## 7 Concluding Remarks  
In the preceding sections, a new thermo-coupled elastoplastic \(J_{2}\)-flow models with evolving elastic behavior has been proposed to achieve an explicit, direct simulation of complicated inelastic behavior of SMPs displaying both the stress softening effect at unloading and the strain recovery effect at heating. The constitutive quantities incorporated in the proposed model are explicitly determined from suitable uniaxial data. Extensive test data for any given shapes of loading and unloading curves as well as the strain-temperature curve may be accurately, automatically fitted by means of single-variable shape functions. It has been demonstrated that all may be done in the scope of classical elastoplasticity and, therefore, that no additional variables need be introduced and treated. Usual implicit, complicated procedures both in searching for suitable forms of constitutive functions and in identifying values of unknown parameters may be bypassed. This study represents a development of the model suggested in a most recent study [40]. Results in this reference are derived for certain usual deformation features of polymeric materials. However, new treatments need further be introduced for the purpose of simulating thermo-mechanical deformation features unique to SMPs. In fact, SMPs exhibit thermal effects and loading-unloading response features that differ in nature from those for usual polymers. As a consequence, the shape functions representing the loading curve and each unloading curve (cf., Figure 1) need be given in other suitable forms and, moreover, both thermal expansion and strain recovery need be considered in a unified manner. Here, new forms of shape functions that apply to deformation features of SMPs are presented in Section 4 and closed-form results with both thermal expansion and strain recovery effects are derived in Section 5. Good agreement with test data is achieved. It should be pointed out that thermo-mechanical deformation features of SMPs may be simulated from various standpoints, as mentioned in the introduction section. Various constitutive models may be established based on additional variables associated with micro-structural mechanisms. Representatives are those based on either transition between micro-structural phases or evolution of macromolecular networks. For instance, transition between an active and a frozen phase is considered in [32] with a volume-fraction variable for the frozen phase, while evolution of a reversible and a permanent network is treated in [36; 37; 38]. It may be noted that the volume-fraction variable has been  
Figure 3: Fitting the data in [26] for the strain recovery effect of a SMP sample. fruitfully used in modeling thermo-mechanical behavior of SMAs with the martensite-to-austenite phase transition mechanism. On the other side, various types of networks are directly related to certain aspects of cross-links and entanglements of chain-like macromolecules in polymeric materials. Irrespective of the fact that the proposed model is straightforward in a sense without assuming any ad hoc variables of micro-structural origins, it may be of significance to explain how it accommodates itself to relevant underlying micro-mechanisms. For instance, the back stress \(\alpha\), which plays an essential role in deriving the thermo-induced plastic flow responsible for the strain recovery effect in Section 5, may be explained based on the notion of a permanent network as introduced in [36; 37; 38]. In general, each macroscopic constitutive quantity introduced in the proposed model may be related to relevant micro-mechanisms via suitable averaged procedures. This respect need be studied in future. It is known that thermo-mechanical responses of SMPs depend on both strain rate and thermo-mechanical history. For example, the loading curve and each unloading curve as shown in Figure 1 will be changed by strain rate and cyclic loading. In particular, cyclic thermo-mechanical loadings may lead to fatigue failure.