## 2 Thermoelastoplastic Approximate Algorithms
### Numerical Integration Scheme
Finally, by comparing the diagrams in Figure 12 by row, it is apparent that the assumptions of JS are somewhat better fulfilled for \(h=2.5\,\mathrm{GPa}\) and \(p_{0}^{*}=1.6\,\mathrm{kN}\) than for the other considered parameter set, even though the residual stress approximation is better for the latter. This indicates that higher violations of the JS assumptions may be tolerable with stronger hardening. To further detail the discussion, we shall examine how the inaccuracies that arise during the load passing step as a consequence of the discussed constraint violations eventually propagate to the residual stress prediction in the course of the relaxation step. To this end, the residual stresses for the cases \(h\in\{5,10\}\,\mathrm{GPa}\) and \(p_{0}^{*}=3.2\,\mathrm{kN}\) are once more shown in Figure 14 alongside the corresponding components of the stress \(\overline{\sigma}_{ij}\) at the end of the loading pass, prior to the relaxation procedure of the JS algorithm. We chose these combinations of \(h\) and \(p_{0}^{*}\) since plastic flow during relaxation still occurs for the \(h=5\,\mathrm{GPa}\) case, while the relaxation is entirely elastic for \(h=10\,\mathrm{GPa}\) (which it is also for all the other studied combinations except \(p_{0}^{*}=3.2\,\mathrm{kN}\) with \(h=2.5\,\mathrm{GPa}\)). Comparing the dashed lines in Figure 14a,b,d,e, the impact of the JS assumptions' violations that we pointed out above indeed manifests in the intermediate results \(\overline{\sigma}_{ij}\), i.e., \(\overline{\sigma}_{\xi\xi}\) and \(\overline{\sigma}_{\xi\xi}\) deviate from the corresponding FE reference solutions particularly for values of \(\eta\) where large violations of the underlying JS assumption on \(\sigma_{\xi\xi}\) were observed during the loading pass. Furthermore, considering all studied cases, the inaccuracy of \(\overline{\sigma}_{ij}\) w.r.t. to the FE results was found to become smaller for larger \(h\), as expected according to better fulfillment of Equation (12). However, these inaccuracies seem somewhat less sensitive w.r.t. \(h\) as the strong dependency of the residual stress prediction's accuracy on \(h\) suggests for the two parameter combinations considered in Figure 11.3 On the other hand, looking at Figure 14c,f, the discrepancy of the accuracy of the predicted residual stresses for \(h=5\,\mathrm{GPa}\) and \(h=10\,\mathrm{GPa}\) may be related to the larger straining during relaxation for \(h=5\,\mathrm{GPa}\) due to the much bigger \(\overline{\xi}_{\xi\xi}\) which is forced to zero and correspondingly drives the transient loading during relaxation; see Equations (34) and (37). This assessment seems to be supported by the remark [PERSON] [8] gave in passing that large magnitudes of relaxation may be the cause of the JS algorithm's instability for weak hardening. However, since the strains are computed from the stress updates in Algorithms 1 and 2, deviations of \(\overline{\xi}_{\xi\xi}\) from the FE reference strain ultimately relate back to inaccurate stresses obtained during the loading pass and thus to the appropriateness of the assumptions Equation (12). Finally, we made similar observations on the stability of the JS algorithm for very small \(h\) as above for the MM method, e.g., the stresses after the loading pass become increasingly oscillatory for \(p_{0}^{*}=1.6\,\mathrm{kN}\) and \(h<0.25\,\mathrm{GPa}\), and no solution could be found within reasonable tolerances in the implicit Algorithm 2, respectively. This agrees with [PERSON]'s investigations of repeated isothermal sliding contact under kinematic hardening behavior [8]. ## 5 Discussion

The properties of the MM and JS approximate algorithms have already been addressed in detail in the previous section alongside the associated numerical studies and results. In this section, we discuss our findings in a broader sense and with respect to applications that have been proposed, in particular for manufacturing processes.