Mixed- and crack-type dynamical problems of electro-magneto-elasticity theory
The authors investigate the solvability of three-dimensional dynamical mixed boundary value problems of electro-magneto-elasticity theory for homogeneous anisotropic bodies with interior cracks. Laplace transform technique, the potential method and the theory of pseudo-differential equations are used to prove the existence, uniqueness and regularity theorems for the considered problem, and to analyze asymptotic properties of solutions near the crack edges and near the intersection of lines carrying different boundary conditions. There is a full coupling between the mechanical and electromagnetic fields. For the considered system of equations, it is shown that the initial conditions for electric and magnetic potentials cannot be specified independently, as they are uniquely determined by the initial conditions for the components of the displacement vector. A concrete mixed initial-boundary value problem is considered for a bounded elastic, electromagnetic solid with a single interior crack. The exterior boundary of the solid is divided into two disjoint parts: (i) a Dirichlet part with prescribed displacement vector, electric potential and magnetic potential; (ii) a Neumann part with prescribed mechanical stress vector, normal components of the electric displacement vector and the magnetic induction vector. Moreover, it is assumed that the electric potential and the normal component of the electric displacement vector as well as the magnetic potential and the normal component of the magnetic induction vector are continuous across the crack surface. Initial conditions are given only for the mechanical displacement vector and its time derivative. Asymptotic expansions of the solution are obtained near the crack edge and at intersections of the lines where the boundary conditions are set.
scientific article; zbMATH DE number 7379136