On the theory of anisotropic flat elasticity

A. P. Soldatov

Contemporary Mathematics. Fundamental Directions, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 3, Tome 60 (2016), pp. 114-163

Résumé

For the Lamé system from the flat anisotropic theory of elasticity, we introduce generalized double-layer potentials in connection with the function-theory approach. These potentials are built both for the translation vector (the solution of the Lamé system) and for the adjoint vector functions describing the stress tensor. The integral representation of these solutions is obtained using the potentials. As a corolary, the first and the second boundary-value problems in various spaces (Hölder, Hardy, and the class of functions just continuous in a closed domain) are reduced to the equivalent system of the Fredholm boundary equations in corresponding spaces. Note that such an approach was developed in [13, 14] for common second-order elliptic systems with constant (higher-order only) coefficients. However, due to important applications, it makes sense to consider this approach in detail directly for the Lamé system. To illustrate these results, in the last two sections we consider the Dirichlet problem with piecewise-constant Lamé coefficients when contact conditions are given on the boundary between two media. This problem is reduced to the equivalent system of the Fredholm boundary equations. The smoothness of kernels of the obtained integral operators is investigated in detail depending on the smoothness of the boundary contours.