## 2 Thermoelastoplastic Approximate Algorithms  
The purpose of this section is two-fold. First, we reformulate the methods of [PERSON] and [PERSON] [6] and of [PERSON] [7]--abbreviated by \"MM\" and \"JS\", respectively--for anisothermal problems in which isotropic strain hardening is accompanied by thermal softening. Then, algorithms and guidelines for the numerical solution of the arising differential equation systems will be provided.  
The thermomechanical problem under consideration is visualized in Figure 1. It consists of a thermoelastoplastic half-space under plane-strain conditions which is loaded by a moving heat flux distribution as well as by distributed normal and tangential surface tractions. The \(\xi\)-\(\eta\)-system (we may drop the out-of-plane coordinates \(\zeta\) and \(z\) indicated in Figure 1 in most of our discussion, since the assumption of plane-strain precludes any gradients in these coordinate directions), within which the distributions are stationary, moves at constant velocity \(v\) in the \(\xi\)-direction w.r.t. the \(x\)-\(y\)-system that is fixed to the material points of the half-space. In this setting, the material derivative of any tensor field \(F\) on the half-space reduces to its convective part within the \(\xi\)-\(\eta\)-system, i.e.,  
\[\dot{F}=\frac{\mathrm{D}F}{\mathrm{D}t}=-\frac{\partial F}{\partial\xi}v\,, \tag{1}\]  
where \(\mathrm{D}(\bullet)/\mathrm{D}t\) denotes the material derivative, so that temporal changes of state in the material \(x\)-\(y\)-system correspond to spatial changes in the \(\xi\)-\(\eta\)-system, and our whole discussion can be continued in the latter coordinate system.  
Assume now that the quasi-stationary solution of the _thermoelastic_ half-space problem is known at least on (a discretization of) a sufficiently large domain \(\Omega^{*}\) in the \(\xi\)-\(\eta\)-plane which contains the support of the load distributions. By sufficiently large, we imply that the stresses and temperature on the complement set of \(\Omega^{*}\) will not suffice to cause or sustain plastic flow of a passing material point, which may be estimated using the chosen yield condition (e.g., Equation (2)) under the assumption of no prior strain hardening and of decaying stresses and temperature beyond the vicinity of the loads. Then, the underlying idea of the MM and JS algorithms is to approximate the _thermoelastoplastic_ response of a material point passing through \(\Omega^{*}\) in a fixed depth \(\eta\) by subjecting it to specific in-plane  
Figure 1: Half-space loaded by a distributed surface heat flux \(q^{*}(\xi)\) and normal and tangential surface tractions \(t^{*}_{\eta\eta}(\xi)\), \(t^{*}_{\xi\eta}(\xi)\). Assuming a quasi-stationary problem, the distributions are constant in time in the \(\xi\)-\(\eta\)-\(\zeta\)-system that moves at constant velocity \(v\) in the \(\xi\)-direction w.r.t. the material coordinate system \(x\)-\(y\)-\(z\). The coordinate axes pointing into the plane are denoted by \(\zeta\) and \(z\), respectively. The assumption of plane-strain implies zero total normal strain along these axes, i.e., \(\varepsilon_{\xi\zeta}=\varepsilon_{\mathrm{zx}}=0\).  
components of the stress tensor \(\sigma^{*}_{ij}\) as well as the temperature \(T^{*}\) from the thermoelastic solution while solving for the remaining stress components by means of classical von Mises plasticity and--in case of the MM algorithm--using an additional kinematic constraint. We will expand on this in detail and provide the full algorithms in Section 2.2, but before, we shall state the equations of von Mises plasticity in the specific form we require.