A thermodynamical formulation for the constitutive modeling of a shape memory alloy with two martensite phases

Abstract

This paper presents a thermodynamical formulation for the one-dimensional constitutive model for shape memory alloys (SMAs) proposed by the authors in Marfia and Rizzoni (Eur J Mech A Solids 40:166–185, 2013) and able to describe the pseudo-elastic and shape memory effects and the martensite detwinning. The model takes into account the asymmetric behavior in tension and compression and the different elastic properties of the three phases considered for the SMA material: austenite, tensile and compressive martensite. A new formulation based on two specific energy potentials, the Helmholtz and the Gibbs free energies, is proposed. For the two potentials an expression is given, depending on the martensite volume fractions taken as internal variables, and incorporating a mixing energy of the three phases as proposed in Frémond (C R Acad Sci Paris 304:239–244, 1987). An original analysis of the non dissipative and dissipative processes is carried out in the general framework of tension-compression asymmetry and different elastic properties of the three phases; in particular, in the dissipative case the non-negativity of the dissipation is used to restrict evolutive processes. The numerical procedure developed in Marfia and Rizzoni (Eur J Mech A Solids 40:166–185, 2013) is applied to time integrate the evolutive equations of the internal variables. Applications are carried out in order to verify the effectiveness of the proposed model and to compare the numerical results of the model with the experimental results, available in the literature.

Appendix: stability analysis in the non dissipative case.

In this Appendix, the stability analysis in the non dissipative case is performed by adopting the following procedure: the signs of $Y^+,$ $Y^-$ and $\Delta Y:= Y^+ - Y^-$ are studied as functions of the stress and the temperature, and the stability of the various phases is established according to Table 1. The computation shows that three cases need to be distinguished, and each case is detailed in each one of the following Subsections.

1.1 Appendix 1: Case $T < T_{0}$ and $\alpha ^+, \alpha ^- > 0$

In this case, three further subcases arise depending on the sign of $\Delta \alpha ,$ defined by relation (27). If $\Delta \alpha \,>\, 0,$ then the following results are obtained.

• if $\sigma = 0$ or $\sigma = - \Delta \epsilon / \Delta \alpha ,$ then it follows that $Y^+ = Y^- >0.$ Thus, condition (24) implies that $\xi _S^++ \xi _S^-= 1,$ i.e. the material is composed only of the phases $S^+$ and $S^-,$ and the austenite phase is absent.

• if $\sigma < - \Delta \epsilon / \Delta \alpha$ or $\sigma >0,$ then it follows that $Y^+ > Y^-$ and $Y^+ > 0.$ Thus, (24) implies that $\xi _S^+= 1, \xi _S^-= 0,$ i.e. only phase $S^+$ is present.

• if $- \Delta \epsilon / \Delta \alpha < \sigma < 0,$ then it follows that $Y^- > Y^+$ and $Y^- > 0.$ Thus, (24) implies that $\xi _S^+= 0, \xi _S^-= 1,$ i.e. only phase $S^-$ is present.

If $\Delta \alpha\, < \,0,$ then the following results are obtained.

• if $\sigma = 0$ or $\sigma = - \Delta \epsilon / \Delta \alpha ,$ then it still follows that $Y^+ = Y^- >0,$ and thus $\xi _S^++ \xi _S^-= 1$ (mixture of $S^+$ and $S^-$).

• if $0 < \sigma < - \Delta \epsilon / \Delta \alpha ,$ then it follows that $Y^+ > Y^-$ and $Y^+ > 0.$ Thus (24) implies that $\xi _S^+= 1, \xi _S^-= 0$ (phase $S^+$).

• if $\sigma < 0$ or $\sigma > - \Delta \epsilon / \Delta \alpha ,$ then it follows that $Y^- > Y^+$ and $Y^- > 0.$ Thus, (24) implies that $\xi _S^+= 0, \xi _S^-= 1$ (phase $S^-$).

If $\Delta \alpha = 0,$ then the following results are obtained.

• if $\sigma = 0,$ then it follows that $Y^+ = Y^- >0,$ and thus $\xi _S^++ \xi _S^-= 1$ (mixture of $S^+$ and $S^-$).

• if $\sigma >0,$ then it follows that $Y^+ > Y^-$ and $Y^+ > 0.$ Thus $\xi _S^+= 1, \xi _S^-= 0$ (phase $S^+$).

• if $\sigma < 0,$ then it follows that $Y^- > Y^+$ and $Y^- > 0.$ Thus, $\xi _S^+= 0, \xi _S^-= 1$ (phase $S^-$).

These results are illustrated in Fig. 8 and they give the portions for $T < T_0$ of the phase diagrams shown in Fig. 2.

Fig. 8

Regions of the stress-temperature plane where $Y^{+}, Y^{-}$ are positive or negative, for $T < T_0$ and accordingly to the sign of $\Delta \alpha$ .Dashed lines correspond to points where $Y^+$ or $Y^-$ vanish, bold continuous lines to points where $Y^+ = Y^-.$

1.2 Appendix 2: Case $T \ge T_{0}$ and $\alpha ^+, \alpha ^- > 0$

In this case, the quantities $\sigma _{R}^{+}(T), \sigma _{L}^{+}(T), \sigma _{R}^{-}(T)$ and $\sigma _{L}^{-}(T)$ defined by relations (31–34) play an important role in view of the factorization (30).

The graphs of $\sigma _{R}^{\pm }(T), \sigma _{L}^{\pm }(T)$ as functions of the temperature are depicted with dash lines in the Figs. 9, 10 and 11, which also show the regions of the $(\sigma , T)$ plane where $Y^+$ and $Y^-$ take positive or negative values in the three cases $\Delta \alpha >0,$ $\Delta \alpha <0,$ and $\Delta \alpha =0.$ In the same Figures, dashed lines are taken to indicate the loci where $Y^+$ and $Y^-$ vanish. The sets of points satisfying the conditions $Y^+ > Y^-, Y^+ < Y^-$ and $Y^+ = Y^-$ are also indicated.

In the study of the signs of $Y^+, Y^-$ and $\Delta Y$ the following nine cases occur, depending on the sign of $\Delta \alpha$ and $\epsilon _L^++ \epsilon _L^-.$

If $\Delta \alpha > 0$ and $\epsilon _L^+> -\epsilon _L^-,$ then the following results are obtained.

• If $\sigma > \sigma _{R}^{+}(T)$ or $\sigma < \min _{T > T_0} \{ \sigma _{C}^{-}(T), \sigma _{L}^{+}(T) \},$ then $Y^+ >0$ and $Y^+ > Y^-.$ Thus, (24) implies that $\xi _S^+= 1, \xi _S^-= 0,$ i.e. only phase $S^+$ is present.

• If $T_0 < T < T_L$ and $\sigma _C^-(T) < \sigma < \sigma _{L}^{-}(T),$ then $Y^- >0$ and $Y^- > Y^+;$ thus (24) implies that $\xi _S^+= 0, \xi _S^-= 1,$ i.e. only phase $S^-$ is present.

• If $\max _{T > T_0} \{ \sigma _L^-(T), \sigma _L^+(T) \} < \sigma < \sigma _{R}^{+}(T),$ then $Y^- , Y^+$ $< 0;$ thus (24) implies that $\xi _S^+= 0 = \xi _S^-,$ i.e. only phase $A$ is present.

• If $\sigma = \sigma _C^-(T)$ and $T_0 <T < T_L,$ then $Y^+ = Y^- >0;$ thus (24) implies $\xi _S^++ \xi _S^-= 1,$ i.e. a mixture of the phases $S^+$ and $S^-$ is present.

• If $\sigma = \sigma _{R}^{+}(T)$ or $T > T_L$ and $\sigma = \sigma _{L}^{+}(T),$ then $Y^+ =0$ and $Y^- < 0;$ thus (24) implies $\xi _S^-= 0 ,$ i.e. a mixture of the phases $S^+$ and $A$ is present.

• If $T_0 < T < T_L$ and $\sigma = \sigma _L^-(T),$ then $Y^- =0$ and $Y^+ < 0;$ thus (24) implies $\xi _S^+= 0 ,$ i.e. a mixture of the phases $S^-$ and $A$ is present.

• If $T = T_0$ and $\sigma = 0$ or if $T = T_L$ and $\sigma = \sigma _L^-(T_L) = \sigma _L^+(T_L) = \sigma _C^-(T_L),$ then $Y^+ = 0 = Y^-;$ thus (24) implies that the point representative of the system lies inside the triangle ${\mathcal {T}},$ i.e. the three phases $S^+,$ $S^-$ and $A$ coexist.

If $\Delta \alpha > 0$ and $\epsilon _L^+< -\epsilon _L^-,$ then the following results are obtained.

• If $\sigma < \sigma _{C}^{-}(T)$ or $\sigma > \max _{T > T_0} \{ \sigma _{C}^{+}(T), \sigma _{R}^{+}(T) \},$ then $Y^+ >0$ and $Y^+ > Y^-;$ thus, (24) implies that $\xi _S^+= 1, \xi _S^-= 0,$ i.e. only phase $S^+$ is present.

• If $\sigma _{C}^{-}(T) < \sigma < \sigma _{L}^{-}(T)$ or $T > T_R$ and $\sigma _R^-(T) < \sigma < \sigma _{C}^{+}(T),$ then $Y^- >0$ and $Y^- > Y^+;$ thus (24) implies that $\xi _S^+= 0, \xi _S^-= 1,$ i.e. only phase $S^-$ is present.

• If $\sigma _{L}^{-}(T) < \sigma < \max _{T > T_0} \{ \sigma _R^-(T), \sigma _R^+(T) \} ,$ then $Y^- , Y^+$ $< 0;$ thus (24) implies that $\xi _S^+= 0 = \xi _S^-,$ i.e. only phase $A$ is present.

• If $\sigma = \sigma _{C}^{-}(T)$ or if $T> T_R$ and $\sigma = \sigma _C^+(T),$ then $Y^+ = Y^- >0;$ thus (24) implies $\xi _S^++ \xi _S^-= 1,$ i.e. a mixture of the phases $S^+$ and $S^-$ is present.

• If $T_0 < T < T_R$ and $\sigma = \sigma _{R}^{+}(T),$ then $Y^+ =0$ and $Y^- < 0;$ thus (24) implies $\xi _S^-= 0 ,$ i.e. a mixture of the phases $S^+$ and $A$ is present.

• If $\sigma = \sigma _L^-(T)$ or if $T > T_R$ and $\sigma = \sigma _R^-(T),$ then $Y^- =0$ and $Y^+ < 0,$ thus (24) implies $\xi _S^+= 0 ,$ i.e. a mixture of the phases $S^-$ and $A$ is present.

• If $T = T_0$ and $\sigma = 0$ or if $T = T_R$ and $\sigma = \sigma _R^+(T_R) = \sigma _R^-(T_R) = \sigma _C^+(T_R),$ then $Y^+ = 0 = Y^-;$ thus (24) implies that the point representative of the system lies inside the triangle ${\mathcal {T}},$ i.e. the three phases $S^+,$ $S^-$ and $A$ coexist.

If $\Delta \alpha > 0$ and $\epsilon _L^+= -\epsilon _L^-,$ then the following results are obtained.

• If $\sigma > \sigma _{R}^{+}(T)$ or if $\sigma < \sigma _{C}^{-}(T),$ then $Y^+ >0$ and $Y^+ > Y^-;$ thus, (24) implies that $\xi _S^+= 1, \xi _S^-= 0,$ i.e. only phase $S^+$ is present.

• If $\sigma _{C}^{-}(T) < \sigma < \sigma _{L}^{-}(T),$ then $Y^- >0$ and $Y^- > Y^+;$ thus (24) implies that $\xi _S^+= 0, \xi _S^-= 1,$ i.e. only phase $S^-$ is present.

• If $\sigma _{L}^{-}(T) < \sigma < \sigma _R^+(T),$ then $Y^- , Y^+ < 0;$ thus (24) implies that $\xi _S^+= 0 = \xi _S^-,$ i.e. only phase $A$ is present.

• If $\sigma = \sigma _{C}^{-}(T),$ then $Y^+ = Y^- >0;$ thus (24) implies $\xi _S^++ \xi _S^-= 1,$ i.e. a mixture of the phases $S^+$ and $S^-$ is present.

• If $\sigma = \sigma _{R}^{+}(T),$ then $Y^+ =0$ and $Y^- < 0;$ thus (24) implies $\xi _S^-= 0 ,$ i.e. a mixture of the phases $S^+$ and $A$ is present.

• If $\sigma = \sigma _L^-(T),$ then $Y^- =0$ and $Y^+ < 0,$ thus (24) implies $\xi _S^+= 0 ,$ i.e. a mixture of the phases $S^-$ and $A$ is present.

• If $T = T_0$ and $\sigma = 0,$ then $Y^+ = 0 = Y^-;$ thus (24) implies that the point representative of the system lies inside the triangle ${\mathcal {T}},$ i.e. the three phases $S^+,$ $S^-$ and $A$ coexist.

If $\Delta \alpha\, <\, 0$ and $\epsilon _L^+> -\epsilon _L^-,$ then the following results are obtained.

• If $\sigma _{R}^{+}(T) < \sigma < \sigma _{C}^{-}(T),$ or if $T>T_L$ and $\sigma _{C}^{+}(T) < \sigma < \sigma _{L}^{-}(T),$ then $Y^+ >0$ and $Y^+ > Y^-.$ Thus, (24) implies that $\xi _S^+= 1, \xi _S^-= 0,$ i.e. only phase $S^+$ is present.

• If $\sigma < \max _{T > T_0} \{ \sigma _{L}^{-}(T), \sigma _C^+(T) \},$ then $Y^- >0$ and $Y^- > Y^+;$ thus (24) implies that $\xi _S^+= 0, \xi _S^-= 1,$ i.e. only phase $S^-$ is present.

• If $T_0 < T < T_L$ and $\sigma _{L}^{-}(T) < \sigma < \sigma _{R}^{+}(T),$ or if $T>T_L$ and $\sigma _{L}^{+}(T) < \sigma < \sigma _{R}^{+}(T),$ then $Y^- , Y^+ < 0;$ thus (24) implies that $\xi _S^+= 0 = \xi _S^-,$ i.e. only phase $A$ is present.

• If $\sigma = \sigma _{C}^{-}(T)$ or $T > T_L$ and $\sigma = \sigma _{C}^{+}(T),$ then $Y^+ = Y^- >0,$ thus (24) implies $\xi _S^++ \xi _S^-= 1,$ i.e. a mixture of the phases $S^+$ and $S^-$ is present.

• If $\sigma = \sigma _{R}^{+}(T),$ of if $T > T_L$ and $\sigma = \sigma _{L}^{+}(T),$ then $Y^+ =0$ and $Y^- < 0,$ thus (24) implies $\xi _S^-= 0 ,$ i.e. a mixture of the phases $S^+$ and $A$ is present.

• If $T_0 < T < T_L$ and $\sigma = \sigma _L^-(T),$ then $Y^- =0$ and $Y^+ < 0,$ thus (24) implies $\xi _S^+= 0 ,$ i.e. a mixture of the phases $S^-$ and $A$ is present.

• If $T = T_0$ and $\sigma = 0,$ or if $T = T_L$ and $\sigma = \sigma _L^-(T_L) = \sigma _L^+(T_L) = \sigma _C^+(T_L),$ then $Y^+ = 0 = Y^-$ and therefore (24) implies that the point representative of the system lies inside the triangle ${\mathcal {T}},$ i.e. the three phases $S^+,$ $S^-$ and $A$ coexist.

If $\Delta \alpha < 0$ and $\epsilon _L^+< -\epsilon _L^-,$ then the following results are obtained.

• If $T_0 < T < T_R$ and $\sigma _R^+(T) < \sigma < \sigma _{C}^{-}(T),$ then $Y^+ >0$ and $Y^+ > Y^-.$ Thus, (24) implies that $\xi _S^+= 1, \xi _S^-= 0,$ i.e. only phase $S^+$ is present.

• If $\sigma < \sigma _{L}^{-}(T)$ or $\sigma > \max _{T > T_0} \{ \sigma _{C}^{-}(T), \sigma _{R}^{-}(T) \},$ then $Y^- >0$ and $Y^- > Y^+;$ thus (24) implies that $\xi _S^+= 0, \xi _S^-= 1,$ i.e. only phase $S^-$ is present.

• If $\sigma _{L}^{-}(T) < \sigma < \min _{T > T_0} \{ \sigma _R^-(T), \sigma _R^+(T) \} ,$ then $Y^- , Y^+$ $< 0;$ thus (24) implies that $\xi _S^+= 0 = \xi _S^-,$ i.e. only phase $A$ is present.

• If $T_0 \le T \le T_R$ and $\sigma = \sigma _C^-(T),$ then $Y^+ = Y^- >0,$ thus (24) implies $\xi _S^++ \xi _S^-= 1,$ i.e. a mixture of the phases $S^+$ and $S^-$ is present.

• If $T_0 \le T \le T_R$ and $\sigma = \sigma _{R}^{+}(T),$ then $Y^+ =0$ and $Y^- < 0,$ thus (24) implies $\xi _S^-= 0 ,$ i.e. a mixture of the phases $S^+$ and $A$ is present.

• If $\sigma = \sigma _L^-(T),$ or if $T> T_R$ and $\sigma = \sigma _R^-(T),$ then $Y^- =0$ and $Y^+ < 0,$ thus (24) implies $\xi _S^+= 0 ,$ i.e. a mixture of the phases $S^-$ and $A$ is present.

• If $T = T_0$ and $\sigma = 0,$ or if $T = T_R$ and $\sigma = \sigma _R^-(T_R) = \sigma _R^+(T_R) = \sigma _C^-(T_R),$ then $Y^+ = 0 = Y^-$ and therefore (24) implies that the point representative of the system lies inside the triangle ${\mathcal {T}},$ i.e. the three phases $S^+,$ $S^-$ and $A$ coexist.

If $\Delta \alpha < 0$ and $\epsilon _L^+= -\epsilon _L^-,$ then the following results are obtained.

• If $\sigma _{R}^{+}(T) < \sigma < \sigma _{C}^{-}(T),$ then $Y^+ >0$ and $Y^+ > Y^-.$ Thus, (24) implies that $\xi _S^+= 1, \xi _S^-= 0,$ i.e. only phase $S^+$ is present.

• If $\sigma > \sigma _C^-(T)$ or if $\sigma < \sigma _{L}^{-}(T),$ then $Y^- >0$ and $Y^- > Y^+;$ thus (24) implies that $\xi _S^+= 0, \xi _S^-= 1,$ i.e. only phase $S^-$ is present.

• If $\sigma _L^-(T) < \sigma < \sigma _{R}^{+}(T),$ then $Y^- , Y^+ < 0;$ thus (24) implies that $\xi _S^+= 0 = \xi _S^-,$ i.e. only phase $A$ is present.

• If $\sigma = \sigma _C^-(T),$ then $Y^+ = Y^- >0,$ thus (24) implies $\xi _S^++ \xi _S^-= 1,$ i.e. a mixture of the phases $S^+$ and $S^-$ is present.

• If $\sigma = \sigma _{R}^{+}(T),$ then $Y^+ =0$ and $Y^- < 0,$ thus (24) implies $\xi _S^-= 0 ,$ i.e. a mixture of the phases $S^+$ and $A$ is present.

• If $\sigma = \sigma _L^-(T),$ then $Y^- =0$ and $Y^+ < 0,$ thus (24) implies $\xi _S^+= 0 ,$ i.e. a mixture of the phases $S^-$ and $A$ is present.

• If $T = T_0$ and $\sigma = 0,$ then $Y^+ = 0 = Y^-$ and therefore (24) implies that the point representative of the system lies inside the triangle ${\mathcal {T}},$ i.e. the three phases $S^+,$ $S^-$ and $A$ coexist.

If $\Delta \alpha = 0$ and $\epsilon _L^+> -\epsilon _L^-,$ then the following results are obtained.

• If $\sigma > \sigma _{R}^{+}(T)$ or if $T > T_L$ and $\sigma _C(T) < \sigma < \sigma _{L}^{+}(T),$ then $Y^+ >0$ and $Y^+ > Y^-.$ Thus, (24) implies that $\xi _S^+= 1, \xi _S^-= 0,$ i.e. only phase $S^+$ is present.

• If $\sigma < \min _{T>T_0} \{ \sigma _C(T) , \sigma _{L}^{-}(T)\},$ then $Y^- >0$ and $Y^- > Y^+;$ thus (24) implies that $\xi _S^+= 0, \xi _S^-= 1,$ i.e. only phase $S^-$ is present.

• If $\max _{T > T_0} \{ \sigma _L^-(T), \sigma _L^+(T) \} < \sigma < \sigma _{R}^{+}(T),$ then $Y^- , Y^+$ $< 0;$ thus (24) implies that $\xi _S^+= 0 = \xi _S^-,$ i.e. only phase $A$ is present.

• If $T > T_L$ and $\sigma = \sigma _C(T),$ then $Y^+ = Y^- >0,$ thus (24) implies $\xi _S^++ \xi _S^-= 1,$ i.e. a mixture of the phases $S^+$ and $S^-$ is present.

• If $\sigma = \sigma _{R}^{+}(T)$ or if $T > T_L$ and $\sigma = \sigma _{L}^{+}(T),$ then $Y^+ =0$ and $Y^- < 0,$ thus (24) implies $\xi _S^-= 0 ,$ i.e. a mixture of the phases $S^+$ and $A$ is present.

• If $T_0 < T < T_L$ and $\sigma = \sigma _L^-(T),$ then $Y^- =0$ and $Y^+ < 0,$ thus (24) implies $\xi _S^+= 0 ,$ i.e. a mixture of the phases $S^-$ and $A$ is present.

• If $T = T_0$ and $\sigma = 0,$ or if $T = T_L$ and $\sigma = \sigma _L^-(T_L) = \sigma _L^+(T_L)=\sigma _C (T_L),$ then $Y^+ = 0 = Y^-$ and therefore (24) implies that the point representative of the system lies inside the triangle ${\mathcal {T}},$ i.e. the three phases $S^+,$ $S^-$ and $A$ coexist.

If $\Delta \alpha = 0$ and $\epsilon _L^+< -\epsilon _L^-,$ then the following results are obtained.

• If $\sigma > \max _{T>T_0} \{ \sigma _{R}^{+}(T). \sigma _C(T) \},$ then $Y^+ >0$ and $Y^+ > Y^-.$ Thus, (24) implies that $\xi _S^+= 1, \xi _S^-= 0,$ i.e. only phase $S^+$ is present.

• If $T > T_R$ and $\sigma _R^-(T) < \sigma < \sigma _{C}(T),$ or if $\sigma < \sigma _L^-(T),$ then $Y^- >0$ and $Y^- > Y^+;$ thus (24) implies that $\xi _S^+= 0, \xi _S^-= 1,$ i.e. only phase $S^-$ is present.

• If $\sigma _{L}^{-}(T) < \sigma < \min _{T > T_0} \{ \sigma _R^-(T), \sigma _R^+(T) \} ,$ then $Y^- , Y^+$ $< 0;$ thus (24) implies that $\xi _S^+= 0 = \xi _S^-,$ i.e. only phase $A$ is present.

• If $T > T_R$ and $\sigma = \sigma _C(T),$ then $Y^+ = Y^- >0,$ thus (24) implies $\xi _S^++ \xi _S^-= 1,$ i.e. a mixture of the phases $S^+$ and $S^-$ is present.

• If $T_0 < T < T_R$ and $\sigma = \sigma _{R}^{+}(T),$ then $Y^+ =0$ and $Y^- < 0,$ thus (24) implies $\xi _S^-= 0 ,$ i.e. a mixture of the phases $S^+$ and $A$ is present.

• If $T > T_R$ and $\sigma = \sigma _R^-(T),$ or if $\sigma = \sigma _L^-(T),$ then $Y^- =0$ and $Y^+ < 0,$ thus (24) implies $\xi _S^+= 0 ,$ i.e. a mixture of the phases $S^-$ and $A$ is present.

• If $T = T_0$ and $\sigma = 0,$ or if $T = T_R$ and $\sigma = \sigma _R^-(T_R) = \sigma _R^+(T_R) = \sigma _C(T_R),$ then $Y^+ = 0 = Y^-$ and therefore (24) implies that the point representative of the system lies inside the triangle ${\mathcal {T}},$ i.e. the three phases $S^+,$ $S^-$ and $A$ coexist.

If $\Delta \alpha = 0$ and $\epsilon _L^+= -\epsilon _L^-,$ then the following results are obtained.

• If $\sigma > \sigma _{R}^{+}(T),$ then $Y^+ >0$ and $Y^+ > Y^-.$ Thus, (24) implies that $\xi _S^+= 1, \xi _S^-= 0,$ i.e. only phase $S^+$ is present.

• If $\sigma < \sigma _{L}^{-}(T),$ then $Y^- >0$ and $Y^- > Y^+;$ thus (24) implies that $\xi _S^+= 0, \xi _S^-= 1,$ i.e. only phase $S^-$ is present.

• If $\sigma _{L}^{-}(T) < \sigma < \sigma _{R}^{+}(T),$ then $Y^- , Y^+ < 0;$ thus (24) implies that $\xi _S^+= 0 = \xi _S^-,$ i.e. only phase $A$ is present.

• If $\sigma = \sigma _{R}^{+}(T),$ then $Y^+ =0$ and $Y^- < 0,$ thus (24) implies $\xi _S^-= 0 ,$ i.e. a mixture of the phases $S^+$ and $A$ is present.

• If $\sigma = \sigma _L^-(T),$ then $Y^- =0$ and $Y^+ < 0,$ thus (24) implies $\xi _S^+= 0 ,$ i.e. a mixture of the phases $S^-$ and $A$ is present.

• If $T = T_0$ and $\sigma = 0,$ then $Y^+ = 0 = Y^-$ and therefore (24) implies that the point representative of the system lies inside the triangle ${\mathcal {T}},$ i.e. the three phases $S^+,$ $S^-$ and $A$ coexist.

Fig. 9

Regions of the stress-temperature plane where $Y^{+}, Y^{-}$ take positive or negative values for $T \ge T_0, \; \Delta \alpha >0$ and accordingly to the sign of $\Delta \epsilon .$ The dashed lines correspond to points where $Y^+$ or $Y^-$ vanish, bold continuous curves are loci where $Y^+ = Y^-.$

Fig. 10

Regions of the stress-temperature plane where $Y^{+}, Y^{-}$ take positive or negative values for $T \ge T_0, \; \Delta \alpha <0$ and accordingly to the sign of $\Delta \epsilon .$ The dashed lines correspond to points where $Y^+$ or $Y^-$ vanish, bold continuous curves are loci where $Y^+ = Y^-.$

Fig. 11

Regions of the stress-temperature plane where $Y^{+}, Y^{-}$ take positive or negative values for $T \ge T_0, \; \Delta \alpha =0$ and accordingly to the sign of $\Delta \epsilon .$ The dashed lines correspond to points where $Y^+$ or $Y^-$ vanish, while at the points on the bold continuous curve the condition $Y^+ = Y^-$ is satisfied

1.3 Appendix 3: Case $\alpha ^+, \alpha ^- = 0$

In this case $Y^+$ and $Y^-$ take the special forms (52), (53). Figure 12a shows the regions of the $(\sigma , T)$ plane where $Y^+$ and $Y^-$ take now positive or negative values. The sets of points where $Y^+$ and $Y^-$ vanish are also shown, together with those satisfying the conditions $Y^+ > Y^-, Y^+ < Y^-$ and $Y^+ = Y^-.$ In particular, the following results are obtained.

• If $T< T_0$ and $\sigma >0,$ or if $T> T_0$ and $\sigma > \sigma _{R}^{+}(T),$ then $Y^+ >0$ and $Y^+ > Y^-.$ Thus, (24) implies that $\xi _S^+= 1, \xi _S^-= 0,$ i.e. only phase $S^+$ is present.

• If $T< T_0$ and $\sigma <0,$ or if $T> T_0$ and $\sigma < \sigma _{L}^{-}(T),$ then $Y^- >0$ and $Y^- > Y^+;$ thus (24) implies that $\xi _S^+= 0, \xi _S^-= 1,$ i.e. only phase $S^-$ is present.

• If $\sigma _{L}^{-}(T) < \sigma < \sigma _{R}^{+}(T),$ then $Y^- , Y^+ < 0;$ thus (24) implies that $\xi _S^+= 0 = \xi _S^-,$ i.e. only phase $A$ is present.

• If $T> T_0$ and $\sigma = \sigma _{R}^{+}(T),$ then $Y^+ =0$ and $Y^- < 0,$ thus (24) implies $\xi _S^-= 0 ,$ i.e. a mixture of the phases $S^+$ and $A$ is present.

• If $T> T_0$ and $\sigma = \sigma _L^-(T),$ then $Y^- =0$ and $Y^+ < 0,$ thus (24) implies $\xi _S^+= 0 ,$ i.e. a mixture of the phases $S^-$ and $A$ is present.

• If $T = T_0$ and $\sigma = 0,$ then $Y^+ = 0 = Y^-$ and therefore (24) implies that the point representative of the system lies inside the triangle ${\mathcal {T}},$ i.e. the three phases $S^+,$ $S^-$ and $A$ coexist.

The corresponding non dissipative phase diagram is represented in Fig. 12b.

Fig. 12

a Regions of the stress-temperature plane for $T \ge T_0$ where $Y^{+}, Y^{-}$ take positive or negative values in the special case of equal elastic moduli of the three phases ($\alpha ^+, \alpha ^- = 0$) and b corresponding non dissipative phase diagram