Energy degradation mode in nonlocal Macro-Meso-Scale damage consistent model for quasi-brittle materials

The accurate simulation of crack initiation and propagation plays an important role in exploring the failure mechanism and safety evaluation of engineering structures. The proposal of the fracture phase-field theory has made a breakthrough in crack simulation. Recently, the nonlocal macro-meso-scale consistent damage (NMMD) models are more efficient in predicting crack initiation and propagation. However, the energy degradation mechanism caused by the cracking phase, which plays a critical role in these models, is still more or less empirically determined. In this paper, assuming that the geometric damage in the NMMD model follows an exponential decay distribution and advocates the equivalence principle of energy dissipation, the intrinsic correlation between the energy degradation mode and the influence domain size is explored. It is of physical interest to find that the faster the energy degradation in the initial stage, the slower the energy degradation in the later stage, requiring a larger range of influence domain. The example of a three-point bending of the notched beam with different energy degradation modes validates this discovery. Moreover, the detailed implementation for the secondary development of the NMMD model in the Abaqus platform is provided. Several two- and threedimensional fracture benchmarks including the dynamic cracking problem and thermal-mechanical coupling problem are verified with the well-established solver in Abaqus.