Three-dimensional problem of perfect plasticity (kinematic equations determining three-dimensional plastic flow for a facet and edge of the Tresca prism)

In the present study a system of partial differential equations which describes kinematic of three-dimensional plastic flow for the states corresponding to an edge of the Tresca prismis obtained. The system includes the Cauchy equations and the compatibility equations formulated for the displacements and strains increments. These equations are then analysed by the aid of the triorthogonal isostatic co-ordinate net. The systemof kinematic equations is shown correctly determines displacements increments and be of the hyperbolic type. Relations for the displacements increments valid along principal stress lines are derived. Kinematic of plane and axial symmetric plastic flow are separately considered for each case. Kinematic equations for states corresponding to a facet of the Tresca prism which are of the less importance are also examined. Slip kinematic on a surface of maximumshear strain rate in perfectly plastic continuous media is studied. Sliding on the surface is shown can be realized only along asymptotic directions and only within hyperbolic zones of the surface (wherein the Gaussian curvature of the surface is negative). Integrable equations along asymptotic lines of the maximum shear strain rate surface for the jumps of tangent velocities are obtained. Kinematic equations corresponding to elliptic zones on a maximum shear strain rate surface (i.e. if the Gaussian curvature of the surface is positive) are derived and analysed.