Tri-axial deformation of a plastic-rigid solid

Abstract

The paper deals with the deformation of an ideal plastic solid that is initially rigid, i.e., elastic deformations are neglected. In addition, the solid flows by means of the mechanism of extended slip, for which the rotation-rate vector field remains continuous and the strain-rate tensor is solenoidal. The Tresca yield criterion applies to such a solid and with an associated flow-rule is represented in a manner that includes both bi-axial and tri-axial states of strain. Two new theorems are proved, and a second-order partial differential equation is derived for the first invariant of the stress tensor (hydrostatic pressure); the analogue of a similar published equation for the bi-axial strain case. To illustrate the methodology, the above theory is applied to the tri-axial problem of a thick metal plate clamped round a circle and deflected by means of pressure. It is shown, from the exact solution, that the errors due to the use of the approximate membrane formula for a clamped thin plate are small, even for a 6 mm thick plate clamped on a circle 100 mm in diameter. Surprisingly, there is an initial thickening of the plate and it is shown that, regardless of the plate thickness, the pressure passes through a maximum at a deflection equal to 1/ √ 3 of the radius of the clamping circle.

Dynamic Transverse Deflection of a Free Mild-Steel Plate

“…We obtain from the second of these equations, by Equation (A.1), of appendix A, and incompressibility, In general, letting V denote the velocity vector, the strain-rates are given by [8] ,…”

“…where R is the current radius of a cylinder, with its generators parallel to the punch axis, cutting the inner edge of the boundary It has been shown [8] that the velocity in the plate in the plastic domain may be obtained from Equation (A.10). By Equation (3.7), since B is a unit vector of constant direction, this equation, in the case of the circular plate and concentric circular punch of radius a, leads to the solution…”

“…Yet the very concept of an equation of state for metals has been criticized by a number of prominent writers. The purpose of the present paper is to set forth the principles necessary to properly analyze the dynamic response of a thin freely supported plate of mild-steel struck at high constant speed by a flat ended rigid punch, assuming the plate to deform by means of the mechanism of extended slip [8][9][10][11][12].…”

“…It will be shown that this boundary-wave travels at one-half of the velocity of elastic waves of rotation in the solid concerned. As in quasistatic deformation by extended slip [8][9][10][11][12], the strain-rate tensor within the plastic domain remains solenoidal.…”

“…Flow by means of extended slip commences, when sufficient cold work has been expended so that these domains have grown suffi-ciently to join, at which stage the yield "curve" of the material exhibits a definite point of yield in loading. The yield curve is generally bi-linear in the case of such solids [8][9][10][11][12] and yield is regulated by the Tresca yield criterion [14] because the surfaces of extended slip are aligned with the surfaces of maximum shear associated with the external constraints and the Tresca criterion requires the maximum shear stress to equal the static shear yield stress of the solid; but we shall amend this criterion so as to encompass the dynamic case directly.…”