Analytical solutions of some internal boundary value problems of elasticity for domains with hyperbolic boundaries

Abstract

An analytical solution of two-dimensional problems of elasticity in the region bounded by a hyperbola in elliptic coordinates is constructed using the method of separation of variables. The stress–strain state of a homogenous isotropic hyperbolic body and that with a hyperbolic cut is studied when there are non-homogenous (non-zero) boundary conditions given on the hyperbolic boundary. The graphs for the numerical results of some test problems are presented.

2D Elastostatic Problems in Parabolic Coordinates

“…Considering the homogeneous boundary conditions of the concrete problem, we will insert φ 1 and φ 2 functions selected from the (14) in the right sides of (15) or (16), and we will expand the left sides in the Fourier series. In both sides expressions which are with identical combinations of trigonometric functions will equate to each other and will receive the infinite system of linear algebraic equations to unknown coefficients A 1n and A 2n of harmonic functions, with its main matrix having a block-diagonal form.…”

“…In order to solve boundary value and boundary-contact problems in the areas with curvilinear border, it is purposeful to examine such problems in the relevant curvilinear coordinate system. Namely, the problems for the regions bounded by a circle or its parts are considered in the polar coordinate system [1][2][3][4], while the problems for the regions bounded by an ellipse or its parts or hyperbola are considered in the elliptic coordinate system [5][6][7][8][9][10][11][12][13], and the problems for the regions with parabolic boundaries are considered in the parabolic coordinate system [14][15][16]. The problems for the regions bounded by the circles with different centers and radiuses are considered in the bipolar coordinate system [17][18][19].…”

Exact solution of some exterior boundary value problems of elasticity in parabolic coordinates

“…Considering the homogeneous boundary conditions of the concrete problem, we will insert u 1 and u 2 functions selected from (12) into equations (13) or 14, and we will expand the left-hand sides in the Fourier series. Both sides of the expressions, which show identical combinations of trigonometric functions, will equate to each other and will receive the infinite system of linear algebraic equations to unknown coefficients A 1n and A 2n of the harmonic functions, with the main matrix having a blockdiagonal form.…”

“…In Zappalorto et al [12], the boundary value problems are formulated according to the complex potential function using parabolic coordinate systems. The present author's earlier work [13] solved the internal boundary value problems for a body bounded by the lines of a parabolic coordinate system.…”