NUMERICAL STUDY OF METAL MATRIX NANOCOMPOSITES USING DISCRETE DISLOCATION APPROACH ELLIOT LAW NATIONAL UNIVERSITY OF SINGAPORE 2011 NUMERICAL STUDY OF METAL MATRIX NANOCOMPOSITES USING DISCRETE DISLOCATION APPROACH ELLIOT LAW (B. Eng. (Civil) (Hons), NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2011 Acknowledgements First and foremost, praise be to God for His goodness and faithfulness in bringing me through this entire study. I thank God for blessing me with wisdom and sustaining me with good health as I worked on this project. I would also like express my deepest thanks to my parents and family members for their unconditional love and support throughout all these years. I would like to express my utmost gratitude to my supervisors, Dr. Pang Sze Dai and Prof. Quek Ser Tong, for their advice, guidance and counsel since my undergraduate years. I have learnt so much from them and it has been such a great pleasure and joy to be able to work with them. I am also very grateful for their care and support; they have inspired me to likewise toward others and to give my best in all my undertakings. I would also like to thank the National University of Singapore (NUS) for supporting me with the Research Scholarship for the entire duration of my study. I would also like to acknowledge my fellow students whom I have had the opportunity to work with throughout the duration of my studies. A special note of appreciation to Mr. Tran Diep Phuoc Thao, Ms. Liu Lihui, Ms. Matilda Loh and Mr. Too Jun Lin for their friendship, encouragement and support; the joy of working with them has made my work more meaningful as well as given me the impetus and motivation to complete this pursuit. i Last but not least, I would like to express my thanks to the staff at the Structural Engineering Laboratory, especially Mdm. Annie Tan (who has since transferred to the Engineering Design and Innovation Centre) and Mr. Ang Beng Oon, for their support in this project. I would also like to acknowledge Dr. Sharon Nai Mui Ling from the Singapore Institute of Manufacturing Technology (SIMTech) for her tremendous assistance in the experimental work conducted for this study. Special thanks also to Assoc. Prof. Manoj Gupta and Dr. Khin Sandar Tun from the Department of Mechanical Engineering at NUS as well as Mr. Lam Kim Song from the Fabrication Support Centre of the same department for their help in the fabrication and machining of the nanocomposite specimens as well as sharing of experimental results on tensile properties and microstructural characteristics. Give thanks to the Lord, for He is good; His love endures forever. (Ps. 107:1) ii Table of Contents i Acknowledgements iii Table of Contents viii Summary List of Tables xi List of Figures xii List of Symbols xxii 1.0 Introduction 1.1 Metal matrix composites – overview 1.2 Numerical studies on metal matrix composites 1.2.1 Metal matrix composite systems 1.2.2 Influence of various microstructural features on mechanical properties of metal matrix composites 1.2.3 Microstructural modelling using representative volume element approach 1.2.4 Boundary conditions for representative volume element 1.3 Metal matrix nanocomposites – overview 1.4 Size effects on mechanical properties of metal matrix composites 13 1.5 Methods of computer simulations for dislocations 15 1.5.1 Atomistic methods 16 1.5.2 Continuum methods 17 1.5.3 Extended finite element method 19 iii 1.5.4 2.0 iv 20 1.6 Numerical simulations of metal matrix nanocomposites 21 1.7 Objective 25 1.8 Scope and limitations 25 1.9 Organization of the study 26 Theoretical framework for composite material model using discrete dislocation method 27 2.1 Dislocations – basic concepts and characteristics 27 2.2 Discrete dislocation formulation 32 2.2.1 Instantaneous state of dislocated body 32 2.2.2 Forces between dislocations 35 2.2.3 Constitutive relations for motion of dislocations 36 2.2.4 Constitutive relations for creation and annihilation of dislocations 39 Analytical solutions for dislocation fields in two-dimensional space 43 2.3.1 Special case of two-dimensional space with horizontal slip planes 45 2.4 Implementation of discrete dislocation formulation for twodimensional unit cell analyses 50 2.5 Computational scheme 54 2.3 3.0 Multi-scale modelling and coupled atomisticcontinuum methods Numerical implementation issues 59 3.1 Computational time-step and efficiency 59 3.1.1 59 Evaluation of boundary nodal forces due to dislocation stress field 3.1.2 3.2 3.3 4.0 Effect of cut-off distance on calculation of dislocation fields 63 3.1.3 Evaluation of dislocation glide force 65 3.1.4 Time-step 67 3.1.5 Tracking of dislocation events and processes 69 Calibration of material parameters for dislocation processes 72 3.2.1 Density and strength of impurities or obstacles 74 3.2.2 Density of dislocation sources 80 3.2.3 Nucleation strength of dislocation sources 82 3.2.4 Calibration procedure 84 Size of representative volume element 90 3.3.1 Density of dislocation sources 94 3.3.2 Strength and density of impurities 98 3.3.3 Change in mean overall response with number of realizations 100 3.3.4 Overall response of composite material 104 Effects of microstructural features and constituent material properties on mechanical response of metal matrix nanocomposites 111 4.1 Inclusion volume fraction 111 4.2 Inclusion size 115 4.3 Inclusion aspect ratio and orientation 118 4.4 Arrangement of inclusions 124 4.5 Material properties of constituent phases 132 4.6 Nature of dislocation pile-ups in metallic nanocomposites 137 v 5.0 Simulation of damage in metal matrix nanocomposites 141 5.1 Damage of inclusions 144 5.1.1 Effects of fracture strength, volume fraction and size of inclusions 145 5.1.2 156 5.2 6.0 vi Matrix damage 160 5.2.1 Effect of void formation in a pure metallic matrix 162 5.2.2 Effect of void formation in a metallic nanocomposite 166 5.2.3 Effect of inclusion volume fraction 170 5.2.4 174 Effect of inclusion arrangement Experimental verification 181 6.1 Experimental method 181 6.1.1 189 Fixture assembly and installation 6.1.2 Testing procedure 191 6.2 Materials processing and specimen fabrication 193 6.3 Experimental results 196 6.3.1 196 6.4 7.0 Effect of inclusion arrangement Hardness and tensile properties 6.3.2 Stress-strain response under shear 196 Comparison with numerical results 201 Conclusions and future work 205 7.1 Conclusions 205 7.2 Recommendations for future work 208 7.2.1 Effect of interfacial zone 208 7.2.2 Consideration of crystallographic details 213 References 217 Appendix A: Modifications to standard Iosipescu shear test fixture for testing of smaller specimens 231 Appendix B: List of publications 235 vii Summary A metal matrix composite (MMC) is a composite material with reinforcement phase which is dispersed within a continuous metallic host to improve the thermomechanical properties of the host metal. Recent experiments show that reducing the reinforcement size to the nanoscale dramatically increases the mechanical strength of MMCs. While extensive numerical studies on the mechanical properties of conventional MMCs have been conducted, only a handful of such studies exist for metal matrix nanocomposites (MMNCs). Numerical simulations are useful for performing virtual experiments on MMNCs to explore effects which are currently difficult to investigate experimentally and to analyse the underlying processes that govern the mechanical response of these materials. Hence, the objective of this study is to investigate the mechanical properties of MMNCs using numerical simulations in order to determine the best combinations of constituent material properties, compositions and microstructure for optimum mechanical performance of these materials. 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Yang, Y, Lan, J. and Li, X. 2004. Study on bulk aluminum matrix nano-composite fabricated by ultrasonic dispersion of nano-sized SiC particles in molten aluminum alloy. Materials Science and Engineering A 380: 378-383. Yao, Z., Wang, J.S., Liu, G.R., Cheng, M. 2004. Improved neighbor list algorithm in molecular simulations using cell decomposition and data sorting method. Computer Physics Communications 161: 27-35. Zhang, Z. and Chen, D. L. 2006. Consideration of Orowan strengthening effect in particulate-reinforced metal matrix nanocomposites: a model for predicting their yield strength. Scripta Materialia 54: 1321-1326. Zhang, J. T., Liu, L. S., Zhai, P. C., Fu, Z. Y. and Zhang, Q. J. 2007a. The prediction of the dynamic responses of ceramic particle reinforced MMCs by using multiparticle computational micro-mechanical model. Composites Science and Technology 67: 2775-2785. Zhang, W. X., Li, L. X. and Wang, T. J. 2007b. Interphase effect on the strengthening behavior of particle-reinforced metal matrix composites. Computational Materials Science 41: 145-155. Zhang, Z. and Chen, D. L. 2008. Contribution of Orowan strengthening effect in particulate-reinforced metal matrix nanocomposites. Materials Science and Engineering A 483-484: 148-152. 230 Appendix A: Modifications to standard Iosipescu shear test fixture for testing of smaller specimens The standard specimen dimensions for the Iosipescu shear test are shown in Figure 6.2 in which the total width is 20 mm while the nominal width of the gauge region is 12 mm. However, the total width of the Mg-ZnO nanocomposite specimen used in this study is restricted to only mm, with the nominal width between the notches reduced proportionally to mm. Hence, additional modifications are made to the test fixture in order to accommodate a smaller specimen without resorting to using a scaled-down fixture which is much more expensive than the standard-size fixture. As shown in Figure 6.7, additional metal pieces are required to reduce the distance between the contact points adjacent to the notches (i.e. reduce the length b shown in Figure 6.4) to prevent premature failure of the smaller specimen due to high bending stresses, as well as to fill in the gap in the grip regions due to the reduced specimen width. Calculations to determine the dimensions of these additional metal pieces are shown here. As shown in Figure 6.4, the maximum bending moment in the specimen is given by M max  Pb (A.1) Since the cross-section of the specimen is rectangular, the elastic section modulus S of the cross-section in the grip region about the (horizontal) bending axis is given by S hd1 (A.2) 231 where h is the thickness of the specimen while d1 is the total width as shown in Figure 6.2. Assuming that the shear stress τ is constant along the cross-section of the specimen in the notched region, the relationship between τ and the applied load P is given by P   hw (A.3) where w is the width of the specimen across the notched region. Therefore, if the grip region remains elastic throughout the duration of the test, the relationship between the maximum bending stress σmax in the grip region and the shear stress τ in the notched region can be determined as follows: M max 3bw  2 S d1  max  (A.4) To prevent premature failure of the grip region, the maximum bending stress σmax must be less than the yield strength σy of the specimen. Assuming that shear strength y y and substituting into Equation (A.4), the following relationship between b and the specimen dimensions can be obtained: bw d1  3  bw d1  0.577 (A.5) For the standard test fixture used in this study, b = 0.5 in ≈ 12.7 mm. Hence, for the standard-size specimen, bw d1 232  12.7  12  0.381  0.577 20 (A.6) Consequently, the standard-size specimen will not fail prematurely due to bending in the grip region with a factor of safety of 1.5. For the smaller specimen, d1 and w are limited to mm and mm respectively. Therefore, the corresponding limit for b is given by b 0.577d1 0.577    5.2 mm w (A.7) Hence, a value of b = 0.15 in ≈ 3.8 mm is suggested for the smaller specimen to prevent bending failure in the grip region. The corresponding factor of safety is approximately 1.35. Based on the suggested value of b and the dimensions of the smaller specimen, the dimensions of the additional metal pieces are determined as shown in Figure A.1. These additional metal pieces are inserted into the test fixture as shown in Figure 6.7 when testing of the smaller specimen is conducted. 0.500 1.625 0.125 0.550 0.550 1.800 0.250 0.075 (Units: inches) Figure A.1 Dimensions for additional metal pieces for testing of smaller specimens. 233 This page is intentionally left blank. 234 Appendix B: List of publications Law, E., Pang, S.D. and Quek, S.T. 2009. Numerical study of metal matrix nanocomposites using discrete dislocation approach. 22nd KKCNN Symposium on Civil Engineering, Chiang Mai, Thailand, October 31 – November 2. Quek, S.T., Law, E. and Pang, S.D. 2010. Mechanical response of metal matrix nanocomposites using discrete dislocation approach. 18th International Conference on Composites or Nano Engineering (ICCE-18), Anchorage, Alaska, USA, July 4-10. Law, E., Pang, S.D. and Quek, S.T. 2010. Effect of thermal residual stress on the mechanical response of metal matrix composites with nanosized reinforcement particles. 23rd KKCNN Symposium on Civil Engineering, Taipei, Taiwan, November 13-15. Law, E., Pang, S.D. and Quek, S.T. 2010. Effect of RVE size in the discrete dislocation simulation of the mechanical response of metal matrix composites with nanosized reinforcement particles. 7th Asian-Australasian Conference on Composite Materials (ACCM-7), Taipei, Taiwan, November 15-18. Law, E., Pang, S.D. and Quek, S.T. 2011. Discrete dislocation analysis of the mechanical response of silicon carbide reinforced aluminum nanocomposites. Composites Part B: Engineering 42(1): 92-98. Law, E., Pang, S.D. and Quek, S.T. 2011. Numerical analysis of the effect of particle arrangement on mechanical behavior and particle damage in metal matrix nanocomposites. 18th International Conference on Composite Materials (ICCM18), Jeju Island, Korea, August 21-26. Law, E., Pang, S.D. and Quek, S.T. 2012. Effects of particle arrangement and particle damage on the mechanical response of metal matrix nanocomposites: a numerical analysis. Acta Materialia 60(1): 8-21. 235 This page is intentionally left blank. 236 [...]... Distribution of dislocations within metallic matrix with τobs = 0.60 GPa and ρnuc of (a) 40 μm-2, and (a) 160 μm-2 82 xiii Figure 3.13 Mean overall response of metallic matrix for different values of τ*nuc with τobs of (a) 0.15 GPa, and (b) 1.20 GPa 83 Figure 3.14 Overall response of metallic matrix with different deformation stages indicated 87 Figure 3.15 Distribution of dislocations within metallic matrix. .. and τobs of (a) 0.15 GPa, and (b) 0.60 GPa 76 Figure 3.9 Mean overall response of metallic matrix for different values of τobs with ρobs of (a) 80 μm-2, and (b) 160 μm-2 78 Figure 3.10 Distribution of dislocations within metallic matrix with τobs = 0.60 GPa and ρobs of (a) 80 μm-2, and (a) 320 μm-2 80 Figure 3.11 Mean overall response of metallic matrix for different values of ρnuc with τobs of (a) 0.15... element in which a dislocation is located 71 Figure 3.6 Overall response of aluminum matrix for different realizations of random dislocation source and impurity distributions 75 Figure 3.7 Mean overall response of metallic matrix for different values of ρobs with τobs of (a) 0.15 GPa, and (b) 1.20 GPa 75 Figure 3.8 Deformation of RVE and distribution of dislocations within metallic matrix with ρobs =... = 200 MPa 159 Figure 5.14 (a) Mean overall response of pure metallic matrix with different values of εfailure, with the corresponding (b) percentage of voids, and (c) density of active dislocations (i.e excluding annihilated dislocations) in the matrix 163 Figure 5.15 Distribution of dislocations in pure metallic matrix with failure strain εfailure of (a) infinity, (b) 0.020 and (c) 0.015 at γave =... Figure 3.17 Mean overall response of metallic matrix for different RVE sizes with τobs = 0.15 GPa and ρnuc of (a) 160 μm-2, and (b) 40 μm-2 95 Figure 3.18 Deformation of RVE and distribution of dislocations within metallic matrix for RVE sizes of (a) 0.5 μm × 0.5 μm and (b) 3 μm × 3 μm with τobs = 0.15 GPa and ρnuc = 160 μm-2 97 Figure 3.19 Mean overall response of metallic matrix for different RVE sizes... and ρnuc of (a) 40 μm-2, and (b) 160 μm-2 98 Figure 3.20 Mean overall response of metallic matrix for different RVE sizes with τobs = 0.15 GPa and ρobs of (a) 80 μm-2, and (b) 320 μm-2 99 Figure 3.21 Mean overall response of metallic matrix for different RVE sizes with τobs = 0.60 GPa and ρobs of (a) 80 μm-2, and (b) 320 μm-2 100 Figure 3.22 Overall response for various realizations of metallic matrix. .. size of 25 nm for different values of εfailure, with the corresponding (b) percentage of voids, and (c) density of active dislocations (i.e excluding annihilated dislocations) in the matrix 175 Figure 5.21 Distribution of dislocations in composite material with regular rectangular arrangement of 2 per cent inclusion volume fraction and inclusion size of 25 nm at γave = 1.05% for different values of εfailure... nucleation strength of dislocation sources τobs Strength of obstacles or impurities xxiii This page is intentionally left blank xxiv 1.0 Introduction This chapter gives an overview of metal matrix composites (MMCs) and metal matrix nanocomposites (MMNCs), the numerical studies which have been performed on these materials, the size effects observed in MMNCs, and the numerical methods for modelling dislocations... values, and (b) maximum number of subdivisions for adaptive quadrature used to evaluate boundary nodal forces due to dislocation stress field 62 Figure 3.3 Overall response for composite material with different cutoff distances rcut used in the evaluation of dislocation fields 64 Figure 3.4 Distribution of dislocations within matrix of composite material for cut-off distance rcut of (a) 100 b, and (b) 10000... 5.16 (a) Mean overall response of composite material with nonclustered random arrangements of 2 per cent inclusion volume fraction and inclusion size of 25 nm for different values of εfailure, with the corresponding (b) percentage of voids, and (c) density of active dislocations (excluding annihilated dislocations) in the matrix 167 xviii Figure 5.17 Distribution of dislocations in composite material . NUMERICAL STUDY OF METAL MATRIX NANOCOMPOSITES USING DISCRETE DISLOCATION APPROACH ELLIOT LAW NATIONAL UNIVERSITY OF SINGAPORE 2011 NUMERICAL STUDY OF METAL. of inclusions 124 4.5 Material properties of constituent phases 132 4.6 Nature of dislocation pile-ups in metallic nanocomposites 137 vi 5.0 Simulation of damage in metal matrix nanocomposites. response of metallic matrix for different values of ρ obs with τ obs of (a) 0.15 GPa, and (b) 1.20 GPa. 75 Figure 3.8 Deformation of RVE and distribution of dislocations within metallic matrix