On the Hyperbolicity of Spatial Equations of Perfect Plasticity in Isostatic Coordinate Net

Abstract

The paper considers the problem of classifying a system of the partial differential equations of three-dimensional problem of the theory of perfect plasticity (for the stressed states corresponding to an edge of the Tresca prism), as well as determining the substitution of independent variables in order to reduce these equations to the analytically simplest Cauchy normal form. The initial system of equations is presented in the isostatic coordinate net and is essentially nonlinear. The criterion of maximum simplicity is formulated for the Cauchy normal form. The coordinate system is found to reduce the initial system to the simplest possible Cauchy normal form. The obtained condition when the system of equations takes the simplest possible normal form, shown in the paper, is stronger than the t‑hyperbolicity condition according to Petrovskii if we take t as the canonical isostatic coordinate which level surfaces form the spatial layers, that are normal to the field of the principal directions corresponding to the greatest (the lowest) principal stress.