Numerical Analysis of Stressed State of an Elastic Strip Plate with Collinear Cracks and Thin-Walled Inclusions at Antiplane Shear Loading

In this paper, we calculate the stress distribution state and S.I.F of the crack tips and dislocations of edges of a long strip elastic rectangular plate (Figure 1). The stress Intensity is essentially decreasing the known strength and durability of structural members and engineering parts. For this reason, the necessity of theoretical investigation of stress concentration zones and the development of the methods which decrease the stress intensities is occurred. One of these methods was proposed in [1], the edges of linear finite crack of elastic infinite plate at the end areas are joined via thin-walled inclusion in the shape of continuously distributed linear and nonlinear deformed springs, meanwhile, the plate is subjected by uniformly distributed tensile remote stress perpendicular to the central line of crack. Taking into account the above-mentioned physical model of inclusions and based on assumptions in [24], the valuable decrease of stress intensity factors (SIF) at the end points of crack can be achieved by the appropriate selection of elastic and geometric characteristics of problem, and this can prevent the crack propagation. Applying Fourier finite sine transformation, the solution of stated problem can be reduced to the solution of singular integral equation (SIE), and, consequently, via the known method [5-7], the solution of singular integral equations can be reduced to the system of linear equations. For the main characteristics of stated problem, such as the SIF, the crack opening, the shear stresses on the edges of the inclusion, and the shear stresses out-of-crack on the line of its location, the obvious equations are obtained, the special cases are considered and for various materials the decreasing trend of S.I.F based on the various shear modulus were shown.


Introduction
In this paper, we calculate the stress distribution state and S.I.F of the crack tips and dislocations of edges of a long strip elastic rectangular plate ( Figure 1). The stress Intensity is essentially decreasing the known strength and durability of structural members and engineering parts. For this reason, the necessity of theoretical investigation of stress concentration zones and the development of the methods which decrease the stress intensities is occurred. One of these methods was proposed in [1], the edges of linear finite crack of elastic infinite plate at the end areas are joined via thin-walled inclusion in the shape of continuously distributed linear and nonlinear deformed springs, meanwhile, the plate is subjected by uniformly distributed tensile remote stress perpendicular to the central line of crack. Taking into account the above-mentioned physical model of inclusions and based on assumptions in [2][3][4], the valuable decrease of stress intensity factors (SIF) at the end points of crack can be achieved by the appropriate selection of elastic and geometric characteristics of problem, and this can prevent the crack propagation. Applying Fourier finite sine transformation, the solution of stated problem can be reduced to the solution of singular integral equation (SIE), and, consequently, via the known method [5][6][7], the solution of singular integral equations can be reduced to the system of linear equations. For the main characteristics of stated problem, such as the SIF, the crack opening, the shear stresses on the edges of the inclusion, and the shear stresses out-of-crack on the line of its location, the obvious equations are obtained, the special cases are considered and for various materials the decreasing trend of S.I.F based on the various shear modulus were shown.
< of the crack. Besides that at the ending areas The edges of the crack are joined by the thin-walled inclusions with the shear modulus G deforming by the Winkler model. Let's assume that the prismatic body Ω subjected to the above-mentioned shear forces is in a state of anti plane deformation in the direction of Oz-axis on the basic plane Oxy . The main rectangle is cross-section of the body Ω with the plane located on this plane Oxy (Figure 1).
It is necessary to determine the dislocation density on the crack edges, SIF, the crack edges opening, the shear contact stresses on the edges of the inclusion, and the shear stresses outside the crack on the line of its location. For convenience in numerical calculation the strip plate is divided to several plates so that each segment has one crack at the center ( Figure 2). zero components of stresses. Therefore, the problem can be mathematically stated as a boundary value problem in the following way: For of the determination of boundary value problem (1), the rectangle D is divided by Ox-axis onto upper rectangles. The following supporting boundary value problems are considered for them.
Where, the sign "+" and "-" are related to the rectangles D + and D − , correspondingly.  .
Taking into account the symmetry of stated problem with respect to -axis

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Consequently, the determination of supporting problem (2) for the rectangle D + can be considered only. Based on the reference [8], the above-mentioned problem can be determined via Fourier finite sine transformation on the variable x .
Therefore, the Fourier inverse transformation has the following expression: Multiplying by ( ) sin nx π  both sides of the differential equation and the border conditions of (2), and integrating it from 0 to  , Fourier finite transformation (5) can be applied to the boundary value problem (2) Where, notation ( ) The boundary value problem (7) can be defined by the following equation:

The Singular Integral Equation
After some simple transformations and calculations according to [1] and [9][10][11] the following equations can be derived: The first integral of equation (10) and through the half of dislocation density on the edges of the crack

Gauss Quadrature Method
Now, as it was mentioned above, the determinative singular integral equations (S.I.E) (10) can be reduced to a system of linear equations and following to the approach represented in [1], the determinative S.I.E (10) can be reduced to the system of Algebraic linear equations as follows:

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The main physical characteristics of the stated problem can be expressed by the determination of the system of linear equations (14). The plate shear stress out of the crack at the interval y=0 , (0, / 2 ) ( / 2 , ) x l a l a l ∈ − ∪ + can be obtained as below: For the dimensionless crack opening, the following equation can be obtained ( )

Numerical Calculation
For numerical calculations we consider a special case of the loading of the rectangular plate. For this case, the crack edges are free of shear forces, and shear concentrated force acting on the horizontal sides of rectangular plate: is a certain Dirac Delta function. In this case (11), as well as, with respect to equation (8) Taking into account the above-mentioned consideration, the can be expressed in the following way [1]: And the function ( ) f r from equation (11) can be obtained in the following way.

Conclusion
Numerical calculations show that the repair of crack tips causes avoiding the singularities and reduces the anti-plane S.I.F KIII , about 50 percent, also for strengthen and stop the crack propagation near to region at the tips, it is not need to use a material with very high shear rigidity value G0 [12]. Meanwhile the crack opening C.O.D decreases about 50 percent, and in addition the shear stresses at the crack tips fall down near to 30 percent. The linear Algebraic system of equation (14) were solved regarding to relations (19)-(21) for the special case of anti-plane shear loading for several metals on a base metal steel for the main plate [13]. The shear moduli of the metals over the steel shear modulus are represented as a ratio on the horizontal axis in ( Figure 5).
The stress intensity factors S.I.F K III that are very important and show the intensity or index of upper limit of shear stresses magnitude calculated through the equation (18) that are shown in (Figure 5), which presents the decreasing trend of S.I.F curve when the ratio of k=G0/G goes up. It also shows that the crack tip repairing by adding another material at the tips to prevent crack propagation, can reduce the S.I.F about 50 percent, means this approach is very effective to avoid the crack propagations in cracked plates, mathematically is a treatment for well-known singularities at the crack tips, defects and holes.