MATH-FRAC

This research project, funded by the National Institute of Higher Mathematics within the Starting Grant research projects, aims to construct a rigorous mathematical framework to model the formation and propagation of cracks in soft solids. We will address these aspects by using the theory of continuum mechanics and, specifically, phase-field models of fracture. While these models are nowadays frequently used in the context of linear elastic materials, robust models for soft matter are still missing. Indeed, the large deformations of the body and the softening behavior of these materials require the introduction of both geometrical and physical nonlinearities in the constitutive laws.

The main idea is that fracture nucleation can be described as the result of mechanical instability. We will investigate the variational structure of the models, by analyzing the possible bifurcations of the system. We will also construct a numerical framework to produce quantitative predictions of the proposed models. This research project paves the ground for the mathematical description of material failure in a wide range of materials. Apart from solving open mathematical problems, this research project will have a high impact on both industrial applications and applied science research, with possible applications ranging from polymeric-based industrial composites to the description of living tissues.