Integrity conditions for elastic-plastic damaged solids subjected to cyclic loading

Integrity conditions for elastic-plastic, isotropically damaged solids with isotropic and kinematic strain hardening as subjected to cyclic loading, are in question. It is assumed that the damage process is coupled with the process of plastic deformation. The shakedown conditions are assumed to be satisfied. A new sufficient condition for shakedown accounting for a mixed isotropic and kinematic hardening is developed. The problem of evaluating limit yield-condition arguments is reduced to a min-max problem. In the case of plain strain, the problem is equivalent to a hyperbolic equation in partial derivatives of the second order. A method for computing the purely elastic damaged response of the solid to the prescribed loading program is proposed. The limit yield condition with specified arguments makes it possible to obtain upper and lower estimates for the local actual limit values of the damage parameter admitted by the yield condition for the given loading program. The estimates lead to necessary and sufficient conditions of integrity. The proposed method is illustrated by an example.