On the thermomechanical modeling of shape memory alloys

In this work, a general inelastic framework for the derivation of general three-dimensional thermomechanical constitutive laws for materials undergoing phase transformations is proposed. The proposed framework is based on the generalized plasticity theory and on some basic elements from the theory of continuum damage mechanics. More specifically, a new elaborate formulation of generalized plasticity theory capable of accommodating the multiple and interacting loading mechanisms, which occur during the phase transformations, is developed. Furthermore, the stiffness variations occurring during phase transformations are taken into account by the proposed framework. For this purpose, the free energy is decomposed into elastic and inelastic parts, not in a conventional way, but in one which resembles the elastic-damage cases. Also, a rate-dependent version of the theory is provided. The concepts presented are applied for the derivation of a three-dimensional thermomechanical constitutive model for Shape Memory Alloy materials. Numerical simulations to show qualitatively the ability of the model to capture the behavior of the shape memory alloys are also presented. Furthermore, the model has been fitted to actual experimental results from the literature.     

Computational modeling of size-dependent superelasticity of shape memory alloys

We propose a nonlocal continuum model to describe the size-dependent superelastic responses observed in recent experiments of shape memory alloys. The modeling approach extends a superelasticity formulation based on the martensitic volume fraction, and combines it with gradient plasticity theories. Size effects are incorporated through two internal length scales, an energetic length scale and a dissipative length scale, which correspond to the gradient terms in the free energy and the dissipation, respectively. We also propose a computational framework based on a variational formulation to solve the coupled governing equations resulting from the nonlocal superelastic model. Within this framework, a robust and scalable algorithm is implemented for large scale three-dimensional problems. A numerical study of the grain boundary constraint effect shows that the model is able to capture the size-dependent stress hysteresis and strain hardening during the loading and unloading cycles in polycrystalline SMAs.     

Temperature analysis of one-dimensional NiTi shape memory alloys under different loading rates and boundary conditions

Superelastic polycrystalline NiTi shape memory alloys under tensile loading accompany the strain localization and propagation phenomena. Experiments showed that the number of moving phase fronts and the mechanical behavior are very sensitive to the loading rate due to the release/absorption of latent heat and the material’s inherent temperature sensitivity of the transformation stress. In this paper, the moving heat source method based on the heat diffusion equation is used to study the temperature evolution of one-dimensional superelastic NiTi specimen under different loading rates and boundary conditions with moving heat sources or a uniform heat source. Comparisons of temperature variations with different boundary conditions show that the heat exchange at the boundaries plays a major role in the nonuniform temperature profile that directly relates to the localized deformation. Analytical relation between the front temperature of a single phase front, the inherent Clausius–Clapeyron relation (sensitivity of the material’s transformation stress with temperature), heat transfer boundary conditions and the loading rate is established to analyze the nucleation of new phase fronts. Finally, the rate-dependent stress hysteresis is also simply discussed by using the results of temperature analyses.     

The effect of notches on the fracture behavior in NiTi shape memory alloys

Tensile fracture experiments have been performed on double-notch plate form specimens with different notch types and sizes. Specimen without notch is also studied. The macro-mechanical responses as well as detail examination of the fracture surface have been carried out. The stress, plastic strain and phase transformation fields are analyzed by finite element (FE) simulations using a pseudoelastic constitutive model which considers the permanent plastic deformation. Experimental results show that different type of notches can influence not only the macro-mechanic pseudoelastic but also plastic behaviors of the specimens. Both notch type and notch size affect the mechanism of crack initiation. Notch size influences the specimen behavior in different way for different type of notches. Most of the experimental observations are interpreted properly by the FE results.     

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.     

Recoverable strains in composite shape memory alloys

New upper bounds are proposed for a generic problem of geometric compatibility, which covers the problem of bounding the effective recoverable strains in composite shape memory alloys (SMAs), such as polycrystalline SMAs or rigidly reinforced SMAs. Both the finite deformation and infinitesimal strain frameworks are considered. The methodology employed is a generalization of a homogenization approach introduced by Milton and Serkov [2000. Bounding the current in nonlinear conducting composites. J. Mech. Phys. Solids 48, 1295-1324] for nonlinear composites in infinitesimal strains. Some analytical and numerical examples are given to illustrate the method.     

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.     

Crack growth resistance of shape memory alloys by means of a cohesive zone model

Crack growth resistance of shape memory alloys (SMAs) is dominated by the transformation zone in the vicinity of the crack tip. In this study, the transformation toughening behavior of a slowly propagating crack in an SMA under plane strain conditions and mode I deformation is numerically investigated. A small-scale transformation zone is assumed. A cohesive zone model is implemented to simulate crack growth within a finite element scheme. Resistance curves are obtained for a range of parameters that specify the cohesive traction-separation constitutive law. It is found that the choice of the cohesive strength t₀ has a great influence on the toughening behavior of the material. Moreover, the reversibility of the transformation can significantly reduce the toughening of the alloy. The shape of the initial transformation zone, as well as that of a growing crack is determined. The effect of the Young's moduli ratio of the martensite and austenite phases is examined.     

Variational prediction of the mechanical behavior of shape memory alloys based on thermal experiments

In this work, we present simulations of shape memory alloys which serve as first examples demonstrating the predicting character of energy-based material models. We begin with a theoretical approach for the derivation of the caloric parts of the Helmholtz free energy. Afterwards, experimental results for DSC measurements are presented. Then, we recall a micromechanical model based on the principle of the minimum of the dissipation potential for the simulation of polycrystalline shape memory alloys. The previously determined caloric parts of the Helmholtz free energy close the set of model parameters without the need of parameter fitting. All quantities are derived directly from experiments. Finally, we compare finite element results for tension tests to experimental data and show that the model identified by thermal measurements can predict mechanically induced phase transformations and thus rationalize global material behavior without any further assumptions.     

Thermomechanical coupling in shape memory alloys under cyclic loadings: Experimental analysis and constitutive modeling

In this paper, we examine the influence of thermomechanical coupling on the behavior of superelastic shape memory alloys subjected to cyclic loading at different loading rates. Special focus is given to the determination of the area of the stress-strain hysteresis loop once the material has achieved a stabilized state. It is found that this area does not evolve monotonically with the loading rate for either transient or asymptotic states. In order to reproduce this observation analytically, a new model is developed based on the ZM model for shape memory alloys which was modified to account for thermomechanical coupling. The model is shown to predict the non-monotonic variation in hysteresis area to good accord. Experimentally observed variations in the temperature of SMA test samples are also correctly reproduced for lower strain rates.     

Hysteresis in shape-memory alloys

We present an overview of hysteresis phenomena in the martensitic transformation, and their relevance in the thermomechanical behaviour of shape-memory alloys. The first part of the paper introduces the concept of hysteresis, and the related phenomena of branching, dissipation and memory. The second part deals with revising some aspects of the thermomechanical behaviour of shape-memory alloys, emphasizing the hysteretic behaviour of single crystals and polycrystals under different driving conditions. The last part of the work is dedicated to the problem of modelling hysteresis phenomena in shape-memory alloys. Our focus is on phenomenological approaches which, as shown in the paper, account for the memory properties observed in hysteretic trajectories and open the possibility of deriving a generic energy balance for systems with hysteresis.     

Modeling the dynamic behavior of shape memory alloys

The paper studies the single degree of freedom vibration of a rigid mass suspended by a thin-walled shape memory alloy tube under torsional loading. The behavior is analyzed for the cases of quasiplasticity (low temperatures) and pseudoelasticity (high temperatures) on the basis of an improved version of the Müller–Achenbach model. To illustrate the strong hysteresis-induced damping capacity and the non-linear vibration characteristics, both, free and forced vibrations are considered in the first part of the paper. This is done on the basis of an isothermal version of the model, while the second part of the paper focuses on the effect of non-constant temperature caused by the rate-dependent release and absorption of latent heats.     

A constitutive relation for pseudo-elastic behaviour in shape memory alloys

A constitutive relation to describe pseudo-elastic deformation in shape memory alloys is presented in this paper. It is capable of describing deformation behaviour of polycrystalline materials under triaxial stress state as well as of monocrystalline materials under one-dimensional condition. Total strain rate is supposed to be composed of elastic strain rate and transformation strain rate. Deformation behaviour of Cu−Zn−Sn alloy and Ti−ni alloy is simulated by use of the proposed constitutive relation. it is shown that simulated results are in a good agreement with experimental data. The project supported by National Natural Science Foundation of China.     

On the origin of Lüders-like deformation of NiTi shape memory alloys

Polycrystalline NiTi shape memory alloys deformed in tension tend to exhibit localization and propagation of deformation in macroscopic shear bands. The propagation of the deformation bands is characterized by a plateau-type stress-strain curve. Such behavior has been reported only for NiTi alloys and only in tension. The reason for this behavior is still unclear although different hypotheses have been proposed in the literature. This article briefly summarizes relevant experimental evidences and offers an explanation based on micromechanics modelling of pseudoelasticity of polycrystalline NiTi.     

Micromechanical modelling of the effect of plastic deformation on the mechanical behaviour in pseudoelastic shape memory alloys

Except for the recoverable strain induced by phase transformation, NiTi alloys are very ductile even in the martensite phase. The purpose of the present paper is to study the influence of permanent deformation, which results from plastic deformation of martensite, on the mechanical behaviour of pseudoelastic NiTi alloys. Based on phenomenological theory of martensitic transformation and crystal plasticity, a new three dimensional micromechanical model is proposed by coupling both the slip and twinning deformation mechanisms. The present model is implemented as User MATerial subroutine (UMAT) into ABAQUS/Standard to study the influences of plastic deformation on the stress and strain fields, and on the evolution of martensite transformation. Results show that with the increasing of plastic deformation the residual strain increases and the phase transformation stress–strain curves from the martensite to austenite become steeper and less obvious. Both characteristics, stabilisation of martensite and impedance of the reverse transformation, due to plastic deformation are captured.     

A three-dimensional phenomenological model for martensite reorientation in shape memory alloys

In this work, we propose a macroscopic phenomenological model that is based on the classical framework of thermodynamics of irreversible processes and accounts for the effect of multiaxial stress states and non-proportional loading histories. The model is able to account for the evolution of both twinned and detwinned martensite. Moreover, reorientation of the product phase according to loading direction is specifically accounted for. Towards this purpose the inelastic strain is split into two contributions deriving, respectively, from creation of detwinned martensite and reorientation of previously existing martensite variants. Computational tests demonstrate the ability of the model to simulate the main aspects of the shape memory response in a one-dimensional setting and some of the features that have been experimentally found in the case of multiaxial non-proportional loading histories. Experimental non-proportional loading paths have also been simulated and a good qualitative agreement between numerical and experimental response is observed.     

Three-dimensional constitutive model for shape memory alloys based on microplane model

A new model for the behavior of polycrystalline shape memory alloys (SMA), based on a statically constrained microplane theory, is proposed. The new model can predict three-dimensional response by superposing the effects of inelastic deformations computed on several planes of different orientation, thus reproducing closely the actual physical behavior of the material. Due to the structure of the microplane algorithm, only a one-dimensional constitutive law is necessary on each plane. In this paper, a simple constitutive law and a robust kinetic expression are used as the local constitutive law on the microplane level. The results for SMA response on the macroscale are promising: simple one-dimensional response is easily reproduced, as are more complex features such as stress-strain subloops and tension-compression asymmetry. A key feature of the new model is its ability to accurately represent the deviation from normality exhibited by SMAs under nonproportional loading paths.     

Three dimensional large deformation analysis of phase transformation in shape memory alloys

Shape memory alloys（SMAs）have been explored as smart materials and used as dampers,actuator elements,and smart sensors.An important character of SMAs is its ability to recover all of its large deformations in mechanical loading-unloading cycles without showing permanent deformation.This paper presents a stress-induced phenomenological constitutive equation for SMAs,which can be used to describe the superelastic hysteresis loops and phase transformation between Martensite and Austenite.The Martensite fraction of SMAs is assumed to be dependent on deviatoric stress tensor.Therefore,phase transformation of SMAs is volume preserving during the phase transformation.The model is implemented in large deformation finite element code and cast in the updated Lagrangian scheme.In order to use the Cauchy stress and the linear strain in constitutive laws,a frame indifferent stress objective rate has to be used.In this paper,the Jaumann stress rate is used.Results of the numerical experiments conducted in this study show that the superelastic hysteresis loops arising with the phase transformation can be effectively captured.     

On the Fremond’s constitutive model for shape memory alloys

The remarkable properties of shape memory alloys have increasing the interest in applications in different areas varying from biomedical to aerospace hardware. Despite the large number of applications, the modeling of SMA is the objective of many researches developed in order to describe all details of its thermomechanical behavior. The present contribution revisits a constitutive model presented by Savi et al. (2002), which is built up on the classical Fremond’s model, in order to contemplate the horizontal enlargement of the stress–strain hysteresis loop. Numerical simulations present qualitative agreement with experimental data, showing pseudoelastic, one-way and two-way shape memory effects.