## 2 Thermoelastoplastic Approximate Algorithms
### Rate-Independent Plasticity for Isotropic Strain Hardening and Thermal Softening  
In this section, we derive the governing equations of classic rate-independent von Mises plasticity with isotropic strain hardening and thermal softening on which the generalized MM and JS algorithms will be based. We assume purely isotropic strain hardening, so that the von Mises yield function may be defined as  
\[f(\sigma_{ij},\epsilon^{\text{eq}},T)=\frac{3}{2}\sigma^{\prime}_{ij}\sigma^{ \prime}_{ij}-(\sigma_{\text{Y}}(\epsilon^{\text{eq}},T))^{2} \tag{2}\]  
where index notation is adopted and superscript primes \((\bullet)^{\prime}\) indicate the deviatoric part of second-order tensors throughout this work, i.e.,  
\[\sigma^{\prime}_{ij}\coloneqq\sigma_{ij}-\frac{1}{3}\sigma_{kl}\delta_{ij} \tag{3}\]  
with \(\delta_{ij}\) denoting the Kronecker symbol. Furthermore, the associative flow rule is assumed to govern the plastic strain,  
\[\varepsilon^{\text{p}}_{ij}\overset{!}{=}\gamma\frac{\partial f}{\partial \sigma_{ij}}=3\gamma\,\sigma^{\prime}_{ij}, \tag{4}\]  
with a positive consistency parameter \(\gamma\in\mathbb{R}_{>0}\) which can be related to the equivalent plastic strain rate \(\varepsilon^{\text{eq}}\),  
\[\varepsilon^{\text{eq}}\coloneqq\sqrt{\frac{2}{3}\varepsilon^{\text{p}^{ \prime}_{ij}}\varepsilon^{\text{p}^{\prime}}_{ij}}\overset{!}{=}2\sigma_{ \text{Y}}(\varepsilon^{\text{eq}},T)\;\gamma\,, \tag{5}\]  
by using the yield condition \(f=0\) in the last step. Solving Equation (5) for \(\gamma\) and inserting back into Equation (4), we obtain the Prandtl-Reu8 form of the flow rule,  
\[\varepsilon^{\text{p}}_{ij}=\left\langle\frac{3\varepsilon^{\text{eq}}}{2 \sigma_{\text{Y}}(\varepsilon^{\text{eq}},T)}\right\rangle\!\sigma^{\prime}_ {ij} \tag{6}\]  
where we introduced the Macaulay bracket \(\left\langle\bullet\right\rangle=\max\{0,\bullet\}\) to guarantee that the plastic strain rate is either zero or is oriented in the direction of the stress deviator. To formulate the problem entirely in stress space, the equivalent plastic strain rate may be eliminated using the consistency condition  
\[0\overset{!}{=}j\overset{!}{=}2\overset{!}{=}3\dot{\sigma}^{\prime}_{ij} \sigma^{\prime}_{ij}-2\sigma_{\text{Y}}(\varepsilon^{\text{eq}},T)\bigg{(} \frac{\partial\sigma_{\text{Y}}}{\partial\varepsilon^{\text{eq}}}\dot{ \varepsilon}^{\text{eq}}+\frac{\partial\sigma_{\text{Y}}}{\partial T}\dot{T} \bigg{)}\,. \tag{7}\]  
Solving Equation (7) for \(\varepsilon^{\text{eq}}\) and inserting into Equation (6) eventually yields  
\[\varepsilon^{\text{p}}_{ij}=\frac{1}{h_{\varepsilon}}\langle\sigma^{\prime}_{ kl}n_{kl}-h_{\text{T}}\dot{T}\rangle n_{ij} \tag{8}\]  
where--for brevity of notation--we defined  
\[h_{\varepsilon}\coloneqq\frac{2}{3}\frac{\partial\sigma_{\text{Y}}}{\partial \varepsilon^{\text{eq}}}\,,\qquad h_{\text{T}}\coloneqq\sqrt{\frac{2}{3}} \frac{\partial\sigma_{\text{Y}}}{\partial T}\qquad\text{and}\qquad n_{ij} \coloneqq\sqrt{\frac{3}{2}}\frac{\sigma^{\prime}_{ij}}{\sigma_{\text{Y}}( \varepsilon^{\text{eq}},T)}\,. \tag{9}\]  
\(h_{\varepsilon}\) and \(h_{\text{T}}\) relate to strain hardening and thermal softening, respectively, while (for \(f=0\)) \(n_{ij}\) represents the outward unit normal to the von Mises yield surface and hence satisfies\(n_{ij}n_{ij}=1\). Note the second summand in Equation (8), which additionally arises due to thermal softening in comparison to the flow rule considered by [PERSON] [8]. Finally, by inserting Equation (8) into the definition of the equivalent plastic strain, Equation (5), we obtain  
\[\epsilon^{\text{eq}}=\sqrt{\frac{2}{3}}\frac{1}{|h_{\epsilon}|}\langle\dot{ \sigma}^{\prime}_{kl}n_{kl}-h_{\text{T}}\dot{T}\rangle\,. \tag{10}\]