An analytical solution of eccentric transverse deflection in rigid-plastic solids

Abstract

For a rigid-plastic solid obeying Tresca's yield criterion which deforms by the mechanism of extended slip, it conforms to the principle of rotation-rate continuity if the yield shear stress remains constant. It has been proved on this basis that the divergence of the strain-rate tensor is zero and the velocity vector field obeys the Laplace equation in the deformation region. This property can lead to two general solutions, one is derived into three-dimensional slip-line field theory; the other is applied to solve the problem of transverse deflection of a clamped plate deflected by a flat-ended rigid punch. For concentric loading conditions explicit analytical solutions can be obtained directly, while for eccentric loading conditions the method of fundamental solutions (MFS) is applied in the deformation region which is seen as a doubly-connected domain obeying the harmonic equation. The punching force needed for the deflection and strain components are then calculated. It has been obtained that the contour lines and the lines of steepest descent are two principle lines of any point in the deformation region, so the method of steepest descent is used to depict the trajectories of the principle lines. Numerical calculations have been made using commercial finite element software ANSYS-LSDYNA. It turns out that the results obtained from numerical calculations fit quite well with the analytical results based on the principle of rotation-rate continuity and MFS. Finally, NURBS curves and Coons surface are used to construct the configurations of slip-planes in this problem, where the normal of slip-planes and the planes on which the two families of slip-lines located are both curved.