The method of lines applied to crack problems including plasticity effect

To seek stress and strain distributions in flawed structures, the governing equations in elasto-plasticity are first derived in terms of displacement increments. The method of lines is then applied to derive the sets of ordinary differential equations with appropriate boundary conditions. Their solutions are sought along continuous lines of a discretized region. They are solved by a combination of power series and modal matrix method. A step by step integration is devised to determine the displacements at nodal points. Non-linear work-hardening is taken care of by the effective stress and strain approach. The cases of uncracked and cracked hollow cylinders of finite length under axisymmetrical loadings are studied in detail. The growth of plastic zone, crack-opening displacement and J-integral along various paths are illustrated.