## 2 Thermoelastoplastic Approximate Algorithms
### Numerical Integration Scheme
To compute the residual stresses, another thermoelastoplastic analysis is conducted, where the loads are relieved and the temperature is equalized in an additional transient simulation step. These FE simulations provide us with reference solutions to assess the quality of results obtained by the methods of Section 2. However, they are not suitable to validate our implementation of the approximate algorithms, since deviations arising from the \"depth-uncoupled\" nature of the MM and JS methods as well as from the assumptions Equations (11) and (12) cannot be distinguished from other potential discrepancies. For this purpose, the two simple FE models shown in Figure 5 were established. They each consist of a single plane-strain finite element with boundary conditions, nodal forces, surface tractions and temperature prescribed specifically so as to impose the kinematic constraints and the elastic stress components defined by Equations (11) and (12). By appropriate definition of the temporal evolution of these loads, we can simulate the thermoelastoplastic behavior of a single material point for arbitrary loading histories under the exact mechanical assumptions of MM and JS using the constitutive models included in Abaqus. This is easily implemented using the UEXTERNALDB and UAMP subroutines to load and apply the desired time series. In particular, by prescribing the solution from the elastic analysis of the \"global\" model seen in Figure 4, validation data for the MM and JS methods can be computed for general half-space problems, as sketched in \[\dot{F}\approx\frac{\Delta F}{\Delta\tau}\,, \tag{22}\]4 An analogous model can be build for the relaxation step of the approximate algorithms, where the strain in the \(\xi\)-direction is enforced by an appropriate displacement boundary condition. \[\dot{F}\approx\frac{\Delta F}{\Delta\tau}\,, \tag{22}\]5 Sketch of the FE model for the half-space problem of \[\dot{F}\approx\frac{\Delta F}{\Delta\tau}\,, \tag{22}\]6 The domain of interest, where thermoelastoplastic material behavior is applied, is shown in green and is discretized with a structured mesh of plane-strain elements. The surrounding gray area serves as thermoelastic embedding and is discretized by an unstructured mesh of plane-strain elements (omitted in the sketch) that progressively coarsens toward the outer boundaries that confine the half-space. ## 4 Numerical Study and Validation of the Algorithms' Implementations

This section covers our numerical studies using the approximate algorithms and discusses them in comparison to reference results computed by the FEM. Specifically, we start by validating the implementations using the models shown in Figure 5 and then move on to elaborate on the accuracy of the approximate methods based on the reference solutions from the FE model seen in \[\dot{F}\approx\frac{\Delta F}{\Delta\tau}\,, \tag{22}\]7

To specify the strain-hardening and thermal-softening behavior for our investigations, we assume the following rate-independent Johnson-Cook type law for the yield strength \(\sigma_{\text{Y}}\):

\[\dot{F}\approx\frac{\Delta F}{\Delta\tau}\,, \tag{22}\]8

Herein, \(\sigma_{\text{Y}}^{0}\) is the initial yield strength below or at the transition temperature \(T_{\text{tr}}\), at which thermal softening is assumed to begin, and prior to strain hardening. The constants \(h\) and \(n\) determine the slope and nonlinearity of the strain-hardening curve for a given temperature. Thermal softening is accounted for by the second factor in Equation (40) and parameterized by the constant exponent \(m\) as well as a function \(\Theta(T)\) defined by

\[\Theta(T)= \begin{cases}0&\text{for}\quad T<T_{\text{t}}\\ \frac{T-T_{\text{t}}}{T_{\text{m}}-T_{\text{t}}}&\text{for}\quad T_{\text{t} }\leq T\leq T_{\text{m}}\\ 1&\text{else}\,,\end{cases}  \tag{41}\]

where \(T_{\text{m}}\) denotes the melting temperature at which the yield strength vanishes.