Damage induced anisotropy is crucial for those initially isotropic materials, e.g., quasi-brittle materials such as concrete, geomaterials, ceramics, etc. The modeling of anisotropic damage is not a straightforward task as that of isotropic one. Despite the substantial research efforts and the noteworthy recent contributions, this problem still remains a challenging issue, among which two major shortcomings of the existing damage models are to be resolved: (i)The damage variables are to a large extent arbitrarily selected without considering their physical meanings, regarding both the nature (scalar, vector, secondorder tensor, etc.) and the number; (ii)Directly using the concept of effective stress and the hypothesis of strain equivalence generally can not guarantee the major symmetry of the derived secant stiffness, leading to non-existence of an elastic potential. Introducing the energy equivalence hypothesis partially solves this problem but the physical definition of the damage variable is lost.
In this paper a novel and rigorous theoretical framework for anisotropic damage model is developed to remedy the foregoing shortcomings. This framework rests on an improved version of
] on (virgin and damaged) secant stiffness tensors and the equivalent thermodynamical considerations. To be more specific, damaged elastic properties are represented in terms of Fourier series expansion of the
modulus orientation distribution functions, where two damage variables respectively characterizing the damage mechanisms in the
spaces are inherently selected. For different degree of approximation, the geometric characters and macroscopic effects of the microdefects (microcracks and microvoids) can be described and well controlled by the selected damage variables. Corresponding to the above method, the equivalent thermodynamical formulations are established based on the concept of
and the hypothesis of
, where the Helmholtz free energy is decomposed into its deviatoric and volumetric components respectively influenced by the selected damage variables. The problems of lacking uniformity and rigor in the selection of damage variables and the incompatibility between physics and thermodynamics resulted from introducing the hypothesis of energy equivalence are thus solved.
To illustrate the proposed framework in modeling anisotropic (orthotropic) and isotropic damage, the special model with a second-order tensor for the deviatoric damage and a scalar for the volumetric damage, as well as the one with two damage scalars are exemplified. Then by considering the deviatoric-volumetric coupling Helmholtz free energy, a restrictive orthotropic damage model with a single second-order damage tensor is presented, demonstrating its capability of describing the shearbulk coupling effects experimentally evidenced in those quasi-brittle materials.