## 5 Large Strain Recovery Effect at Heating
### Strain Recovery Effect  
Equation (51) supplies a closed-form solution for the Hencky strain in a thermo-mechanical process consisting of an isothermal loading-unloading process at constant temperature \(T_{0}\) and then a pure heating process from \(T_{0}\) to a higher temperature \(T\), in which a thermo-induced plastic flow is induced from \(T_{0}^{*}\) to \(T\). This result has been derived for a general multi-axial case. The Hencky strain given is composed of three parts, namely, the pre-strain \(h_{0}\) at \(T_{0}\), the thermo-elastic volumetric strain \((\Theta-\Theta_{0})\,\mathbf{I}\), as well as the part from the thermo-induced plastic flow. The thermo-elastic expansion part is usually very small, but it may become appreciable in a process through the glass transition in some cases. Generally, the thermo-elastic expansion effect over a temperature range may be simulated by choosing a suitable form of the temperature-dependent quantity \(\Theta(T)\).  
With negligible thermo-elastic expansion, it may be demonstrated [40] that the minimum of the magnitude of the strain \(\mathbf{h}\) as given in Equation (51) is always smaller than the initial value \(|\mathbf{h}_{0}|\) at \(T_{0}\). This suggests that generally the thermo-induced plastic flow in a process of pure heating always leads to a partial recovery of any given pre-strain. As such, the thermal recovery at pure heating is derived as a direct, natural consequence of the thermo-coupled elastoplasticity model established in Section 2. In particular, it may be deduced that the complete recovery of any given pre-strain emerges whenever  
\[\alpha_{0}=c_{0}\mathbf{h}_{0} \tag{53}\]  
with the \(c_{0}\) a constant. In particular, the above co-axiality condition is identically fulfilled in the uniaxial strain case.  
Now, the Prager modulus \(c\) is taken to be of the following form [40]:  
\[c=\frac{1}{2}(c_{0}+c_{0}^{*})-\frac{1}{2}(c_{0}-c_{0}^{*})\tanh\left[\lambda \frac{T-T_{0}}{T_{0}^{*}-T_{0}}\right]\,.\]  
In the above, \(c_{0}^{*}<c_{0}\) and \(\lambda>0\) is a large dimensionless parameter. Then, the Hencky strain for \(T\geq T_{0}^{*}\) is given by (c.f., [40])  
\[\begin{cases}\mathbf{h}=\left(\frac{c_{0}}{c_{0}}\frac{r}{r_{0}^{*}}-\frac{c_{0}}{ c_{0}}+1\right)\mathbf{h}_{0}\,,\\ r_{0}^{*}=r|_{T=T_{0}^{*}}=\sqrt{1.5}c_{0}|\mathbf{h}_{0}|\,.\end{cases} \tag{54}\]This result applies to a general multiaxial case. Given any form of the uniaxial temperature-strain relation, \(h=h(T)\), the temperature-dependent stress limit \(r\) is thus determined, as shown below:  
\[r=c_{0}^{*}\left[h(T)-h_{0}\right]+c_{0}h_{0}\,. \tag{55}\]  
With the above stress limit, the proposed model exactly reproduces the strain-temperature relation \(h=h(T)\) and, accordingly, any given uniaxial data for strain recovery may be automatically, accurately fitted.  
An example for the stress limit \(r\) with strain recovery is given as follows:  
\[r=r_{0}^{*}\left[1-\frac{c_{0}^{*}}{c_{0}}\tanh\lambda\frac{T-T_{0}^{*}}{T_{m} -T_{0}^{*}}\right]\,, \tag{56}\]  
where both the temperature \(T_{0}^{*}>T_{0}\) at the initial yielding (cf. Equation (42)) and the temperature \(T_{m}>T_{0}^{*}\) may be of any given values. The corresponding strain-temperature function is given by  
\[h=h_{0}\left[1-\tanh\lambda\frac{T-T_{0}^{*}}{T_{m}-T_{0}^{*}}\right]\,. \tag{57}\]  
For a fairly large \(\lambda\), any given large pre-strain, \(h_{0}\), is almost completely recovered at a temperature slightly higher than \(T_{m}\), as will be seen in the numerical example in the next section.