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The continuum damage model should be regularized to ensure mesh-insensitive results in simulations of strain localization, e.g., for concrete cracking under tension. The paper confronts the conventional gradient damage model with its upgrade including a variable internal length scale. In these models, the Helmholtz free energy depends additionally on an averaged strain measure and its gradient. In the formulation for dynamics the equations of motion are discretized simultaneously with an averaging equation. If gradient regularization is employed with a constant internal length parameter, then an artificially expanded damage zone can occur in the strain softening analysis. This broadening effect can be inhibited by a gradient activity function. The localizing character of the gradient activity has physical motivation—the nonlocal interactions in the fracture zone are reduced with the damage growth. The internal length can decrease exponentially or as a cosine function. After presentation of the theory, including the free energy definition, the finite element analyses of three different examples connected with tensile cracking in concrete are discussed: static tension of a double-edge-notched specimen, dynamic direct tension for a configuration without or with a reinforcing bar and tension of an L-shaped specimen under static and dynamic loading.

Survey of Localizing Gradient Damage in Static and Dynamic Tension of Concrete