SCATTERING OF SH-WAVES BY A GRIFFITH CRACK IN A LONG STRIP AT ASYMMETRIC POSITION

The scattering of SH-waves by a Griffith crack in an infinitely 
long elastic strip situated at an asymmetric position 
has been analyzed. Applying Fourier transform, the mixed boundary 
value problem has been reduced to the solution of dual integral 
equations which finally has been reduced to the solution of a 
Fredholm integral equation of second kind. The numerical 
values of stress intensity factor, crack opening displacement, and 
scattered field outside the crack have been illustrated 
graphically to show the effect of asymmetry of the crack position.


Introduction
Cracks or inclusions are present essentially in most of the structural materials, either as natural defects or as a result of fabrication processes.The diffraction of elastic waves becomes more practical when boundaries are present in the medium.Great attention has been given to the study of diffraction of elastic waves by cracks situated at asymmetric position .Loeber and Sih [5] and Mal [6] have studied the problem of diffraction of elastic waves by a Griffith crack in an infinite medium.The problem of finite crack at the interface of two elastic half-spaces has been discussed by Srivastava et al. [11] and Boström [2].Finite crack perpendicular to the surface of the infinitely long elastic strip has been studied by Chen [3] and by Srivastava et al. [10].Shindo et al. [9] considered the problem of impact response of a finite crack in an orthotropic strip.Crack in inhomogeneous elastic strip has been analyzed by Sarkar et al. [8].Matysiak and Pauk [7] studied edge crack in an elastic layer resting Winkler foundation.Recently, Birinci and Erdol [1] analyzed the problem of a layered composite containing a crack in its lower loaded-by-rigid stamp.
In our note, we have treated the diffraction of SH-waves by a crack situated at asymmetric position in an infinitely long elastic strip which has not been considered yet.This type of situation arises in almost all cases of fabrication processes in construction technology.Applying the Fourier transform, the mixed boundary value problem has been converted to the solution of dual integral equations.The dual integral equations have been finally reduced to a Fredholm integral equation of second kind by applying Abel's transform.Expressions for the stress intensity factor and crack opening displacement have been plotted to show the asymmetry of the crack position.Also stress (scattered field) outside the crack has been calculated and shown by three-dimensional graph.

Formulation
Consider the boundary value problem of interaction of SH-waves by Griffith crack situated at the asymmetric position in an infinitely long elastic strip The crack is located in the region −a < x 1 < a, −∞ < z 1 < ∞, y 1 = 0. Normalizing all the lengths with respect to a and putting the location of the crack becomes 2.1) referring to a Cartesian coordinate system (x, y,z).Let a plane harmonic SH-wave originating at y = −∞ impinge on crack normally to the x-axis.The only nonvanishing z-component of displacement which is independent of z is w(x, y,z) = W(x, y)e −iωt .Now our problem reduces to the solution of the equation where is subject to the boundary conditions ) Henceforth, the time factor e −iwt which is common to all field variables would be omitted in the sequal.
The nonvanishing stresses are ) The solution of (2.2) can be taken as where Therefore the expressions of stresses are (2.12) (2.14) Now, from boundary conditions (2.4) and (2.5), we obtain the following dual integral equations for the determination of the unknown function where (2.17)

Method of solution
In order to reduce the dual integral equations (2.15) and (2.16) to a single Fredholm integral equation, we assume that so that (2.16) is automatically satisfied.Now, (2.15) can be written as where where Using contour integration technique [11], the integral L 1 (u,t) can be converted to the following finite integral: Now, putting b = c = h, we can find the following result for Griffith crack in a symmetric position: (3.7)

Quantities of physical interest
The shear stress τ yz (x, y) in the plane z = 0 in the neighborhood of the crack can be found from (2.12) and is given by Substituting the values of B(ζ) and C(ζ) from (2.14) and using (3.1), the expression for the stress can be presented as Defining the stress intensity factor K by we obtain Now, crack opening displacement can be obtained from (2.10) as which on substitution of the value of A(ξ) from (3.1) takes the form Scattered field τ yz (x, y) for x > 1, z > 0 is calculated from (2.12), (2.14), and (3.1) and is represented by the expression where m = b + c.

Numerical results and discussion
The method of Fox and Goodwin [4] has been used to solve the integral equation (3.4) numerically for different values of dimensionless frequency k 2 and b, c which indicate the asymmetry in position of the crack.The integral in (3.4) has been represented by a quadrature formula involving values of the desired function g(t) at pivotal points inside the specified range of integration, and then converted to a set of simultaneous linear algebraic equations.The solution of linear algebraic equations gives a first approximation to the pivotal values of g(t), which has been improved by the use of difference-correction technique.
After solving the integral equation (3.4) numerically, the stress intensity factor (SIF) K and crack opening displacement (COD) µ∆W(x,0)/τ 0 a have been calculated numerically and plotted separately against k 2 and dimensionless distance x (0 < x < 1), respectively, for different values of b and c to show the effect of positional symmetry of the crack.
In  decreasing with increases in x as well as y and near boundary, it shows wave-like nature due to boundary effect.

. 3 )
Substituting the value of B(ζ) and C(ζ) from (2.14) and with the help of (3.1), (3.2) can S. Ghosh et al. 2341 be converted to the following Fredholm integral equation of second kind: