## 2 Thermoelastoplastic Approximate Algorithms
### Numerical Integration Scheme
The complete set of constitutive parameters is collected in Table 1. Note that in addition to the parameters introduced before, we included the parameters needed to solve the linear heat conduction problem during the thermoelastic analysis for the input solution \(\sigma_{ij}^{*}\), \(T^{*}\), at the bottom of the table. ### Validation of the Approximate Algorithms' Implementation

For the scope of this section, we consider synthetically defined time evolutions for the prescribed loads on the FE models seen in Figure 5. This simplifies the discussion and the increment size selection for the integration in the approximate algorithms, but more importantly, it allows us to easily assert that the thermoelastic input solution indeed complies with the additional kinematic constraint \(\varepsilon_{\xi\xi}=0\) underlying the MM method. Therefore, it is guaranteed that the validation is done under circumstances where the MM and JS approaches constitute no approximation but _exact_ methods from a mechanical viewpoint; i.e., any deviations from the FE results must be of numerical or implementational origin. Figure 5: FE models used to validate our implementation of the approximation schemes’ transient step. \(\tau\) denotes the (pseudo-)time during the transient analysis. The bar connecting the bearings at the top two nodes in subfigure (**a**) represents a multi-point constraint that enforces equal displacement in the \(\xi\)-direction. \(\sigma_{\eta\eta}^{*}(\tau)\) and \(\sigma_{\xi\eta}^{*}(\tau)\) are prescribed by surface traction definitions in Abaqus while three nodal forces are defined to enforce \(\sigma_{\xi\xi}^{*}(\tau)\) for the JS case. To expand on this point, note that the plane-strain constraint \(\varepsilon_{\zeta\zeta}=0\) by virtue of the thermoelastic extension of [PERSON]'s law dictates

\[\sigma_{\zeta\zeta}^{*}=\
u\left(\sigma_{\zeta\zeta}^{*}+\sigma_{\eta\eta}^{*} \right)-\alpha ET^{*} \tag{42}\]

and additionally enforcing \(\varepsilon_{\zeta\zeta}=0\) then leaves us with only one of the two in-plane normal stresses as independent component, i.e., the relation

\[\sigma_{\zeta\zeta}^{*}=\frac{\
u(1-\
u)\sigma_{\eta\eta}^{*}-\alpha E(\
u+1)T ^{*}}{1-\
u^{2}} \tag{43}\]

must be taken into account when defining an elastic input solution for the MM algorithm. The FE analysis of the model in Figure 5a will of course result in a compliant \(\xi\xi\)-component of the stress tensor by itself, but for the approximate algorithm, we must completely define \(\sigma_{ij}^{*}(\tau)\) ourselves beforehand. Since the constitutive behavior is rate-independent, we consider a step of unit pseudo-time \(\tau\in[0,1]\) over which we ramp up the temperature \(T^{*}(\tau)\) linearly from 0 to 700 \({}^{\circ}\)C. This leads to a maximum reduction of the yield stress of around 40% due to thermal softening as per the second factor in Equation (40). Then, we consider a complete loading-unloading cycle for each individual load sketched in Figure 5 in a separate analysis by prescribing a hat function for the load's temporal evolution: For both the JS and the MM algorithm, this comprises two analyses in each of which either \(\sigma_{\eta\eta}^{*}(\tau)\) or \(\sigma_{\xi\eta}^{*}(\tau)\) is prescribed by a hat function. An additional analysis in which \(\sigma_{\xi\xi}^{*}(\tau)\) assumes a hat-shaped loading-unloading cycle is carried out for the JS method. Figures 6 and 7 show the stresses obtained by these analyses, where we used explicit integration (Algorithm 1) with a timestep of \(\Delta\tau=1\times 10^{-3}\). We omit displaying the stresses we correspondingly obtained by implicit integration (Algorithm 2), as they are virtually identical to the results of Algorithm 1 shown here. Note the hat-shaped cycles of the prescribed stress components and that we adjusted the corresponding peak values so that the magnitudes of equivalent plastic strain after the loading cycle were similar. Furthermore, by comparing Figure 6a,b, the expected equal behavior w.r.t.