Functional Analysis and Applications Seminar 

11/09/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
David Natroshvili, Georgian Technical University, Tbilisi, Georgia

Application of pseudodifferential equations to mixed interface crack problems for composite structures

We investigate three-dimensional oscillation interface crack problems (ICP) for metallic-piezoelectric composite bodies with regard to thermal effects. We give a mathematical formulation of the physical problem when the metallic and piezoelectric bodies are bounded along some proper parts of their boundaries where interface cracks occur. By the potential method the ICP is reduced to an equivalent strongly elliptic system of pseudodifferential equations on manifolds with boundary. We study the solvability of this system in different function spaces and prove uniqueness and existence theorems for the original ICP. We analyse the regularity properties of the corresponding thermo-mechanical and electric fields near the crack edges and near the curves where different type boundary conditions collide. In particular, we characterize the stress singularity exponents and show that they can be explicitly calculated with the help of the principal homogeneous symbol matrices of the corresponding pseudodifferential operators. We expose some numerical calculations which demonstrate that the stress singularity exponents essentially depend on the material parameters.