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

Abstract

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.

Ability of localizing gradient damage to determine size effect in concrete beams

“…All distributions of damage for the CGD model are quite spread. Therefore, the problem of too strongly broadened damage zone in the CGD model is confirmed, see also (Geers, 1997;Poh and Sun, 2017;Wosatko, 2022;Zhang et al, 2021).…”

“…More details of different variants of the gradient damage model can be found in many works, e.g. (Geers, 1997;Peerlings et al, 2004;Poh and Sun 2017;Negi et al, 2021;Wosatko, 2022). Below only the most crucial elements of the theory are recalled.…”

“…When the LGD model is employed, the gradient activity is a function of damage ω. It is illustrated by Poh and Sun (2017), Wosatko (2022) that the influence of nonlocal interactions in the localization region should decrease with the increase of damage. It is observed that the crack band width gradually reduces and the model tends to the local one, so its localizing character reveals.…”

“…Moreover, together with the mesh density growth, the differences between the diagrams decline. It is known that the LGD model requires a well-refined discretization (Wosatko, 2022) or a smart mesh densification near the expected cracking region (Negi et al, 2021). Indeed, three meshes are sufficient to show the mesh-independent results for the CGD model.…”

“…Nowadays, the LGD model is employed in many applications, e.g. it can be coupled with three-surface cap plasticity, as derived by Zhao et al (2023) or used for simulations with impact loading, see (Wosatko, 2022).…”