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

Abstract

The present work, by using the method of the separation of variables, states and analytically (exactly) solves the external boundary value problems of elastic equilibrium of the homogeneous isotropic body bounded by the parabola, when normal or tangential stresses are given on a parabolic border. Using MATLAB software, the numerical results and constructed graphs of the mentioned boundary value problems are obtained.

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

“…By means of D , K , u , v functions, the equilibrium equations and Hooke’s law in the elliptic coordinate system will be written as follows [10,15,16]:…”

“…For the region, whose boundary or part of boundary is a curved line, to solve boundary value and boundary-contact problems, it is useful to examine these problems in the relevant curvilinear coordinate system. For example, the problems for the regions bounded by a circle or its parts are considered in the polar coordinate system [1–4], while the problems for the regions bounded by an ellipse or its parts are considered in the elliptic coordinate system [5–8], and the problems for the regions with parabolic boundaries are considered in the parabolic coordinate system [9–10]. The problems for the regions bounded by the circles with different centers and radiuses are considered in the bipolar coordinate system [11–13].…”

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].…”