Primal-Dual Constrained Optimization and Triboloy of Cracks

The classical model of a crack assumes that no interaction between the crack surfaces does occur, thus ignoring the surface effects essential for tribology. To explain the change of geometry and topology of cracks, on our believe, tribological arguments of the crack surfaces should be taken into consideration. The main goal of the project consists in the construction and analysis of proper mathematical tools for the description of dissipative processes of interaction between the crack surfaces. This subject belongs to the fields of mathematical modeling, constrained optimization, and scientific computing in application to problems with cracks. From a mathematical point of view, the keyword of the project is called as a primal-dual optimization. The reason concerns the fact that dual state variables play the key role to express an interaction of the crack surfaces in the function terms. However, the principal difficulty here deals with a nonsmooth character of the corresponding cost functionals, thus requiring a proper regularization which must be consistent mathematically and physically. Numerical experiments should support our theoretical investigations planned in this project. Thus, our multifield approach is directed to the following subjects: 1. Mathematical modeling of an interaction between the crack surfaces accounting the dissipative processes of friction, which are caused by a continuous contact as well as a discrete contact of rough surfaces of the crack. 2. Primal-dual analysis of constrained optimization problems with respect to their well-posedness properties and numerical realization, with applications to the problems for cracks subject to frictional contact conditions. 3. Time-evolution problems of a crack quasi-static propagation, which imply global optimization with respect to shape parameters of the crack. The novelty of the project consists in primal-dual mathematical formulations modelling interaction processes at cracks and exploiting methods of non-smooth analysis. The expected results should advance our understanding important for applications to fracture mechanics and structure design.

The research leaded under the project consists in the mathematical modelling of interaction processes at cracks. In contrast to the classical, linearized model of a crack ignoring the surface effects, we take tribological arguments of the crack surfaces into consideration. In fact, in practice there are observed complex phenomena including a crack curving, kinking, branching, as well as twisting and cusping of the crack front, which cannot be explained within linearized formulations. In the project we accounted for dissipative phenomena of contact due to cohesion, plasticity, and alike with the hemi-variational principles. The shape optimization approach allowed us to gain an insight into time-evolution of cracks during their propagation under processes of interaction between the crack surfaces. Exploiting primal-dual mathematical formulations and methods of non-smooth and non-convex analysis we investigated well-posedness properties of the constrained optimization problems. Appropriate semi-smooth algorithms are developed and analysed with respect to the numerical realization. Numerical experiments support our theoretical investigations in this project. The obtained results advance the concept of cracks and our understanding important for applications to fracture mechanics, geophysics, and structure design.