The appropriate modeling of damage induced anisotropy and unilateral effectsdue to macrocracks evolution are rather crucial to predict the nonlinearresponses of concrete material and structures. With the introduction ofscalar, vector, second-order and fourth-order tensors, or even higher damageinternal variables, continuum damage mechanics (CDM) is capable ofmacroscopically describing the influences of microcracks evolution on thenonlinear behavior of concrete and therefore has been widely adopted in theliteratures. From the physically motivated viewpoint, the terminology``damage'' is directly characterized as the degradation of the stiffness orequivalently, the increase of the compliance. Therefore, the fourth-ordertensor and more specifically, the material stiffness (or the materialcompliance, or the change relative to the initial value), is perhaps themost rational candidate as the damage variable. Accordingly, the so-calledelastic degradation model which introduces the degradation strain rate waspreferred in the modeling of damage induced anisotropy. To describe theunilateral effects upon cyclic loadings, the decomposition of rank-twovariables (e.g. stress, strain, effective stress, etc.) into the positiveand negative components along with the corresponding fourth-order projectionoperators is generally adopted. Nevertheless, the expressions for theprojection operators are not unique, and all the existing CDM models formicrocracks induced anisotropy and unilateral effects imply athermodynamically consistent result, i.e. the non-zero energy generationsupon damage unloading when applied to describe the unilateral effects.In this paper based on the framework of irreversible thermodynamics and thetheory of internal variables, a thermodynamically consistent anisotropicdamage model for concrete is proposed. To model the unilateral effects, thestress tensor is decomposed into its positive and negative components. Theincrements of the intrinsic compliances under purely positive and purelynegative stress states are adopted as the internal damage variables todescribe the microcracks evolution on the macroscopic nonlinear behavior ofconcrete. The conditions for zero spurious energy dissipations are presentedand then new expressions for projection operators are introduced toguarantee the thermodynamical consistency. The damage evolution laws arepostulated by analogy to the flow rules in classical plasticity and theplastic-type rate constitutive relation is derived. The numericalimplementation of the proposed model including the back-Euler method basedstress updating algorithm and the algorithmic consistent tangent moduli arealso presented in details. Finally, the model is specialized to concretewith appropriate evolution laws and is applied to numerically simulate thestress-strain relations of several typical loading cases. The obtainednumerical predictions agree with the experimental data fairly well,demonstrating the validity of proposed model. It is worth to note that,neither the principle of strain equivalence nor the hypothesis of strainenergy equivalence is required in the developed model.
