The paper is devoted to constitutive models for shape memory alloys (SMA), in essence, based on SMA model proposed by D. Lagoudas et al. [Int. J. Plast. 32–33, 155–183 (2012; doi:10.1016/j.ijplas.2011.10.009)], but new variables are involved. The proposed model takes into account several thermomecanical mechanisms, which are introduced in terms of specific variables associated, as follows:
transformation-induced plasticity (TRIP) (through the \(\epsilon^{pt}\) variable); two-way shape memory effect (TWSME) (via the internal stress tensor \(\boldsymbol{\beta}\)); plastic yield (using the variable \(\epsilon^{py}\)); tension-compression asymmetry.
The second law of thermodynamics is considered in the Clausius-Duhem form, but in terms of the Gibbs free energy.
In what follows we discuss some aspects related to the constitutive model proposed in this paper.
In the paper within the small strain framework the total strain \(\epsilon\) is decomposed (Formula (2.1)) into \(\epsilon = \mathcal{S}{\sigma} + \boldsymbol{\alpha} (T- T_0) + \epsilon^{t}+ \epsilon^{pt} +\epsilon^{py},\) where \(\epsilon^{t}\) is the transformation strain. The Gibbs free energy is expressed (Formula (2.2)) like in [Lagoudas et al., loc. cit.] in terms of \(G^M\) and \(G^A\), which represent the thermal elastic Gibbs free energy of martensite, and austenite, respectively, and the free energy of mixing, \(G^{\mathrm{mix}},\) namely \(G = \xi^{d} G^M + (1 - \xi^{d})G^A + G^{\mathrm{mix}},\) were \(\xi^{d}\) denotes the martensitic volume fraction. In the paper \(G,\) \(G^M\) and \(G^A\) are written to be dependent on 8 variables, among them 5 variables are those which enter the strain decomposition (2.1), together with \(\boldsymbol{\beta}, \xi^{d},\) and the strain tensor \(\epsilon.\) We remark that the variables \({\sigma}\) and \(\epsilon\) are not independent, due to the relationship (2.1), and thus the presence of \(\epsilon\) is not justified in the presence of \({\sigma}.\) Here \(G^{\mathrm{mix}}\) is function of \(\xi^{d}.\) If we look at Formula (2.3), giving rise to \(G^M\) and \(G^A,\) a new variable, \(g^p\) called yield hardening, appears. The variable \(g^p\) is given by a differential-type equation (2.4), dependent on \(\epsilon.\) Thus the Gibbs free energy function depends on the history of \(\epsilon,\) via the variable \(g^p,\) and not explicit on the current value of \(\epsilon.\) Formula (2.9) gives the time derivative of Gibbs function, but the formula contains the time derivatives of certain functions as well as the differential, as for instance \(dT\) and \(d \sigma.\) The term \(\dot{\epsilon}: \partial_{\epsilon} G\) appears in all Formulae (2.8)–(2.16), and there exists a mention that \(\dot{\epsilon}\) refers to plastic yield, on page 4, and to effective plastic yield strain on page 5. On the other hand the authors state that the Kuhn-Tucker conditions are applied to Formulae (2.15)–(2.16), and as a consequence Formula (2.18) holds. The Kuhn-Tucker conditions are generally formulated within the plasticity to characterize the plastic factor or the so-called plastic multiplier, related to certain plastic yield surface. The authors introduce in an disjointed manner, via Formulae (2.21) and (2.22), together with (2.19) and (2.22), the equation of the transformation and plastic yield surface, for which the numerical algorithms are formulated. The authors state that the material parameters of the model are determined through uniaxial tests, and that the proposed model is verified through the experiments on SMA tubes.