1. Introduction

The problem solution is reduced to definition of two analytic functions and satisfying the boundary conditions. After some mathematical transformations and reasonings, we get two systems of infinite linear algebraic equations with respect to the expansion coefficients and of the functions and .

In order to illustrate the obtained solution, numerical examples are considered.

The centre of the hole coincides with the origin of coordinates , the linear cracks are arranged symmetrically along the axis , the coordinates of the end points of the cracks are denoted by (fig.1). The domain occupied by a body consists of a “lower” half-plane bounded by a straight line and contour of the hole (an ellipse with two linear cracks).

The problem on definition of stresses in domain occupied by an elastic medium has a great value in such a statement in the practice of mining, tunnel building (metrobuilding) and also in many problems of hydro engineering.

2. The Problem for a Weighty Half-plane

On the base of the known principle of mechanics – independence of actions of applied forces, the problem may be divided into two separate cases, the problem for a weighty

half-plane, and a problem for a weightless half-plane under the action of uniformly distributed pressure .

Figure 1. Cross-section of a half-plane with the elliptic holes and two cracks.

In the domain of a half plane bounded by the straight line and contour of holes (ellipse with two linear cracks) the stress components and satisfy the differential equilibrium equations and compatibility conditions (for the first case of the weighty half-plane).

; ;

(2.1)

where is density of elastic medium, g is free fall acceleration.

The boundary conditions:

on the linear boundary L₂ (2.2)

on the contour of the hole , (2.3)

where is normal to and directed from the interior of the domain to the outside.

Each of the stresses and may be represented in the form of the sum [3:9]:

;;

(2.4)

where: and are stresses in solid (without holes) weighty half-plane with the indicated above acting forces (uniformly distributed pressure on the segment of the linear boundary ) and are some special solutions of differential equations (2.1).

The constituents and are additional stresses stipulated by the availability of the hole in medium that weakens it.

Since the components and satisfy equations (2.1), (as special solution of these systems of equations), we can take them in the form[3;9]:

;

(2.5)

Here, is a constant dependent on medium’s property, so called a distance piece factor; for a considerable depth, is very close to a unit.

The integration constants and should be defined from the condition of inversion of the components and in and , respectively on the linear boundary L₂. In (2.5), by accepting , we get

(2.6)

Thus, the components and will take the form:

;

(2.7)

The stress components and satisfy the homogeneous system of equilibrium equations and compatibility equations (2.1) assuming that they vanish at infinity and on the linear boundary .

So, we have:

; on (2.8)

on . (2.9)

Since the contour is free from external forces, then for releasing the stresses and from this contour we should apply the forces with opposite sign and .

In the present case, we are interested mainly in the stress state in the vicinity of the contour (ellipse with two linear cracks) including at the end points of the cracks Therewith, taking into account the conditions (for considerable depth of medium), not making any series error, we can reject exact satisfaction of boundary condition (2.8) and in expression (2.9) neglect the quantity contained in parenthesis of the right hand side. Thus, condition (2.9) is reduced to the form:

(2.10)

We can introduce the functions and of the complex variable that are regular in domain and connected with stress components and by Kolosov – Muskheleshvili – Sherman formulae:

;

(2.11)

Taking into account these expressions, boundary conditions (2.10) are transformed to the following form:

on (2.12)

3. Problem for a Weightless Half-Plane under the Action of Uniformly Distributed Pressure P

In the second case, when a uniformly – distributed load of intensity acts on the linear boundary on the segment , the stresses and for an entire weightless half-plane are determined by the expressions [1;6]:

;

;

(3.1)

where ;

; ;

and are the angles under which the loaded segment from the point of a half-plane is seen, and are the distances from the point to the end of the segment (i.e. to the point and on the linear boundary ).

In the present case, the additional stresses and , stipulated by availability of a hole satisfy the following boundary conditions (similar to the first case, i.e. formulae 2.4). Each of the stresses and are represented in the form of the sum

(3.2)

Similar to the first problem, the stress components and are defined by analytic functions by the known formulae (see. 2.11). Then, boundary conditions (3.2) are transformed also to the following form

(3.3)

In equations (2.12) and (3.3), the variable is an affix of the points of the contour as for a weighty half-plane (the first problem):

;

;

(3.4)

At a considerable depth , then we get ;

For a weightless half-plane, under the action the uniformly-distributed load of intensity P, on the segment , of the boundary L₂ , the right side of equation (3.3) has the form [1;6;7]:

;

(3.5)

It is seen that boundary conditions for determining additional stresses at both variants of active external forces are reduced to determining two analytic functions and (by comparing 2.12 and 3.3 it is seen that the left side are same in both cases, the right side has the form of 3.4 and 3.5, respectively).

For both variants, the desired functions and are representable in the form [3-10].

;

(3.6)

Here, and are analytic functions regular everywhere in the lower half-plane, with a rectilinear boundary , and are holomorphic functions in domain outside of the hole (ellipse with two rectilinear cracks).

The functions and regular outside of the contour L₁ (outside of the hole) are representable in the form [1;6;9;10]

;

(3.7)

Assuming that the coefficients and in expansions (3.7) are known, by the Muskheleshvili method (the method based on the use of a Cauchy type integral) on the linear boundary from the boundary condition 2.12 or from 3.3 for we find the functions and

(3.8)

We should recall that the following expressions hold on the linear boundary :; ; ; ; ; ; .

Now, pass to boundary conditions on the contour of the hole .

The exterior of the contour (ellipse with two rectilinear cracks) is mapped onto the exterior of a unit circle by means of the function [4;5].

; ;

(3.9)

where;

.

The quantities are determined according to[4;5] from the following condition

(3.10)

where ;

The function inverse to the mapping function (3.9) is found in the form [4;5]:

(3.11)

Where; ;

The quantities are determined similar to the quantities replacing in (3.10) by , moreover will be defined from the recurrent formula (3.11) having taken instead of .

For an ellipse, the first six quantities , are found and arranged in the table (see. [4;5]).

Allowing for expansions (3.6), (3.7), (3.8) and (3.9), boundary condition (2.12) or (3.3) on the contour of the hole is reduced to the form (after some mathematical transformations and reasonings, passing to a new variable , and taking into account that holds on a unit circle):

(3.12)

where ; for a weighty half-plane.

, for a weightless half-plane under the action of a uniformly distributed load of intensity on the segment of the rectilinear boundary L₀.

In equation (3.12) having equated the coefficients under the same degrees of , we get the following two systems of infinite algebraic equations for determining the coefficients :

(3.13)

(3.14)

where:

;

The values of the quantities are not cited here because of their bulky form.

Solving jointly systems of equations (3.13) and (3.14) (having taken some first terms from each system), the coefficients and are determined. Analytic functions and are determined for each of indicated problems respectively, from formula (3.7).

Having known the values of the functions and , the additional stresses and are determined by formula (2.11).

The found stress components , and ; for each of the mentioned variants of active loads are given below (calculated at typical points of the holes ; for , where is the length of the aperture and also at the points , for ).

On the base of force action independence principle, by the superposition method we can determine stress state under simultaneous action of loads indicated in both cases:

;

;

(3.15)

where , and belong to the first, , and to the second problem.

4. Numerical Results

4.1. Uniformly Distributed Pressure of Intensity Acts on a Rectilinear Boundary of a Weightless Half-Plane on the Segment

For an entire (without hole) half-plane, this problem has been solved in [1;6;7;8]. Using the obtained results for an entire half-plane by the method indicated above, i.e. solving the system of equations (3.13 and 3.14) for the right side in the form (3.12), by the known formulae (2.11) we find additional stresses stipulated by the availability of a hole. The value of complete – total stresses for different variants for dimensions of a hole and depth are the followings: (for typical points of the hole)

Variant I: ; ;

; the point «» in the draft

;

at the point «».

;

for at the point ;; for ;; .

for at the point ;;.

Variant II: ; ; ;at the point ; ; ;

For ; ; ; at the point ; ; ;

Variant III: ; ; at the points

i.e. at considerable depths the additional stresses equal to zero (influence of a hole and load on a rectilinear boundary almost doesn’t influence the stress distribution).

4.2. A weighty Half-Plane Weakened by a Hole (Ellipse with Two Rectilinear Cracks) is Subjected Only to the Action of Gravity

The contours and are free from external forces. Solving the system of equations for the right side in the form (3.12), we define the desired coefficients and for different relative dimensions. Then, by formula (2.11) we find additional (as the result, total stresses) stresses: and at the typical points of the hole (since we are interested only in stress concentration at the points of hole’s contour). Below we give the found values of these stresses for different variants:

Variant I: for ; ;; ;for ; ;.

Variant II: ;; for ; ; при; ; .

Variant III: ; (distance piece factor equals to one), then at all the points of the contour we have

; .

5. Conclusions

The quantities of additional and total stresses on the of a rectilinear aperture (i.e. at the end points of the crack and in the middle of this crack when ) considerably increase when the crack of approaches to the linear contour

As calculations show, such type stresses have a particular local character and as a rule, they rapidly drop (weaken) when they are removed from the vertex of the crack, where they have maximal value. When , ; the internal contour will become an ellipse, and the problem on definition of additional stresses will be similar to that in the paper[9]. Therefore, here we don’t cite the results.

Notes

¹The method elaborated by N. I. Muskheleshvili [6] is based on the use of a Cauchy type integral. For example: