A finite element model for shape memory alloys considering thermomechanical couplings at large strains

In the present work we propose a new thermomechanically coupled material model for shape memory alloys (SMA) which describes two important phenomena typical for the material behaviour of shape memory alloys: pseudoelasticity as well as the shape memory effect. The constitutive equations are derived in the framework of large strains since the martensitic phase transformation involves inelastic deformations up to 8%, or even up to 20% if the plastic deformation after the phase transformation is taken into account. Therefore, we apply a multiplicative split of the deformation gradient into elastic and inelastic parts, the latter concerning the martensitic phase transformation. An extended phase transformation function has been considered to include the tension–compression asymmetry particularly typical for textured SMA samples. In order to apply the concept in the simulation of complex structures, it is implemented into a finite element code. This implementation is based on an innovative integration scheme for the existing evolution equations and a monolithic solution algorithm for the coupled mechanical and thermal fields. The coupling effect is accurately investigated in several numerical examples including pseudoelasticity as well as the free and the suppressed shape memory effect. Finally, the model is used to simulate the shape memory effect in a medical foot staple which interacts with a bone segment.     

Molecular dynamics simulation study of microstructure evolution during cyclic martensitic transformations

Shape memory alloys (SMA) exhibit a number of features which are not easily explained by equilibrium thermodynamics, including hysteresis in the phase transformation and “reverse” shape memory in the high symmetry phase. Processing can change these features: repeated cycling can “train” the reverse shape memory effect, while changing the amount of hysteresis and other functional properties. These effects are likely to be due to formations of localised defects and these can be studied by atomistic methods. Here we present a molecular dynamics simulation study of such behaviour employing a two-dimensional, binary Lennard-Jones model. Our atomistic model exhibits a symmetry breaking, displacive phase transition from a high temperature, entropically stabilised, austenite-like phase to a low temperature martensite-like phase. The simulations show transformations in this model material proceed by non-diffusive nucleation and growth processes and produce distinct microstructures. We observe the generation of persistent lattice defects during forward-and-reverse transformations which serve as nucleation centres in subsequent transformation processes. These defects interfere the temporal and spatial progression of transformations and thereby affect subsequent product morphologies. During cyclic transformations we observe accumulations of lattice defects so as to establish new microstructural elements which represent a memory of the previous morphologies. These new elements are self-organised and they provide a basis of the reversible shape memory effect in the model material.     

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 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.     

Phase transitions and thermodynamics for the shape memory alloy AuZn

A model for shape memory alloys described by a intermediate pattern between a first and a second order phase transition is studied. Moreover, by the thermodynamic compatibility of the model, we provide suitable restrictions on the potentials of the Ginzburg-Landau system. Finally, we present numerical simulations of this shape memory model, which are in good agreement with experimental data.     

Couplage dans les alliages à mémoire de formeCoupling in shape memory alloys.

This Note concerns the study of the micromechanic behavior of shape memory alloys. The advantage of this model permits the coupling between the martensitic transformation and microstructural evolution observed after cycling. The model makes it possible to obtain consecutives equations, which explain at the same time, mechanical properties and the changing structures during the transformation. It provides original physical results on the global behavior of shape memory alloys. To cite this article: A. Alhamany et al., C. R. Mecanique 332 (2004).     

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.      

Magnetic field-induced martensitic phase transformation in magnetic shape memory alloys: Modeling and experiments

In this work, a continuum based model of the magnetic field induced phase transformation (FIPT) for magnetic shape memory alloys (MSMA) is developed. Hysteretic material behaviors are considered through the introduction of internal state variables. A Gibbs free energy is proposed using group invariant theory and the coupled constitutive equations are derived in a thermodynamically consistent way. An experimental procedure of FIPT in NiMnCoIn MSMA single crystals, which can operate under high blocking stress, is described. The model is then reduced to a 1-D form and the material parameter identification from the experimental results is discussed. Model predictions of magneto-thermo-mechanical loading conditions are presented and compared to experiments.     

On modelling phase propagation in SMAs by a Maxwellian thermo-viscoelastic approach

In this paper we propose a Maxwellian thermo-viscoelastic approach to the problem of phase transformation in shape memory alloys. An explicit temperature-dependent non-monotone piecewise linear stress–strain relation is considered and the corresponding free energy function is used to establish the heat propagation equation. The heat exchange between the material and its environment is also taken into account. The numerical simulations of three end-displacement rates lead to serrated hysteresis loops and result in inhomogeneous deformations and exothermic/endothermic behavior during loading/unloading tests. It is shown that the influence of the strain rate on the size and shape of the hysteresis loop is due in fact to the heat exchange between the bar and its environment. The predictions are compared qualitatively with experimental results.     

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     

On non-monotonic rate dependence of stress hysteresis of superelastic shape memory alloy bars

This paper presents a simple thermo-mechanical model to explain and quantify the observed strain-rate dependence of the stress hysteresis of shape memory alloys (SMAs) bars/strips during stress-induced forward/reverse phase transition with latent heat release/absorption. By solving the convective heat transfer equation and employing the temperature dependence of the SMA’s transformation stresses, we are able to prove that the stress hysteresis depends non-monotonically on the applied strain rate with a peak appearing at an intermediate strain rate. We further showed that such a non-monotonic rate dependence is governed by the competition of phase-transition time (or latent-heat release/absorption time) and the time of heat exchange with the environment, and that the hysteresis peak is achieved when the two time scales become comparable. A bell-shaped scaling law of the rate dependence is derived, agreeing quantitatively well with the results of experiments.     

Steady states of austenitic-martensitic domains in the Ginzburg-Landau theory of shape memory alloys

In this paper we investigate mathematically F. Falk's one-dimensional Oinzburg-Landau model for the martensitic phase transitions in shape memory alloys. In particular, we are interested in possible steady state configurations, i.e. we look for distributions for the austenitic and martensitic phases remaining constant in time while the outside temperature is maintained constant, and no body forces, distributed heat sources or boundary stresses are applied.     

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.     

Numerical simulation and experimental investigation of the elastocaloric cooling effect in sputter-deposited TiNiCuCo thin films

The exploitation of the elastocaloric effect in superelastic shape memory alloys (SMA) for cooling applications shows a promising energy efficiency potential but requires a better understanding of the non-homogeneous martensitic phase transformation. Temperature profiles on sputter-deposited superelastic \({\mathrm {Ti_{55.2}Ni_{29.3}Cu_{12.7}Co_{2.8}}}\) shape memory alloy thin films show localized release and absorption of heat during phase transformation induced by tensile deformation with a strong rate dependence. In this paper, a model for the simulation of the thermo-mechanically coupled transformation behavior of superelastic SMA is proposed and its capability to reproduce the mechanical and thermal responses observed during experiments is shown. The procedure for experiment and simulation is designed such that a significant temperature change from the initial temperature is obtained to allow potential cooling applications. The simulation of non-local effects is enabled by the use of a model based on the one-dimensional Müller–Achenbach–Seelecke model, extended by 3D mechanisms such as lateral contraction and by non-local interaction, leading to localization effects. It is implemented into the finite element software COMSOL Multiphysics, and comparisons of numerical and experimental results show that the model is capable of reproducing the localized transformation behavior with the same strain rate dependency. Additionally to the thermal and the mechanical behavior, the quantitative prediction of cooling performance with the presented model is shown.     

超弹性形状记忆合金管单向拉伸试验的数值模拟

NiTi形状记忆合金具有很强的超弹性行为,这种超弹性行为是由于材料在应力作用下发生可逆的马氏体相变所引起。最近Sun和Lee^[4]在NiTi形状记忆合金管单向拉伸试验中观测到,应力诱导马氏体相变具有螺旋带状的形貌特征,本文对此作了数值模拟研究。采用包含应变软化效应的三线性本构关系,建立了NiTi形状记忆合金管的三维有限元模型。通过迭代计算,成功地再现了试验中所观察到的螺旋状相变带从形成到长大的全过程。数值计算结果表明,产生这一独特现象的力学机制,在于NiTi形状记忆合金管在拉伸状态下出现的局部变形失稳极其传播。     

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 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.