DAMAGE IN QUASI-BRITTLE MATERIALS FROM A TWO-SCALE PERSPECTIVE: FROM GEOMETRIC DISCONTINUITY TO FREE ENERGY REDUCTION

The construction of multiscale damage evolution law is one of the key ingredients in damage mechanics for transitioning from phenomenological to a rational and solid foundation. However, few have noticed that there are actually two types of damage variables: one representing the geometrical discontinuity of materials, and the other representing damage in terms of free energy reduction (i.e., degradation of mechanical behavior of materials). The key of solid failure mechanics lies in how to achieve the conversion from geometric damage to energetic damage. In the current paper, the damage evolution law and the conversion from geometric damage to energy dissipation, i.e., the degree of mechanical degradation, is quantitively described from a two-scale point of view, and the rational foundation of the nonlocal macro-meso-scale damage (NMMD) model is further consolidated. In the present paper, the continuum is considered as a set of material points. Material points within the influence domain interact with others, forming material point pairs, and thus a meso structure is attached to each material point. Under external loading, the material points move, leading to the deformation of material point pairs. When some geometric deformation quantity (e.g., the positive elongation quantity) of material point pairs exceeds a critical value, the damage starts to develop in mesoscopic material point pairs. The progressive fracture of mesoscopic material point pairs leads to the changes in the degree of discontinuity, and finally results in the changes in the topology of macroscopic continuum. Therefore, the topologic damage can be naturally defined as the weighted summation of mesoscopic damage in point pairs within the influence domain, and can be adopted to describe the discontinuity of macroscopic solid. The topologic damage is intrinsically a damage variable in the geometric sense. On the other hand, the damage evolution leads to the dissipation of free energy, resulting in degradation of mechanical properties of materials. As the macroscopic energy dissipation is the summation of the mesoscopic dissipation energy in material point pairs induced by mesoscopic damage evolution, the damage in the energetic sense or mechanical sense can be evaluated by the ratio of total dissipated mesoscopic energy in material point pairs and total elastic free energy. Therefore, the conversion from geometric damage to energy dissipation is conducted on the meso scale via the material point pairs rather than the macro scale. Consequently, the empirical energetic degradation function in continuum damage mechanics or phase-field fracture model is not needed anymore in the current model, and the map between the topologic damage and energetic degradation factor is determined by the actual deformation state rather than a fixed function. As a result, this model might have a better adaptivity for the fracture problems with complex strain/stress states. The numerical results indicate that the current model can not only capture the whole process of crack initiation and propagation, but also quantitatively describe the load-deformation curve under loading. Compared with the original NMMD model where the transition from geometric damage to energy dissipation is conducted on the macro scale, the proposed model can better capture the details of the experimental results.