Micromechanical modeling of stress-induced phase transformations. Part 2. Computational algorithms and examples

Based on the theory developed in Part 1 of this paper [Levitas, V.I., Ozsoy, I.B., 2008. Micromechanical modeling of stress-induced phase transformations. Part 1. Thermodynamics and kinetics of coupled interface propagation and reorientation. Int. J. Plasticity. doi:10.1016/j.ijplas.2008.02.004], various non-trivial examples of microstructure evolution under complex multiaxial loading are presented. For the case without interface rotation, the effect of the athermal thresholds for austenite (A)–martensite (M) and martensitic variant M_I–variant M_II interfaces and loading paths on stress–strain curves and phase transformations was studied. For coupled interface propagation and rotation, two types of numerical simulations were carried out. The tetragonal–orthorhombic transformation has been studied under general three-dimensional interface orientation and zero athermal threshold. The cubic–tetragonal transformation was treated with allowing for an athermal threshold and interface reorientation within a plane. The effect of the athermal threshold, the number of martensitic variants and an interface orientation in the embryo was studied in detail. It was found that an instability in the interface normal leads to a jump-like interface reorientation that has the following features of the energetics of a first-order transformation: there are multiple energy minima versus interface orientation that are separated by an energy barrier; positions of minima do not change during loading but their depth varies; when the barrier disappears (i.e. one of the minima transforms to the local saddle or maximum points), the system rapidly evolves toward another stable orientation. Depending on the loading and material parameters, we observed a large continuous change in interface orientation, a jump in interface reorientation, a jump in volume fractions and stresses, an expected stress relaxation during the phase transition and unexpected stress growth during the transition because of large change in elastic moduli.     

Interfacial energy and dissipation in martensitic phase transformations. Part II: Size effects in pseudoelasticity

This paper is a continuation of the Part I (H. Petryk, S. Stupkiewicz, Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory. J. Mech. Phys. Solids, 2010, doi:10.1016/j.jmps.2009.11.003). A fully three-dimensional model of an evolving martensitic microstructure is examined, taking into account size effects due to the interfacial energy and also dissipation related to annihilation of interfaces. The elastic micro-strain energy at microstructured interfaces is determined with the help of finite element computations and is approximated analytically. Three interface levels are examined: of grain boundaries attained by parallel martensite plates, of interfaces between austenite and twinned martensite, and of twin interfaces within the martensite phase. Minimization of the incremental energy supply, being the sum of the increments in the free energy and dissipation of the bulk and interfacial type at all levels, is used as the evolution rule, based on the theory presented in Part I. An example of the formation and evolution of a rank-three laminated microstructure of finite characteristic dimensions in a pseudoelastic CuAlNi shape memory alloy is examined quantitatively.     

Size effects in martensitic microstructures: Finite-strain phase field model versus sharp-interface approach

A finite-strain phase field model for martensitic phase transformation and twinning in shape memory alloys is developed and confronted with the corresponding sharp-interface approach extended to interfacial energy effects. The model is set in the energy framework so that the kinetic equations and conditions of mechanical equilibrium are fully defined by specifying the free energy and dissipation potentials. The free energy density involves the bulk and interfacial energy contributions, the latter describing the energy of diffuse interfaces in a manner typical for phase-field approaches. To ensure volume preservation during martensite reorientation at finite deformation within a diffuse interface, it is proposed to apply linear mixing of the logarithmic transformation strains. The physically different nature of phase interfaces and twin boundaries in the martensitic phase is reflected by introducing two order-parameters in a hierarchical manner, one as the reference volume fraction of austenite, and thus of the whole martensite, and the second as the volume fraction of one variant of martensite in the martensitic phase only. The microstructure evolution problem is given a variational formulation in terms of incremental fields of displacement and order parameters, with unilateral constraints on volume fractions explicitly enforced by applying the augmented Lagrangian method. As an application, size-dependent microstructures with diffuse interfaces are calculated for the cubic-to-orthorhombic transformation in a CuAlNi shape memory alloy and compared with the sharp-interface microstructures with interfacial energy effects.     

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.     

Evolution of Phase Boundaries by Configurational Forces

An initial boundary value problem modeling the evolution of phase interfaces in materials showing martensitic transformations is studied. The model, which is derived rigorously from a sharp interface model with phase interfaces driven by configurational forces and which generalizes that model, consists of the equations of linear elasticity coupled with a nonlinear partial differential equation of hyperbolic character governing the evolution of the order parameter. It is proved that in one space dimension, global solutions exist for which the order parameter belongs to the space of functions of bounded variation. Other models for interface motion by martensitic transformations and by interface diffusion are suggested.     

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。     

Structural changes without stable intermediate state in inelastic material. Part II. Applications to displacive and diffusional–displacive phase transformations,strain-induced chemical reactions and ductile fracture

A number of simple examples demonstrate the applicability of the general theory developed in Part I of this paper to various structural changes in solids, namely to displacive and generalized second-order phase transformations, twinning and reorientation of crystal lattice, ductile fracture, and strain-induced chemical reactions. The theory is extended to diffusional-displacive phase transitions. The following problems for elastic and elastoplastic materials are solved analytically: displacive and diffusional–displacive phase transitions in a spherical particle inside the space under external pressure, martensitic phase transition and twinning in ellipsoidal inclusion under applied shear stress, spherical void nucleation, crack propagation in a similar framework to the Dugdale model for a plane stress state. In most cases explicit expressions for the thermodynamic and kinetic conditions for structural changes and the geometric parameters of the nucleus are obtained and analyzed. The following typical cases in the determination of the geometric parameters of nucleus are found: solely from the principle of the minimum of transformation time and the kinetic equation without any constraints or with thermodynamic constraint; from the principle of the minimum of transformation mass and the thermodynamic criterion of structural changes (thermodynamically admissible nucleus); as an interatomic distance. For diffusional–displacive phase transformations, additional variants are related to the necessity to consider the diffusion equation (for the diffusion-controlled transformation) and constraints related to the maximum and minimum possible volume fraction of solute atoms. The nucleation kinetics for various nucleus geometries is compared.     

Interface propagation and microstructure evolution in phase field models of stress-induced martensitic phase transformations

Analytical solutions for diffuse interface propagation are found for two recently developed Landau potentials that account for the phenomenology of stress-induced martensitic phase transformations. The solutions include the interface profile and velocity as a function of temperature and stress tensor. An instability in the interface propagation near lattice instability conditions is studied numerically. The effect of material inertia is approximately included. Two methods for introducing an athermal interface friction in phase field models are discussed. In the first method an analytic expression defines the location of the diffuse interface, and the rate of change of the order parameters is required to vanish if the driving force is below a threshold. As an alternative and more physical approach, we demonstrate that the introduction of spatially oscillatory stress fields due to crystal defects and the Peierls barrier, or to a jump in chemical energy, reproduces the effect of an athermal threshold. Finite element simulations of microstructure evolution with and without an athermal threshold are performed. In the presence of spatially oscillatory fields the evolution self-arrests in realistic stationary microstructures, thus the system does not converge to an unphysical single-phase final state, and rate-independent temperature- and stress-induced phase transformation hysteresis are exhibited.     

Multiple bifurcations and local energy minimizers in thermoelastic martensitic transformations

Thermoelastic martensitic transformations in shape memory alloys can be modeled on the basis of nonlinear elastic theory.Microstructures of fine phase mixtures are local energy minimizers of the total energy.Using a one-dimensional effective model,we have shown that such microstructures are inhomogeneous solutions of the nonlinear Euler-Lagrange equation and can appear upon loading or unloading to certain critical conditions,the bifurcation conditions.A hybrid numerical method is utilized to calculate the inhomogeneous solutions with a large number of interfaces.The characteristics of the solutions are clarified by three parameters:the number of interfaces,the interface thickness,and the oscillating amplitude.Approximated analytical expressions are obtained for the interface and inhomogeneity energies through the numerical solutions.     

Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory

A model of evolving martensitic microstructures is formulated that incorporates the interfacial energy and dissipation on three different scales corresponding to the grain boundaries attained by martensite plates, the interfaces between austenite and martensite plates, and the twin interfaces within martensite plates. Three different time scales are also considered in order to clarify the meaning of rate-independent dissipation related to instabilities at more refined temporal and spatial scales. On the slowest time scale, the process of deformation and martensitic phase transformation is investigated as being composed of segments of smooth quasi-static evolution separated by sudden jumps associated with creation or annihilation of interfaces. A general evolution rule is used in the form of minimization of the incremental energy supply to the whole compound thermodynamic system, including the rate-independent dissipation. Close relationship is shown between the evolution rule and the thermodynamic condition for stability of equilibrium, with no a priori assumption on convexity of the dissipation function. It is demonstrated that the extension of the minimum principle from the first-order rates to small but finite increments requires a separate symmetry restriction imposed on the state derivative of the dissipation function. Formulae for the dissipation associated with annihilation of interfaces are proposed that exhibit limited path-independence and satisfy that symmetry requirement. By exploiting the incremental energy minimization rule with the help of the transport theorems, local propagation conditions are derived for both planar and curved phase transformation fronts. The theory serves as a basis for the algorithm for calculation of the stress-induced evolution of martensitic microstructures along with their characteristic dimensions and related hysteresis loops in shape memory alloys; the examples are given in Part II of the paper.     

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 thermodynamical constitutive model for shape memory materials. Part II. The SMA composite material

The phenomenological SMA equations developed in Part I are used in this second paper to derive the free energy and dissipation of a SMA composite material. The derivation consists of solving a boundary value problem formulated over a mesoscale representative volume element, followed by an averaging procedure to obtain the macroscopic composite constitutive equations. Explicit equations are derived for the transformation tensors that relate the composite transformation strain rate to the phase transformation rate in the fiber and matrix. Some key findings for the two-way SME in a SMA fiber/elastomer matrix composite are that processing-induced residual stresses alter the composite austenite start and martensite start temperatures, as well as the amount of composite strain recovered during a complete cycle of temperature and fiber martensite volume fraction. Relative to the two-way SME response of stiff-matrix composites, it was found that compliant-matrix composites: (1) complete the phase transformation over a narrower temperature range; (2) exhibit greater transformation strain during the reverse transformation; and (3) undergo an incomplete strain cycle during a complete cycle of temperature and fiber martensite volume fraction. Due to the interaction of the fiber and matrix during transformation, macroscopic proportional stressing of the composite results in non-proportional fiber stressing, which in turn causes a small amount of martensitic reorientation to occur simultaneously with the transformation.     

Modelling and numerical simulation of martensitic transformation in shape memory alloys

We consider the evolution of martensitic fine structures in shape memory alloys which undergo an isothermal phase-transformation. This process is modelled on a microscopical, continuum-mechanical level by partial differential equations. Here a homogeneous degree-1 dissipation potential is involved which can reflect specific energies needed for rate-independent phase transformations. An interface energy is incorporated by a nonlocal term, and hard-device loading is considered. After setting up the model and specifying its energy balance properties, three-dimensional numerical experiments for the cubic-to-tetragonal transformation in an InTl single crystal are presented which demonstrate geometrical/material interactions under tensile and shear loading.Received: 27 June 2002, Accepted: 18 March 2003, Published online: 27 June 2003PACS: 81.30Kf     

The computation of the dynamics of the martensitic transformation

We present numerical computations for the dynamics of the development of twinned martensitic microstructure and for the propagation of the austeniticfinely twinned martensitic interface. Our computations approximated a threedimensional model for the dynamics of the In-20.7 at% T1 alloy which used the stress tensor derived from the Ericksen-James energy density.     

Influence of the loading path on the strength of fiber-reinforced composites subjected to transverse compression and shear

The influence of the loading path on the failure locus of a composite lamina subjected to transverse compression and out-of-plane shear is analyzed through computational micromechanics. This is carried out using the finite element simulation of a representative volume element of the microstructure, which takes into account explicitly fiber and matrix spatial distribution within the lamina. In addition, the actual failure mechanisms (plastic deformation of the matrix and interface decohesion) are included in the simulations through the corresponding constitutive models. Two different interface strength values were chosen to explore the limiting cases of composites with strong or weak interfaces. It was found that failure locus was independent of the loading path for the three cases analyzed (pseudo-radial, compression followed by shear and shear followed by compression) in the composites with strong and weak interfaces. This result was attributed to the fact that the dominant failure mechanism in each material was the same in transverse compression and in shear. Failure is also controlled by the same mechanisms under a combination of both stresses and the failure locus depended mainly on the magnitude of the stresses that trigger fracture rather than in the loading path to reach the critical condition.     

Path dependence and multiaxial behavior of a polycrystalline NiTi alloy within the pseudoelastic and pseudoplastic temperature regimes

Several multiaxial experiments on polycrystalline NiTi have been conducted within a wide temperature range. In this vein, the pseudoelastic as well as the pseudoplastic behavior are investigated within the distinct temperature regimes. Isothermal and temperature varying thermomechanical loading paths are applied by means of an active temperature control in order to characterize the path dependence of pseudoelasticity and the multiaxial one-way effect of the alloy. The main focus is on the determination of the dependence of the loading sequence, the related non-linearity of the material and the combined material interaction, e.g., referring to reorientation processes for complex loading paths with respect to pseudoelasticity and the one-way effect. Isothermal tension/compression/torsion experiments are performed on an austenitic microstructure spanning all four quadrants of the axial/torsional strain subspace. In this regard, it is deduced in the course of this contribution that the apparently qualitatively different material behavior for different strain paths in the pseudoelastic temperature regime might be explained by the axial/torsional and tension/compression asymmetry. Furthermore, some multidimensional axial/torsional stress controlled experiments are realized with loading on a martensitic and unloading being implemented both on martensitic and austenitic microstructures. Here, the peculiarity of the one-way effect referring to apparently different transformation temperatures is ascribed to the loading history of the specimen material and to differently oriented martensite variants. In order to elucidate these effects, potential explanations for the pseudoelastic path dependence and the non-linearity in the material behavior with reference to the multiaxial one-way effect are presented.     

马氏体相变微结构演化过程的模拟与分析

实验中观察到形状记忆合金在应力诱发马氏体相变过程中,出现多界面的微结构,马氏体相会逐渐长大变粗,同时会出现由马氏体形核造成的应力突然降低.用多阱的弹性能函数来刻画此相变与微结构演化过程,发现相变时会出现多界面的微结构且伴随着马氏体相的形核至奥氏体相的消失过程,出现了界面数先增后减的变化,同时应力会出现跳跃而不连续.相对应的动力学模型的有限差分的计算结果同样显示形核时出现了多界面的微结构并伴随着应力的大幅振荡,随着载荷的增加界面位置随之移动,使得马氏体相区域逐渐长大.理论分析与数值模拟的结果较好地刻画了实验中观察到的马氏体相变过程中的形核,产生多界面,再到马氏体逐渐长大这一微结构的演化过程.