﻿ On a Problem of an Infinite Plate with a Curvilinear Hole inside the Unit Circle

Applied Mathematics
Vol.06 No.01(2015), Article ID:53637,14 pages
10.4236/am.2015.61020

On a Problem of an Infinite Plate with a Curvilinear Hole inside the Unit Circle

F. S. Bayones, B. M. Alharbi

Department of Mathematics, Faculty of Science, Taif University, Taif, KSA

Received 8 January 2015; accepted 26 January 2015; published 29 January 2015

ABSTRACT

In this work, we used the complex variable methods to derive the Goursat functions for the first and second fundamental problem of an infinite plate with a curvilinear hole C. The hole is mapped in the domain inside a unit circle by means of the rational mapping function. Many special cases are discussed and established of these functions. Also, many applications and examples are considered. The results indicate that the infinite plate with a curvilinear hole inside the unit circle is very pronounced.

Keywords:

Complex Variable Method, An Infinite Plate, Curvilinear Hole, Conformal Mapping, Goursat Functions

1. Introduction

Many intangible phenomena can be found in nature-like magnetic field, electricity and heat. These phenomena cannot be presented mathematically in the real plane. The complex plane plays an important role in presenting these intangible phenomena. Also, many mathematical problems cannot be solved in the real plane; their solutions can be found in the complex plane.

The considerable mathematical difficulties which arise during any attempt to solve plane elastic problems necessitate the search for practical methods of solution. The first use and development of the methods of complex function theory in two-dimensional elastic problems were made by Muskhelishvili (see [1] ), and their ideas were expounded in their latter books (see [2] - [4] ). The development of the theory was based on the complex representation of the general solution of the equations of the plane theory of elasticity. This complex representation has been found very useful for the effective solution of the plane elastic problems.

Contact and mixed problems in the theory of elasticity have been recognized as a rich and challenging subject for study (see Popov [5] , Sabbah [6] and Atkin and Fox [7] ). These problems can be established from the initial value problems or from the boundary value problems, or from the mixed problems (see Colton and Kress [8] and Abdou [9] ). Also, many different methods are established for solving the contact and mixed problems in elastic and thermoelastic problems; the books edited by Noda [10] , Hetnarski [11] , Parkus [12] and Popov [5] contain many different methods to solve the problems in the theory of elasticity in one, two and three dimensions.

Several authors wrote about the boundary value problems and their applications in many different sciences (see [7] [13] - [15] ). Form these problems, we established contact and mixed problems (see [8] [16] ). Complex variable method used to express the solutions of these problems in the form of power series applied Laurent’s theorem (see [8] [17] - [19] ). The extensive literature on the topic is now available and we can only mention a few recent interesting investigations in [20] -[24] .

The first and second fundamental problems in the plane theory of elasticity are equivalent to finding analytic functions and of one complex argument.

These functions satisfy the boundary conditions

(1)

where and are two analytic functions; t denotes the affix of a point on the boundary. In the first fundamental problem, is a given function of stresses, while in the second fundamental problem

(2)

And is a given function of the displacement; and are called the Lame constants.

Let the complex potentials and take the form

(3)

(4)

where X, Y are the components of the resultant vector of all external forces acting on the boundary and are constants; generally complex functions are single-valued analytic functions within the region inside the unit circle and.

Take the conformal mapping which mapped the domain of the curvilinear hole on the domain inside a unit circle by the rational function

(5)

and does not vanish or become infinite to conform the curvilinear hole of an infinite elastic plate onto the domain inside a unit circle i.e.

(6)

2. Conformal Mapping

Consider the rational mapping on the domain inside a unit circle by the rational function

(7)

where, m and n are complex number, Equation (7) must satisfy the condition Equation (6).

For determining the tax parameters and, we put, in Equation (7) to get

(8)

Then

(9)

(10)

Also,

To obtain the critical points, we consider

(11)

this linear equation of three order, the roots of this equation must be under 1.

The following graphs give the different shapes of the rational mapping (7), see Figure 1.

3. The Components of Stresses

It is known that, the components of stresses are given by, see [1]

(12)

(13)

Hence, we have

(14)

(15)

and

(16)

4. Goursat Functions

To obtain the tow complex potential functions (Goursat functions) by using the conformal mapping (7) in the

boundary condition (6). We write the expression in the form,

(17)

where,

(18)

Figure 1. The different shapes of the rational mapping (7).

is a regular function for.

In order to separate the singularity, we use the definition of mapping, to have

(19)

The term in the are has no singular point while has a singularity at.

where

(20)

To determine form Equation (19), we can write the form

(21)

By using the residues in this equating we have

(22)

Using Equation (3) and Equation (4) in Equation (1), we get

(23)

where

(24)

(25)

(26)

Assume that the function with its derivatives must satisfy the Holder condition. Our aim is to determine the functions and for the various boundary value problems. For this multiply both sides of

Equation (23) by, where is any point in the interior of and integral over the circle, we obtain

(27)

Using Equations (24)-(26) in Equation (27) then applying the properties of Cauchy integral, to have

(28)

and

(29)

(30)

Also,

(31)

where,

(32)

From the above, Equation (27) becomes

(33)

To determined, where are complex constants, differentiating Equation (33) with respect to and substituting in Equation (29), we get

(34)

Substituting Equation (18) in Equation (34), then using the properties of Cauchy integral and applying the reside theorem at the singular points, we obtain

(35)

where

(36)

The last equation can be written in the form

(37)

where,

(38)

taking the complex conjugate of Equation (37), we get

(39)

form Equation (37) and Equation (39), we have

(40)

To obtain the complex function we have form Equation (23) after substituting the expression of and, and taking the complex conjugate of the resulting equation after using the expression of

to yields,

(41)

where,

(42)

and calculate sum residue, we obtain multiplying both sides of Equation (41) by, where is any

point in the interior of and integrating over the circle, then using the properties of Cauchy’s integral and calculating the sum residue, we obtain

(43)

where,

(44)

and

(45)

5. Special Cases

Now, we are in a position to consider several cases:

1) Let, we get the mapping function represent of the hole is an ellipse, see Figure 2

(46)

by let

Figure 2. The different shapes of the rational mapping for special cases.

Then (33) and (43) becomes

(47)

Also,

(48)

where

2) For, we get the mapping function represent of the hole is an ellipse, see Figure 2

(49)

then

Then (33) and (43) becomes

(50)

(51)

where

3) Let, we get the mapping function represent of the hole is an ellipse, see Figure 2

(52)

Then (33) and (43) becomes

(53)

(54)

4) Let, where we get the mapping function represent of the hole is an ellipse, see Figure 2

(55)

Then (33) and (43) becomes

(56)

Also,

(57)

where

5) Let,we get the mapping function represent of the hole is an ellipse, see Figure 2

(58)

Then (33) and (43) becomes

(59)

Also,

(60)

6. Applications

In this section we study some applications:

1) For and, we have the case of infinite plate stretched at in-

finity by the application of a uniform tensile stress of intensity, making an angle with the x-axis. The plate weakened by the curvilinear hole which is free from stresses (see Figure 3, Figure 4 (n1 = 0.001, n2 = 0.002l, m1 = 0.025, m2 = 0.03I, c = 2, p = 0.25)). Then the functions in (33) and (43) become

(61)

(62)

(63)

(64)

(65)

Figure 3. The relation between components of stresses and the angle made on the x-axis.

Figure 4. The ratio of vertical to horizontal stresses.

(66)

where

2) For and, where is a real constant (see Figure 5, Figure 6 ()).

Then the functions in (33) and (43) become

(67)

(68)

(69)

(70)

(71)

Figure 5. The relation between components of stresses and the angle made on the x-axis.

Figure 6. The ratio of vertical to horizontal stresses.

(72)

(73)

(74)

where

3) For (see Figure 7, Figure 8 ()). Then the functions in (33) and (43) become

(75)

(76)

(77)

Figure 7. The relation between components of stresses and the angle made on the x-axis.

Figure 8. The ratio of vertical to horizontal stresses.

(78)

(79)

(80)

where

References

1. Muskhelishvili, N.I. (1953) Some Basic Problems of Mathematical Theory of Elasticity. Noordroof, Holland.
2. Spiegel. M.R. (1964) Theory and Problems of Complex Variables, Schaum’s Outline Series. McGraw-Hill, New York.
3. Rubenfeld, L.A. (1985) A First Course in Applied Complex Variables. John Wiley & Sons, New York.
4. Bieberbach, L. (1953) Conformal Mapping. Chelsea Publishing Company, New York.
5. Popov, G.Ya. (1982) Contact Problems for a Linearly Deformable Functions. Odessa, Kiev.
6. Sabbah, A.S., Abdou, M.A. and Ismail, A.S. (2002) An Infinite Plate with a Curvilinear Hole and Flowing Heat. Proc. Math. Phys. Soc. Egypt.
7. Atkin, R.J. and Fox, N. (1990) An Introduction to the Theory of Elasticity. Longman, Harlow.
8. Colton, D. and Kress, R. (1983) Integral Equation Method in Scattering Theory. John Wiley, New York.
9. Abdou, M.A. (2003) On Asymptotic Method for Fredholm-Volterra Integral Equation of the Second Kind in Contact Problems. Journal of Computational and Applied Mathematics, 154, 431-446. http://dx.doi.org/10.1016/S0377-0427(02)00862-2
10. Noda, N. and Hetnarski Yoshinobu Tanigowa, R.B. (2003) Thermal Stresses. Taylor and Francis, UK.
11. Hetnarski, R.B. (2004) Mathematical Theory of Elasticity. Taylor and Francis, London.
12. Parkus, H. (1976) Thermoelasticity. Springer-Verlag , New York.
13. Gakhov, F.D. (1966) Boundary Value Problems. General Publishing Company, Ltd., Canada
14. Ciarlet, P.G., Schultz, M.H. and Varga, R.S. (1967) Numerical Methods of High-Order Accuracy for Nonlinear Boundary Value Problems. I. One Dimensional Problem. Numerische Mathematik, 9, 394-430.
15. Zebib, A. (1984) A Chebyshev Method for the Solution of Boundary Value Problems. Journal of Computational Physics, 53, 443-455. http://dx.doi.org/10.1016/0021-9991(84)90070-6
16. Saito, S. and Yamamto, M. (1989) Boundary Value Problems of Quasilinear Ordinary Differential Systems on a Finite Interval. Math. Japon., 34, 447-458.
17. Abdou, M.A. (1994) First and Second Fundamental Problems for an Elastic Infinite Plate with a Curvilinear Hole. Alex. Eng. J., 33, 227-233.
18. Abdou, M.A. (2002) Fundamental Problems for an Infinite Plate with a Curvilinear Hole Having Infinite Poles. Applied Mathematics and Computation, 125, 177-193.
19. Abdou, M.A. and Khamis, A.K. (2000) On a Problem of an Infinite Plate with a Curvilinear Hole Having Three Poles and Arbitrary Shape. Bulletin of the Calcutta Mathematical Society, 92, 309-322.
20. Abd-Alla, A.M., Abo-Dahab, S.M. and Bayones, F.S. (2013) Propagation of Rayleigh Waves in Magneto-Thermo- Elastic Half-Space of a Homogeneous Orthotropic Material under the Effect of Rotation, Initial Stress and Gravity Field. Journal of Vibration and Control, 19, 1395-1420. http://dx.doi.org/10.1177/1077546312444912
21. Abd-Alla, A.M. and Abo-Dahab, S.M. (2012) Effect of Rotation and Initial Stress on an Infinite Generalized Magneto- Thermoelastic Diffusion Body with a Spherical Cavity. Journal of Thermal Stresses, 35, 892-912. http://dx.doi.org/10.1080/01495739.2012.720209
22. Abd-Alla, A.M., Abd-Alla, A.N. and Zeidan, N.A. (2000) Thermal Stresses in a Non-Homogeneous Orthotropic Elastic Multilayered Cylinder. Journal of Thermal Stresses, 23, 413-428.
23. Abd-Alla, A.M., Mahmoud, S.R., Abo-Dahab, S.M. and Helmy, M.I. (2011) Propagation of S-Wave in a Non-Homo- geneous Anisotropic Incompressible and Initially Stressed Medium under Influence of Gravity Field. Applied Mathematics and Computation, 217, 4321-4332. http://dx.doi.org/10.1016/j.amc.2010.10.029
24. Abd-Alla, A.M., Mahmoud, S.R. and AL-Shehri, N.A. (2011) Effect of the Rotation on a Non-Homogeneous Infinite Cylinder of Orthotropic Material. Applied Mathematics and Computation, 217, 8914-8922.