On the localization of discontinuities in calculations of incompressible elastic media dynamics

The problem of algorithmic localization of closely spaced discontinuities of strains is considered in the present paper by the example of one-dimensional cylindrical rupture surfaces (shock waves). These shock waves are generated in a cylindrical layer of an incompressible elastic medium through the twisting impact action in the presence of preliminary antiplanar strain. We assume that strains (preliminary and acquired in the course of propagation of boundary perturbation in the environment) are finite, and the Almansi tensor is used as their measure. It is shown that the boundary shock perturbation causes in a medium two front surfaces of strain discontinuity: the plane-polarized shock wave of load, which increases the preliminary antiplane shear, and the neutral circularly polarized wave, which changes the shift direction in accordance with the impact produced. Velocity values, with which the discontinuity surfaces propagate in the elastic medium, are determined. The propagation velocity of a plane-polarized shock wave of load depends on the pre-strain in the medium and the intensity of the boundary impact. The propagation velocity of the shock wave of circular polarization (neutral shock wave) is completely determined by pre-strain in the medium. In order to determine the position of the discontinuity surfaces and calculate the intensity of discontinuities at each time step, we are building special frontline ray expansions, which are embedded in the finite-difference calculation scheme. These expansions are written for grid points outside the frontal region. A way of constructing frontline ray expansions behind the surface of the strain discontinuity is specified. It is based on the recurrence properties of geometric and kinematical compatibility conditions. A numerical algorithm and a program for calculating the displacement fields and a component of the Cauchy stress tensor are constructed for the specified problem. Results of the computational experiment for the rubber-like material with the specified properties are obtained.