987) applied nonlocal theory and solved surface wave, crack, and screw dislocation problems. Zhou (998, 999) used nonlocal theory for solving crack problems without considering stress singularity at crack tip. Zeng et al. (6) 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. (7) 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 (7) 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 (7) studied wave propagation in laminated

55 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.

56 .. NONOCA CONTINUM THEORY Basic equations of conservation using nonlocal theory can be written as: Mass: d dt ρ ρdv = + ( ρvk), k = (-) t V Momenta: dvk tlk, l + ρ ( f k ) = (-) dt t kl = t lk (-) Energy: θ & η h hˆ = (-5) Entropy (Clausius-Duhem): h ˆ θ bˆ (-6) where ρ, ν, t kl, f k, η, h, ĥ, θ and bˆ 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 (-) represents the conservation of mass and indicates that mass is invariant under motion. Equation (-), 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, ). The stress tensor, t kl t, in the nonlocal theory is given as ( x) = α( x x ) σ kl ( x dυ (-7) kl ) υ where x is the reference point in the body and α ( x x ) is the nonlocal kernel function, which depends on the internal characteristic length. σ kl is the macroscopic elastic stress tensor at any point x calculated from following equations:

57 σ = (-8) ij Cijklε kl ε = u + u ) / (-9) kl ( k, l l, k where ε kl, C ijkl 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 t ij (x). The key for calculating nonlocal stress is the nonlocal kernel function α ( x x ), which depends on the internal characteristic length and has the dimensions of length -. Therefore, the stresses in nonlocal theory are influenced by the internal characteristic length, a, and the characteristic length ratio, a / l (l is an external characteristic length) (Eringen, 98). The kernel function has the following characteristics: υ α( x ) dυ = (-) α x x ) = α at x = x (-) ( max lim α ( x x ) = δ ( x x ) (-) a Equation (-) describes that the kernel function is normalized over the volume of the body. It can be seen from Equation (-) that when the internal small scale, a, approaches zero, the nonlocal theory reverts to classical elasticity. Eringen (98, ) (presented α as a Green s function of a linear differential operator: α ( x x ) = δ ( x x ) (-) The nonlocal stress function, Equation (-7), can be reduced to a simple differential equation using the differential operator from Equation (-): tij = σ (-) ij 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 (-) can be written as (Eringen, ): ( ε + γ...) t = σ kl kl (-5)

58 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: ε = ε a ε ; γ = γ a, γ (-6), 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., 8; Duan et al., 7; Zhang et al., 5; Peddieson et al., ; Wang et al., 8; Reddy, 7; Reddy and Pang, 8; Kumar et al., 8; Eringen, ; Eringen, 98; Zhang et al., 6). Duan et al. (7) proposedε ranges between and 9 depending on single-walled carbon nanotube (SWCNT) aspect ratio and boundary conditions. Zhang et al. (5) calculated small scale parameter to be about.8. Zhang et al. (6) evaluatedε between.56 and. for different chiral angles of SWCNTs. Wang and Hu (5) proposed ε =. 88. Eringen (98) determined ε =. 9 and ε =. lead to a close match with the longitudinal and Rayleigh surface wave atomic dispersion curve respectively. Wang and Wang (7) proposed ε =. 88 using the gradient method. Wang (5) estimated ε a <. nm in SWCNT by investigation of wave propagation with wave frequency value greater than Hz. The above discussion shows that the value of ε (small scale parameter) is of the order of. While there is very little information on value of 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.... First Order Approximation Equation(-5) can be approximated by considering only the first term ( ε term) (Eringen, ), leading to: 5

59 d d ( ε ( + )) t kl = σ dx dy kl (-7) 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 d dy ( ε ) t = σ (-8) For a constant σ, the solution to Equation (-8) is given as y y t = σ + Acosh( ) + B sinh( ) (-9) ε ε The constants A, B, and ε are obtained from boundary conditions; however, because of symmetry, B=. The appropriate boundary condition is obtained when the resultant force and moment derived from Equation (-9) 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 (-9) is an unacceptable solution and the governing equation given by Equation (-8) is not appropriate for this application.... Second Order Approximation In order to consider the second approximation of Equation (-5), only the second term of Equation (-5) is retained. That is, d dy ( + γ ) t = σ (-) The analytical solution to Equation (-) satisfying the symmetry condition is given by y y t = σ + Acosh( ) cos( ) (-) γ γ 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 6

60 Resultant force in nonlocal theory Resultant force in elasticity y y tkl dy = σ kl dy + Acosh( ) cos( ) dy (-) γ γ where is the cross-sectional dimension of RVE in the y-direction (Shown in Figure (-5)). The resultant force in average elasticity and the nonlocal theory model in Equation (-) should be equal. Therefore, the remaining term in Equation (-) is equal to zero: Acosh( y ) cos( γ y ) dy = γ (-) By integrating Equation (-) we obtain F = cos( )sinh( ) + sin( )cosh( ) = (-) γ γ γ γ Solving Equation (-) we can find several values of γ in each length scale for CNT composites. Roots of Equation (-) give the values of. γ As shown in Figure (-), the first four roots of Equation (-) are.6, 7.775,.79, and 6.668, respectively. Furthermore, the values of function F around larger /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: y y tkl ydy = σ kl ydy + Acosh( ) cos( ) ydy (-5) γ γ where the first term on right hand side of Equation (-5) is the first moment of forces in average elasticity. For a combination of γ and from Figure (-), the variable A can be calculated as 7

61 A = t ydy σ kl y y cosh cos ydy γ γ kl (-6) Using the FEM results for the RVE, we can evaluate the first moment of forces using the above equations F c γ F γ F γ F γ Figure (-). Values of /γ for CNT composites using Equation (-)... NUMERICA EXAMPE A rectangular RVE of CNT composite (Figure (-)) 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 (iu and Chen, ). The geometry of RVE is as follows: length = nm, inner 8

62 radius of CNT =.6 nm, and thickness of CNT =. 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: E CNT = nn/nm, ν CNT =.; Matrix: =,,, nn/nm, ν =.; E m 5 m Figure (-). A representative volume element (RVE) for CNT composite The D FE model of RVE shown in Figure (-) 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 nm to obtain nano-scale resolution. The CNT is modeled as hollow cylinder with thickness of. nm; the element size for CNT is. nm in cross-section and 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 midplane, stress, and strain in the RVE are obtained as shown in Figure (-). (a) 9

63 (b) (c) Figure (-): FE results for RVE of CNT composite, (a) Displacement of mid-plane (b) longitudinal stress, and (c) longitudinal strain Figure (-): Nonlocal stress distribution of CNT composite for the first three roots Different roots of Equation (-) present different curvatures for nonlocal stress distributions. Figure (-) 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. 5

64 As shown in Figure (-), 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 (-5). Sym. O Sym. = Figure (-5): Cross section of CNT composite Figure (-5) shows the cross-section of CNT composite and because of symmetry only one quarter of CNT is considered for showing results. Figure (-6) compares stress distributions in CNT composite using nonlocal theory, average elasticity and FEM. The values of matrix modulus is varied from E m = 5 GPa to E m = GPa. 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 = to (as expected) whereas the stress from nonlocal theory varies similar to the variation of γ (lowest root) in Figure (-). Furthermore, as shown in Figure (-6(c)) and (-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,,, and nn/nm (Figure -6 (a)-(d)) are equal to 85,, 8, and 8 nn, respectively. 5

65 (a) (b) (c) (d) Figure (-6): Stress distribution in CNT composite using nonlocal theory, average elasticity and FEM for various values of matrix modulus (a) E m = 5 nn/nm, (b) E m = nn/nm, (c) E m = nn/nm, (d) E m = nn/nm The first moment (integral of stress times distance) computed from the stress distributions (Figure (-6)) show nearly identical values for FEM and nonlocal theory, but the average elasticity results are different (Figure (-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 5

66 function of strain in the entire domain which is similar to the behavior of atoms in nano-scale structures. (a) (b) (c) (d) Figure (-7): The first moment of stress in CNT composite using FEM, average elasticity, and nonlocal theory for various values of matrix modulus (a) = 5 nn/nm, (b) E = nn/nm, (c) E = nn/nm, (d) = nn/nm E m m m E m.. 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, 7). 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 5

67 effects in large scale structures. The approximation of the nonlocal equation of motion in an isotropic material is given as (Eringen, ). ( λ + μ). u μ u ( ε + γ...) ρ& u& = (-7) where λ, μ are ame 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 (-7) is obtained by using Fourier transforms to simplify the nonlocal equations and matching the dispersion curves of nonlocal theory and atomic model.... First order approximation Using the first order approximation suggested by Eringen (98,987), the equation of motion for wave propagation in a nonlocal elastic solid is written as ( λ + μ). u μ u ( ε ) ρ& u& = (-8) 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, 98,987) ω c k ω c T k = = ( + ε ( + ε k k ) ) (-9) (-) Here ω and k are frequency and wave number whereas c and c T are compression and shear wave velocities, respectively.... Second order approximation It was shown in section.. 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 (-7) is given as ( λ + μ). u μ u ( + γ ) ρ& u& = (-) 5

68 The corresponding dispersion relations are obtained as ω c k ω c T k = = ( + γ ( + γ k k ) ) (-) (-) Equations (-) and (-) show the dispersion relation using the second order approximation. The small parameters, ε and/or γ, defined in Equation (-6) are constant appropriate to each material. Several efforts have been made for evaluating the small parameters for SWCNT. Zhang et al. (6) used the MD simulation for evaluating ε, and they found.56 ε. for different chiral angles of SWCNTs. Zhang et al. (5) evaluated ε. 8 by comparing molecular mechanics simulation with nonlocal buckling analysis of SWCNT. Wang and Hu (5) proposed ε =.88 by using second-order strain gradient in elasticity theory and MD simulation. Eringen (98) determined that using the first order approximation of nonlocal theory, with ε =.9a where a is the lattice parameter, leads to a close match with the atomic model. As reported in Eringen (98, 987), the maximum deviation between the first order approximation of nonlocal theory and the Born-von Karman theory is less than 6%. Wang and Wang (7) compared the gradient method with nonlocal first order method ( ε =. 85). 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 ( γ ) is evaluated by matching the dispersion curve of the proposed nonlocal theory with lattice dynamic simulation. For ε =.9a and γ =.5a, Figure (-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. 55

69 Figure (-8): Dispersion curve for the Born-von Karman lattice dynamic and nonlocal theory..5. CONCUSIONS 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: ) The computed stress/strain considers the effect of the entire domain, which produces more accurate results especially for nano-scale structures. ) 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. ) The dispersion curve obtained from nonlocal theory is very close to the Born-von Karman atomic model. 56

70 CHAPTER 5. EFFECTS OF NONOCA SMA SCAE PARAMETERS ON BEHAVIOR OF CARBON NANOTUBE BEAMS 5.. INTRODUCTION Modeling Carbon nanotube (CNT) reinforced composites using nonlocal theory is described in Chapter. Recently researchers have applied nonlocal theory for modeling of characteristic nanodevices 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 (u et al., 7; Peddieson et al., ; Wang et al., 8). Comprehensive studies on modeling CNT composite beams using nonlocal theory including analytical solutions for bending, vibration and buckling of beams were reported by Reddy (7) and Reddy and Pang (8). 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. (8) also studied the buckling of CNTs using similar nonlocal one dimensional continua with Euler-Bernoulli approach. Heireche et al. (8) and u et al. (7) 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 (6) 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. (6) 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 57

71 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.. NONOCA EUER-BERNOUI BEAM MODE The equilibrium of forces and moments in the vertical direction of an infinitesimal element of a beam structure is given as (Wang, 5): t w A x V = ρ (5-) V x M = (5-) 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 = A da z M σ (5-) x w = z ε (5-) 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, 8): t x w m t w m q x M = + (5-5) where q is the transverse force per unit length. The mass inertias m and m are defined as: = = A A da z m da m, ρ ρ (5-6) Substituting Equations (5-) and (5-5) into the nonlocal constitutive relation Equation (-5) leads to (Reddy, 8): + = 6 x q x t w m t x w m q t x w m t w m x w EI M γ ε (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:

72 w w w EI m + ε m q x x x t x t 6 w w q w w γ m m + q= m m x x t t x x t x t (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. d dx d w EI ε q + γ dx d q + = q dx (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. Q M E E d w dq d q = EI ε + γ = q( ) d c dx η η x dx dx d w d q EI ε q( x) + γ = q( η) dηd c x c x ξ dx dx = ξ (5-) (5-) θ dw dq x = ε q( η) dη γ q( η) dηdξdα c cx c dx EI dx x ξ η x = (5-) w x x ε q( η) dηdξ γ q( x) q( η) dηdξdαdβ c c cx c EI x ξ x ξ η α 6 = (5-) where c, c, c, c 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: q ( x) = q q x) = q ( πx.sin (5-) (5-5) 59

73 q( x) = q q( x) = q. π + x.. ( x ) +.) (5-6) π (5-7) Equations (5-) and (5-5) imply uniformly distributed load and sinusoidal distributed load, respectively. Equations (5-6) and (5-7) 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 nanolength scale which is considered as dimension of atoms or molecules. Equation (5-6) is used for simply supported, clamped, propped cantilever beams whereas Equation (5-7) is used for cantilever beam. Reddy and Pang (8) have presented solutions of Equations ((5-)-(5-)) 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 Simply supported beam Boundary conditions for a nonlocal simply supported beam are: d q ε γ d w w=, M = EI q( x) + = at x =, dx dx The constants of integration are obtained for different load conditions: a) Uniformly distributed load q (5-8) By applying Equation (5-8), the constants of integration for a nonlocal simply supported beam under uniform distributed load condition are obtained as: c = q, c =, c = ε q + q, c = γ q Thereore, the deflection and bending moment are obtained as: (5-9) [ x( x)( + x + )] q w( x) = x EI ε (5-) 6

74 ( x ) q M ( x) = x (5-) Equations (5-) and (5-) 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 (8). The maximum deflection and bending moment occur in the middle of the beam (x=/) given by w [ 5 + ε ] q max = 8 8EI (5-) q M max = 8 (5-) Equation (5-) 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: ( ε π + ), q q c =, c =, c = c = π π (5-) The deflection and bending moment are obtained as: ( ε π + γ ) q πx w ( x) = sin π + EIπ (5-5) q M( x) = π πx sin (5-6) Equations (5-5) and (5-6) show that both small scale parameters contribute to the displacement whereas neither of the small scale parameters contribute to the bending moment. The maximum x = deflection and bending moment at are: ( ε π + γ ) q w max = π + EIπ (5-7) 6

75 q M max = π (5-8) Equation (5-7) 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 (5-6) load condition are:.58q c =.89886arctan(5 ) q, c =, c = arctan(5 ) c = q arctan(5 ) +.8ε arctan(5 ) (5-9) The corresponding bending moment for simply supported beam under the approximated point load is given as: M ( x) = q.59n(5 + ) +.59.arctan(5).8x.arctan( x 5) arctan( x 5) +.59n( + x + 5 ) (5-) Equation (5-) 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., the effect of nonlocal small scale parameters increases the deflection of nonlocal beam Clamped beam Boundary conditions for a nonlocal beam clamped at both ends are defined as: dw w =, θ = = at x =, dx The constants of integration are obtained for different loads conditions: (5-) a) Uniformly distributed load q The constants of integration are obtained as: 6

76 c = q, c = q ε +, c =, c = γ q (5-) The deflection and bending moment are derived as: ( x ) qx w( x) = EI q M ( x) = 6x 6x + ε + ( ) (5-) (5-) Equation (5-) 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=/ which is equal to: w max = q 8EI (5-5) 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: M M max max q = q = ( ε ) x = ( + ε ) x =, (5-6) 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: c γ qπ ( + ε π + π γ ), c =, q q =, c = c π π (5-7) = The deflection and bending moment are given as: 6

77 ( ε π + γ π + ) πx q sin w( x) = EIπ πx + πx q πx M( x) = π sin + ε π + γ π π (5-8) (5-9) Equations (5-8) and (5-9) show that both small scale parameters contribute to the displacement and bending moment. The maximum deflection and positive bending moment at x=/ are: q w max = π + EIπ (5-) M q ( π )( ε π + γ ) ( π ) ε π γ π ) max = π and the maximum (negative) bending moment occurring at both ends of the beam is: ( + ε π + γ ) x (5-) q M max = π =, π (5-) Equations (5-) and (5-) show that the nonlocal small scale parameters increase the deflection and negative bending moment of the nonlocal beam. Equations (5-) 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 (5-6) load condition are given as: c c c 6 = q (.8arctan(5) ) q arctan(5)( ε + 8.ε = ) ε + 5.8q.58q =, c 6 = (5-) 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 6

78 parameter. The numerical results for the point load conditions are presented in section 5., however the analytical results were not presented here because they are very large Cantilever beam Boundary conditions for a nonlocal cantilever beam are written as: w =, dw θ = = dx at x = M =, V = at x = (5-) The constants of integration are obtained based on the applied load conditions as follows. a) Uniformly distributed load q The constants of integration for nonlocal cantilever beam under uniformly distributed load are obtained as: c = q, c = q, c =, c = γ q (5-5) The deflection and bending moment are given as: ( 6 + x ) qx w( x) = x ε EI q M ( x) = ( x ) (5-6) (5-7) Equation (5-6) 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=, whereas the maximum bending moment occurs at x=: w ( ε ) q max = 8EI (5-8) M max q = (5-9) 65

79 Equation (5-8) 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: = c q q γ qπ =, c =, c =, c π π (5-5) The deflection and bending moment are given as: πx ( 6ε π + 6γ π + 6 ) πx sin + x ( x ) q w( x) = 6EIπ π (5-5) q πx M ( x) = πx + sin π π (5-5) Equation (5-5) 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: ( ( π ) ε π γ π ) x q w max = = π EI (5-5) M q = x π (5-5) max = Equation (5-5) 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 (5-7) as an approximation of a point load condition are: 66

80 c c =.8arctan( ) q q = 8 +,, + c =.59 q c n(.58 q = + + ) +.8arctan( ) q, (5-55) The bending moment is obtained by using Equations (5-), (5-7) and (5-55). (.8( x )arctan( x + ) +.59n( + x x )) M ( x) = q + (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 Propped cantilever beam Boundary conditions for a nonlocal propped cantilever beam at both ends of the beam are: w =, dw θ = = dx M =, w = at at x = x = (5-57) The constants of integration are obtained using different load conditions as follows. a) Uniformly distributed load q The constants of integration of a nonlocal propped cantilever beam under a uniformly distributed load are obtained as: c q = ( ε + 5 ) q, c = ( ε ), c =, c = γ q 8 8 (5-58) The deflection and bending moment are given as: ( x)( + )) ε q x w( x) = x 8EI ( x)( + )) q M ( x) = x 8 ε (5-59) (5-6) 67

81 Equations (5-59) and (5-6) 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 (5-6) and (5-6), respectively: x x max max 6ε ε + 96ε (deflection) = (bendingmoment) = ε 8 (5-6) (5-6) 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 (8). Reddy and Pang (8) 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 (5-6) and (5-6) 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=. The maximum deflection and bending moment at their corresponding locations are derived as: w max q = 786 EI where κ = ( 6ε + 5 κ ) ( ε ε + ε κ κ ) 96ε + ε + (5-6) ( 6ε ε ) 9q M max = M q 8 ( + ε ) max = (5-6) (5-65) Equations (5-6), (5-6) 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: 68

82 69 ( ) ( ),,, = = + + = = c q c q c q c π γ γ π π ε π γ π π π ε π (5-66) The deflection and bending moment are given as: ( ) = π π π π π γ π ε π sin ) ( x x x x EI q x w (5-67) = 5 5 sin ) ( x x x x q x M γ π π ε γ π π ε π π π (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: x.5756 ) (deflection max = (5-69) π π γ π π ε π + + = max ( arccos moment) (bending x (5-7) The location of maximum negative bending moment is x=. 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: ( ) max. EI q W + + = π γ π ε π (5-7) ) ( max q M + + = γ π π ε π (5-7) 8 max where arccos 9 q M + + = = γ π π ε η π η η η π π (5-7)

83 The nonlocal small scale parameters affect beam deflection and bending moment. Equation (5-7) 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 (5-6) as an approximation for point load are obtained as: q c = q + c q + c arctan(5) ε ε 77.65ε.676 ( 5 + ) n(5 + ) ( ) ( ) q = arctan(5) ε ε ε + ( 5 + ) n(5 ( 5 + ) + ) 5.8q = ( ), + c.5q = 5 + q q ( 5 + ) ( ) (5-7) It is shown in Section 5. that both small scale parameters contribute to deflection and bending moment of nonlocal propped cantilever beam under a point load. 5.. NUMERICA RESUTS Numerical results are presented for a carbon nanotube. The material properties of the carbon nanotube are the same as Reddy and Pang (8). ρ = kg/m, a =. nm 9 πd 8 E = GPa, ν =.9, d =. m, I = =.9 m 6 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-) shows the results using normalized length. In Figure (5-), x is the location of the displacement, ε=ε a (nm) is the first nonlocal small 7

84 scale parameter and w is the deflection of the beam. As shown in Figure (5-), 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. w(x) (nm) ocal elasticity ε a.5 (nm) ε a. (nm) ε a.5 (nm) ε a. (nm) a) Coordinate, x/ w(x) (nm) 6 5 ocal elasticity ε a.5 (nm) ε a. (nm) ε a.5 (nm) ε a. (nm) b) Coordinate, x/ 7

85 w(x) (nm) ocal elasticity ε a.5 (nm) ε a. (nm) ε a.5 (nm) ε a. (nm).5 c) Coordinate, x/ Figure (5-): 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-), (5-8) and (5-6) indicate that only the first small scale parameter affects the deflection of the nonlocal beam. Figure (5-) shows the ratio of maximum deflection of nonlocal beam to that of local (classical) elasticity beam with the variation of small scale parameter. 7

86 w max (nonlocal)/ w max (local) Simply Supported Clamped Cantilever Propped Cantilever ε a (nm) Figure (5-): Ratio of maximum deflection of nonlocal beam to maximum deflection in local elasticity under uniform load distribution As shown in Figure (5-), 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. M max (nonlocal)/ M max (local) Simply Supported Cantilever Propped Cantilever Clamped ε a (nm) Figure (5-): Ratio of maximum bending moment of nonlocal beam to maximum bending moment in local elasticity under uniform load distribution 7

87 As shown in Figure (5-), 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-), 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-) shows that the first small scale parameter has a significant effect on the positive bending moment of nonlocal clamped and propped cantilever beams. M max+ (nonlocal) /M max+ (local) Simply Supported Clamped Propped Cantilever ε a (nm) Figure (5-): 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 (5-7), (5-) and (5-7), 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. 7

88 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 (5-8) and (5-5), the 75

89 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) a) b) 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 76

90 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. M max (nonlocal)/ M max (local) Simply Supported Cantilever Clamped Prepped Cantilever ε a (nm) 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. 77

91 a) b) c) d) Figure (5-): 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-), 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.. CONCUSIONS 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 78