Mathematical simulation of elastoplastic deformation in cubic materials with an account of anisotropic bulk compressibility

Abstract

It was shown for the first time that when modelling the deformation of materials with cubic symmetry (at full stress), the rotation of the computational axes leads to the identification of anisotropic volumetric compressibility. Loading of the materials with cubic symmetry of properties in the directions not coincided with the main directions (for example, 011) allows one to detect 75% cases of the auxetic single crystals (i.e. with negative Poisson’s ratio). In these cases, the negative volumetric compressibility has anisotropy, in contrast to the volumetric compressibility calculated along the crystallographic axes for cubic materials. Anisotropy of the volumetric compressibility leads to anisotropy of velocities of propagation of body waves. This paper considers several elastoplastic problems for a cubic material with different orientations of a coordinate system about its crystallographic axes. The behaviour of such a material under dynamic loads is modelled with an account of anisotropic bulk compressibility to provide the same anisotropy of bulk wave velocities in the elastic and plastic ranges and a uniform pressure function at the elastic-to-plastic strain transition. For each orientation of the coordinate system and respective planes, different values at the indicatrices of elastic constants are specified, and this specifies different deformation processes in cubic materials. Such an effect is demonstrated by solving three problems in three-dimensional (3D) statements approximating the following processes: (1) one-dimensional elastoplastic deformation in a thin target impacted by a thin plate; (2) uniform compression in a spherical body under pulsed hydrostatic pressure; and (3) 3D elastoplastic deformation in a cylindrical body striking a rigid target in view of anisotropic bulk compressibility. The problems were solved numerically using original programs based on the finite element method modified by GR Johnson for impact problems. Solving problems in a 3D formulation makes it possible to take into account the dependences of the direction of the elastic and plastic characteristics of the material, as well as the velocities of propagation of elastic and plastic waves from that direction. The simulation results suggest that for cubic materials, changing the orientation of two coordinate axes in a plane changes the strains along all three axes, including those perpendicular to this plane. It is concluded that anisotropic bulk compressibility in cubic materials should be allowed for by mathematical models of their elastic and plastic deformation. We demonstrate that the orientation of a computational coordinate system for cubic materials should be in those directions in which their deformation is analysed in each particular case.