## 5 Large Strain Recovery Effect at Heating
### Thermo-Induced Plastic Flow at Heating  
We next treat a process of pure heating with the temperature \(T\) changing from \(T_{0}\) to \(T_{*}\). In this case, the stress limit \(r\) relies merely on the temperature \(T\). Usually, \(r\) should decrease with increasing temperature. Namely,  
\[r^{\prime}(T)=\frac{dr}{dT}<0\,. \tag{39}\]  
With an initial temperature \(T_{0}\) held fixed, a pre-strain or plastic strain is induced in a SMP sample via a loading-unloading process at constant temperature \(T_{0}\), as indicated in Section 5.1. Let  
\[f(T)=\frac{1}{2}|\mathbf{\alpha}_{0}|^{2}-\frac{1}{3}(r(T))^{2}\,. \tag{40}\]  
The unloading at \(T=T_{0}\) leads to  
\[f(T_{0})=\frac{1}{2}|\mathbf{\alpha}_{0}|^{2}-\frac{1}{3}(r(T_{0}))^{2}<0\,. \tag{41}\]  
Consider a heating process with \(\dot{T}>0\). On account of Equation (39), the \(f(T)\) increases with increasing temperature and hence it may increase from the initial value \(f(T_{0})<0\) at \(T_{0}\) to zero at a certain temperature \(T_{0}^{*}\), namely,  
\[f(T_{0}^{*})=\frac{1}{2}|\mathbf{\alpha}_{0}|^{2}-\frac{1}{3}(r(T_{0}^{*}))^{2}=0\,. \tag{42}\]  
Moreover, with \(\tilde{\mathbf{\tau}}=\mathbf{O}\) and \(\dot{T}>0\) as well as Equation (40) we deduce that  
\[\frac{1}{\tilde{h}}\left(\frac{\partial f}{\partial\tilde{\mathbf{\tau}}}\dot{ \tilde{\mathbf{\tau}}}\dot{\tilde{\mathbf{\tau}}}\log+\frac{\partial f}{\partial T}\dot {T}\right)=-(cr)^{-1}r^{\prime}(T)\dot{T}>0\,. \tag{43}\]  
Hence, from Equation (8)\({}_{1}\) it follows that the loading condition is met. Thus, plastic flow may be induced at pure heating and emerges for \(T\geq T_{0}^{*}\).  
By setting \(\mathbf{\tau}=\mathbf{O}\) in the rate equations given in Section 2, the governing equations for the thermo-induced plastic flow as indicated above may be derived as follows:  
\[\mathbf{D}=\mathbf{D}^{\varepsilon}+\mathbf{D}^{p}\,, \tag{44}\]  
\[\mathbf{D}^{\varepsilon}=\Theta^{\prime}(T)\dot{T}\mathbf{I}\,, \tag{45}\]  
\[\mathbf{D}^{p}=r^{\prime}(T)\frac{\dot{T}}{cr}\,\mathbf{\alpha}\,, \tag{46}\]  
\[\dot{\mathbf{\alpha}}^{\log}=\frac{1}{r}f^{\prime}(T)\dot{T}\,\mathbf{\alpha}\,, \tag{47}\]  
\[\dot{\tilde{\mathbf{\kappa}}}=\frac{2}{3}c^{-1}rr^{\prime}(T)\dot{T}\,, \tag{48}\]  
with the initial conditions at \(T=T_{0}^{*}\) given below:  
\[\begin{cases}F|_{T=T_{0}^{*}}=e^{\Theta_{0}^{*}-\Theta_{0}}F_{0},\ \Theta_{0}^{*}=\Theta(T_{0}^{*}),\ \Theta_{0}=\Theta(T_{0})\,,\\ \mathbf{\alpha}|_{T=T_{0}^{*}}=\mathbf{\alpha}_{0},\ \dot{\kappa}|_{T=T_{0}^{*}}=\delta_{0},\ r |_{T=T_{0}^{*}}=\sqrt{15}|\mathbf{\alpha}_{0}|\,.\end{cases} \tag{49}\]In deriving Equation (49), the following fact has been used: only thermo-elastic volumetric deformation is induced from \(T_{0}\) to \(T_{0}^{*}\).  
A closed-form solution may be derived from the above equations. Results are as follows:  
\[\alpha=\sqrt{\frac{2}{3}}\,\frac{\alpha_{0}}{|\alpha_{0}|}\,r(T),\quad T\geq T_ {0}^{*}\;, \tag{50}\]  
for the back stress, and  
\[\begin{cases}\mathbf{F}=e^{\Theta-\Theta_{0}}\mathbf{F}_{0}\,,\\ \mathbf{h}=(\Theta-\Theta_{0})\,\mathbf{I}+\mathbf{h}_{0}+\sqrt{\frac{2}{3}}\,\frac{\alpha _{0}}{|\alpha_{0}|}\int_{T_{0}^{*}}^{T}c^{-1}r^{\prime}(T)dT,\quad T\geq T_{0 }^{*}\;,\end{cases} \tag{51}\]  
for the deformation gradient and Hencky's logarithmic strain, and, finally,  
\[\tilde{\kappa}=\tilde{\kappa}_{0}+\frac{2}{3}\int_{T_{0}^{*}}^{T}c^{-1}rr^{ \prime}(T)dT,\quad T\geq T_{0}^{*}\;, \tag{52}\]  
for the effective plastic work.