KINEMATICALLY DETERMINED AXIALLY-SYMMETRICAL PLASTIC FLOWS OF METALS AND GRANULAR-MATERIALS

In certain plastic regimes for axially symmetric flows of rigid-perfectly plastic solids obeying Tresca's yield condition and associated flow rule, the velocity equations uncouple from the stress equations and are kinematically determined. Moreover, the same equations apply for axially-symmetric deformations of an incompressible granular material which deforms according to the Coulomb-Mohr yield condition and the so-called 'double-shearing theory'. Here we derive a new family of exact solutions of these equations and the underlying first-order ordinary differential equation is shown to be exactly integrable. When the integration constant B is set to zero, two simple families of exact solutions are obtained; one is well known while the other appears not to have been given previously. Moreover, we note that these two simple families of exact solutions can be combined to present an additional exact solution which is continuous across the join z = 0. When the constant B is non-zero, the solution involves the roots of a quartic equation and there appears to be no simple expressions for the associated velocity field. For the new solution arising from B = 0, a partial analysis of the corresponding stress field is presented assuming Tresca's yield condition, and some simple closed expressions, which incorporate gravity, are obtained. The stress-field calculations assuming a Coulomb-Mohr yield criterion are far more elaborate and are not attempted here. Stream paths for both the previously known exact solution and the new solution are displayed.