On the formulation of anisotropic degradation using a pseudo-logarithmic damage tensor

In spite of its appeal, anisotropic damage is being introduced in the constitutive equations of engineering materials at a slow pace. One of the main reasons is the difficulty of establishing general evolution laws. This originates from the lack of physical meaning of the thermodynamic forces conjugate to the damage variables, which finally constitute the space in which loading functions and 'damage rules' are defined. In this article, the authors propose a new 'pseudo-logarithmic' rate of damage, which has the advantage of exhibiting a simple and meaningful conjugate force with very convenient properties. A main advantage is the physical interpretation of the corresponding "damage rule", which clearly separates the effects of its volumetric part, responsible for isotropic degradation, from its deviatoric part, responsible for anisotropic effects. This new concept is applied to a second-order tensor secant formulation, which is developed using traditional concepts of Continuum Damage Mechanics within the general theoretical framework of elastic degradation and damage recently proposed by the authors. As the first example problem, a 'generalized pseudo-Rankine' damage model is developed. Depending on the value of a single parameter, the loading surface may assume shapes which vary between a pi-plane cut-off and a Rankine criterion, causing a transition from a purely isotropic to a highly anisotropic degradation response. A simple closed-form solution is obtained for uniaxial tension, which allows to interpret the remaining material parameters in terms of the traditional peak strength sigma_peak and fracture energy per unit volume g_f. The possibilities opened with the new approach are illustrated with a numerical tension/shear test with significant rotation of principal strains. The results exhibit complex features, such as loss of coaxiality and secondary stress peaks, that agree with results obtained with more complicated models of the multiple fixed crack type.