Presence of Heat on an Infinite Plate with a Curvilinear Hole Having Two Poles

In the present paper Cauchy integral methods have been applied to derive exact and expressions for Goursat’s function for the first and second fundamental problems of isotropic homogeneous perforated infinite elastic media in the presence of uniform flow of heat. For this, we considered the problem of a thin infinite plate of specific thickness with a curvilinear hole where the origins lie in the hole is conformally mapped outside a unit circle by means of a specific rational mapping. Moreover, the three stress components xx σ , yy σ and xy σ of the boundary value problem in the thermoelasticity plane are obtained. Many special cases of the conformal mapping and four applications for different cases are discussed and many main results are derived from the work.


Introduction
The boundary value problems for isotropic performed plates have been discussed by several authors.Some authors used Laurant's theorem [1]- [8] to express each complex potential as a power series, and others [9]- [16] used complex variable method of Cauchy integral.The extensive literature on the isotropic is now available and we can only mention a few recent interesting investigations in refs [17]- [26].
In thermoelastic problems for elastic media, the first and second boundary value problems are equivalent to two finite analytic functions ( ) z φ and ( ) z ψ of one complex argument z x iy = + .These functions must satisfy the boundary condition, ( ) ( ) ( ) ( ) where t denotes the affix of a point on the boundary.In terms of ( ) does not vanish or become infinite for 1 ζ > , where the infinite region is outside a unit circle γ .For the first fundamental boun- dary value problems or the called stress boundary value problems, 1 f t is a given function of stress.
While for f t is a given function of displacement called the thermal conductivity; we have the second fundamental boundary value problems called the displacement boundary value problems.
The complex potential functions ( ) 1 t φ and ( ) 1 t ψ take the following forms, see [23] ( ) ( ) ( ) and, where, , X Y are the components of the resultant vector of all external forces acting on the boundary and , * Γ Γ are complex constants.Generally the two complex functions ( ) φ ζ and ( ) ψ ζ are single value analytic func- tions within the region outside the unit circle γ and ( ) ( ) In [27], Muskhelishvili used the ra- tional mapping, ( ) for solving the problem of stretching of an infinite plate weakened by an elliptic hole.Sokolonikoff [3] used the same rational mapping of Equation ( 4) to solve the problem of elliptical ring, where the Laurant's theorem is used.This transformation of Equation ( 4) conformally maps the infinite domain bounded internally by an ellipse onto the domain outside the unit circle 1 ζ = in the ζ -plane.The application of the Hilbert problem is used by Muskhelishvili [27] to discuss the case of a stretched infinite plate weakened by a circular cut.England [1] considered an infinite plate which is weakened by a hypotrochoid hole, conformally mapped onto a unit circle 1 ζ = by the transformation mapping ( ) where ( ) does not vanish or become infinite outside the unit circle γ , and solve the boundary value prob- lems.
The main reason for interest in this mapping is that the general shape of the hypotrochoids is curvilinear polygons, for n = 1 ellipse, for n = 2 a curvilinear triangle, for n = 3 a curvilinear square… etc.In previous papers (Abdou et al. [9]- [15] [28] [29]), the complex variable method has been applied to solve the first and second fundamental problems for the same domain of the infinite plate with general curvilinear hole c conformally mapped on the domain outside a unit circle by using respectively, the rational mapping functions and In this paper, the complex variable method has been applied to solve the first and second fundamental prob-lems for the same previous domain of the infinite plate with a general curvilinear hole C, with three poles and presence of heat, conformally mapped on the domain outside a unit circle γ by the rational mapping functions ( ) ( )( ) where 0 c > , flowing uniformly in the direction of the negative y-axis, where the increasing temperature Θ is assumed to be constant across the thickness of the layer, i.e.

( )
, and q is the constant temperature gradient.The uniform flow of heat is distributed by the presence of an insulated curvilinear hole C. The heat equation satisfies the relations, where n is the unit vector perpendicular to the surface.Neglecting the variation of the strain and the stress with respect to the thickness of the layer, the thermoelastic potential function Φ satisfies the formula (see [14]), ( ) where α is a scalar which presents the coefficient of the thermal expansion and ν is Poisson's ratio.Assume that the force of the layer is free of applied loads.In this case, formula (1) for the first and second boundary value problems respectively takes the following forms, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) where the applied stresses ( ) X s and ( ) Y s are prescribed on the boundary of the plane; s is the length meas- ured from an arbitrary point; u and v are the displacement components; G is the shear modulus and Φ represents the thermoelastic potential function.Also, here the applied stresses ( ) X s and ( ) Y s must satisfy the following (see [14]), ( ) ( ) where , xx xy σ σ and yy σ are the components of stresses which are given by the following relations, ( ) ( ) where ( ) is the coefficient of heat transfer.
The rational mapping ( ) ( ) the last formula represents the first and second boundary value problems in the ζ -plane.
In this study, we use the rational mapping (10) to map the curvilinear hole C of the infinite viscoelastic fluid layer outside a unit circle γ .Then, we use the properties of Cauchy integral to obtain the two complex potential functions (Goursat functions).After that, we determine the components of stresses.We establish many applications and special cases from this work.

The Rational Mapping
The mapping function (10)

Method of Solution
In this section, we use the complex variable method to obtain the two complex functions (Goursat functions) ( ) φ ζ and ( ) ψ ζ .Moreover, the three stress components xy σ , xx σ and yy σ will be completely determined.
The solution of Equation ( 9) is given by, By substituting Equation (19) in Equation ( 11) and using the definition of 2 ∇ Φ in polar coordinates the thermoelastic potential function takes the form, Also, the stresses components can be adapted in the forms, Im .
By using Equation (19) and Equation (20), Equations ( 21)-( 23) becomes and, where, After determining the Goursat functions the components of stress are completely determined.
Using Equation (2) and Equation (3) in Equation ( 18), we get using Equation (29) in Equation (31), we have for generality, we get where, using Equations ( 35)-(39) in Equation (40) then applying the properties of Cauchy integral to have ) The formula (41) represents the integro-differential equation of the second kind with Cauchy kernel.

Special Cases
Now, we are in a position to consider several cases: 1) For 1 0 n = , we have the mapping discussed by Abdou and Kkhar El-din [9], see Figure 2.
3) For 1 2 0, 0 n n = = , we have the mapping ( ) . The main reasons of interest in this mapping is that the general shapes of the hypotrochids are curvilinear polygons, for we have a curvilinear triangle see Abdou [28].For 3 =  a curvilinear square, see Abdou and Badr [29], see Figure 4.

Some Applications
In this section, we assume different values of the given function in the first or second fundamental boundary value problems.Then, we obtain the expression of Goursat functions.After that, the components of stresses can be calculated directly.45) and (50) take the form ( ) where, This application discusses the first fundamental boundary value problem of an infinite layer stretched at infinity by the application of a uniform tensile stress of intensity heat in the negative direction of y-axis.This layer is weakened by a curvilinear hole C which is free from stress.2) When the study is in the normal layer, we have the following shapes for the stress components, see Figure 7 and Figure 8.

Application 2: Bi-axial tension
For ( ) Goursat's functions in Equations ( 45) and (50) take the form ( ) where, ( ) ( ) This application discusses the first fundamental boundary value problem of an infinite layer stretched at infinity by the application of a uniform tensile stress of intensity heat in the negative direction of y-axis.This layer is weakened by a curvilinear hole C which is free from stress.
Goursat's functions in Equations ( 45), (50) take the form   This application discusses the first fundamental boundary value problem of an infinite layer stretched at infinity by the application of a uniform tensile stress of intensity heat in the negative direction of y-axis.This layer is weakened by a curvilinear hole C which is free from stress.
where,  This application discusses the first fundamental boundary value problem of an infinite layer stretched at infinity by the application of a uniform tensile stress of intensity heat in the negative direction of y-axis.This layer is weakened by a curvilinear hole C which is free from stress.

Conclusions
From the previous discussions we have the following results: 1) We find that the effect of heat is very clear; we find that the values of components stress are reduced with existence of heat, while at absence of heat we find the values of components of stresses are increasing.
2) With increasing angle and with absence ( ) 4) The components of stresses effected are clear by increasing the values of  .

1, 2
, , p =   , m and n are real parameters restricted such that ( ) z ζ ′ does not vanish or become infinite outside the unit circle γ .Consider a heat qy Θ = cω ζ = maps the boundary C of the given region occupied by the middle plane of the layer in the z-plane onto the unit circle γ in the ζ -plane.Curvilinear coordinates ( ) , ρ θ are thus intro- duced into the z-plane, which are the maps of the polar conditions in the ζ -plane as given by e iθ maps the curvilinear hole C in z-plane onto the domain of the outside unit circle in ζ -plane under the conditions that ( ) ω ζ ′ does not vanish or become infinite outside the unit circle γ .The following graphs give the different shapes of the rational mapping (10), see Figure 1.

Figure 1 .
Figure 1.The different shapes of the rational mapping (10).

F
σ with its derivatives must satisfy Hölder condition.Our aim is to determine the functions ( ) φ ζ and ( ) ψ ζ for the various boundary value problems.For this multiply both sides of Equ- any point in the interior of γ an integral over the circle, we obtain

Figure 4 .
Figure 4.The different shapes of the rational mapping in (3).

1. Application 1 :
When the curvilinear is not allowed to rotate.At 1 =  , the rigid curvilinear kernel is restrained in its original position by a couple which is not sufficient to rotate the kernel.

Figure 5 and
components xx σ , yy σ and xy σ are obtained in large forms calculated by computer and illustrated in the following two cases: 1) In the thermoelasticity layer, we have the following shapes for the stress components by using the substitutions 0Figure 6.

Figure 6 .Figure 7 .
Figure 6.The ratio of vertical to horizontal stresses.

Figure 8 .
Figure 8.The ratio of vertical to horizontal stresses.

,Figure 9 and
the stress components xx σ , yy σ and xy σ are obtained in large forms calculated by computer and illustrated in the following two cases: 1) In the thermoelasticity layer, we have the following shapes for the stress components by using the substitutions 0Figure 10.

Figure 10 .
Figure 10.The ratio of vertical to horizontal stresses.2)When the study is in the normal layer, we have the following shapes for the stress components, see Figure11and Figure12.3.Application 3: When a couple with a given moment acting on the curvilinear hole and the stresses vanishing at infinityFor

Figure 12 .
Figure 12.The ratio of vertical to horizontal stresses.

Figure 13 and 4 . Application 4 :
components xx σ , yy σ and xy σ are obtained in large forms calculated by computer and illustrated in the following two cases: 1) In the thermoelasticity layer, we have the following shapes for the stress components by using the substitutions 0Figure 14. 2) When the study is in the normal layer, we have the following shapes for the stress components, see Figure 15 and Figure 16.The external force acts on the center of the curvilinear hole Γ = = , Then, Goursat, s functions in Equations (45), (50) take the form

Figure 14 .Figure 15 .
Figure 14.The ratio of vertical to horizontal stresses.

Figure 17 and
components xx σ , yy σ and xy σ are ob- tained in large forms calculated by computer and illustrated in the following two cases: 1) In the thermoelasticity layer, we have the following shapes for the stress components by using the substitutions 0Figure 18. 2) When the study is in the normal layer, we have the following shapes for the stress components, see Figure 19 and Figure 20.

Figure 16 .Figure 17 .
Figure 16.The ratio of vertical to horizontal stresses.

Figure 18 .Figure 19 .
Figure 18.The ratio of vertical to horizontal stresses.

Figure 20 .
Figure 20.The ratio of vertical to horizontal stresses.