An extended shakedown theory for elastic-plastic-damage material models

Internal variable elastic-plastic-damage, or elastic-damage, material models endowed with free energy are considered. Referring to a structure of such a material subjected to loads varying inside a given domain, the classical notion of (elastic) shakedown is widened to signify that the structure eventually responds to the loads in an elastic manner after certain (finite) amounts of plastic strain and/or damage have been produced. For structures fulfilling an ad-hoc D-stability requisite, an extended shakedown theorem is presented as a generalization of the classical Melan theorem to nonlinear elasticity and damage - besides nonlinear hardening. For common materials exhibiting linear elastic behaviour for constant damage, the extended Melan theorem saves its classical format, but the elastic stress response to the loads must be a damaged response computed on the basis of a trial time-independent damage field; more particularly, in the case of elastic-damage materials, no self-stresses are considered. The impending inadaptation collapse modes at the shakedown limit are studied under periodic loads showing that reverse damage is not allowed and thus either damage ceases after some transient phase or it continues afterwards in a ratchetting collapse mode until fracture occurs. The greater computational difficulties posed by the extended Melan theorem are observed and a few numerical applications are presented.