Study of deflected mode of ellipse and ellipse weakened with crack

Abstract

The paper considers the boundary value problems of elasticity for an ellipse and ellipse with a crack when tangential stress is applied to the ellipse boundary. The mathematical models of these problems are obtained by setting the relevant problems for a semi-ellipse: a) the continuity conditions for the problem solution are given at the linear border, b) the continuity conditions for the problem solution are given at the portion of the linear boundary, beyond the focuses, with the tangential stresses given on the section between the focuses. So, a semi-ellipse can be bound as a whole ellipse, with the continuity conditions of the solution on the section between its focuses met in one case (when there is no crack) and not met in another case (when there is a crack, which is affected by the tangential stress). The problem solution for the cracked ellipse is reduced to the solution of the internal and external problems of elasticity, which are solved quite simply by the method of separation of variables.

2D Elastostatic Problems in Parabolic Coordinates

“…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].…”