Localization Analysis in the Plastic Regime of Concrete Damage-Plastic Models

This presentation explores the conditions for strain localization within the plastic regime of the Concrete Damage Plasticity Model (CDPM2 [1]), prior to the initiation of material damage. While localization in concrete is typically associated with the softening branch—where damage evolution leads to a loss of material stability—non-associated plasticity models can exhibit discontinuous bifurcations even during the hardening phase. This study focuses on the theoretical and numerical assessment of the localization tensor to determine the boundaries of stable plastic flow. The analysis reveals that localization can indeed occur within the plastic regime, primarily as a consequence of the non-associated flow rule. For a given stress state and set of material parameters, the critical hardening modulus required to maintain a positive determinant of the localization (acoustic) tensor is analytically evaluated. Subsequently, the stress space is discretized to systematically evaluate this critical hardening modulus across stress states that can be reached before the onset of damage. The results demonstrate that when all other material parameters are held fixed, the hardening modulus parameter $H_p$ must be selected within a specific admissible interval to entirely prevent localization during the loading process. Furthermore, the study investigates the sensitivity of this allowed interval for $H_p$ to variations in the remaining material parameters, showing how certain parameter combinations restrict or altogether eliminate the stable domain. The results provide essential guidelines for the robust calibration of CDPM2, ensuring that the model remains stable during the hardening phase and that any subsequent localization is physically representative of the damage process rather than a premature mathematical instability of the plastic formulation.

References

1. P. Grassl, D. Xenos, U. Nyström, R. Rempling, and K. Gylltoft, CDPM2: A damage-plasticity approach to modelling the failure of concrete, International Journal of Solids and Structures, 50(24):3805–3816, 2013.