NSFC Grant - Free discontinuity problems and applications in fracture mechanics

Fracture mechanics is one of the oldest fields in material science, which still has many unanswered questions. It studies complex physical phenomena such as elasticity, plasticity, stress and strain, friction and non-penetration of crack-faces, etc. The most challenging problem though seems to be the analysis of the situation at the crack-front or, in two dimensional models, crack-tip. This is where one expects the crack to grow further and this is where one wants to predict crack's behavior. The major breakthrough in this field was achieved by an English aeronautical engineer Alan Arnold Griffith, who was the first to describe rigorously the stress of the fracture and to explain certain effects considered as contradicting the theory by his time. In 1998 Gilles Francfort and Jean-Jacques Marigo suggested a rigorous mathematical model based on Griffith's concepts. This model is considered as one of the most reliable and realistic for brittle fracture. Many theoretical questions are still open in this field and the strong singularity of the problem does not allow the developing of precise and effective numerical analysis.

The use of the advanced techniques from the theory of free boundary problems leads to new regularity results, which are of the significant importance for both, the methodology of studying free discontinuity problems, and development of realistic and effective simulations for real-live applications.