## 1 Introduction
### Constitutive Modeling for Numerical Investigations of the TemcoreTM Process  
Since it is not feasible to inspect the transient stresses shown in Figure 2 by in-process measurements, numerical simulations play an important role in understanding the generation of residual stresses during the heat treatment and thereby potentially attain further optimization of the process parameters. Given the requirement to resolve the macro-geometry of the considered component within the spatial discretization, constitutive modeling for heat treatment simulations is often based on component-scale continuum formulations where the microstructure is accounted for only be means of volume proportions of the existing phases [5; 6; 19; 20]. In the simplest scenario, this approach typically involves solving the following initial boundary value problems (IBVPs):  
* The transient non-homogeneous isotropic heat equation governing heat conduction in the spatial domain \(\Omega\in\mathbb{R}^{n}\) spanned by the solid and the temporal domain \([t_{0},t_{\rm e}]\subset\mathbb{R}\) with prescribed surface convection on \(\partial\Omega\) and a known initial temperature field \(T_{0}(\mathbf{x})\)[21]: \[\rho c_{\rm P}\frac{\partial T(\mathbf{x},t)}{\partial t} =\
abla\cdot(\lambda\
abla T(\mathbf{x},t))+\dot{q} \forall(\mathbf{x},t)\in\Omega\times[t_{0},t_{\rm e}]\] (1) \[-\lambda\
abla T(\mathbf{x},t) =h(T(\mathbf{x},t)-T_{\infty}(\mathbf{x},t)) \forall(\mathbf{x},t)\in\partial\Omega\times[t_{0},t_{\rm e}]\] (2) \[T(\mathbf{x},t) =T_{0}(\mathbf{x}) \forall(\mathbf{x},t)\in\Omega\times\{t_{0}\}\] (3) Here, \(\rho,c_{\rm P}\), and \(\lambda\) denote the density, specific heat capacity, and thermal conductivity of the compound, respectively, \(T_{\infty}\) the coolant temperature, \(h\) the heat transfer coefficient, and \(\dot{q}\) the internal heat generation, e.g., due to latent heat associated with phase transformations or plastic dissipation. * The mechanical compatibility and (quasi-)static equilibrium equations [21] where geometric linearity and negligence of body forces are typically suitable assumptions for heat treatment simulations [20]: \[\epsilon(\mathbf{x},t)=\frac{1}{2}\Big{(}\
abla\mathbf{u}(\mathbf{x },t)+(\
abla\mathbf{u}(\mathbf{x},t))^{T}\Big{)},\quad\
abla\cdot\boldsymbol{ \sigma}(\mathbf{x},t)=\mathbf{0} \forall(\mathbf{x},t)\in\Omega\times[t_{0},t_{\rm e}]\] (4) \[\boldsymbol{\sigma}(\mathbf{x},t)=\boldsymbol{\sigma}(\epsilon( \mathbf{x},t),t) \forall(\mathbf{x},t)\in\Omega\times[t_{0},t_{\rm e}]\] (5) In this system, \(\boldsymbol{\sigma}\) denotes the Cauchy stress tensor, \(\epsilon\) the linear strain tensor, and \(\mathbf{u}\) the displacement field. Equations (4) and (5) are usually complemented with displacement- or traction-free boundary conditions on a partition \(\partial\Omega_{\mathbf{u}}\cup\partial\Omega_{\boldsymbol{\sigma}}=\partial\Omega\) of the solid's boundary (the outward normal of which is denoted as \(\mathbf{n}\)) as well as stress- and displacement-free initial conditions: \[\mathbf{u}(\mathbf{x},t)=\mathbf{0} \forall(\mathbf{x},t)\in\partial\Omega_{\mathbf{u}}\times[t_{0},t_{ \rm e}],\quad\boldsymbol{\sigma}(\mathbf{x},t)\cdot\mathbf{n}(\mathbf{x},t)= \mathbf{0} \forall(\mathbf{x},t)\in\partial\Omega_{\boldsymbol{\sigma}}\times[t_{0},t_ {\rm e}]\] (6) \[\mathbf{u}(\mathbf{x},t)=\mathbf{0} \forall(\mathbf{x},t)\in\Omega\times\{t_{0}\}, \boldsymbol{\sigma}(\mathbf{x},t)=\mathbf{0} \forall(\mathbf{x},t)\in\Omega\times\{t_{0}\}\] (7) To complete the description of the mechanical IBVP, a constitutive model that particularizes Equation (5) must be specified, where the assumption of isotropic thermo-elasto-plastic behavior according to the classical flow theory of plasticity is almost universal in literature [20].