Scaling Laws and Multiscale Approach in the Mechanics of Heterogeneous and Disordered Materials

The present paper is a review of research carried out on scaling laws and multiscaling approach in the mechanics of heterogeneous and disordered materials in the last two decades, especially at the Politecnio di Torino. The subject encompasses theoretical, numerical and experimental aspects. The research followed two main directions. The first one concerns the implementation and the development of the cohesive crack model, which has been shown to be able to simulate experiments on concrete like materials and structures. It is referred to as the dimensional analysis approach, since it succeeds in capturing the ductile-to-brittle transition by increasing the structural size owing to the different physical dimensions of two material parameters: the tensile strength and the fracture energy. The second research direction aims at capturing the size-scale effects of quasibrittle materials, which show fractal patterns in the failure process. This approach is referred to as the renormalization group (or fractal) approach and leads to a scale-invariant fractal cohesive crack model. This model is able to predict the size effects even in tests where the classical approach fails, e.g., the direct tension test. Within this framework and introducing the fractional calculus, it is shown how the Principle of Virtual Work can be rewritten in its fractional form, thus obtaining a scaling law not only for the tensile strength and the fracture energy, but also for the critical strain.