MODELING THE COUPLING BETWEEN MARTENSITIC PHASE TRANSFORMATION AND PLASTICITY IN SHAPE MEMORY ALLOYS DISSERTATION

Transcription

1 MODELING THE COUPLING BETWEEN MARTENSITIC PHASE TRANSFORMATION AND PLASTICITY IN SHAPE MEMORY ALLOYS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Sivom Manchiraju, B. Tech., M.S. Graduate Program in Materials Science and Engineering The Ohio State University 2011 Dissertation Committee: Professor Peter Anderson, Advisor Professor Michael Mills Professor Yunzhi Wang Dr. Ronald Noebe

2 Copyrighted by Sivom Manchiraju 2011

3 ABSTRACT The thermo-mechanical response of NiTi shape memory alloys (SMAs) is predominantly dictated by two inelastic deformation processes martensitic phase transformation and plastic deformation. This thesis presents a new microstructural finite element (MFE) model that couples these processes and anisotropic elasticity. The coupling occurs via the stress redistribution induced by each mechanism. The approach includes three key improvements to the literature. First, transformation and plasticity are modeled at a crystallographic level and can occur simultaneously. Second, a rigorous large-strain finite element formulation is used, thereby capturing texture development (crystal rotation). Third, the formulation adopts recent first principle calculations of monoclinic martensite stiffness. The model is calibrated to experimental data for polycrystalline NiTi (49.9 at% Ni). Inputs include anisotropic elastic properties, texture, and DSC data as well as a subset of pseudoelastic and load-biased thermal cycling data. This calibration process provides updated material values namely, larger self-hardening between similar martensite plates. It is then assessed against additional pseudoelastic and load-biased thermal cycling experimental data and neutron diffraction measurements of martensite texture evolution. Several experimental trends are captured in particular, the transformation strain during thermal cycling monotonically increases with increasing bias stress, reaching a peak and then decreasing due to intervention of plasticity a trend which ii

4 existing MFE models are unable to capture. Plasticity is also shown to enhance stressinduced martensite formation during loading and generate retained martensite upon unloading. The simulations even enable a quantitative connection between deformation processing and two-way shape memory effect. Some experimental trends are not captured in particular, the ratcheting of macrostrain with repeated thermal cycling. This may reflect a model limitation that transformation-plasticity coupling is captured on a coarse (grain) scale but not fine (martensitic plate) scale. Lastly, Crystallographic Theory of Martensite (CTM) and micromechanics-based modeling is applied to analyze recent TEM observations. In particular, the observation of sub-micron dislocation loops is explained in terms of the large stress generated by the phase transformation at the variant (sub-micron) scale. Second, the observation of atypical compound twin related martensite variants in TEM foils is explained in terms of the loss of constraint produced by free-surfaces. iii

5 DEDICATED TO MY PARENTS Anand Swaroop Manchiraju and Lakshmi Raja Rajeshwari iv

6 ACKNOWLEDGEMENTS During my long graduate student life, I owe my gratitude to many people who have helped me directly or indirectly in completion of this thesis and in having a wonderful time during this phase of life. I would like to acknowledge my deepest thanks to all of them. Some of them whom, I would like to point out are My Ph.D. advisor Prof. Peter Anderson: who is the driving force behind this thesis. His guidance, encouragement, patience and above all his teaching have made this thesis possible. He supported me at a crucial juncture of my graduate student life and opened new frontiers of research to me. I sincerely thank him for this. Prof. Michael Mills: who has been like a co-advisor to me, especially when it comes to experimental aspects of this work. He introduced and got me interested in experimental aspects of materials science. His constant inputs have kept me honest with my modeling and have given me new directions in research. Prof. Yunzhi Wang: for being on my committee, providing guidance and encouragement and teaching Phase Field Method. Dr. Ronald Noebe: for being on my committee, providing exciting experimental data to work on and above all, keeping me honest with my modeling. My labmates: Lin Li, John Carpenter, Harshad Paranjape, Mike Gram and Xiang Chen for providing intellectual atmosphere conducive for research and for many discussions both technical and non-technical, which made the lab a fun place to work. v

7 I would also like to thank Prof. Raj Vaidyanathan and his group (Dr. Shipeng Qiu and Othmane Benafan) at University of Central Florida for the neutron diffraction data, Dr. Santo Padula and Darrell Gaydosh at NASA GRC for the experimental data of polycrystalline NiTi, Prof. Gunther Eggeler at Uni-Bochum, Germany for the in-situ TEM experimental data, Dr. Dave Norfleet and Dr. Peter Sarosi (OSU) for the experimental data for NiTi micropillar, Dr. Nick Hatcher (Northwestern University) for the anisotropic elastic moduli, Prof. Myoung-Gyu Lee (Pohang University of Science and Technology) for texture analysis and Prof. William Clark (OSU) for CTM discussions. I would also like to thank my friends with whom I had many fruitful technical discussions- Sanket Sarkar, Jayesh Jain, Dakshinamurthy Valiveti, Himanshu Bhatnagar, Deepu Joseph, Anand Srivastava, Abhijit Tiwari, Gayatri Venkataramani, Kedar Kirane, Yash Bhandari as well as many others, with whom I may not have had any technical discussions but who made the stay at OSU a pleasant experience and hence deserve my special thanks. Most importantly, I would like to thank my parents, my brother and my entire family for their unwavering support and encouragement. I also acknowledge the financial support from National Aeronautics and Space Administration (Grant no. NNX08AB49A) and the computational support from the Ohio Super Computer Center (Grant no. PAS676). Their support is gratefully acknowledged. vi

8 VITA Born Machilipatanam, India B.Tech., Mechanical Engineering, Jawaharlal Nehru Technological University, Hyderabad, India M.S., Mechanical Engineering, Michigan State University, E. Lansing, Michigan, USA University Fellow, The Ohio State University, Columbus, Ohio Present.. Graduate Research Associate, The Ohio State University, Columbus, Ohio JOURNAL PUBLICATIONS S. Manchiraju, P. M. Anderson, 2010, Coupling between martensite phase transformations and plasticity: A microstructure-based finite element model, Int. J. Plasticity, 26, p D. M. Norfleet, P. M. Sarosi, S. Manchiraju, M. F. X. Wagner, M. D. Uchic, P. M. Anderson, M. J. Mills, 2009, Transformation-induced plasticity during pseudoelastic deformation in Ni-Ti microcrystals, Acta Mater., 57, p S. Manchiraju, K.Kirane and S. Ghosh, 2008, Dual-time scale crystal plasticity FE model for cyclic deformation of Ti alloys, Journal of Computer-Aided Materials Design, 14, p S. Manchiraju, M. Asai and S. Ghosh, 2007, A dual-time scale finite element model for simulating cyclic deformation of polycrystalline alloys, Journal of Strain Analysis for Engineering Design, 42, p D.Liu, S. Manchiraju, D.Templeton, B. Raju, 2005, Finite element simulations of composite vehicle structures under impact loading, SAE Transactions J. Materials and Manufacturing, 114, p vii

9 FIELDS OF STUDY Major Field: Materials Science and Engineering Studies in: Mechanical Behavior of Materials, Continuum Mechanics and Finite Element Method viii

10 TABLE OF CONTENTS Abstract. Dedication. Acknowledgments... Vita... List of Tables List of Figures Page ii iv v vii xiii xiv CHAPTERS: 1 Introduction SMA Response under Thermo-Mechanical Loading Stress Free Temperature Cycling Load Biased Temperature Cycling Isothermal Deformation at θ>θ AF Isothermal Deformation at θ<θ MF Continuum Level Models for SMA Phenomenological Models Microstructure Based Models Objectives and Scope Dissertation Outline ix

11 2 Constitutive Relations and Finite Element Model Constitutive Relations Numerical Implementation of Constitutive Relations Constitutive Relations for Martensite Detwinning at θ<θ MF Model Application to Solutionized 50.9 at.% Ni-Ti Finite Element Geometry for NiTi Polycrystals Material Parameters: Calibration to Single Crystal Data Response of an Initially Stress-Free Random Polycrystal Pseudoelastic Tensile Response at = AF + 24 K Thermal cycling response under load bias Effect of Pre-Straining Randomly Oriented Polycrystals Summary Model Application to Hot-Worked 49.9 at.% Ni-Ti Experimental Characterization Material System and History Texture Analysis Differential Scanning Calorimetry Determination of the Stress-Strain Response Above θ AF Load-Biased Thermal Cycling In-Situ Neutron Diffraction Polycrystalline Simulations x

12 4.2.1 Discretization at the Grain Scale and Texture Specification Predeformation E p(pre) in the Austenitic State Isothermal Deformation Testing and Post Heating Stress-Biased Thermal Cycling In-Situ Neutron Diffraction Simulations Calibration Of Material Parameters Elastic-Thermal, P el-thermal = {C A, C M, A th-a, A th-m } Transformation, P trans = {θ T, λ T, f c and h tu } Determination of Structure of h tu Austenite Plasticity, P plastic = { γ 0,m, g 0 s, g sat, h 0, Q, a, E p(pre) } and T Update Model Assessment Isothermal Deformation Response Thermal Cycling Response Effect of Bias Stress and Prestrain on Transformation Strain and Critical Temperatures Plastic Strain Enhancement Due to Phase Transformation Discrepancy in Open Loop Strain Assumptions Concerning Martensite Elastic Moduli and Thermal Expansion Coefficients Texture Evolution.. 73 xi

13 4.5 Model Calibration and Assessment for Isothermal ζ ε Response at θ<θ MF Summary Modeling Micron Scale Single Crystal SMA Response CTM based modeling Micromechanics Based Model for Stress Field Calculation Pseudoelastic Compression of NiTi Micro-Pillar Experimental Observations Analytic Modeling of the Preferred Martensite Plate Analytic Modeling of Stressed Slip Systems near a Martensite Plate FEM Analysis of the Pseudoelastic Compression of Pillar Martensitic Transformation during Tensile loading and In-situ TEM analysis Experimental Observations Analytical Modeling for Favored Variants Model Application to Tensile Tests Along [1 1 1] and [1 1 2 ] Summary Conclusions and Future Work References Appendix A Appendix B xii

14 LIST OF TABLES Table Page 3.1 Material parameters in the constitutive relations calibrated for solutionized Ti-50.9 at% Ni Calibrated properties for hot rolled/hot drawn polycrystalline 49.9 at% Ni-Ti (55 wt% Ni) Analysis of Potential Twinning Modes and Martensitic Variants Legend for the Austenite slip systems shown in Figs. 5.4 and xiii

15 Figure LIST OF FIGURES Page 1.1 Illustration of shape memory effect and microstructure at different stages of thermo-mechanical loading of a Shape Memory Alloy (SMA). Shape recovery is complete in the absence of plastic deformation. The transformation temperatures θ AF and θ MF are defined in Section A schematic of SMA response showing inelastic deformation due to austenite to martensite transformation. Plastic deformation is absent in this schematic. Plastic deformation can change this schematic response significantly (see Chapters 3-4); (a) Isothermal axial macroscopic stress vs. macroscopic axial strain E response at temperature θ>θ AF ; (b) Isobaric axial macrostrain E vs. temperature θ during thermal cycling under a bias stress (a) Finite element representation of a polycrystal consisting of a array of elements, each representing a grain. The shading indicates the grain-to-grain variation in maximum Schmid factor for crystal slip; (b) inelastic processes at a material point involving (i) formation of martensite plate type t with deformation gradient S t Trans, invariant plane normal m t Trans, displacement vector b t Trans, and volume fraction v t ; and (ii) plastic deformation on slip system s with deformation gradient S s Slip, invariant plane normal m s Slip, displacement vector b s Slip, and activity t Room temperature (θ AF + 24 K) macrostress 33 vs. macrostrain E 33 response in uniaxial compression for solutionized Ti 50.9at%Ni single crystals oriented along [111] vs. [210]. Experimental measurements are from [12] and the fitted predictions are from the finite element model with the material parameters in Table Room temperature (θ AF + 24 K) macrostress 33 vs. macrostrain E 33 response in uniaxial compression for solutionized Ti 50.9at%Ni single crystals oriented along (a) [100], (b) [110], (c) [211], and (d) [123]. Experimental measurements are from [12] and the finite element model predictions are based on the material parameters in Table 3.1. The maximum Schmid factors for transformation and crystal slip ( ζ, ζ ) are shown for each case T max S max xiv

16 3.3 Predicted tensile macrostress 33 vs. macrostrain E 33 response for an initially stress-free, random polycrystal of solutionized Ti 50.9 at% Ni at room temperature ( AF + 24 K), assuming either (i) transformation + plasticity mechanisms or (ii) transformation only mechanism. The average martensite volume fraction (v M ) is indicated at various points Predicted distributions of (a) remnant martensite volume fraction v M and (b) P remnant equivalent plastic strain for the transformation + plasticity case in Fig. 3.3, after unloading to point C (Fig. 3.3), and (c) remnant equivalent P plastic strain for the plasticity only case, after unloading (not shown in Fig. 3.3). Maxima and minima are listed and type A, B, and C sites are identified Predicted (a) local martensite volume fraction v M and (b) local slip activity = vs. local strain 33 at sites A, B, and C 1 in Fig The analysis corresponds to the transformation + plasticity case in Figs. 3.3 and Predicted remnant martensite volume fraction v M vs. remnant equivalent P plastic strain at 5832 integration points throughout the polycrystal, after unload. The analysis corresponds to the transformation + plasticity case in Figs. 3.3 and 3.4. Approximate trend lines for A and B type sites are shown Predicted (a) macrostrain E 33 vs. temperature for an initially stress-free, random polycrystal of solutionized Ti 50.9 at% Ni, subjected to a constant tensile stress bias 33 bias = 300, 500, 550, or 600 MPa; and (b) macro martensite volume fraction vs. temperature for 33 bias = 600 MPa. The results in (a) are for the transformation + plasticity case, where E trans is the difference in macro strain E 33 at the lower cycle temperature and at the upper cycle temperature at the end of thermal cycle, E p cycle is the accumulated plastic strain during the cooling + heating cycle, and the symbols ( ) show the evolution in temperatures ( MS, MF, AS, AF ). The results in (b) are for transformation + plasticity and transformation only cases and show the increased hysteretic width for the former case Predicted (a) 1 st cycle and (b) 2 nd cycle macrostress 33 vs. macrostrain E 33 response at room temperature ( AF + 24 K), for a random polycrystal of solutionized Ti 50.9 at% Ni subjected to different pre-strains: 6%, 6%, and 0 (no pre-strain). The pre-strain is imposed by heating the sample and deforming it in the austenitic state, then cooling to 298 K, and setting the reference macrostrain E 33 = 0 prior to loading. The circles indicate the approximate onset of forward (A M) transformation during loading and reverse (M A) transformation during unloading 53 xv

17 3.9 Predicted fraction of material points undergoing transformation vs. macrostress 33 for the pre-strain = 6% and 0% cases in Fig The 1 st cycle responses (solid curves) begin at the symbols and the second cycle responses (dashed curves) begin at the symbols. The symbols indicate the approximate onset stress MS for the forward (A M) transformation for various cases Predicted macro volume fraction of martensite v M vs. macrostrain E 33 during cyclic loading, for the pre-strain = 6%, 6%, and 0 (no pre-strain) cases in Fig The 1 st cycle responses begin at the symbols and the 2 nd cycles responses begin at the symbols Predicted macrostrain E 33 and macro volume fraction of martensite v M vs. temperature during a cooling + heating cycle, for a random polycrystal of solutionized Ti 50.9 at% Ni subjected to 6% pre-strain. The pre-strain is imposed by heating the sample and deforming it in the austenitic state. The symbols indicate the start of the cooling+ heating cycle Axial macrostrain E vs. temperature θ under an axial bias stress Σ bias showing (a) experimental data for the 49.9 at% Ni-Ti alloy and (b) calibrated model results. The inserts show the transformation strain E T vs. Σ bias, where E T is defined by feature F7 in (a). The calibrated model parameters are obtained by fitting features F1-F14 as summarized in Table Axial pole figures showing texture for polycrystalline austenite (a) experiments using HIPPO diffractometer (hot-worked 55 wt% Ni-Ti) and (b) simulations obtained by fitting to experimental texture results from SMARTS assuming axisymmetry. A strong (111) texture and weak (100) texture is observed Axial macroscopic stress Σ vs. macroscopic axial strain E at different test temperatures (θ 0 = 130 and 215 ºC) showing (a) experimental data for the 49.9 at% Ni-Ti alloy and (b) calibrated model results. E unload and E post-heat are the macrostrains after unloading and after a 600 ºC post heat treatment of the unloaded sample, respectively. The calibrated model parameters are obtained by fitting features F1-F14 as summarized in Table (a) Axial macrostrain E vs. temperature θ during thermal cycling with an axial stress bias bias =50 MPa and (b) axial macrostress vs. macrostrain E at test temperature 0 = 130 ºC. The experimental data is for the polycrystalline 49.9 at% Ni-Ti alloy. The calibrated model results use the fitted parameters as summarized in Table 4.1. Other simulation results in (a) use the fitted parameters except with the martensite interaction matrix h tu of Patoor et al. [31] or with E p(pre) = 0. The E p(pre) = 0 cases in (b) are nearly xvi

18 coincident and use (g s 0, h0 ) = (235 MPa, 500 MPa) vs. (250 MPa, 50 MPa) case (the former is slightly higher) Axial macrostrain E vs. temperature during thermal cycling with zero bias stress. The experimental data is for the polycrystalline 49.9 at% Ni-Ti alloy. The calibrated model parameters are obtained by fitting features F1-F14 as summarized in Table 4.1. The predicted two-way effect occurs because E p(pre) = 0.7% Axial transformation strain E T vs. axial bias stress bias during thermal cycling between min = 30 ºC and max = 165 ºC (lower experimental curve) vs. 200 ºC (upper experimental curve). The experimental data is for the polycrystalline 49.9 at% Ni-Ti alloy. The calibrated model parameters are obtained by fitting features F1-F14 as summarized in Table 4.1. Also shown is the calibrated model result using the martensite interaction matrix h tu of Patoor et al. [31]. The model results are insensitive to max = 165 vs. 200 ºC Axial plastic macrostrain E p vs. temperature during thermal cycling with an axial bias stress bias = 400 MPa. The calibrated model parameters are obtained by fitting features F1-F14 for the polycrystalline 49.9 at% Ni-Ti alloy, as summarized in Table 4.1. Also shown is the calibrated model result with plasticity only, meaning that phase transformations are not permitted Axial macrostrain E vs. temperature during thermal cycling with an axial bias stress bias = 400 MPa, showing predictions of the calibrated model, the calibrated model with martensite elastic stiffness C M ~ ½ C A, and the calibrated model with C M and C A from Wagner and Windl [66]. A fourth case uses C M = C M(Isotropic) and overlaps the calibrated model result, where C M(Isotropic) is an isotropic matrix with Young s modulus E and elastic shear modulus G given by Hill averages of moduli of table Evolution of volume fraction of martensite contributing to diffraction of (100) and (011) planes parallel and perpendicular to the loading axis during temperature cycling of the polycrystalline 49.9 at% Ni-Ti under a compressive bias stress of bias =-150 MPa. The inserts on the right shows the experimental intensity vs. d- spacing at = 130 ºC, with small retained martensite peaks confirming the model predictions Normalized neutron diffracted intensities from (a) (100) and (b) (011) martensite planes that are parallel to the loading axis, as a function of bias stress bias. The intensities are normalized by the intensity at bias =100 MPa. The experimental results are measured at min = 30 ºC, following thermal xvii

19 cycling of the polycrystalline 49.9 at% Ni-Ti to max = 230 ºC. The calibrated model result uses parameters summarized in Table 4.1. Also shown is the calibrated model result using the martensite interaction matrix h tu of Patoor et al. [31] Axial macroscopic stress Σ vs. macroscopic axial strain E at temperature θ 0 = 22 ºC of an initially self-accommodated martensite microstructure of the polycrystalline 49.9 at% Ni-Ti under tensile and compressive loads. The calibrated model (gray) is able to capture the hardening and the tensioncompression asymmetry observed in the experiments (black) Martensite plate geometry formed by martensite variants k and m. The plate is modeled as a cuboid with dimensions 2a, 2b, 2c, where a = b = 20c. x 3 = ±c are invariant planes with a crystallographic normal m (k,m), x 2 is the orthogonalized shear direction given by b (k,m) (b (k,m) m (k,m) )m (k,m), where b (k,m) is the transformation displacement vector. x 1 is orthogonal to x 2 and x 3. The interface between variants k and m has the crystallographic normal n (k,m) and displacement vector a (k,m) Comparison of the crystallographic theory of martensitic transformations for NiTi to TEM observations from the preliminary micropillar testing. Shown are the predictions for the plate type T4 in Table 5.1. The line labeled invariant plane shows the predicted intersection of the invariant plane with the [001] plane of the image. The line labeled twin interface shows the intersection of the predicted martensite-martensite interface with the [001] plane of the image The normalized shear stress ζ 23 /E vs. normalized position x 1 /a with x 2 =x 3 = 0, for a martensite plate with dimensions a = b = 20c, assuming homogeneous, isotropic elastic properties. The results from the FFT based micromechanics code agree well with the analytical solution by Chiu [84] Spatial distribution of (i) the most stressed slip system and (ii) corresponding resolved shear stress (in MPa) on three planes (a) x 1 =1.0125a ; (b) x 2 =1.0125b and (c) x 3 =1.05c that are just outside the faces of a martensite plate (T4, k = 8, m = 5 in Table 5.1) as shown in Figure 5.5. Slip system numbers correspond to those listed in Table Spatial distribution of slip systems with a resolved shear stress exceeding 1500 MPa for a stress axis of [780], on planes (a) x 1 =1.0125a, (b) x 2 =1.0125b, and (c) x 3 =1.05c located just outside the faces of a martensite plate. These calculations assume at4 type plate with (k, m) = (8, 5), as identified in Table 5.1. Slip system numbers correspond to those listed in Table xviii

20 5.6 The spatial distribution of total volume fraction of martensite formed at the peak load during the pseudoelastic compression of the pillar using FEM model developed in Chapters 2-4. Also shown in the insert on the right, is the martensite volume fraction on the plane used to make the TEM foil (shown in Fig. 5.2) TEM image from [87] of B19 martensite in B2 austenite matrix of NiTi. The interface between the twinned martensite structure and B2 makes an angle 130 with the [110] B2 and is close to the angle predicted (126 ) by the model with the two variants listed in eq. (5.10) present in the ratio of 50:50 in the martensite microstructure TEM image from [87] of Martensite variants formed during [110] tension. Modeling results and the electron diffraction pattern (Fig. 5.9) of region SAD 1 confirms that two compound twin related variants of martensite are formed with twin plane (110) Electron diffraction pattern for the SAD1 (shown in figure 5.8) (a) TEM experimental diffraction pattern from [87] (b) Simulated kinematic diffraction pattern obtained by overlaying the diffraction pattern of the two variants given by eq. (5.10) in proper orientation relationship. 117 xix

21 CHAPTER 1: INTRODUCTION Shape memory alloys (SMAs) get their name because of their ability to remember their shape upon deformation. SMAs, when deformed below a critical temperature, seem to deform plastically with permanent deformation upon unloading. However, the material when heated above a certain critical temperature recovers its original shape as illustrated schematically in Figure 1.1, as if it remembers the shape prior to the deformation. This phenomenon is called shape memory effect. SMAs exhibit another interesting and technologically important thermo-mechanical property called pseudoelasticity. SMAs when subjected to suitable thermo-mechanical load deform by inelastic strain, which they recover back upon the removal of the load, as if the strains were elastic. Thus, the name pseudoelaticity. Figure 1.2a and Figure 1.2b show the reversible nature of the inelastic strain in SMA by schematically showing their stressstrain response at a suitably chosen constant temperature and strain-temperature response under a constant bias stress, respectively. Because of these special thermo-mechanical properties, SMA applications are quite diverse, ranging from actuators and sensors in aerospace and automobile applications to stents and vascular implants in medical applications [1-3]. SMAs derive these remarkable properties by undergoing martensitic phase transformation. Martensite phase transformation is a solid-solid transformation where material under suitable thermo-mechanical loading transforms from one crystal 1

22 structure to the other. This transformation is diffusion-less (chemical composition remains the same), sudden and displacive (by coordinated movement of atoms). This transformation is a result of material seeking to minimize it s energy and hence can be understood conceptually by looking at the free energy of the material. In the typical operating temperature range of a SMA, the alloy can exist in two phases with different crystal structure. The phase which has minimum energy at high temperature is called the austenite phase and the phase which has minimum energy at lower temperature is called the martensite phase. Under stress free conditions, at temperature θ>θ 0, the Helmholtz energy is minimized by the Austenite lattice, at temperature θ=θ 0, the energy of the austenite crystal structure and the martensite structure are the same and at even lower temperatures, θ<θ 0, martensite lattice minimizes the energy. Austenite has greater symmetry and martensite has lower symmetry. Thus there are more than one symmetry related martensite variants with total number of variants determined by the point group symmetries of austenite and martensite [4]. The transformation between the austenite phase and any of the martensite variant phases or between two martensite variants results in inelastic strain. Plastic deformation due to dislocation motion is another major contributor to inelastic strain in SMA. Moreover, inelastic strain due to plasticity is irrecoverable. Thus, SMAs deform by two major inelastic deformation mechanisms martensitic transformation and plasticity. The coupling of phase transformations and plasticity is an important phenomenon that enables training and optimization of SMA performance. However, it can produce undesirable characteristics such as accumulation of remnant deformation, reduced work output, and early fatigue failure during repeated thermo- 2

23 mechanical cycling. This coupling exists, in part, because both inelastic mechanisms phase transformations and plasticity alter the internal stress state and this in turn alters the driving force for each mechanism. The response of SMA under different thermomechanical loading conditions and the coupling observed between martensitic phase transformation and plasticity is briefly reviewed next. 1.1 SMA Response under Thermo-Mechanical Loading Stress Free Temperature Cycling SMA exists in austenite phase at high temperature. As the temperature θ, is lowered, to a critical temperature, martensite start temperature θ MS the austenite (A) phase starts to convert into martensite (M) phase and this A M transformation is complete when the temperature reaches the martensite finish temperature θ MF with material transformed entirely into the martensitic phase. Upon heating, when the temperature is raised to the austenite start temperature θ AS, the martensite starts to convert back into austenite and the material returns back into austenite phase completely at austenite finish temperature θ AF. The hysteresis is present due to the motion of the interfaces. Miyazaki et al. [5] observe that the θ MS temperature is lowered, i.e. martensitic transformation is hindered with repeated temperature cycling in solutionized NiTi. Moreover, they observe dislocations being generated during cycling. However, in aged Ni-rich NiTi, wherein dislocation motion is difficult, θ MS remains unchanged with cycling. Thus it is concluded that plastic deformation can hinder martensitic transformation under stress-free temperature cycling. In the absence of residual stresses in the SMA, under stress-free temperature cycling, all variants of martensite are formed in almost equal proportion and they self 3

24 accommodate, resulting in no net strain during the temperature cycling. However, if the SMA is pre-strained at elevated temperature (θ >> θ AF ), where plasticity in austenite dominates, it induces residual stress field in the austenite. Miller and Lagoudas [6] show that during thermal cycling of Ti 50 at%ni SMA under no load the temperature cycling can induce a tensile vs. compressive strain, depending on the pre-strain history Load Biased Temperature Cycling When temperature is cycled in the presence of a bias load, the martensite formed is no-longer self-accommodating and a net strain is achieved through the transformation. The difference in the strain at high temperature and the low temperature during this temperature cycling is called the transformation strain and is an important parameter for actuator applications of SMA. The transformation strain gradually increases with increasing bias stress [7-9]. Increasing bias stress also increases the transformation temperatures due to increased contribution of elastic work to the driving force for transformation. The coupling due to plasticity causes the transformation strain to reach a maximum and then to start decreasing with increased bias stress. Moreover, the width of the strain vs. temperature hysteresis loop increases [10]. Hamilton et al. [10] observe dislocations at the austenite-martensite interface in single crystal NiTi. The interpretation is that plastic strain accommodates and thus stabilizes the martensite phase and hence increasing the hysteretic width Isothermal Deformation at θ>θ AF When SMA is deformed at constant temperature θ>θ AF the stress induces martensitic phase transformation. The stress can also drive plastic deformation especially 4

25 as θ increases, owing to the Clausius-Clapeyron relation [11]. Numerous macroscopic observations qualitatively describe the nature of coupling between the two processes. Macroscopic (~cm or larger) single and polycrystalline samples subjected to simple tension or compression often exhibit remnant strain after unloading [11-15]. TEM analysis shows dislocations and martensite at the unloaded state [13]. Subsequent heating to greater than austenite finish temperature ( AF ) frequently reduces but does not eliminate the remnant strain. The reduction is attributed to the removal of stabilized martensite and the remaining portion is often attributed to plasticity. This remaining portion of strain typically increases with test temperature and also for crystal orientations and textures (e.g., <111> in compression) with a comparatively large resolved shear stress on austenite slip systems [11, 12]. Again, the interpretation is that plastic strain accommodates and thus stabilizes the martensite phase. Macroscopic observations also show a pronounced effect of pre-strain at elevated temperature (θ >> θ AF ), where plasticity in austenite dominates. In particular, compressive pre-straining of Ti wt%ni increases the linearity and hardening in subsequent compressive stress-strain response at θ > θ AF [16]. The interpretation is that pre-straining induces a distribution of internal stress that biases the macroscopic stress for transformation. Microscopic observations reveal that plasticity and transformation are coupled over a range of length scales. On a polycrystalline scale, Gall et al. [12] show that upon loading, the grain-to-grain variation in crystal orientation will favor plasticity in some grains and transformation in others. At even smaller length scales, Norfleet et al. [17] shows that during pseudoelastic loading, dislocations are formed by the local stress generated by martensite twin variants and further, these dislocations index to an austenite 5

26 slip system. Austenite slip is also supported by Sehitoglu et al. [18], who note that the yield strength in austenite is ~40% smaller than that for martensite, for a solutionized Ti 51.5 at%ni at room temperature Isothermal Deformation at θ<θ MF At θ<θ MF, in a stress free condition, material is in martensite phase. When deformed, the fraction of martensite plates favored by the external stress increase at the expense of the less favored martensite. This deformation mechanism is called reorientation of the martensite. In the second stage of the deformation, the martensite plates which are composed of two twin related martensite variants, detwin and individual variants which are most favored by external stress keep growing. The detwinning process can generate dislocations as the twinned structure responsible for coherency is no longer present. These observations have been confirmed by ex-situ TEM analysis of the microstructure by Liu et al. [19]. The process of reorientation and detwinning continue with almost no increase in the external stress and the plateau region of the stress-strain curve characterizes this process. Upon unloading, there is remnant strain which can be recovered back when heated to austenite phase, if plastic deformation is absent. 1.2 Continuum Level Models for SMA The continuum level models for SMA can be broadly categorized by the microstructural scale of the constitutive relations. 6

27 1.2.1 Phenomenological Models Models that use non-crystallographic description of the inelastic deformation mechanisms are characterized in this group. For modeling transformation, these models typically track the total volume fraction of martensite. To capture the response of SMA under temperature cycling, more sophisticated models track the volume fractions of stress-induced vs. thermally-induced martensite via state variables [20, 21] or adopt a functional dependence of transformation strain on total martensite volume fraction and stress [22]. Most models completely ignore plastic deformation. However, some use a phenomenological description of the coupling between plasticity and transformation [23-29]. For example, the amount of retained martensite after unloading is sometimes viewed as a phenomenological function of equivalent plastic strain [27]. Overall, such models tend to be computationally efficient but they are not robust, since both the deformation mechanisms and the coupling function between them are limited to specific loading conditions, temperatures, or alloy compositions Microstructure Based Models Models in this category use crystallographic description of the deformation processes. The polycrystalline SMA is modeled by a collection of individual grains. For modeling polycrystalline behavior, capturing three interactions between different grains of a polycrystalline aggregate, between different martensite variants within each grain and finally between martensite variants and plasticity is important. These interactions are over a range of length scales and hence some approximations have to be made based on computational costs involved in modeling a macroscopic response. 7

28 The grain-to-grain interaction is modeled by either using a self-consistent model e.g. [30-34] wherein the strain compatibility is enforced in a weak sense or using a finite element model with each grain modeled explicitly using finite elements e.g. [35-38] where the strain compatibility is enforced strictly. For modeling martensitic transformation, typically these models track the volume fraction of habit plane variants of the martensite within a material point. Hence the interaction between martensite variants within a material point is typically approximated through phenomenological relations. Some models ignore this interaction all together [35-37] while others formulate interaction relations between individual twinned habit plane variants (hpv) [30, 31, 34] or between hpv groups [33], or between individual lattice correspondence variants (lcv) [32]. Recent advances in finite element modeling have pushed toward smaller scales in an effort to capture fundamental transformation processes. Thamburaja et al. [39] employ spatial gradient terms in free energy to develop a non-local description of detwinning. A Landau-type description of free energy [40-42] has been implemented in FEM, and micromechanics models have been developed to track the growth and reorientation of the austenite-martensite interface [43]. However, these models [30-43] do not model plastic deformation and hence are incapable of modeling many of the features of experimental deformation response of SMA, as outlined in the Section 1.1, which are primarily due to the coupling of martensitic transformation and plasticity. A fundamental advance in modeling of SMA is to couple these mechanisms via the stress redistribution caused by one mechanism on the other. Such models use a crystallographic description of transformation. Some use stress redistribution as the only 8

29 coupling mechanism [44-46] and others add phenomenological coupling functions to capture additional effects, such as the effect of dislocation structure on martensite formation [47-52]. These more fundamental approaches adopt a range of constitutive descriptions and scales. For example, Levitas et al. [51, 52] study the propagation of one transformation front in an elasto-plastic medium, appropriate at smaller length scales. Other models [44-49, 53, 54] track only the volume fraction of martensite habit plane/bain strain variants, thus smearing out the detailed configuration. This scale is adopted by Thamburaja [45] and Pan et al. [44] to study how martensite detwinning and reorientation couples with plasticity in the martensite phase. The J 2 -von Mises description of plasticity used in these prior models is advanced by Wang et al. [46]. Wang et al. [46] adopt a crystal-based description of martensite plasticity that becomes active only after the austenite-martensite transformation is complete. Thus simultaneous operation of plasticity and transformation is still missing in Wang et al. [46]. 1.3 Objectives and Scope The objective of this thesis is to study the interplay between martensitic transformation and plastic deformation in SMA. At macroscale, this is done by formulating and numerically implementing a microstructure based finite element (MFE) material model which couples the martensitic phase transformation with plastic deformation. This work is unique in that it adopts crystallographic descriptions of both processes and allows for their simultaneous operation. The coupling is captured through the grain-to-grain redistribution of stress caused by both plasticity and phase 9

30 transformation, so that each mechanism affects the driving force of the other. Though, the formalism is generic and is applicable to other SMA, this thesis only applies the model to NiTi SMA. By suitable thermo-mechanical processing, the martensitic transformation in NiTi can be restricted to Austenite (B2) to monoclinic (B19 ) martensite. This thesis focuses on this particular class of NiTi SMA. Also, it incorporates crystallographic slip systems in austenite rather than martensite, as supported by recent studies of solutionized Ti 50.7 at%ni crystals following pseudoelastic loading by Norfleet et al., [17]. The model developed is assessed against a wide gamut of experimental results for widely used NiTi shape memory alloy. Several experimental trends are captured in particular, the dependence of transformation strain on the bias stress, plasticity resulting in enhancement of stress-induced martensite formation during loading and generation of retained martensite upon unloading, deformation processing resulting in increased strain hardening in pseudoelastic response and two-way-shape memory effect trends which existing MFE models are unable to capture. Some experimental trends are not captured in particular, the ratcheting of macrostrain with repeated thermal cycling. This may reflect a model limitation that transformation-plasticity coupling is captured on a coarse (grain) scale but not fine (martensitic plate) scale. Moreover, the model is restricted to smaller pseudoelastic strains (~6% for NiTi SMA), where experimental evidence suggests that martensite detwinning is small since reorientation and detwinning of martensite are not explicitly included in the constitutive relation. At microscale, crystallographic theory of martensite (CTM) and micromechanics-based modeling is applied to analyze recent Transmission Electron Microscopy (TEM) observations. 10

31 1.4 Dissertation Outline Chapter 2 to follow describes the constitutive relation that couples plasticity and transformation at a material point. The constitutive relations as well as their numerical implementation in commercial finite element code ABAQUS [55] is described. A ratedependent crystal plasticity flow law from Peirce et al. [56] is employed with a rateindependent, crystallographic law from Thamburaja and Anand [36] for the forward and backward austenite-martensite transformation. The scale is smeared out so that the volume fractions of various martensite habit plates are predicted, but not their detailed spatial arrangement. Large deformation formalism is used and analytical jacobian required for solution in an implicit numerical integration scheme is derived. Chapter 3 qualitatively assesses the MFE model predictions with a gamut of experimental responses. The model is first calibrated to experimental data for solutionized single crystals of Ti 50.9 at%ni. Then, the predicted response of initially stress-free, randomly oriented polycrystals under isothermal tension-testing at temperature > AF and during thermal cycling with a bias stress is assessed against experimental response of polycrystalline NiTi for different compositions. The predicted effects of elevated temperature pre-straining on subsequent pseudoelastic and thermalcycling response of the polycrystalline NiTi is assessed next. The results of this qualitative assessment are summarized in the end. Chapter 4 assesses the MFE model predictions quantitatively to experimental response of polycrystalline NiTi ( 49.9at% Ni). The model is rigorously calibrated against the inputs that include anisotropic elastic properties, texture, and DSC data as well as a subset of isothermal deformation and load-biased thermal cycling data. The 11

32 model is assessed against additional experimental data, including neutron diffraction measurements of martensite texture evolution. Several experimental trends are captured in particular, the transformation strain during thermal cycling monotonically increases and reaches a peak with increasing bias stress. This is achieved, in part, by modifying the martensite hardening matrix proposed by Patoor et al. [31]. Some experimental trends are not captured in particular, the ratcheting of macrostrain with continued thermal cycling. This may reflect a model limitation that transformationplasticity coupling is captured on a coarse (grain) scale but not fine (martensitic plate) scale. Chapter 5, then moves into a much smaller length scale of individual variants of martensite to understand the fundamentals of martensitic phase transformation and it s coupling with plasticity. The modeling framework relies on the crystallographic theory of martensite (CTM) and micromechanics-based stress analysis. The model is used to explain two recent TEM experimental observations of dislocations and martensite twins. In particular, the observation of sub-micron dislocation loops is explained in terms of the large stress generated by the phase transformation at the variant (sub-micron) scale. Also, the observation of compound twin related martensite variants during in-situ TEM tensile tests is explained in terms of the loss of constraint produced by free-surfaces. Dissertation conclusions are discussed in Chapter 6. The contribution to the literature is discussed in terms of model development and application. The short comings of the model shed light towards the future work required to understand the deformation behavior of shape memory alloys. 12

33 Macrostress (MPa) Macrostrain E Deformation at θ< θ MF Heating to θ> θ AF Twinned self-accomodating martensite at θ< θ MF Stress free cooling to θ< θ MF Detwinned martensite at θ< θ MF Austenite at θ> θ AF Figure 1.1: Illustration of shape memory effect and microstructure at different stages of thermo-mechanical loading of a Shape Memory Alloy (SMA). Shape recovery is complete in the absence of plastic deformation. The transformation temperatures θ AF and θ MF are defined in Section (a) (b) θ> θ AF Macrostrain E Figure 1.2: A schematic of SMA response showing inelastic deformation due to austenite to martensite transformation. Plastic deformation is absent in this schematic. Plastic deformation can change this schematic response significantly (see Chapters 3-4); (a) Isothermal axial macroscopic stress vs. macroscopic axial strain E response at temperature θ>θ AF ; (b) Isobaric axial macrostrain E vs. temperature during thermal cycling under a bias stress Σ bias = 300MPa Temperature (K) 13

34 CHAPTER 2: CONSTITUTIVE RELATIONS AND FINITE ELEMENT MODEL Constitutive relations relate the total deformation increment at a material point to the current microstructure, stress, and stress increment at that point. The deformation includes contributions from elasticity, thermal expansion, plasticity, and phase transformation from austenite to martensite or vice-versa. Though the constitutive relations developed are generic and can be applied to any SMA, this thesis concentrates on a particular SMA- NiTi which is suitably processed to restrict the martensitic phase transformation to Austenite (B2) to Martensite (B19 ) phase. Thus the crystallographic description for phase transformation and the description of elastic anisotropy use lattice structure and symmetry of B2 and B19. Also, it incorporates crystallographic slip systems in austenite rather than martensite, as supported by recent studies of solutionized Ti 50.7 at%ni crystals following pseudoelastic loading by Norfleet et al., [17]. Norfleet et al., [17] shows that during pseudoelastic loading, dislocations are formed by the local stress generated by martensite twin variants and further, these dislocations index to an austenite slip system. Austenite slip is also supported by Sehitoglu et al. [18], who note that the yield strength in austenite is ~40% smaller than that for martensite, for a solutionized Ti 51.5 at% Ni at room temperature. Constitutive relations from [57, 58] are summarized in this Chapter. 14

35 The microstructural information at a material point includes the crystallographic orientation of the parent austenite phase, the volume fractions of candidate martensitic plates, and the activity on each of the candidate slip systems. This is depicted in Figure 2.1a, where the shading denotes the grain-to-grain variation in crystallographic orientation, and in Figure 2.1b, where a material point is assumed to have a distribution of slip activities and martensite plates. However, the exact internal distribution of such activities and plates and their interaction are not modeled explicitly; rather, only the average volume fraction v t of each plate type t and the average activity γ s (representing a measure of net shear) on each slip systems is represented. However, our aim is to capture the interaction between material points in single or polycrystalline samples, thereby tracking the inhomogeneous development of slip activity and volume fraction with position. Large deformation formalism is used in the constitutive relations. Section 2.1 discusses the constitutive relations and Section 2.2 discusses the numerical implementation of the constitutive relations in commercial finite element code ABAQUS using an implicit time integration (unconditionally stable) scheme. 2.1 Constitutive Relation Following [59], an infinitesimal vector dx in the undeformed (reference) configuration is distorted to a vector dx = F dx in the deformed configuration, where the deformation gradient e inel F = F F (2.1) 15

36 is multiplicatively decomposed into an elastic part (F e ) and an inelastic-thermal part (F inel ), where the elastic strain is E e =½(F et F e I). Typically, the time rate of change F of the deformation gradient and time rate of change of temperature may be imposed. As outlined in Section 2.2, the relation inel inel inel F L F (2.2) is needed to determine the corresponding Cauchy stress T, where the inelastic velocity gradient is NS NT NT inel Slip Trans va s s vt t ; va 1 vm and vm vt s 1 t 1 t 1 L S S (2.3) The fundamental material parameters needed in Eqs. (2.2) and (2.3) are the volume fraction v A of austenite, the volume fraction v t of each possible type (t = 1 to N T ) of martensite plate which sums to the total martensite volume fraction v M, the amount of slip s from each possible (s = 1 to N S ) slip system, the local orientation of the parent austenite phase to the macroscopic loading axes, the anisotropic elastic moduli C A and C M of the austenite phase and martensite plates, and the corresponding thermal expansion matrices A th-a and A th-m. Effective parameters are employed in Eq. (2.3) to describe the ensemble of martensite plates and austenite. In particular, the effective anisotropic elastic moduli and thermal expansion coefficients are specified by a rule-of-mixtures approach. The aggregate thermal expansion coefficient A th : A th v A v A (2.4) A th -A M th -M 16

37 and effective elastic moduli C is rigorously calculated using Crystallographic Theory of Martensite (CTM) and rule-of-mixtures type averaging as: C v C A A NT t 1 v C t M( t) wherec M( t) v t, i C M( t, i) v t, j C M( t, j) (2.5) Each habit plate variant t of martensite has a local volume fraction v t and elastic modulus C M(t) given by the average moduli C M(t,i), C M(t,j) and volume fractions v t,i, v t,j (= 1 v t,i ) of the correspondence variants i and j within plate t. For practical purposes, the components are referred to the austenite crystal basis. This requires the rotation operators Q t,i and Q t,j to relate a direction in the martensite crystal basis of variants i and j, respectively, to the crystal basis of austenite from which plate t formed. They are obtained by solving eq. (17) from the CTM [60] t, i ij ij LC- i t, j ij LC-j Q R R R and Q R R (2.6) where R LC-i and R LC-j are the lattice correspondence rotations between martensite variants i and j and austenite, respectively, R ij is the rotation required to form a twin plane between variants i and j, and R ij is the rotation required to form a habit plane with the austenite [60, 61]. Although the thermal expansion coefficient for martensite plates can be constructed in a similar fashion, the isotropic form A th-m = M I is adopted with the supposition that the primary effect of anisotropy stems from the elastic moduli. This assumption is discussed further in Section and justified by the small magnitude of thermal inelastic strain (~0.1%) during thermal cycling [62]. 17

38 Also, the effective L inel for the ensemble is approximated in terms of the time rate of change s of slip on each austenite system s, weighted over the volume fraction v A of austenite, plus the time rate of change vt of volume fraction of each martensite plate type t. As depicted in Fig. 2.1b, Slip Slip Slip S b m s s s is the deformation gradient produced by a unit slip on system s, where Slip b s and Slip m s are the Burgers vector and normal to the slip plane. Likewise, Trans Trans Trans S b m t t t is the deformation gradient produced by the transformation of austenite to a type t martensite plate, where transformation direction and Trans m t is the habit plane normal [4]. Trans b t is the average In principle, the slip rate s and transformation rate v t in Eq. (2.3) may be functions of stress, stress rate, temperature, temperature rate, and internal distribution of slip and plates. The well-known crystal plasticity formulation by Peirce et al. [56] 1 s m Slip et e * Slip s 0 sign( s) ; s bs F F T m s (2.7) g s is used, where s is the resolved shear stress on slip system s, 0 is a reference slip rate, m is the rate sensitivity, and T* is the symmetric Piola-Kirchoff stress. The slip system hardness g s for an arbitrary slip system s evolves as N S a g 1 and 0 1- r gs hr Q Q sr r hr h g r 1 sat (2.8) The variables in Eq. (2.8) are explained in Table

39 Following the formalism of Thamburaja and Anand [36], the transformation rate is viewed as sufficiently fast to ensure that the energetic driving force f t for formation of plate type t is always bounded by a critical value f c, f f ( v,, ) f c t t T c (2.9) Thus, if the Cauchy stress T or temperature are incremented as to make f t > f c, the volume fraction v t instantly changes via the forward or reverse austenite martensite transformation to ensure the consistency condition f t = f c. When f t is within the bounds, v t does not change. The driving force f t is defined as the work conjugate to the volume fraction v t of plate type t at a material point. To implement this, the Helmholtz free energy per unit reference volume is defined as: e 1 e e e T R ( E,, vt ) E C E ( 0) Ath C E ( T) vm 2 T N 1 T NT vt htuvu 2 t 1 u 1 (2.10) The first term is the contribution of the elastic strain energy, the second is the work done by the stress field through the thermal strain, the third is the chemical free energy associated with the phase transformation, and the last is the interaction energy between martensite plates. The chemical free energy is constructed to vary linearly with, where the equilibrium transformation temperature T is the average of the martensite and austenite start temperatures, i.e., θ T = (θ MS + θ AS )/2, and λ T is the latent heat per unit reference volume. The interaction energy is assumed to vary quadratically with plate 19

40 volume fraction. Plate interaction energy can be viewed in terms of the local stress exerted by one plate on another, generating additional work not captured by the first term. The elements h tu and their effect on the thermo-mechanical response of SMA along with other approximations of interaction energy are discussed in detail in Section The integral energy conservation law for an arbitrary material volume P R with boundary P R and normal N in the reference configuration is given as (Thamburaja [45], Eq. 17): d dt 1 x x dv b x r dv + PN x da (2.11) P R R R R R 2 P R P R The temperature θ is assumed to be spatially uniform throughout this thesis. Accounting for the spatially varying temperature field will bring in additional hardening in the stressstrain response and transformation over a wider range of temperatures during the temperature cycling. However, these effects are minimized by imposing a low strain rate (0.001/sec) during isothermal loading and a low temperature change rate (0.08 o C/sec) during temperature cycling. Thus, justifies this simplifying assumption. ρ R is the mass density, x is the velocity, ε R is the specific internal energy per unit reference volume, b is the body force per unit mass, r R is the heat generated per unit reference volume per unit time and P is the first Piola- Kirchoff stress. The time derivative is denoted by a superscript dot ( ) throughout this thesis and the subscript R denotes quantities defined in the reference configuration. Applying the conservation laws for mass and linear momentum, Eq. (2.11) can be reduced to the local form of energy conservation: 20

41 P F- R r R = 0 (2.12) The local form of the Clausius-Duhem inequality in the reference configurations is: R r R (2.13) where R is the specific entropy per unit reference volume. Combining Eqs. (2.12) and (2.13) and using the Legendre transformation R R R yields an alternate form of the Clausius-Duhem inequality, P F- (2.14) R R 0 where ψ R is the Helmholtz free energy per unit reference volume. Using the kinematic relations: T e 1 e et P det( F) TF, det( ) * T F F T F e T e e 1 D {F } E {F } and e e 1 e inel inel 1 e 1 } } } F = F {F F F {F {F F the first term in Eq. (2.14) becomes: Using the chain rule, * e et e * inel P F T E F F T L (2.15) NT e R e R R R E,, vt E e E v t= 1 t Substituting Eqs. (2.15), (2.16) and (3) into Eq. (2.14) gives: vt (2.16) 21

42 N T * R e R et e * trans R T e E R F F T St vt E v t= 1 t NS et e * slip va F F T Ss s 0 s= 1 (2.17) e The fields E,, vt and s are independent of one another and can be arbitrarily varied. This yields the prescriptions for the Cauchy stress T and the symmetric Piola-Kirchoff stress T * : e 1 e et R e det( ) * * T F F T F where T C E A e th ( 0) E (2.18) Similarly, the energetic driving force to increase volume fraction v t is NT Trans et e * Trans f T t ( vt, T, ) bt F F T mt ( T) h tuvu (2.19) T u Numerical Implementation of Constitutive Relations The numerical scheme has the following objective: to supply updated values of (T, F inel, v t, s, g s ) at some new time = t +, given (T, F inel, v t, s, g s ) at time t and the proposed new deformation gradient F( ) and temperature ( ). Also required are the Jacobian matrices T/ E and T/. This is achieved by programming a two-level iterative scheme into the user- defined subroutine UMAT in the commercial finite element code ABAQUS [55]. The numerical implementation follows Eqs. (15)-(29) of Kalidindi et al. [63], using the constitutive relations in Section 2.1. Integrating Eq. (2.3) gives 22

43 NS NT inel Slip Trans inel A s s t t s 1 t 1 F ( ) I + v ( ) S + v ( ) S F ( t) (2.20) Using Eqs. (2.1) and (2.20), the symmetric Piola-Kirchoff stress T * in Eq. (2.18) can be expressed in terms of a trial stress T *el, which assumes the increment in deformation to be purely elastic, plus correction terms due to incremental plastic strain and martensite volume fractions: for which NS NT * *el Slip Trans ( ) va s ( ) s vt ( ) t s 1 t 1 T T C C (2.21) *el 1 inel -T T inel -1 T C A - I - Ath 0 ; ( t) ( ) ( ) ( t) 2 A = F F F F (2.22) 1 1 C = C 2 S S ; C = C 2 S S Slip Slip Slip T Trans Trans Trans T s s s t t t (2.23) A two-level iterative scheme is used to determine s ( ) and v t ( ) such that the resulting T * ( ) from Eq. (2.21) is consistent with the rate-dependent plasticity law (Eq. 2.7) relating T and s, and the rate-independent transformation law (Eq. 2.9) relating T and v t. For iteration n = 0, the solution ( s, v t, T * ) (0) = ( s (t), 0, T * (0)) is proposed, where s (t) is from the previous time step increment and T * (0) = T * ( s (t), 0) is computed from Eq. (2.21). In general, this solution does not satisfy Eqs. (2.7) and (2.9). For iteration n = 1, a new solution ( s, v t, T * ) (1) is proposed, where the pair ( v t(1), T * (0)) satisfies the transformation law (Eq. 2.9), the pair ( s(1), T * ( s(0), v t(1) )) satisfies the plasticity law (Eq. 2.7), and T * (1) = T * ( s(1), v t(1) ) from Eq. (2.21). This process is 23

44 repeated for n iterations until the solution ( s, v t, T * ) (n) converges to an acceptable tolerance. The numerical procedure to satisfy Eq. (2.9) is adopted from Thamburaja and Anand [36] and that to satisfy Eq. (2.7) is described in Kalidindi et al. [63]. The material Jacobian matrix T/ E is derived following Balasubramanian [64]. T/ is required if temperature varies with time. A detailed derivation is provided in Appendix A. This implicit time integration scheme for the constitutive relations is an improvement over explicit time integration used in literature [35-37] for modeling rate-independent transformation because of its unconditional numerical stability. 2.3 Constitutive Relations for Martensite detwinning at θ<θ MF The constitutive relations developed in sections model the SMA inelastic response due to austenite to martensite transformation and back transformation of martensite to austenite (A M) and plasticity in austenite phase. SMA can also undergo inelastic deformation by detwinning and reorientation of martensite phase [19]. This section extends the constitutive model developed to a special case of martensite detwinning response: martensite deformation at temperature θ < θ MF, the temperature at which the material is in 100% martensite phase. Under load, the martensite variants reorient and detwin to produce inelastic strain. Since martensite is stable at this temperature, unlike the pseudoelastic deformation the inelastic strain is not recovered upon unloading. Since the detwinning and reorientation of the twinned habit plane variant (hpv) martensite plates is the source for inelasticity, the volume fractions v t (t=1 to N T ) of all the hpv martensite plates in eq. (2.3) are first converted into the volume fractions ξ i (i=1 to 24

45 N lcv ) of lattice correspondence variant (lcv) martensite using Crystallographic Theory of Martensite (CTM). Specifically, ξ i are obtained by solving eq. (17) from the CTM [60] which gives the volume fractions v t,i, v t,j (= 1 v t,i ) of the correspondence variants i and j within plate t. Then, the total volume fraction ξ i of each of the correspondence variants i, with the total number of correspondence variants being N lcv (=12 for B2 B19 transformation) is calculated by summing over all hpv plates N T as NT ξ i = vt,i t=1 (2.24) The detwinning and reorientation of martensite variants result in a rate of change ξ i in lcv volume fraction ξ i which leads to inelastic deformation gradient F inel. Similar to section 2.1, F inel can be calculated by integrating the inelastic velocity gradient L inel, which for martensite detwinning is given by: Nlcv i i 1 inel Trans L U i (2.25) where Trans Ui is the Bain deformation matrix for the lcv martensite. For B2 B19 martensitic transformation in NiTi, table 4.3 of [4] prescribes Trans U i. The relation between the rate of change of volume fractions ξ i of lcv and the stress T * is explained next. Both reorientation and detwinning are processes that result in the conversion of martensite lcv j into lcv i for (i, j =1 to N lcv ). This conversion changes the volume fraction 25

46 of both lcvs. The net change in volume fraction of lcv i, ξ i is given as the sum of all such conversions: Nlcv i ij with ij ji and ii 0 j 1 (2.26) where ξ ij denotes the rate of change of volume fraction of lcv i due to the conversion of lcv j to lcv i and ξ ji denotes the rate of change of volume fraction of lcv j due to the same conversion. The second and third equality in eq. (2.26) ensures that the net volume fraction Nlcv i i 1 =1. This conversion and hence the rate of change of volume fraction ξ ij, is driven by the driving force f ij. Following the derivation of eq. (2.19), the driving force f ij is given as: et e * Trans Trans fij F F T : Ui U j (2.27) Following Thamburaja [45], the rate ξ ij is viewed as sufficiently fast to ensure that the energetic driving force f ij which drives the conversion of martensite lcv j to lcv i is always bounded by a critical value f cdtw, fcdtw fij fcdtw (2.28) Thus, if the Cauchy stress T is incremented as to make f ij > f cdtw, the volume fractions of martensite lcvs i and j instantly change via the forward or reverse conversion of lcv i 26

47 lcv j to ensure the consistency condition f ij = f cdtw. When f ij is within the bounds, ξ ij =0 and no conversion between the two lcvs takes place. The only additional material parameter to be calibrated in this extension of the model from sections is the critical barrier for the initiation of the detwinning f cdtw. 27

48 x 3 x 1 x 2 B.Cs: u 3 (x 3 =0)=0 u 1 (x 1 =x 2 =x 3 =0)=0 u 2 (x 1 =x 2 =x 3 =0)=0 u 2 (x 1 =1,x 2 =x 3 =0)=0 (a) Figure 2.1: (a) Finite element representation of a polycrystal consisting of a array of elements, each representing a grain. The shading indicates the grain-to-grain variation in maximum Schmid factor for crystal slip; (b) inelastic processes at a material point involving (i) formation of martensite plate type t with deformation gradient S t Trans, invariant plane normal m t Trans, displacement vector b t Trans, and volume fraction v t ; and (ii) plastic deformation on slip system s with deformation gradient S s Slip, invariant plane normal m s Slip, displacement vector b s Slip, and activity t. 28

49 CHAPTER 3: MODEL APPLICATION TO SOLUTIONIZED 50.9 AT.% NI-TI The constitutive model developed in Chapter 2 is applied to model the thermomechanical response of solutionized 50.9 at. % Ni-Ti. Ni-Ti is by far the most widely used SMA. This particular alloy composition is chosen for model assessment because of availability of experimental data of single crystal deformation response [12]. The chosen alloy undergoes austenite (B2) to monoclinic (B19 ) transformation. Single crystal experimental deformation response can be used to determine the material parameters in the constitutive model in a very convenient way. This Chapter summarizes the results of [57]. The finite element model used is described in Section 3.1. The calibration of the model is discussed in Section 3.2. In the absence of experimental data for polycrystalline solutionized 50.9 at. % Ni-Ti, the predictions of polycrystal behavior are assessed qualitatively against experimental response of NiTi at other compositions. In Section 3.3, the predicted response of initially stress-free, randomly oriented polycrystals under isothermal tension-testing at temperature > AF and during thermal cycling with a bias stress is assessed against experimental response of solutionized NiTi for different compositions. The predicted effects of elevated temperature pre-straining on subsequent pseudoelastic and thermal-cycling response of polycrystalline NiTi is assessed in Section 3.4. The results of this qualitative assessment are summarized in Section

50 3.1 Finite Element Geometry for NiTi Polycrystals All calculations use the geometry in Fig. 2.1a, in which a polycrystal is modeled by a cube assembly of 8-node 3D brick elements (C3D8 for isothermal analyses and C3D8T for thermo-mechanical analyses), with each element representing a grain with an assigned random crystal orientation of the austenite (B2) phase. Two types of analyses are considered. For a typical isothermal analysis, the displacement u 3 =0 on the bottom face (x 3 = 0) and u 3 = E33 t on the top face (x 3 = 1), where the macroscopic strain rate E 33 = 10 4 /s are prescribed. Zero traction is imposed on all other degrees of freedom except those needed to prevent rigid body rotation. For a typical thermo-mechanical analysis, a distribution of constant stress 33 is imposed on the top face, u 3 =0 on the bottom face, and the temperature θ is cycled in a spatially uniform manner. The number of grains, N grain = 729, is considered sufficient since the predicted macroscopic response does not change significantly for N grain > Material Parameters: Calibration To Single Crystal Data Table 3.1 shows the adopted material parameters for solutionized Ti-50.9 at% Ni single crystals. They are partitioned into three groups. Elastic-thermal properties G el-thermal = {C A, C M, A th-a, A th-m } are taken from the literature. The cubic elastic moduli for austenite (B2) is taken from Brill et al.[65]. For the martensite elastic stiffness, the approximation C M = C A is made. This is based on recent first principle calculations that show the elastic moduli of the B19 phase is greater than the B2 phase [66, 67]. This is viewed as more realistic than the assumption 30

51 C M ~ ½C A often adopted in the SMA finite element literature [35, 36, 46]. The monoclinic elastic stiffness for martensite predicted by DFT calculations from [67] are used in Chapter 4 and the effect of the assumptions on martensite elastic stiffness made in this Chapter and elsewhere in literature is assessed. The thermal expansion properties are assumed to be of the isotropic form A th-i = i I (i = A or M), with A and M values reported in Table 3.1. These values are taken from the experiments by Boyd and Lagoudas [22]. The results to follow are not sensitive to the thermal expansion properties since the thermal strains are orders of magnitude smaller than the transformation strains. The primary temperature effect enters through the structural driving force (3 rd term) on the right-hand side of Eq. (2.10). Transformation parameters G trans = { T, hcom, hincom, T, fc} are determined from the experiments of Gall et al. [12] for solutionized Ti-50.9 at%ni. Specifically, DSC measurements of AS and MS are used to specify T = ( AS + MS )/2 = 257 K. The remaining parameters are calibrated from room temperature compressive stress-strain data for [210] oriented single crystals, which are reported to deform by martensitic transformation with negligible plastic deformation. Of the 192 possible plate types predicted by the crystallographic theory of martensite [60], N T = 24 is chosen, as frequently adopted in the literature [34-38, 46]. Moreover, the chosen plates are of type-ii twinned martensite variants and have been observed experimentally [17]. Additional plates increase the computational time for the solution of the transformation criterion (Eq. 2.9) in Section 2.1 and do not substantially alter the transformation surface in stress space. The values of Trans b t and Trans m t (t = 1 to 24) are provided in Table 3.1 of 31

52 Thamburaja and Anand [36]. The structure of the hardening matrix h tu in Eq. (2.10) is defined following [30, 31] h tu h h if det( ε ε ) 0 com T( t) T( u) if det( ε ε ) 0 inc T( t) T( u) (3.1) Where in plates are grouped into compatible and incompatible modes determined by det( ε ε ), (=0 for compatible) where T(t) is the transformation strain induced by T( t) T( u) plate type t. The consequence of using this structure for h tu is discussed in detail in Section The parameters λ T and f c are determined by matching the experimental stress at the onset of the forward and back transformations in a [210] single crystal. The ratio h com : h inc = 1:4 as proposed by Gall and Sehitoglu [34] for NiTi SMA and the magnitude is then selected to fit the experimental hardening. Figure 3.1 shows the resulting comparison between the experimental [210] compressive response and the model using the fitted parameters in Table 3.1. Although the match between the forward and backward transformation stress is good, the model predicts a sharp transition to elastic response at > 7% strain and underestimates the remnant strain (points A and A in Fig. 3.1). These discrepancies may be due to continued inelastic deformation in the experiment, possibly from reorientation or detwinning of martensite plates. Such mechanisms are not included in the constitutive relation. A consequence is that the fitted constitutive relation is not expected to be applicable at > 6% strain. 0 0 s 0 sat Austenite plasticity parameters G plastic = {, m, g, h, Q,a,g } are calibrated from the room temperature compressive stress-strain response of [111] oriented single 32

53 crystals, which are observed to undergo predominantly plastic deformation in the austenite phase. Specifically, N S = 12 is used, with 6 <100>/{110} and 6 <100>/{010} slip systems, as reported for the austenite (B2) phase by Chumlyakov et al. [68]. In the absence of enough experimental data to determine the plasticity parameters, a low strainrate sensitivity m = 0.02, reference strain rate 0 = 0.001/sec (> E 33 = 10-4 ), and Q = 1.4 are assumed following Kalidindi et al. [63]. The large hardening in the [111] experimental response dictates a large saturation hardness, g sat = 900 MPa. Figure 3.1 shows the resulting comparison between the [111] experimental response and the model using the fitted parameters in Table 3.1. In this case, g, a, and h 0 (see Eqs. 2.7 and 2.8) are chosen to match the experimental remnant strain upon unloading (point B). However, the model cannot achieve the large experimental values of yield strength and strain hardening without ruining the successful match in remnant strain at point B. During loading, the fitted model shows a transition from plasticity-dominated to transformation-dominated flow at point C. During unloading, it has an abrupt back transformation compared to the gradual trend in the experimental result. Figure 3.2 assesses the ability of the fitted model to capture the isothermal stressstrain response for single crystals in other orientations: [100], [110], [211], and [123]. The largest Schmid factors for transformation for these respective cases are 0 s T ζ max = 0.38, 0.40, 0.41, 0.44 and those for crystalline slip are S max ζ = 0, 0.50, 0.47, and Here, max = max[(b * a)(m * a)], where a is a unit vector along the compressive axis and (b *, m * ) are unit vectors parallel to the (b, m) for the appropriate transformation or slip 33

54 systems. The model reasonably captures the onset of transformation, but it predicts an abrupt transition compared to the rounded experimental traces. However, such abrupt transitions are reported for micron-scale samples of solutionized, Ti 50.7 at%ni, loaded near [110] by Norfleet et al. [17]. Overall, the model tends to underestimate the stress for forward transformation, the amount of hardening, and the amount of remnant strain upon unloading. The disagreement in the [110] and [123] cases might be attributed to the larger peak strain in the experiments. However, this does not explain the severe discrepancy in the [211] case, which has an intermediate ratio of T ζ max / S max ζ compared to the [110] and [123] cases. Overall, the calibrated model captures the approximate orientation-dependence of the onset of transformation/plasticity, but it underestimates the remnant strain observed in the single crystal data. This may be a consequence of the smeared out representation of martensite plates at a material point, so that the fine-scale plasticity at the austenitemartensite interface is not captured. Moreover, the large strains imposed in the experiments may induce reorientation and detwinning of martensite processes not explicitly included in the constitutive relation. For this reason, the model is restricted to smaller pseudoelastic strains, where experimental evidence suggests that martensite detwinning is small. Specifically, single crystal tests by Gall et al. [69] show that at small strains, the transformation is consistent with internally twinned rather than detwinned martensite. Recent transmission electron microscopy studies confirm this, for solutionized NiTi samples under uniaxial compression [17]. Thus, the model is a useful description, but only at smaller imposed strain (<6%). 34

55 3.3 Response of An Initially Stress-Free Random Polycrystal The finite element model with parameters listed in Table 3.1 is used to predict the behavior of a solutionized Ti 50.9 at%ni polycrystal with random grain orientations and an initially stress-free state. Inserting T, T, h in, h incom, and f c from Table 3.1 into Eq. (2.19) yields a stress-free austenite finish temperature AF = 274 K and stress-free martensite finish temperature MF = 229 K. This compares well with the respective experimental values, AF = 280 K and MF = 231 K from Gall et al. [12]. Two types of studies are considered: tensile loading/unloading at = AF +24 K; and thermal cycling between 370 K and 220 K under a constant tensile stress (load bias) Pseudoelastic Tensile Response at = AF + 24 K Three cases are used to study the interaction between plasticity and transformation during tensile loading/unloading of a randomly oriented polycrystal at a macrostrain rate of 10-4 /s and temperature = AF + 24 K: (i) a transformation only case for which plasticity is turned off by making 0 = 0 in Eq. (2.7); (ii) a transformation + plasticity case using the 0, f c values in Table 3.1 and the same macro strain history as above; and (iii) a plasticity only case for which transformation is turned off by making f c in Eq. (2.9) very large (10,000 MPa), with the same applied traction history as (ii). Several observations are made: Polycrystals display more rounded transitions, hardening, and incomplete transformation vs. single crystals. The evidence for this is shown by the transformation + plasticity stress-strain trace in Fig. 3.3, where the width of the elastic-inelastic transition 35

56 is ~1% macrostrain, the subsequent hardening rate is ~2 GPa, and v M does not reach 1, even at 7% macrostrain. The last prediction is consistent with observations by Miyazaki et al. [15] that transformation continues into Stage II. The other inelastic-elastic and elastic-inelastic transitions are also gradual. In contrast, the predicted single crystal responses all have abrupt transitions, negligible hardening, and v M 1 at sufficiently large strain. These differences are manifestations of the constraints imposed by neighboring grains in the polycrystal. Plasticity tends to enhance the overall martensite volume fraction at a given stress. The evidence for this is provided by comparing the two curves in Fig. 3.3 at three different equi-stress locations. Upon loading, a comparison of points A and A shows that the transformation + plasticity case has a martensite volume fraction v M = 0.46 vs for the transformation only case. During unloading, points B and B show that the transformation + plasticity case has v M = 0.56 vs for the transformation only case. After complete unloading, points C and C show that the transformation + plasticity case has a retained martensite volume fraction v M = 0.03 vs. 0 for the transformation only case. These results can be rationalized for the loading and unloading cases separately. In the former, there are A sites that transform first, e.g., near stress concentrations in favorably oriented grains. If transformation only is enforced, A sites are expected to transform and shed stress to B sites. In the transformation + plasticity case, A sites may also plastically deform, thereby increasing the load shedding to B sites. This load shedding serves to increase the macroscopic v M in the stressed polycrystal. 36

57 The unloading case may also be understood in terms of A and B sites. For the transformation only case, unloading decreases v M more or less in the same spatial sequence that it formed; that is, v M decreases at B sites, followed by A sites. For the transformation + plasticity case, plasticity serves to decrease the tension at A sites and increase it in neighboring B sites. The net effect of this redistribution is to suppress the overall decrease in v M. At unload, plasticity stabilizes martensite plates formed during loading and also new plates formed during unloading. This concept is reinforced by the distribution of remnant martensite and plastic strain (Figures 3.4a and b, respectively) and the evolution of martensite and slip at specific sites (Figures 3.5a and b, respectively). In particular, the distribution of remnant effective plastic strain ( P ) in Fig. 3.4b identifies three types of site: an A site characterized by a large P (= 0.031), a B site characterized by P = 0, and sites C 1 and C 2 characterized by intermediate values, P = and 0.013, respectively. Figure 3.4a shows that the A and B sites have retained v M. However, Fig. 3.5a shows that the A site has a retained alternate martensite not formed during loading, while the B site retains primary martensite formed during loading. In particular, note how the unloading curve for the A site reaches a minimum at point (+), signifying removal of the martensite formed during loading. Continued unloading then increases v M, through the formation of new plates driven by a local compressive residual stress. In contrast the B site curve contains no crossover or reversal. Plasticity tends to suppress martensite formation at the same site. The total slip ( ) traces in Fig. 3.5b reveal that when the slope d /d 33 is larger, the corresponding slope 37

58 dv M /d 33 in Fig. 3.5a is smaller. The open circles in the A site curves demarcate regions where d /d 33 begins to decrease and dv M /d 33 begins to increase. A similar situation occurs for the C 1 site and also the C 2 site (not shown). The amounts of local retained martensite and plasticity do not always correlate. The scatter plot in Figure 3.6 shows an overall lack of correlation between the local remnant martensite volume fraction (v M ) and local remnant equivalent plastic strain ( P ). Four categories of sites are identified. Type A sites typically have the largest values of and an approximate correlation is v M ~ 20( P 0.01) ± Type B sites tend to have smaller P values and are viewed as recipients of load shedding. These points alleviate the load shedding through transformation or plasticity and an approximate bounding curve in this region is v M ~ 20(0.01 P P ) ± The minimum in the scatter plot suggests that points with P ~ 0.01 do not tend to retain martensite. For A sites, P ~ 0.01 may be too small to induce new martensite plates upon unloading and for B sites, P ~ 0.01 may be sufficiently large to alleviate the effects of load shedding. The B site results highlighted in Figs. 3.4 and 3.5 show that retained martensite can occur at material points that have no plasticity because of plastic deformation of neighbors. This is in contrast to assumptions in some phenomenological models [27, 70] which ignore neighborhood effects. Polycrystalline constraints and transformation induce inhomogeneous plastic deformation. Each grain in the polycrystal is predicted to undergo transformation before 38

59 yield if loaded in simple tension. However, in polycrystalline form, grains are generally observed to deform by both plasticity and transformation. This is a manifestation of the complex stress states generated by polycrystalline constraints. Further, the remnant effective plastic strain P in the transformation + plasticity case (Fig. 3.4b) is much more inhomogeneous than for the plasticity only (Fig. 3.4c) case. The relatively small transformation hardening values (h com and h incom in Table 3.1) make transformation an effective means to localize plasticity Thermal Cycling Response under Load Bias Initially stress-free, randomly-oriented polycrystalline samples are first loaded to a tensile bias stress bias = 300, 500, 550, or 600 MPa and then cycled at constant stress from = 370 to 200 K at 0.15 K/s and then back to 370 K at 0.15 K/s. An upper cycle temperature 370 K >> θ AF (274 K) and a lower cycle temperature 200 K< θ MF (229 K) are chosen so that the transformation is complete during the thermal cycling. Some key observations are: An increase in bias stress increases MS, MF, AS, and AF, but the increase is not uniform. The plots of macro-strain E 33 vs. temperature in Figure 3.7a clearly show this nonlinear behavior for an initially stress-free polycrystal. A particular trend is that the difference, AF AS, increases with bias stress, indicating a larger temperature range over which the M A transformation occurs. This can be attributed partly to the increased contribution of the stress work, T T, which produces a grain-to-grain variation in the transformation temperature in a stressed polycrystal. However, Figure 3.7b shows the evolution of macroscopic v M with cooling and heating at bias = 600 MPa, for the 39

60 transformation only and transformation + plasticity cases. During initial cooling, MS is controlled by the first-to-transform A sites and MF is controlled by the last-totransform B sites. Upon heating, AS is presumably controlled by the reverse transformation at B sites and AF by the reverse transformation A sites. A comparison of the two curves shows that the plasticity developed during the cycle produces a modest decrease in AS and a large increase in AF. AF increases more strongly with bias than AS, thereby widening thermal hysteresis loops as observed experimentally by Hamilton et al. [10]. The decrease in AS due to plasticity is consistent with the experimental observation of decrease in AS during no-load thermal cycling of solutionized NiTi by Miyazaki et. al. [5]. These observations are consistent with plasticity that sheds load from B to A sites, thus stabilizing martensite at A sites. Thermal cycling can enhance plasticity in stressed polycrystals. Figure 3.7a shows that a cooling/heating cycle together with a large bias produces cyclic plastic strain, E p cycle, in the direction of the bias stress. Figure 3.7b confirms that this strain is plastic for bias = 600 MPa, since upon heating, the martensite volume fraction returns to zero at the completion of the cycle. Figure 3.7a also shows the transformation strain, E trans, defined as the difference in macro strain E 33 between the lower and upper cycle temperatures. E trans reaches a peak in the vicinity of bias = 550 MPa. This peak is produced by the competition between transformation and plasticity. In particular, increasing bias serves to favor martensite plates that provide a large contribution to E trans. However, increasing bias can also promote local plasticity during the austenite-to- 40

61 martensite transformation during heating. This dominates at sufficiently large bias, thereby decreasing E trans. 3.4 Effect of Pre-Straining Randomly Oriented Polycrystals The predictions thus far show that an initial stress-free polycrystal can develop a residual stress state through stress or thermal cycling, provided plastic deformation is induced. Likewise, experiments show that stress or thermal cycling can alter shape memory response. Further, pre-straining a NiTi shape memory alloy at elevated temperature in the austenitic state is observed to harden and narrow hysteretic stressstrain loops during subsequent room temperature (θ > θ AF ) testing [16]. To examine this, the room temperature tensile stress-strain response at 298 K ( AF + 24 K) is predicted for samples that have been pre-strained in the austenitic state to ~6% in either tension or compression. To accomplish this, the temperature is increased to an arbitrarily high value (1000 K) and the polycrystal is strained such that upon unloading, the macroscopic plastic strain is 6% (or 6%). The temperature is then decreased to 298 K during which the residual stress can potentially stabilize martensite. After cooling, the new reference macroscopic sample strain is set to zero, and two stressstrain cycles are applied at a strain rate magnitude of 10-4 /s and a peak stress of 710 MPa. The following observations are made: Austenitic pre-straining is predicted to increase hardening and linearity in subsequent pseudoelastic stress-strain response. The 1 st cycle stress-strain response in Figure 3.8a shows increased hardening for both pre-tensile and pre-compressive straining. Pre-straining of either sign significantly lowers the onset stress for 41

62 transformation during loading (note A and B dots), suggesting that at some material locations, the residual and applied stress work together to induce transformation. During unloading, the critical stress for reverse transformation is increased by tensile prestraining and decreased by compressive pre-straining (note dots). The overall effect in either case is to collapse the characteristic flag-shaped hysteretic response into a smaller, more linear shape a trend similar to experiments by Rathod et al. [16]. Austenitic pre-straining followed by cooling induces remnant martensite. In particular, Fig. 3.8a shows that the 1 st cycle induces remnant strains of 0.7% and 1.2% in the pre-tension and pre-compression samples, respectively. The calculations confirm that the remnant strains are accomplished by a redistribution of martensite with no increment in plastic strain. In contrast, the no pre-strain case has a remnant strain from both retained martensite and plasticity. The 2 nd stress-strain cycle differs from the 1 st primarily in the initial loading portion. A comparison of Figs. 3.8a and 3.8b shows that for pre-strained samples, the initial loading portion of the 2 nd cycle becomes steeper and therefore more resistant to stress-induced transformation. This is reflected by an increase in the threshold stress for forward transformation (points A). In contrast, the no pre-strain case shows a decrease (points B). For all cases, the relatively large plastic strain hardening in this model a result of fitting to the [111] tensile data (Section 3.2) contributes to a shakedown in cyclic response, so that negligible shape change or remnant strain is produced in subsequent cycles. The increased hardening in pre-strained samples correlates with a larger range of applied stress over which transformation occurs. This is supported by Figure 3.9, 42

63 which shows the fraction of material points undergoing transformation as a function of applied stress. Overall, the transformation start stress MS is smaller for the pre-strained cases than the no pre-strained case. Similarly MS is smaller for cycle 2 vs. cycle 1 of the no pre-strain case. These results suggest that plastic strain broadens the range of applied stress over which transformation occurs, by inducing tensile (and compressive) regions that lower (and increase) the macroscopic stress for transformation. Austenitic pre-straining is capable of locking in unique distributions of retained martensite. Figure 3.10 shows the macroscopic martensite volume fraction vs. macro strain for different pre-strain cases. The 6% case has a narrower hysteresis in this space, operating over the range 0.3 < v M < The +6% case operates over a more narrow range of 0.3 < v M < 0.75, yet a larger change in macro-strain is achieved. The pronounced nonzero values of v M at zero macrostress are indicative of a significant residual stress state that stabilizes martensite. In particular, the stress-free cooling regions in Fig show that the martensite formed during cooling can either decrease or increase E 33, depending on the sign of the pre-strain. Austenitic pre-straining is predicted to induce a two-way shape memory effect. Figure 3.11 shows the predicted evolution of macrostrain E 33 and martensite volume fraction v M vs. temperature. The large range of over which the transformation occurs is indicative of an internal stress state. The expansion of the sample during heating indicates that the residual stress state favors martensite with a compressive contribution to E 33. This well-known experimental result is consistent with Miller and Lagoudas [6], 43

64 who also observe a negative two-way strain for samples with tensile pre-strain in the austenite state. 3.5 Summary The microstructure-based finite element approach developed in Chapter 2 is used to study the thermal-mechanical response of NiTi shape memory alloys. The constitutive relation and numerical method of solution are unique in that they couple thermal expansion, anisotropic elasticity, a rate-dependent crystal plasticity approach by Kalidindi et al. [63], and a rate-independent B2-to-B19 and B19 -to-b2 phase transformation formalism by Thamburaja and Anand [36]. An important ingredient is that the simultaneous competition between transformation and plasticity is captured on a crystallographic level. This enables fitting to single crystal data (solutionized Ti at% Ni) and subsequent study of polycrystalline response that includes the grain-tograin variation in elastic, plastic, and phase-transformation properties. The model is suitable at smaller imposed strains, where martensite detwinning is not expected to dominate. Overall, the predictions are qualitatively consistent with several experimental results and they provide insight to the coupling of plasticity and phase transformations in the NiTi system. Several observations are made: Single crystal behavior: The orientation-dependent, compressive stress-strain response of solutionized Ti at% Ni single crystals is captured qualitatively, but the model underestimates experimental values of remnant strain for several orientations, especially when the peak tensile strain exceeds 6%. The discrepancy could be due, in part, to reorientation and detwinning of martensite plates processes not captured in the 44

65 constitutive law. Also the local, fine-scale interaction between crystallographic slip and individual martensite plates is not modeled at a fundamental level but rather averaged over a crystallographic volume element in which discrete plates and plastic regions are smeared out. Pseudoelastic stress-strain response of an initially stress-free, randomly-oriented polycrystal: The model predicts a typical flag-shape response as observed experimentally. Compared to single crystals, polycrystals are predicted to have more rounded elastic-inelastic transitions, increased hardening, and incomplete transformation from B2 to B19 phase under stress. During loading, plasticity competes with phase transformations at individual material points but for the overall polycrystal, plasticity enhances martensite volume fraction at a given applied stress. Upon unloading, plasticity is observed to induce retained martensite as observed experimentally. In turn, the transformation process induces more inhomogeneous plastic deformation than if it were not present at all. Response of an initially stress-free, randomly-oriented polycrystal to thermal cycling under a load bias: The model predicts a strain-temperature hysteresis typical of experiments. An increase in bias stress is shown to increase the martensite start and finish temperatures during cooling and the austenite start and finish temperatures during heating. At sufficiently large bias stress, plastic deformation is predicted. A consequence is that the transformation strain observed upon heating attains a maximum with respect to the bias stress. Effect of elevated temperature pre-straining on polycrystalline response: Tensile pre-straining is predicted to distort the pseudoelastic hysteretic response in a trend similar 45

66 to experiments, although the predicted magnitude of distortion is less than observed experimentally. Pre-straining (both tensile and compressive) is shown to lock in different distributions of retained martensite. During cyclic loading, the variation of martensite volume fraction with macrostrain has a unique trace that depends on the pre-strain history. If a tensile pre-strained sample is then cycled thermally at zero bias stress, a negative two-way shape memory effect is predicted as observed experimentally. Most of the above observations have been captured for the first time by a microstructure based FEM model for NiTi SMA. However, because of lack of experimental data, the model is assessed only qualitatively. In Chapter 4, the model is applied to polycrystalline NiTi (49.9at% Ni) and the model predictions are assessed quantitatively against a wide gamut of experimental results. Table 3.1: Material parameters in the constitutive relations calibrated for solutionized Ti-50.9 at% Ni. Elastic and Thermal properties (G el-thermal ) Phase transformation Parameters (G trans ) Austenite plasticity Parameters (G plastic ) Elastic constants [65]: Austenite: C A 11 = 130 GPa, C A 22 = 98, C A 44 = 34 GPa; Martensite: Same as Austenite, C M = C A Thermal expansion coefficient [22]: Austenite: α A = /K ; Martensite: α M = /K Equilibrium transformation temperature: T = 257 K Latent Heat of transformation per unit volume: T =130 MPa Coefficients of hardening Matrix (refer Eq. 16): h com = C A 44 /16000, h incom = C A 44 /4000 Critical energy barrier per unit volume for transformation: f c = 8.4 MPa Reference rate of shear: 0 = 0.002/sec Strain rate exponent: m = 0.02 Initial hardness of slip system s: g 0 s = 320 MPa Self hardening coefficient: h 0 = 500 MPa Ratio of self to latent hardening: Q = 1.4 Hardening exponent: a = Saturation hardness: g sat = 900 MPa 46

67 Macro Stress, 33 (MPa) Simulation Experiment [111] C [210] A A' B Macro Strain, E 33 Figure 3.1: Room temperature (θ AF + 24 K) macrostress 33 vs. macrostrain E 33 response in uniaxial compression for solutionized Ti 50.9 at% Ni single crystals oriented along [111] vs. [210]. Experimental measurements are from [12] and the fitted predictions are from the finite element model with the material parameters in Table (MPa) ( a) T max =0.38 S max = Simulation Experiment [100] E (MPa) ( b) T max =0.4 S max = [110] E (MPa) (c) T max =0.41 T S max =0.47 max = S max = [211] E (MPa) (d) Figure 3.2: Room temperature (θ AF + 24 K) macrostress 33 vs. macrostrain E 33 response in uniaxial compression for solutionized Ti 50.9 at% Ni single crystals oriented along (a) [100], (b) [110], (c) [211], and (d) [123]. Experimental measurements are from [12] and the finite element model predictions are based on the material parameters in Table 3.1. The maximum Schmid factors for transformation and crystal slip ( ζ, ζ shown for each case. 47 [123] E 33 T max S max ) are

68 Macro Stress, 33 (MPa) Stage I A':0.38 A'':0.51 A:0.46 Transformation only B':0.52 B'':0.65 Stage II M =0.92 M = B: Transformation + Plasticity C':0 C: Macro Strain, E Figure 3.3: Predicted tensile macrostress 33 vs. macrostrain E 33 response for an initially stress-free, random polycrystal of solutionized Ti 50.9 at% Ni at room temperature ( AF + 24 K), assuming either (i) transformation + plasticity mechanisms or (ii) transformation only mechanism. The average martensite volume fraction (v M ) is indicated at various points. 48

69 C 2 :0.1 C 1 :0 C 2 :0.013 C 1 :0.015 A:0.4 B:0.06 A:0.031 B:0 (a) ν M (Trans.+Plast.) max: 0.58 min: 0 (b) ε p (Trans.+Plast.) max: 0.39 min: 0 C 2 :0.005 C 1 :0.007 A:0.007 B:0.002 (c) ε p (Plast. only) max: 0.23 min: 0 Figure 3.4: Predicted distributions of (a) remnant martensite volume fraction v M and (b) P remnant equivalent plastic strain for the transformation + plasticity case in Fig. 3.3, P after unloading to point C (Fig. 3.3), and (c) remnant equivalent plastic strain for the plasticity only case, after unloading (not shown in Fig. 3.3). Maxima and minima are listed and type A, B, and C sites are identified. 49

70 1 Site: B C 1 A Martensite Volume Fraction, M B site: A site: Remnant Remnant Primary Alternate Martensite Martensite + ( a) Local Strain, Site: A Total Slip, C ( b B ) Local Strain, 33 Figure 3.5: Predicted (a) local martensite volume fraction v M and (b) local slip activity = vs. local strain 33 at sites A, B, and C 1 in Fig The analysis corresponds to the transformation + plasticity case in Figs. 3.3 and

71 Remnant Martensite Volume Fraction, M B sites A sites Remnant Equivalent Plastic strain, p Figure 3.6: Predicted remnant martensite volume fraction v M vs. remnant equivalent P plastic strain at 5832 integration points throughout the polycrystal, after unload. The analysis corresponds to the transformation + plasticity case in Figs. 3.3 and 3.4. Approximate trend lines for A and B type sites are shown. 51

72 Macro Strain, E ( a) MF AS Bias 33 (MPa) E trans (%) AF E P cycle Macro Martensite Volume Fraction, M MS Temperature, (K) 1 ( b) Transf. only Decrease in AS Transf.+Plasticity Increase in AF Temperature, (K) Figure 3.7: Predicted (a) macrostrain E 33 vs. temperature for an initially stress-free, random polycrystal of solutionized Ti 50.9 at% Ni, subjected to a constant tensile stress bias bias 33 = 300, 500, 550, or 600 MPa; and (b) macro martensite volume fraction vs. temperature for bias 33 = 600 MPa. The results in (a) are for the transformation + plasticity case, where E trans is the difference in macro strain E 33 at the lower cycle temperature and at the upper cycle temperature at the end of thermal cycle, E p cycle is the accumulated plastic strain during the cooling + heating cycle, and the symbols ( ) show the evolution in temperatures ( MS, MF, AS, AF ). The results in (b) are for transformation + plasticity and transformation only cases and show the increased hysteretic width for the former case. 52

73 Macro Stress, 33 (MPa) Macro Stress, 33 (MPa) Pre-strain: 6% -6% B A A 100 ( a) 1 st Cycle Macro Strain, E Pre-strain: 6% -6% B A A 100 ( b) 2 nd Cycle Macro Strain, E 33 Figure 3.8: Predicted (a) 1 st cycle and (b) 2 nd cycle macrostress 33 vs. macrostrain E 33 response at room temperature ( AF + 24 K), for a random polycrystal of solutionized Ti 50.9 at% Ni subjected to different pre-strains: 6%, 6%, and 0 (no pre-strain). The pre-strain is imposed by heating the sample and deforming it in the austenitic state, then cooling to 298 K, and setting the reference macrostrain E 33 = 0 prior to loading. The circles indicate the approximate onset of forward (A M) transformation during loading and reverse (M A) transformation during unloading. 53

74 Pre-strain: 6% 0% 1 st 6% 2 nd 2 nd 0% 1 st MS Figure 3.9: Predicted fraction of material points undergoing transformation vs. macrostress 33 for the pre-strain = 6% and 0% cases in Fig The 1 st cycle responses (solid curves) begin at the symbols and the second cycle responses (dashed curves) begin at the symbols. The symbols indicate the approximate onset stress MS for the forward (A M) transformation for various cases. Macro Martensite Volume Fraction, M Stress free cooling -6% Pre-strain: 6% Macro Strain, E 33 Figure 3.10: Predicted macro volume fraction of martensite v M vs. macrostrain E 33 during cyclic loading, for the pre-strain = 6%, 6%, and 0 (no pre-strain) cases in Fig The 1 st cycle responses begin at the symbols and the 2 nd cycles responses begin at the symbols. 0% 54

75 Macro Strain, E M E Temperature, (K) Figure 3.11: Predicted macrostrain E 33 and macro volume fraction of martensite v M vs. temperature during a cooling + heating cycle, for a random polycrystal of solutionized Ti 50.9 at% Ni subjected to 6% pre-strain. The pre-strain is imposed by heating the sample and deforming it in the austenitic state. The symbols indicate the start of the cooling+ heating cycle Macro Martensite Volume Fraction, M 55

76 CHAPTER 4: MODEL APPLICATION TO HOT-WORKED 49.9 AT.% NI-TI The model developed in Chapter 2 is now quantitatively assessed against the experimental response for hot-worked 49.9 at.% NiTi. An important challenge in this assessment is to capture the actuation response [7-9]. A typical situation is shown in Figure 4.1a, where the material is subjected to a constant tensile bias stress bias. Upon heating, the SMA contracts by a macroscopic transformation strain E T (see feature F7), thus performing work. A key trend is the dependence of E T on bias (see inset to Fig. 4.1a). The macrostrain (E) vs. temperature ( ) traces in Fig. 4.1a show the hysteretic width H (feature F6), widths A-M (feature F8) and M-A for the austenite-to-martensite and martensite-to-austenite transformations, respectively, and the open loop strain per thermal cycle, E cycle. These quantities typically depend on bias, composition, processing history, and microstructural features such as single/polycrystalline orientation/texture, precipitate morphology, dislocation substructure, and internal stress state. Some particular challenges are to capture the initial, gradual increase in E T with bias, the attainment of a peak E T at moderate bias, and the decrease in E T at larger bias (see inset, Fig. 4.1a). The gradual increase in E T tests model assumptions about martensite interactions, (see last term to the right of eq. (2.10)). These can be formulated between individual twinned habit plane variants (hpv) [30, 31, 34, 38, 46], hpv groups [33], or individual lattice correspondence variants (lcv) [32]. 56

77 An implementation based on individual martensite hpv by Patoor et al. [31] is often adopted (e.g., [34, 38, 46]), but the predicted E T increases too abruptly with bias [32]. The peak in E T with bias tests model assumptions concerning plasticity. A principal goal and contribution of this Chapter is to determine whether the stateof-the-art modeling approach developed in Chapter 2 can capture key thermal cycling features shown in Fig. 4.1a, in addition to other behavior. In addition, the Patoor et al. [31] formalism for hpv-hpv interaction as well as hpv-crystal plasticity interaction are assessed. The model is rigorously calibrated against the inputs that include anisotropic elastic properties, texture, and DSC data as well as a subset of recent isothermal deformation and load-biased thermal cycling data. This allows for inclusion of recently reported elastic constants for monoclinic martensite [67] and it incorporates the appropriate invariant-plane condition between austenite and martensite a feature not satisfied in a recent self-consistent approach to incorporate anisotropic elasticity [71]. The model results are assessed against additional experimental data, including recent neutron diffraction measurements of martensite texture evolution. This kind of rigorous assessment is unique for MFE models for NiTi SMA, particularly the model comparison with in-situ neutron diffraction during thermal cycling is not reported elsewhere in the literature. Several experimental trends are captured in particular, the transformation strain during thermal cycling monotonically increases and reaches a peak with increasing bias stress. This is achieved, in part, by modifying the martensite hardening matrix proposed by Patoor et al. [31]. Some experimental trends are not captured in particular, the ratcheting of macrostrain with continued thermal cycling. This may reflect a model 57

78 limitation that transformation-plasticity coupling is captured on a coarse (grain) scale but not fine (martensitic plate) scale. This Chapter summarizes the results of [58]. Section 4.1 describes experimental data for the polycrystalline 49.9at %Ni-Ti alloy, performed in support of the modeling activity. Section 4.2 describes the finite element (FE) model. Section 4.3 presents the process for calibrating the model parameters to a subset of experimental data in Section 4.1, and Section 4.4 assesses predictions of the calibrated model to additional data in Section 4.1. Section 4.5 provides an overall assessment of model capabilities and conclusions. 4.1 Experimental Characterization Material System and History The binary NiTi alloy, containing 49.9 at.% (55 wt.%) Ni, used in this study is described in detail elsewhere [9]. The alloy was produced by Special Metals, New Hartford, NY and was supplied in the form of 10 mm diameter rods in the hot-rolled/hotdrawn and hot-straightened condition, but the specific thermo-mechanical processing history is proprietary. The alloy is single phase with a dynamically recrystallized and equiaxed grain structure and average grain size of ~40 m. All experimental measurements on as-machined specimens are preceded by two no-load, or stress-free, thermal cycles from 30 to 200 C at a heating/cooling rate of approximately 20 C/min. These thermal cycles are used to reduce internal stress due to the original material processing procedures and subsequent machining operations. 58

79 4.1.2 Texture Analysis The as-received alloy exhibits a very weak, recrystallization texture. Figure 4.2a shows the experimentally determined (100), (110) and (111) pole figures. These results were obtained at Los Alamos National Laboratory (LANL) via neutron diffraction at elevated temperature, where the austenite phase is stable using High Pressure-Preferred Orientation (HIPPO) diffractometer Differential Scanning Calorimetry DSC calorimetric tests were performed to measure the heat of transformation, Q DSC, as well as martensite start temperature, θ MS, martensite finish temperature, θ MF, austenite start temperature, θ AS, and austenite finish temperature, θ AF [9]. Key results as used in this study are determined from n = 7 DSC sampling size and are summarized in Section Determination of the Stress-Strain Response Above θ AF The NiTi alloy does not exhibit anything resembling, flag-like pseudoelastic behavior, as typically observed above θ AF in Ni-rich NiTi alloys [11, 12, 18, 34-37, 69]. However, a small amount of stabilized martensite apparently forms when the alloy is deformed plastically above θ AF. Figure 4.3a shows the uniaxial tension stress-strain response at θ 0 = 130 C, for maximum imposed axial strains E max = 0.02 and 0.04, and also at 215 C for E max = A small axial strain rate of 10-4 /s is used to help ensure isothermal conditions. After E max is attained, the sample is unloaded isothermally and then heated to 600 C under no load. During heating, the macrostrain E decreases due to recovery (transformation) of stabilized martensite to austenite. It is then cooled back to θ 0. 59

80 Recovery is confirmed by noting that the E-θ response during initial cooling is consistent with the thermal contraction of austenite [62]. The strain E post-heat (feature F13, Fig. 4.3a) is measured at θ 0 and is expected to be primarily due to plastic deformation Load-Biased Thermal Cycling Figure 4.1a shows the macrostrain-temperature (E-θ) response of NiTi at different values of uniaxial bias stress (Σ bias ). The stress level is achieved by imposing a 10-4 /s axial strain rate using strain control until the desired stress is reached at constant temperature, θ min = 30 ºC, where martensite is stable. The controller is then switched into load control and the stress is held constant. This is followed by thermal cycling up to θ max = 200 ºC and back to θ min = 30 ºC at constant Σ bias. The second thermal cycle responses are shown in Fig. 1a, as well as the transformation strain E T vs Σ bias (inset) In-Situ Neutron Diffraction Neutron diffraction spectra are obtained during stress-biased thermal cycling, using the protocol in Section These measurements are obtained in time-of-flight mode using the Spectrometer for MAterials Research at Temperature and Stress (SMARTS) facility at LANL. Two detector banks furnish the diffracted intensity vs. wavelength from two groups of crystallographic planes: those parallel and those perpendicular to the loading axis. From this data, the evolution of martensite and austenite (volume fraction and orientation) can be determined during thermal cycling. Additional details can be found in [9] and the references therein. 60

81 4.2 Polycrystalline Simulations A brief description of the finite element representation of the polycrystal and boundary conditions is provided. The FE representation is similar to the one outlined in Chapter 2 and Chapter 3 with additional matching of experimental texture as well as new boundary conditions appropriate for simulating new experimental data Discretization at the Grain Scale and Texture Specification The polycrystal is modeled by a cube assembly of 8-node 3D brick elements (C3D8 for isothermal analyses and C3D8T for thermo-mechanical analyses) using the commercial finite element (FE) software ABAQUS [55]. Each element represents a grain with an assigned crystal orientation of the austenite (B2) phase. The texture analysis software POPLA [72] is used to obtain an initial texture that is statistically equivalent to that in the experiments. Due to the late availability of HIPPO data, the texture was initially fitted with SMARTS data with an assumption of axisymmetry. The assumption was later verified with the texture from HIPPO which shows qualitatively similar texture with strong (111) texture and relatively weak (100) texture along the hot-working direction. Figures 4.2a and 4.2b show the initial experimentally determined texture for the as-received NiTi and the simulated version of this texture used as input for the model, respectively. The number of grains, N grain = 343, is considered sufficient since the macroscopic response does not change significantly for N grain >

82 4.2.2 Predeformation E p(pre) in the Austenitic State Even though the material was hot-worked and all experimental measurements on as-machined specimens are preceded by two no-load or stress-free thermal cycles, some level of internal stress can still be present in the samples. The simulations introduce internal stress via predeformation of the polycrystal in the austenitic state. This is achieved by heating to 300 ºC to stabilize the austenite phase, then uniaxial loading at a strain rate of 10-4 /s to achieve a macroscopic plastic axial prestrain of E p(pre), then unloading. The sample is then cooled to the desired test temperature θ 0. E p(pre) is determined by calibration to select experimental data (Section 4.1) Isothermal Deformation Testing and Post Heating The polycrystal is strained to E max at some constant temperature θ 0 > θ AF and then unloaded. This is achieved via prescribed normal displacement rates and zero shear tractions on the top and bottom surfaces, equivalent to ±10-4 /s axial strain rate. The transverse faces are traction-free. The macrostrain E unload after unloading is partitioned into plastic and transformation contributions by integration of the local plastic and transformation strains over the polycrystal. The simulations assume spatially uniform, isothermal conditions. This is considered reasonable due to the small imposed strain rate, which minimizes local heating and heat transfer rates associated with the austenitemartensite phase transformation. Figure 4.3b shows results using the calibrated model Stress-Biased Thermal Cycling The polycrystal is heated to the maximum cycling temperature θ max = 190 ºC, then the bias stress Σ bias is imposed using an axial strain rate of 10-4 /s. The sample is then 62

83 cycled between θ min = 30 ºC and θ max at ±0.08 ºC/s. The simulations assume a spatially uniform temperature, which is reasonable given the small heating/cooling rate. A difference is that the simulations begin the load application and cycling at θ max, compared to θ min for the experimental data. However, previous experiments [8] and the current results show that this does not affect key parameters such as transformation strain. Figure 4.1b shows results using the calibrated model In-Situ Neutron Diffraction Simulations Comparison to experimental data requires the simulated output to undergo three post-processing steps. First, the predicted volume fractions of martensite habit plane variants are converted into volume fractions of martensite correspondence variants and their orientation using the Crystallographic Theory of Martensite (CTM) [4] in particular, Eq. 17 of [60] is solved. Second, the martensite volume fractions v N and v N must be determined, where N denotes the plane normal of interest (e.g., [100] or [011]) and and denote whether N is parallel or perpendicular to the loading axis. The third step is to normalize the diffracted intensities I(N) of plane N for different loading cases with the reference intensity I ref (N) from a reference test case such that the intensity dependence upon martensite volume fractions v N is isolated as I( N) I ref ( N) vn (4.1) v N ref Load biased thermal cycling under a bias stress of 100 MPa is chosen as the reference. 63

84 4.3 Calibration Of Material Parameters Table 4.1 shows the material parameters used to simulate the 49.9 at% (55wt%)Ni-Ti SMA. They are adopted from published literature values or calibrated to differential scanning calorimetry (DSC) data and select isothermal deformation and thermal cycling tests performed specifically to support this modeling effort. These various features and the calibration process are described in Sections Elastic-Thermal, P el-thermal = {C A, C M, A th-a, A th-m } The anisotropic elastic stiffness elements C A for the cubic (B2) austenite phase and C M for the monoclinic (B19 ) martensite phase are adopted from density functional theory calculations (Feature F2, Table 4.1) by Hatcher et al. [67]. Isotropic thermal expansion properties are assumed with A and M taken from the neutron diffraction experiments (Feature F3, Table 4.1) of Qiu et al. [62]. The use of recently computed elastic moduli corrects frequent assumptions in the literature that C M ~ ½C A e.g. [35, 37] or C M C A e.g. [22, 34, 38, 46, 47, 73]. These effects, including differences in moduli from Hatcher et al. [26] vs. Wagner et al. [41], are discussed in Section Transformation, P trans = {θ T, λ T, f c and h tu } This study models N T = 24 type-ii twinned martensite plate types, similar to Chapter 3. The rationale for the assumption is discussed in Section 3.2. The transformation parameters P trans are obtained from a combination of DSC (no load), isothermal deformation tests, and thermal cycling data at bias = 50 MPa. The DSC data (averaged over 7 samples) gives θ MF, θ MS, θ AS and θ AF = 46, 71, 86 and 109 ± 2 ºC, respectively, for this 49at.% Ni-Ti alloy. These values (Feature F4) furnish the 64

85 transformation temperature θ T = (θ MS + θ AS )/2 = K (standard deviation = 2 K). The calibrated simulations use a latent heat of transformation T = 140 MJ/m 3 as detailed in Section 4.3. Initially, an average of Q DSC (Feature F5) for the forward A M and reverse M A heats of transformation is used, yielding λ T Q DSC = MJ/m 3 (standard deviation=4.7 MJ/m 3 ). This is viewed as an initial guess since Q DSC includes contributions from elastic or defect energies associated with the transformation. The critical driving force f c for transformation is obtained from f c M A A M T 2 T H 2 T (4.2) A M M A which is obtained by writing the forward (f t = f c ) and backward (f t = f c ) critical conditions in extended form using Eq and then taking the difference between them. Eq. (4.2) furnishes f c /λ T =0.051, based on an average hysteretic width (Feature F6) H = 36.3 K (standard deviation = 1.15 K) for thermal cycling at Σ bias = 50 MPa (see Fig. 4.1a). This Σ bias is selected to minimize plasticity; a similar f c /λ T value occurs with Σ bias = 0 data. Formally, H reflects an average width over the range = 0.1 to 0.9, where is the fraction of transformation strain Determination of structure of h tu The structure of the martensite plate interaction matrix, h tu, in Eq is obtained by calibrating the simulations to give E T 1% (Feature F7) and A M 25 ºC (Feature F8) at Σ bias =50 MPa. A M is the decrease ( MF - MS ) in temperature to complete the A M transformation. The outcome is that h tu has three independent values: 65

86 h self ( C A(44) / 400) if t u h tu h com ( 0) if t u anddet( T (t) T (u) ) 0 h inc ( C A(44) /12000) if t u anddet( T (t) T (u) ) 0 (4.3) This structure is a modification to the popular form proposed by Patoor et al. [31], where h self = h com = C A(44) /3000 MPa and h inc = C A(44) /750MPa are used for NiTi [34, 38, 46]. The rationale for the modification is that simulations employing the Patoor et al. [31] structure (curve h tu(patoor), Figure 4.4a) give E T ~ 4% for Σ bias = 50 MPa and thus over predict experimental values. This over-prediction occurs regardless of the h com and h inc values or transformation-related parameters (e.g. ). Niclaeys et al. [32] also report over prediction of transformation strain when h tu = h tu(patoor) is used. An underlying reason is that martensite plates with maximal transformation strain along the loading axis dominate over other plate types, because the stress work (term 1 in f c, Eq. 2.19) initially renders the driving force f + largest for this favored (+) plate type. As long as h com < h inc, the favored (+) plates experience less hardening than non-favored ( ) plates, because f + / v + < f / v +. Thus, all existing MFE models in the literature, which either ignore the plate-plate interaction by setting h tu =0 [35-37] or use the approximation by Patoor et al. [31] [30, 31, 34, 38, 46], or use hpv groups [33], or individual lattice correspondence variants (lcv) [32] to determine the compatibility between plates favor, formation of highly textured martensite. The martensite plates favored by the external bias stress dominating over the non-favored ( ) (self accommodating) plates even at the small bias stress. Indeed, compatibility constraints between neighboring grains should encourage formation of multiple plate types, but it is insufficient to match experimental data. The modification sets h self >> h com, h inc to suppress formation of a single plate type and results 66

87 in a rather weak texture of martensite with plates self-accommodating one another when subjected to temperature cycling under small bias stress. The calibration is guided by application of Eq to an idealized case of two populations of plate types: a favored (+) type that renders a transformation strain E T(max) if v + 1 and a non favored (self-accommodating) ( ) collection of plate types that renders E T ~ 0 if v 1. Taking the difference between the critical force conditions f + = f c and f = f c furnishes S b i a h T s 2 m a x 1 h T T 2 m a x 1 (4.4) T where S + bias is the stress work to form favored plates, and h + = h ++ h + and h = h h + are differences in hardening elements. Thus, increases in the differences (h self h com ) and (h self h inc ) are expected to decrease E T. Eq also furnishes M S M F T h h (S b i a h s ) h T h h and h s e l f m a x MS (4.5) S The second of Eqs. 4.5 suggests that h self can be determined from the difference, max MS (Feature F9), from isothermal deformation tests (Fig. 4.3a). Further, Eq. (4.4) and the first of Eqs. (4.5) suggest that the differences, h self h com and h self h inc, can be determined by capturing Features F7 and F8 for thermal cycling at low bias stress (Fig. 4.1a). The final results reported in Eq. (4.3) and Table 4.1 are obtained as a best fit. 67

88 4.3.3 Austenite Plasticity, P plastic = { γ 0,m, g 0 s, g sat, h 0, Q, a, E p(pre) } and T update These parameters are obtained from literature values and additional isothermal deformation data. In particular, 6 {110}/<100> and 6 {001}/<100> austenitic slip systems are considered [68]. The rate sensitivity m = 0.02, reference strain rate γ 0 = 10-3 /s ( E 10-4 /s), Q = 1.4, a = and g sat = 900 MPa (see Eqs. 7 and 8) are adopted from prior calibration of the model to single crystal pseudoelastic compression tests of Chapter 3 (Feature F10, and Table 3.1), using loading orientations that enhance plasticity. In reality, there is limited data to determine these parameters accurately, but the simulations are less sensitive to them. The remaining plasticity parameters are the threshold resolved shear stress g s 0 for plastic flow, initial strain hardening h0, and plastic pre-deformation 0 E p(pre). Since transformation and plasticity are coupled, a best fit of g s = 272 MPa, h0 = 50 MPa, and E p(pre) = 0.7% and λ T = 140 MJ/m 3 (updated from an initial guess MJ/m 3 ) is obtained by the calibration. The calibration procedure involves matching MS, max, and E post-heat = for E max = 0.02 (Features F11-13, Figure 4.4b). The updated, smaller T decreases MS and max to better match the data, but g s 0 and h0 are indeterminant. For example, the same 0 curve (E p(pre) = 0, Fig. 4b) is obtained for ( g s, h0 ) = (250 MPa, 50 MPa) vs. (235 MPa, 500 MPa). The first case is selected since it better matches E post-heat = (Feature F14, Fig. 4.3a). The second case under predicts the value. A final issue is that curve E p(pre) = 0 (Fig. 4.4b) over predicts MS (Feature F11). Section 3.4 demonstrates that compressive prestraining decreases MS and increases 68

89 0 hardening. The calibrated model result (Fig. 4.4b) adopts E p(pre) = 0.7% and ( g s, h0 ) = (272 MPa, 50 MPa) as a best match. A larger compressive prestrain does not improve the match. This pre-deformation results in a multi-axial and spatially varying residual stress distribution with stress in the loading direction varying from -345 MPa to 272 MPa. These magnitudes are a substantial fraction of MS (feature F11) in Fig. 4.3a. 4.4 Model Assessment Isothermal Deformation Response Overall, the calibrated model captures the monotonic loading portion for different E max and, but certain features of the unloading response are not reproduced. In particular, the predicted monotonic loading for = 215 ºC (Fig. 4.3b) and max for = 130 ºC (Pt A, Fig. 4.3b) reproduce the experimental trends. This includes the successful prediction of an upward shift in -E response with increasing. The calibrated model gives a Clausius-Clapeyron slope dσ/dθ= 6.2 MPa/K for the most favorably oriented martensite plates, consistent with literature [34, 74]. An increase in θ from 130 to 215 ºC should therefore increase the transformation stress by ΔΣ = 627 MPa, yet the experiments and simulations show ΔΣ ~ 100 MPa. The results suggest that transformation is promoted by complex, multi-axial stress states inherent in polycrystals, and the interaction with concurrent plastic deformation observed in this material. The discrepancy between the experimental and simulated unloading paths suggests that the reverse (M A) transformation is complex. In particular, the experimental unloading paths are relatively linear (Fig. 4.3a), indicating that any stress-induced martensite is apparently locked in place by the concurrent plastic deformation during 69

90 loading and is only recovered upon heating to elevated temperature as the plastic deformation is relaxed. In contrast, the simulation ( = 130 ºC, Fig. 4.3b) is nonlinear, with elastic response followed by reverse transformation. Also, there is a discrepancy in the recovered strain from the 600 ºC post heat treatment with simulations recovering smaller strain compared to experiments. A possible implication is that the f c values for the M A transformation may not be constant throughout the polycrystal, as assumed in the simulations, and they may depend on the local amount of plastic deformation Thermal Cycling Response Overall, several experimental trends are captured including the textured nature of martensite, but there are quantitative discrepancies Effect of Bias Stress and Prestrain on Transformation Strain and Critical Temperatures Consistent with experiments, the predicted transformation strain E T increases with bias, as do the critical temperatures AS, AF, MS, MF, and temperature difference MS MF (Fig. 4.1). Figure 4.5 shows that the response for thermal cycling at bias = 0 is qualitatively captured. This demonstrates that a plastic predeformation E p(pre) = 0.7% creates a positive E T, by inducing an internal stress state that biases martensite plate formation. Conversely, a negative E T is generated for E p(pre) > 0. Figure 4.6 shows that the simulations capture the increase and then decrease in E T with bias. This is due primarily to the new proposed structure for h tu (Eq. 11) and the coupling of crystal plasticity and transformation (Eq. 2.3). These features cause E T to increase gradually with 70

91 bias due to the formation of favored (+) martensite. For bias > 350 MPa, the model predicts a decrease in E T with bias due to the onset of open loop strain, E cycle (see Fig. 4.8). The predicted effect of E p(pre) = 0.7% vs. 0 on the E T bias response in Fig. 6 is negligible. For comparison, simulations using the Patoor et al. [31] structure for h tu substantially overestimate E T at small bias (Curve h tu(patoor), Fig. 4.6) and predict E T to be relatively independent of Σ bias in the single crystal limit, contrary to these and previous experimental results [10] Plastic Strain Enhancement Due to Phase Transformation Simulations of thermal cycling at large bias (400 MPa) reveal that transformation enhances the macroscopic plastic strain E p. Figure 4.7 shows rather modest increases in E p with thermal cycling for the plasticity only case where transformation is turned off. In contrast, E p is larger in the calibrated model case, where both transformation and plasticity are present. The largest increases in E p occur during both cooling and heating, at a stage when v M is small (see regions 1 and 2, Fig. 4.7). The phenomenon is reminiscent of ratcheting of E p during stress-biased thermal cycling of composites with a large thermal strain mismatch between constituents [75]. It also consistent with results from Section showing that plasticity aids martensite transformation during pseudoelastic loading to some macroscopic stress Discrepancy in Open Loop Strain An important quantitative discrepancy is the underestimate of incremental strain per thermal cycle, E cycle, or open loop strain (Fig. 4.1). Although transformation is predicted to enhance plasticity, the simulations shake down to negligible values of 71

92 E cycle after a few (2-3) cycles compared to experiments that show continuous ratcheting for 100 s of cycles. Moreover, the simulations predict noticeable E cycle values only at large bias (> 350 MPa) while experiments show noticeable values even at small bias (50 MPa). This discrepancy may be due to the nature of the aggregate constitutive relation developed in Chapter 2, and the assumption that the rate of slip activity ( γ s ) and rate of martensite formation ( v t ) are both computed from the average Piola-Kirchoff stress T * for the aggregate, thereby constituting an isostress approach to the aggregate. In reality, the local stress field around martensite plates can be sufficiently large to drive local plasticity, as evidenced by recent transmission electron microscopy of dislocation content in micron-scale, pseudoelastically-compressed single crystals [17] and load-biased thermal cycling of larger single crystals [10] Assumptions Concerning Martensite Elastic Moduli and Thermal Expansion Coefficients The simulations reveal several aspects concerning elastic moduli and thermal expansion coefficients. First, Figure 4.8 shows that the predicted transformation strain E T during thermal cycling increases by about 1% larger if a common assumption that C M ~ ½ C A is used (e.g., [35, 37]). Both experiments [76] and first principle calculations [66, 67] show that martensite is in fact stiffer than the austenite and any assumption of C M C A is erroneous. A more compliant martensite increases E T by increasing the elastic strain and texturing in the martensite; the elastic strain contribution is deduced from the unloading traces from points A to A in Fig The results also suggest that good approximations can be obtained by modeling martensite as elastically isotropic (C M = 72

93 C M(isotropic) ), obtained by approximating martensite with isotropic stiffness elements with Young s modulus and the shear modulus given by the Hill s averages, for Young s modulus E h (180 GPa) and shear modulus G h. In particular, Fig. 4.8 shows that the anisotropic and C M = C M(isotropic) cases essentially overlap, with E h = 180 GPa (180 GPa) and shear modulus G h = 69 GPa computed from the Hatcher et al. data [67]. If instead the anisotropic moduli from Wagner et al. [66] are used, then smaller values E h = 122 GPa and G h = 45 GPa occur, but also C A is smaller. The net effect is to shift the response upward ~0.3% (Fig. 4.8), leaving E T essentially unchanged. The observation that martensite can be represented appropriately by isotropic elastic moduli suggests that it is also reasonable to use isotropic thermal expansion coefficients, at least for aggregate scale simulations. Thus, the adoption of isotropic descriptions for A th-a and A th-m in Section 4.1 appears reasonable Texture Evolution The calibrated simulations show substantial promise in capturing the nature of textured martensite formation during stress-biased thermal cycling. For bias = 150 MPa, in-situ neutron diffraction [77] at 130 ºC during heating reveals peaks from martensite with plane normals N = [011] to the loading axis and also N = [100] to the loading axis. Cases N = [011] and N = [100] are not observed. The calibrated model predicts the same results at 130 ºC. Figure 4.9 shows the evolution of the volume fraction of martensite that contributes to the diffraction intensities for N = [100] and N = [011] in both and to the loading axis. At 130 ºC, there is 5% total retained martensite (not shown in figure 4.9), some of which has N = [011] and N = [100], but none of which 73

94 has N = [011] or N = [100] as shown in Figure 4.9. This complements earlier work by Gao et al. [78] comparing pole figures from a micromechanics model to neutron diffraction data, albeit under pseudoelastic loading. The simulations also capture the effect of bias on martensite texture during thermal cycling. Figures 4.10a and 4.10b show that the normalized intensities of [100] and [011] respectively, measured at min = 30 ºC, monotonically increases with bias. Intensities are normalized by the intensities obtained at bias =100 MPa. The calibrated model predicts a similar trend a decreasing intensity of [011] and an increasing intensity of [100] with increasing bias. However, the simulations based on the hardening matrix of Patoor et al. [31] (see discussion following Eq. 4.3) underestimates the martensite texture development with increasing bias because even at a low bias, it predicts a highly textured martensite, leaving little room for additional texturing with increasing bias. The calibrated model results do depend slightly on E p(pre) because it induces a multiaxial residual stress state that can aid or oppose the bias stress, depending on the location. The net effect is that the predicted the normalized intensity [100] in Fig. 4.10a shifts downward by ~0.03 for E p(pre) = 0 (compared to 0.7%) while [011] in Fig. 4.10b remains largely unchanged. 4.5 Model Calibration and Assessment for Isothermal σ-ε Response at θ<θ MF While the thesis mainly concentrates on mechanical response of NiTi under loadbias temperature cycling and isothermal stress-strain response at temperature θ>θ AF, the model is capable of capturing the isothermal deformation response of martensite at temperature θ<θ MF. This section first determines the only additional material parameter 74

95 f cdtw, required for the extended model presented in section 2.3 and then assesses the model predictions with additional experimental data. Figure 4.11 shows the experimental tensile and compressive stress- strain response of martensite at 22 ºC. The initial stress-free martensite microstructure is obtained by lowering the temperature of the material to 22 ºC under stress free conditions. A selfaccommodating martensite structure is expected to form, barring the small amount of texturing due to residual stresses. A macroscopic strain rate of /sec is used to impose the strain to this martensite structure. The model starts off with cooling the calibrated pre-strained austenite phase from section 4.3 from 180 ºC to 22 ºC. The resulting hpv martensite plate volume fractions are then converted into lcv martensite volume fractions using eq. (2.24). Once the initial martensite structure is obtained, the model is subjected to strain at a rate of /sec. The material parameter f cdtw (critical barrier for initiation of detwinning, eq. (2.27)) is calibrated by matching the experimental stress at the initiation of the inelastic response (the initial plateau stress in fig. 4.11) for the tensile loading case. The model is able to capture the subsequent hardening response in the tensile loading case. This hardening arises due to the exhaustion of martensite variants that are oriented to easily detwin. Since the model does not incorporate the plasticity in the martensite phase, it does not capture the second plateau in the stress-strain response observed at large strains (> 7%). The model also shows a stiffer elastic response of martensite. This might be because of use of elastic moduli of martensite from DFT calculations at 0 K. 75

96 Figure 4.11 also compares the model predictions of the compressive response with the experiments. The model successfully captures the tension-compression asymmetry with enhanced hardening in the compressive response. This asymmetry is captured automatically due to the crystallographic description of the inelastic deformation processes used in the model. 4.6 Summary Several key elements are required to capture the thermal cycling and isothermal deformation trends for polycrystalline NiTi ( 49.9at.% Ni). Among the most critical features is the coupling of deformation and transformation processes at the crystallographic (granular) scale. Also required is a calibration procedure by which unique values of important plasticity and transformation constitutive parameters are determined from experimental data. This work shows that with an appropriate set of data it is possible to calibrate a model that captures (i) the effect of temperature on isothermal deformation loading response; (ii) the effect of bias stress on critical temperatures, transformation strain, and martensite texture evolution during thermal cycling, and (iii) the effect of deformation processing in the austenitic state on the two-way effect and uniaxial loading at > AF. In addition to the coupling of transformation and deformation mechanisms, two key modifications to this model are the rigorous incorporation of anisotropic elasticity and an augmented form of the martensite hardening matrix h tu (Eq. 4.3). The first allows incorporation of elastic moduli for martensite determined from recent density functional theory (DFT) calculations. This corrects a frequent assumption that the martensite moduli 76

97 are ~ ½ those of austenite. This assumption should be avoided and at a minimum, simulations should adopt the Hill averages of the anisotropic (DFT) moduli of martensite. The second modification involving h tu (Eq. 4.3) increases the self-hardening term, thereby suppressing the formation of a dominant habit plane variant (plate). This is required to capture the gradual increase in transformation strain and martensite texture with increasing bias stress during thermal cycling, while coupling with plasticity is required to capture the decrease in transformation strain at high bias stress. This rigorous model development, calibration, and assessment also provides insight to current model deficiencies. Although the simulations capture the compatibility constraints between grains and the coupling between plasticity and transformation on the grain scale, the aggregate nature of the approach does not capture detailed compatibility and coupling issues at the martensite plate scale. This coupling may be due to stress redistribution in the vicinity of plates [17] as well as the inherent effects of dislocation structure on the critical driving force for martensite formation (f c, Eq. 2.9). A better understanding at this scale may reveal the nature of the relatively linear unloading paths and strain recovery during subsequent post-heat treatment (Fig. 4.1a). It may also provide insight to why thermal cycling experiments show strain ratcheting at such small (50 MPa) bias stress. In contrast, the present grain-scale simulations predict some plasticity at this bias stress, but the cyclic response stabilizes after a few cycles with no subsequent ratcheting. Much larger (> 350 MPa) bias stress is required to predict ratcheting. The present model also does not capture the observed dependence of transformation strain on upper cycle temperature (Fig. 4.6) [9]. The model uses 24 type II twinned habit plane variants commonly used in the literature instead of the full 192 type I and type II 77

98 martensite predicted by crystallographic theory of martensite [60], an assumption which can also affect the predicted martensite texture. Given the recent observations of complex dislocation structures induced by transformation [10, 17, 79] and complex arrays of martensite plates, it is remarkable that an aggregate-based, grain-scale model is so successful. 78

99 Table 4.1: Calibrated properties for hot rolled/hot drawn polycrystalline 49.9at%Ni-Ti (55wt% Ni) Austenite Structure, See Section Property Value Calibration Feature Texture Pole figure (see Fig. 2b) F1: Fig 4.2a Elastic and Thermal, P el-thermal = {C A, C M, A th-a, A th-m }, See Section Property Value Calibration Feature C A (GPa) C 11 = 183 C 12 = 146 C 44 = 46 F2: DFT calculations [67] C M (GPa) C 11 = 249 C 12 = 129 C 13 = F2: DFT calculations [67] 107 C 15 = 15 C 22 = 245 C 23 = 125 C 25 = 3 C 33 = 212 C 35 = 1 C 44 = 87 C 46 = 4 C 55 = 66 C 66 = 86 Hill Averages: E = 180, G = 69 α A (/K) F3: Neutron diffraction [62] α M (/K) F4: Neutron diffraction [62] Phase Transformation, P trans = {θ T, λ T, f c and h tu }, See Section Property Value Calibration T (K) F5: DSC data T (MJ/m 3 ) 140 initial guess: F11-13: Fig. 4.3a F5: DSC data f c (MPa) T (= 7.14) F6: Fig. 4.1a and Eq. (4.2) h tu (MPa) h com = 0; h inc = C A(44) /12000; h self = C A(44) /400 F7-8: Fig. 4.1a. F9: Fig. 4.3a and Eq. (4.3) See also Table 2 [34] Austenite Plasticity, P plastic = { 0, g sat, g s0, h 0, Q, a, E p(pre) }, See Section Property Value Calibration γ 0 (/s) F10: Table 3.1 M 0.02 F10: Table 3.1 Q 1.4 F10: Table 3.1 A F10: Table 3.1 g sat (MPa) 900 F10: Table 3.1 g 0 s (MPa) 272 F11-14: Fig. 4.3a h 0 (MPa) 50 F11-14: Fig. 4.3a E p(pre) (%) 0.7 F11-14: Fig. 4.3a 79

100 Macrostrain E Macrostrain E (a) experiments 0.04 E T 0.1 Σ bias (MPa) E T 0.02 Engineering Strain, E 33 E cycle bias (MPa) Bias (MPa) F7: E T Temperature, F6: ( o C) F8: H A-M Temperature (C) (b) calibrated model 0.04 E T Engineering Strain, E 33 E Σ bias (MPa) E T 0.02 bias (MPa) Bias (MPa) Temperature, 100 ( o C) 150 Temperature, ( o C) Temperature (C) Figure 4.1: Axial macrostrain E vs. temperature θ under an axial bias stress Σ bias showing (a) experimental data for the 49.9 at% Ni-Ti alloy and (b) calibrated model results. The inserts show the transformation strain E T vs. Σ bias, where E T is defined by feature F7 in (a). The calibrated model parameters are obtained by fitting features F1-F14 as summarized in Table

101 Max: 2.0 {100} Max: 1.8 Min: 0.4 Min: 0.11 Max: 1.6 {110} Max: 1.5 Min: 0.71 Min: 0.82 Max: 1.6 {111} Max: 2.0 Min: 0.68 Min: 0.61 (a) Experiments (b) simulations Figure 4.2: Axial pole figures showing texture for polycrystalline austenite (a) experiments using HIPPO diffractometer (hot-worked 55 wt% Ni-Ti) and (b) simulations obtained by fitting to experimental texture results from SMARTS assuming axisymmetry. A strong (111) texture and weak (100) texture is observed 81

102 Engineering Macrostress Stress, 33 (MPa) Engineering Macrostress Stress, 33 (MPa) (a) experiments F9 130 C F11: MS E max F12: max θ 0 = 215 C 130 C Engineering Strain, E E unload Macrostrain 33 E F13: E post-heat F14: E post-heat (b) calibrated model θ 0 = 215 C 130 C A C Engineering Strain, E 33 Macrostrain E Figure 4.3: Axial macroscopic stress Σ vs. macroscopic axial strain E at different test temperatures (θ 0 = 130 and 215 ºC) showing (a) experimental data for the 49.9 at% Ni-Ti alloy and (b) calibrated model results. E unload and E post-heat are the macrostrains after unloading and after a 600 ºC post heat treatment of the unloaded sample, respectively. The calibrated model parameters are obtained by fitting features F1-F14 as summarized in Table

103 Engineering Stress, 33 (MPa) Macrostress (MPa) Macrostrain E Engineering Strain, E E p(pre) = 0 h tu(patoor) Σ bias = 50 MPa calibrated model experiment Temperature, ( o C) (a) Temperature (C) E p(pre) = 0 calibrated model F11: MS experiment F12: max 100 = 130 C Engineering Strain, E (b) F13: 33 E post-heat Macrostrain E Figure 4.4: (a) Axial macrostrain E vs. temperature θ during thermal cycling with an axial stress bias bias =50 MPa and (b) axial macrostress vs. macrostrain E at test temperature 0 = 130 ºC. The experimental data is for the polycrystalline 49.9 at% Ni-Ti alloy. The calibrated model results use the fitted parameters as summarized in Table 4.1. Other simulation results in (a) use the fitted parameters except with the martensite interaction matrix h tu of Patoor et al. [31] or with E p(pre) = 0. The E p(pre) = 0 cases in (b) are nearly coincident and use (g s 0, h0 ) = (235 MPa, 500 MPa) vs. (250 MPa, 50 MPa) case (the former is slightly higher). 83

104 Macrostrain E x calibrated model Engineering Strain, E experiment Σ bias = Temperature, ( o C) Temperature (C) Figure 4.5: Axial macrostrain E vs. temperature during thermal cycling with zero bias stress. The experimental data is for the polycrystalline 49.9 at% Ni-Ti alloy. The calibrated model parameters are obtained by fitting features F1-F14 as summarized in Table 4.1. The predicted two-way effect occurs because E p(pre) = 0.7%. Transformation Strain, strain E T E T experiment h tu(patoor) calibrated model Bias Stress, Bias (MPa) Bias Stress bias (MPa) Figure 4.6: Axial transformation strain E T vs. axial bias stress bias during thermal cycling between min = 30 ºC and max = 165 ºC (lower experimental curve) vs. 200 ºC (upper experimental curve). The experimental data is for the polycrystalline 49.9 at% Ni-Ti alloy. The calibrated model parameters are obtained by fitting features F1-F14 as summarized in Table 4.1. Also shown is the calibrated model result using the martensite interaction matrix h tu of Patoor et al. [31]. The model results are insensitive to max = 165 vs. 200 ºC. 84

105 Plastic macrostrain E p Plastic Strain, E P x Σ Bias =400 MPa 0.1 v M = region 1 plasticity only Temperature, ( o C) Temperature (C) calibrated model region 2 Figure 4.7: Axial plastic macrostrain E p vs. temperature during thermal cycling with an axial bias stress bias = 400 MPa. The calibrated model parameters are obtained by fitting features F1-F14 for the polycrystalline 49.9 at% Ni-Ti alloy, as summarized in Table 4.1. Also shown is the calibrated model result with plasticity only, meaning that phase transformations are not permitted. Engineering Strain, E 33 Macrostrain E A A calibrated model C M = ½ C A (cubic) C M =C [Wagner and Windl] Temperature, ( o C) Temperature (C) E cycle Figure 4.8: Axial macrostrain E vs. temperature during thermal cycling with an axial bias stress bias = 400 MPa, showing predictions of the calibrated model, the calibrated model with martensite elastic stiffness C M ~ ½ C A, and the calibrated model with C M and C A from Wagner and Windl [66]. A fourth case uses C M = C M(Isotropic) and overlaps the calibrated model result, where C M(Isotropic) is an isotropic matrix with Young s modulus E and elastic shear modulus G given by Hill averages of moduli of table

106 Diffracting Martensite Volume Fraction v M (011) Σ Bias =-150 MPa v M (100) Detects plane normal to loading direction v M (100) v M (011) v M (100) Retained M (Observed experimentally): v M (011) v M (100) Detects plane normal to loading direction Temperature, ( o C) Figure 4.9: Evolution of volume fraction of martensite contributing to diffraction of (100) and (011) planes parallel and perpendicular to the loading axis during temperature cycling of the polycrystalline 49.9 at% Ni-Ti under a compressive bias stress of bias =- 150 MPa. The inserts on the right shows the experimental intensity vs. d- spacing at = 130 ºC, with small retained martensite peaks confirming the model predictions. 86

107 1.2 calibrated model Normalized IntensityI [100] (a) experiment h tu(patoor) Bias Stress bias (MPa) Normalized Intensity I [011] (b) experiment h tu(patoor) calibrated model Bias Stress bias (MPa) Figure 4.10: Normalized neutron diffracted intensities from (a) (100) and (b) (011) martensite planes that are parallel to the loading axis, as a function of bias stress bias. The intensities are normalized by the intensity at bias =100 MPa. The experimental results are measured at min = 30 ºC, following thermal cycling of the polycrystalline 49.9 at% Ni-Ti to max = 230 ºC. The calibrated model result uses parameters summarized in Table 4.1. Also shown is the calibrated model result using the martensite interaction matrix h tu of Patoor et al. [31]. 87

108 Macrostress (MPa) Calibrated Model Compression Exp. Calibrated Model Tension Exp Macrostrain E Figure 4.11: Axial macroscopic stress Σ vs. macroscopic axial strain E at temperature θ 0 = 22 ºC of an initially self-accommodated martensite microstructure of the polycrystalline 49.9 at% Ni-Ti under tensile and compressive loads. The calibrated model (gray) is able to capture the hardening and the tension-compression asymmetry observed in the experiments (black). 88

109 5. MODELING MICRON SCALE SINGLE CRYSTAL SMA RESPONSE One of the principal shortcomings of the model developed in Chapters 2-4 has been its inability to capture interactions at the martensite plate scale. There are compatibility and coupling issues at the martensite plate scale between transforming martensite variants themselves and between martensite variants and dislocations. This coupling may be due to stress redistribution in the vicinity of plates [17] as well as due to the inherent effects of dislocation structure on the critical driving force for martensite formation (f c, Eq. 2.9). How the microstructure and the defect substructure evolve can affect the thermomechanical response of SMA. Chapters 1-4 highlighted experimental observations which the model is not capable of capturing, e.g. the observance of open loop strain as well as ratcheting of strain during temperature cycling at as low a bias stress as 50 MPa (see Fig. 4.4a and [9]), dependence of the response of SMA on the upper cycle temperature of the thermal cycling [9], and observance of dislocation generation during no-load temperature cycling [15]. To understand these fundamental aspects of martensitic phase transformation and plasticity, experimental work in the micron length scale of individual martensite variants is required. This Chapter presents two such state-of-the-art experimental efforts done by our collaborators at The Ohio State University, USA and at Ruhr-Universität Bochum, Germany. Norfleet et al. [17] studied the pseudoelastic compression response of micron 89

110 scale pillars of solutionized NiTi SMA, while the research group of Prof. G. Eggeler at Ruhr-Universität Bochum [80] studied the pseudoelastic martensitic transformation under tensile load in-situ in TEM. The experimental work is complemented by modeling using Crystallographic Theory of Martensite (CTM) and micromechanics based stress-field calculation. This modeling is more suitable for the length scale under consideration than the homogenized macroscale model presented in Chapter 2-4. The modeling framework is presented first with the CTM model discussed in Section 5.1 and the micromechanics based model discussed in Section 5.2. Section 5.3 discusses the salient experimental observations of pseudoelastic compression of NiTi micro pillars and model application to explain some of the experimental observations. Section 5.4 discusses the salient experimental observations of martensitic transformation under tensile load under in-situ TEM in NiTi and the model application to explain some of the observations. The Chapter is concluded in Section 5.5 summarizing the modeling results. 5.1 CTM Based Modeling Crystallographic Theory of Martensite (CTM) [4] can be used to identify the most favored martensite variants to form under simple tension or compression loading in single crystals. The crystallographic details, e.g. the interface between the variants and between the variants and austenite, and the twin relations of the variants formed, can also be determined. The theory stipulates any material vector v at the interface between Austenite and martensite must not stretch or rotate relative to the surrounding B2 matrix. Thus, the deformation gradient F inside the martensite relative to the austenite should satisfy 90

111 ( F I) v 0 (5.1) This ensures that the long-range elastic strain energy due to the transformation is zero. Eq. (5.1) is commonly referred as the invariant plane condition. Solutionized NiTi undergoes a one step B2-to-B19 (cubic-to-monoclinic I) transformation with 12 possible variants (k=1 to 12) of B19 martensite with deformation matrix F (k) [60] inside the variant k relative to the austenite. In a bulk NiTi, eq. (5.1) is not satisfied for F = F (k). However, F = F av(k,m) an average deformation gradient, given by F av(k,m) =λ F (k) + (1- λ) F (m) = I + b (m,k) m (k,m) (5.2) induced by martensite variants k and m weighted by λ can satisfy eq. (5.1). Thus, in bulk NiTi, austenite transforms into a twinned martensite plate (habit plane variant) of variants k and m such that their average deformation gradient F av(k,m) satisfies the invariant plane condition of eq. (5.1). Figure 5.1 shows a schematic of such a martensitic plate, which is a weighted combination of two martensite variants, k and m. They are arranged in alternating parallel laths that are typically a few nanometers in thickness substantially smaller than the plate dimensions, 2c «2a 2b. The crystallographic normal to the plate is m (k,m) and the transformation induces an average displacement vector b (k,m) of a material point in the plate, relative to another material point positioned a unit distance along the m (k,m) direction. The term average is used in the sense of smearing out the individual martensite variants into a continuum with a uniform deformation gradient given in eq. (5.2). In 91

112 reality, large local stress magnitudes are generated near individual B2-martensite variant interfaces. The twin interface between laths has a crystallographic normal n (k,m) and displacement vector a (k,m), defined analogously to b (k,m). For the B2-to-B19' transformation, there are 12 possible martensite variants (k, m = 1 to 12) and 192 possible plates specified by m (k,m), b (k,m) and corresponding a (k,m) and n (k,m). A MATLAB code which calculates these 192 plate types for NiTi taking deformation of individual variants F (k) as input is shown in the Appendix B. However, at samples of smaller length scales, there are significant deviations from the bulk response of transformation. In particular, Bhattacharya and James [81] show that for thin films of austenite, the elastic strain energy due to out of plane mismatch between austenite and martensite phases at the interface is negligible. Thus the invariant plane condition in eq. (5.1) is relaxed to an invariant line condition. Thus, only a material vector v lying in the film with normal n film needs to be invariant and eq. (5.1) relaxes to ( F I) v 0 v such that v n 0 (5.3) film Bhattacharya and James [81] show that eq. (5.3) has a solution if n cof( F T F I) n 0 (5.4) film film Thus, for any martensite variant k with deformation gradient F = F (k) and a film with normal n film, if eq. (5.3) is satisfied, then the single martensite variant k can form an interface with the austenite in the limit of a thin film. Furthermore, the invariant line vector v at the interface and in the case of multiple variants, the twin interface plane n (k,m) between the variant k and m can be calculated following [4]. 92

113 The most favored martensitic variants to form under tension/compression along loading direction e L are the variants that result in maximum transformation strain along e L. The full transformation strain ε * is given as: ε * T 0.5( F F I) (5.5) * ε LL and hence the transformation strain in the loading direction, ε * LL is given as * T * LL L L ε ( e ) ( ε e ) (5.6) Where the deformation gradient F = F av(k,m) for martensitic transformation in bulk NiTi and F = F (k) when the thin film approximation is enforced. Once the crystallographic details of the variants formed are determined, the kinematic electron diffraction pattern can be predicted using the lattice parameters of B19 reported in Otsuka and Ren [82]. 5.2 Micromechanics Based Model for Stress Field Calculation Once the variants most favored by the external stress are determined, the stress field produced by these variants in the austenite matrix can be calculated using micromechanics theory by Eshelby [83]. The martensite variant is treated as an inclusion of volume within which there is a transformation strain ε * given by eq. (5.5). This inclusion is assumed to be present in a homogeneous body D which is free from any external force or surface constraints. The goal is to find a displacement field u(x) and the resulting σ(x) that satisfies the mechanical equilibrium: σ ij, j 0 ( i 1,2,3) (5.7) and boundary conditions at the free surface σn 0 ( 1,2,3) ij j i where n j is the normal to the free surface of the body D. By additively decomposing the total strain ε(x) into 93

114 elastic strain ε e (x) and transformation strain ε *, ε(x) = ε e (x) + ε * (x) ; applying straindisplacement relation ij (x) = 0.5( u i / x j + u j / x i ), to relate the total strain ε(x) to displacement field u(x); and finally applying the Hooke s law to relate the stress σ(x) to elastic strain ε e (x) through elastic modulus C as ij (x) = C ijkl ε e kl(x), eq. (5.7) can be expressed with displacement u(x) as the primary variable as: * ijkl k, lj ijkl kl, j C u C ε (5.7) The components C ijkl of the elastic moduli are approximated to be the same for both the B2 and martensite phases. Closed form solutions to Eq. (5.7) are provided for particular inclusion shapes [83-85]. The stress field from transformation regions of arbitrary shape is obtained by introducing a scalar function (x) that = 1 inside the transformed region and = 0 otherwise. This order parameter can be expressed as a Fourier series (ξ1x1 ξ2x2 ξ3x 3) η( ) η(ξ ξ ξ ) e i x (5.8) ξ1, ξ2, ξ3 1, 2, 3 where the Fourier coefficients (x 1,x 2,x 3 ) are determined by the shape of the transformation region and the dimensions (L 1, L 2, L 3 ) of the periodic cell containing the transformation region. Eq. (5.7) is solved for u(x) using the Fast Fourier Transform (FFT) method [84] and then the corresponding stress field σ(x) is calculated. Two approaches are taken to model the transformation. The plate approach takes to be a plate with a uniform, volume average transformation deformation gradient (Eq. 5.2). The variant approach uses several transformation volumes, each corresponding to an individual martensite variant in the plate. 94

115 5.3 Pseudoelastic Compression of NiTi Micro-Pillar Experimental Observations The details of the experimental response can be found in Norfleet et al. [17] and only the main highlights are presented here. [110] oriented microcrystals of solutionized 50.7 at%ni-ti were prepared by focused ion machining and then tested in compression to investigate the stress-induced B2-to-B19' transformation at room temperature in the pseudoelastic regime. Two different sized micro pillars of diameters 5 and 20 μm are studied. The pillars are loaded to a total compressive strain of 2.5-3%. The stress-strain response shows the typical flag shaped curve with a sharp onset of the transformation, consistent with little prior plasticity, but some remnant strain upon unloading. A TEM foil with foil normal [001] is cut after the pseudoelastic test for transmission electron microscopy defect analysis. Post mortem scanning transmission electron microscopy reveals no apparent retained martensite but rather a macroscopic band of dislocation activity (shown in Figure 5.2) within which are planar arrays of ~100 nm dislocation loops involving a single a<010>{101} slip system indexed to austenite cubic basis. The result that much of the +a[010] dislocation content lies on (10 1 ) planes is rather surprising since the <100>/{110} family has a lower Schmid factor (0.35) compared to the <100>/{100} family (0.50). Hence, the observed dislocations are not on a slip system with the highest Schmid factor, with respect to the macroscopic applied stress. Moreover, because of the distinct band of dislocations, it is seems that the band of dislocation is at a location where transformation took place, since martensite is known to form distinct interface (habit planes) with austenite. However, this has to be verified through modeling 95

116 as the TEM analysis was post-mortem and no retained martensite was observed in the foil Analytic Modeling of the Preferred Martensite Plate The procedure outlined in Section 5.1 is used to identify the most likely twinning modes (plates) to form, based on: (1) the macroscopic forward transformation strain; (2) a ranking of the modes that generate the largest compressive strain using eq. (5.6); and (3) the orientations of the band of residual dislocation content and planar arrays of dislocations. The first aspect is addressed by noting that for the two-cycle 5μm micropillar, the 1 st forward cycle imparts a macroscopic compressive transformation strain of ~1.65%. This strain could be achieved if the transformation strain in the band ~5%, since STEM images shown in Fig. 5.2 show the volume fraction of the band to be ~1/3. The second aspect is pursued by ranking the compressive axial strain produced by each of the 192 possible twinning modes, using Eq. (5.6). Table 5.1 lists the 32 twinning modes that produce an axial [110] compressive strain >5%. These are arranged into 8 classes T1 to T8 each associated with a mode (A, B, or C) and type (I or II) following a convention in Table 7.3 of Bhattacharya [4]. Each class is comprised of 4 crystallographic permutations of m (k,m) and b (k,m), with axial compressive transformation strains ranging from 5.2% to 5.6%. The consistency with the macroscopic forward transformation strain suggests activation of these highly favored twinning modes. The final aspect is pursued by superimposing the predictions for the T4 class and a STEM image from the 5 m two-cycle sample, as shown in Fig The projected value = 31º fits remarkably well. In principal, the = 30º prediction for the T1 class also fits 96

117 well. However, the T4 class has a twin interface orientation ( = 9º) that aligns well with the linear, nearly vertical lines in Figure 5.2, which are traces of the dislocation loop arrays. In contrast, the T1 class has poor alignment ( = 38º). Thus, the T4 class is viewed as most consistent. In fact, when the slight misorientation in the experiments is accounted for that puts the compression axis close to [780] rather than [110], the T4 combinations (k,m) = (5,8) and (8,5) produce the maximum axial strain, relative to all 192 possible martensite plate types Analytic Modeling of Stressed Slip Systems near a Martensite Plate The stress state around the T4 class of plates may explain why dislocations are observed on a lower Schmid factor (0.35) system, (10 1 )/[010]. The Fast Fourier Transform method (Section 5.2) is used with the package FFTW [86]. The periodic cell dimensions L 1 = 4a = L 2 = 4b and L 3 = 4c sufficiently isolate the stress fields of individual plates, and sufficient accuracy is obtained with a three-dimensional grid of points. The thin plate geometry in Figure 5.1 is used, with the invariant planes at x 3 = ±c, where c = a/20 = b/20. The Cartesian coordinate system has x 3 m (k,m) and x 2 b (k,m) (b (k,m) m (k,m) )m (k,m). Two representations are employed to study the stress field. In the plate approach, the transformation strain in the plate has a uniform value 0.5(b m + m b). Isotropic elastic moduli (E= 148 GPa, =0.3) are also assumed. Figure 5.3 shows that the normalized stress distribution 23 (x 1 )/E from the FFT approach agrees well with the analytical solution by Chiu [85] for an isolated cuboidal inclusion with a uniform 97

118 transformation strain. This validates the FFT method, but the plate approach is unable to capture plasticity at the individual variant scale, as revealed by the STEM images. In the variant approach, the stress field around a plate composed of an arbitrary number of (k, m) = (4, 1) variant pairs is computed using the FFT approach. The B2-to- B19' transformation produces a strain 0.5(U T i U i I), where U 1 and U 4 (5.9) are referred to the austenite cubic basis, with = , = , = and = [8]. The austenite elastic moduli are C 11 =130 GPa, C 22 =98 GPa and C 44 =34 GPa [65], with the same values assumed for martensite, as discussed in Chapter 3. A macroscopic compressive transformation stress of 600 MPa is superimposed as observed for the forward transformation stress in the experiments. Figure 5.4 shows the most stressed austenite slip systems just outside the martensite plate (Fig. 5.1). Figures 5.4a, 5.4b, and 5.4c show results on the cuts x 1 = a, x 2 = 1.05b, and x 3 = 1.05c, respectively. A type (i) plot indicates the favored austenite {100}/<001> or {110}/<001> slip system with the largest resolved shear stress. A type (ii) plot shows the corresponding resolved shear stress (in MPa) on the favored system. The stress is clearly localized along contours parallel to the underlying twin interfaces. Figure 5.5 shows magnified views of the same three cuts in Fig. 5.4, except that a filter is applied to show only slip systems for which the resolved shear stress exceeds 1500 MPa. A white background indicates regions where no slip system meets this criterion. Although the 1500 MPa criterion is arbitrary, it serves as an effective filter to 98

119 identify the most likely spatial regions for slip activity and the particular slip systems involved. A value >1500 MPa would shrink the predicted bands of slip activity and a smaller value would enlarge them, eventually reproducing the type (i) plots in Fig Three significant observations are made from Fig First, the active slip systems in the vicinity of the plate do not include (100)/[010], even though it has the largest Schmid factor (0.50). Second, the experimentally observed (10 1 )/[010] system is predicted to be dominant on two faces of the plate. Third, the predicted regions for (10 1 )/[010] and (101)/[010] activity are aligned to the twin interfaces in the martensite plates, correlating well with the STEM images of planar loop arrays (Fig. 5.2). Two additional comments pertain to the scope of the variant approach. Fig. 5.4 and Fig. 5.5 show the results based on the T4 (4,1) plate mode in Table 5.1. However, the rightmost column in Table 5.1 shows that the other T4 modes always favor slip systems (10 1 )/[010] and (101)/[010] over the (100)/[010] system. Finally, these key results from the variant approach hold for other plate shapes, such as an ellipsoid-shaped inclusion and a cuboid-shaped inclusion with hemispherical ends FEM Analysis of the Pseudoelastic Compression of Pillar For the completion sake, this section shows the application of microstuctural FEM model developed in Chapters 2-4 to the pseudoelastic pillar compression. The material parameters for the solutionized 50.9 at % Ni-Ti from Table 3.1 are adopted in this study, with the exception of transformation temperature θ T taken as 237 K, because of the unavailability of experimental data to calibrate the model for the solutionized 50.7 at % Ni-Ti. The compositional change is expected to result in a decrease in transformation 99

120 temperature of about 20 K [17]. The FEM results presented in this section are compared qualitatively with the experiments. A more rigorous calibration is not expected to change this qualitative comparison. Figure 5.6 shows the model geometry C3D8 elements are to model the pillar while 1080 C3D8 elements are used to model the base of the pillar. The platen on top is given a downward velocity so that the pillar deforms at an average strain rate of /sec. Perfect alignment between platen and the pillar is assumed. The pillar compression direction is assumed to be [780], accounting for the slight misorientation from the [110] pillar observed in the experiments [17]. As the pillar is compressed, the martensitic transformation initiates from the base of the pillar- owing to the stress concentration due to the change in the cross sectional area at the base of the pillar. Formation of martensite breaks the symmetry along the loading direction due to lattice rotations and slowly a band of transforming martensite is formed in the pillar. Fig. 5.6 shows the spatial distribution of the total volume fraction v M of martensite at peak load. The homogenized nature of the model results in a diffused band of martensite with none of the elements showing a complete transformation of austenite into the martensite. However, formation of the band of martensite supports the TEM observation of band of dislocations formed. Moreover, the band is composed of martensite plate types of labeled T4 in the Table T1, also consistent with modeling in section However, when the activities in the slip-systems are analyzed, the (100)/[010] slip system, which has the largest schmid factor (0.5) along the loading direction is observed to cause the plastic strain and not the (10 1 )/[010] slip system observed in the experiments. This again shows the FEM model s inability to capture the 100

121 stress-field at the vicinity of individual martensite plates and local plasticity at the austenite-martensite interface which is observed in the experiments and is captured by the micromechanics based model. 5.4 Martensitic Transformation during Tensile loading and In-Situ TEM analysis Experimental Observations The details of the experimental results can be found in [87] and only the main highlights are presented here. The experiments used solutionized and aged NiTi that undergoes a one step B2-to-B19 (cubic-to-monoclinic I). The martensitic transformation is observed in-situ in a TEM by applying a tensile load onto the sample along the [110] direction of austenite in a cubic reference frame, henceforth referred by subscript B2 next to the direction. The TEM foil has a foil normal [ 110] B2 and is made thin so that it is electron transparent. The TEM image of B19 martensite phase growing in the austenite matrix during the test is shown in Figure 5.7. The interface between the twinned martensite structure and B2 makes an angle 130 with the [110] B2. A high resolution TEM image shows twinned martensite structure in Figure 5.8. Notable in the figure is the twin interface plane normal which is parallel to the loading direction [110] B2. A selected area diffraction pattern (SAD) of one of the regions labeled SAD1 in Fig. 5.8 is shown in Figure 5.9a which suggests a compound twinning relationship between the martensite variants formed with twin plane normal being (110) B2. This is rather unusual since, none of the 192 plate types formed for the B2 to B19 transformation is composed of compound twin related variants [60]. The CTM modeling framework presented in section 5.1 is used to understand these experimental observations. 101

122 5.4.2 Analytical Modeling for Favored Variants The procedure outlined in Section 5.1 is used to identify the favored variants of martensite. First, the phenomenological theory for bulk NiTi is applied. The maximum transformation strain ε * LL in the loading direction is calculated using eq. (5.6) with F = F av(k,m) of eq. (5.2) for the 192 plate types for B2-B19 transformation. A maximum * ε LL = 5.05% is predicted for the plates (habit plane variants, hpv) with type II twin relations. The predicted hpv have the twin interface plane n (k,m) =(± ) B2 or ( ±0.58) B2 both inconsistent with the experimental results which shows the variants to be in a compound twin relation with twin interface plane n (k,m) =(1 1 0) B2. Though, there are hpvs whose constituent variants have the experimentally observed twin interface plane n (k,m) =(1 1 0) B2 ; they are of type-i twins and results in much smaller axial strain of * ε LL = 2.47%, hence not expected to be favored by the external stress. Furthermore, for NiTi, any two martensite variants k and m which are compound twin related, have an average deformation F av(k,m) which does not satisfy the invariant plane condition eq. (5.1). Thus, the phenomenological theory for bulk NiTi cannot account for the compound twinning observed in the experiments. The TEM foil being very thin in the foil normal n film =[ 110 ] B2 direction, phenomenological theory for thin films is applied next. It is found that 10 out of 12 B19 martensite variants satisfy eq. (5.3) the invariant line condition. Thus, in the limit of thin film with normal n film =[ 110] B2, any of these 10 variants can form an interface with the austenite with negligible long range elastic stress. Moreover, variants k=9 and m=10 102

123 with deformation gradient (the right Cauchy stretch tensor) F (k) in austenite cubic coordinate system given by: α δ ε α δ - ε 9 10 F δ α ε and F δ α - ε (5.10) ε ε γ - ε - ε γ with α= , γ= δ=0.058 and ε= (Bhattacharya [4]), produce the maximum ε * LL = 8.75%. Thus, the austenite is expected to transform into variants k=9 and m=10. Moreover, these two variants form a compound twin with twin plane n (9,10) =(1 1 0) B2 consistent with the TEM observation shown in Fig To further verify the consistency of the predicted variants with the experimental TEM observations, the kinematic diffraction pattern of the predicted variants is compared with the experimental electron diffraction pattern in Fig. 5.9a. The diffraction pattern is of zone axis [0 1 0] B19 of the martensite phase in martensite crystal reference frame (denoted by subscript B19 ). The two variants are related by a 180 o rotation about [0 0 1] B19. The simulated diffraction pattern in Figure 5.9b is obtained by overlaying the kinematic diffraction pattern of the two variants of martensite at the proper orientation relationship. Thus, the predicted diffraction pattern and the twin interface plane n (9,10) =(1 1 0) B2 are both consistent with the experimental observations in the limit of thin film. Next, an attempt is made to estimate the volume fraction λ of the two favored variants k=9 and 10. In the bulk NiTi, the volume fraction of the variants is fixed by the invariant plane condition. However, in the limit of thin film, with the constraint relaxing to invariant line, each variant can individually make an invariant line with the austenite. 103

124 However, the TEM results show that the martensite structure alternates between the two variants giving rise to many narrowly separated twin interfaces planes n (9,10) =(1 1 0) B2. Even though both variants result in the same ε * LL = 8.75%, formation of multiple narrowly separated twin interfaces is remarkable. To address this, an average deformation gradient F av = λ F (9) + (1- λ)f (10) is constructed and the resulting transformation strain * ε is calculated using eq. (5.5). The resulting strain * ε is transformed in the orthogonal coordinate system {e L, n film, e L n film } along the loading direction e L, film normal direction n film and the transverse direction (e L n film ). It is postulated that the volume fraction λ is the one volume fraction which maximizes plane (e L - e L n film ) components of * ε LL but minimizes all other in- * ε. It is assumed that this would result in a small elastic energy. The two predicted variants produce identical direct components of * ε, but opposing in-plane shear. Thus, at λ=0.5, they cancel each other s in-plane shear component and minimize the mismatch between * ε and the austenite and hence the elastic energy. Thus, the martensite microstructure observed is of alternating variants. This predicted volume fraction is checked for consistency with the TEM observations. Fig. 5.7 shows the growing martensite variants in the austenite. The interface between the austenite and martensite is at approximately 130 o counterclockwise with the [110] B2 direction. The presumption of the modeling is that the interface is an invariant line v given in eq. (5.3). Following Bhattacharya and James [81] the invariant line v for the two predicted variants is calculated as v (9) =[ ] B2 with an angle of 149 o counterclockwise with the [110] direction and v (10) =[ ] B2 with an angle of 106 o counter-clockwise with the [110] B2 direction. However, the 104

125 average composite structure of the two variants with deformation gradient F av = 0.5 F (9) +(1-0.5)F (10) results in an average invariant line v (9,10) =[ ] with an angle 126 o counter-clockwise with the [110] B2 direction very close to the TEM observation of 130 o Model Application to Tensile Tests along [1 1 1] and [1 1 2 ] This section follows the Section and applies the model to similar experimental results in different tensile loading directions- in particular [1 1 1] B2 and [1 1 2 ] B2 directions. Experiments show martensite microstructure very similar to the fig. 5.8 (martensite under [1 1 0] B2 tension) with the interface normal parallel to [1 1 0] B2. The electron diffraction for the [1 1 1] B2 case is exactly the same as the [ 1 1 0] B2 case shown in fig. 5.9 while the [1 1 2 ] B2 case do not show any twinning in the electron diffraction. The model application (with the thin film approximation) to [1 1 1] B2 predicts three variants of martensite each producing * ε LL =10.27 %. However, the calculated twin interface plane normal between any two of the predicted variants is not the experimentally observed (1 1 0) B2 plane normal. Instead the twin planes are either of {221} type or ( 110) B2 or (1 0 1) B2 or ( 0 1 1) B2 depending upon the two variants selected out of the three predicted ones. Moreover, the twinning is either of Type I or Type II type and not the compound twin type, which is consistent with the experimental compound diffraction pattern. 105

126 Similarly, the model application (with the thin film approximation) to [1 1 2 ] B2 predicts two variants of martensite each producing ε * LL =9.95 %. However, again the two predicted variants either are in a Type I twin relation with twin plane ( 110) B2 or in a Type II twin relation with twin plane (2 2 1) B2 inconsistent with experimental observation of interface plane (1 1 0) B2. The exact reason for the failure of the model is under investigation. Probable reasons may include the stress-field generated by the precipitate having an effect in the variant selection, in addition to the external stress an effect currently ignored in the model. 5.5 Summary Pseudoelastic compression test along [110] B2 direction for single crystal solutionized 50.7 at%ni-ti and post-mortem TEM defect analysis show: A sharp elastic/transformation transition, suggesting that little plasticity occurs prior to the onset of the martensitic transformation. Type a[0 10]/(10 1 ) dislocations are observed both inside and at the periphery of the transformation zone. Those within the transformation zone are arranged as arrays of dislocation loops, while those at the periphery have distinctive arrowhead morphology. Since the dislocations index to rational directions in the austenite cubic system, these dislocations are due to plastic deformation of austenite. This observation has formed the rationale for the modeling of plasticity in the austenite phase in Chapters 2-4. CTM model along with micromechanics based stress calculation model are applied to explain the TEM observations: 106

127 Four twinning modes that likely produced the remnant bands of dislocations are identified from 192 possible modes. The twin interface predicted for these four twinning modes aligns with the observed planar arrays of dislocation loops, suggesting that loop formation may be driven by local stress fields at the scale of individual variants. A variant level micromechanics model reveals that the observed (10 1 )/[010] slip system is highly stressed by the four likely twinning modes, in B2 regions that are just outside the martensite plate and parallel to twin interfaces. Further consideration of the small misorientation (4 ) from the [110] deformation axis suggests that the most highly stressed martensitic plate system is activated. Application of the crystallographic theory of martensite transformations to experimental STEM images offers a powerful method to study the coupled interaction between transformation and plasticity. This preliminary theoryexperiment approach suggests that fine-scale plasticity is produced on a scale and orientation commensurate with individual martensite phases. In-situ TEM tensile test along [110] B2 direction for single crystal solutionized and aged NiTi show: Austenite transforms into multiple variants of martensite which are compound twin related with the twinning plane being (110) B2. A fine microstructure of alternatively varying variant is observed as shown in Fig The interface between the martensite plate and the austenite makes an angle of 130 o with the tensile direction [110] B2. Modeling work follows crystallographic theory of martensite (CTM). First, the theory for 107

128 bulk NiTi is applied. Results are inconsistent with the above mentioned TEM observations. In particular, a compound twinned microstructure of martensite cannot form an invariant plane with austenite and hence is not expected to be present. When CTM is applied in conjunction with thin film approximation (eq. (5.3)), which relaxes the invariant plane condition to invariant line, the predicted results are consistent with the TEM observations. In particular- A single variant of martensite can form an interface with austenite with the relaxed invariant line condition. This is in agreement with previous in-situ TEM results of Tirry and Schryvers [88] as well, who observe individual martensite variants with no twinning to grow in the austenite matrix. The external imposed stress favors two martensite variants to form. The two variants are compound twin related with the twinning plane (110) B2. The simulated kinematic diffraction pattern is also in agreement with the experimental electron diffraction pattern. When the two favored variants are in a ratio of 50:50 in the microstructure, they maximize the transformation strain in the loading direction and minimize all other in-plane strain components. Such an arrangement is desirable for a low elastic energy. The invariant line predicted by such an arrangement is also in agreement with the TEM observation of the austenite-martensite interface that makes an angle of 126 o with the tensile direction [110] B2 compared to experimental observation of 130 o. 108

129 Table 5.1: Analysis of Potential Twinning Modes and Martensitic Variants Case (1) T1 Mode B, Type I T2 Mode B, Type I T3 Mode B, Type II axial (2) (k, m) (3) -5.2% (1,4), (4,1) (5,8), (8,5) -5.6% (1,4), (4,1) (5,8), (8,5) -5.6% (1,4), (4,1) (5,8), (8,5) m (k,m) b (k,m) (4) (5) Max stressed Slip system (6) ( 0.91, (-0.05, -30º +38º , -0.11, ±0.33) ±0.05) (-0.25, (-0.11, 0.91, -0.05, ±0.33) ±0.05) (-0.35, -0.83, ±0.43) (-0.83, -0.35, ±0.43) (-0.38, -0.77, ±0.51) (0.77, -0.38, ±0.51) (-0.12, -0.03, ±0.04) (-0.03, 0.12, ±0.04) (0.12, -0.02, ±0.05) (-0.02, 0.12, ±0.05) -68º -38º -71º -9º Continued See Figure 5.1 for a description of m, b, n, and a. 1 Mode and type are terms used in the phenomenological theory of martensite, e.g., see Table 7.3 in [4]. 2 As predicted by Eq. (5.6) in Section k and m refer to particular martensite variants as defined in Table 4.3 in [4]. 4 the counterclockwise angle between the [110] axis and the invariant plane, projected on the [001] plane. 5 the counterclockwise angle between the [110] axis and the twin interface, projected on the [001] plane. 6 Obtained from the micromechanics calculations of Section

130 Table 5.1 continued T4 Mode B, Type II T5 Mode C, Type I T6 Mode C, Type I T7 Mode C, Type II T8 Mode C, Type II -5.3% (5,8), (-0.22, 0.89, 0.40) -5.2% (1,5), (4,8) (5,1), (8,4) -5.6% (1,5), (4,8) (5,1), (8,4) -5.6% (1,5), (4,8) (5,1), (8,4) -5.3% (1,5), (4,8) (8,5), (-0.22, 0.89, -0.40) (1,4), (0.89, -0.22, 0.40) (4,1) (0.89, -0.22, -0.40) (5,1), (8,4) (0.0, -0.81, ±0.58) (-0.81, 0.0. ±0.58) (0.86, 0.28, ±0.43) (0.28, 0.86, ±0.43) (0.91, 0.20, ±0.36) (0.20, 0.91, ±0.36) (0.04, -0.89, ±0.46) (-0.89, 0.04, ±0.46) (-0.10, -0.06, 0.06) (-0.10, -0.06, -0.06) (-0.06, -0.10, 0.06) (-0.06, -0.10, -0.06) (0.09, 0.04, ±0.04) (0.04, 0.09, ±0.04) (0.0, -0.09, ±0.06) (-0.09, 0.0, ±0.06) (0.0, -0.1, ±0.05) (-0.01, 0.0, ±0.05) (0.1, 0.03, ±0.04) (0.03, 0.10, ±0.04) -31º +9º x 2 x 3 & x 1 x 3 plane: (101)/[010] x 1 x 2 plane: (101)/[010] -31º +9º x 2 x 3 & x 1 x 3 plane: (101)/[010] x 1 x 2 plane: (101)/[010] -31º +9º x 2 x 3 & x 1 x 3 plane: (101)/[010] x 1 x 2 plane: (101)/[010] -31º +9º x 2 x 3 & x 1 x 3 plane: (101)/[010] x 1 x 2 plane: (101)/[010] -45º º +57 º 0-90º -42º +90º 110

131 Label Austenite Slip System 1 (010)/[100 ] and equivalent (100)/[010] 2 (001)/[100 ] and equivalent (100)/[001] 3 (001)/[010 ] and equivalent (010)/[001] 4 (011)/[100] and equivalent (110)/[001] 5 (01 1 )/[100] and equivalent ( 110)/[001] 6 (101)/[010] 7 (10 1 )/[010] Table 5.2: Legend for the austenite slip systems shown in Figs. 5.4 and 5.5 Variant k Variant m x 3 m a n 2c 2a x 1 x 2 2b Figure 5.1: Martensite plate geometry formed by martensite variants k and m. The plate is modeled as a cuboid with dimensions 2a, 2b, 2c, where a = b = 20c. x 3 = ±c are invariant planes with a crystallographic normal m (k,m), x 2 is the orthogonalized shear direction given by b (k,m) (b (k,m) m (k,m) )m (k,m), where b (k,m) is the transformation displacement vector. x 1 is orthogonal to x 2 and x 3. The interface between variants k and m has the crystallographic normal n (k,m) and displacement vector a (k,m). 111

132 ζ 23 /E 23 /E FFT Solution Analytical Solution FFT Solution Anlytical Solution x /a 1 1 x 1 /a 1 Figure 5.2: Comparison of the crystallographic theory of martensitic transformations for NiTi to TEM observations from the preliminary micropillar testing. Shown are the predictions for the plate type T4 in Table 5.1. The line labeled invariant plane shows the predicted intersection of the invariant plane with the [001] plane of the image. The line labeled twin interface shows the intersection of the predicted martensite-martensite interface with the [001] plane of the image. Figure 5.3: The normalized shear stress ζ 23 /E vs. normalized position x 1 /a with x 2 =x 3 = 0, for a martensite plate with dimensions a = b = 20c, assuming homogeneous, isotropic elastic properties. The results from the FFT based micromechanics code agree well with the analytical solution by Chiu [85]. 112

133 x 32 x 2 1 (i) x 3 /a 3 (ii) x 3 /a x 3 /a 3 x 3 /a x /a x 2 /a x /a x /a x /a x /a (a) x =1.0125a (b) x 2 =1.0125b (c) x 3 =1.05c x 2 /a x /a Legend (010)/[100] (favored by external load) (MPa) Figure 5.4: Spatial distribution of (i) the most stressed slip system and (ii) corresponding resolved shear stress (in MPa) on three planes (a) x 1 =1.0125a ; (b) x 2 =1.0125b and (c) x 3 =1.05c that are just outside the faces of a martensite plate (T4, k = 8, m = 5 in Table 5.1) as shown in Figure 5.5. Slip system numbers correspond to those listed in Table

134 Legend (101)/[010] (observed in TEM) Other {101}/<010> Other {010}/<100> x 3 m x 2 /a 2 (011)/[100] other {010}/<100> 1 (010)/[100] (favored by external load) 2c x /a (c) x 3 =1.05c (011)/[100] 1 2a x 1 x 2 2b (001)/[010] (101)/[010] (observed in TEM) x 3 /a x /a (a) x 1 =1.0125a x /a (b) x 2 =1.0125b Figure 5.5: Spatial distribution of slip systems with a resolved shear stress exceeding 1500 MPa for a stress axis of [780], on planes (a) x 1 =1.0125a, (b) x 2 =1.0125b, and (c) x 3 =1.05c located just outside the faces of a martensite plate. These calculations assume at4 type plate with (k, m) = (8, 5), as identified in Table 5.1. Slip system numbers correspond to those listed in Table 5.2. x 3 /a (101)/[010] (observed in TEM)

135 TEM Foil Plane Figure 5.6: The spatial distribution of total volume fraction of martensite formed at the peak load during the pseudoelastic compression of the pillar using FEM model developed in Chapters 2-4. Also shown in the insert on the right is the martensite volume fraction on the plane used to make the TEM foil (shown in Fig. 5.2). 115

136 Average Interface 130 [110] [0 0 1] Interface for F 9 Interface for F 10 1 column Figure 5.7: TEM image from [87] of B19 martensite in B2 austenite matrix of NiTi. The interface between the twinned martensite structure and B2 makes an angle 130 with the [110] B2 and is close to the angle predicted (126 ) by the model when the two variants listed in eq. (5.10) are present in the ratio of 50:50 in the martensite microstructure Figure 5.8: TEM image from [87] of Martensite variants formed during [110] tension. Modeling results and the electron diffraction pattern (fig. 5.9) of region SAD 1 confirms that two compound twin related variants of martensite are formed with twin plane (110). 116

137 6.8 (a) (b) Figure 5.9: Electron diffraction pattern for the SAD1 (shown in fig. 5.8) (a) TEM experimental diffraction pattern from [87] (b) Simulated kinematic diffraction pattern obtained by overlaying the diffraction pattern of the two variants given by eq. (5.10) in proper orientation relationship. 117