## 2 Thermoelastoplastic Approximate Algorithms
### Approximate Plane-Strain Rolling and Sliding Contact Algorithms  
As noted above, the core idea of the MM and JS approaches is to enforce _some_ of the transient in-plane stresses from the thermoelastic solution onto a material point passing under the surface loads and then to solve for the other stress components using the equations of von Mises plasticity in conjunction with specific kinematic constraints. Figure 2a illustrates this procedure for a material point moving--at a fixed depth \(\eta\) below the surface loads--with velocity \(v\) in the negative \(\xi\)-direction through the domain \(\Omega^{*}\). During this process, plane-strain dictates \(\varepsilon_{\zeta\zeta}\equiv 0\). Additionally, the specific assumptions underlying both algorithms are the following [7,8]:  
MM algorithm: \[\sigma_{\eta\eta}\overset{!}{=}\sigma^{*}_{\eta\eta}\,,\quad \sigma_{\zeta\eta}\overset{!}{=}\sigma^{*}_{\zeta\eta}\,,\quad\varepsilon_{ \zeta\xi}\overset{!}{=}0\,,\quad T\overset{!}{=}T^{*}\] (11) JS algorithm: \[\sigma_{\zeta\xi}\overset{!}{=}\sigma^{*}_{\zeta\xi}\,,\quad \sigma_{\eta\eta}\overset{!}{=}\sigma^{*}_{\eta\eta}\,,\quad\sigma_{\zeta\eta }\overset{!}{=}\sigma^{*}_{\zeta\eta}\,,\quad\quad\quad\quad\quad T \overset{!}{=}T^{*}\,,\] (12)  
where we adopt the convention to indicate quantities known from the thermoelastic solution by a superscript asterisk \((\bullet)^{*}\) henceforth. We postpone the discussion of these assumptions, which reportedly enable good approximations in a variety of applications [6, 7, 8, 9, 10], to Section 4 and note that the MM method requires the determination of \(\sigma_{\zeta\xi}\) and \(\sigma_{\zeta\zeta}\) using \(\varepsilon_{\xi\xi}=\varepsilon_{\zeta\zeta}=0\) whereas only \(\sigma_{\zeta\zeta}\) must be computed using \(\varepsilon_{\zeta\zeta}=0\) for the JS case. To compute the unknown stress components, let us consider a material point moving below the surface loads as shown in Figure 2a and assume an elastoplastic response, i.e., \(f=0\) and \(\dot{\sigma}^{\prime}_{ij}n_{ij}-h_{\text{T}}\dot{T}\geq 0\) with \(f\), \(n_{ij}\) and \(h_{\text{T}}\) defined as in Section 2.1. Then, the total strain rate is given as the sum of elastic, plastic, and thermal strains,  
\[\dot{\epsilon}_{ij}=\underbrace{\frac{1}{E}\Big{(}\dot{\sigma}_{ij}-\
u\big{(} \dot{\sigma}_{kk}\dot{\sigma}_{ij}-\dot{\sigma}_{ij}\big{)}\Big{)}}_{=\dot{ \epsilon}^{\prime}_{ij}}+\underbrace{\frac{1}{h_{\epsilon}}\big{(}\dot{\sigma }^{\prime}_{kl}n_{kl}-h_{\text{T}}\dot{T}\big{)}n_{ij}}_{=\dot{\epsilon}^{ \prime}_{ij}}+\underbrace{\alpha\dot{T}\delta_{ij}}_{=\dot{\epsilon}^{\text{ th}}_{ij}}\,, \tag{13}\]  
where the isotropic Hooke's law, the flow rule Equation (8) and linear isotropic thermal expansion have been employed for the elastic, plastic and thermal strain rates, respectively. \(E\), \(\
u\), and \(\alpha\) denote [PERSON]'s modulus, [PERSON]'s ratio and the coefficient of thermal expansion. To treat both methods and their respective conditions, Equations (11) and (12), in a unified formalism, we now adopt the notation proposed by [PERSON] [8], so that the MM  
Figure 2: Illustration of the transient step of the residual stress approximation scheme as a continuous process (**a**) and as a corresponding discrete increment (**b**). The green dots in (**b**) represent the spatial discretization \(\Omega^{*}_{\text{d}}\) of the domain \(\Omega^{*}\) on which temperature and thermoelastic stresses are assumed to be given as input to the thermoelastoplastic approximation scheme. and JS schemes may be recovered by setting \(\Psi=0\) or \(\Psi=1\), respectively, in the derivations to follow.