## 2 Thermoelastoplastic Approximate Algorithms
### Numerical Integration Scheme
This is well below the level of strain hardening for which we found the residual stress approximations to be accurate in loading regimes within which the FE analyses yield reasonable residual stress distributions. However, there are few reports [13; 14; 15] of successful applications of the hybrid and JS algorithms for cutting operations with this workpiece material. It is difficult to draw a conclusion on this circumstance for two reasons: Firstly, the cited reports lack sufficient specifications of the assumed hardening law (and the blending function in case the hybrid scheme is used), which apparently is not uncommon in applied literature, unfortunately. Secondly, the loading they prescribe to the half-space's surface cannot be reconstructed easily, since it results from extensive cutting force models that usually require in-process measurement data as input. However, parameters for the Johnson-Cook law are provided in the recent study of [PERSON] et al. [17], corresponding to \(h\approx 650\,\mathrm{GPa}\) and \(n=0.45\) within our notation, and indeed, highly oscillatory residual stress approximations are reported even though the thermomechanical properties, the temperature, and the elastic input stresses evidently change smoothly along the depth. It should be noted that [PERSON] et al. [17] use a slightly modified version of [PERSON]'s hybrid algorithm; however, their formulation lacks a consistent treatment of thermal softening as proposed in this paper (which is also the case in other earlier studies, e.g., [10; 11]). Thus, they effectively set \(h_{\mathrm{T}}=0\) within our notation, in opposition to the Johnson-Cook model they consider, which at least for our parameter set noticeably impacts the agreement of the stress response with control results obtained by the mechanically equivalent FE analyses described in Section 4.1. As mentioned above, the approximate algorithms were extended to incorporate thermal softening in this work. We demonstrated that many of the observations of [PERSON] and [PERSON] [7] and [PERSON] [8] regarding the accuracy of the algorithms w.r.t. the hardening carry over to anisothermal problems and isotropic hardening. However, it should be noted that thermal softening poses an additional challenge for the approximate algorithms, since it weakens the stabilization of plastic flow through strain hardening in case of simultaneous local heating. Thus, it might be that larger strain hardening is required to obtain accurate results as when assuming a temperature-independent yield stress, especially since the general approach of treating individual depths in an uncoupled, non-compatible way also prevents stress redistribution. Finally, we considered both an implicit and an explicit scheme for the integration in the loading pass step. In our investigation of the half-space problem, we asserted sufficiently small step sizes to obtain comparable stresses with both integration schemes. However, as discussed in Section 4.1, much larger step sizes may be used with the implicit algorithm without compromising on accuracy, as is expected from an implicit integration method in a path-dependent problem. This proves useful in performance critical scenarios, as considered e.g., by [PERSON] et al. [4], in which the additional effort of a Newton iteration might thereby be balanced against the need to compute the thermoelastic input solution on a much finer discretization, which certainly comes at a cost, especially if double convolution integrals are evaluated numerically over the whole domain for each grid point as is commonly done [9; 11; 13; 14; 15; 16; 17; 23]. ## 6 Conclusions

In this work, the approximate elastoplastic rolling/sliding contact algorithms of Jiang and Sehitoglu [7] (JS), McDowell and Moyar [6] (MM), and McDowell [8] (hybrid) have been reformulated and extended to allow for nonlinear isotropic strain hardening with thermal softening.