Conditions for shakedown of damaged elastic plastic bodies

Conditions for shakedown of damaged elastic plastic bodies with internal variables were investigated. The damage was assumed to be brittle and isotropic. The damage evolution equation was integrated, so that a relation between the damage parameter and the damage energy release rate was obtained. The relation provides an opportunity to transfer from the effective stress tensor components to the nominal ones in the yield function. The problem of deriving the plastic adaptation conditions was reduced to a min-max problem of mathematical programming. The notion of the core of the limit yield surface was introduced. As a result, a lower estimate for limit value of the parameter was obtained, and necessary conditions for general (plastic and damage) adaptation are formulated. The notion of the effective limit stress was introduced for the case where the yield function was an homogeneous function of the first order in the stress tensor components, and strain hardening is isotropic. Lower estimates for the damage parameter and the parameter of strain hardening were obtained, and necessary conditions for general adaptation were derived. A modification of the material model was proposed such that the damage process ceases along with the process of plastic deformation. The developed method was expounded by an example.