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

Abstract

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.

Introduction

Shape memory alloys (SMAs) are metallic alloys that can undergo martensitic phase transformations as a result of applied thermomechanical loads and are capable of recovering apparently permanent strains when heated above a certain temperature. At high temperatures the crystal lattice is in a high symmetry, parent austenitic phase, while at lower temperatures a low symmetry martensitic phase appears (Otsuka and Wayman, 1999, Otsuka and Ren, 2005). The key characteristic of all SMAs is the occurrence of a martensitic phase transformation from the austenitic phase to the different variants of the low temperature, low symmetry martensitic phase. The martensitic transformation is a shear-dominant diffusionless solid-state phase transformation occurring by nucleation and growth of the martensitic phase from the parent austenitic phase (Olson and Cohen, 1982). What make SMAs remarkably different from other materials are primarily the shape memory effect (SME) and pseudoelasticity, which are associated with the specific way the phase transformation occurs.

When a shape memory alloy undergoes a martensitic phase transformation, it transforms from the parent phase to one or more of the different variants of the martensitic phase (Otsuka and Wayman, 1999). In the absence of applied stresses, the variants of the martensitic phase usually arrange themselves in a self-accommodating manner through twinning, resulting in no observable macroscopic shape change. By applying mechanical loading the martensitic variants are forced to reorient (detwin) into a single variant leading to large macroscopic inelastic strains. After heating above a certain temperature, the martensitic phase returns to the austenitic phase, and the inelastic strains are recovered. This behavior is known as the SME (Otsuka and Wayman, 1999). Pseudoelasticity is observed when the martensitic phase transformation is induced by applied thermomechanical loading of the austenitic phase in which case detwinned martensite is directly produced from austenite. The process is again associated with large inelastic (transformation) strains which are recovered upon unloading due to the reverse phase transformation (Wayman, 1983, Otsuka and Wayman, 1999). The extensive list of alloys exhibiting SME and pseudoelasticity includes the Ni–Ti alloys, and many copper-, iron-, silver- and gold-based alloys (Nishiyama, 1978).

During the last two decades the area of constitutive modeling of polycrystalline SMAs has been a topic of many research publications and significant advancements have been reported. One major class of SMA constitutive models is the phenomenological one, which relies on continuum thermodynamics with internal state variables to account for the changes in the microstructure due to phase transformation (Tanaka et al., 1992, Patoor et al., 1988, Ortin and Planes, 1988, Ortin and Planes, 1989, Berveiller et al., 1991, Liang and Rogers, 1992, Sun et al., 1991, Sun and Hwang, 1993a, Sun and Hwang, 1993b, Graesser and Cozzarelli, 1991, Brinson, 1993, Raniecki and Lexcellent, 1994, Lagoudas et al., 1996, Marketz et al., 1995, Leclercq and Lexcellent, 1996, Juhasz et al., 2002, Bo and Lagoudas, 1999a, Bo and Lagoudas, 1999b, Bo and Lagoudas, 1999c, Lagoudas and Bo, 1999, Lexcellent et al., 2000, Lagoudas and Entchev, 2004, Auricchio et al., 2007). These type of models usually assume a macroscopic energy function that depends on state and internal variables used to describe the degree of phase transformation. Evolution equations are then postulated for the internal variables. Most phenomenological constitutive models adopt such a thermodynamic structure and select the martensitic volume fraction as an internal state variable to account, on the average, for the influence of the microstructure.

The early constitutive models (Tanaka, 1986, Tanaka et al., 1986, Tanaka et al., 1995, Sato and Tanaka, 1988, Liang and Rogers, 1990, Liang and Rogers, 1992, Brinson, 1993, Boyd and Lagoudas, 1994, Boyd and Lagoudas, 1996a, Boyd and Lagoudas, 1996b) have been used to derive the pseudoelastic response of SMAs and their main focus is the hardening function selected to model the stress–strain response during the stress-induced martensitic phase transformation. A unified framework for these early constitutive models has been presented by Lagoudas et al. (1996). Further improvements in the accuracy of SMAs models were achieved by Raniecki and Lexcellent, 1998, Qidwai and Lagoudas, 2000b, Lexcellent et al., 2002, who proposed different transformation functions in order to capture the asymmetric response that SMAs exhibit in tension and compression. Qidwai and Lagoudas (2000b) also studied the implications of the principle of maximum dissipation during phase transformation on the transformation surfaces and evolution equations for the martensitic volume fraction. These models are suitable for stress-induced martensitic transformations or for SMAs which have already been trained to exhibit the two-way shape memory effect (TWSME) (for details on the TWSME, see Otsuka and Wayman, 1999). Modifications, such as making the total amount of transformation strain dependent on the stress, have been introduced by some authors (Bo and Lagoudas, 1999a, Bo and Lagoudas, 1999b, Bo and Lagoudas, 1999c, Lagoudas and Bo, 1999, Lagoudas and Entchev, 2004) in order to account for certain aspects of the SME. These changes however do not make an explicit distinction between twinned and detwinned martensite and thus are only suitable for SME related thermomechanical loading paths.

The analysis of the existing models and their comparison to the experimental results has shown that current SMA constitutive models which take into account the development of stress-induced martensite have reached a high level of sophistication. However, such models generally lack the ability to handle other loading paths involving detwinning and reorientation of martensite in conjunction with the pseudoelastic response. Therefore, there is need for a 3D constitutive model that can accurately capture not only the material response during pseudoelastic and SME loading paths, but also loading paths that involve co-existence of all the three material phases – austenite, twinned (self-accommodated) martensite and detwinned martensite. Such a model should also be implemented numerically and tested on a comprehensive set of model problems. This will allow to perform numerical simulations of problems of varying engineering difficulty, such as the actuation of SMA micro-grips (Kohl et al., 2002), the cooling/heating cycles in the manufacturing and deployment of biomedical devices (Jung et al., 2004), temperature actuated flow regulating devices (Popov, 2005) and fuel powered SMA actuators (Jun et al., 2007), to name a few.

The early attempts to combine the pseudoelastic material response with detwinning of martensite were done in one dimension by Brinson, 1993, Brinson and Lammering, 1993, Boyd and Lagoudas, 1993. These models used two internal state variables to model pseudoelasticity and detwinning. In addition, Brinson (1993) used a uniaxial phase diagram in stress–temperature space which conveniently defines the thermodynamically stable domains for the three phases and the possible transformations between them. The work was further refined by Bekker and Brinson, 1997, Bekker and Brinson, 1998 who incorporated minor loops for the pseudoelastic transformation. However, this basic phase diagram does not account for certain loading paths, especially those that traverse the regions where the three phases can co-exist.

Three-dimensional thermodynamics based models of combined detwinning and pseudoelasticity have been proposed by Leclercq and Lexcellent, 1996, Lagoudas and Shu, 1999, Juhasz et al., 2002. The models of Leclercq and Lexcellent, 1996, Lagoudas and Shu, 1999 used two scalar volume fractions for twinned and detwinned martensite. While formulated in 3D, they were implemented and tested only on 1D examples. Furthermore, complex loading paths which involve a mixture of the three phases were also not tested. The model of Juhasz et al. (2002) used the entire transformation strain as a tensorial internal variable instead of the volume fraction of detwinned martensite. All three models used phase diagrams which were based on the work of Brinson (1993). While attempts were made to overcome some of its basic limitations, the current work attempts to present an extended phase diagram which refines existing concepts and also incorporates new experimental results.

In this paper, a three-dimensional, thermodynamics based model with three internal variables is formulated for the simultaneous modeling of pseudoelasticity and detwinning of self-accommodated martensite in polycrystalline SMAs. The model is consistent with an extended uniaxial phase diagram. The novel characteristics of this model are: (i) integration into the phase diagram of new experimental results which demonstrate that twinned and detwinned martensite transform to austenite at different temperatures; (ii) refinement of the phase diagram with respect to loading paths that involve a mixture of the three phases; (iii) the use of three independent internal variables (in contrast to the usual two, typically used in this class of models) which provides a new approach to modelling the training of SMA materials and the associated evolution of the phase diagram; (iv) numerical implementation which tests complex loading paths, including ones that involve a mixture of the three phases.

The paper begins with experimental results which demonstrate that, at zero stress, twinned and detwinned martensite transform to austenite at different temperatures (Section 2). The phase diagram is constructed in Section 3 based on these observations, as well as a careful reexamination of published experimental data on detwinning of twinned martensite and the conversion of twinned martensite to austenite. The 3-D constitutive model is presented in Section 4. A discussion of how to identify the material parameters used in the model from experimentally observable quantities is given in Section 4.5. Finally, the numerical implementation of the model into a displacement based finite element method (FEM) code is presented in Section 5 and numerical examples are given in Section 6.

To simplify the presentation, throughout this paper the three phases are denoted by A, M^t and M^d for austenite, twinned martensite and detwinned martensite, respectively. The five possible phase transformations are denoted by $A\to M^{\text{t}}$, $A\to M^{\text{d}}$, $M^{\text{t}}\to A$, $M^{\text{d}}\to A$ and $M^{\text{t}}\to M^{\text{d}}$ for austenite to twinned martensite, austenite to detwinned martensite, twinned martensite to austenite, detwinned martensite to austenite and twinned to detwinned martensite, respectively. The detwinning of twinned martensite $M^{\text{t}}\to M^{\text{d}}$ does not involve phase transformation and is, in fact, an inelastic deformation process of reorientation of martensitic variants (cf. e.g. Otsuka and Wayman, 1999). For the sake of simplicity, the collective term transformations is applied to it whenever the distinction is not important. Note also that the transformation $M^{\text{d}}\to M^{\text{t}}$ from detwinned to twinned martensite (the so-called rubber-like effect, cf. Otsuka and Wayman, 1999) is not thermodynamically stable and it is not considered. Finally, the critical start and finish transformation temperatures at zero stress level (cf. Fig. 3, Fig. 4) are denoted as follows: M_s and M_f for the $A\to M^{\text{t}}$ transformation, $A_{\text{s}}^{\text{t}}$ and $A_{\text{f}}^{\text{t}}$ for the $M^{\text{t}}\to A$ transformation and $A_{\text{s}}^{\text{d}}$ and $A_{\text{f}}^{\text{d}}$ for the $M^{\text{d}}\to A$ transformation. The clarification that these temperatures are at zero stress level will be omitted, and only the term transformation temperatures will be used.