An `extended' volumetric/deviatoric formulation of anisotropic damage based on a pseudo-log rate. (Q1867082)

It is well-known that the so-called ``basic'' formulation, introducing a representation of anisotropic elastic degradation through a second-order symmetric damage tensor and introducing the damage evolution laws, describes a Valanis-type restricted form of orthotropic damage combined with evolution laws defined in terms of a pseudo-logarithmic damage rate. In the present paper, the ``basic'' formulation is extended to provide a more general description of orthotropic damage for initially isotropic materials. The extension is based on the volumetric/deviatoric decomposition of the underlying isotropic undamaged stiffness and compliance. The new developments focus on the following constitutive aspects: ``extended'' secant relatons of pure elastic damage based on damage-effect tensors decomposed into products of isotropic and anisotropic parts; the definition of pseudo-logarithmic rate of damage and of the relevant conjugate force in the ``extended'' framework, general damage evolution laws in pseudo-logarithmic space and resulting tangent stiffness. A major new feature of the model is the introduction of a path parameter, allowing to assign different weights to bulk and shear damage components. In this case, two isotropic elastic constants and three evolving eigenvalues of the second-order damage tensor form a set of six parameters available to characterize the nine engineering material parameters entering the secant stiffness and compliance relations. A second important characteristic of the model are the underlying product-type decompositions of damage and damage-effect tensors into isotropic and anisotropic parts, which reflect themselves in a similar decomposition of effects of the consequent secant moduli. The product decompositions also allow inversion of the tensors involved, which preserve a dual structure between stiffness- and compliance-based versions of the formulation. A pseudo-logarithmic rate of damage is also introduced as in the ``basic'' formulation, which allows to derive a convenient thermodynamic force useful to determine the loading functions and damage evolution rules. The present ``extended'' formulation is closed by the derivation of the tangent stiffness.