## 2 Thermoelastoplastic Approximate Algorithms
### Numerical Integration Scheme
In the following, a top-down approach will be taken to guide the presentation and discussion of the numerical results, i.e., we consider the residual stresses right away. These are usually the quantities of interest in applications and accordingly the only ones presented in most of the corresponding literature. #### 4.2.1 MM Method

The residual stresses predicted by the MM method are shown in Figure 11. Comparing the predictions with the corresponding reference solutions computed by the \"global\" FE model (shown in Figure 4), it is immediately obvious that the approximations are at best of qualitative nature for the cases studied here. In particular, the MM implementation fails to accurately predict the location of the compressive residual stress peak as well as the change to tensile stresses in the immediate subsurface. Looking only at the predicted compressive zone below \(\eta=0.1\,\mathrm{mm}\), it is apparent that the compressive peak magnitudes are approximated best in case of \(h=10\,\mathrm{GPa}\) for both studied load amplitudes \(p_{0}^{*}\). Furthermore, with \(h=2.5\,\mathrm{GPa}\), the peak approximation is much worse for the milder loading \(p_{0}^{*}=1.6\,\mathrm{kN}\) than for \(p_{0}^{*}=3.2\,\mathrm{kN}\). This indicates a dependency of the prediction's accuracy on a combination of loading intensity and hardening behavior. Within the studied parameter space, it seems that larger mechanical loading and strain hardening are advantageous, which--regarding the influence of the loading intensity--agrees with the assessment of [PERSON] [8] that the MM scheme is unrealistic in case of predominately elastic loading cycles. Figure 10: Yield stress as a function of equivalent plastic strain for \(h\in\{2.5,5,10\}\) GPa and three different temperatures. We investigated the sharp reversal of the residual stress approximations near the surface by using once more the \"local\" FE model of Figure 5a to confirm our implementation's prediction with a mechanically equivalent FE analysis on a subset of the depths and parameter combinations seen in Figure 11 and found no disagreement. Thus, it seems to us that the poor performance of the MM method in the cases we studied is linked to its inherent mechanical assumptions, as shown in Equation (11). Indeed, enforcing \(\varepsilon_{\xi\xi}=\varepsilon_{\xi\xi}=0\) in conjunction with the isotropic constitutive behavior implies equal mechanical response in the \(\xi\)- and \(\zeta\)-directions. Accordingly, the predicted residual stress components \(\sigma_{\xi\xi}^{\tau}\) and \(\sigma_{\xi\xi}^{\tau}\) are equal, which by itself disagrees in particular in the subsurface tensile region with the FE results; see Figure 11. This deficiency of the MM method was already noted by [PERSON] and [PERSON] [7], who study isothermal problems and linear kinematic hardening, and--in accordance with our results--also observed too shallow plastic zones predicted by the MM approach. A straightforward way to elaborate on the justification of the algorithm's underlying assumptions is to consider the \"true\" transient stress and strain components of a material point that passes the loaded region as computed by an elastoplastic FE analysis of the half-space (Figure 4) and to compare them to the respective values prescribed in the models of MM and JS. In particular, we may thus directly examine and compare the validity of the assumptions \(\varepsilon_{\xi\xi}=0\) (MM) and \(\sigma_{\xi\xi}^{\tau}=\sigma_{\xi\xi}^{\tau}\) (JS). Figure 12 displays how the in-plane components of \(\sigma_{ij}\) obtained by a transient elastoplastic FE analysis deviate from the respective components of the elastic input solution \(\sigma_{ij}^{\tau}\) for two of the studied combinations of \(p_{0}^{*}\) and \(h\), which correspond to a \"worst-case\" scenario (magenta curves in Figure 11a,b, as well as a \"best-case\" scenario (black curves in Figure 11c,d). Additionally, the elastoplastic FE solution for \(\varepsilon_{\xi\xi}\) is shown.