 Engineering, 2010, 2, 485-495 doi:10.4236/eng.2010.27064 Published Online July 2010 (http://www.SciRP.org/journal/eng) Copyright © 2010 SciRes. ENG 485Dynamic Stress Intensity Factors for Three Parallel Cracks in an Infinite Plate Subject to Harmonic Stress Waves Shouetsu Itou Department of Mechanical Engineering, Kanagawa University, Yokohama, Japan E-mail: itous001@kanagawa-u.ac.jp Received February 25, 2010; revised April 9, 2010; accepted April 22, 2010 Abstract 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. Keywords: Three Cracks, Time-Harmonic Problem, Stress Intensity Factor, Integral Equation, The Schmidt Method 1. Introduction A time-harmonic solution for stresses around a crack in an infinite plate was reported by Loeber and Sih . 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 . Adopting a somewhat different approach, the same problem was studied independently by Mal . The corresponding three-dimensional solu-tions for a penny-shaped crack have been obtained by Sih and Loeber [4,5] and by Mal . 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 . Later, the Mode III solution was given for two collinear cracks by Itou . As for three collinear cracks, Mode I solution s were determined under the condition that time-harmonic normal traction is applied to the surfaces of the cracks . 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 . 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 . 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 . Ayatollahi and Fariborz provided the analysis of multiple curved cracks in an infinite plane under in-plane time harmonic loads . 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 inten-sity factors dur ing passage of time-harmonic wave s [1 4 ] . A peak value of the dynamic stress intensity factor for collinear cracks in an infinite elastic plate is generally about 1.20-1.60 times larger than that of the corresponding static v alue peakiKstaticiK. However, in the paper , 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 2ahpeakiK staticiK ratio is 4.16 for 1.0ha . It was also shown that similar results appear for two parallel cracks in an infinite orthotropic plate subjected to incident time-harmonic stress waves . From this fact, it is expected that peak staticiiKK ratio will be very large for three parallel cracks during passage of the time harmonic stress waves. S. ITOU 486 In investigating the peak staticiiKK ratio for three par-allel 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 . The author has developed a Fortran program to obtain the unknowns in four dual infinite series [15,17]. How-ever, it is very difficult to write a Fortran program that is capable of solvin g the unknown s 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 prob-lem 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 nor-mal to the cracks. The boundary co nditions were redu ced to dual integral equations with use of the Fourier trans-form 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 progr am has been developed to calculate the stress intensity factors for several crack configurations numerically. 2. Fundamental Equations Consider a crack in an infinite plate located along the xaxis from to at , with respect to the rectangular coordinates aa0y)(,xy; an upper crack from b to at ; a lower crack from to at ; and incident time-harmonic stress waves pro- pagating normally to the cracks, as shown in Figure 1. For convenience, is referred to as layer 1); is referred to as layer 2); is re-ferred to as the upper half-plane 3); and is re-ferred to as the lower half-plane 4). Let and be defined as the by21hy0c1hy*uch*v2hhy1h0yy2x and components of the displace-ment, respectively. If the displacement components and are expressed by two functions y*u*v*,,xyt and ,,*xyt such that ** ****,uxyv x  y (1) the equations of motion reduce to the following forms: 2*22*222 *22*2 2*222*21,1LTxyc txyc t    (2) where is time. The dilatational wave velocity tLc and the shear wave velocity under plane stress con-ditions can be given as follows: Tc221 ,Lc2Lc  (3) 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 *2*2222*22**2*2222*22**2*2*2222*222222yy Lxx LxyL 2xctyctxyxyxyxct      (4) with 2221LTcc (5) The incident stress waves that propagate through the infinite plate parallel to the –axis in the negative di-rection can be expressed as follows: y**exp exp0incyy LLLincxypitypiyctcc (6) where is a constant and p is the circular fre-quency. Substituting the following relations  **,, exp ,,, expxyti txyti t (7) into Equation (2) results in 22222222220,0xyxy   (8) with  2,11TLTLTLT Tcccc ccc c  (9) Hereafter, the time factor exp it is omitted from the equations for convenience. Hence, displacements and stresses are expressed, respectively, by the following: ,uxvxyy (10) Copyright © 2010 SciRes. ENG S. ITOU487 22222 22*22 222222222222222yy Lxx Lxy Lxcxyycxxy xc     y (11) The boundary conditions for this problem can be ex-pressed as 1313 1,at,yyyy xyxyyh x (12) 1212,at0,yyyy xy xyyx (13) 2424 2,at,yyyy xyxyyhx (14) 1111exp,0 at,yyL xypihc yhx b (15) 31 3110,0, at,uu vvyhbx (16) 11,0at0,yy xypy xa (17) 12 120,0, at0,uuvvy ax (18) 2222exp,0at,yyL xypihc yhx c(19) 24 2420,0, at,uuvvy hcx  (20) 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). 1, 2ii 3. Analysis To obtain a solution, the following Fourier transforms are introduced:    exp ,()1 2expffxixdxfxif ixd  (21) Applying Equ aiton (18) to Equation (8), we obtain : 222222220,0ddyddy   (22) In the Fourier transform domain, the displacements and stresses are denoted, respectively, by the forms ,ui ddyvddyi   (23) 22222 22222212212212yyxxxyid dyddy iddid dy      y (24) For the layer , the solutions of Equation (22) have the following forms: 1, 2i22 2222 22sinh coshsinh coshii iii iAyB yCyD   y (25) where ,,,ii iABD 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 : 3344,,,CDCD22 2233 322 2233 3exp expexp expCyCiDyDi yy(26) 22 2244 422 2244 4exp expexp expCyCiDyDi   yy (27) The stresses and displacements can be expressed by twelve unknowns: 111,,ABC, and . Using Equations (12), (13) and (14), which are valid for 1222233,,,,,,,DABCDCD4C4Dx, the twelve unknowns are reduced to six unknowns, yielding the following relations: 21 111 121 131142 1521621 211 221 2312422522631 311 321 331342 3523631 411 421 431 4424524641511 521 531542 552iCAfBfiCfi DfAfBfiDAfBfiCfiDfAfBfCAfBfiCfiDfAfBfiDAfBfiCfi DfAfBfCAfBfiCfiDfAfB     5641 611621 631 64265266fiDAfBfiCfi DfAfBf (28) where the expressions of the known functions ,1,2,,6ijfij have been omitted. To satisfy the boundary conditions (16), (18) and (20), the differences of the displacements are expanded as follows: 113 1sin 2sinfor0forbb nnuuanxbxbbx (29) 113 1cos21sinfor0forbb nnvv bnxbxbbx (30) 112 1sin 2sinfor0foraa nnuuc nxaxaax (31) Copyright © 2010 SciRes. ENG S. ITOU 488 112 1cos21sinfor0foraannvv dnxaxax a (32) 124 1sin2sinfor0forccnnuuenxcxcx c (33) 124 1cos(21) sinfor0forcc nnvvf nxcxcx c(34) where and ,,,,nnnnnabcde nf are unknowns, and the subscripts and indicate the values at ,ab c1,yh and , respectively. The Fourier trans-forms of Equations (29)-(34) are expressed by 0y2yh  13 2113 211221bb nnnbb nnniuuanJ bvvbnJ b   (35)   12 2112 211221aa nnnaa nnniuucnJ avvdnJ a  (36)   24 2124 211221cc nnncc nnniuuenJcvvfnJc   (37) where nJ is the Bessel function. However, the variables on the left-hand sides of Equa-tions (35), (36) and (37) can be expressed in terms of the unknowns 11 112,, ,,ABiCiDA and : 2B13111112 113 114215216131 211 221 231 24225226bbbbiuuAh BhiCh iDhAhBhvv AhBhiChiDhAhBh   (38) 121311 321 331342 35236121 411 421 431 44245246aaaaiu uAh Bh iChiDhAhBhvv AhBhiChiDhAhBh  (39) 24151152 153 154255256241 611621 631 64265266cccciuuAh Bh iCh iDhAhBhvv AhBhiChiDhAhBh  (40) where the expressions of the known functions ,ijhij have been omitted. 1, 2,, 6Equating Equations (35), (36) and (37) with Equations. (38), (39) and (40), the unknowns 1,A1,B1,iC1,iD 2A and can be indicated by the unknowns and 2Bne,na,nbc ,n,ndnf. For example, the unknown 1A is ex-pressed as:       11121121 231 21141 251 2161 21221221221nnnnnnnnnnnnnnnnnnAanHJb bnHJbcnHJa dnHJaen HJcfn HJac     (41) where ijH is the cofactor of the element , and ijh is given by ,, 1,2,,6ijhij (42) Consequently, stresses that satisfy the boundary con-ditions (12), (13), (14), (16), (18) and (20) can be ex-pressed in terms of the unknowns and ,na,nb,nc,nd,ndnf. For example, stress 1yy at is of the form 1yh            1211 012210132014210152016212c(2 1)/cos(2 )/cos21 cos2cos(2 1)/connyy hnnnnnnnnnnnnnnnanQJb xdbnQJbxdcnQJa xddnQJ axdenQJcxdfnQJcos   01snxd (43) where the expressions of the known functions iQ Copyright © 2010 SciRes. ENG S. ITOU489 1,2,, 6ihave been omitted. Finally, the remaining boundary conditions (15), (17) and (19), which must be satisfied inside the cracks, reduce to the forms:    1115nnnnaFxeF1213 1411 1116 111 fornnnn nnnn nnnnnnbFxcFx dFxxfFxu xxb    (44)  2125nnnnaFxeF2223 2411 1126 211 fornnnn nnnn nnnnnnbFxcFx dFxxfFxu xxb   (45)   3135nnnnaF xeF32 33 3411 1136 311 fornnnnnnnn nnnnnnbF xcF xdF xxfFxu xxa   (46)   4145nnnnaFxeF4243 4411 1146 411 ()fornnnn nnnn nnnnnnbFxcFx dFxxfFxu xxa   (47)   5155nnnnaF xeF52 53 5411 1156 511 fornnnnnnnn nnnnnnbF xcF xdF xxfFxu xxc  (48)    6165nnnnaF xeF6263 6411 1166 611 fornnnn nnnn nnnnnnbF xcF xdF xxfFxu xxc   (49) where the functions are known. For example, ,1,2,,6nijFxij,11nFx 12nFx66n and Fx are ex-pressed as   111 202cosnnFxn QJbxd (50)   222 1022 1221cos21sinnLnLnQQJ b12cosFxxdQ bxnxb   (51)   66362 1021 sinnnFxnQJc xd  (52) where the constant 2LQ is given by 22LLQQ (53) where L is a larger value of . The functions iux are denoted by the equations:  112345 26exp ,0,,0,exp ,0ux pihuxux puxuxpihux   (54) The unknowns and,,,,nnn nnabcde nf in Equations (44), (45), (46), (47), (48) and (49) can now be solved using the Schmidt method described in Appendix A. 4. Stress Intensity Factors Once the unknown coefficients and ,,,,nnnnnabcde nf 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 rela-tions    01/2 1/222 221/2 1/222 22cos, sinsin ,cosnnnnnJax xdax ax x anax axx anforax   (55) the stress intensity factors can be expressed as 111(121121(171lim 221 1lim 221hyy hxbnnLnhxy hxbnnLnKxbbn QKxbanQ b))b (56) 011(0)161021(0)211lim2 ()21(1)lim 221yyxannLnxyxannLnKxadn QKxacnQ aa (57) Copyright © 2010 SciRes. ENG S. ITOU 490 21423012242351lim 221 1lim 221hyy hxcnnLnhxy hxcnnLnKxcfnQKxcenQ cc (58) where the constants are given by expressions taking the similar form as Equation (53). 7,16,21,30,35iLQi 5. Numerical Examples The dynamic stress intensity factors were calculated nu-merically with quadruplex precision using a Fortran pro-gram, during the operation of which, overflow and un-derflow do not occur within the range to . Numerical calculations were performed for a Poisson’s ratio 5500105500100.25. The semi-infinite integrals, which appear in the known functions ()nijFx must be evaluated numerically. It can be verified that the numerical integra-tions have been performed satisfactorily because the in-tegrands decay rapidly as the integration variable (,1,2, ,6)ij increases. To solve the unknown coefficients and ,na,nb,nc,ndnenf, the Schmidt method has been ap-plied by truncating the infinite series in Equations (44), (45), (46), (47), (48) and (49) by summing from 1n 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 giv-en in Equations (56), (57) and (58) are calculated for 1.0ba ca and 122.5ha ha; these are plot-ted with respect to Tac in Figure 2. The straight dashed lines on the left-hand side of Figure 2 indicate the corresponding static values given by Ishida . Figure 3 shows the values for 121.0ha ha. The dynamic stress intensity factors for two cracks in an infi-nite plate have been solved by Takakuda . In the present study, the same problem has been reworked, and the results are plotted in Figures 4 and 5 for 12.5ha and 1.0, respectively, where 1.0ba is assumed, and denotes the distance between the two parallel cracks. For the case of two parallel cracks, the distance in Figure 1 is considered to be infinite. 1h2h Figure 1. Geometry and coordinate system. Figure 2. Stress intensity factors ,11hK,12hK,01K,02K 21hK and 22hK for122.5ha ha and ba ca . 1.0 6. Discussion In the previous paper , time-harmonic stresses are solved for three cracks in an infinite elastic plate during the passage of time-harmonic stress waves. Two collin-ear 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 Copyright © 2010 SciRes. ENG S. ITOU491 Figure 3. Stress intensity factors ,11hK,12hK,01K,02K 21hK and 22hK for121.0ha ha and ba ca . 1.0 Figure 4. Stress intensity factors ,11hK,12hK01K and 02K for 12.5ha and 1.0ba. (For the case of two parallel cracks, the distance in Figure 1 is considered to be infinite). 2h 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 de-scribed 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 bound-ary conditions with respect to the three cracks are re-duced to six sets of an infinite series. Each of the two Figure 5. Stress intensity factors ,11hK,12hK01K and 02K for 11.0ha and 1.0ba. (For the case of two parallel cracks, the distance in Figure 1 is considered to be infinite). 2h sets of an infinite series must satisfy the boundary condi-tions inside the lower, middle and upper cracks, respec-tively. Therefore, the Schmidt method is newly extended in the present paper to determine the six sets of unknown coefficients as described in section 1. In this study, the minimum value of 12ha ha used during numerical calculations was 1.0. If the calcu-lations were performed for lower values of 1/ha2ha, the resulting peak value of 01Kpa would be significantly larger than 12.8. There is scope for fur-ther inquiry into whether or not the absolute values of the stress intensity factors would increase in the case of four parallel cracks in an infinite elastic plate. The author considers that the peak value of 01Kpa increases as the number of parallel cracks increases. If a time- harmonic load is applied to materials weakened by many parallel cracks, we note that the dynamic stress intensity factors would have a greater value than that for the cor-responding static solution. 7. 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 0.9Tac similar to that shown by Takakuda in the two-cracks case . The value Copyright © 2010 SciRes. ENG S. ITOU Copyright © 2010 SciRes. ENG 492 01Kpa is 12.8 for 121.0ha ha. 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.  S. Itou, “Dynamic Stress Concentration around Two Co-planar Griffith Cracks in an Infinite Elastic Medium,” ASME Journal of Applied Mechanics, Vol. 45, No. 12, 1978, pp. 803-806.  S. Itou, “Diffraction of an Antiplane Shear Wave by Two Coplanar Griffith Cracks in an Infinite Elastic Medium,” International Journal of Solids and Structures, Vol. 16, No. 12, 1980, pp. 1147- 1153. 2) For 122.5ha ha (Figure 2), the slope to the peak value of 01Kpa is comparatively gentle. However, for 121.0ha ha (Figure 3), the curve rises steeply to the maximum value of 01Kpa , after which it declines equally steeply. The curve exhib-its a sharp peak at the critical circular frequency near 0.9Tac.  S. Itou, “Dynamic Stresses around Two Cracks Placed Symmetrically to a Large Crack,” International Journal of Fracture, Vol. 75, No. 3, 1996, pp. 261-271.  K. Takakuda, “Scattering of Plane Harmonic Waves by Cracks (in Japanese),” Transactions of Japan Society of Mechanical Engineeris, Series A, Vol. 48, No. 432, 1982, pp. 1014-1020.  H. So and J. Y. Huang, “Determination of Dynamic Stress Intensity Factors of Two Finite Cracks at Arbi-trary Positions by Dislocation Model,” International Journal of Engineering Science, Vol. 26, No. 2, 1988, pp. 111-119. 3) In static solutions, the stress intensity factors for three equal-length cracks decrease slightly as 1ha 2ha decreases. However, this is accompanied by a significant increase in the peak value of 01Kpa.  S. A. Meguid and X. D. Wang, “On the Dynamic Interac-tion between a Microdefect and a Main Crack,” Pro-ceedings of the Royal Society of London, Series A, Vol. 448, No. 1934, 1995, pp. 449-464. 8. References  M. Ayatollahi and S. J. Fariborz, “Elastodynamic Analy-sis of a Plane Weakened by Several Cracks,” Interna-tional Journal of Solids and Structures, Vol. 46, No. 7-8, 2009, pp. 1743-1754.  J. F. Loeber and G. C. Sih, “Diffraction of Antiplane Shear by a Finite Crack,” Journal of the Acoustical Soci-ety of America, Vol. 44, No. 1, 1968, pp. 90-98.  S. Itou and H. Haliding, “Dynamic Stress Intensity Fac-tors around Three Cracks in an Infinite Elastic Plane Subjected to Time-Hharmonic Stress Waves,” Interna-tional Journal of Fracture, Vol. 83, No. 4, 1997, pp. 379- 391.  G. C. Sih and J. F. Loeber, “Wave Propagation in an Elastic Solid with a Line of Discontinuity or Finite Crack,” Quarterly of Applied Mathematics, Vol. 27, No. 2, 1969, pp. 193-213.  S. Itou and H. Haliding, “Dynamic Stress Intensity Fac-tors around Two Parallel Cracks in an Infinite-Orth- otropic Plane Subjected to Incident Harmonic Stress Wa v es, ” International Journal of Solids and Structures, Vol. 34, No. 9, 1997, pp. 1145-1165.  A. K. Mal, “Interaction of Elastic Waves with a Griffith Crack,” International Journal of Engineering Science, Vol. 8, No. 9, 1970, pp. 763-776.  G. C. Sih and J. F. Loeber, “Torsional Vibration of an Elastic Solid Containing a Penny-Shaped Crack,” Journal of the Acoustical Society of America, Vol. 44, No. 5, 1968, pp. 1237-1245.  P. M. Morse and H. Feshbach, “Methods of Theoretical Physics,” McGraw-Hill, New York, Vol. 1, 1958.  S. Itou, “Axisymmetric Slipless Indentation of an Infinite Elastic Hollow Cylinder,” Bulletin of the Calcutta Ma-thematical Society, Vol. 68, No. 17, 1976, pp. 157-165.  G. C. Sih and J. F. Loeber, “Normal Compression and Radial Shear Waves Scattering at a Penny-Shaped Crack in an Elastic Solid,” Journal of the Acoustical Society of America, Vol. 46, No. 3B, 1969, pp. 711-721.  M. Ishida, “Elastic Analysis of Cracks and Stress In-tensity Factors (in Japanese),” Fracture Mechanics and Strength of Materials, Baifuukan Press, Tokyo, Vol. 2, 1976.  A. K. Mal, “Interaction of Elastic Waves with a Penny- Shaped Crack,” International Journal of Engineering Science, Vol. 8, No. 5, 1970, pp. 381-388. S. ITOU 493 Appendix A For convenience, Equations (42) through (47) can be rewritten as   11 12131411 111516111 fornn nnnnnnnn nnnn nnnnaFx bFxcFx dFxeFxfFxu xxb    (A.1)    21 22232411 112526 211 fornn nnnnnnnn nnnn nnnnaFx bFxcFxdFxeFxfFxu xxb   (A.2)   31 32333411 113536 311 fornn nnnnnnnn nnnn nnnnaF xbFxcF xdFxeFxfFxu xxa   (A.3)    41 42434411 114546 411 fornn nnnnnnnn nnnn nnnnaFx bFxcFxdFxeFxfFxu xxa   (A.4)   51 52535411 115556 511 fornn nnnnnnnn nnnn nnnnaF xbFxcF xdFxeFxfFxu xxc   (A.5)   61 62636411 116566 611 fornn nnnnnnnn nnnn nnnnaF xbFxcFxdFxeFxfFxu xxc   (A.6) A set of functions that satisfy the orthogonal-ity condition ()nGx 200,bbmnnmnn nGxGx dxIIGx dx  (A.7) can be constructed from a given set of arbitrary function, say 65()nFx, such that 651nninnniiGxPPFx (A.8) where is the cofactor of the element of , which is defined as inPind'nD 11 12121 65 6501',ncnininnnddddDdFxddnFxdx (A.9) Representing the fifth series in Equation (A.6) by the orthogonal series with coefficients ()nGx n, the fol-lowing relationships are derivable:   65116616211 164 6611nnn nnnnn nnnnnn nnn nnnneFxG xuxaF xbF xcF xdF xfF x   63 (A.10) The second equality yields     6600162 63001164 6600111forccnn iniinccin iiniiiccin iiniiiGxuxdxa GxFxdxIbGxFxdx cGxFxdxdGxFxdx fGxFxdxxc 1 (A.11) and considering Equation (A.8), the first equality shows that 011111abc dnniniiniiniinii niiiiiieabcd ff   (A.12) with       060610620630640660,,,,,cnjnjjn jj jcnjanij ijn jj jcnjbnij ijn jj jcnjcnij ijn jj jcnjdnij ijn jj jcnjfnij ijn jj jPGxuxdxPIPGxF xdxPIPGxF xdxPIPGxF xdxPIPGxF xdxPIPGxF xdxPI (A.13) Copyright © 2010 SciRes. ENG S. ITOU 494 Substituting Equation (A.12) into Equation (A.5), the equality now becomes   **51 525311 1***5456 511fornn nnnnnn nnn nnnnaF xbFxcF xdF xfF xuxxc   * (A.14) with       *51 51551*52 52551*53 53551*54 54551*56 56551*055 551,,,,,ann iniibnn iniicnn iniidnn iniifnn iniiiiiFxFx FxFxFx FxFxFx FxFxFx FxFxFx Fxux uxFx (A.15) Using the same procedure, the orthogonal function *()nHx is constructed from *56 ()nFx as **561nninnniiHxQQFx (A.16) where is the cofactor of the element inQing of , which is defined as ''nD 11 121211**56 560'' ,nnnncin ingg ggDggngFxFxdx (A.17) Using Equations (A.14) and (A.16), the coefficient nf can be expressed by and as follows: ,,nnnabcnd01111abnniniiniiicdiniiniiifacdb (A.18) with      0**50**510**520**530**5402*0,,,,,cnjnjjn jj jcnjanij ijn jj jcnjbnij ijn jj jcnjcnij ijn jj jcnjdnij ijn jj jcnnQHxu xdxQJQHxF xdxQJQHxF xdxQJQHxF xdxQJQHxF xdxQJJHxdx (A.19) Substituting Equation (A18) into Equation (A12), we obtain the following relation: 0***11**11abnnini iniiicdiniiniiieabcd  (A.20) with 0* 001*1*1*1*1,,,,fnn jnjjaa afninijinjjbb bfniniji njjcc cfniniji njjdd dniniji njjf (A.21) Replacing the coefficients and nenf in Equations (A.1), (A.2), (A.3) and (A.4) with Equations (A.18) and (A.20), the equality becomes   **11 121311 1**14 11fornn nnnnnn nnnnaF xbFxcF xdFxuxxb  * (A.22)   ** *21 222311 1**24 21fornn nnnnnnnnnnaF xbF xcFxdFxuxxb    (A.23) Copyright © 2010 SciRes. ENG S. ITOU Copyright © 2010 SciRes. ENG 495  **31 323311 1**34 31fornn nnnnnn nnnnaF xbFxcF xdFxuxxa   * (A.24)   **41 4243111**44 41fornn nnnnnnnnnnaF xbF xcFxdFxu xxa   * (A.25) with      **11 1115161**12 1515161**13 1315161**14 1415161*0*11 151,,,,aann iniiniibbnn iniiniiccnn iniiniiddnn iniiniiii iiFxFxFx FxFxFxFx FxFxFxFxFxFxFxFx FxuxuxF x   016iFx(A.26)     **31 3135361**32 3535361**33 3335361**34 3435361*0*33 351,,,,aann iniiniibbnn iniiniiccnn iniiniiddnn iniiniiiiiiFxFxFx FxFxFxFx FxFxFxFx FxFxFxFx Fxux uxF x     036iFx(A.28)    **21 2125261**22 2525261**23 2325261**24 2425261*0*22 251,,,,aann iniiniibbnn iniiniiccnn iniiniiddnn iniiniiiiiiFxFxFxFxFxFxFx FxFxFxFx FxFxFxFx Fxux uxF x    026iFx(A.27)      **41 4145461**42 4545461**43 4345461**44 4445461*0*44 451,,,,aann iniiniibbnn iniiniiccnn iniiniiddnn iniiniiiiiiFxFxFxFxFxFxFxFxFxFxFx FxFxFxFxFxux uxF x   046iFx(A.29) Equations (A.2 2) , (A. 23 ), (A.24) and (A.25 ) have been already solved for and in Ref. [17, 1 5] . ,,nnnabcnd