Thermoelastic Wheel - Rail Contact Problem with Temperature Dependent Friction Coefficient

Abstract

The paper deals with the numerical solution of wheel-raił rolling contact problems with the temperature field and wear as additional components. We shall consider the contact of a wheel with an elastic raił resting on a rigid foundation. It is assumed that the friction between the bodies is described by the Coulomb law. Moreover we assume a frictional heat generation and heat transfer across the contact sur face as well as Archard 's law of wear in contact zone. The friction forces and the heat flux depend on the friction coeflicient. This coeflicient is assumed to depend on temperature. In the paper quasistatic approach to solve this contact problem is employed. This approach is based on an assumption that for the observer moving with the rolling body the displacement of the supporting foundation is independent on time. The system is described by the elliptic variational inequality governing the displacement and the parabolic equation governing the heat flow. In order to solve numerically this system we will decouple it into mechanical and thermal parts. Finite element method is used as a discretization method. Using duality theory and the regularized relation between tangent and norma! components of the contact stress we formulate the mechanical part of this problem as an optimization problem with respect to the norma! contact stress. Pschenichnyj method combined with Newton method are used to solve numerically th.is discretized optimization problem. Next for calculated displacement field the thermal part of the system is solved using Newton method. Numerical examples showing the influence of the temperature dependent friction coefficient on the contact !ocal traction and the length of the contact zone are provided.

Thermoelastic rolling contact problem with temperature dependent friction

“…Denote by u = (u 1 , u 2 ), u = u(x, t), x ∈ Ω, t ∈ (0, T ), T > 0, a displacement of the strip and by θ = θ(x, t) the absolute temperature of the strip. The displacement u and the temperature θ of the strip satisfiy the system of equations [2,3,5,9,10] …”

“…The systems O x 1 x 2 and Ox 1 x 2 are related by x 1 = x 1 − V t and x 2 = x 2 . Assuming that for the observer moving with a wheel the displacement u(x 1 , x 2 ) of the strip does not depend on time [1,2,7] it implies u (x 1 , x 2 )= u (x 1 − V t, x 2 , t) = 0, therefore we obtain…”

“…This vector is a priori unknown and is given by conditions of contact and friction. These conditions on Γ C × (0, T ) take a form [1,2,4,6,7,9,10] …”

“…Let Ω denotes now the moving part of the strip seen by the observer. Taking into account (5) quasistatic approximation of the problem (1) -(3) takes the form [2]: find functions u and θ dependent on x ∈ Ω satisfying…”

“…This vector is a priori unknown and is given by conditions of contact and friction. These conditions on Γ C × (0, T ) take a form [1,2,4,6,7,9,10] Friction coefficient µ = µ(θ) is dependent on temperature θ. g r denotes the gap between bodies in contact. w = w(x, t) denotes an internal state variable to model the distance between the bodies due to wear [9] and satisfies the Archard law…”