First Boundary Value Problem of Electroelasticity for a Transversally Isotropic Plane with Curvilinear Cuts

The first boundary value problem of electroelasticity for a transversally isotropic plane with curvilinear cuts is investigated. The solvability of a system of singular integral equations is proved by using the potential method and the theory of singular integral equations. In this paper the first boundary value problem of electroelasticity is investigated for a transversally isotropic plane with curvilinear cuts. The second boundary value problem of electroelasticity for this kind medium was solved in [1] using the theory of analytic functions and singular integral equations. The boundary value problems of electroelasticity for transversally isotropic media were studied in [2] (Ch. VI), while the boundary value problems of elasticity for anisotropic media with cuts were considered in [3], [4]. The uniqueness theorems for boundary value problems of electroelasticity are given in [5], [6]. Here we shall be concerned with the plane problem of electroelasticity (it is assumed that the second component of the threedimensional displacement vector is equal to zero, while the components u1, u3 and the electrostatic potential u4 depend only on the variables x1 and x3 [2]). Let an electroelastic transversally isotropic plane be weakened by curvilinear cuts lj = ajbj , j = 1, . . . , p. Assume that lj , j = 1, . . . , p, are simple nonintersecting open Lyapunov arcs. The direction from aj to bj is taken as the positive one on lj . The normal to lj will be drawn to the right relative to motion in the positive direction. Denote by D− the infinite plane with curvilinear cuts lj , j = 1, . . . , p, l = p ∪ j=1 lj . Suppose that D− if filled with some material. 1991 Mathematics Subject Classification. 73R05.