On the mathematical and thermodynamical descriptions of strain equivalence based anisotropic damage model

Despite the substantial research efforts and the noteworthy contributions, the modeling of damage induced anisotropy remains an unsolved and challenging issue. A mathematically rigorous and thermodynamically consistent framework for strain equivalence based anisotropic damage model is developed in this paper. The mathematical description rests on an improved representation theorem on stiffness. More specifically, to determine the stiffness that is expressed in the form of an irreducible decomposition, two decoupled orientation distribution functions for the shear and bulk moduli are introduced and then represented in terms of the Fourier series expansions. Upon such observations, two independent damage variables are then naturally selected to characterize the distinct damage mechanisms in the deviatoric and volumetric spaces. Based on the principle of strain equivalence and the concept of effective stress, the equivalent thermodynamical description is established, which constitutes the necessary supplement to its mathematical counterpart and facilitates the postulation of consistent damage evolution laws. For those initially isotropic materials, the isotropic, orthotropic and anisotropic (microplane-type) damages as well as the shear–bulk coupling effects, are rigorously modeled within the mathematical and thermodynamical descriptions, respectively, while the thermodynamical consideration of the microcrack closure–reopening effects is also presented. Together with the appropriately postulated damage loading function and the consistently derived evolution laws for the internal variables, a new orthotropic damage model characterized by product-type symmetrized effective stress and secant stiffness, is developed with application to concrete modeling under mode-I failure. The numerical predictions under uniaxial tension are also presented, preliminarily illustrating the capability of the model. To gain further insight, the overall theoretical aspect of the improved representation theorem and the secant stiffness of the developed orthotropic damage model, are comparatively discussed, from which the advantages of the proposed framework are identified.