Physically consistent nonlocal macro–meso-scale damage model for quasi-brittle materials: A unified multiscale perspective

Abstract

Despite great efforts in capturing the damage evolution law in multiscale approaches in damage mechanics, much less attention has been paid to the conversion from geometric damage to energy dissipation. In the present paper, the newly proposed nonlocal macro–meso-scale damage (NMMD) model is physically consolidated and a multiscale point of view is consistently adopted in both the damage evolution and energy dissipation. In this model, for each macroscopic material point a mesoscopic structure is attached such that all the point pairs composed of this point and the points in its influence domain are connected to it when the material is intact. The deformation field under loading will result in deformation of point pairs, and thus mesoscopic damage related to a point pair will occur if the deformation index of this point pair exceeds the prescribed threshold. Such mesoscopic change among point pairs connected to a material point within its influence domain has two-fold consequences: geometrically the accumulation of mesoscopic damage will result in macroscopic discontinuity, and simultaneously the free energy will be dissipated, leading to degradation of mechanical behavior of quasi-brittle materials. In other words, the damage as a metric of the geometric discontinuity will lead to the damage in the sense of energy dissipation and degradation of mechanical behavior for quasi-brittle materials, which can be physically captured by the ratio of summation of all the dissipated mesoscopic free energy in point pairs to the free energy of intact material within the influence domain. This energy-based macroscopic damage could then be inserted into the framework of continuum damage mechanics. Therefore, the conversion from geometric damage to energetic damage is implemented on the mesoscale instead of the macroscale, and thus a macroscopic energetic degradation function is not needed. In addition, the structured strain which is suitable for quasi-brittle materials can be adopted to determine the deformation index of point pairs. Correspondingly, the macroscopic free energy can be split into dissipative and non-dissipative parts. In this way, several inconsistencies in the previous NMMD models are remedied. The influence of mesoscopic model parameters is investigated in depth, and several numerical examples including mode-I, mixed-mode and compressive splitting fracture problems of quasi-brittle materials are carried out. Numerical results indicate that the proposed model can not only well capture the cracking process with or without initial cracks but also quantitatively predict the load–deformation curves without mesh size sensitivity. Moreover, due to the inherent physical consistency, in mixed-mode cracking problems the proposed model performs better than the previous NMMD model. Problems to be further studied are also discussed.