CONSTITUTIVE MODELING OF ROLLED SHAPE MEMORY ALLOY SHEETS TAKING INTO ACCOUNT PRE-TEXTURE AND ANISOTROPIC HARDENING

The drawing or rolling process endows polycrystal shape memory alloy with a crys- tallographic texture, which can result in macroscopic anisotropy. The main purpose of this work is to develop a constitutive model to predict the thermomechanical behavior of shape memory alloy sheets, which accounts for the crystallographic texture. The total macroscopic strain is decom- posed into elastic strain and macro-transformation strain under isothermal condition. Considering the transformation strain in local grains and the orientation distribution function of crystallo- graphic texture, the macro-transformation strain and the effective elastic modulus of textured polycrystal shape memory alloy are developed by using tensor expressions. The kinetic equation is established to calculate the volume fraction of the martensite transformation under given stress. Furthermore, the Hill＇s quadratic model is developed for anisotropic transformation hardening of textured SMA sheets. All the calculation results are in good agreement with experimental data, which show that the present model can accurately describe the macro-anisotropic behaviors of textured shape memory alloy sheets.     

Anisotropy of martensitic transformations in modeling of shape memory alloy polycrystals

Based on the knowledge of the anisotropy associated with the martensitic transformations obtained from tension/compression experiments with oriented CuAlNi single crystals, a simple constant stress averaging approach is employed to model the SMA polycrystal deformation behaviors. Only elastic and inelastic strains due to the martensitic transformation, variant reorientations in the martensite phase and martensite to martensite transformations in thermomechanical loads are considered. The model starts from theoretical calculation of the stress-temperature transformation conditions and their orientation dependence from basic crystallographic and material attributes of the martensitic transformations. Results of the simulations of the NiTi, NiAl, and Cu-based SMA polycrystals in stress–strain tests are shown. It follows that SMA polycrystals, even with randomly oriented grains, typically exhibit tension/compression asymmetry of the shape of the pseudoelastic σ−ε curves in transformation strain, transformation stress, hysteresis widths, character of the pseudoelastic flow and in the slope of temperature dependence of the transformation stresses. It is concluded that some macroscopic features of the SMA polycrystal behaviors originate directly from the crystallography of the undergoing MT's. The model shows clearly the crystallographic origin of these phenomena by providing a link from the crystallographic and material attributes of martensitic transformations towards the macroscopic σ−ε−T behaviors of SMA polycrystals.     

Stress-Induced Phase Transformations in Shape-Memory Polycrystals

Shape-memory alloys undergo a solid-to-solid phase transformation involving a change of crystal structure. We examine model problems in the scalar setting motivated by the situation when this transformation is induced by the application of stress in a polycrystalline material made of numerous grains of the same crystalline solid with varying orientations. We show that the onset of transformation in a granular polycrystal with homogeneous elasticity is in fact predicted accurately by the so-called Sachs bound based on the ansatz of uniform stress. We also present a simple example where the onset of phase transformation is given by the Sachs bound, and the extent of phase transformation is given by the constant strain Taylor bound. Finally we discuss the stress–strain relations of the general problem using Milton–Serkov bounds.     

Propagation and scattering of ultrasonic waves in polycrystals with arbitrary crystallite and macroscopic texture symmetries

A general ultrasonic attenuation model for a polycrystal with arbitrary macroscopic texture and triclinic ellipsoidal grains is described with proper accounting for the anisotropic Green’s function for the reference medium. The texture and the ellipsoidal grain frames in the model are independent and the wave propagation direction is arbitrary. The attenuation coefficients are obtained in the Born approximation accompanied by the Rayleigh and stochastic asymptotes. The scattering model displays statistical anisotropy due to two independent factors: (1) shape of the oriented grains and (2) preferred crystallographic orientation of the grains leading to macroscopic anisotropy of the homogenized reference medium. The model is applicable to most single phase polycrystalline materials that may occur as a result of thermomechanical manufacturing processes leading to different macrotextures and elongated-shaped grains. It predicts the strength of ultrasonic scattering and its dependence on frequency and propagation direction as a function of grain shape, grain crystallographic symmetry and macroscopic texture parameters and provides the texture-induced dependence of macroscopic ultrasonic velocity on propagation angle. It considers proper wave polarizations due to macroscopic anisotropy and scattering-induced transformations of waves with different polarizations. Competing effects of grain shape and texture on the attenuation are observed. In contrast to the macroscopically isotropic case, where in the stochastic regime the attenuation is highest in the direction of the longest ellipsoidal axis of the grain, the wave attenuation in the elongation direction may be suppressed or amplified by the texture with different effects on the quasilongitudinal and quasitransverse waves. The frequency behavior is also interestingly affected by texture: a hump in the total attenuation coefficient is found for the fast quasitransverse wave which is purely the result of macroscopic anisotropy and the existence of two quasitransverse waves; this hump is not observed in the macroscopically isotropic case. Striking differences of the texture effect on the directional dependences of the attenuation coefficients are found at low versus high frequencies.     

铁电单晶Pb(Mg1/3Nb2/3)O3-0.32PbTiO3的实验及本构模型研究

对压力作用下沿[001]晶向极化的Pb(Mg弛豫型铁电单晶;极化旋转(相变);黏塑性模型;本构;细观力学国家自然科学基金，教育部全国优秀博士学位论文作者专项基金2006-11-06对压力作用下沿[001]晶向极化的Pb(Mg1/3Nb2/3)O3-0.32PbTiO3(PMN0.32PT)弛豫型铁电单晶的应力应变行为进行了实验研究,实验结果表明铁电单晶〈001〉晶向的应力应变行为和铁电多晶有本质的不同,是传统的电畴翻转机理所难以解释的,所提出的极化旋转(相变)模型合理地解释了实验中观察到的现象;基于提出的极化旋转(相变)模型,采用细观力学方法建立了铁电单晶的细观本构模型.在模型中采用黏塑性公式描述铁电单晶可能的8个相变系统的相变行为.为了验证模型的可靠性,用该模型模拟了铁电单晶〈001〉晶向的应力应变实验曲线.计算表明,该模型能较好地模拟铁电单晶〈001〉晶向的相变行为.     

Mesoscale kinetics produces martensitic microstructure

We present molecular dynamics (MD) simulations of a martensitic phase transformation studying post-transformation microstructure and moving austenite-martensite interfaces. Unlike in energy-minimisation theories, the transformation dynamics dominate the martensite morphology. We use a binary Lennard-Jones potential to describe a square-to-hexagonal transformation by shear-and-shuffle. The high-T stable square lattice and low-T hexagonal lattice represent austenite and martensite, giving four martensitic variants. Compatible twin variants have no lattice misfit and zero interfacial energies which makes our model directly comparable with the crystallographic theory of martensite. Although our dynamical interpretation is different to previous work, our MD simulations exhibit very similar martensitic morphologies to real materials. We observe the nucleation of wedge-shaped, twinned martensite plates, plate growth at narrow, travelling transformation zones, subsonic transformation waves, elastic precursors inducing secondary nucleations and the formation of martensitic domains. Martensite is produced within narrow transformation zones where atoms change their lattice sites in a co-operative manner so as to form crystallographic layers. These motions produce inertia forces on the mesoscopic length-scale which induce the formation of twin variants in the subsequent layers to transform.     

A generalized self-consistent polycrystal model for the yield strength of nanocrystalline materials

Inspired by recent molecular dynamic simulations of nanocrystalline solids, a generalized self-consistent polycrystal model is proposed to study the transition of yield strength of polycrystalline metals as the grain size decreases from the traditional coarse grain to the nanometer scale. These atomic simulations revealed that a significant portion of atoms resides in the grain boundaries and the plastic flow of the grain-boundary region is responsible for the unique characteristics displayed by such materials. The proposed model takes each oriented grain and its immediate grain boundary to form a pair, which in turn is embedded in the infinite effective medium with a property representing the orientational average of all these pairs. We make use of the linear comparison composite to determine the nonlinear behavior of the nanocrystalline polycrystal through the concept of secant moduli. To this end an auxiliary problem of Christensen and Lo (J. Mech. Phys. Solids 27 (1979) 315) superimposed on the eigenstrain field of Luo and Weng (Mech. Mater. 6 (1987) 347) is first considered, and then the nonlinear elastoplastic polycrystal problem is addressed. The plastic flow of each grain is calculated from its crystallographic slips, but the plastic behavior of the grain-boundary phase is modeled as that of an amorphous material. The calculated yield stress for Cu is found to follow the classic Hall-Petch relation initially, but as the gain size decreases it begins to depart from it. The yield strength eventually attains a maximum at a critical grain size and then the Hall-Petch slope turns negative in the nano-range. It is also found that, when the Hall-Petch relation is observed, the plastic behavior of the polycrystal is governed by crystallographic slips in the grains, but when the slope is negative it is governed by the grain boundaries. During the transition both grains and grain boundaries contribute competitively.     

Anisotropic yield criterion for polycrystalline metals using texture and crystal symmetries

An anisotropic yield criterion for polycrystalline metals which uses texture data and takes advantage of crystal symmetries is presented. A linear transformation is developed to map an anisotropic yield surface for a polycrystal to an appropriate isotropic yield surface. The transformation developed reflects the symmetry of the material being modeled. First, the transformation is determined. Then, information regarding the orientation distribution (texture) of the crystals in a polycrystalline aggregate is used to determine, via averaging, the transformation for the polycrystal. The transformation, along with appropriate isotropic yield surface, provides a phenomenological approach to modeling yield, yet accounts for microstructural texture. The approach reduces to the Hill (1950) anisotropic plasticity theory under certain conditions. The yield surfaces and R-values for various face-centered-cubic ( fcc) polycrystalline textures are computed by this method. Results compare favorably with those given by other theories, and with experiment. The method proves to have the computational efficiency of phenomenological approaches to modeling yield, while effectively incorporating the physics of more complex crystallographic approaches.     

Experimental and numerical determinations of the initial surface of phase transformation under biaxial loading in some polycrystalline shape-memory alloys

Biaxial proportional loading such as tension (compression)–internal pressure and bi-compression tests are performed on a Cu-Zn-Al and Cu-Al-Be shape memory polycrystals. These tests lead to the experimental determination of the initial surface of phase transformation (austenite→martensite) in the principal stress space (σ₁,σ₂). A first “micro–macro” modeling is performed as follows. Lattice measurements of the cubic austenite and the monoclinic martensite cells are used to determine the “nature” of the phase transformation, i.e. an exact interface between the parent phase and an untwinned martensite variant. The yield surface is obtained by a simple (Sachs constant stress) averaging procedure assuming random texture. A second modeling, performed in the context of the thermodynamics of irreversible processes, consists of a phenomenological approach at the scale of the polycrystal. These two models fit the experimental phase transformation surface well.     

Numerical modelling of the plasticity induced during diffusive transformation. Case of a cubic array of nuclei

The experimental work of Taleb and Petit-Grostabussiat [Taleb, L., Petit-Grostabussiat, S., 2002. Elastoplasticity and phase transformations in ferrous alloys: some discrepancies between experiments and modeling. J. Phys. IV 12 (11), 187–194; Taleb, L., Petit, S., 2006. New investigations on transformation induced plasticity and its interaction with classical plasticity. Int. J. Plasticity 22 (1), 110–130] has shown evidence that the evolution of TRansformation Induced Plasticity (TRIP) in a low carbon steel (16MND5) could be significantly influenced by the loading history of the parent phase, for a martensitic as well as a bainitic transformation. Furthermore, estimates from the Leblond model – one of the few micromechanical models currently found in different Finite Element (FE) softwares – have appeared to be in disagreement with experiments in these cases where the parent phase has been strain hardened. This has motivated the development of alternative approaches based on FE computations. This paper presents our first investigations about simulations of diffusive transformations with FE in an idealized case: the parent and the product phase are considered as two homogeneous materials with given elastoplastic properties and density; the transformation takes place at the same instant at predefined elements constituting the nuclei; then it progresses at a uniform rate by changing the material properties of the layer of elements surrounding the nuclei. In the basic configuration of modelling, the volume of discretization stands for a unit cell of a periodic cellular array, with a single central nucleus. In a more complex configuration, which is introduced shortly here and to be presented in details in the paper under preparation [Barbe, F., Quey, R., Taleb, L., Souza de Cursi, E., 2006. Numerical modelling of the plasticity induced during diffusive transformation. Case of a random instantaneous array of nuclei, in preparation], the volume of computation contains few to several nuclei at random locations. For both configurations, results in terms of effective (mean) TRIP as a function of the volume fraction of product phase are in correct quantitative and qualitative agreement with experimental results.     

A study of microstructural length scale effects on the behaviour of FCC polycrystals using strain gradient concepts

Grain size is a critically important aspect of polycrystalline materials and experimental observations on Cu and Al polycrystals have shown that a Hall–Petch-type phenomenon does exist at the onset of plastic deformation. In this work, a parametric study is conducted to investigate the effect of microstructural and deformation-related length scales on the behaviour of such FCC polycrystals. It relies on a recently proposed non-local dislocation-mechanics based crystallographic theory to describe the evolution of dislocation mean spacings within each grain, and on finite element techniques to incorporate explicitly grain interaction effects. Polycrystals are modeled as representative volume elements (RVEs) containing up to 64 randomly oriented grains. Predictions obtained from RVEs of Cu polycrystals with different grain sizes are shown to be consistent with experimental data. Furthermore, mesh sensitivity studies revealed that, when there is a predominance of geometrically necessary dislocations relative to statistically stored dislocations, the polycrystal response becomes increasingly mesh sensitive. This was found to occur especially during the early stages of deformation in polycrystals with small grains.     

A micromechanics-inspired constitutive model for shape-memory alloys that accounts for initiation and saturation of phase transformation

A constitutive model to describe macroscopic elastic and transformation behaviors of polycrystalline shape-memory alloys is formulated using an internal variable thermodynamic framework. In a departure from prior phenomenological models, the proposed model treats initiation, growth kinetics, and saturation of transformation distinctly, consistent with physics revealed by recent multi-scale experiments and theoretical studies. Specifically, the proposed approach captures the macroscopic manifestations of three micromechanial facts, even though microstructures are not explicitly modeled: (1) Individual grains with favorable orientations and stresses for transformation are the first to nucleate martensite, and the local nucleation strain is relatively large. (2) Then, transformation interfaces propagate according to growth kinetics to traverse networks of grains, while previously formed martensite may reorient. (3) Ultimately, transformation saturates prior to 100% completion as some unfavorably-oriented grains do not transform; thus the total transformation strain of a polycrystal is modest relative to the initial, local nucleation strain. The proposed formulation also accounts for tension–compression asymmetry, processing anisotropy, and the distinction between stress-induced and temperature-induced transformations. Consequently, the model describes thermoelastic responses of shape-memory alloys subject to complex, multi-axial thermo-mechanical loadings. These abilities are demonstrated through detailed comparisons of simulations with experiments.     

An experimental study on grain deformation and interactions in an Al-0.5%Mg multicrystal

Heterogeneous plastic deformation behavior of a coarse-grained Al-0.5%Mg multicrystal was investigated experimentally at the individual grain level. A flat uniaxial tensile specimen consisting of a single layer of millimeter-sized grains was deformed quasi-statically up to an axial strain of 15% at room temperature. The initial local crystallographic orientations of the grains and their evolutions after 5, 12, and 15% plastic strains were measured by electron backscattered diffraction pattern analysis in a scanning electron microscope. The local in-plane plastic strains and rigid body rotations of the grains were measured by correlation of digital optical video images of the specimen surface acquired during the tensile test. It is found that both intergranular and intragranular plastic deformation fields in the aluminum multicrystal specimen under uniaxial tension are highly heterogeneous. Single or double sets of slip-plane traces were predominantly observed on the electro-polished surfaces of the millimeter-sized grains after deformation. The active slip systems associated with these observed slip-plane traces were identified based on the grain orientation after deformation, the Schmid factor, and grain interactions in terms of the slip-plane trace morphology at grain boundaries. It is found that the aluminum multicrystal obeys neither the Sachs nor the Taylor polycrystal deformation models but deforms heterogeneously to favor easy slip transmission and accommodation among the grains.     

Investigating the limits of polycrystal plasticity modeling

A material model which describes the rate-dependent crystallographic slip of FCC metals has been implemented into a quasistatic, large deformation, nonlinear finite element code developed at Sandia National Laboratories. The resultant microstructure based elastic–plastic deformation model has successfully performed simulations of realistic looking 3-D polycrystalline microstructures generated using a Potts-model approach. These simulations have been as large as 50,000 elements composed of 200 randomly oriented grains. This type of model tracks grain orientation and predicts the evolution of sub-grains on an element by element basis during deformation of a polycrystal. Simulations using this model generate a large body of informative results, but they have shortcomings. This paper attempts to examine detailed results provided by large scale highly resolved polycrystal plasticity modeling through a series of analyses. The analyses are designed to isolate issues such as rate of texture evolution, the effect of mesh refinement and comparison with experimental data. Specific model limitations can be identified with lack of a characteristic length scale and oversimplified grain boundaries within the modeling framework.     

A multiscale thermomechanical model for cubic to tetragonal martensitic phase transformations

We develop a multiscale thermomechanical model to analyze martensitic phase transformations from a cubic crystalline lattice to a tetragonal crystalline lattice. The model is intended for simulating the thermomechanical response of single-crystal grains of austenite. Based on the geometrically nonlinear theory of martensitic transformations, we incorporate microstructural effects from several subgrain length scales. The effective stiffness tensor at the grain level is obtained through an averaging scheme, and preserves crystallographic information from the lattice scale as well as the influence of volumetric changes due to the transformation. The model further incorporates a transformation criterion that includes a surface energy term, which takes into account the creation of interfaces between martensite and austenite. These effects, which are often neglected in martensitic transformation models, thus appear explicitly in the expression of the transformation driving force that controls the onset and evolution of the transformation. In the derivation of the transformation driving force, we clarify the relations between different combinations of thermodynamic potentials and state variables. The predictions of the model are illustrated by analyzing the response of a phase-changing material subjected to various types of deformations. Although the model is developed for cubic to tetragonal transformations, it can be adapted to simulate martensitic transformations for other crystalline structures.     

Effect of microstructure on the hardening and softening behavIors of polycrystalline shape memory alloys Part I: Micromechanics constitutive modeling

The effects of microstructure and its evolution on the macroscopic superelastic stress-strain response of polycrystalline Shape Memory Alloy (SMA) are studied by a microstructure-based constitutive model developed in this paper. The model is established on the following basis: (1) the transformation conditions of the unconstrained single crystal SMA microdomain (to be distinguished from the bulk single crystal), which serve as the local criterion for the derivation of overall transformation yield conditions of the polycrystal; (2) the micro- to macro-transition scheme by which the connection between the polycrystal aggregates and the single crystal microdomain is established and the macroscopic transformation conditions of the polycrystal SMA are derived; (3) the quantitative incorporation of three microstructure factors (i.e., nucleation, growth and orientation distribution of martensite) into the modeling. These microstructural factors are intrinsic of specific polycrystal SMA systems and the role of each factor in the macroscopic constitutive response is quantitatively modeled. It is demonstrated that the interplay of these factors will result in different macroscopic transformation kinematics and kinetics which are responsible for the observed macroscopic stress-strain hardening or softening response, the latter will lead to the localization and propagation of transformation bands in TiNi SMA. The project supported by the Research Grant Committee (RGC) of Hong Kong SAR, the National Natural Science Foundation of China and the Provincial Natural Foundation of Jiangxi Province of China     

Finite element simulation of a polycrystalline ferroelectric based on a multidomain single crystal switching model

In this paper, we compute the constitutive behavior of a ferroelectric ceramic by a plane strain finite element model, where each element represents a single grain in the polycrystal. The properties of a grain are described by the microscopic model for switching in multidomain single crystals of ferroelectric materials presented by Huber et al. [J. Mech. Phys. Solids 47 (1999) 1663]. The poling behavior of the polycrystal is obtained by employing the finite element formulation for electromechanical boundary value problems developed by Landis [Int. J. Numer. Meth. Eng. 55 (2002) 613]. In particular, we address the influence of the single grain properties and the interaction between grains, respectively.     

动态载荷下剪切变形局部化、微结构演化与剪切断裂研究进展

总结和评述了近年来金属与合金变形局部化的形成、微结构演化与剪切断裂方面作者和相关的研究工作成果. 材料包括低碳钢, SS304不锈钢, Fe-15%Ni-15%Cr单晶, Al-Li合金,α-Ti和Ti-6Al-4V, Al/SiCp复合材料等.综述内容主要包括:采用改进的Hopkinson扭杆装置,对剪切变形局部化形成、发展和演化过程进行了实验观察与数值模拟;采用"侧剖"与"对接"等定点方法制备电子显微镜薄膜试样,对剪切带内相变与再结晶、非晶转变、旋涡结构等动态变形现象,以及与宏观动态力学行为对应的位错胞的形成、发展和坍塌等微结构特征进行了观测;提出了应变和应变率同时作为剪切带形成的两个必要条件的直接实验证据;在剪切带内发现了α'$-马氏体相变现象,以及相变产物与母体之间的晶体学关系;通过位错单滑移或交滑移等微观剪切最后发展成为宏观剪切的机制;对剪切带内再结晶结构的观测和对再结晶动力学本构关系的定量描述;对剪切带特别是``白色'腐蚀带(或相变带)的形成机制的分析和新的解释,指出 ``白色'是带内亚结构取向趋于一致,其在光学或扫描显微镜下很难辨认这些亚结构的取向差所致,并非表明剪切带内一定发生了相变;通过截断实验和实时跟踪观测发现,剪切带内微裂纹的萌生与聚合是材料承载能力骤然下降并导致最后断裂的主控因素.此外,本文对近年来在准静态和循环加载下材料的局部化形变与剪切断裂的实验结果予以简要评述,指出其微观机制与动态载荷下的截然不同, 是由位错的平面滑移所控制的,与热效应无关的等温变形.      

Thermomechanical behavior of shape memory alloy under complex loading conditions

The thermo-mechanical behavior of polycrystalline shape memory alloy (SMA) under multi-axial loading with varying temperature conditions has been studied by experiments. Recently the research has been extended theoretically and a mechanical model of polycrystalline SMA and the corresponding mesoscopic constitutive equations have been developed. The model presented in this paper is constructed on the basis of the crystal plasticity and the deformation mechanism of SMA. The variants in the crystal grains and the orientations of crystal grains in the polycrystal are considered in the proposed model; the constitutive equations are derived on the basis of the proposed model. The volume fraction of the martensite variants in the transformation process and the influence of the stress state on the transformation process are also considered. Some calculated results obtained by the constitutive equations are presented and compared with the experimental results. It is found that the deformation behavior of SMA under complex loading conditions can be well reproduced by the calculation of the constitutive equations.     

Modeling texture evolution in equal channel angular extrusion using crystal plasticity finite element models

A new approach to modeling crystallographic texture evolution in Equal Channel Angular Extrusion (ECAE) is presented in this paper. The proposed approach utilizes an elastic–viscoplastic single crystal constitutive model implemented in a finite element framework. A representative volume element of the polycrystal is subjected to boundary conditions that simulate the approximate deformation history experienced by different regions of the sample (at different through-thickness depths) in both Route A and Route C processing. The proposed approach aims to capture the influence of the complex interactions that ensue among the constituent individual crystals of a polycrystal in controlling the texture evolution in the sample, while capturing the boundary conditions inherent to ECAE deformation. The predictions from the proposed approach are compared against previously reported experimental measurements in ECAE of copper. It is observed that the proposed approach provides significantly better agreement with the measurements when compared against previously reported model predictions.