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

This paper covered the study of the boundary value problem for isotropic homogeneous perforated infinite elastic media in presence of unifo rm flow of heat. For this, we considered the problem of a thin infinite p late of specific thickness with a curvilinear hole where the origin lie inside the hole is conformally mapped outside a unit circle by means of a specific rational mapping . The complex variable method has been applied and it transforms the problem to the integro-differential equation with Cauchy kernel that can be solved to find two complex potential functions which called Gaursat functions. Moreover, the three stress components for the boundary value problem in the thermoelasticity plane are obtained. Many special cases of the conformal mapping and three applicat ions for different cases are discussed and many main results derived from the work.


Introduction
Several authors wrote about the boundary value problems and their applications in many d ifferent sciences, see Gakhov [1], Ciarlet et al. [2], Zebib [3] and Saito and Yamamto [4]. 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.
In thermoelastic problems for elastic media, the first and second boundary value problems are equivalent to finding two analytic functions and of one comp lex argument . These functions must satisfy the boundary conditions, (1.1) where denotes the affix of a point on the boundary. In terms of , , does not vanish or become infin ite fo r , the infin ite region mapped to outside a unit circle . For the first fundamental boundary value problem or it called the stress boundary value problem we have and is a given function of stress. While for and is a given function of displacement which called the thermal conductivity then eq.(1.1) called the second fundamental boundary value problems or the displacement boundary value problems. The books written by Noda et al. [10], Hetnarski and Ignaczak [11], Parkus [12] and Popov [13] contain many different methods to solve the problems in the theory of elasticity in one, two and three dimensions.
The comp lex potential functions and take the following form, see Parkus [12] (1.2) and, (1.3) where, are the components of the resultant vector of all external forces acting on the boundary and are complex constants. Generally the two co mp lex functions and are single value analytic functions within the region outside the unit circle and .
xx yy xy , , In [14], Muskhelishvili used the rational mapping, is real nu mber, (1.4) to solving the problem of infin ite plate weakened by an elliptic hole.
El-Sirafy and Abdou in [15], used the rational mapping, (1.5) to solve the first and second fundamental problems fo r the infinite plate with general curvilinear hole confrmally mapped on the domain outside a unit circle.
The rational mapping, (1.6) 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, is studied completely by England [16].
In this paper, we consider the boundary value problem for isotropic homogeneous perforated infinite elastic media in presence of uniform flow of heat. Then, we use the mo re general shape of conformal mapping to obtain the complex potential functions for the p roblem in the form integro-differential equation with singular kennel. Many special cases are obtained and several applicat ions are discussed fro m the work.
This study is useful for researchers who work on the studies of petroleum tubes industry or water or gas. It also benefits the physics scientists who work on the study of the ozone hole.

Formulation of the Problem
Consider a thin infinite plate of thickness with a curvilinear hole , where the origin lie inside the hole is conformally mapped on the domain outside a unit circle by means of a rat ional mapping, does not vanish or become infinite outside the unit circle . If a temperature d istribution is following uniformly in the d irection o f the negative a xis, where the increasing a temperature distribution is assumed to be constant a cross the thickness of the plate, i.e.
, and is the constant temperature gradient. The uniform flo w of heat is distributed by the presence of an insulated curvilinear hole . The heat equation satisfies the relation, where is the unit vector perpendicular to the surface. Neglecting the variation of the strain and the stress with respect to the thickness of the plate, the thermoelastic potential satisfies the formula, see Parkus [12] (2.4) where is a scalar which present the coefficient of the thermal expansion and is Poisson's ratio. Assume the force of the plate is free of applied loads.
In this case, the formu la (1.1) fo r the first and second boundary value problems respectively take the following form, where the applied stresses and are prescribed on the boundary of the plane is the length measured fro m an arbitrary point, and are the displacement co mponents, is the shear modulus. Also, here the applied stresses and must satisfy the following, see Parkus [12] (2.7) The last formula represent the first and second boundary value problems in plane.

The Rational Mapping
The rational mapping (2.1) maps the curvilinear hole in plane onto the domain of outside unit circle in plane under the condition does not vanish or beco me infinite outside the unit circle . The following graphs give the different shapes of the rational mapping (2.1).

Method of Solution
In this section, we use the complex variable method to obtain the two complex functions, Goursat functions, and . Moreover, the three stress components , and will be co mplete determined.

The Stress Components
The solution of Eq. (2.2) is given by,  After determine the Goursat functions the components of stress are completely determined.

Goursat Functions
To obtain the two complex potential functions, Goursat functions by using the conformal mapping (2.1) in the boundary condition(1.1), we write the expression in the form, where, , (4.12) and, is a regular function for .
has a singularity at and .
Using   (4.23) The formula (4.22) represents the integro-differential equation of second kind with Cauchy kernel. The references, Fedotov [17], Hanyga and Seredyńska [18], Bavula [19] and AL-Jawary and Wrobel [20], contain many different methods to solve this kind of the equations analytically and numerically in one, two and three dimensions.
To obtain the integral terms of Eq. (4.29) The last equation can be written in the form, ,

Some Applications
In this section, we assume different values of the given functions in the first or second fundamental boundary value problems. Then, we obtain the exp ression of Goursat functions. After that, the components of stresses can be calculated directly.

Application 1: Curvilinear Hole for an Infinite Pl ate
Subjected to Uniform Tensile Stress and Fl owing He at. , the stress components , and are obtained in large forms calculated by computer and illustrated in the following two cases : (i) When the study is in the normal plate, we have the following shapes for the stress components, see Fig.(12) and Fig.(13).

Conclusions
Fro m the previous discussion s we have the following result (i) The solution of the boundary value problems fo r isotropic homogeneous infinite elastic media in plane reduce to obtain the two co mplex potential functions, Gaursat functions, in plane by using conformal mapping. (ii) The conformal mapping , where , for mapped infinite reg ion to out side a unit circle .
(iii) When we applied the conformal mapping the boundary value problems reduce to the integro-differentail equation with discontinuous kernel.
(iv) Cauchy method is the best method to solving the integro-differentail equation with Cauchy kernel and obtaining the two complex functions and directly.
(v) The co mponents of stress and is completely determine and p lotting after obtaining the two complex functions. (ix) All of the previous works in this paper is considered as special cases from this study.