^{1}

^{*}

^{2}

^{*}

Complex variables method has been used to solve the first and second fundamental problems for an infinite plate weakened by a generalized curvilinear hole *C*. The curvilinear hole is conformally mapped on the domain outside or inside a unit circle *γ* using a general rational mapping function with complex constants. Many special and new cases are derived from this work. Some of the work of the previous authors in this domain will be considered as special cases of this paper. Also the interesting cases when the shape of the hole takes different famous shapes are included. The components of stresses for some examples are obtained.

The boundary value problems for isotropic homogeneous performed infinite plates have been discussed by several authors: see Colton and Kress [

It is worth mentioning that Exadaktylos and Stavropoulou [

It is known that, see Muskhelishvili [

where,

In terms of the rational mapping function

where X, Y are the components of the resultant vector of all external forces acting on the boundary and

In the absence of body forces, Muskhelishvili [

In this work, the complex variables method will be applied to solve the first and second fundamental problems for an infinite plate with a generalized curvilinear hole C conformally mapped on the domain outside a unit circle

where

Also, many applications for the first and second fundamental problems are considered and the components of stress and strain have been obtained and plotted to investigate their physical meaning. Moreover, computer work using maple 9.5 has been used in applications to give the shapes of holes and curves of stresses with some calculations of stresses at their important points.

The physical interest of the mapping (6) comes from its special cases and its different shapes of holes that can be obtained, see Figures 1-6.

From the rational mapping we can discuss the following:

1) The number of the holes corners is subjected to

2) The shape of the hole depending on the values of n’s and m’s.

3) Entering none zero values of the complex constants m and d never gives symmetric graphs. While, entering zero values for all imaginary parts of both m and d, we get symmetric shapes around the x-axis. On the other hand, entering zero values for all real parts of both m and d, we get symmetric shape around the y-axis.

4) The complex constant m works on circling the shape from the symmetry situation and the circling angle is given by

5) Using the rational mapping function

6) The complex constant d works on expanding the corners of the hole shape.

In this section, we use the transformation mapping (6) in the boundary conditions (1), and complex variables method, Cauchy method, to obtain a closed form expression for the Goursat functions

where

and

Using (7) in the boundary conditions (1) and on

where

and

The function

where

and the complex constant b, will be determined, is given by

Differentiating (15) with respect to z, then using the result in (17), the complex constant b takes the form

where

Also, the function

where

The two formulas (15) and (19) are representing the Goursat functions for the first and second fundamental problems for an infinite elastic plate weakened by The two formulas (15) and (19) are representing the Goursat functions for the first and second fundamental problems for an infinite elastic plate weakened by a generalized curvilinear hole C, that can be transformed outside a unit circle g by the rational mapping (6).

An important new case for discussion is using the transformation mapping

This mapping function, when

Here, we discuss the following:

1) By considering the reality of the constants of the mapping (1.6), the Goursat functions, in this case, are agree with work of Abdou and Khar-Eldin [

2) When

_{J} are complex constants (23)

The Goursat functions, in this case, become

The results of the two formulas (24) and (25) are in agreement with the work of Abdou and Asseri [

3) When

The Goursat functions, in this case, of the two formulas (15) and (19) agree with the all results of Abdou and Asseri [

4) In the mapping function (20) if we let m = 0, then for finite expansion, we will have the following mapping function

with the corresponding Goursat functions

The three Formulas (27)-(29) are equivalent to those derived by Exadaktylos and Stavropoulou [

5) Also, in (27), if we allow the index inside the summation sign to take the form

1) For

The complex constant b has been determined by Equation (18) and its value was calculated by using Maple 9.5. Here, we have the Goursat functions for an infinite plate weakened by a curvilinear hole C which is free from stresses. The plate stretched at infinity by the application of a uniform tensile stress of intensity P, making an angle q with the x-axis.

For_{xx}, s_{yy}, s_{xy} and the angle q are considered in Figures 7-9.

2) For

Thus, (32) and (33) give the solution of the first fundamental problem for an isotropic infinite plate with a curvilinear hole, when there are no external forces and the edge of the hole is subject to a uniform pressure P.

If in application (2) we write

For _{xx} ,s_{yy} ,s_{xy} and the angle q, using Maple 9.5 are considered in Figures 10-12.

3) For

where

Here, we have the case of uni-directional tension of an infinite plate with a rigid curvilinear centre. The constant e, which represents the angle of rotation, can be determined from the condition that the resultant moment of the forces, acting on the curvilinear centre from the surrounding material, must vanish i.e.

Hence, we have

where

For θ_{xx}, s_{yy}, s_{xy} and the angle q are considered in Figures 13-15.

From the previous results, we can establish the following Case (1): In the case of Bi-axial tension, we have

Hence, we get

where

The complex constant b has been determined by Equation (18) and its value was calculated by using Maple 12. For n = 0.1 + 0.1i, m = 0.2 − 0.2i, d = 1 + i, P = 1/4 and c = 2. The relation θ between the stress components s_{xx}, s_{yy}, s_{xy} and the angle q are considered in Figures 16-18.

Case (2): When the curvilinear centre not allowed to rotate, i.e. when

For _{xx} ,s_{yy} , s_{xy} and the angle q , using Maple 12 are considered in Figures 19-21.

4) When the force acts on the centre of the curvilinear kernel and the stresses vanish at infinity. In this case the kernel can not be rotate and it remains in its original position. Hence, we get

where

Therefore, we have the solution of the second fundamental problem in the case when (X, Y) acts on the centre of curvilinear hole.

For_{xx},s_{yy}, s_{xy} and the angle q, using Maple 9.5 are considered in Figures 22-24.

From the previous work the following discussion and results can be concluded 1) In the theory of two-dimensional linear elasticity one of the most useful techniques for the solution of the boundary value problem for a region weakened by a curvilinear hole is to transform the region into a simpler shape to get the solution directly without difficulties.

2) The transformation mapping

3) The physical interest of the using mapping transform comes from its different shapes of holes it treats and different directions it takes. This mapping function deals with famous shapes of tunnels, thereon it is useful in studying the stresses around tunnels. In underground engineering the tunnel is assumed to be driven in a homogeneous, isotropic, linear elastic and pre-stressed geometrical situation. Also, the tunnel is considered to be deep enough such that the stress distribution before excavation is homogeneous. Excavating underground openings in soils and rocks is done for several purposes and in multi-sizes. At least, excavation of the opening will cause the soil or rock to deform elastically. The excavation in soil or rock is a complicated, dangerous and expensive process. The mechanics of this can be very complex. However, the use of conformal mapping that allows us to study stresses and strains around a unit circle makes it useful for engineers and easier for mathematicians.

4) The complex variables method (Cauchy method) is considered one of the best methods for solving the integro differential equation, boundary value problem, of Equation (1) and obtaining the two complex potential functions, Goursat functions,

5) The stress is an internal force whereas positive values of it mean that stress is in the positive direction, i.e. stress acts as a tension force. On the other side, negative values of stress mean that the stress is in the negative direction, i.e. stress acts as a press force.

6) The most important issue deduced from mapping the stress components is that

7) When