An ‘extended’ formulation of isotropic and anisotropic damage with evolution laws in pseudo-log space

Considerable effort has been devoted in recent years to the formulation of elastic degradation and damage. A large portion of damage models, however, still adhere to variations of the traditional ``(1-D)-type'' scalar formulation. In previous papers, the authors presented a unified framework for elastic degradation and damage which is similar in terminology and notation to the well-known theory of elasto-plasticity. The traditional scalar damage model, in which only E decreases while nu remains constant, was easily recovered as the first example of application. In the present paper, the unified theory is briefly revisited, and new formulations for elastic degradation of initially isotropic materials are presented. In a first step, the `basic' anisotropic formulation based on product-type symmetrization of effective stresses and energy equivalence is recovered. A new pseudo-log damage rate is introduced, that makes possible a convenient simple form of the conjugate force and a physical interpretation of the ``damage rule''. In a second step, an `extended' anisotropic damage formulation is presented with a second-order damage tensor that remains isochoric, plus two scalar damage variables. The extended formulation encompasses a more general type of isotropic degradation with simultaneous evolution of the two elastic parameters. Examples of application include the `extended' associated scalar damage model, and a `pseudo-Rankine' anisotropic model which exhibits convenient properties and looks very promising for practical calculations.