Non‐classical interface problems for piecewise homogeneous anisotropic elastic bodies

Abstract

Communicated by E. MeisterTwo non-classical model interface problems for piecewise homogeneous anisotropic bodies are studied. In both problems on the contact surface jumps of the normal components of displacement and stress vectors are given. In addition, in the first problem (Problem H) the tangent components of the displacement vectors are given from both sides of the contact surface, while in the second one (Problem G ) the tangent components of the stress vectors are prescribed on the same surface. The existence and uniqueness theorems are proved by means of the boundary integral equation method, and representations of solutions by single layer potentials are established. In the investigation the general approach of regularization of the first kind of integral equations is worked out for the case of two-dimensional closed smooth manifolds. An equivalent global regularizer operator is constructed explicitly in the form of a singular integro-differential operator.

Fundamental and singular solutions of Lamé equations for media with arbitrary elastic anisotropy

“…Routine procedure, similar to one exploited for obtaining (17), (19), produces the kernel W in the form of multipolar series:…”

“…where in contrast to (17), (19), the series on the right-hand side of (24) contains harmonics of the even order only, because of positive homogeneity of the amplitude with respect to the ξ-variable. Matrix coefficients W nk are determined by integration of the amplitude (23) on the unit sphere in R 3 similarly to obtaining V nk .…”

The 2-D Neumann problem in an exterior domain bounded by closed and open curves

“…We obtain the following statement. ( ), then due to lemma 1 the solution (s) of (4), (8) satisfies (2d) and therefore the function (7) satisfies conditions at infinity (2c). On the basis of the Theorem 2 we arrive at the final result.…”

“…The modern researches are mostly devoted to the generalized solvability in the Sobolev and Besov spaces and to the extension of classical results to the case of Lipshitz boundary. Recent advances in problems for the Laplace equation, wave propagation and the elasticity theory are presented in [1,2,7,8,11,14,16] (see also references in these papers. )…”

Mixed interface problems for anisotropic elastic bodies

“…Classical and nonclassical mathematical problems for isotropic piecewise homogeneous bodies are studied in [1,2,3] by means of potential methods, while similar problems for anisotropic piecewise homogeneous bodies are investigated in [4,5].…”