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Concrete materials show a very complex macroscopic deformation behavior under tension and compression, accompanied by crack opening and crack closing phenomena under cyclic loading. The continuum damage mechanics offers a promising framework for the description of the damage deformation behavior. This paper proposes a continuum damage model, which is formulated based on energy equivalence using a unified equivalent strain. The evolution of isotropic damage is governed by two independent history variables to describe the crack opening and closing behavior, i.e., unilateral behavior, of concrete. The evolution of damage and inelastic strains are described by a single damage function and a modified failure surface, respectively. Moreover, the implicit gradient method is applied to the equivalent strains to achieve proper localization of deformation. The stiffness recovery and crack opening/closing mechanisms are simulated considering the thermodynamically consistent framework. Validation of numerical results with experimental data and the previous models demonstrates the efficiency of the model to simulate concrete behavior under monotonic and cyclic/reverse loadings.

A macroscopic gradient-enhanced damage model for deformation behavior of concrete under cyclic loadings