Dynamic Stress Intensity Factors for Three Parallel Cracks in an Infinite Plate Subject to Harmonic Stress Waves

Dynamic stresses around three parallel cracks in an infinite elastic plate that is subjected to incident time-harmonic stress waves normal to the cracks have been solved. Using the Fourier transform technique, the boundary conditions are reduced to six simultaneous integral equations. To solve these equations, the differences of displacements inside the cracks are expanded in a series. The unknown coefficients in those series are solved using the Schmidt method such that the conditions inside the cracks are satisfied. Numerical calculations are carried out for some crack configurations.


Introduction
A time-harmonic solution for stresses around a crack in an infinite plate was reported by Loeber and Sih [1].In their study, they obtained the Mode III dynamic stress intensity factor during the passage of a time-harmonic anti-plane shear wave.Subsequently, they also solved the crack problem for a compression wave and a vertically polarized shear wave [2].Adopting a somewhat different approach, the same problem was studied independently by Mal [3].The corresponding three-dimensional solutions for a penny-shaped crack have been obtained by Sih and Loeber [4,5] and by Mal [6].
Materials are generally weakened by some cracks.Therefore, it is of interest to reveal the mutual effect of the cracks on the dynamic stress intensity factors.Itou solved the dynamic stresses around two collinear cracks in which a self-equilibrated system of pressure is varied harmonically with time [7].Later, the Mode III solution was given for two collinear cracks by Itou [8].As for three collinear cracks, Mode I solutions were determined under the condition that time-harmonic normal traction is applied to the surfaces of the cracks [9].
Materials are occasionally weakened by some parallel cracks.Takakuda solved the time-harmonic problem for two parallel cracks in an infinite plane subjected to waves that impinge perpendicular to the cracks [10].So and Huang analyzed the Mode III stress intensity factor around two cracks located in arbitrary positions in an infinite medium subjected to incident SH waves [11].Meguid and Wang cleared the effect of the existence of an arbitrarily located and oriented micro defect on the dynamic stress intensity factors for a finite main crack subjected to a plane incident wave [12].Ayatollahi and Fariborz provided the analysis of multiple curved cracks in an infinite plane under in-plane time harmonic loads [13].Itou and Haliding assumed that two small collinear cracks are situated symmetrically above a main crack in an infinite plate and provided the dynamic stress intensity factors during passage of time-harmonic waves [14].
A peak value of the dynamic stress intensity factor for collinear cracks in an infinite elastic plate is generally about 1.20-1.60times larger than that of the corresponding static value . However, in the paper [10], it was found that a peak value of the dynamic stress intensity factor for two parallel cracks is significantly larger than those for the collinear cracks.For example, for an infinite plate containing two parallel cracks of length separated by a distance , the 2 a h peak i K static i K ratio is 4.16 for 1.0 h a  [10].It was also shown that similar results appear for two parallel cracks in an infinite orthotropic plate subjected to incident time-harmonic stress waves [15].From this fact, it is expected that peak static i i K K ratio will be very large for three parallel cracks during passage of the time harmonic stress waves.
In investigating the peak static i i K K ratio for three parallel cracks, the mixed boundary value conditions are reduced to six dual integral equations.It has been shown that the integral equations can be converted into six sets of an infinite series and that the unknown coefficients in the series can be solved using the Schmidt method [16].The author has developed a Fortran program to obtain the unknowns in four dual infinite series [15,17].However, it is very difficult to write a Fortran program that is capable of solving the unknowns in six sets of an infinite series, making it difficult to solve the time-harmonic problem for three parallel cracks.The present author decided to solve the time-harmonic dynamic crack problem for three parallel cracks in an infinite elastic plate because it is of importance to provide the dynamic stress intensity factors in fracture mechanics.
In this study, time-harmonic stresses are solved for three parallel cracks in an infinite elastic plate during the passage of time-harmonic stress waves propagating normal to the cracks.The boundary conditions were reduced to dual integral equations with use of the Fourier transform technique.In order to solve these equations, the differences between the crack surface displacements are expanded to a series of functions that are equal to zero outside the cracks.The Schmidt method is modified so as to solve for the unknown coefficients in six sets of an infinite series.
A Fortran program has been developed to calculate the stress intensity factors for several crack configurations numerically.

Fundamental Equations
Consider a crack in an infinite plate located along the x  axis from to at , with respect to the rectangular coordinates x y ; an upper crack from b  to at ; a lower crack from to at ; and incident time-harmonic stress waves propagating normally to the cracks, as shown in Figure 1.For convenience, is referred to as layer 1); is referred to as layer 2); is referred to as the upper half-plane 3); and is referred to as the lower half-plane 4).Let and be defined as the x and components of the displacement, respectively.If the displacement components and are expressed by two functions the equations of motion reduce to the following forms: where is time.The dilatational wave velocity t L c and the shear wave velocity under plane stress conditions can be given as follows: where  is the modulus of rigidity,  is Poisson's ratio, and  is the density of the material.
The stresses can be expressed by the equations with The incident stress waves that propagate through the infinite plate parallel to the -axis in the negative direction can be expressed as follows: where is a constant and p  is the circular frequency.
Substituting the following relations Hereafter, the time factor  is omitted from the equations for convenience.Hence, displacements and stresses are expressed, respectively, by the following: The boundary conditions for this problem can be expressed as   , 0 a t 0 , where the subscript indicates the layer , the subscript 3 indicates the upper half-plane (3), and the subscript 4 indicates the lower half-plane (4).

Analysis
To obtain a solution, the following Fourier transforms are introduced: Applying Equaiton (18) to Equation (8), we obtain: In the Fourier transform domain, the displacements and stresses are denoted, respectively, by the forms For the layer , the solutions of Equation ( 22) have the following forms: where , , , are unknown coefficients.For the upper half-plane (3) and the lower half-plane (4), the solutions of Equation ( 22) have the following forms in terms of the unknown coefficients : The stresses and displacements can be expressed by twelve unknowns: 1 1 1 , , A B C , and .Using Equations ( 12), ( 13) and ( 14), which are valid for , , , , , , , x   , the twelve unknowns are reduced to six unknowns, yielding the following relations: where the expressions of the known functions   , 1,2, ,6 have been omitted.
To satisfy the boundary conditions ( 16), ( 18) and (20), the differences of the displacements are expanded as follows: where and , , , ,  29)-( 34) are expressed by where However, the variables on the left-hand sides of Equations (35), ( 36) and (37) can be expressed in terms of the unknowns 1           where the expressions of the known functions where ij H is the cofactor of the element , and Consequently, stresses that satisfy the boundary conditions ( 12), ( 13), ( 14), ( 16), ( 18) and (20) can be expressed in terms of the unknowns and where the expressions of the known functions have been omitted.Finally, the remaining boundary conditions ( 15), ( 17) and ( 19), which must be satisfied inside the cracks, reduce to the forms: for for for where the functions are known.
For example, where the constant 2L Q is given by where L  is a larger value of  .The functions   i u x are denoted by the equations: exp , 0, , 0, exp , 0 The unknowns and , , , ,  44), ( 45), ( 46), ( 47), ( 48) and ( 49) can now be solved using the Schmidt method described in Appendix A.

Stress Intensity Factors
Once the unknown coefficients and , , , , a b c d e n f have been solved, all the stresses and displacements can likewise be solved.In fracture mechanics, it is important to determine the stress intensity factors defined from the stresses in the region near the crack ends.Using the relations the stress intensity factors can be expressed as where the constants are given by expressions taking the similar form as Equation ( 53).

Numerical Examples
The dynamic stress intensity factors were calculated numerically with quadruplex precision using a Fortran program, during the operation of which, overflow and underflow do not occur within the range to .Numerical calculations were performed for a Poisson's ratio   For the case of two parallel cracks, the distance in

Discussion
In the previous paper [14], time-harmonic stresses are solved for three cracks in an infinite elastic plate during the passage of time-harmonic stress waves.Two collinear cracks are situated symmetrically on either side of the main crack.The mixed boundary conditions with respect to the three cracks are reduced to four sets of an infinite series.The two sets of an infinite series are derived from the boundary conditions inside the main crack, while the other two sets are derived from those inside one of the upper small cracks.The method to solve the unknown coefficients in the infinite series has been already described in [15,17].
In the present paper, time-harmonic stresses are solved for an infinite elastic plate weakened by three parallel cracks.As the three cracks are not collinear, the boundary conditions with respect to the three cracks are reduced to six sets of an infinite series.Each of the two

Conclusions
Based on the numerical calculations outlined above, and with reference to Figures 2 through 5, the following conclusions are reached: 1) There is a critical circular frequency in the three cracks case near 0.9 1.0 h a h a   .For two parallel cracks, the corresponding value is 5.87.It can be seen that the presence of the third crack has a significant effect upon the dynamic stress intensity factor around the center crack between the upper and lower parallel cracks.
[7] S. 2) For 1 2 2.5 h a h a   (Figure 2), the slope to the peak value of However, for 1 2 1.0 h a h a   (Figure 3), the curve rises steeply to the maximum value of after which it declines equally steeply.The curve exhibits a sharp peak at the critical circular frequency near 0.9 T a c   .
[9] S. 3) In static solutions, the stress intensity factors for three equal-length cracks decrease slightly as 1 h a  2 h a   decreases.However, this is accompanied by a significant increase in the peak value of

Appendix A
For convenience, Equations ( 42) through (47) can be rewritten as for A set of functions that satisfy the orthogonality condition can be constructed from a given set of arbitrary function, say 65 ( ) where is the cofactor of the element of , which is defined as (A.10) The second equality yields (A.11) and considering Equation (A.8), the first equality shows that for Using the same procedure, the orthogonal function * ( ) Substituting Equation (A18) into Equation (A12), we obtain the following relation: Replacing the coefficients and The Fourier transforms of Equations (

in Figure 2 .
can be verified that the numerical integrations have been performed satisfactorily because the integrands decay rapidly as the integration variable Schmidt method has been applied by truncating the infinite series in Equations (44), (45), (46), (47), (48) and (49) by summing from 1 n  to .It has been verified that the values for the left-hand side of the equations coincide with those for the right-hand side with acceptable accuracy.8 nThe absolute values of the stress intensity factors given in Equations (56), (57) and (58) are calculated for 1The straight dashed lines on the left-hand side of Figure2indicate the corresponding static values given by Ishida[18].

Figure 3
stress intensity factors for two cracks in an infinite plate have been solved by Takakuda[10].In the present study, the same problem has been reworked, and the results are plotted in Figures4 and 5 denotes the distance between the two parallel cracks.