## 2 Thermoelastoplastic Approximate Algorithms
### Approximate Plane-Strain Rolling and Sliding Contact Algorithms
Then, the conditions \(\dot{\epsilon}_{\xi\xi}=0\) (MM, \(\Psi=0\)) and \(\dot{\sigma}_{\xi\xi}=\dot{\sigma}_{\xi\xi}^{*}\) (JS, \(\Psi=1\)) can be summarized to the single equation  
\[\dot{\epsilon}_{\xi\xi}= \frac{1}{E}\Big{(}\dot{\sigma}_{\xi\xi}-\
u\Big{(}\dot{\sigma}_{ \zeta\xi}+\dot{\sigma}_{\eta\eta}^{*}\Big{)}\Big{)}+\frac{1}{h_{\xi}}\Big{(} \dot{\sigma}_{\xi\xi}n_{\xi\xi}+\dot{\sigma}_{\zeta\xi}n_{\zeta\xi}+\dot{\sigma }_{\eta\eta}^{*}n_{\eta\eta}+\dot{\sigma}_{\xi\eta}^{*}n_{\xi\eta}-h_{\rm T} \dot{T}^{*}\Big{)}n_{\xi\xi}+\alpha\dot{T}^{*}\] \[\overset{!}{=} \Psi\left(\frac{1}{E}\Big{(}\dot{\sigma}_{\xi\xi}^{*}-\
u\Big{(} \dot{\sigma}_{\zeta\xi}+\dot{\sigma}_{\eta\eta}^{*}\Big{)}\Big{)}+\frac{1}{h_ {\xi}}\Big{(}\dot{\sigma}_{\xi\xi}^{*}n_{\xi\xi}+\dot{\sigma}_{\zeta\xi}n_{ \zeta\xi}+\dot{\sigma}_{\eta\eta}^{*}n_{\eta\eta}+\dot{\sigma}_{\xi\eta}^{*}n_{ \xi\eta}-h_{\rm T}\dot{T}^{*}\Big{)}n_{\xi\xi}+\alpha\dot{T}^{*}\right). \tag{14}\]  
Note that \(\dot{\sigma}_{\xi\xi}\) is replaced by \(\dot{\sigma}_{\xi\xi}^{*}\) in the otherwise identical sum in the second line. The plane-strain kinematic constraint yields  
\[0\overset{!}{=}\dot{\epsilon}_{\zeta\xi}=\frac{1}{E}\Big{(}\dot{\sigma}_{\zeta \zeta}-\
u\Big{(}\dot{\sigma}_{\xi\xi}+\dot{\sigma}_{\eta\eta}^{*}\Big{)} \Big{)}+\frac{1}{h_{\xi}}\Big{(}\dot{\sigma}_{\xi\xi}n_{\xi\xi}+\dot{\sigma}_ {\zeta\xi}n_{\zeta\xi}+\dot{\sigma}_{\eta\eta}^{*}n_{\eta\eta}+\dot{\sigma}_{ \zeta\eta}^{*}n_{\xi\eta}-h_{\rm T}\dot{T}^{*}\Big{)}n_{\zeta\zeta}+\alpha \dot{T}^{*}\,. \tag{15}\]  
For an _elastic_ response, i.e., if either \(f<0\) or \(f=0\) and \(\dot{\sigma}_{ij}^{\prime}n_{ij}-h_{\rm T}\dot{T}<0\), we can obtain the corresponding equations by simply dropping all terms related to the plastic strain rate \(\dot{\epsilon}_{ij}^{\rm P}\) from Equations (13)-(15). Ideally, the differential equation system (14) and (15)--or its particularization for elastic behavior--should at each arbitrary but fixed depth \(\eta\) be integrated w.r.t. \(\xi\) from a stress-free initial state to a stress free stationary state of the _elastic_ solution, i.e., within bounds \(\xi=\dot{\xi}_{+\infty}^{*}\) and \(\xi=\dot{\xi}_{-\infty}^{*}\leq\dot{\xi}_{+\infty}^{*}\) sufficiently far away from the surface loads so that the stress \(\sigma_{ij}^{*}\) of the elastic solution has decayed to zero. Thus, the residual stress approximation \(\sigma_{ij}^{\rm P}\) obtained by such an integration will by virtue of Equations (11) and (12) satisfy the conditions on the stresses that are required for homogeneity in the \(\xi\)-direction in a plane strain half-space problem [5],  
\[\sigma_{\eta\eta}^{*}\overset{!}{=}0\qquad\text{and}\qquad\sigma_{\xi\eta}^{ *}\overset{!}{=}0\,. \tag{16}\]  
Additionally, as noted by [PERSON] and [PERSON] [5], the residual strains have to satisfy the conditions  
\[\varepsilon_{\xi\xi}^{*}\overset{!}{=}0\qquad\text{and}\qquad\varepsilon_{ \zeta\xi}^{*}\overset{!}{=}0\,. \tag{17}\]  
The condition on the residual strain \(\varepsilon_{\xi\xi}^{*}\) will only be guaranteed for the MM approach but not for the method of JS; compare Equations (11) and (12). Furthermore, since the JS approach enforces \(\sigma_{\xi\xi}=\sigma_{\xi\xi}^{*}\), the corresponding residual stress component will generally not be approximated well but vanish after a complete pass below the loaded region. Finally, if we chose to integrate only to \(\dot{\xi}_{-}^{*}>\dot{\xi}_{-\infty}^{*}\) where the stress \(\sigma_{ij}^{*}\) is not yet fully decayed to zero, but already sufficiently so as to not sustain any further plastic response, there will in any case be some violation of Equation (16).