## 2 Thermoelastoplastic Approximate Algorithms
### Numerical Integration Scheme
Our investigations indicate that the MM algorithm does not perform well for isotropic hardening and anisothermal problems of the kind we studied. Hence, the JS algorithm appears to be preferable. Indeed, this assessment seems consistent with the choice of a number of authors [9, 11, 13, 23] to employ the JS method for manufacturing applications in which elevated temperatures occur, while we had difficulties finding literature that relies on the method of MM for similar problems. However, other authors [10, 14, 15, 16, 17] have used the hybrid scheme proposed by [PERSON] [8], with some even employing a similar Johnson-Cook-type hardening law [10]. This hybrid scheme essentially blends between the MM and JS methods using a scalar blending factor \(0\leq\Psi\leq 1\) to formally \"unify\" both based on purely heuristic considerations. Since we adopted this idea of [PERSON] to consolidate our notation, the solution schemes we proposed cover the hybrid algorithm as well. While [PERSON] [8] provided convincing evidence for the suitability of this purely heuristic approach within the application he studied and carefully adjusted the functional dependency of the blending factor on the hardening slope based on auxiliary FE calculations, it is not obvious that his conclusions and the proposed blending function carry over from the isothermal loading, the linear kinematic hardening, and the parameter space he considered for anisothermal problems with nonlinear isotropic hardening. In fact, judging from the poor predictions we obtained with the MM assumptions for all considered hardening slopes, there seems to be little incentive to choose the hybrid scheme over the plain JS method if sufficiently strong strain hardening behavior may be assumed. On the other hand, the blending function \(\Psi(h_{\epsilon})\) proposed by [PERSON] [8] specifically shifts from the JS assumptions for \(h_{\epsilon}\to\infty\) to the MM assumptions for \(h_{\epsilon}\to 0\) in a gradual way. Thus, one might assume that the large overestimation of the residual stresses that we saw for the JS method in case of weak strain hardening could be shadowed by blending over to the MM assumptions for smaller \(h_{\epsilon}\), at least for sufficient loading (where MM performed better) and not too small \(h_{\epsilon}\) (for which the MM algorithm is unstable). To check this, we reevaluated the half-space problem using the hybrid algorithm with theblending function proposed by [PERSON] [8] for all loading and hardening parameter combinations we considered in this work. While the JS algorithm's overestimation of the tensile residual stress peaks can indeed be somewhat alleviated in the two cases of weaker strain hardening we considered, this comes at the cost of underestimating the tensile peaks for stronger strain hardening, where the JS method works very well, which might be more critical in applications. Furthermore, the residual stresses in the compressive zone generally were largely underestimated, and the large near-surface errors we discussed for the MM method appeared in the hybrid scheme, too. In conclusion, we found no systematic improvement of the predictions using the hybrid algorithm over the JS algorithm but--on the contrary--much worse approximations in the parameter range in which the JS scheme was demonstrated to work well. It should be noted though that the blending might be modified to bias the JS method and thus improve this situation. However, this is only practical if correct reference residual stress solutions computed under equivalent mechanical assumptions are at hand. We demonstrated that a certain minimum amount of strain hardening, depending also on the loading, is necessary to obtain good approximations. This seems to restrict the usefulness of the discussed algorithms for materials with weak hardening. As an example, the strain-hardening behavior of Ti-6 Al-4V Grade 5 is adequately modeled by the Johnson-Cook-type law considered in this work for \(h\approx 185\,\mathrm{GPa}\) and \(n\approx 0.3\).