Spurious energy dissipation/generation in stiffness recovery models for elastic degradation and damage

number of constitutive models have been proposed in recent years for elastic degradation and damage, many of which include procedures for the recovery of stiffness upon closure of tensile microcracks. Most of these recovery procedures are based on the decomposition of stress or strain into positive (tensile) and negative (compressive) components, which are incorporated in the elastic formulation taking recourse to fourth-order positive and negative projection operators. Due to the non-dissipative nature of microcrack closure-reopening for a certain fixed state of degradation, the recovery formulation should possess a well-defined energy potential along the line of hyperelasticity, which conserves energy upon closed-loop load histories. This condition seems to have escaped the rapidly expanding literature on damage mechanics, i.e. closure formulations have not been verified in this regards. In the paper, the (lack of) energy conservation is examined in terms of the spurious dissipation rate, which is developed for a relatively general class of recovery models. They include the positive-negative projection operators and the bimodular formulations with different stiffnesses for tension and compression. It is shown that under proportional loading in strain or stress, all these formulations are energy conservative. Under non-proportional loading, howeve:r, they are only conservative in conjunction with isotropic degradation, and they exhibit spurious dissipation-generation when anisotropic degradation is considered and the load history involves rotation of principal directions. Copyright 0 1996 Elsevier Science Ltd.