Plastic deformation and constitutive modeling of magnesium based nanocomposites

PLASTIC DEFORMATION AND CONSTITUTIVE MODELING OF MAGNESIUM-BASED NANOCOMPOSITES CHEN YANG (B.Eng., Sichuan University M.Eng., Sichuan University M.Eng. National University of Singapore) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2014 Declaration I hereby declare that this thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Chen Yang 2015 Acknowledgements Acknowledgements I would like to gratefully thank Prof. Victor P.W. Shim and Prof. Manoj Gupta for their invaluable guidance, continuous support and encouragement in this work. In particular, I wish to express my sincere gratitude to Prof. Shim for his patience, insightful advice and efforts in maintaining a wonderful research environment. His serious and positive attitude on research, communication and life inspires me far beyond this work. I would like to thank all the members in the Impact Mechanics Laboratory. Thanks to the laboratory officers, Mr. Jow Low Chee Wah and Mr. Alvin Goh Tiong Lai for their technical support in experimental work. As for my friends and colleagues, I would like to thank Dr. Guo Yangbo, Dr. Gao Guangfa, Dr. Nguyen Quy Bau, Dr. Tan Long Bin for their kind assistance with experiments and valuable discussions on modelling, my thanks also to Dr. Jing Lin, Dr. Ma Dongfang, Dr. Kianoosh Marandi, Ms. Xu Juan, Dr. Habib Pouriayevali, Mr. Saeid Arabnejad Khanooki, Mr. Nader Hamzavi, and Mr. Emmanuel Tapie for their help in matters relating to work and life in the laboratory. Special thanks to professors who taught courses I took, for sharing their knowledge and time to give me a better understanding of the theories related to my research. I would also like to thank the National University of Singapore for providing a research scholarship to support my pursuit of graduate studies. Last, but by no means least, I would like to express my sincere gratitude to my family for their love, unending care, constant support and encouragement. ii   Table of Contents Table of Contents Declaration . i  Acknowledgements ii  Summary vi  List of Figures . viii  List of Tables xiii  Nomenclature xiv  Chapter – Introduction . 1  Chapter – Literature Review . 5  2.1 Development of magnesium based metal matrix composites . 5  2.2  Deformation mechanisms in magnesium 7  2.2.1  Slip and twinning systems in magnesium . 8  2.2.2  Introduction to deformation twinning . 10  2.2.3  CRSS for different slip and twinning systems in magnesium . 13  2.3  Mechanical behavior of magnesium and its alloys . 15  2.3.1  Mechanical behavior of pure magnesium single crystals 15  2.3.2  Mechanical behavior of magnesium alloys . 18  2.3.3  Mechanical behavior of magnesium alloys under dynamic loading . 23  2.4  Constitutive modeling using crystal plasticity theory . 29  Chapter – Experiments . 32  3.1  Materials Used . 32  3.2  Primary processing (DMD) . 34  3.3  Hot extrusion . 34  3.4  Microstructure characterization . 36  3.5  Density measurement 36  3.6  X-ray diffraction studies 37  3.7  Quasi-static tension and compression tests . 37  iii   Table of Contents 3.7.1  Tensile tests . 37  3.7.2  Compression tests 38  3.8  Dynamic mechanical tests using Split Hopkinson Bar (SHB) Devices 40  3.8.1  Tensile tests . 40  3.8.2  Compression tests 42  3.9  Fractography 42  Chapter – Mechanical properties of magnesium nanocomposites under quasistatic and dynamic loading 44  4.1  Introduction . 44  4.2  Experimental results for tensile properties and discussion . 45  4.2.1 Macrostructure . 45  4.2.2 Microstructures . 46  4.2.3 Texture change during quasi-static and dynamic tensile tests 48  4.2.4 Quasi-static tensile mechanical properties 53  4.2.5 Dynamic tensile mechanical properties 56  4.2.6 Fractography for tension tests 63  4.3  Results for compressive properties and discussion . 65  4.3.1 Texture change during quasi-static and dynamic compression tests 65  4.3.2 Quasi-static compressive mechanical properties 68  4.3.3 Dynamic compressive mechanical properties . 71  4.3.4 Competition between grain size and strain rate effect on flow stress 78  4.3.5 4.4  Fractography for compression tests 82  Conclusions . 85  Chapter – Constitutive modeling . 87  5.1  Introduction . 87  5.2  Slip and twin systems 89  iv   Table of Contents 5.3  Constitutive model 92  5.3.1  Constitutive law for slip and twin . 94  5.3.2  Evolution of twin volume fractions . 99  5.4  Numerical implementation 101  5.4.1  Finite element model . 101  5.4.2 Model validation 102  5.4.3  Calibration of material parameters for constitutive model 105  5.4.4  Initial texture in simulations 107  5.4.5  Simulation of tensile loading  results and discussion 109  5.4.6  Simulation of compressive loading  results and discussion 115  5.4.7 Numerical tests for three simple textures in monolithic AZ31 120  5.5 Conclusions . 124  Chapter – Conclusions and recommendations for future work . 125  6.1  Conclusions . 125  6.2  Recommendations for future work 127  Bibliography . 129  Appendix A. Time-integration procedure for crystal plasticity constitutive model 148    v   Summary Summary The focus of this research effort is to investigate the response of AZ31 Mg alloy based nanocomposites to tensile and compressive plastic deformation and to develop a crystal plasticity model to capture the essential characteristics of the behaviour observed. AZ31 Mg alloy, reinforced by different volume fractions of 50-nm Al2O3 nanoparticles (1v%, 1.4v% and 3v%), were fabricated by a disintegrated melt deposition technique, followed by hot extrusion. The tensile and compressive mechanical behaviour of these materials at strain rates spanning 10-4 to 103 s-1 was investigated. Compared to monolithic AZ31, the nanocomposites display significantly increased yield stress and ultimate stress for both low and high rate tension, indicating the positive influence of nanoparticles. For each type of material, the strain to failure increases with strain rate for both tension and compression. The ductility of nanocomposites is significantly higher than that of monolithic AZ31, by 49% and 33% respectively for low and high rate tensile loading; this is attributed to the addition of nanoparticles. In contrast, there is no obvious increase in ductility for the nanocomposites subjected to compression; this difference is attributed to the activation of {10 12}  10 1  tension twinning under compression. Such twinning generates many sites for crack initiation and propagation. Consequently, the influence of nanoparticles on ductility is diminished by the activation of tension twinning during compressive deformation. A rate-dependent crystal plasticity model is developed to investigate the mechanical responses of AZ31-based nanocomposites, and implemented in vi   Summary ABAQUS/Explicit (2010) finite element software via a user-defined material subroutine (VUMAT). The effect of the nanoparticles is captured by incorporating a term describing the interaction between the nanoparticles and slip/twinning in the hardening evolution laws for slip/twinning. The simulation results match the experimental stress-strain curves closely, and show that the addition of nanoparticles does not change the average relative degree of activity of slip and twinning during deformation. vii   List of Figures List of Figures Fig. 2-1 Different types of metal matrix composites [36] Fig. 2-2 Hexagonal close-packed structure: a unit cell of the lattice and a hexagonal cell showing the arragement of atoms [38]. Fig. 2-3 Plastic deformation modes in a hexagonal-close packed structure: (a) Basal  a  slip systems, (b) prismatic  a  slip systems, (c) pyramidal  c  a  slip systems and (d) tensile twin [47]. . Fig. 2-4 (a) Depiction of twinning in a localized region in a crystal (b) Crystallographic twinning elements [51] 11 Fig. 2-5 Variation of twinning shear with axial ratio. For the seven hexagonal close packed metals, a filled symbol indicates that twinning is an active mode [52]. . 12 Fig. 2-6 Experimental values of CRSS for slip and twin modes in pure Mg single crystals [55] . 13 Fig. 2-7 Stress-strain curves for pure magnesium single crystals compressed along seven different directions. A - G denote initial crystal orientations [60]. . 16 Fig. 2-8 Temperature-dependence of the critical resolved shear stresses for basal and prismatic slip in magnesium and titanium [62]. . 18 Fig. 2-9 Stress-strain curves for pure magnesium single crystals compressed along (a) C direction; (b) D direction [60]. . 20 Fig. 2-10 Hall-Petch plots of the flow stress at 0.2% offset strain [65] 20 Fig. 2-11 Mechanical response of AZ31 for simple compression and tension at room temperature and a constant strain rate of 10-3 s-1 [69]. 21 Fig. 2-12 Nominal stress-strain relations for the annealed AZ31 alloy followed by ECAE, and the same alloy by direct extrusion [11]. . 22 Fig. 2-13 Plastic flow curves for out of plane compression at various temperatures [75]. . 23 viii   List of Figures Fig. 2-14 Compressive stress-strain response of AZ31, showing the influence of anisotropy effect on strain rate dependence [77] . 24 Fig. 2-15 Comparison of strain rate effect on stress at 5% plastic strain at (a) room temperature and (b) elevated temperatures [79] 26 Fig. 3-1 Flow of experimental work . 33 Fig. 3-2 Array of holes in AZ31 disk to contain particles 33 Fig. 3-3 Schematic diagram of disintegrated melt deposition technique [103] 35 Fig. 3-4 Quasi-static (a) Tensile and (b) compressive test specimens (unit: mm) 38 Fig. 3-5 Quasi-static tension and compression tests using an Instron 8874 universal testing machine 39 Fig. 3-6 Geometry of specimen for dynamic tensile tests 41 Fig. 3-7 Typical stress wave recorded during dynamic tests 41 Fig. 3-8 Schematic diagram of (a) the tensile and (b) compressive SHB devices 43 Fig. 4-1 Macrographs of (a) a cast ingot and (b) an extruded rod. . 45 Fig. 4-2 Optical micrographs showing grain characteristics of (a) AZ31; (b) AZ31/1.0Al2O3; (c) AZ31/1.4Al2O3; (d) AZ31/3.0Al2O3 . 47 Fig. 4-3 SEM micrograph showing the nanoparticle distribution of AZ31/1.4Al2O3 (a) low magnification; (b) white spots are the Al2O3 nanoparticles. . 48 Fig. 4-4 X-ray diffraction patterns of the samples before tests for (a) AZ31; (b) AZ31/1.0 vol%Al2O3 . 50 Fig. 4-5 X-ray diffraction patterns for three types of samples (after extrusion, after quasi-static tensile tests and after dynamic tensile tests) for materials: (a) AZ31; (b) AZ31/1.0 vol%Al2O3; (c) AZ31/1.4 vol%Al2O3; (d) AZ31/3.0 vol%Al2O3. 52 Fig. 4-6 True stress-strain curves of quasi-static tensile tests on AZ31 and its nanocomposites at room temperature 54 ix   Bibliography 58. 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Time-integration procedure for crystal plasticity constitutive model The time-integration procedure for the proposed rate-dependent single crystal constitutive model is presented. Here let t denote the current time; t is a small finite time increment, and   t  t the next instant of time. The algorithm is as follows: Given: (1) {F(t ), F( ), F P (t ), T(t )} ; (2) {m0k , n0k } , n0k is the unit normal to the slip/twin plane, and m0k the unit vector that identifies the slip/twin direction, where k denotes the k -th slip/twin system; (3) the rotation tensor Q(t ) , which rotates the crystal coordinates to coincide with the polycrystalline sample fixed coordinates; (4) the accumulated shear due to twinning f tw (t ) , where  implies the  -th twin system. Calculate: (a) {F P ( ), T( )} ; (b) ftw ( ) ; (c) the rotation tensor Q ( ) . Step 1. Determine the elastic deformation gradient F e (t ) and elastic strain Ee (t ) : F e (t )  F(t )F P (t ) 1   (A-1) Ce (t )  F e (t )T F e (t )   (A-2) e (C (t )  I )   (A-3) Ee (t )  Step 2. Calculate the stress, T* (t ) : T* (t )  Ee (t )   (A-4) Step 3. Calculate the resolved shear stress on the k -th slip/twin system,   (t ) : 148   Appendix A Time-integration procedure for crystal plasticity constitutive model   (t )  m0k  (Ce (t )T* (t )) n0k   (A-5) Step 4. Calculate the plastic shearing rate on the k -th slip/twin system,  k :   0 k k s m k sign( k )    (A-6) Step 5. Update the plastic deformation gradient, F P ( ) : N sl Ntw i  F ( )  {1    i (t )(m0i  n0i )t    (t )(m0  n0 )t}F P (t )   P (A-7) Step 6. Normalize F P ( ) by: F P ( )  [det F P ( )]1/3 F P ( )   (A-8) Step 7. Determine the elastic deformation gradient F e ( ) and the elastic strain, Ee ( ) : F e ( )  F( )F P ( ) 1   (A-9) Ce ( )  F e ( )T F e ( )   (A-10) Ee ( )  (Ce ( )  I )   (A-11) Step 8. Calculate the stress, T* ( ) :   T* ( )  Ee ( )   (A-12) Step 9. Determine the Cauchy stress, T( ) : T( )  [det F ( )]1 F e ( )T* ( )F e ( )T   (A-13) Step 10. Update the strength of the k -th slip/twin system, s k ( ) : s k ( )  s k (t )  s k t   (A-14) Step 11. Update the twin fraction, ftw ( ) : ftw ( )  ftw (t )  ftw t   Step 12. Check the lattice reorientation. 149   (A-15) Appendix A Time-integration procedure for crystal plasticity constitutive model If f tw  f cr , where f cr is the critical strain for lattice rotation, set Q( )  Q (t )( R tw )T , where Rtw is rotation matrix for the tension twin. 150   [...]... method to produce magnesium- based nanocomposites for the study undertaken Although there has been a lot of research, especially in recent years, on the deformation response and constitutive modelling of magnesium alloys, e.g AZ31 [26, 27], most of the research efforts on magnesium- based nanocomposites focus on their synthesis Studies on the deformation behaviour of magnesium- based nanocomposites, which... tensile specimens of (a) monolithic AZ31 and (b) AZ31/1.4% Al2O3 at strains of 0%, 4% and 10% 111 Fig 5-13 Relative degree of activation of slip and twinning in monolithic AZ31 during quasi-static tensile deformation 113 Fig 5-14 Relative degree of activation of slip and twinning in AZ31/1.4% Al2O3 during quasi- static tensile deformation 114 Fig 5-15 Comparison of simulated and experimental... zinc and cadmium follow the FCC case [32] For magnesium and its alloys, plastic deformation of room temperature is primarily related to slip and twinning Their SRS is closely linked slip and twinning activated during deformation [33] Dynamic loads, arising from accidental collisions or foreign-object impact, must be considered in the design of vehicles and aircraft, and therefore, an understanding of. .. is presented and comprises four parts The first part describes development of magnesium- based metal matrix composites The second introduces deformation mechanisms in magnesium The mechanical behaviour of magnesium single crystals and its alloys is presented in the third part, and the last part is related to the development of 3   Chapter 1 Introduction constitutive modelling using crystal plasticity... diagrams of three representative textures 121 xii   List of Figures Fig 5-20 Stress-strain responses for three types of texture under unixial tension and compression 122 Fig 5-21 Average relative activity of slip and twin modes for three types of textures under unixial tension and compression 123 xiii   List of Tables List of Tables Table 2-1 Some physical properties of magnesium. .. with the influence of strain rate and texture are also investigated A constitutive model based on crystal plasticity, which includes the effects of nanoparticles, is formulated in Chapter 5 For magnesium, dislocation slip and twinning are the main deformation mechanisms considered in this crystal plasticity model The effect of nanoparticles is reflected in the hardening laws for slip and twinning The... dislocation theory show that the physical mechanism of plastic deformation of crystalline material is associated with the movement of dislocations Various slip and twinning systems have been identified via research over the years, to be activated during plastic deformation of magnesium [41] Basal slip and pyramidal twinning were first reported in 1939 by Beck as the deformation mechanism at low homologous temperatures... slip and {10 12} twinning systems The exceptions for orientations C and D (Fig.2-9) in magnesium- lithium crystals arise because Mg-4% Li displays a much lower post-yield strength than that of pure magnesium and Mg0.5%Th This is the result of activation of prismatic {10 10}  1210  slip, which is not observed in either pure magnesium or Mg-0.5%Th Compared to single crystals, the plastic deformation of. .. al [43], Reed-Hill and Robertson [44], as well as Yoshinaga and Horiuchi [45] Following this, Obara, Yoshinaga et al [46] and Ando and Tonda [47] identified pyramidal  c  a  slip as another active deformation mechanism at low homologous temperatures Fig 2-3 shows the different slip and twin systems Based on the von Mises criterion [48], activation of five independent plastic deformation systems... slip and twinning systems in magnesium The activation of slip and twinning systems are generally related to the CRSS When the resolved shear stress on certain slip or twinning plane is equal to or larger than the value of its CRSS, this slip or twinning system will be activated and contribute to plastic deformation of the crystalline material [52, 58] The experimental values of the CRSS for the slip and . PLASTIC DEFORMATION AND CONSTITUTIVE MODELING OF MAGNESIUM- BASED NANOCOMPOSITES CHEN YANG (B.Eng., Sichuan University M.Eng., Sichuan University M.Eng. National University of Singapore). and twinning systems in magnesium 13 2.3 Mechanical behavior of magnesium and its alloys 15 2.3.1 Mechanical behavior of pure magnesium single crystals 15 2.3.2 Mechanical behavior of magnesium. produce magnesium- based nanocomposites for the study undertaken. Although there has been a lot of research, especially in recent years, on the deformation response and constitutive modelling of magnesium