An analytical method for the spherical stress wave equation in linear hardening materials

An analytical method for the spherical stress wave equation in linear hardening materials under impact loading is given in this paper. In this method, the elastic, unloading and plastically loading wave equations are solved, separately. In the elastic region, the solutions under different wave velocity ratio conditions are given. In the unloading region and the plastically loading region, the general solutions of the equations are given and the ordinary differential equations of the arbitrary functions are obtained. And then the exact solutions of the equations are given. The stress contour diagrams in the medium under typical loading conditions are given by the analytical method. The correctness of the analytical method is verified by comparing with the numerical results. On the stress contour diagrams, the discontinuities of the analytical results are clearer than that of the numerical results. A parametric study is conducted to show the effects of the parameters on the behaviour of stress distributions. This method can be useful for the rapid calculation of the elastic-plastic spherical stress waves in engineering and the theoretical analysing and law studying of the elastic-plastic spherical stress wave researches.