## 2 Thermoelastoplastic Approximate Algorithms
### Numerical Integration Scheme
Determine \(\Delta\sigma_{\zeta\zeta}^{-}\) and \(\Delta\sigma_{\zeta\zeta}^{-}\) by solving the linear equation system: \[\left(1+\frac{E}{h_{\epsilon}^{+}}n_{\zeta\zeta}^{+}n_{\zeta\zeta}^{+} \right)\Delta\sigma_{\zeta\zeta} +(1-\Psi)\left(-\
u+\frac{E}{h_{\epsilon}^{+}}n_{\zeta\zeta}^{+}n_{ \zeta\zeta}^{+}\right)\Delta\sigma_{\zeta\zeta}^{*} \left(1+\frac{E}{h_{\epsilon}^{+}}n_{\zeta\zeta}^{+}n_{\zeta\zeta}^{+} \right)\Delta\sigma_{\zeta\zeta}^{*}\] \[+(\Psi-1)\left(\left(-\
u+\frac{E}{h_{\epsilon}^{+}}n_{\zeta\zeta} ^{+}n_{\eta\eta}^{+}\right)\Delta\sigma_{\eta\eta}^{*}+\frac{2E}{h_{\epsilon} ^{+}}n_{\zeta\zeta}^{+}n_{\zeta\zeta}^{+}\right)\Delta\sigma_{\zeta\zeta}^{*} -\left(\frac{Eh_{\Gamma}^{+}}{h_{\epsilon}^{+}}n_{\zeta\zeta}^{+}-\alpha E \right)\Delta T^{*}\right)\] (25) \[\left(-\
u+\frac{E}{h_{\epsilon}^{+}}n_{\zeta\zeta}^{+}n_{\zeta \zeta}^{+}\right)\Delta\sigma_{\zeta\zeta} +\left(1+\frac{E}{h_{\epsilon}^{+}}n_{\zeta\zeta}^{+}n_{ \zeta\zeta}^{+}\right)\Delta\sigma_{\zeta\zeta} = \left(\
u-\frac{E}{h_{\epsilon}^{+}}n_{\eta\eta}^{+}n_{\zeta\zeta}^{ +}\right)\Delta\sigma_{\eta\eta}^{*}\] \[-\frac{2E}{h_{\epsilon}^{+}}n_{\zeta\zeta}^{+}n_{\zeta\zeta}^{+} \Delta\sigma_{\zeta\eta}^{*}+\left(\frac{Eh_{\Gamma}^{+}}{h_{\epsilon}^{+}}n_ {\zeta\zeta}^{+}-\alpha E\right)\Delta T^{*}\] (26)
2. Update the stresses \(\sigma_{ij}^{-}=\sigma_{ij}^{+}+\Delta\sigma_{ij}\) using the solution from the previous step and \(\Delta\sigma_{\eta\eta}=\Delta\sigma_{\eta\eta}^{*}\) as well as \(\Delta\sigma_{\zeta\eta}=\Delta\sigma_{\zeta\eta}^{*}\). 3. Compute approximations of \(n_{ij}^{-}\), \(h_{\epsilon}^{-}\), and \(h_{\Gamma}^{-}\) by Equation (9) with \(\epsilon^{\text{eq}}=\epsilon^{\text{eq}+}\), \(T=T^{*-}\). 4. Update the equivalent plastic strain, using \(n_{ij}^{-}\), \(h_{\epsilon}^{-}\) and \(h_{\Gamma}^{-}\) from the previous step: \[\epsilon^{\text{eq}-} \overset{\eqref{eq:eq:eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eq_eqeq_eq_eq_eqeq_eq_eq_eq_eq_eqeq_eq_eq_eqeq_eq_eqeq_eq_eqeq_eq_eq_eqeq_eq_eqeq_eq_eqeq_eq_eqeq_eqeq_eqeq_eqeq_eqeq_eqeqeq_eqNote that we may reuse this operator \(P\) to generically denote the solution of Equations (23) and (24) for arbitrary given increments--or even absolute values--of the stress and the temperature.

For the relaxation step, a predictor may analogously be derived by setting \(\Delta\varepsilon_{ij}^{\text{P}}=0\) in the discretized versions of Equations (19) and (20), in which the rates prescribed by Equation (18) reduce to the increments

\[\Delta\sigma_{\eta\eta}^{\text{R}}=-\overline{\sigma}_{\eta\eta}M^{-1}\,,\quad \Delta\sigma_{\zeta\eta}^{\text{R}}=-\overline{\sigma}_{\xi\xi}M^{-1}\,,\quad \Delta\varepsilon_{\zeta\xi}^{\text{R}}=-\overline{\xi}_{\xi\xi}M^{-1}\,\,\, \text{and}\quad\Delta T^{\text{R}}=-\overline{T}M^{-1}\,. \tag{34}\]

Again, the predictor stress increment \(\Delta\sigma_{ij}^{\text{prR}}\) is obtained by solving the resulting linear equation system,

\[\Delta\sigma_{\zeta\xi}^{\text{prR}} =\frac{1}{1-\
u}\Bigg{(}\
u\Delta\sigma_{\eta\eta}^{\text{R}}+ \frac{E\Delta\varepsilon_{\zeta\xi}^{\text{R}}}{\
u+1}-\alpha E\Delta T^{ \text{R}}\Bigg{)}\,,\qquad\Delta\sigma_{\eta\eta}^{\text{prR}}=\Delta\sigma_{ \eta\eta}^{\text{R}}\,, \tag{35}\] \[\dot{F}\approx\frac{\Delta F}{\Delta\tau}\,, \tag{22}\]0

Note that this predictor is constant throughout the relaxation.