Mathematical model of the elastoplastic shell collapse, taking into account the possible development of the process instability

The mathematical model for the subsequent numerical study of the shaped charge liner collapse affected by external surface forces simulating an explosive load is presented. The basic liner was considered as an originally cylindrical compressible elastoplastic shell within the framework of a two-dimensional flat nonstationary problem of continuum mechanics. To ensure the rationality of the modeling and numerical calculation at the initial time the design fragment was discriminated in the liner by central beams. Deformation of the fragment being a part of the shell was taken into account by the boundary conditions of cyclic repeatability in the tangential direction. For numerical solving the well-known Wilkins Lagrangian method was used, which was refined in terms of the relations describing the mechanical behavior of an elastoplastic medium. Additionally, a self-developed grid adjustment procedure was used, excluding the appearance of highly elongated cells in the calculation. The instability of the shell deformation was initiated by harmonic surface perturbations, initially assigned on the outer or inner surfaces. The degree of instability was assessed by the deviation of the disturbed surface (or the boundary of the so-called stream-forming layer) from the cylindrical one. The used finite-difference algorithms are implemented in the form of appropriate calculation programs. A number of computational verification measures was performed proving the viability of the developed mathematical model and the possibility of its further use

Spherically Symmetric Shock Waves in Materials with a Nonlinear Stress-Strain Dependence

The paper considers the dynamic deformation features of constructional materials with nonlinear stress-strain dependence. For the one-dimensional shock waves with nonzero curvature arising in constructions under dynamic loading the propagation regularities are studied on the basis of the matched asymptotic expansions method. In the nonstationary problem with the longitudinal spherical shock wave the relations for simultaneous consideration of dynamic properties in the outer and inner problem of the perturbation method are obtained. The solution in the front-line area is constructed on the basis of the evolution equation different from ones for a plane longitudinal wave. The need for a solving of an additional ODE system for matching outer and inner expansions is shown. It is obtained that the outer solution asymptotics in the spherically symmetric problem contains waves reflected from the leading front in contrast to the solution behavior behind the front of the plane shock wave.