MATHEMATICAL AND THERMODYNAMICAL REPRESENTATIONS OF ANISOTROPIC DAMAGE MODELS FOR CONCRETE

Despite the substantial research efforts in the mesoscopic damage mechanics, continuum damage mechanics and microplane theory, etc., the modeling of damage induced anisotropy remains a challenging issue. Firstly, to (partially) solve the exhibited problems and to investigate the interrelations among these models, a mathematical representation of continuum damage model was established based on the improved stiffness representation theorem. The orientation distribution functions (ODFs) for the macroscopic bulk and shear moduli were introduced, and the corresponding macroscopic volumetric and deviatoric damage ODFs were expanded into the forms of Fourier serials, from which the coefficients in the irreducible form of stiffness tensor can be uniquely determined. Based on the above approach, the isotropic and anisotropic damage models, in which a damage scalar was used to describe the isotropic volumetric performance whereas an additional damage scalar or a second-order damage tensor was postulated to represent the isotropic or orthotropic deviatoric behavior, respectively, were derived without the introduction of the phenomenological assumptions (e.g. the strain equivalence, the energy equivalence, etc.). Secondly, the equivalent thermodynamical representation of continuum damage model was developed within the framework of irreversible thermodynamics furnished with the theory of internal variables. The general expression of the Helmholtz free energy potential was proposed and the consistent evolution laws of the involved damage variables were derived from the postulate of maximum damage dissipation. As illustrative examples, the above isotropic and orthotropic damage models were re-derived from the thermodynamical representation. Finally, to investigate the influences of the microstructural changes on the macroscopic nonlinear material behavior, second-order and fourth-order fabric tensors were proposed to develop the mathematical and thermodynamical representations for the orthotropic and anisotropic microplane models. The interrelations of physical quantities on the macroscopic and microplane levels, such as the damage variables and the conjugated damage energy release rates, the damage dissipations and the evolution laws of the damage variables, were systematically obtained. The proposed approach can be used to develop the multiscale damage model for concrete in the near future.