Spherical elastic-plastic wave solutions (Q1095707)

Two closed-form radially symmetric elastic-plastic wave solutions, corresponding to spherical cavity loadings, are compared to numerical solutions constructed by a finite difference code. An infinitesimal strain theory, with plastic incompressibility, a Mises yield condition and either perfectly plastic flow or parabolic work hardening, allows wave function representations in both elastically and plastically deforming regions. The solutions examined are for (a) the loading problem in which the cavity wall pressure is raised instantaneously to a value \(p_ 0\), sufficient to induce plastic yielding, then maintained at \(p_ 0\), and (b) the unloading problem when the wall pressure is released continuously after the instantaneous step to \(p_ 0\). In the former case there is a region of continuous plastic loading extending from the cavity wall to the diverging plastic discontinuity front which persists for a finite time until the plastic discontinuity is annulled. In the latter case, provided that the initial pressure decrease rate exceeds a critical value, the region behind the diverging plastic discontinuity front is purely elastic. In both cases the numerical solution smears the plastic discontinuity, and while the maintained plastic loading in case (a) allows the maximum pressure to be attained over a finite distance, this is prevented by the elastic unloading in case (b) and the following release wave is significantly distorted. It is shown that suitable ramping of the applied load produces an excellent analytic-numeric agreement unless the continuous unloading is too rapid.