应力作用下NiTi形状记忆合金微结构演化的相场模拟及其本征应变率敏感性

基于Ginzburg-Landau动力学控制方程建立了NiTi形状记忆合金非等温相场模型,实现了对NiTi合金内应力诱导马氏体相变的数值模拟。同时将晶界能密度引入系统局部自由能密度,从而考虑多晶系统中晶界的重要作用。数值计算了单晶和多晶NiTi形状记忆合金在单轴机械载荷作用下微结构的动态演化过程和宏观力学行为,并重点研究了晶粒尺寸为60 nm的NiTi纳米多晶在低应变率下（0.000 5～15 s-1）力学行为的本征应变率敏感性。研究结果表明,单晶NiTi合金系统高温拉伸-卸载过程中马氏体相变均匀发生,未形成奥氏体-马氏体界面。而纳米多晶系统在加载阶段出现了马氏体带的形成-扩展现象,在卸载阶段出现了马氏体带的收缩-消失现象。相同外载作用过程中,NiTi单晶系统的宏观应力-应变曲线具有更大的滞回环面积,拥有更优的超弹性变形能力。计算结果显示,在中低应变率下纳米晶NiTi形状记忆合金应力-应变关系表现出较明显的应变率相关性,应变率升高导致材料相变应力提升。这一应变率相关性主要源于相场模型中外加载荷速率与马氏体空间演化速度的相互竞争关系。     

A 2D finite element based on a nonlocal constitutive model describing localization and propagation of phase transformation in shape memory alloy thin structures

Here, the effects of localization and propagation of martensitic phase transformation on the response of SMA thin structures subjected to thermo-mechanical loadings are investigated using nonlocal constitutive model in conjunction with finite element method. The governing equations are derived based on variational principle considering thermo-mechanical equilibrium and the spatial distribution of the nonlocal volume fraction of martensite during transformation. The nonlocal volume fraction of martensite is defined as a weighted average of the local volume fraction of martensite over a domain characterized by an internal length parameter. The local version of the thermo-mechanical behavior model derived from micromechanics considers the local volume fraction of martensite and the mean transformation strain. A 4-noded quadrilateral plane stress element with three degrees of freedom per node accounting for in-plane displacements and the nonlocal volume fraction of martensite is developed. Numerical simulations are conducted to bring out the influence of material and geometrical heterogeneities (perturbations/defects) on the localization and propagation of phase transformation in SMA thin structures. Also, a sensitivity analysis of the material response due to the localization and the other related model parameters is carried out. The detailed investigation done here clearly shows that the localization of phase transformation has significant effect on the response of shape memory alloys.     

Three-dimensional finite element simulations of ferroelectric polycrystals under electrical and mechanical loading

Complex, non-linear, irreversible, hysteretic behavior of polycrystalline ferroelectric materials under a combined electro-mechanical loading is a result of domain wall motion, causing simultaneous expansion and contraction of unlike domains, grain sub-divisions that have distinct spontaneous polarization and strain. In this paper, a 3-dimensional finite element method is used to simulate such a polycrystalline ferroelectric under electrical and mechanical loading. A constitutive law due to Huber et al. [1999. A constitutive model for ferroelectric polycrystals. J. Mech. Phys. Solids 47, 1663-1697] for switching by domain wall motion in multidomain ferroelectric single crystals is employed in our model to represent each grain, and the finite element method is used to solve the governing conditions of mechanical equilibrium and Gauss's law. The results provide the average behavior for the polycrystalline ceramic. We compare the outcomes predicted by this model with the available experimental data for various electromechanical loading conditions. The qualitative features of ferroelectric switching are predicted well, including hysteresis and butterfly loops, the effect on them of mechanical compression, and the response of the polycrystal to non-proportional electrical loading.     

Defect redistribution within a continuum grain boundary plasticity model

The mechanical response of polycrystalline metals is significantly affected by the behaviour of grain boundaries, in particular when these interfaces constitute a relatively large fraction of the material volume. One of the current challenges in the modelling of grain boundaries at a continuum (polycrystalline) scale is the incorporation of the many different interaction mechanisms between dislocations and grain boundaries, as identified from fine-scale experiments and simulations. In this paper, the objective is to develop a model that accounts for the redistribution of the defects along the grain boundary in the context of gradient crystal plasticity. The proposed model incorporates the nonlocal relaxation of the grain boundary net defect density. A numerical study on a bicrystal specimen in simple shear is carried out, showing that the spreading of the defect content has a clear influence on the macroscopic response, as well as on the microscopic fields. This work provides a basis that enables a more thorough analysis of the plasticity of polycrystalline metals at the continuum level, where the plasticity at grain boundaries matters.     

A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite

A 3-D constitutive model for polycrystalline shape memory alloys (SMAs), based on a modified phase transformation diagram, is presented. The model takes into account both direct conversion of austenite into detwinned martensite as well as the detwinning of self-accommodated martensite. This model is suitable for performing numerical simulations on SMA materials undergoing complex thermomechanical loading paths in stress–temperature space. The model is based on thermodynamic potentials and utilizes three internal variables to predict the phase transformation and detwinning of martensite in polycrystalline SMAs. Complementing the theoretical developments, experimental data are presented showing that the phase transformation temperatures for the self-accommodated martensite to austenite and detwinned martensite to austenite transformations are different. Determination of some of the SMA material parameters from such experimental data is also discussed. The paper concludes with several numerical examples of boundary value problems with complex thermomechanical loading paths which demonstrate the capabilities of the model.     

A multivariant micromechanical model for SMAs Part 2. Polycrystal model

An averaging scheme is developed to simulate the behavior of a polycrystalline shape memory alloy (SMA) specimen using the Multivariant Micromechanics approach. An untextured polycrystalline specimen is assumed to be formed by a number of randomly oriented single crystal grains. The previously developed Multivariant technique is used to model the response of each single crystal grain subjected to its stress field seen in the polycrystalline sample. Using spherical grains, the Eshelby–Kröner approach is used to formulate the interaction between grains and to determine the stress state in each individual grain. This model successfully captures the basic features of SMA polycrystalline response to loading and temperature. In addition, comparison is made to recent experimental data with fully triaxial load states. Reasonable qualitative agreement is obtained and some issues related to crystallography of the material model are addressed.     

SIMULATIONS OF MECHANICAL BEHAVIOR OF POLYCRYSTALLINE COPPER WITH NANO-TWINS

Mechanical behavior and microstructure evolution of polycrystalline copper with nano-twins were investigated in the present work by finite element simulations. The fracture of grain boundaries are described by a cohesive interface constitutive model based on the strain gradient plasticity theory. A systematic study of the strength and ductility for different grain sizes and twin lamellae distributions is performed. The results show that the material strength and ductility strongly depend on the grain size and the distribution of twin lamellae microstructures in the polycrystalline copper.     

Development of inter- and intragranular stresses during switching of ferroelectric polycrystals

Ferroelectrics are crystalline inorganic materials consisting of domains with different directions of spontaneous polarization. By application of sufficiently high electric fields, these domains can switch into a common direction, thus making the material piezoelectric. Due to ferroelasticity, the domains can be also switched into different states by the application of mechanical stress. In polycrystalline materials, as used in most applications, electric and stress fields interact so as to maintain compatibility. We study the influence of grain-to-grain interactions on the overall and local switching behavior and in particular the induced stresses inside grains and across grain boundaries. The behavior inside each grain is represented by the single-crystal model of [Huber, J.E., Fleck, N.A., Landis, C.M., McMeeking, R.M., 1999. A constitutive model for ferroelectric polycrystals. Journal of the Mechanics and Physics of Solids 47 (8), 1663–1697] and the polycrystal response is obtained through a two-dimensional multi-grain model in which grains are represented individually. We investigate the effect of random grain orientations, both in the plane of consideration and in three directions, and compare plane strain with plane stress conditions. It is found that the overall piezoelectric response under electric loading is not dependent only on the intra- and intergranular stresses in the plane but is also significantly affected by stresses in through-thickness direction.     

Multi-grain analysis versus self-consistent estimates of ferroelectric polycrystals

Ferroelectrics are polycrystalline materials consisting of intragranular regions with different polarization directions, called domains. The domains can be switched into different states by the application of an electric field or mechanical stress. We study the influence of grain-to-grain interactions on the overall and local switching behavior. The behavior inside each grain is represented by the micromechanics model of Huber et al. [1999. A constitutive model for ferroelectric polycrystals. J. Mech. Phys. Solids 47 (8), 1663-1697]. The predictions of a self-consistent model of the polycrystal response are compared with those of a multi-grain model in which grains are represented individually. In one flavor of the multi-grain model, each grain is represented by a single finite element, while in the other the fields inside each grain are captured in more detail through a fine discretization. Different electrical and mechanical loading situations are investigated. It is found that the overall response is only mildly dependent on the accuracy with which grain-to-grain interactions are captured, while the distribution of grain-average stresses is quite sensitive to the resolution of the intragranular fields.     

关于铁电铁弹材料的自然构形

铁电换弹材料常在极化下工作,其性能受材料的微结构直接影响。铁电材料在自极化过程中自变形和自极化方向的不唯一性,以及多晶材料中存在各种不同方向晶界的不可避免性,使得其自极化稳定构形具有极其复杂的结构。根据铁电铁材料单晶特征,提出在自极化过程中该材料能量密度是变形梯度和电位移向量的非凸函数。并且从能量角度出发,导出角电铁弹材料的自极化稳定构形所应满足的必要条件;得出在自然条件下（即无外力和无外加电场作     

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.     

Modelling of plasticity and damage in a polycrystalline microstructure

The mesomechanical behavior of a polycrystalline microstructure subjected to monotonic and cyclic loadings is investigated. The analysis is based on a Voronoi polygonization strategy for generation of grains embedded in a contiguous matrix. The main emphasis is to investigate the interaction between the microconstituents and the failure processes along grain boundaries. A rational interface theory based on damage development coupled to inelastic slip and dilatation is developed. The theory uses the interface width as a constitutive parameter, which regularizes the theory of LEMAITRE [1992], that is restricted to perfect bond between grain and matrix. In a series of FE-analyses parameter variations were performed: The unit cell size (as compared to the average grain diameter), the grain-matrix area ratio, the interface width and constitutive parameters. It appears that the composite behavior can be designed as brittle or ductile solely depending on the strength and rate of damage development in the interfaces. A localization band was detected, and its orientation is in the range that is predicted within continuum theory.     

Modelling of localization and propagation of phase transformation in superelastic SMA by a gradient nonlocal approach

In this work, a nonlocal phenomenological behavior model is proposed in order to describe the localization and propagation of stress-induced martensite transformation in shape memory alloy (SMA) wires and thin films. It is a nonlocal extension of an existing local model that was derived from a micromechanical-inspired Gibbs free energy expression. The proposed model uses, besides the local field of the internal variable, namely the martensite volume fraction, a nonlocal counterpart. This latter acts as an additional degree of freedom, which is determined by solving an additional partial differential equation (PDE), derived so as to be equivalent to the integral definition of a nonlocal quantity. This PDE involves an internal length parameter, dictating the global scale at which the nonlocal interactions of the underlying micromechanisms are manifested during phase transformation. Moreover, to account for the unstable softening behavior, the transformation yield force parameter is considered as a gradually decreasing function of the martensite fraction. Possible material and geometric imperfections that are responsible for localization initiation are also considered in this analysis. The obtained constitutive equations are implemented in the Abaqus® finite element code in one and two dimensions. This requires the development of specific finite elements having the nonlocal volume fraction variable as an additional degree of freedom. This implementation is achieved through the UEL user’s subroutine. The effect of martensitic localization on the superelastic global behavior of SMA wire and holed thin plate, subjected to tension loading, is analyzed. Numerical results show that the developed tool correctly captures the commonly observed unstable superelastic behavior characterized by nucleation and propagation of martensitic phase. In particular, they show the influence of the internal length parameter, appearing in the nonlocal model, on the size of the localization area and the stress nucleation peak.     

Modeling dynamic plasticity and spall fracture in high density polycrystalline alloys

The dynamic thermomechanical response of a tungsten heavy alloy is investigated via modeling and numerical simulation. The material of study consists of relatively stiff pure tungsten grains embedded within a more ductile matrix alloy comprised of tungsten, nickel, and iron. Constitutive models implemented for each phase account for finite deformation, heat conduction, plastic anisotropy, strain-rate dependence of flow stress, thermal softening, and thermoelastic coupling. The potentially nonlinear volumetric response in tungsten at large pressures is addressed by a pressure-dependent effective bulk modulus. Our framework also provides a quantitative prediction of the total dislocation density, associated with cumulative strain hardening in each phase, and enables calculation of the fraction of plastic dissipation converted into heat energy. Cohesive failure models are employed to represent intergranular fracture at grain and phase boundaries. Dynamic finite element simulations illustrate the response of realistic volume elements of the polycrystalline microstructure subjected to compressive impact loadings, ultimately resulting in spallation of the material. The relative effects of mixed-mode interfacial failure criteria, thermally-dependent fracture strengths, and grain shapes and orientations upon spall behavior are weighed, with interfacial properties exerting a somewhat larger influence on the average pressure supported by the volume element than grain shapes and initial lattice orientations within the bulk material. Spatially resolved profiles of particle velocities at the free surfaces of the volume elements indicate the degree to which the incident and reflected stress waves are altered by the heterogeneous microstructure.     

A micromechanical model for pretextured polycrystalline shape-memory alloys including elastic anisotropy

We present a micromechanical model for polycrystalline shape-memory alloys which is capable of reproducing important aspects of the material behavior such as pseudoelasticity, pseudoplasticity, tension–compression asymmetry and the influence of texture inhomogeneities which may occur from the production process of components or specimens. Our model is based on the optimization of the material’s free energy density and uses a dissipation ansatz which is homogeneous of first order. Considering the full anisotropic material properties of both the austenite and the martensite phase, we compute the evolution of the orientation distributions of austenite and martensite as internal variables of our model.      

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.     

A Micromechanics-based Finite Element Model for the Constitutive Behavior of Polycrystalline Ferromagnets

A micromechanics-based finite element model for the constitutive behavior of polycrystalline ferromagnets is developed. In the model, the polycrystalline solid is assumed to comprise numerous single crystals with randomly distributed crystallographic orientations, and the single crystals, in turn, consist of ferromagnetic domains, each of which is represented by a cubic element. The dipole directions of the domains are randomly assigned to simulate the crystallographic nature of ferromagnetic polycrystals. A switching criterion for the domains is specified at the microscopic level. The macroscopic constitutive behavior is obtained by averaging the microscopic/local behavior of each domain. The developed model has been applied to the simulation of a ferromagnetic material. With appropriate material parameters adopted, hysteresis loops of the predicted magnetic induction versus magnetic field and those of the strain versus magnetic field are shown to agree well with experimental observations.The project supported by the National Natural Science Foundation of China (90205030, 10472088, 10425210), the National Basic Research Program of China (2006CB601202) and the State Administration of the Foreign Experts Affairs Through the “111” Project (B06024) The English text was polished by Yunming Chen.     

随机晶界分布和连通性对晶间破坏行为的影响

采用随机有限元方法研究了两个具有相同不均匀材料属性和相同随机晶界含量但不同随机晶界分布的特殊多晶体样本在准静态拉伸下的晶间破坏行为,发现随机晶界连通性强的样本的断裂能比较低．在不同随机晶界含量下分别构造的50个晶界随机分布的样本的计算结果表明,断裂能与平均晶界连通数的关联并不明显,甚至还出现了相反的结果,即在相同随机晶界含量下,平均晶界连通数大的计算样本的断裂能反而大．     

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.