On the thermomechanical modeling of shape memory alloys

Abstract

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.

Introduction

Shape memory alloy materials (SMAs) are a unique class of metal alloys which can be deformed severely and afterwards recover their original shape after a thermomechanical cycle (shape memory effect), or a stress cycle within some appropriate temperature regimes (pseudoelasticity). The mechanisms of this recovery are either a diffusionless transformation between the austenite phase (which is a highly ordered phase and is also called the parent phase) and the martensite phase (which is a less ordered one) or the reorientation (detwinning) of martensite variants. Detailed exposures to the physics of the subject may be found in Warlimont et al. [1], Duerig et al. [2], Shaw and Kyriakides [3]. In these references at least six different manifestations of the phase transformations are reported; namely pseudoelasticity by transformation and/or reorientation (also called ferroelasticity), shape memory effect by transformation and/or reorientation, two-way shape memory effect and R-phase transformations. This extremely complex behavior along with the increased use of the SMAs in innovative applications in many engineering fields (e.g. [2], [4], [5]) results in a greater need for a better understanding of these materials. Due to recent rapid advances in computer technology, complex constitutive representations can be considered, since their numerical implementation is no longer intractable, no matter how complex they may be.

Recently, several researchers have started to develop constitutive models for SMAs. The approaches used for the mathematical description of the behavior of these materials usually fall in one of the following categories: (a) Constitutive models based on solid state physics, (b) models based on the irreversible thermodynamics of solids and (c) models based on plastic flow theory.

In the area of solid-state physics there are mainly three approaches currently used for the study of martensitic transformations. These are the first-principle electron energy band calculations or (harmonic) phonon energy calculations; the neutron scattering techniques and the non-linear, non-local, elastic field theory (Landau–Ginzburg theory). In the case of the first-principle and phonon calculations, Ho and Harmon [6] and Ye et al. [7], among others, have applied first-principle methods to the first-order transforming materials (first principle means that no adjustable parameters enter into the calculations). The information obtained from this approach can be very important for the development of Landau–Ginzburg models. At present, these calculations are limited by the power of the available computers, which determines the maximum atoms per unit cell which can be studied (30–50 on modern supercomputers).

The neutron scattering techniques are used in order to provide direct information on the nature of the microscopic lattice distortions that produce the lattice correspondence.

The basic idea of the Landau–Ginburg theory is that out of all the complexities of statistical mechanics one can reduce the behavior of a system undergoing a phase transformation to that of a few order parameters (i.e. parameters that give a measure of the transformation development), governed by a free energy function, which depends on temperature, stress and these parameters. Examples of modeling the martensitic transformations through a Landau–Ginzburg theory may be found in Lindgard [8]. Chu and Moran [9] have implemented numerically this approach to model phase transformations in a class of non-linear elastic materials. A very interesting application of the Landau–Ginzburg theory has been proposed by Falk and Konopka [10], who furnish a free energy expression valid for alloys undergoing a cubic-to-monoclinic phase transition (NiTi or Cu-based SMAs).

The above briefly reviewed models are inconvenient for large-scale computations because of the difficulties in establishing the parameters which are entering the expression of the free energy function.

Also, in the realm of the so-called non-equilibrium (or irreversible) thermodynamics, among others, Tanaka and Iwasaki [11], Liang and Rogers [12], Brinson [13], Ivshin and Pence [14], [15] have proposed models based on the use of a set of thermomechanical equations describing the kinetics of the martensitic transformations. The constitutive equations are developed in a non-linear manner on the basis of a free energy driving force and the laws of thermodynamics. Boyd and Lagoudas [16] have extended the thermomechanical approach by Ortin and Planes [17], [18], Patoor et al. [19], Raniecki and Lexcellent [20] and Sun and Hwang [21], [22] in order to account for more general loading conditions such as non-proportional loading and combined isotropic and kinematic hardening. Although these non-linear models seem to be theoretically attractive, they may be more difficult to be dealt with numerically, especially in two- or three-dimensional problems, compared to plasticity-based models. The reason is that a lot of work has been put recently in the return mapping algorithms of plasticity-based models, both in their purely algorithmic as well as in their, mathematical aspects, resulting in the development of robust algorithms, well suited for finite element applications. It should be mentioned though that there is an important difference between the return mapping algorithms of classical plasticity—in which we are dealing with elastic and inelastic domains that are connected—and of non-classical plasticity theories like the one which will be employed here. This issue will be discussed in Remark 3 in Section 4 of this paper. The above-mentioned models in many cases are derived in a one-dimensional framework and then are extended into multi-dimensional models. A major difficulty in extending the one-dimensional into multi-dimensional models derives from the lack of multi-axial experimental data.

Another approach, and a potential remedy to some of those difficulties, is the employment of classical plastic-flow theories (e.g. [5], [23]) which besides having the advantage of relatively easier extension into three-dimensional formulations can provide us with robust algorithms for large-scale computations. It should be clear, because the previous group is called the “irreversible thermodynamics” group of models, that plasticity-based models are also thermodynamically consistent since they follow the general “thermodynamically consistent” plasticity framework. Although these models, by employing a suitably formulated hardening law, are able to simulate some aspects of the experimentally observed behavior, they possess some crucial deficiencies. First, unloading and reloading (up to the unloading point) are considered as elastic, which means that in the case of unloading and reloading—or in general, in the case of cyclic loading—the proposed models will not capture the hysteretic behavior of the SMA material. Moreover, temperature effects are not taken into account. An alternative approach has been proposed by Lubliner and Auricchio [24] (see also [25]) who developed a three-dimensional thermomechanical constitutive model, based on generalized plasticity theory [26], [27]. Generalized plasticity is a general theory of rate-independent inelastic behavior which is physically motivated by loading–unloading irreversibility and is mathematically founded on set theory and topology. The general mathematical foundation provides the theory the ability to deal with “non-standard” cases such as non-connected elastic domains.

The main objective of this work is to extend the previous work by Lubliner and Auricchio [24] providing a general theoretical framework, which in turn may constitute a basis for the derivation of constitutive models which can capture more aspects of the behavior of SMAs with fewer simplifying assumptions than in the existing models. The new framework accommodates stiffness variations, the existence of multiple and possibly interacting loading mechanisms during phase transformations and rate of loading effects. Furthermore, the capabilities of this model are shown not only in qualitative simulations but in simulations of actual experimental data as well. As it was mentioned earlier, the martensite transformation is diffusionless, during which there is no interchange in the position of neighboring atoms but atom movements resulting in changes in the crystal structure [28, pp. 278–279]. The martensite formation has been explained by a shear mechanism and by other researchers by a sequence of two shear mechanisms [28, pp. 279–280]. The shear mechanism can take place either by twinning or by slip depending on the composition and on the thermodynamical conditions [28, p. 280]. Although in Ref. [28] mainly martensite transformation in steel is described, the authors discuss efforts for the development of a general theory of the crystallography of martensite transformation. The crystallographic mechanisms of the martensite formation in NiTi are similar, i.e. slip or twinning, as in the alloys described in the book of Smallman and Bishop [29]. It is therefore micromechanically justified the use of a phenomenological theory from the family of plasticity theories.

This work is organized as follows: In Section 2 the proposed theoretical framework is presented in its most general form and within the concept of the multi-surface plasticity. In Section 3 the presented concepts are applied for a derivation of a three-dimensional thermo-mechanical constitutive model. Finally, in Section 4 the performance of the model is validated by (a) numerical simulations accounting for both monotonic and cyclic loading and (b) by comparing the prediction capabilities of the model against a series of experimental results.