Utah State University DigitalCommons@USU All Graduate Theses and Dissertations Graduate Studies 12-2008 Variational Asymptotic Micromechanics Modeling of Composite Materials Tian Tang Follow this and additional works at: https://digitalcommons.usu.edu/etd Part of the Applied Mechanics Commons Recommended Citation Tang, Tian, "Variational Asymptotic Micromechanics Modeling of Composite Materials" (2008). All Graduate Theses and Dissertations. 72. https://digitalcommons.usu.edu/etd/72 This Dissertation is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. VARIATIONAL ASYMPTOTIC MICROMECHANICS MODELING OF COMPOSITE MATERIALS by Tian Tang A dissertation submitted in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mechanical Engineering Approved: Dr. Wenbin Yu Dr. Leijun Li Major Professor Committee Member Dr. Brent E. Stucker Dr. Thomas H. Fronk Committee Member Committee Member Dr. David E. Richardson Dr. Byron R. Burnham Committee Member Dean of Graduate Studies UTAH STATE UNIVERSITY Logan, Utah 2008 ii Copyright °c Tian Tang 2008 All Rights Reserved iii Abstract Variational Asymptotic Micromechanics Modeling of Composite Materials by Tian Tang, Doctor of Philosophy Utah State University, 2008 Major Professor: Dr. Wenbin Yu Department: Mechanical and Aerospace Engineering The issue of accurately determining the e®ective properties of composite materials has received the attention of numerous researchers in the last few decades and continues to be in the forefront of material research. Micromechanics models have been proven to be very useful tools for design and analysis of composite materials. In the present work, a ver- satile micromechanics modeling framework, namely, the Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH), has been invented and various micromechancis models have been constructed in light of this novel framework. Considering the periodicity as a small parameter, we can formulate the variational statements of the unit cell through an asymptotic expansion of the energy functional. It is shown that the governing di®eren- tial equations and periodic boundary conditions of mathematical homogenization theories (MHT) can be reproduced from this variational statement. Finally, we employed the ¯nite element method to solve the numerical solution of the constrained minimization problem. If the local ¯elds within the unit cell are of interest, the proposed models can also accurately recover those ¯elds based on the global behavior. In comparison to other existing models, the advantages of VAMUCH are: (1) it invokes only two essential assumptions within the concept of micromechanics for heterogeneous material with identi¯able unit cells; (2) it has an inherent variational nature and its numerical implementation is shown to be straightfor- ward; (3) it calculates the di®erent material properties in di®erent directions simultaneously, which is more e±cient than those approaches requiring multiple runs under di®erent loading conditions; and (4) it calculates the e®ective properties and the local ¯elds directly with the iv same accuracy as the fluctuation functions. No postprocessing calculations such as stress averaging and strain averaging are needed. The present theory is implemented in the computer program VAMUCH, a versatile engineering code for the homogenization of heterogeneous materials. This new microme- chanics modeling approach has been successfully applied to predict the e®ective properties of composite materials including elastic properties, coe±cients of thermal expansion, and speci¯c heat and the e®ective properties of piezoelectric and electro-magneto-elastic com- posites. This approach has also been extended to the prediction of the nonlinear response of multiphase composites. Numerous examples have been utilized to clearly demonstrate its application and accuracy as a general-purpose micromechanical analysis tool. (280 pages) v Acknowledgments I am deeply grateful and indebted to my advisor, Professor Wenbin Yu, for his con- sistent guidance, support, and encouragement during my graduate studies at Utah State University. Without his help and patience, this work would not have been possible to reach its ¯nal stage. In addition to introducing me to the subject of micromechanics, the most important skill that he helped me obtain was how to be an e®ective researcher. I would like to thank my committee members, Dr. Brent E. Stucker, Dr. Leijun Li, Dr. Thomas H. Fronk, and Dr. David E. Richardson, for their guidance, criticisms, and insightful comments on my dissertation. I would like to thank our academic advisor, Bonnie Ogden, for her enthusiasm and great help. I would also like to thank our department head, Dr. Byard Wood, for his encourage- ments and writing the recommendation letter for my application for a dissertation fellow- ship. This work is supported, in part, by the National Science Foundation under Grant DMI- 0522908 and the State of Utah Community/University Research Initiative Grant. This support is gratefully acknowledged. Special thanks go to Dr. Harald Berger from Otto-von-Guericke-University of Magde- burg, Germany, for technical discussions and kindly providing the ANSYS macro ¯les and the data of the e®ective properties of PZT-7A ¯ber reinforced composites. Finally, I would like to express particular gratitude to my wife, Yanqing Yang, for her endless loving, caring, support, and encouragements. Tian Tang vi Contents Page Abstract ::::::::::::::::::::::::::::::::::::::::::::::::::::::: iii Acknowledgments ::::::::::::::::::::::::::::::::::::::::::::::: v List of Tables ::::::::::::::::::::::::::::::::::::::::::::::::::: ix List of Figures :::::::::::::::::::::::::::::::::::::::::::::::::: xi 1 Introduction ::::::::::::::::::::::::::::::::::::::::::::::::: 1 1.1 Literature Review on Micromechanics . 2 1.1.1 Volume Averaging and E®ective Properties . 2 1.1.2 Rigorous Bounds . 5 1.1.3 Analytical Methods . 10 1.1.4 Numerical Methods . 14 1.2 Motivation and Objective . 16 1.3 Outline of the Dissertation . 17 2 A Multiphysics Micromechanics Model of Smart Materials Using the Variational Asymptotic Method :::::::::::::::::::::::::::::::::::: 25 2.1 Introduction . 25 2.2 Theoretical Formulation . 28 2.3 Finite Element Implementation . 34 2.4 Numerical Results . 37 2.4.1 Two-phase composites . 38 2.4.2 Three-phase composites . 47 2.5 Conclusion . 47 3 Variational Asymptotic Homogenization of Heterogeneous Electromagne- toelastic Materials ::::::::::::::::::::::::::::::::::::::::::::::: 50 3.1 Introduction . 50 3.2 Theoretical Formulation . 53 3.3 Finite Element Implementation . 59 3.4 Numerical Examples . 62 3.4.1 Two-phase Electromagnetoelastic Composites . 63 3.4.2 Three-phase Electromagnetoelastic Composites . 64 3.5 Conclusion . 75 4 Variational Asymptotic Micromechanics Modeling of Heterogeneous Piezo- electric Materials :::::::::::::::::::::::::::::::::::::::::::::::: 79 4.1 Introduction . 79 4.2 Piezoelectricity and Piezoelectric Composites . 82 4.3 Theoretical Formulation . 82 4.4 Finite Element Implementation . 87 4.5 Numerical Examples . 90 4.5.1 Predict E®ective Properties of Composites . 91 4.5.2 Predict Local Fields . 94 vii 4.6 Conclusion . 97 5 A Variational Asymptotic Micromechanics Model for Predicting Conduc- tivity of Composite Materials :::::::::::::::::::::::::::::::::::::: 107 5.1 Introduction . 107 5.2 Theoretical Formulation . 109 5.3 An Illustrative Example . 113 5.4 Finite Element Implementation . 115 5.5 Numerical Examples . 118 5.5.1 E®ective Thermal Conductivity of Fiber Reinforced Composites . 119 5.5.2 E®ective Thermal Conductivity of Particle Reinforced Composites . 122 5.5.3 Recovery of Local Heat Flux . 126 5.6 Conclusion . 127 6 Asymptotical Approach to Initial Yielding Surface and Elastoplasticity of Metal Matrix Composites ::::::::::::::::::::::::::::::::::::::::: 132 6.1 Introduction . 132 6.2 Theoretical Formulation . 134 6.3 Finite Element Implementation . 138 6.4 VAMUCH Prediction of Initial Yielding Surface . 141 6.5 VAMUCH for the Simulation of the Elastoplastic Behavior of the Composite 141 6.6 Validation Examples . 142 6.6.1 Material Properties . 142 6.6.2 Binary Composites . 143 6.6.3 Fiber Reinforced Composites . 144 6.7 Conclusion . 147 7 Micromechanics Modeling of the Nonlinear Behavior of Electrostrictive Multiphase Composites ::::::::::::::::::::::::::::::::::::::::::: 153 7.1 Introduction . 153 7.2 Basic Equations . 155 7.3 Theoretical Formulation . 156 7.4 Finite Element Implementation . 160 7.5 Numerical Examples . 164 7.6 Conclusions . 167 8 Conclusions and Recommendations of Future Work ::::::::::::::::: 170 8.1 Conclusions . 170 8.1.1 Di®erences from the View Point of Theoreticians . 170 8.1.2 Di®erences from the View Point of Practicing Engineers . 173 8.2 Recommendations of Future Work . 174 Appendices ::::::::::::::::::::::::::::::::::::::::::::::::::::: 176 Appendix A Variational Asymptotic Method for Unit Cell Homogenization of Periodically Heterogeneous Materials . 177 A.1 Introduction . 177 A.2 A Variational Statement for Unit Cells . 181 A.3 Variational Asymptotic Method For Unit Cell Homogenization . 184 A.4 Finite Element Implementation of VAMUCH . 186 A.5 Validation of VAMUCH . 190 A.5.1 Binary Composites . 191 A.5.2 Fiber Reinforced Composites . 193 A.5.3