## 2 Thermoelastoplastic Approximate Algorithms
### Numerical Integration Scheme
Indeed, as can be seen from Figure 9, significant plastic flow occurs during relaxation while good agreement of the stresses with the FE results is verified. Since explicit integration is employed for the relaxation (Algorithm 3), the considerations of the previous paragraph on the accuracy of strains apply equally to the moderate deviations observed here. In conclusion, we have provided FE-based validation for each implementational branch of the solution scheme seen in Figure 3 at this point and shall turn our attention to actual half-space problems in the next section. ### Analysis of the Approximation Quality for Half-Space Problems

To study the approximate algorithms in the context of the half-space problem shown in Figure 1, we choose the particular distributions

\[t^{*}_{\eta\eta}(\xi)= \begin{cases}\frac{2\eta_{0}^{*}}{\pi b^{2}}\sqrt{b^{2}- \xi^{2}}&\text{for}\quad-b\leq\xi\leq b\\ 0&\text{else}\end{cases} \qquad\text{ and }\qquad\quad t^{*}_{\xi\eta}(\xi)=-\mu\,t^{*}_{\xi \xi}(\xi) \tag{44}\]

for the surface loads and for the heat flux distribution, we set

\[q^{*}(\xi)= \begin{cases}q_{0}^{*}&\text{for}\quad-b\leq\xi\leq b\\ 0&\text{else}.\end{cases}  \tag{45}\]

Herein, \(b\) denotes the half-width of the loaded area w.r.t. the \(\xi\)-\(\eta\)-system and \(p_{0}^{*}\), \(q_{0}^{*}\) and \(\mu\) are positive constants. The elliptical pressure distribution in Equation (44) stems from Hertzian contact theory and is very commonly postulated in rolling and sliding contact research. Assuming proportional tangential tractions is known to be somewhat simplistic for sliding contact problems due to the occurrence of both stick and slip regions, but this assumption is widespread nonetheless [7; 8]. The authors are aware of existing analytic expressions for the elastic stress solution corresponding to these particular surface tractions [21] as well as the integral given by [PERSON] and [PERSON] [22] for the temperature field associated with Equation (45). However, it seems that no integral-free, closed-form solution exists for the thermoelastic problem. Therefore, the elastic input solution \(\sigma^{*}_{ij}\) will be computed by the FEM as described in Section 3, although faster approximate methods [4] or even heuristic approaches [1] are favorable in applications. Figure 9: Stress, strain, and equivalent plastic strain evolution according to the FE analysis and the JS algorithm during both the loading cycles seen in Figure 5(a) and the subsequent relaxation step. Algorithms 2 and 3 were used. The load parameters are chosen as \(b=1\,\mathrm{mm}\), \(q_{0}^{*}=1250\,\mathrm{kW}\,\mathrm{m}^{-2}\) and \(\mu=0.2\). Furthermore, two levels of intensity are considered for the mechanical loading, \(p_{0}^{*}=1.6\,\mathrm{kN}\) and \(p_{0}^{*}=3.2\,\mathrm{kN}\), so that we may study the algorithms over a wide range of very small to moderately large plastic strains. To extend the range of hardening behavior considered in the investigation, we assume \(h\in\{2.5,5,10\}\) GPa with \(\sigma_{Y}^{0}=920\,\mathrm{MPa}\) and \(n=0.5\). Figure 10 displays the flow curves within the relevant domains of temperature and equivalent plastic strain. Note that the slope of the yield stress w.r.t. the equivalent plastic strain is initially vertical according to Equation (40) if \(n<1\) and therefore, \(h_{\varepsilon}=\infty\) for \(\varepsilon^{\mathrm{eq}}=0\), which is regularized to a large finite value when the explicit Algorithm 1 is employed. Otherwise, \(h_{\varepsilon}^{+}=\infty\) in the first elastoplastic increment implies \(\Delta\varepsilon_{ij}^{\mathrm{p}}=0\) and \(\varepsilon^{\mathrm{eq}-}=\varepsilon^{\mathrm{eq}+}=0\), so that by induction, it is easily seen that plastic flow would be suppressed within the explicit scheme for this particular choice of hardening behavior. By contrast, the implicit Algorithm 2 requires no special treatment for unbounded \(h_{\varepsilon}\).