A unified theory of elastic degradation and damage based on a loading surface

A number of new models with stiffness degradation have been proposed in recent literature in the small strain regime. However, most of these works represent specific formulations, each using its own terminology, notation and assumptions, and relatively little effort has been spent so far towards achieving a common theoretical framework similar for instance to the theory of elastoplasticity. Moreover, most of the existing damage models are presented with intensive recourse to abstract thermodynamics concepts, and they combine stiffness degradation with plasticity, which (though being ultimately necessary to represent the actual material behavior) makes it much more difficult to isolate, analyse and understand the properties of the formulation for elastic stiffness degradation. As a contribution in this field, this paper presents a unifying theoretical framework to describe a class of models for elastic stiffness degradation based on the concept of loading surface. The derivation includes two consecutive steps: first, the constitutive framework for elastic-degrading models with evolution laws which are expressed directly in terms of the secant stiffness (or compliance) tensor, and second the elastic-damage models, in which the scant stiffness (or compliance) is assumed to depend on a reduced set of damage variables with clearer physical meaning and simpler evolution laws. Whenever possible, terminology is borrowed from the classical formulation of elastoplasticity, and thermodynamic concepts are introduced only as needed. Both stress-based and strain-based developments are compared throughout the paper, and the concept of associativity is reanalysed and generalized within the new unified framework of elastic degradation. The most significant scalar damage models found in the literature are reinterpreted in the context of this unified theory. Finally, a general expression is obtained for the tangential stiffness operator of associated scalar models (stress- and strain-based) of the (1-D) type, that includes all the models considered as particular cases. More general damage formulations [scalar non-(1-D), vectorial, tensorial] are reviewed and discussed systematically in a sequel paper.