An upper bound to the free energy of n-variant polycrystalline shape memory alloys

In the last two decades, the problem of computing the elastic energy of phase transforming materials has been studied by a variety of research groups. Due to the non-quasiconvexity of the underlying multi-well landscape, different relaxation methods have been used in order to estimate the quasiconvex envelope of the energy density, for which no explicit expression is known at present.This paper combines a recently developed lamination bound for monocrystalline shape memory alloys which relies on martensitic twinned microstructures with the work of Smyshlyaev and Willis [1998a. A ‘non-local’ variational approach to the elastic energy minimization of martensitic polycrystals. Proc. R. Soc. London A 454, 1573–1613]. As a result, a lamination upper bound for n-variant polycrystalline martensitic materials is obtained.The lamination bound is then compared with Reuß- and Taylor-type estimates. While, for given volume fractions, good agreement of lamination upper and convexification lower bounds is obtained, a comparison using energy-minimizing volume fractions computed from the various bounds yields larger differences. Finally, we also investigate the influence of the polycrystal's texture. For a strong ellipsoidal texture, we observe even better agreement of upper and lower bounds than for the case of isotropic statistics.     

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.     

About lamination upper and convexification lower bounds on the free energy of monoclinic shape memory alloys in the context of <Emphasis Type="Italic">T</Emphasis><Subscript>3</Subscript>-configurations and R-phase formation

This work is concerned with different estimates of the quasiconvexification of multi-well energy landscapes of NiTi shape memory alloys, which models the overall behavior of the material. Within the setting of the geometrically linear theory of elasticity, we consider a formula of the quasiconvexification which involves the so-called energy of mixing.We are interested in lower and upper bounds on the energy of mixing in order to get a better understanding of the quasiconvexification. The lower bound on the energy of mixing is obtained by convexification; it is also called Sachs or Reuß lower bound. The upper bound on the energy of mixing is based on second-order lamination. In particular, we are interested in the difference between the lower and upper bounds. Our numerical simulations show that the difference is in fact of the order of 1% and less in martensitic NiTi, even though both bounds on the energy of mixing were rather expected to differ more significantly. Hence, in various circumstances it may be justified to simply work with the convexification of the multi-well energy, which is relatively easy to deal with, or with the lamination upper bound, which always corresponds to a physically realistic microstructure, as an estimate of the quasiconvexification. In order to obtain a potentially large difference between upper and lower bound, we consider the bounds along paths in strain space which involve incompatible strains. In monoclinic shape memory alloys, three-tuples of pairwise incompatible strains play a special role since they form so-called T ₃-configurations, originally discussed in a stress-free setting. In this work, we therefore consider in particular numerical simulations along paths in strain space which are related to these T ₃-configurations. Interestingly, we observe that the second-order lamination upper bound along such paths is related to the geometry of the T ₃-configurations. In addition to the purely martensitic regime, we also consider the influence of adding R-phase variants to the microstructure. Adding single variants of R-phase is shown to be energetically favorable in a compatible martensitic setting. However, the combination of several R-phase variants with compatible or incompatible martensite yields significant differences between the bounds considered.     

Coupling between martensitic phase transformations and plasticity: A microstructure-based finite element model

A microstructural finite element (MFE) model is developed to capture the interaction between martensitic transformations and plasticity in NiTi shape memory alloys (SMAs). The interaction is modeled through the grain-to-grain redistribution of stress caused by both plasticity and phase transformation, so that each mechanism affects the driving force of the other. A unique feature is that both processes are modeled at a crystallographic level and are allowed to operate simultaneously. The model is calibrated to pseudoelastic data for select single crystals of Ti–50.9at.%Ni. For polycrystals, plasticity is predicted to enhance the overall martensite volume fraction at a given applied stress. Upon unloading, residual stress can induce remnant (retained) martensite. For thermal cycling under load bias, plasticity is observed to limit the net transformation strain/cycle and increase the hysteretic width. Deformation processing, via plastic pre-straining at elevated temperature, is shown to dramatically alter subsequent pseudoelastic response, as well as induce two-way shape memory behavior during no-load thermal cycling. Overall, the model is suitable at smaller imposed strains, where martensite detwinning is not expected to dominate.     

Finite element simulations of martensitic phase transitions and microstructures based on a strain softening model

A new approach for modeling multivariant martensitic phase transitions (PT) and martensitic microstructure (MM) in elastic materials is proposed. It is based on a thermomechanical model for PT that includes strain softening and the corresponding strain localization during PT. Mesh sensitivity in numerical simulations is avoided by using rate-dependent constitutive equations in the model. Due to strain softening, a microstructure comprised of pure martensitic and austenitic domains separated by narrow transition zones is obtained as the solution of the corresponding boundary value problem. In contrast to Landau-Ginzburg models, which are limited in practice to nanoscale specimens, this new phase field model is valid for scales greater than 100 nm and without upper bound. A finite element algorithm for the solution of elastic problems with multivariant martensitic PT is developed and implemented into the software ABAQUS. Simulated microstructures in elastic single crystals and polycrystals under uniaxial loading are in qualitative agreement with those observed experimentally.     

A variational approach to the theory of the elastic behaviour of polycrystals

Variational principles for anisotropic and nonhomogeneous elasticity, established by the authors in a previous paper, have been applied to the derivation of lower and upper bounds for the elastic moduli of polycrystals in terms of the moduli of the constituting crystals. The results hold for arbitrary crystal shapes. Explicit results tor cubic polycrystals showed that the present bounds are a considerable improvement of the well-known Voigt and Reuss bounds. Good agreement with experimental results has been obtained.     

Thermal expansion of polycrystalline aggregates: I. Exact analysis

An exact relation is developed between the thermal expansion coefficient and the bulk modulus of statistically isotropic polycrystalline aggregates composed of crystals of hexagonal, tetragonal or trigonal symmetry. This relation is exploited to derive simple close bounds for the thermal expansion coefficient in terms of single crystal properties. Comparison of bounds to experimentally obtained expansion coefficients shows fair to very good agreement.     

Hill–Mandel condition and bounds on lower symmetry elastic crystals

Despite advances in contemporary micromechanics, there is a void in the literature on a versatile method for estimating the effective properties of polycrystals comprising of highly anisotropic single crystals belonging to lower symmetry class. Basing on variational principles in elasticity and the Hill–Mandel homogenization condition, we propose a versatile methodology to fill this void. It is demonstrated that the bounds obtained using the Hill–Mandel condition are tighter than the Voigt and Reuss [1], [2] bounds, the Hashin–Shtrikman [3] bounds as well as a recently proposed self-consistent estimate by Kube and Arguelles [4] even for polycrystals with highly anisotropic single crystals.     

The role of intergranular constraint on the stress-induced martensitic transformation in textured polycrystalline NiTi

The influences of grain boundaries and relative grain misorientations on stress-induced martensitic transformations in NiTi are studied using unique experiments and finite element modeling. Tensile and compressive mechanical tests reveal that polycrystalline NiTi with a dominant <111> fiber texture and single crystal NiTi oriented along the [111] direction exhibit nearly identical stress–strain curves during a stress-induced martensitic transformation. Micro-mechanical finite element simulations of fiber textured polycrystals and single crystals undergoing a multi-variant martensitic transformation confirm the relative indifference of the macroscopic transformation attributes to the presence of grain boundaries. On the microscale, the finite element simulations further reveal that the insensitivity of the transformation to intergranular constraint is linked to the local stress disturbance created by transforming grains. The transformation of grains that are favorably oriented with respect to the loading axis creates local stresses that invariably assist the transformation in neighboring grains, effectively lowering the influence of grain misorientations and boundaries on the macroscopic transformation behavior.     

A non-convex lower bound on the effective energy of polycrystalline shape memory alloys

This paper addresses the estimation of the effective free energy of polycrystalline shape memory alloys, in the framework of nonlinear elasticity and infinitesimal strains. The translation method is combined with a Hashin-Shtrikman type variational formulation to provide rigorous lower bounds on the effective free energy. Those bounds incorporate both intra-grain compatibility conditions (resulting in a non-convex bound) and inter-grain constraints (by taking one- and two-point statistics into account). Some examples are given to compare the results obtained with other bounds from the literature.     

Rate-independent crystalline and polycrystalline plasticity,application to FCC materials

This paper deals with the simulation of the mechanical response and texture evolution of cubic crystals and polycrystals for a rate-independent elastic–plastic constitutive law. No viscous effects are considered. An algorithm is introduced to treat the difficult case of multi-surface plasticity. This algorithm allows the computation of the mechanical response of a single crystal. The corresponding yield surface is made of the intersection of several hyper-planes in the stress space. The problem of the multiplicity of the slip systems is solved thanks to a pseudo-inversion method. Self and latent hardening are taken into account. In order to compute the response of a polycrystal, a Taylor homogenization scheme is used. The stress–strain response of single crystals and polycrystals is computed for various loading cases. The texture evolution predicted for compression, plane strain compression and simple shear are compared with the results given by a visco-plastic polycrystalline model.     

Bounds on the Elastic Moduli of Completely Random Two-Dimensional Polycrystals

Explicit bounds on the elastic moduli of completely random planar polycrystals, the shape and crystalline orientations of the constituent grains of which are uncorrelated, are derived and calculated for a number of crystals of general two-dimensional anisotropy. The bounds on the elastic two-dimensional bulk modulus happen to coincide with the simple third order (in anisotropy contrast) bounds for the subclass of idealistic circular cell polycrystals. The bounds on the shear modulus are close to the much simpler bounds for circular cell polycrystals, which approximate aggregates of equiaxed grains.     

Study of transformation plasticity in tetragonal zirconia polycrystals by moiré interferometry

Transformation plasticity in ceria-stabilized tetragonal zirconia polycrystals due to the stress-induced tetragonal-to-monoclinic martensitic transformation under tension and bending is studied by moiré interferometry. The whole fringe patterns includingu fields andv fields are acquired. According to these patterns, the distributions of transformation plasticity in transformation zones are obtained, and the phenomenon of plastic flow localization for transformation is revealed. The above work provides a significant experimental foundation for establishing transformation constitutive relations The project supported by the National Natural Science Foundation of China     

纳米单晶NiTi合金单程形状记忆效应的分子动力学模拟

采用基于第二近邻修正型嵌入原子势的分子动力学方法研究了纳米单晶NiTi合金的单程形状记忆效应,详细阐明了温度诱发马氏体相变和应力诱发马氏体重定向过程中纳米单晶的变形行为和微结构演化,进一步分析了加/卸载速率对NiTi合金单程形状记忆效应的影响。结果表明,NiTi纳米单晶在应力加载过程中发生马氏体重定向,卸载后存在残余应变;当加热到奥氏体转变结束温度以上时,马氏体逆相变为奥氏体相,残余应变逐渐减小,但未完全回复;随着应力加载速率的增加,重定向临界应力和模量逐渐增加;再次降温过程中不同加载速率下的原子结构演化各不相同。     

纳米单晶NiTi合金单程形状记忆效应的分子动力学模拟

采用基于第二近邻修正型嵌入原子势的分子动力学方法研究了纳米单晶NiTi合金的单程形状记忆效应,详细阐明了温度诱发马氏体相变和应力诱发马氏体重定向过程中纳米单晶的变形行为和微结构演化,进一步分析了加/卸载速率对NiTi合金单程形状记忆效应的影响。结果表明,NiTi纳米单晶在应力加载过程中发生马氏体重定向,卸载后存在残余应变;当加热到奥氏体转变结束温度以上时,马氏体逆相变为奥氏体相,残余应变逐渐减小,但未完全回复;随着应力加载速率的增加,重定向临界应力和模量逐渐增加;再次降温过程中不同加载速率下的原子结构演化各不相同。     

On the plastic wave speeds in rate-independent elastic–plastic materials with anisotropic elasticity

This paper presents a theoretical study of the speeds of plastic waves in rate-independent elastic–plastic materials with anisotropic elasticity. It is shown that for a given propagation direction the plastic wave speeds are equal to or lower than the corresponding elastic speeds, and a simple expression is provided for the bound on the difference between the elastic and the plastic wave speeds. The bound is given as a function of the plastic modulus and the magnitude of a vector defined by the current stress state and the propagation direction. For elastic–plastic materials with cubic symmetry and with tetragonal symmetry, the upper and lower bounds on the plastic wave speeds are obtained without numerically solving an eigenvalue problem. Numerical examples of materials with cubic symmetry (copper) and with tetragonal symmetry (tin) are presented as a validation of the proposed bounds. The lower bound proposed here on the minimum plastic wave speed may also be used as an efficient alternative to the bifurcation analysis at early stages of plastic deformation for the determination of the loss of ellipticity.     

The role of texture in tension–compression asymmetry in polycrystalline NiTi

The purpose of the present study is to thoroughly understand the stress–strain behavior of polycrystalline NiTi deformed under tension versus compression. To do this, a micro-mechanical model is used which incorporates single crystal constitutive relationships and experimentally measured polycrystalline texture into the self-consistent formulation. For the first time it is quantitatively demonstrated that texture measurements coupled with a micro-mechanical model can accurately predict tension/compression asymmetry in NiTi shape memory alloys. The predicted critical transformation stress levels and transformation stress–strain slopes under both tensile and compressive loading are consistent with experimental results. For textured polycrystalline NiTi deformed under tension it is demonstrated that the martensite evolution is very abrupt, consistent with the Luders type deformation experimentally observed. The abrupt transformation under tension is attributed to the fact that the majority of the grains are oriented along the [111] crystallographic direction, which is soft under tensile loading. Since single crystals of the [111] orientation are hard under compression it is also demonstrated that under compression the martensite in textured polycrystalline NiTi evolves relatively slower.     

Relaxation in crystal plasticity with three active slip systems

We study a variational model for finite crystal plasticity in the limit of rigid elasticity. We focus on the case of three distinct slip systems whose slip directions lie in one plane and are rotated by 120° with respect to each other, with linear self-hardening and infinite latent hardening, in the sense that each material point has to deform in single slip. Under these conditions, plastic deformation is accompanied by the formation of fine-scale structures, in which activity along the different slip systems localizes in different areas. The quasiconvex envelope of the energy density, which describes the macroscopic material behavior, is determined in a regime from small up to intermediate strains, and upper and lower bounds are provided for large strains. Finally sufficient conditions are given under which the lamination convex envelope of an extended-valued energy density is an upper bound for its quasiconvex envelope.     

An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations

Modeling the energetic behavior of martensitic (phase transforming) materials usually leads to non quasiconvex energy formulations. For this reason, researchers often employ quasiconvex relaxation methods to improve the character of the formulation. Unfortunately, explicit expressions for the relaxed free energy density for multi-variant martensitic materials are typically not available. Thus, some researchers have employed a Reuβ-like convex lower bound, which neglects compatibility constraints, as an estimate on the free energy of mixing. To be confident with such a technique, one needs a measure of the quality of the lower bound. In this paper, we seek such a measure by comparing the Reuβ-like lower bound to an upper bound. The upper bound is constructed upon assumptions on the type of microstructures that form in such alloys. In particular, we consider lamination type microstructures which form by temperature- or stress-induced transformation in monoclinic and orthorhombic Copper-based alloys with cubic austenitic symmetry. Our results display a striking congruence of upper and lower bounds in the most relevant cases.     

Fast texture updates in fcc polycrystal plasticity based on a linear active-set-estimate of the lattice spin

The paper presents a framework for a fast computational estimate of the texture evolution in fcc polycrystals. The underlying basis is a Taylor-type approach of rigid-plasticity for fcc crystals, where the decision of slip activity is based on a purely geometric argument in terms of the rate of total deformation. This approach allows a purely kinematic estimate of the texture evolution completely independent of the hardening response of the single crystal grains. We show that such a geometric estimate of the orientation microstructure provides very good approximations when compared with Taylor-type models of rigid viscoplasticity or elasto-plasticity of fcc crystals. It is shown for several deformation modes of polycrystalline aggregates that the geometric approach predicts the texture evolution very well in the range of large strains. In contrast to standard approaches for the analysis of texture, the proposed new method is extremely robust and very fast even for a high number of discrete grain orientations. As a consequence, such an approach provides a perfect base for large scale computations of anisotropy development in polycrystals.