On the nonexistence of certain solutions for damage mechanics models

The objective of this paper is to demonstrate that elastic and rigid plastic boundary value problems in damage mechanics may not have a solution. Two classes of damage mechanics models are considered. The constitutive equations of one class of models consist of a pressure-independent yield criterion, its associated flow rule, Hooke’s law and a damage evolution equation. The damage parameter enters the yield criterion. Therefore, these models are partly coupled. The constitutive equations of the other class are the constitutive equations of the classical rigid perfectly plastic model (or a rigid viscoplastic model) supplemented with an empirical ductile fracture criterion. The viscoplastic model contains a saturation stress. These models are uncoupled. In the case of partly coupled models, a simple boundary value problem is formulated and solved. It is shown that the solution breaks down for certain values of input parameters. In the case of uncoupled models, it is shown that empirical ductile fracture criteria are not compatible with solution behavior in the vicinity of maximum friction surfaces. An approach to formulate a new type of empirical ductile damage models is outlined.