Shape memory behaviour: modelling within continuum thermomechanics

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Abstract

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.

Introduction

Shape memory alloys exhibit a thermomechanical behaviour which cannot be observed in other materials (see e.g. Funakubo, 1987; Otsuka and Wayman, 1998): pseudoelasticity, pseudoplasticity as well as one- and two-way shape memory effects. These effects are based on two basic deformation mechanisms occurring in the microstructure, namely the stress- and temperature-induced martensitic (diffusionless) phase transitions and the orientation of the martensite twins. Both deformation mechanisms are connected with temperature changes and dissipation. However, the thermomechanical coupling effects during the martensitic phase transitions are much more pronounced in comparison to the case of the orientation of the martensite twins (see e.g. Shaw and Kyriakides, 1995; Helm and Haupt, 2001).

Due to the interesting material properties of shape memory alloys, the development of exceptional new products is possible. To support the engineering applications, a number of material models were developed. Some of these (see e.g. Bertram, 1982; Graesser and Cozzarelli, 1994) are not formulated in the context of thermomechanics, although the material behaviour is strongly temperature-dependent. Other theories are confined to the one-dimensional material behaviour (see e.g. Falk, 1983; Liang and Rogers, 1990), or they are limited to represent pseudoelasticity only (see e.g. Raniecki et al., 1992; Auricchio and Taylor, 1997). Mostly, rate-dependent effects are disregarded (see e.g. Boyd and Lagoudas, 1996).

In the present paper, a thermomechanically consistent material model is proposed to represent the three-dimensional material behaviour of shape memory alloys. This includes the one-way shape memory effect, the two-way shape memory effect due to external loads as well as the pseudoelastic and the pseudoplastic material behaviour. The transition range between pseudoelasticity and pseudoplasticity is also represented. Furthermore, the constitutive model is able to depict viscous characteristics of the material. As it is outlined in Section 2, the model is based on a free energy function depending on the temperature and internal variables. In order to model the energy storage during the martensitic phase transition, the free energy function is formulated according to a mixture theory. For the description of the history-dependence, evolution equations for the internal variables are set up. One internal variable is the fraction of martensite; further internal variables are the inelastic strain tensor and a strain-like variable describing internal stress fields (residual stresses). The viscous behaviour of shape memory materials, which is established experimentally (see Helm and Haupt (2001) for NiTi alloys) during the martensitic phase transition and the orientation of the martensite twins, is modelled by means of an inelastic multiplier of Perzyna-type (cf. Perzyna, 1963). Of course, it is difficult to observe the viscous phenomena in the case of pseudoelasticity because the effects of viscosity interact with the thermomechanical coupling phenomena. However, in the case of pseudoplasticity distinct rate-dependent effects like stress relaxation as well as creep occur. Thermomechanical coupling phenomena play only a minor role. In this case, phenomena of viscosity can be separately identified.

The proposed model describes the material behaviour of shape memory alloys under the assumption of small deformations. In addition, the elastic part of the material behaviour is assumed to be isotropic. This corresponds to a polycrystalline material structure with a large number of grains. As a consequence, effects of texture cannot be represented within the present constitutive theory.

Shape memory alloys show strong thermomechanical coupling phenomena. Consequently, it is necessary that the material model complies with the 2nd law of thermodynamic. Here, we prove the thermomechanical consistency in the sense of the Clausius–Duhem inequality, which is a special formulation of the 2nd Law of thermodynamics. In Section 3 some numerical solutions of the constitutive model in the context of isothermal and non-isothermal strain and stress processes are presented. The numerical simulations demonstrate that the developed theory is able to represent the exceptional behaviour of these so-called smart materials.

Section snippets

Phenomenological material model

In this section the constitutive model is formulated in the framework of phenomenological continuum thermomechanics. First, the basic structure of the material model is developed on the basis of a free energy function and observing the Clausius–Duhem inequality. Thereafter, the free energy and the evolution equations for the internal variables are set up in detail. Finally, the thermomechanical consistency of the complete material model is proven, and the thermomechanical coupling phenomena are

Numerical studies

In the remainder part of the article numerical simulations of uniaxial as well as biaxial stress–strain processes under strain-, stress-, and temperature-control are outlined. The material model, a system of ordinary differential equations, is solved numerically by means of an explicit Runge–Kutta method. In order to compare the predictions of the model with the experimental data presented in Helm (2001) and Helm and Haupt (2001), the initial-value problem is solved for a rod under simple

Conclusions

In this article, a phenomenological material model to represent the multiaxial shape memory behaviour is proposed. The material model is based on a free energy function and evolution equations for internal variables. The internal variables have a definite physical meaning in relation to processes occurring in the microstructure. Furthermore, all material parameters are regarded as temperature-dependent. In particular, the evolution equations for the internal stresses allow us to represent the

Acknowledgements

We gratefully acknowledge the support of this work by the Deutsche Forschungsgemeinschaft (DFG).

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