The Boundary State Method in Solving the Anisotropic Elasticity Theory Problems for a Multi-Connected Flat Region

Abstract

The energy method of boundary states is used to solve plane problems of the theory of elasticity for an anisotropic body, which, in the general case, has non-circular cutouts. The body is in a flat state of tension.

The basis of the method is the concept of spaces of internal and boundary states, which are Hilbert and are conjugated by an isomorphism. The space of internal States includes components of the displacement vector, components of strain and stress tensors. The scalar product for the space of internal States expresses the internal energy of elastic deformation. The boundary state space includes displacements of the body boundary points and forces at the boundary. The scalar product in the space of boundary States expresses the action of surface forces. The General solution of the plane Lechnitzky problem for an anisotropic body is used to construct the basis of internal States, with usage of analytical functions for a multi-connected medium. Afterwards, the orthogonalization of the state spaces bases is performed with the decomposition of the desired elastic state into a Fourier series, where the basis States act as elements of the decomposition. Using the forces specified on the boundary, Fourier coefficients are derived and the solution is constructed.

The first basic problem for circular and rectangular plates with two circular notches is solved. A graphic illustration of the results is given and conclusions are drawn.