Anisotropic damage evolution laws deduced from normality structure
基于正则结构的各向异性损伤演化律

Generally speaking, damaging is always anisotropic even for the initially isotropic materials. This paper deals with anisotropic damage evolution laws, whose complex tensorial and high-degree nonlinear properties make them the most elusive parts of such anisotropic models and hinder their practical application. Most damage evolution laws in existing phenomenological damage models can be covered by the linear irreversible thermodynamics which is also termed as the phenomenological equations. The essential problem is to identify the conditions of the phenomenological equation and the specific form of the damage characteristic tensor J for a solid weakened by preexisting microcracks. The essential problem cannot be solved in phenomenological damage models, where the phenomenological equation is taken as a prerequisite rather than a conclusion. Even if the phenomenological equation really holds true, irreversible thermodynamics or current available experimental data is not enough to determine the damage characteristic tensors J uniquely. In this paper, the basic internal variables include numerous vector form variables which correspond to microcracks. The crack tensor is taken as the averaging measurement of the basic internal variables. The generalized crack tenor and brittle indexes are introduced to characterize the damage potential function and damage characteristic tensor. Rice's kinetic rate laws of local internal variables, with each rate being stress dependent only via its conjugate thermodynamic force, are corner stones of the normality structure. In this paper, it is revealed that the phenomenological equations emerges from normality structure if all the kinetic rate laws are homogeneous functions of the same degree in their conjugate forces. Since in most cases the relationship between propagation rate of cracks and their energy release rate can be covered by power laws, the homogeneous condition is fulfilled naturally. Therefore, it is concluded that the phenomenological equations are appropriate form for damage evolution laws. The damage characteristic tensor possesses the same symmetry and positive definiteness as the elasticity tensor. Based on the deduced analytic damage characteristic tensor, it is clear that the fourth-order identity tensor is not suitable to be taken as a damage characteristic tensor. The damage characteristic tensor depend on not only the current microstructural parameters but also the current conjugate forces.