Variational formulation and stability analysis of a three dimensional superelastic model for shape memory alloys

This paper presents a variational framework for the three-dimensional macroscopic modelling of superelastic shape memory alloys in an isothermal setting. Phase transformation is accounted through a unique second order tensorial internal variable, acting as the transformation strain. Postulating the total strain energy density as the sum of a free energy and a dissipated energy, the model depends on two material scalar functions of the norm of the transformation strain and a material scalar constant. Appropriate calibration of these material functions allows to render a wide range of constitutive behaviours including stress-softening and stress-hardening. The quasi-static evolution problem of a domain is formulated in terms of two physical principles based on the total energy of the system: a stability criterion, which selects the local minima of the total energy, and an energy balance condition, which ensures the consistency of the evolution of the total energy with respect to the external loadings. The local phase transformation laws in terms of Kuhn–Tucker relations are deduced from the first-order stability condition and the energy balance condition.The response of the model is illustrated with a numerical traction–torsion test performed on a thin-walled cylinder. Evolutions of homogeneous states are given for proportional and non-proportional loadings. Influence of the stress-hardening/softening properties on the evolution of the transformation domain is emphasized. Finally, in view of an identification process, the issue of stability of homogeneous states in a multi-dimensional setting is answered based on the study of second-order derivative of the total energy. Explicit necessary and sufficient conditions of stability are provided.     

Viscoplastic constitutive modeling with one scalar state variable

State variables have been used to represent the material resistance to plastic deformation in the recent development of the viscoplastic constitutive equations. In a previous paper, an experimental method was suggested to identify the relative roles to be played by the scalar state variable (drag stress) and the tensorial state variable (back stress) in the state-variable based constitutive equations. The results on 2618-T61 aluminum alloy tested at 200°C suggested that the scalar state variable alone should be enough to model the experimental results of 2618-T61 aluminum alloy.

In the current work, an early version of the viscoplastic constitutive equation proposed by Bodner and Partom, which was formulated with one scalar state variable, was adopted to model the experimental results of 2618-T61 aluminum. Experiments included creep tests under stepwise loadings, controlled-strain-rate tests, and creep tests under nonproportional loadings. A constitutive equation based on strain hardening approach, which was developed in an earlier work, was also studied.

In order to improve the results of the Bodner and Partom's model, a recovery term which was an explicit function of the time exposed to the test temperature was suggested for the scalar state variable. Aging was discussed as one possible softening mechanism for the current material.     

磁性形状记忆合金力磁耦合行为的唯象模型研究

本文基于热力学及耗散概念,推导了一种磁性形状记忆合金（MSMA）的力磁耦合三维唯象本构关系。采用内变量模拟微观相结构及磁结构的演化,考虑了马氏体相变过程及马氏体重定向过程。同时,本文摒弃了传统的单畴假设,运用双畴模型来模拟循环加载下的磁畴结构演化。根据文献中的二维加载情况的实验数据确定本模型的参数。数值模拟结果表明：本文的模型可以很好的捕捉磁性形状记忆合金的形状记忆效应、应变及磁化响应的滞后性。应变及磁化响应模拟结果明显比文献中的理论模型更加吻合实验数据,尤其是低磁场时尤为明显。     

Shape memory behaviour: modelling within continuum thermomechanics

A phenomenological material model to represent the multiaxial material behaviour of shape memory alloys is proposed. The material model is able to represent the main effects of shape memory alloys: the one-way shape memory effect, the two-way shape memory effect due to external loads, the pseudoelastic and pseudoplastic behaviour as well as the transition range between pseudoelasticity and pseudoplasticity.The material model is based on a free energy function and evolution equations for internal variables. By means of the free energy function, the energy storage during thermomechanical processes is described. Evolution equations for internal variables, e.g. the inelastic strain tensor or the fraction of martensite are formulated to represent the dissipative material behaviour. In order to distinguish between different deformation mechanisms, case distinctions are introduced into the evolution equations. Thermomechanical consistency is ensured in the sense that the constitutive model satisfies the Clausius–Duhem inequality.Finally, some numerical solutions of the constitutive equations for isothermal and non-isothermal strain and stress processes demonstrate that the various phenomena of the material behaviour are well represented. This applies for uniaxial processes and for non-proportional loadings as well.     

A 3D finite strain phenomenological constitutive model for shape memory alloys considering martensite reorientation

Most devices based on shape memory alloys experience both finite deformations and non-proportional loading conditions in engineering applications. This motivates the development of constitutive models considering finite strain as well as martensite variant reorientation. To this end, in the present article, based on the principles of continuum thermodynamics with internal variables, a three-dimensional finite strain phenomenological constitutive model is proposed taking its basis from the recent model in the small strain regime proposed by Panico and Brinson (J Mech Phys Solids 55:2491–2511, 2007). In the finite strain constitutive model derivation, a multiplicative decomposition of the deformation gradient into elastic and inelastic parts, together with an additive decomposition of the inelastic strain rate tensor into transformation and reorientation parts is adopted. Moreover, it is shown that, when linearized, the proposed model reduces exactly to the original small strain model.     

A micromechanical continuum model for the tensile behavior of shape memory metal nanowires

We have previously discovered a novel shape memory effect and pseudoelastic behavior in single-crystalline face-centered-cubic metal (Cu, Ni, and Au) nanowires. Under tensile loading and unloading, these wires can undergo recoverable elongations of up to 50%, well beyond the recoverable strains of 5-8% typical for most bulk shape memory alloys. This phenomenon only exists at the nanoscale and is associated with a reversible lattice reorientation driven by the high surface-stress-induced internal stresses. We present here a micromechanical continuum model for the unique tensile behavior of these nanowires. Based on the first law of thermodynamics, this model decomposes the lattice reorientation process into two parts: a reversible, smooth transition between a series of phase-equilibrium states and a superimposed irreversible, dissipative twin boundary propagation process. The reversible part is modeled within the framework of strain energy functions with multiple local minima. The irreversible, dissipative nature of the twin boundary propagation is due to the ruggedness of strain energy curves associated with dislocation nucleation, glide, and annihilation. The model captures the major characteristics of the unique behavior due to lattice reorientation and accounts for the size and temperature effects, yielding results that are in excellent agreement with the results of molecular dynamics simulations.     

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（0.0005～15 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。     

Numerical simulation of shape memory alloys structures using interior-point methods

This paper is concerned with the numerical simulation of quasi-static evolutions in shape memory alloys structures. In micromechanical models of shape memory alloys, the phase transformation is described by an internal variable representing the volume fractions of the different variants of martensite. In contrast with the framework of standard plasticity, that internal variable is physically constrained to satisfy a set of inequalities at each point. From a structural simulation point of view, handling such local constraints in the evolution problem is not obvious. Robustness and computational time are particularly sensitive issues. In this paper is proposed a new approach which essentially consists in reformulating the non-linear incremental problem as a linear complementarity problem. That approach notably calls for an easy implementation in FEM codes. Comparison with some experimental and numerical results from literature shows the relevancy of the proposed approach.     

Computational modeling of porous shape memory alloys

In this study the mesoscopic behavior of porous shape memory alloys has been simulated with particular attention to the mechanical response under cyclic loading conditions. A recently developed constitutive law, accounting for full martensite reorientation as well as phase transformation, was implemented into the commercial finite element code ABAQUS. Due to stress concentrations in a porous microstructure, the constitutive law was enhanced to account for the development of permanent inelasticity in the shape memory matrix. With this simulation method, the complex interaction between porosity, local phase transformation and macroscale response has been evaluated. The results have implications for use of porous SMAs in biomedical and structural applications.     

Hysteresis in magnetic shape memory composites: Modeling and simulation

Magnetic shape memory alloys are characterized by the coupling between the reorientation of structural variants and the rearrangement of magnetic domains. This permits to control the shape change via an external magnetic field, at least in single crystals. Composite materials with single-crystalline particles embedded in a softer matrix have been proposed as a way to overcome the blocking of the reorientation at grain boundaries.We investigate hysteresis phenomena for small NiMnGa single crystals embedded in a polymer matrix for slowly varying magnetic fields. The evolution of the microstructure is studied within the rate-independent variational framework proposed by Mielke and Theil (1999). The underlying variational model incorporates linearized elasticity, micromagnetism, stray field and a dissipation term proportional to the volume swept by the twin boundary. The time discretization is based on an incremental minimization of the sum of energy and dissipation. A backtracking approach is employed to approximately ensure the global minimality condition.We illustrate and discuss the influence of the particle geometry (volume fraction, shape, arrangement) and the polymer elastic parameters on the observed hysteresis and compare with recent experimental results.     

A three-dimensional model of the thermomechanical behavior of shape memory alloys

A new macro-scale model of shape memory alloys is developed within the framework of generalized standard materials with internal constraints [Moumni, Z., 1995. Sur la modélisation du changement de phase à l’état solide. Ph.D. Thesis, École Nationale Supérieure des Ponts et Chaussées]. It is shown that the introduction of two state variables: the martensite volume fraction and the martensite orientation strain tensor, is sufficient to account for several effects exhibited by SMAs subject to thermomechanical loading, namely: self-accommodation, orientation and reorientation or martensite, as well as superelasticity and one-way shape memory. These phenomena are simulated using the same set of constitutive equations and evolution laws. A simple procedure for identifying the parameters of the model is described in detail and a validation against experimental data is conducted. The model is then used to analyze a 3D SMA structure representing a superelastic stent. Cyclic loading and other pertaining phenomena like training and two-way shape memory are considered in the second part of this paper.     

A thermodynamical constitutive model for shape memory materials. Part I. The monolithic shape memory alloy

Pseudoelasticity and the shape memory effect (SME) due to martensitic transformation and reorientation of polycrystalline shape memory alloy (SMA) materials are modeled using a free energy function and a dissipation potential. Three different cases are considered, based on the number of internal state variables in the free energy: (1) austenite plus a variable number of martensite variants; (2) austenite plus two types of martensite; and (3) austenite and one type of martensite. Each model accounts for three-dimensional simultaneous transformation and reorientation. The single-martensite model was chosen for detailed study because of its simplicity and its ease of experimental verification. Closed form equations are derived for the damping capacity and the actuator efficiency of converting heat into work. The first law of thermodynamics is used to demonstrate that significantly more work is required to complete the adiabatic transformation than the isothermal transformation. Also, as the hardening due to the austenite/martensite misfit stresses approaches zero, the transformation approaches the isothermal, infinite specific heat conditions of a first-order transformation. In a second paper, the single-martensite model is used in a mesomechanical derivation of the constitutive equations of an active composite with an SMA phase.     

A CONSTITUTIVE MODEL FOR TRANSFORMATION, REORIENTATION AND PLASTIC DEFORMATION OF SHAPE MEMORY ALLOYS

A constitutive model is developed for the transformation, reorientation and plastic deformation of shape memory alloys (SMAs). It is based on the concept that an SMA is a mixture composed of austenite and martensite, the volume fraction of each phase is transformable with the change of applied thermal-mechanical loading, and the constitutive behavior of the SMA is the combination of the individual behavior of its two phases. The deformation of the martensite is separated into elastic, thermal, reorientation and plastic parts, and that of the austenite is separated into elastic, thermal and plastic parts. Making use of the Tanaka’s transformation rule modified by taking into account the effect of plastic deformation, the constitutive model of the SMA is obtained. The ferroelasticity, pseudoelasticity and shape memory effect of SMA Au-47.5 at.%Cd, and the pseudoelasticity and shape memory effect as well as plastic deformation and its effect of an NiTi SMA, are analyzed and compared with experimental results.     

A NEW MODEL OF SHAPE MEMORY ALLOYS

A new constitutive model of shape memory alloys ( SMAs) based on Tanaka ' s martensite fraction exponential expression is produced. This new model can present recoverable shape memory strain during different phase transformation, and reflect the action of martensite reorientation . Also it can overcome the defect of Tanaka ' s Model when the SMAs ' microstructure is fully martensite . The model is very simple and suitable for using , and the correct behavior of the model is proved by test.     

Micromechanical prediction of the two-way shape memory effect in shape memory alloy composites

The two-way shape memory effect in monolithic shape memory alloys has been widely investigated both theoretically and experimentally. In the present study, this effect is analyzed for shape memory alloy composites by employing a micromechanical model. To this end, the responses of polymeric matrix and metal matrix unidirectional composites with embedded shape memory alloy fibers are determined. For the polymeric matrix composite, the effect of axial, transverse and shear loadings as well as the fiber volume fraction on the resulting two-way shape memory behavior are studied. The local distributions of stresses among the shape memory alloy fiber and epoxy matrix in the low- and high-temperature shapes of the composite are also investigated. Two training procedures that generate the two-way shape memory effect in the metal matrix composite are offered. The present analysis shows that the two-way shape memory effect in the chosen type of metal matrix composite is not as useful as in the polymeric matrix one. Finally, for a polymeric matrix composite that is subjected to a transverse normal loading, the effect of imperfect bonding between the shape memory alloy fibers and the neighboring matrix is investigated.     

Differential and integrated form consistency in 1-D phenomenological models for shape memory alloy constitutive behavior

Shape memory alloys are being explored increasingly for developing smart structures and devices in aerospace, automotive and other application areas. The material behavior is highly nonlinear with coupled thermomechanical response involving temperature and/or stress induced phase transformations. Modeling the constitutive behavior of these materials poses several challenges and a few phenomenological models exist that provide a quick and reasonable approach to assess their behavior. Due to phenomenological approach, several assumptions are made in order to simplify the model and some of them introduce inconsistencies or anomalies into the model. In this paper, a frequently used approach, namely, Brinson [Brinson, L.C., 1993. One dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable. J. Intell. Mater. Syst. Struct. 4(2), 229–242.] model, is investigated. The constitutive equation is usually expressed at the outset in the differential form and the integrated form of the same is obtained. It is shown that the two forms of equations are not consistent in the Brinson [Brinson, L.C., 1993. One dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable. J. Intell. Mater. Syst. Struct. 4(2), 229–242.] model, given the assumed form of material functions. In the present work, the nature and implications of the inconsistency are highlighted. The cause of incompatibility is the inconsistent material definitions. A modified consistent constitutive model is proposed by redefining the material function which satisfies the compatibility condition. The advantages in using the proposed modified model are highlighted with numerical case studies involving hysteretic stress–strain behavior.     

Phase proportioning in CuAlBe shape memory alloys during thermomechanical loadings using electric resistance variation

The microstructure of shape memory alloys changes with the thermomechanical history of the material. During thermomechanical loading, austenite, thermally-induced martensite or stress-induced martensite can be simultaneously present in the material. In applications integrating SMA parts, utilization conditions seriously affect the microstructure and can generate macroscopic strain or stress. Consequently, during thermomechanical loadings, it is important to be able to proportion the different phases and consequently to understand the kinetic transformation. This is very useful in the development of constitutive equations. This study shows, by a series of tests, that the proposed experimental method, based on the measurement of the variation of electric resistance of CuAlBe wires, permits to determine the volume fraction of the different phases present in the material (i.e., austenite, stress-induced martensite and thermally-induced martensite). The proposed method is applied to the most common thermomechanical behavior met in engineering applications of shape memory alloys: pseudoelasticity, pseudoplasticity, recovery-stress and stress-assisted two-way shape memory effect. The proportioning method based on a mixture law integrating the resistivity of pure phases present in the SMA is first performed on different two-phase mixture cases and then applied to a three phase mixture case.     

Stability and Elastic Properties of the Stress-Free B2 (CsCl-type) Crystal for the Morse Pair Potential Model

Solid-to-solid martensitic phase transformations are responsible for the remarkable behavior of shape memory alloys. There is currently a need for shape memory alloys with improved corrosion, fatigue, and other properties. The development of new accurate models of martensitic phase transformations based on the material’s atomic composition and crystal structure would lead to the ability to computationally discover new improved shape memory alloys. This paper explores the Effective Interaction Potential method for modeling the material behavior of shape memory alloys. In particular, an extensive parameter study of the Morse pair potential model of the stress-free B2 cubic crystal is performed. Results for the stability, potential energy, current unit cell volume, instantaneous bulk modulus, and the two instantaneous cubic shear moduli are presented and discussed. It is found that an Effective Interaction Potential model based on the Morse potential is appropriate for modeling transformations between the B2 cubic structure and the B19 orthorhombic structure, but is not likely to be capable of simulating the B2 cubic to B19′ monoclinic transformation found in the popular shape memory alloy NiTi. In fact, this conclusion may be extended to all types of pair interaction potential models.      

A non-local implicit gradient-enhanced model for unstable behaviors of pseudoelastic shape memory alloys in tensile loading

In this paper, a gradient-enhanced 3-D phenomenological model for shape memory alloys using the non-local theory is developed based on a 1-D constitutive model. The method utilizes a non-local field variable in its constitutive framework with an implicit gradient formulation in order to achieve results independent of the finite element discretization. An efficient numerical approach to implement the non-local gradient-enhanced model in finite element codes is proposed. The model is used to simulate stress drop at the onset of transformation, and its performance is evaluated using different experimental data. The potential of the presented numerical approach for behavior of shape memory alloys in eliminating mesh-dependent simulations is validated by conducting various localization problems. The numerical results show that the developed model can simulate the observed unstable behaviors such as stress drop and deviation of local strain from global strain during nucleation and propagation of martensitic phase.