Thermoelastic Problem for an Infinite Plate with a Curvilinear Hole Having Finite Poles

Abstract

In the present paper, Muskhelishvili's complex variable method of solving two-dimensional elasticity problems has been applied to derive exact expressions for Goursat's functions for the boundary value problems of the infinite plate weakened by a hole having many poles and arbitrary shape which is conformally mapped n the dorhain outside a unit circle by means of general rational mapping function. Also the components of stress are obtained, when a stationary heat is flowing uniformly in the perpendicular direction of the hole. Some applications are considered. The interesting cases when the shape of the hole takes different shapes are included as special cases.

Closed Forms of Goursat Functions in Presence of Heat for Curvilinear Holes

“…In the papers, El-Sirafy and Abdou [12], Abdou and Khar-El din [13], Abdou and Badr [14], Abdou and Khamis [15], and Abdou [16], many rational mappings are used to solve the first and second fundamental problems of an infinite plate with a curvilinear hole, using Cauchy complex variable method, where the two complex Goursat functions are obtained. Moreover, in all previous works the coefficients of the rational mappings were real.…”

Complex Potential Functions and Integro-Differential Equation in Elastic Media Problem in Presence of Heat

“…is used by Abdou and Khamis [8] to obtain the solution of the problem of an infinite plate with a curvilinear hole having three poles and they take the circle shape, ellipse shape and square shape as a special cases of the holes. Also, the conformal mapping function, , (…”

“…Fro m these problems, we established contact problems and mixed problems in the theory of elasticity , see Co lton and Kress [5], and Abdou [6]. In Abdou [6,7], Abdou and Kha mis [8] and Abdou and Khar-Eldin [9],Co mp lex variab le method is used to express the solutions of these problems in the form of power series by applied Laurant's theorem. On the other hand, some of authors applied the complex variab le method to obtain the solutions in the form t wo co mplex potential functions which called Gaursat functions by using the properties of Cauchy integrals on the circle, or on any region mapped to outside a unit circle by means of a general rat ional mapping where, doesn't vanish or become infinite outside the unit circle.…”

An Infinite Elastic Plate Weakened by a Generalized Curvilinear Hole and Goursat Functions

“…Some authors used Laurent's theorem to express the solution in the series form, see England [5], Parkus [6] and Kalandiya [7]. Others used complex variables method of Cauchy integrals to express the solution of the boundary value problems in the form of two complex potential functions, Goursat functions, by using many rational mappings, see Muskhelishvili [8], El-Sirafy and Abdou [9], Abdou and Khar-Eldin [10], Abdou and Khamis [11], Abdou [12] and Abdou et al [13]. In all previous works, the coefficients of the rational mappings were real.…”