Meso-scale physical modeling of energetic degradation function in the nonlocal macro-meso-scale consistent damage model for quasi-brittle materials

Abstract

Abstract The modeling of crack initiation and propagation is of vital importance in the design and safety evaluation of structural components and engineering structures. Great advances have been achieved in the crack simulation during the past decades. Although the phase-field theory and the newly proposed nonlocal macro-meso-scale consistent damage (NMMD) model can predict the crack initiation, propagation and load-deformation curves for quasi-brittle materials correctly, the energetic degradation function, which bridges the energy-based damage and the topologic damage and plays an important role in these models, is still more or less empirically determined. In the present paper, a physical interpretation and quantitative modeling of the energetic degradation function via the embedded meso-scale mechanism of damage in the NMMD model is proposed. For this purpose, the NMMD model is firstly outlined. In this model, the meso-scale damage is firstly defined based on the degradation of bonds of material point pairs according to the irreversible elongation. The macro-scale topologic damage is then evaluated by averaging the meso-scale damage in the influence domain. It is then inserted into the thermodynamically consistent framework of continuum damage mechanics via the energetic degradation function bridging the topologic damage and the energy-based damage. To physically determine the energetic degradation function, the Helmholtz free energy of the damaged and undamaged materials is evaluated from the meso-scale mechanism of damage. In addition, it is theoretically proved that such determined energetic degradation function is invariant against the meso-scale parameters, including the radius of influence domain and the critical elongation quantity, and is weakly related to the strain state. Four numerical examples involving mode-I and mix-mode failures of isotropic quasi-brittle materials under static loading are studied to verify the NMMD model with the physically determined energetic degradation function. It is demonstrated that by the proposed approach there is no need to prescribe the initial crack and potential propagation path, the crack propagation can be captured automatically and the mesh size sensitivity is circumvented. Problems to be further studied are also discussed.