# Power Dissipation and Wear Modeling in Wheel-Rail Contact
## 2 Formulation of Wheel-Rail Contact Problem
### Thermo-Elsto-Plastic Contact Model  
Let us formulate the rolling contact problem on the strip \(\Omega\). In the subsequent sections, we shall formulate rail material elasto-plastic deformation, friction, heat flow and wear models. Let us introduce the notation. We denote the strip displacement and its absolute temperature by \(u\) and \(\theta\), respectively. Both functions are dependent on spatial variable \(x=(x_{1},x_{2})\in\Omega\) and time variable \(t\in[0,T]\), and \(T>0\) is a given constant, i.e., we shall write \(u=u(x,t)=(u_{1},u_{2})\) and \(\theta=\theta(x,t)\). The strip deformation under moving loading is characterized in terms of the Cauchy stress tensor and linearized strain tensor [33] denoted, respectively, by \(\sigma=\{\sigma_{ij}\},i,j=1,2\), \(\sigma_{ij}=\sigma_{ji}\) and \(\varepsilon^{e}(u)=\{\varepsilon^{e}_{ij}(u)\}\). The Cauchy stress tensor and the elastic strain tensor are related by Hooke's law. Figure 1: Rolling contact problem. Consider the elasto-plastic deformation \(u\) of the rail strip under the loading of the moving wheel. Linear and kinematic hardening in this plastic deformation [33; 34] are taken into account. The generalized plastic strain \((\varepsilon^{p},\xi)\) and the generalized plastic stress \((\sigma,\chi)\) tensors [33] describe this transition process. In the framework of the small plasticity model [4; 32], the plastic strain \(\varepsilon^{p}(u)\) of the rail strip is the difference between the total strain \(\varepsilon(u)\) and the elastic strain \(\varepsilon^{e}(u)\):  
\[\varepsilon^{p}(u)=\varepsilon(u)-\varepsilon^{e}(u). \tag{1}\]  
The elastic strain \(\varepsilon^{e}(u)\) satisfies [PERSON]'s law [4]. The plastic strain trace equals zero in the domain \(\Omega\)[33]. Scalar internal variable \(\xi\) and back stress tensor \(\chi\) describe, respectively, hardening behavior and internal force appearing during hardening. This force translates the initial yield surface. Strain hardening is a process of making the metal stronger and harder due to plastic deformation. This phenomenon is characterized by the hardening tensor \(H\) defined as the proportionality coefficient relating in domain \(\Omega\) functions \(\xi\) and \(\chi\):  
\[\chi=-H\xi. \tag{2}\]  
We shall use the von Mises criterion to determine the rail material yield stress, distinguishing elastic and plastic ranges of stress. Let \(\varphi\) denote the smooth von Mises yield function defined as follows:  
\[\tilde{\varphi}=\frac{1}{2}\sqrt{\sigma^{D}:\sigma^{D}},\ \ \sigma^{D}\ \text{ deviatoric component of}\ \sigma \tag{3}\]  
For \(\tilde{\varphi}<0\) and \(\tilde{\varphi}>0\), respectively, the rail strip material is in an elastic and plastic deformation state [32; 35]. For \(\tilde{\varphi}=0\), the yield state is reached. The generalized plastic stresses \((\sigma,\chi)\) ensuring \(\varphi\leq 0\) are assembled in the set of admissible stresses \(\mathcal{K}\)[33], i.e.,  
\[\mathcal{K}=\{(\sigma,\chi):\tilde{\varphi}(\sigma,\chi)\leq 0\}.\]  
The plastic deformation process is governed either by the principle of virtual plastic work or by the principle of the associative flow rule [21]. These principles relate generalized plastic strains and stresses as well as the [PERSON] yield criterion. The associative flow rule governing the evolution of the generalized plastic strain \((\varepsilon^{p}(u),\xi)\) can be written in the following form [33]: there exists a scalar \(\phi\geq 0\) such that the time increment of the generalized plastic strain \((de^{p},d\xi)\) is proportional to the gradient \(\
abla\varphi\) of the yield function \(\varphi\) with respect to a spatial variable \(x\), and moreover, complementarity conditions are satisfied:  
\[(de^{p},d\xi)=\phi\
abla\varphi(\sigma,\chi)\ \ \text{and}\ \ \phi\geq 0,\ \ \varphi(\sigma,\chi)\leq 0,\ \ \phi\varphi(\sigma,\chi)=0. \tag{4}\]  
Recall from [33] the interpretation of condition (4), called the consistency condition. For \(\varphi=0\), plastic loading occurs if \(d\varphi=0\).