Nonlinear model of ice surface softening during sliding taking into account spatial inhomogeneity of strain, stress and temperature

The model of ice surface softening is represented by a system of three one-dimensional partial differential parabolic equations, taking into account the spatial inhomogeneity. Using one-mode and adiabatic approximations, an analytical soliton solution of a one-dimensional Ginzburg-Landau differential equation for the spatial normal distribution of shear strain to the ice surface is obtained. The analytical form of the numerical procedure for solving the equations, including initial and boundary conditions, is written on the basis of an explicit two-layer difference scheme. The distributions of time and stationary values of static friction force, kinetic friction force and temperature are constructed. Two cases were considered: 1) the upper and lower surfaces move with equal velocities in opposite directions; 2) the upper surface moves along the stationary lower surface. The dependencies of stress, strain and temperature on the coordinate in the normal direction to the surface are determined for different time series. It is shown that a stationary distribution of friction forces and temperature along the thickness of the near-surface ice layer is established with time. The values of the kinetic and static friction forces in the near-surface ice layer increase monotonically with distance from the friction surfaces, while the coordinate dependence of the temperature has a nonmonotonic appearance. The stationary values of the static friction force in the near-surface ice layer decrease with increasing temperature of the friction surfaces, indicating that the surface transforms to a more liquid-like state, while the coordinate dependence has a monotonically increasing form.