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The dynamic response of an edge crack in a functionally graded orthotropic strip
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 32 (2005) 385–400 www.elsevier.com/locate/mechrescom
The dynamic response of an edge crack in a functionally graded orthotropic strip Li-Cheng Guo *, Lin-Zhi Wu, Tao Zeng Center for Composite Materials, Harbin Institute of Technology, Harbin 150001, P.R. China Available online 10 March 2005
Abstract The dynamic response of a functionally graded orthotropic strip with an edge crack perpendicular to the boundaries is studied. The material properties are assumed to vary continuously along the thickness direction. Laplace and Fourier transforms are applied to reduce the problem to a singular integral equation. Numerical results are presented to illustrate the influences of parameters such as the nonhomogeneity constant and geometry parameters on the dynamic stress intensity factors (SIFs). 2005 Elsevier Ltd. All rights reserved. Keywords: Functionally graded orthotropic strip; Edge crack; Dynamic stress intensity factors
1. Introduction Functionally graded materials (FGMs) are composites applicable to many fields for their significant advantages over the discretely layered materials. The gradual variation of material properties in FGMs is quite effective to decrease the thermal and residual stresses and enhance the bonding strength. Therefore, FGMs have great potential in the fields including semiconductor and sensor materials, wear and corrosion resistant coating, and thermal gradient structures. Since the FGM structures usually work under critical situations and are frequently subjected to dynamic loads, assessing the dynamic fracture behavior of FGMs is an important aspect for the material design. Great efforts have been made to study the static fracture behavior of FGMs. Delale and Erdogan (1983) are among the first to study the fracture behavior of FGMs. Erdogan (1985) studied the anti-plane shear problem with a discontinuous derivative of the elastic property. Delale and Erdogan (1988a,b) considered *
Corresponding author. E-mail address: [email protected] (L.-C. Guo).
0093-6413/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2005.02.003
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the interface crack problems of FGMs. Chen and Erdogan (1996) investigated the interface crack problem between the functionally graded ceramic coating and the homogeneous substrate. The generalized mixedmode problem for a crack with an arbitrary orientation in FGMs was discussed by Konda and Erdogan (1994). Wang et al. (2003) studied the anti-plane fracture of a FGM strip with a crack parallel to boundaries. In their paper, the shear modulus is considered as a class of functional forms for which the equilibrium equation has an analytical solution. Compared with static problems, studies on the dynamic behaviors of FGMs are relatively limited. Babaei and Lukasiewicz (1998) studied the dynamic stress intensity factors for a mode-III crack lying in a functionally graded material. They found that the stress intensity factors vary with the crack length to layer thickness ratio. An asymptotic expansion of the stress field around a crack propagating at constant velocity in a FGM is developed by Parameswaran and Shukla (1999). Their analysis revealed that the crack tip stress field retains the inverse square root singularity and only the higher order terms in the expansion are influenced by the material nonhomogeneity. Chalivendra et al. (2002) studied the generalized elastic solution for an arbitrarily propagating crack in FGMs through an asymptotic analysis. With an approximate method by dividing the material into a finite number of layers with different homogeneous properties, Wang et al. (1999a,b, 2000a,b, 2002) analyzed the dynamic crack problems of functionally graded materials with general material properties. Meguid et al. (2002) and Jiang and Wang (2002) investigated the dynamic crack propagation problems in FGMs, respectively. Meguid et al. (2002) found that the gradient distribution of the mechanical properties has important effects on the local stress distribution significantly and the fracture behavior of the solid. Ma et al. (2002) studied the scattering of time harmonic stress wave by two collinear cracks in FGMs. under anti-plane loading conditions. Jain and Shukla (2004) investigated the displacements, strains and stresses associated with propagating cracks in materials with continuously varying properties. In their paper, crack tip stress, strain and displacement fields for a propagating crack along the direction of property gradation in FGMs were obtained through an asymptotic analysis coupled with a displacement potential approach. The impact problem of a functionally graded coating-substrate system with a crack vertical to the coating was dealt with by Guo et al. (2004). On the other hand, experimental studies of the dynamic fracture of FGMs were conducted by Parameswaran and Shukla (1998), Butcher et al. (1999); Rousseau and Tippur (2001a,b, 2002) and Shukla and Jain (2004). Parameswaran and Shukla (1998) investigated the FGMs with a discrete property variation by using photoelasticity technique. During the impact loading experiments of Rousseau and Tippur (2002), the optical method of coherent gradient sensing (CGS) is used in conjunction with the high-speed photography for recording instantaneous deformation fields. In spite of these efforts, the understanding of the dynamic fracture on FGMs is still very limited. Especially, very few papers are published on the dynamic crack problem of a functionally graded strip with an edge crack perpendicular to boundaries. It is very significant to study this kind of crack problems since the geometry can be used as an approximation to a number of structural components and laboratory specimens (Kaya and Erdogan, 1987). This kind of edge crack is also a typical crack initiation for a functionally grade plate (Noda, 1999). The static crack problem of functionally graded isotropic strip with a crack perpendicular to the boundaries has been investigated by Erdogan and Wu (1996, 1997), Kadioglu et al. (1998) and Wang et al. (2002). Itou (1980) studied the dynamic problem for an internal crack in a homogeneous isotropic strip. As for the dynamic crack problem of a functionally graded orthotropic strip, only the internal (embedded) crack has been studied by Chen et al. (2002). Comparing with the internal crack problem of functionally graded orthotropic strip, the singular terms in the singular integral equation for the edge crack problem is different and the dynamic behavior is more complex. In this paper, the transient response of a functionally graded orthotropic strip containing an edge crack perpendicular to the boundaries is investigated theorectically. The time inversion is accomplished by using the Laplace inversion technique developed by Miller and Guy (1966). Numerical results are presented to illustrate the influences of the nonhomogeneity constant and geometry parameters on the dynamic SIFs.
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2. Problem formulation Consider the crack subjected to an impact load as depicted in Fig. 1. The strip is infinite along the y-axis and has a thickness h along the x-axis. The crack is perpendicular to the boundaries of the strip. To compare the present results with those given by other researchers, the following formulations are applicable to the edge crack as well as the internal crack. The material properties are defined as follows: c11 ðxÞ ¼ c110 edx ;
c12 ðxÞ ¼ c120 edx ;
c22 ðxÞ ¼ c220 edx ;
c66 ðxÞ ¼ c660 edx
qðxÞ ¼ q0 edx
ð2Þ
where, c110, c120, c220, c660, q0 and d are constants. The constitutive relation for the functionally graded orthotropic medium is given by 8 ov r ¼ c11 ðxÞ ou þ c12 ðxÞ oy > > ox < xx ov ryy ¼ c12 ðxÞ ou þ c22 ðxÞ oy ox > > : r ¼ c ðxÞðou þ ovÞ xy
66
ð1Þ
oy
ð3Þ
ox
The mixed boundary conditions of the problem in Fig. 1 can be written as rxx ð0; y; tÞ ¼ 0;
rxy ð0; y; tÞ ¼ 0;
1 < y < 1
ð4a; bÞ
rxx ðh; y; tÞ ¼ 0;
rxy ðh; y; tÞ ¼ 0;
1 < y < 1
ð5a; bÞ
rxy ðx; 0; tÞ ¼ 0;
0
ryy ðx; 0; tÞ ¼ r0 H ðtÞ; vðx; 0; tÞ ¼ 0;
a
0 < x < a; b < x < h
where r0 is constant and H(t) denotes the Heaviside unit step function. Substituting Eq. (3) into the equilibrium equation 8 2 < orxx þ orxy ¼ qðxÞ o uðx;y;tÞ ox oy ot2 : orxy þ oryy ¼ qðxÞ o2 vðx;y;tÞ ox oy ot2
Fig. 1. The dynamic edge crack in a functionally graded orthotropic strip.
ð6Þ ð7Þ ð8Þ
ð9Þ
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and applying Laplace transform and Fourier transform to Eq. (9), the displacement expressions can be obtained as 8 4 4 R1 P R1 P > > 1 > u ðx; y; pÞ ¼ 2p E1j ðs; pÞA1j ek1j yisx ds þ p2 1 E2j ðs; pÞA2j cosðsyÞ ds > 1 < j¼1 j¼1 ð10Þ 4 4 > > > v ðx; y; pÞ ¼ 1 R 1 P A1j ek1j yisx ds þ 2 R 1 P A2j sinðsyÞ ds > : 2p 1 p 1 j¼1
j¼1
where the superscript * denotes the Laplace transform and p is the variable in the time transform domain and Emj (m = 1, 2, j = 1, . . . , 4) are shown in Appendix A. According to the constitutive relation, the stress components are obtained as 8 " # 4 4 > R1 P R1 P > dx 1 k yisx 2 > 1j > rxx ðx; y; pÞ ¼ e 2p 1 B1j ðs; pÞA1j e ds þ p 1 B2j ðs; pÞA2j cosðsyÞ ds > > > j¼1 j¼1 > > > > " # > > 4 4 > R1 P R P < dx 1 k1j yisx 2 1 ryy ðx; y; pÞ ¼ e 2p 1 C 1j ðs; pÞA1j e ds þ p 1 C 2j ðs; pÞA2j cosðsyÞ ds ð11Þ j¼1 j¼1 > > > > " # > > 4 4 > R1 P R P > dx 1 k1j yisx 2 1 > rxy ðx; y; pÞ ¼ e 2p 1 D1j ðs; pÞA1j e ds þ p 1 D2j ðs; pÞA2j sinðsyÞ ds > > > j¼1 j¼1 > > : where the coefficients Bmj, Cmj, Dmj (m = 1, 2, j = 1, . . . , 4) are shown in Appendix A and the characteristic roots kmj(m = 1, 2) are determined by the following expressions: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X11 1 X211 4X12 ðj ¼ 1; . . . ; 4Þ k1j ¼ ð12Þ 2 2 " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1 2 d d 2X21 2 X221 4X22 ðj ¼ 1; . . . ; 4Þ ð13Þ k2j ¼ 2 where Xij are given in Appendix A. Without loss of generality, kmj (m = 1, 2) can be arranged so that Re(kmj) P 0 (j = 1, 2) and Re(kmj) < 0(j = 3, 4). For the symmetry of the structure about x-axis only the half part y 6 0 is considered. Since the stresses vanish when y ! 1, We have A1j ¼ 0
ðj ¼ 3; 4Þ
ð14Þ
Introduce the following auxiliary function: gðx; pÞ ¼
o v ðx; 0; pÞ ox
ð15Þ
By using Eqs. (6), (8), (10) and (11) and applying Fourier transform to (15), we can obtain the following relation: Z b gðu; pÞeisu du ðj ¼ 1; 2Þ ð16Þ A1j ¼ q1j a
where q11 ¼
ðs þ ik12 E12 Þ ; ðk12 E12 k11 E11 Þs
q12 ¼
ðs þ ik11 E11 Þ ðk12 E12 k11 E11 Þs
ð17Þ
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By solving the linear algebraic equations obtained from conditions (4) and (5), we have Z b A2 ¼ X 1 ðj ¼ 1; . . . ; 4Þ 1 Rðu; s; pÞgðu; pÞ du
389
ð18Þ
a
where A2 ¼ f A21 2 6 6 X1 ¼ 6 4
A22
A24 gT
A23
ð19Þ
B21 D21 =c660
B22 D22 =c660
B23 D23 =c660
B24 D24 =c660
k21 h
k22 h
k23 h
k24 h
B21 e D21 ek21 h =c660
B22 e D22 ek22 h =c660
Rðu; s; pÞ ¼ f R1 ðu; s; pÞ 1 R1 ðu; a; pÞ ¼ 2p
Z
1
2 X
1
j¼1
1 R2 ðu; a; pÞ ¼ 2pc660 1 R3 ðu; a; pÞ ¼ 2p
R2 ðu; s; pÞ
Z
Z
1 R4 ðu; a; pÞ ¼ 2pc660
Z
ð22aÞ
! k1j B1j q1j eisðuhÞ ds a2 þ k21j
ð22cÞ
j¼1
j¼1
ð21Þ
ð22bÞ
1
1
R3 ðu; s; pÞ R4 ðu; s; pÞ gT
ð20Þ
! a D1j q1j eisu ds a2 þ k21j
2 X
2 X
B24 e D24 ek24 h =c660
7 7 7 5
! k1j B1j q1j eisu ds a2 þ k21j
1
1
B23 e D23 ek23 h =c660
3
1
2 X
1
j¼1
! a D1j q1j eisðuhÞ ds a2 þ k21j
ð22dÞ
It should be noted that a in expressions ((22a)–(22d)) correspond to s in expression (21). In the derivation of expressions Ri(u, s, p), the following relations are used: Z 1 n cosðmaÞena da ¼ 2 ; ReðnÞ > 0 ð23aÞ m þ n2 0 Z 1 m sinðmaÞena da ¼ 2 ; ReðnÞ > 0 ð23bÞ m þ n2 0 By using the theory of residues, Ri(u, s, p) can be simplified as Ri1 ðu; s; pÞeuk21 þ Ri2 ðu; s; pÞeuk22 þ Ri3 ðu; s; pÞ ði ¼ 1; 2Þ Ri ðu; s; pÞ ¼ Ri1 ðu; s; pÞeðhuÞk23 þ Ri2 ðu; s; pÞeðhuÞk24 þ Ri3 ðu; s; pÞ ði ¼ 3; 4Þ
ð24Þ
where Ri1(u, s, p), Ri2(u, s, p) and Ri3(u, s, p) are given in Appendix A. Now, the six unknowns A11, A12, A21, . . . , A24 have been expressed by the unknown auxiliary function. Applying Laplace transform to the boundary condition (7) and substituting Eqs. (14), (16), (18) and (11) into it, the following integral equation can be obtained: Z b ½h1 ðu; x; pÞ þ h2 ðu; x; pÞgðu; pÞ du ¼ edx r0 =p ð25Þ a
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where h1 ðu; x; pÞ ¼ lim y!0
h2 ðu; x; pÞ ¼ lim y!0
K 1 ðy; s; pÞ ¼
2 X
1 2p 2 p
Z
þ1
K 1 ðy; s; pÞeisðxuÞ ds
ð26aÞ
K 2 ðu; x; s; pÞ cosðsyÞ ds
ð26bÞ
1
Z
þ1 0
C 1j q1j ek1j y
ð27aÞ
j¼1 k2j x K 2 ðu; x; s; pÞ ¼ X 3 X 1 1 Rðu; s; pÞe
X 3 ¼ ½ C 21 ek21 x
C 22 ek22 x
C 23 ek23 x
ð27bÞ C 24 ek24 x
ð28Þ
To derive the singular integral equation, an asymptotic analysis is needed. When y ! 0 and s ! 1, the asymptotic form of K1(y, s, p) can be expressed as K 11 ð0; s; pÞ ¼ w11 þ w12 =s
ð29Þ
Here w11 ¼
ic660 ½c2120 þ 16c2220 p211 p212 þ 4c120 c220 ðp211 þ p212 Þ 8c220 ðc120 þ c660 Þp11 p12 ðp11 þ p12 Þ
w12 ¼ c660 d c3120 16c2220 c660 p211 p212 þ c2120 ½3c660 4c220 ðp211 þ p212 Þ þ 4c120 c220 ½12c220 p211 p212 þ c660 ðp211 þ p212 Þ =½16c220 ðc120 þ c660 Þ2 p11 p12 ðp11 þ p12 Þ
ð30aÞ
ð30bÞ
where p11 and p12 are given in Appendix A. For verification, consider the functionally graded isotropic materials with the shear modulus l(x) = l0edx. From Eqs. (29) and (30), we have K 11 ð0; s; pÞ ¼
4l0 i 2l0 d 1 þ 1þk 1þk s
ð31Þ
It can be found that K11(0, s, p) is not related to the Laplace variable p and the first term in Eq. (31) is coincident with K11 given by Erdogan and Wu (1997). It should be noted that a second-order asymptotic term w12/s is kept in expression (29) to deal with the oscillation in the integrand (Shbeeb et al., 1999a,b). By adding and subtracting K11(0, s, p) from the integrand, h1 (u, x, p) can be obtained as 1 Imðx11 Þ þ k 1 ðu; x; pÞ h1 ðu; x; pÞ ¼ ð32Þ p ux k 1 ðu; x; pÞ ¼
Z
þ1
½K 1 ð0; s; pÞ K 11 ð0; s; pÞeisðuxÞ ds
ð33Þ
1
For the functionally graded isotropic materials, h2(u, x, p) for an internal crack (0 < a < b < h) are bounded in the interval a 6 (u, x) 6 b, but h2(u, x, p) for an edge crack (a = 0 or b = h) should include singular terms. Let us analyze the singularity of h2(u, x, p) and give the asymptotic behavior of K2(u, x, s, p) when s ! 1 and a ! 0. Through lengthy manipulation, the asymptotic form of K2(u, x, s, p) can be obtained as K 21 ðu; x; s; pÞ ¼ w21 euk21 þxk23 þ w22 euk21 þxk24 þ w23 euk22 þxk23 þ w24 euk22 þxk24
ð34Þ
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391
where w21 ¼
ðB21 D22 B22 D21 ÞC 23 D24 R11 þ ðB22 D21 B21 D22 ÞB24 C 23 R21 ðB22 D21 B21 D22 ÞðB24 D23 B23 D24 Þ
ð35aÞ
w22 ¼
ðB22 D21 B21 D22 ÞC 24 D23 R11 þ ðB21 D22 B22 D21 ÞB23 C 24 R21 ðB22 D21 B21 D22 ÞðB24 D23 B23 D24 Þ
ð35bÞ
w23 ¼
ðB21 D22 B22 D21 ÞC 23 D24 R12 þ ðB22 D21 B21 D22 ÞB24 C 23 R22 ðB22 D21 B21 D22 ÞðB24 D23 B23 D24 Þ
ð35cÞ
w24 ¼
ðB22 D21 B21 D22 ÞC 24 D23 R12 þ ðB21 D22 B22 D21 ÞB23 C 24 R22 ðB22 D21 B21 D22 ÞðB24 D23 B23 D24 Þ
ð35dÞ
To simplify the expression K21(u, x, s, p), k21, . . . , k24 need to be expanded in the asymptotic forms (as s ! 1) k21 ¼ p21 s þ p23 ;
k22 ¼ p22 s þ p24 ;
k23 ¼ p21 s þ p24 ;
k24 ¼ p22 s þ p23
ð36Þ
where p21, . . . , p24 are constants given in Appendix A. Consequently, expression (34) can be manipulated as K 21 ðu; x; s; pÞ ¼ w21 eg1 ep21 ðuþxÞs þ w22 eg2 eðp21 uþp22 xÞs þ w23 eg3 eðp22 uþp21 xÞs þ w24 eg4 ep22 ðuþxÞs
ð37Þ
where g1 ¼ p23 u þ p24 x;
g2 ¼ p23 u þ p23 x;
g3 ¼ p24 u þ p24 x;
g4 ¼ p24 u þ p23 x
ð38Þ
From (26b), (27b) and (28), h2(u, x, p) can be expressed as 2 h2 ðu; x; pÞ ¼ ½k s ðu; x; pÞ þ k b ðu; x; pÞ p where Z þ1 k b ðu; x; pÞ ¼ ½K 2 ðu; x; s; pÞ K 21 ðu; x; s; pÞ ds
ð39Þ
ð40Þ
0
k s ðu; x; pÞ ¼
Z
1
K 21 ðu; x; s; pÞ ds
ð41Þ
0
To check the expression (37), we still consider the functionally graded isotropic materials with the shear modulus l(x) = l0edx. The following expressions can be obtained: 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1@ ðk 3Þd2 A d ð42Þ p21 ¼ p22 ¼ 1; p23;24 ¼ 2 1þk After some manipulations,K21(u, x, s, p) can be simplified as K 21 ðu; x; sÞ ¼ K11 eðuþxÞs
ð43Þ
where K11 ¼
2l0 ½2 þ ð3u xÞs þ 2uxs2 1þk
ð44Þ
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L.-C. Guo et al. / Mechanics Research Communications 32 (2005) 385–400
k s ðu; xÞ ¼
Z
1 0
" # 2l0 1 6x 4x2 þ K 21 ðu; x; s; pÞ ds ¼ 1 þ k u þ x ðu þ xÞ2 ðu þ xÞ3
ð45Þ
It can be found that the forms of K21(u, x, s, p) and ks(u, x, p) for isotropic FGMs are the same as those obtained by Erdogan and Wu (1997). For the functionally graded orthotropic strip, we have K 21 ðu; x; sÞ ¼ K21 ep21 ðuþxÞs þ K22 eðp21 uþp22 xÞs þ K23 eðp22 uþp21 xÞs þ K24 ep22 ðuþxÞs
ð46Þ
where K21, K22, K23 and K24 are constants shown in Appendix A. Note that K21, . . . , K24 are not related to the variable s. Therefore k s ðu; xÞ ¼
K21 K22 K23 K24 þ þ þ p21 ðu þ xÞ ðp21 u þ p22 xÞ ðp22 u þ p21 xÞ p22 ðu þ xÞ
Finally, by substituting Eqs. (32), (38) and (46) into Eq. (25) we have Z 1 b Imðx11 Þ þ k 1 ðu; xÞ þ 2k s ðu; xÞ þ 2k b ðu; xÞ gðuÞ du ¼ edx r0 =p p a ux
ð47Þ
ð48Þ
Since a second-order asymptotic term w12/s is kept in Eq. (29), for the need to resolve Eq. (48) numerically, referencing the procedure outlined by Shbeeb et al. (1999a,b), k1(u, x) can be manipulated as Z 1h Z U 1 w12 i 1 2Imðh1 Þi 2iIm w11 þ ½2Reðh1 Þ 2Reðw11 Þ k 1 ðu; xÞ ¼ i sin½sðu xÞ ds þ 2p 0 2p 0 s Z 1h 1 w12 i 2Reðh1 Þ 2Re w11 þ cos½sðu xÞ ds þ cos½sðu xÞ ds 2p U s Z Imðw12 Þ u x 1 1 cos½sðu xÞ þ ds ð49Þ Reðw12 Þ þ 2 ju xj p U s The unknown auxiliary functions in Eq. (48) will be solved numerically.
3. Numerical solutions The singular integral Eq. (48) can be solved by the method (Erdogan and Gupta, 1972; Kadioglu et al., 1998). To reduce (48) to a standard form, let us normalize the interval by setting ba bþa ðr; sÞ þ 2 2 For the internal crack, the solution of Eq. (48) may be written as ðu; xÞ ¼
ð50Þ
f ðr; pÞ gðr; pÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 r2 Þ
ð51Þ
For the edge crack, the solution of Eq. (48) may be written as f ðr; pÞ gðr; pÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 rÞ The stress intensity factor of the crack tips can be defined as: for the internal crack rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ba da k 1 ða; pÞ ¼ lim 2ða xÞryy ðx; 0; pÞ ¼ Imðw11 Þe f ð1; pÞ x!a 2
ð52Þ
ð53aÞ
L.-C. Guo et al. / Mechanics Research Communications 32 (2005) 385–400
k 1 ðb; pÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ lim 2ðx bÞryy ðx; 0; pÞ ¼ Imðw11 Þedb y!b
rffiffiffiffiffiffiffiffiffiffiffi ba f ð1; pÞ 2
393
ð53bÞ
for the edge crack. k 1 ðb; pÞ ¼ lim y!b
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2ðx bÞryy ðx; 0; pÞ ¼ Imðw11 Þedb bf ð1; pÞ
ð54Þ
The Laplace inversion of Eqs. (53) and (54) is conducted numerically by the method provided by Miller and Guy (1966).
4. Results and discussions pffiffiffi pffiffiffiffiffiffiffi For convenience, the dynamic SIFs are normalized by k 0 ¼ r0 a0 h for the internal crack and k 0 ¼ r0 b for the edge crack, where the normalized half crack length a0 = (b a)/2h. The wave velocities are defined pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi as c1 ¼ c110 =q0 and c2 ¼ c660 =q0 . To verify the validity of the present work, two examples are given. Firstly, we compare the normalized dynamic stress intensity factors of a central crack in a homogeneous isotropic strip, namely dh = ln 1.0 = 0 and (b + a)/2h = 0.5, with the results given by Itou (1980). Due to the symmetry, the SIFs of two crack tips are equal, which is verified very well by Fig. 2. The curves in Fig. 2 are very similar to those provided by Itou (1980). Another example is provided in Fig. 3 which gives some results for a central crack in a functionally graded orthotropic strip. Assume the half crack length a0 = 0.3 and the nonhomogeneity constant dh = ln 0.2, ln 1.0 and ln 10, respectively. The results corresponding to Material 0 listed in Table 1 are displayed in Fig. 3. It can be found that the results in Fig. 3 are similar to those given by Chen et al. (2002). Subsequently, let us consider the edge crack problem (a = 0, a0 = b/2h) in a functionally graded orthotropic strip when the crack is subjected to a surface impact r0H(t). Two kinds of orthotropic materials listed in Table 1, namely Material 1 and Material 2, are chosen for the numerical calculations. Fig. 4 shows the influence of nonhomogeneity constant dh on the dynamic SIFs, when b/h = 0.3. It is shown that the peak value of k1 (b, t)/k0 increases and occurs at an earlier time with the increasing of dh. Note that dh < 0 means that the crack side is stiffer than the other side of the strip, while dh > 0 means the crack
a0=0.35
1.5
a0=0.3
1.0
k1(a,t)/k0
0.5
k1(b,t)/k0
δh=0 0.0
0
2
4
6
8
10
Fig. 2. The dynamic SIFs of the homogeneous isotropic strip with a central internal crack, dh = 0.
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L.-C. Guo et al. / Mechanics Research Communications 32 (2005) 385–400 2.0
1.5
1.0
δh=ln0.2 δh=ln1.0 δ h=ln10.0
0.5
a0=0.3 0.0 0
6
4
2
8
10
Fig. 3. The dynamic SIFs of the functionally graded orthotropic strip with a central internal crack, a0 = 0.3.
Table 1 Mechanical properties of the functionally graded orthotropic strip Material
c110(N/m2)
c120(N/m2)
c220(N/m2)
c660(N/m2)
q0(kg/m3)
0 1 2
1.578 · 1010 1.578 · 1010 1.048 · 1010
3.248 · 109 3.248 · 109 3.248 · 109
1.048 · 1010 1.048 · 1010 1.578 · 1010
7.070 · 109 7.070 · 109 7.070 · 109
1.58 · 103 7.069 · 103 7.069 · 103
2.4
2.0
b/h=0.3
1.6
1.2
δh=ln0.1 δh=ln0.2 δh=ln1.0 δh=ln5.0
0.8
0.4
0.0
0
5
10
15
20
Fig. 4. The dynamic SIFs for Material 1 with an edge crack, b/h = 0.3.
side is less stiffer than the other side. Therefore, the peak value of normalized dynamic SIFs are greater when the edge crack lies in the less stiffer side of the strip. Form Fig. 5, a similar characteristic can be found. Fig. 6
L.-C. Guo et al. / Mechanics Research Communications 32 (2005) 385–400
395
depicts the influence of the normalized crack length on the dynamic SIFs when dh = ln 5.0. It can be seen that the peak value of k1(b, t)/k0 increases with the increasing of the crack length. By comparing the results for the edge crack problem with those for the internal crack problem given by Chen et al. (2002). It can be found that: for the internal crack the dynamic SIFs increase quickly with time and then keep approximately stable after reaching a peak, while the dynamic SIFs for the edge crack also increase quickly with time but then exhibit a more obvious oscillation after reaching a peak. It is for the reason that the strip with an edge crack, compared with the structure with a central crack, loses symmetry, and the interaction between the waves and the finite boundaries would be stronger. Therefore, the dynamic behavior of the edge crack problem is more complex than that of the internal crack.
2.5
b/ h= 0. 3 2.0
1.5
δh=ln0.1 δh=ln0.2 δh=ln1.0 δh=ln5.0
1.0
0.5
0.0 0
5
10
15
20
Fig. 5. The dynamic SIFs for an edge crack in Material 2, b/h = 0.3.
2.5
2.0
1.5
1.0
b/h=0.2 b/h=0.3
b/h=0.5
0.5
δh=ln5.0 0.0 0
5
10
15
20
Fig. 6. The dynamic SIFs for Material 1 with an edge crack, dh = ln5.0.
396
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5. Conclusions In this paper, the dynamic response of a functionally graded orthotropic strip with an edge crack perpendicular to boundaries is studied. An asymptotic analysis is conducted. It can be found that the singular terms in the integral equation are different from those corresponding to the internal crack problem. The numerical examples are presented to illustrate the influences of the nonhomogeneity constant and the geometry parameters on the dynamic stress intensity factors. It is found that the dynamic behavior of the functionally graded orthotropic strip with an edge crack is more complex than that of the corresponding internal crack problem. Since the geometry structure can be used as an approximation to a number of structural components and laboratory specimens, it is necessary to conduct further researches about this problem.
Acknowledgments This work is supported by the National Science Foundation for Excellent Young Investigators (No. 10325208), China Postdoctoral Science Foundation and NSFC (No. 10432030).
Appendix A ½c660 sðs þ idÞ c220 k21j þ p2 q0 i E1j ¼ ; j ¼ 1; . . . ; 4 ½c120 s þ c660 ðs þ idÞk1j 8 > < B1j ¼ c120 k1j ic110 E1j s C 1j ¼ c220 k1j ic120 E1j s ; j ¼ 1; . . . ; 4 > : D1j ¼ c660 ðis þ k1j E1j Þ E2j ¼
c220 s2 þ c660 k2j ðd þ k2j Þ p2 q0 ; s½c120 k2j þ c660 ðd þ k2j Þ
8 > < B2j ¼ c120 s þ c110 k2j E2j C 2j ¼ c220 s þ c120 k2j E2j ; > : D2j ¼ c660 ðk2j sE2j Þ
j ¼ 1; . . . ; 4
j ¼ 1; . . . ; 4
X11 ¼
sðs þ idÞðc2120 c110 c220 Þ þ c120 c660 ð2s2 þ 2isd d2 Þ ðc220 þ c660 Þp2 q0 c220 c660
X12 ¼
c110 c660 s2 ðs þ idÞ þ ðc110 þ c660 Þðs þ idÞp2 q0 s þ p4 q20 c220 c660
ðA:1Þ
ðA:2Þ
ðA:3Þ
ðA:4Þ
ðA:5aÞ
2
ðc2120 c110 c220 þ 2c120 c660 Þs2 ðc110 þ c660 Þp2 q0 c110 c660
c220 2 c120 2 2 ½ðc220 þ c660 Þs2 þ p2 q0 p2 q0 X22 ¼ s þ d s þ c110 c110 c110 c660
X21 ¼
ðA:5bÞ
ðA:6aÞ
ðA:6bÞ
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R11 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þs2 k21 c110 ðk21 k22 Þðk21 k23 Þðk21 k24 Þ
ðA:7aÞ
R12 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þs2 k22 c110 ðk22 k21 Þðk22 k23 Þðk22 k24 Þ
ðA:7bÞ
R13 ðs; u; pÞ ¼
c120 p2 q0 ðc660 s2 þ p2 q0 Þ 2c110 c660 k21 k22 k23 k24
ðA:7cÞ
R21 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þsk21 ðd þ k21 Þ c110 c660 ðk21 k22 Þðk21 k23 Þðk21 k24 Þ
ðA:8aÞ
R22 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þsk22 ðd þ k22 Þ c110 c660 ðk21 k22 Þðk22 k23 Þðk22 k24 Þ
ðA:8bÞ
R23 ðs; u; pÞ ¼
c120 p2 dq0 s 2c110 c660 k21 k22 k23 k24
ðA:8cÞ
R31 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þs2 k23 c110 ðk21 k23 Þðk23 k22 Þðk23 k24 Þ
ðA:9aÞ
R32 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þs2 k24 c110 ðk21 k24 Þðk24 k22 Þðk24 k23 Þ
ðA:9bÞ
R33 ðs; u; pÞ ¼ R13 ðs; u; pÞ
ðA:9cÞ
R41 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þsk23 ðd þ k23 Þ c110 c660 ðk21 k23 Þðk23 k22 Þðk23 k24 Þ
ðA:10aÞ
R42 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þsk24 ðd þ k24 Þ c110 c660 ðk21 k24 Þðk24 k22 Þðk24 k23 Þ
ðA:10bÞ
R43 ðs; u; pÞ ¼ R23 ðs; u; pÞ pffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n11 C11 þ C12 4 pffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n C11 C12 p12 ¼ 4 12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n11 ¼ sign½Reð C11 þ C12 Þ p11 ¼
ðA:10cÞ ðA:11aÞ
ðA:11bÞ ðA:12aÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n12 ¼ sign½Reð C11 C12 Þ
ðA:12bÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where n11 and n12 give 1 or 1 depending on whether the real parts of C11 þ C12 and C11 C12 are negative, or positive. C11 ¼
c110 c220 c120 ðc120 þ 2c660 Þ c220 c660
ðA:13aÞ
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C12 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðc2120 c110 c220 Þ½ðc120 þ 2c660 Þ c110 c220 =ðc220 c660 Þ
ðA:13bÞ
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p21 ¼ n21 C21 þ C22 2
ðA:14aÞ
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p22 ¼ n22 C21 C22 2
ðA:14bÞ
p23 ¼ lim ðk21 p21 sÞ
ðA:14cÞ
p24 ¼ lim ðk22 p22 sÞ
ðA:14dÞ
s!1
s!1
n21 ¼ sign½Reð
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C21 þ C22 Þ
ðA:15aÞ
n22 ¼ sign½Reð
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C21 C22 Þ
ðA:15bÞ
c110 c220 c120 ðc120 þ 2c660 Þ c110 c660 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 C22 ¼ ðc2120 c110 c220 Þ½ðc120 þ 2c660 Þ c110 c220 =ðc110 c660 Þ
C21 ¼
Here n21 and n22 take 1 or 1, which depend on whether the real parts of negative or positive.
ðA:16aÞ ðA:16bÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C21 þ C22 and C21 C22 are
K21 ¼ ðc2120 c110 c220 Þp21 ðc220 þ c120 p221 Þfc220 ðc660 þ c110 p21 p22 Þ þ p22 ½c2120 p21 þ c120 c660 ðp21 p22 Þ þ c120 c660 p21 p222 g=K
ðA:17aÞ
K22 ¼ ðc2120 c110 c220 Þ½c220 ðc220 þ c120 p221 Þ þ p221 ðc2120 þ c110 c660 p221 Þp22 ðc220 þ c120 p222 Þ=K
ðA:17bÞ
K23 ¼ ðc2120 c110 c220 Þp21 ðc220 þ c120 p221 Þ½c220 ðc660 þ c110 p222 Þ p222 ðc2120 þ c110 c660 p222 Þ=K
ðA:17cÞ
K24 ¼ ðc2120 c110 c220 Þp22 ðc220 þ c120 p222 Þfc220 ðc660 þ c110 p21 p22 Þ þ p21 ½c2120 p22 þ c110 c660 p21 p22 þ c120 c660 ðp22 p21 Þg=K
ðA:17dÞ
2
K ¼ f2c110 ðp21 p22 Þ ðp21 þ p22 Þ½c3120 p21 p22 þ c2120 ðc220 c660 p21 p22 Þ þ c120 ðc110 c660 p221 p222 þ c220 ðc660 þ c110 p21 p22 ÞÞ þ c110 c220 ðc220 þ c660 ðp221 þ p21 p22 þ p222 ÞÞg
ðA:18Þ
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Mechanics Research Communications 32 (2005) 385–400 www.elsevier.com/locate/mechrescom
The dynamic response of an edge crack in a functionally graded orthotropic strip Li-Cheng Guo *, Lin-Zhi Wu, Tao Zeng Center for Composite Materials, Harbin Institute of Technology, Harbin 150001, P.R. China Available online 10 March 2005
Abstract The dynamic response of a functionally graded orthotropic strip with an edge crack perpendicular to the boundaries is studied. The material properties are assumed to vary continuously along the thickness direction. Laplace and Fourier transforms are applied to reduce the problem to a singular integral equation. Numerical results are presented to illustrate the influences of parameters such as the nonhomogeneity constant and geometry parameters on the dynamic stress intensity factors (SIFs). 2005 Elsevier Ltd. All rights reserved. Keywords: Functionally graded orthotropic strip; Edge crack; Dynamic stress intensity factors
1. Introduction Functionally graded materials (FGMs) are composites applicable to many fields for their significant advantages over the discretely layered materials. The gradual variation of material properties in FGMs is quite effective to decrease the thermal and residual stresses and enhance the bonding strength. Therefore, FGMs have great potential in the fields including semiconductor and sensor materials, wear and corrosion resistant coating, and thermal gradient structures. Since the FGM structures usually work under critical situations and are frequently subjected to dynamic loads, assessing the dynamic fracture behavior of FGMs is an important aspect for the material design. Great efforts have been made to study the static fracture behavior of FGMs. Delale and Erdogan (1983) are among the first to study the fracture behavior of FGMs. Erdogan (1985) studied the anti-plane shear problem with a discontinuous derivative of the elastic property. Delale and Erdogan (1988a,b) considered *
Corresponding author. E-mail address: [email protected] (L.-C. Guo).
0093-6413/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2005.02.003
386
L.-C. Guo et al. / Mechanics Research Communications 32 (2005) 385–400
the interface crack problems of FGMs. Chen and Erdogan (1996) investigated the interface crack problem between the functionally graded ceramic coating and the homogeneous substrate. The generalized mixedmode problem for a crack with an arbitrary orientation in FGMs was discussed by Konda and Erdogan (1994). Wang et al. (2003) studied the anti-plane fracture of a FGM strip with a crack parallel to boundaries. In their paper, the shear modulus is considered as a class of functional forms for which the equilibrium equation has an analytical solution. Compared with static problems, studies on the dynamic behaviors of FGMs are relatively limited. Babaei and Lukasiewicz (1998) studied the dynamic stress intensity factors for a mode-III crack lying in a functionally graded material. They found that the stress intensity factors vary with the crack length to layer thickness ratio. An asymptotic expansion of the stress field around a crack propagating at constant velocity in a FGM is developed by Parameswaran and Shukla (1999). Their analysis revealed that the crack tip stress field retains the inverse square root singularity and only the higher order terms in the expansion are influenced by the material nonhomogeneity. Chalivendra et al. (2002) studied the generalized elastic solution for an arbitrarily propagating crack in FGMs through an asymptotic analysis. With an approximate method by dividing the material into a finite number of layers with different homogeneous properties, Wang et al. (1999a,b, 2000a,b, 2002) analyzed the dynamic crack problems of functionally graded materials with general material properties. Meguid et al. (2002) and Jiang and Wang (2002) investigated the dynamic crack propagation problems in FGMs, respectively. Meguid et al. (2002) found that the gradient distribution of the mechanical properties has important effects on the local stress distribution significantly and the fracture behavior of the solid. Ma et al. (2002) studied the scattering of time harmonic stress wave by two collinear cracks in FGMs. under anti-plane loading conditions. Jain and Shukla (2004) investigated the displacements, strains and stresses associated with propagating cracks in materials with continuously varying properties. In their paper, crack tip stress, strain and displacement fields for a propagating crack along the direction of property gradation in FGMs were obtained through an asymptotic analysis coupled with a displacement potential approach. The impact problem of a functionally graded coating-substrate system with a crack vertical to the coating was dealt with by Guo et al. (2004). On the other hand, experimental studies of the dynamic fracture of FGMs were conducted by Parameswaran and Shukla (1998), Butcher et al. (1999); Rousseau and Tippur (2001a,b, 2002) and Shukla and Jain (2004). Parameswaran and Shukla (1998) investigated the FGMs with a discrete property variation by using photoelasticity technique. During the impact loading experiments of Rousseau and Tippur (2002), the optical method of coherent gradient sensing (CGS) is used in conjunction with the high-speed photography for recording instantaneous deformation fields. In spite of these efforts, the understanding of the dynamic fracture on FGMs is still very limited. Especially, very few papers are published on the dynamic crack problem of a functionally graded strip with an edge crack perpendicular to boundaries. It is very significant to study this kind of crack problems since the geometry can be used as an approximation to a number of structural components and laboratory specimens (Kaya and Erdogan, 1987). This kind of edge crack is also a typical crack initiation for a functionally grade plate (Noda, 1999). The static crack problem of functionally graded isotropic strip with a crack perpendicular to the boundaries has been investigated by Erdogan and Wu (1996, 1997), Kadioglu et al. (1998) and Wang et al. (2002). Itou (1980) studied the dynamic problem for an internal crack in a homogeneous isotropic strip. As for the dynamic crack problem of a functionally graded orthotropic strip, only the internal (embedded) crack has been studied by Chen et al. (2002). Comparing with the internal crack problem of functionally graded orthotropic strip, the singular terms in the singular integral equation for the edge crack problem is different and the dynamic behavior is more complex. In this paper, the transient response of a functionally graded orthotropic strip containing an edge crack perpendicular to the boundaries is investigated theorectically. The time inversion is accomplished by using the Laplace inversion technique developed by Miller and Guy (1966). Numerical results are presented to illustrate the influences of the nonhomogeneity constant and geometry parameters on the dynamic SIFs.
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2. Problem formulation Consider the crack subjected to an impact load as depicted in Fig. 1. The strip is infinite along the y-axis and has a thickness h along the x-axis. The crack is perpendicular to the boundaries of the strip. To compare the present results with those given by other researchers, the following formulations are applicable to the edge crack as well as the internal crack. The material properties are defined as follows: c11 ðxÞ ¼ c110 edx ;
c12 ðxÞ ¼ c120 edx ;
c22 ðxÞ ¼ c220 edx ;
c66 ðxÞ ¼ c660 edx
qðxÞ ¼ q0 edx
ð2Þ
where, c110, c120, c220, c660, q0 and d are constants. The constitutive relation for the functionally graded orthotropic medium is given by 8 ov r ¼ c11 ðxÞ ou þ c12 ðxÞ oy > > ox < xx ov ryy ¼ c12 ðxÞ ou þ c22 ðxÞ oy ox > > : r ¼ c ðxÞðou þ ovÞ xy
66
ð1Þ
oy
ð3Þ
ox
The mixed boundary conditions of the problem in Fig. 1 can be written as rxx ð0; y; tÞ ¼ 0;
rxy ð0; y; tÞ ¼ 0;
1 < y < 1
ð4a; bÞ
rxx ðh; y; tÞ ¼ 0;
rxy ðh; y; tÞ ¼ 0;
1 < y < 1
ð5a; bÞ
rxy ðx; 0; tÞ ¼ 0;
0
ryy ðx; 0; tÞ ¼ r0 H ðtÞ; vðx; 0; tÞ ¼ 0;
a
0 < x < a; b < x < h
where r0 is constant and H(t) denotes the Heaviside unit step function. Substituting Eq. (3) into the equilibrium equation 8 2 < orxx þ orxy ¼ qðxÞ o uðx;y;tÞ ox oy ot2 : orxy þ oryy ¼ qðxÞ o2 vðx;y;tÞ ox oy ot2
Fig. 1. The dynamic edge crack in a functionally graded orthotropic strip.
ð6Þ ð7Þ ð8Þ
ð9Þ
388
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and applying Laplace transform and Fourier transform to Eq. (9), the displacement expressions can be obtained as 8 4 4 R1 P R1 P > > 1 > u ðx; y; pÞ ¼ 2p E1j ðs; pÞA1j ek1j yisx ds þ p2 1 E2j ðs; pÞA2j cosðsyÞ ds > 1 < j¼1 j¼1 ð10Þ 4 4 > > > v ðx; y; pÞ ¼ 1 R 1 P A1j ek1j yisx ds þ 2 R 1 P A2j sinðsyÞ ds > : 2p 1 p 1 j¼1
j¼1
where the superscript * denotes the Laplace transform and p is the variable in the time transform domain and Emj (m = 1, 2, j = 1, . . . , 4) are shown in Appendix A. According to the constitutive relation, the stress components are obtained as 8 " # 4 4 > R1 P R1 P > dx 1 k yisx 2 > 1j > rxx ðx; y; pÞ ¼ e 2p 1 B1j ðs; pÞA1j e ds þ p 1 B2j ðs; pÞA2j cosðsyÞ ds > > > j¼1 j¼1 > > > > " # > > 4 4 > R1 P R P < dx 1 k1j yisx 2 1 ryy ðx; y; pÞ ¼ e 2p 1 C 1j ðs; pÞA1j e ds þ p 1 C 2j ðs; pÞA2j cosðsyÞ ds ð11Þ j¼1 j¼1 > > > > " # > > 4 4 > R1 P R P > dx 1 k1j yisx 2 1 > rxy ðx; y; pÞ ¼ e 2p 1 D1j ðs; pÞA1j e ds þ p 1 D2j ðs; pÞA2j sinðsyÞ ds > > > j¼1 j¼1 > > : where the coefficients Bmj, Cmj, Dmj (m = 1, 2, j = 1, . . . , 4) are shown in Appendix A and the characteristic roots kmj(m = 1, 2) are determined by the following expressions: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X11 1 X211 4X12 ðj ¼ 1; . . . ; 4Þ k1j ¼ ð12Þ 2 2 " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1 2 d d 2X21 2 X221 4X22 ðj ¼ 1; . . . ; 4Þ ð13Þ k2j ¼ 2 where Xij are given in Appendix A. Without loss of generality, kmj (m = 1, 2) can be arranged so that Re(kmj) P 0 (j = 1, 2) and Re(kmj) < 0(j = 3, 4). For the symmetry of the structure about x-axis only the half part y 6 0 is considered. Since the stresses vanish when y ! 1, We have A1j ¼ 0
ðj ¼ 3; 4Þ
ð14Þ
Introduce the following auxiliary function: gðx; pÞ ¼
o v ðx; 0; pÞ ox
ð15Þ
By using Eqs. (6), (8), (10) and (11) and applying Fourier transform to (15), we can obtain the following relation: Z b gðu; pÞeisu du ðj ¼ 1; 2Þ ð16Þ A1j ¼ q1j a
where q11 ¼
ðs þ ik12 E12 Þ ; ðk12 E12 k11 E11 Þs
q12 ¼
ðs þ ik11 E11 Þ ðk12 E12 k11 E11 Þs
ð17Þ
L.-C. Guo et al. / Mechanics Research Communications 32 (2005) 385–400
By solving the linear algebraic equations obtained from conditions (4) and (5), we have Z b A2 ¼ X 1 ðj ¼ 1; . . . ; 4Þ 1 Rðu; s; pÞgðu; pÞ du
389
ð18Þ
a
where A2 ¼ f A21 2 6 6 X1 ¼ 6 4
A22
A24 gT
A23
ð19Þ
B21 D21 =c660
B22 D22 =c660
B23 D23 =c660
B24 D24 =c660
k21 h
k22 h
k23 h
k24 h
B21 e D21 ek21 h =c660
B22 e D22 ek22 h =c660
Rðu; s; pÞ ¼ f R1 ðu; s; pÞ 1 R1 ðu; a; pÞ ¼ 2p
Z
1
2 X
1
j¼1
1 R2 ðu; a; pÞ ¼ 2pc660 1 R3 ðu; a; pÞ ¼ 2p
R2 ðu; s; pÞ
Z
Z
1 R4 ðu; a; pÞ ¼ 2pc660
Z
ð22aÞ
! k1j B1j q1j eisðuhÞ ds a2 þ k21j
ð22cÞ
j¼1
j¼1
ð21Þ
ð22bÞ
1
1
R3 ðu; s; pÞ R4 ðu; s; pÞ gT
ð20Þ
! a D1j q1j eisu ds a2 þ k21j
2 X
2 X
B24 e D24 ek24 h =c660
7 7 7 5
! k1j B1j q1j eisu ds a2 þ k21j
1
1
B23 e D23 ek23 h =c660
3
1
2 X
1
j¼1
! a D1j q1j eisðuhÞ ds a2 þ k21j
ð22dÞ
It should be noted that a in expressions ((22a)–(22d)) correspond to s in expression (21). In the derivation of expressions Ri(u, s, p), the following relations are used: Z 1 n cosðmaÞena da ¼ 2 ; ReðnÞ > 0 ð23aÞ m þ n2 0 Z 1 m sinðmaÞena da ¼ 2 ; ReðnÞ > 0 ð23bÞ m þ n2 0 By using the theory of residues, Ri(u, s, p) can be simplified as Ri1 ðu; s; pÞeuk21 þ Ri2 ðu; s; pÞeuk22 þ Ri3 ðu; s; pÞ ði ¼ 1; 2Þ Ri ðu; s; pÞ ¼ Ri1 ðu; s; pÞeðhuÞk23 þ Ri2 ðu; s; pÞeðhuÞk24 þ Ri3 ðu; s; pÞ ði ¼ 3; 4Þ
ð24Þ
where Ri1(u, s, p), Ri2(u, s, p) and Ri3(u, s, p) are given in Appendix A. Now, the six unknowns A11, A12, A21, . . . , A24 have been expressed by the unknown auxiliary function. Applying Laplace transform to the boundary condition (7) and substituting Eqs. (14), (16), (18) and (11) into it, the following integral equation can be obtained: Z b ½h1 ðu; x; pÞ þ h2 ðu; x; pÞgðu; pÞ du ¼ edx r0 =p ð25Þ a
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where h1 ðu; x; pÞ ¼ lim y!0
h2 ðu; x; pÞ ¼ lim y!0
K 1 ðy; s; pÞ ¼
2 X
1 2p 2 p
Z
þ1
K 1 ðy; s; pÞeisðxuÞ ds
ð26aÞ
K 2 ðu; x; s; pÞ cosðsyÞ ds
ð26bÞ
1
Z
þ1 0
C 1j q1j ek1j y
ð27aÞ
j¼1 k2j x K 2 ðu; x; s; pÞ ¼ X 3 X 1 1 Rðu; s; pÞe
X 3 ¼ ½ C 21 ek21 x
C 22 ek22 x
C 23 ek23 x
ð27bÞ C 24 ek24 x
ð28Þ
To derive the singular integral equation, an asymptotic analysis is needed. When y ! 0 and s ! 1, the asymptotic form of K1(y, s, p) can be expressed as K 11 ð0; s; pÞ ¼ w11 þ w12 =s
ð29Þ
Here w11 ¼
ic660 ½c2120 þ 16c2220 p211 p212 þ 4c120 c220 ðp211 þ p212 Þ 8c220 ðc120 þ c660 Þp11 p12 ðp11 þ p12 Þ
w12 ¼ c660 d c3120 16c2220 c660 p211 p212 þ c2120 ½3c660 4c220 ðp211 þ p212 Þ þ 4c120 c220 ½12c220 p211 p212 þ c660 ðp211 þ p212 Þ =½16c220 ðc120 þ c660 Þ2 p11 p12 ðp11 þ p12 Þ
ð30aÞ
ð30bÞ
where p11 and p12 are given in Appendix A. For verification, consider the functionally graded isotropic materials with the shear modulus l(x) = l0edx. From Eqs. (29) and (30), we have K 11 ð0; s; pÞ ¼
4l0 i 2l0 d 1 þ 1þk 1þk s
ð31Þ
It can be found that K11(0, s, p) is not related to the Laplace variable p and the first term in Eq. (31) is coincident with K11 given by Erdogan and Wu (1997). It should be noted that a second-order asymptotic term w12/s is kept in expression (29) to deal with the oscillation in the integrand (Shbeeb et al., 1999a,b). By adding and subtracting K11(0, s, p) from the integrand, h1 (u, x, p) can be obtained as 1 Imðx11 Þ þ k 1 ðu; x; pÞ h1 ðu; x; pÞ ¼ ð32Þ p ux k 1 ðu; x; pÞ ¼
Z
þ1
½K 1 ð0; s; pÞ K 11 ð0; s; pÞeisðuxÞ ds
ð33Þ
1
For the functionally graded isotropic materials, h2(u, x, p) for an internal crack (0 < a < b < h) are bounded in the interval a 6 (u, x) 6 b, but h2(u, x, p) for an edge crack (a = 0 or b = h) should include singular terms. Let us analyze the singularity of h2(u, x, p) and give the asymptotic behavior of K2(u, x, s, p) when s ! 1 and a ! 0. Through lengthy manipulation, the asymptotic form of K2(u, x, s, p) can be obtained as K 21 ðu; x; s; pÞ ¼ w21 euk21 þxk23 þ w22 euk21 þxk24 þ w23 euk22 þxk23 þ w24 euk22 þxk24
ð34Þ
L.-C. Guo et al. / Mechanics Research Communications 32 (2005) 385–400
391
where w21 ¼
ðB21 D22 B22 D21 ÞC 23 D24 R11 þ ðB22 D21 B21 D22 ÞB24 C 23 R21 ðB22 D21 B21 D22 ÞðB24 D23 B23 D24 Þ
ð35aÞ
w22 ¼
ðB22 D21 B21 D22 ÞC 24 D23 R11 þ ðB21 D22 B22 D21 ÞB23 C 24 R21 ðB22 D21 B21 D22 ÞðB24 D23 B23 D24 Þ
ð35bÞ
w23 ¼
ðB21 D22 B22 D21 ÞC 23 D24 R12 þ ðB22 D21 B21 D22 ÞB24 C 23 R22 ðB22 D21 B21 D22 ÞðB24 D23 B23 D24 Þ
ð35cÞ
w24 ¼
ðB22 D21 B21 D22 ÞC 24 D23 R12 þ ðB21 D22 B22 D21 ÞB23 C 24 R22 ðB22 D21 B21 D22 ÞðB24 D23 B23 D24 Þ
ð35dÞ
To simplify the expression K21(u, x, s, p), k21, . . . , k24 need to be expanded in the asymptotic forms (as s ! 1) k21 ¼ p21 s þ p23 ;
k22 ¼ p22 s þ p24 ;
k23 ¼ p21 s þ p24 ;
k24 ¼ p22 s þ p23
ð36Þ
where p21, . . . , p24 are constants given in Appendix A. Consequently, expression (34) can be manipulated as K 21 ðu; x; s; pÞ ¼ w21 eg1 ep21 ðuþxÞs þ w22 eg2 eðp21 uþp22 xÞs þ w23 eg3 eðp22 uþp21 xÞs þ w24 eg4 ep22 ðuþxÞs
ð37Þ
where g1 ¼ p23 u þ p24 x;
g2 ¼ p23 u þ p23 x;
g3 ¼ p24 u þ p24 x;
g4 ¼ p24 u þ p23 x
ð38Þ
From (26b), (27b) and (28), h2(u, x, p) can be expressed as 2 h2 ðu; x; pÞ ¼ ½k s ðu; x; pÞ þ k b ðu; x; pÞ p where Z þ1 k b ðu; x; pÞ ¼ ½K 2 ðu; x; s; pÞ K 21 ðu; x; s; pÞ ds
ð39Þ
ð40Þ
0
k s ðu; x; pÞ ¼
Z
1
K 21 ðu; x; s; pÞ ds
ð41Þ
0
To check the expression (37), we still consider the functionally graded isotropic materials with the shear modulus l(x) = l0edx. The following expressions can be obtained: 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1@ ðk 3Þd2 A d ð42Þ p21 ¼ p22 ¼ 1; p23;24 ¼ 2 1þk After some manipulations,K21(u, x, s, p) can be simplified as K 21 ðu; x; sÞ ¼ K11 eðuþxÞs
ð43Þ
where K11 ¼
2l0 ½2 þ ð3u xÞs þ 2uxs2 1þk
ð44Þ
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L.-C. Guo et al. / Mechanics Research Communications 32 (2005) 385–400
k s ðu; xÞ ¼
Z
1 0
" # 2l0 1 6x 4x2 þ K 21 ðu; x; s; pÞ ds ¼ 1 þ k u þ x ðu þ xÞ2 ðu þ xÞ3
ð45Þ
It can be found that the forms of K21(u, x, s, p) and ks(u, x, p) for isotropic FGMs are the same as those obtained by Erdogan and Wu (1997). For the functionally graded orthotropic strip, we have K 21 ðu; x; sÞ ¼ K21 ep21 ðuþxÞs þ K22 eðp21 uþp22 xÞs þ K23 eðp22 uþp21 xÞs þ K24 ep22 ðuþxÞs
ð46Þ
where K21, K22, K23 and K24 are constants shown in Appendix A. Note that K21, . . . , K24 are not related to the variable s. Therefore k s ðu; xÞ ¼
K21 K22 K23 K24 þ þ þ p21 ðu þ xÞ ðp21 u þ p22 xÞ ðp22 u þ p21 xÞ p22 ðu þ xÞ
Finally, by substituting Eqs. (32), (38) and (46) into Eq. (25) we have Z 1 b Imðx11 Þ þ k 1 ðu; xÞ þ 2k s ðu; xÞ þ 2k b ðu; xÞ gðuÞ du ¼ edx r0 =p p a ux
ð47Þ
ð48Þ
Since a second-order asymptotic term w12/s is kept in Eq. (29), for the need to resolve Eq. (48) numerically, referencing the procedure outlined by Shbeeb et al. (1999a,b), k1(u, x) can be manipulated as Z 1h Z U 1 w12 i 1 2Imðh1 Þi 2iIm w11 þ ½2Reðh1 Þ 2Reðw11 Þ k 1 ðu; xÞ ¼ i sin½sðu xÞ ds þ 2p 0 2p 0 s Z 1h 1 w12 i 2Reðh1 Þ 2Re w11 þ cos½sðu xÞ ds þ cos½sðu xÞ ds 2p U s Z Imðw12 Þ u x 1 1 cos½sðu xÞ þ ds ð49Þ Reðw12 Þ þ 2 ju xj p U s The unknown auxiliary functions in Eq. (48) will be solved numerically.
3. Numerical solutions The singular integral Eq. (48) can be solved by the method (Erdogan and Gupta, 1972; Kadioglu et al., 1998). To reduce (48) to a standard form, let us normalize the interval by setting ba bþa ðr; sÞ þ 2 2 For the internal crack, the solution of Eq. (48) may be written as ðu; xÞ ¼
ð50Þ
f ðr; pÞ gðr; pÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 r2 Þ
ð51Þ
For the edge crack, the solution of Eq. (48) may be written as f ðr; pÞ gðr; pÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 rÞ The stress intensity factor of the crack tips can be defined as: for the internal crack rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ba da k 1 ða; pÞ ¼ lim 2ða xÞryy ðx; 0; pÞ ¼ Imðw11 Þe f ð1; pÞ x!a 2
ð52Þ
ð53aÞ
L.-C. Guo et al. / Mechanics Research Communications 32 (2005) 385–400
k 1 ðb; pÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ lim 2ðx bÞryy ðx; 0; pÞ ¼ Imðw11 Þedb y!b
rffiffiffiffiffiffiffiffiffiffiffi ba f ð1; pÞ 2
393
ð53bÞ
for the edge crack. k 1 ðb; pÞ ¼ lim y!b
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2ðx bÞryy ðx; 0; pÞ ¼ Imðw11 Þedb bf ð1; pÞ
ð54Þ
The Laplace inversion of Eqs. (53) and (54) is conducted numerically by the method provided by Miller and Guy (1966).
4. Results and discussions pffiffiffi pffiffiffiffiffiffiffi For convenience, the dynamic SIFs are normalized by k 0 ¼ r0 a0 h for the internal crack and k 0 ¼ r0 b for the edge crack, where the normalized half crack length a0 = (b a)/2h. The wave velocities are defined pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi as c1 ¼ c110 =q0 and c2 ¼ c660 =q0 . To verify the validity of the present work, two examples are given. Firstly, we compare the normalized dynamic stress intensity factors of a central crack in a homogeneous isotropic strip, namely dh = ln 1.0 = 0 and (b + a)/2h = 0.5, with the results given by Itou (1980). Due to the symmetry, the SIFs of two crack tips are equal, which is verified very well by Fig. 2. The curves in Fig. 2 are very similar to those provided by Itou (1980). Another example is provided in Fig. 3 which gives some results for a central crack in a functionally graded orthotropic strip. Assume the half crack length a0 = 0.3 and the nonhomogeneity constant dh = ln 0.2, ln 1.0 and ln 10, respectively. The results corresponding to Material 0 listed in Table 1 are displayed in Fig. 3. It can be found that the results in Fig. 3 are similar to those given by Chen et al. (2002). Subsequently, let us consider the edge crack problem (a = 0, a0 = b/2h) in a functionally graded orthotropic strip when the crack is subjected to a surface impact r0H(t). Two kinds of orthotropic materials listed in Table 1, namely Material 1 and Material 2, are chosen for the numerical calculations. Fig. 4 shows the influence of nonhomogeneity constant dh on the dynamic SIFs, when b/h = 0.3. It is shown that the peak value of k1 (b, t)/k0 increases and occurs at an earlier time with the increasing of dh. Note that dh < 0 means that the crack side is stiffer than the other side of the strip, while dh > 0 means the crack
a0=0.35
1.5
a0=0.3
1.0
k1(a,t)/k0
0.5
k1(b,t)/k0
δh=0 0.0
0
2
4
6
8
10
Fig. 2. The dynamic SIFs of the homogeneous isotropic strip with a central internal crack, dh = 0.
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L.-C. Guo et al. / Mechanics Research Communications 32 (2005) 385–400 2.0
1.5
1.0
δh=ln0.2 δh=ln1.0 δ h=ln10.0
0.5
a0=0.3 0.0 0
6
4
2
8
10
Fig. 3. The dynamic SIFs of the functionally graded orthotropic strip with a central internal crack, a0 = 0.3.
Table 1 Mechanical properties of the functionally graded orthotropic strip Material
c110(N/m2)
c120(N/m2)
c220(N/m2)
c660(N/m2)
q0(kg/m3)
0 1 2
1.578 · 1010 1.578 · 1010 1.048 · 1010
3.248 · 109 3.248 · 109 3.248 · 109
1.048 · 1010 1.048 · 1010 1.578 · 1010
7.070 · 109 7.070 · 109 7.070 · 109
1.58 · 103 7.069 · 103 7.069 · 103
2.4
2.0
b/h=0.3
1.6
1.2
δh=ln0.1 δh=ln0.2 δh=ln1.0 δh=ln5.0
0.8
0.4
0.0
0
5
10
15
20
Fig. 4. The dynamic SIFs for Material 1 with an edge crack, b/h = 0.3.
side is less stiffer than the other side. Therefore, the peak value of normalized dynamic SIFs are greater when the edge crack lies in the less stiffer side of the strip. Form Fig. 5, a similar characteristic can be found. Fig. 6
L.-C. Guo et al. / Mechanics Research Communications 32 (2005) 385–400
395
depicts the influence of the normalized crack length on the dynamic SIFs when dh = ln 5.0. It can be seen that the peak value of k1(b, t)/k0 increases with the increasing of the crack length. By comparing the results for the edge crack problem with those for the internal crack problem given by Chen et al. (2002). It can be found that: for the internal crack the dynamic SIFs increase quickly with time and then keep approximately stable after reaching a peak, while the dynamic SIFs for the edge crack also increase quickly with time but then exhibit a more obvious oscillation after reaching a peak. It is for the reason that the strip with an edge crack, compared with the structure with a central crack, loses symmetry, and the interaction between the waves and the finite boundaries would be stronger. Therefore, the dynamic behavior of the edge crack problem is more complex than that of the internal crack.
2.5
b/ h= 0. 3 2.0
1.5
δh=ln0.1 δh=ln0.2 δh=ln1.0 δh=ln5.0
1.0
0.5
0.0 0
5
10
15
20
Fig. 5. The dynamic SIFs for an edge crack in Material 2, b/h = 0.3.
2.5
2.0
1.5
1.0
b/h=0.2 b/h=0.3
b/h=0.5
0.5
δh=ln5.0 0.0 0
5
10
15
20
Fig. 6. The dynamic SIFs for Material 1 with an edge crack, dh = ln5.0.
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L.-C. Guo et al. / Mechanics Research Communications 32 (2005) 385–400
5. Conclusions In this paper, the dynamic response of a functionally graded orthotropic strip with an edge crack perpendicular to boundaries is studied. An asymptotic analysis is conducted. It can be found that the singular terms in the integral equation are different from those corresponding to the internal crack problem. The numerical examples are presented to illustrate the influences of the nonhomogeneity constant and the geometry parameters on the dynamic stress intensity factors. It is found that the dynamic behavior of the functionally graded orthotropic strip with an edge crack is more complex than that of the corresponding internal crack problem. Since the geometry structure can be used as an approximation to a number of structural components and laboratory specimens, it is necessary to conduct further researches about this problem.
Acknowledgments This work is supported by the National Science Foundation for Excellent Young Investigators (No. 10325208), China Postdoctoral Science Foundation and NSFC (No. 10432030).
Appendix A ½c660 sðs þ idÞ c220 k21j þ p2 q0 i E1j ¼ ; j ¼ 1; . . . ; 4 ½c120 s þ c660 ðs þ idÞk1j 8 > < B1j ¼ c120 k1j ic110 E1j s C 1j ¼ c220 k1j ic120 E1j s ; j ¼ 1; . . . ; 4 > : D1j ¼ c660 ðis þ k1j E1j Þ E2j ¼
c220 s2 þ c660 k2j ðd þ k2j Þ p2 q0 ; s½c120 k2j þ c660 ðd þ k2j Þ
8 > < B2j ¼ c120 s þ c110 k2j E2j C 2j ¼ c220 s þ c120 k2j E2j ; > : D2j ¼ c660 ðk2j sE2j Þ
j ¼ 1; . . . ; 4
j ¼ 1; . . . ; 4
X11 ¼
sðs þ idÞðc2120 c110 c220 Þ þ c120 c660 ð2s2 þ 2isd d2 Þ ðc220 þ c660 Þp2 q0 c220 c660
X12 ¼
c110 c660 s2 ðs þ idÞ þ ðc110 þ c660 Þðs þ idÞp2 q0 s þ p4 q20 c220 c660
ðA:1Þ
ðA:2Þ
ðA:3Þ
ðA:4Þ
ðA:5aÞ
2
ðc2120 c110 c220 þ 2c120 c660 Þs2 ðc110 þ c660 Þp2 q0 c110 c660
c220 2 c120 2 2 ½ðc220 þ c660 Þs2 þ p2 q0 p2 q0 X22 ¼ s þ d s þ c110 c110 c110 c660
X21 ¼
ðA:5bÞ
ðA:6aÞ
ðA:6bÞ
L.-C. Guo et al. / Mechanics Research Communications 32 (2005) 385–400
397
R11 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þs2 k21 c110 ðk21 k22 Þðk21 k23 Þðk21 k24 Þ
ðA:7aÞ
R12 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þs2 k22 c110 ðk22 k21 Þðk22 k23 Þðk22 k24 Þ
ðA:7bÞ
R13 ðs; u; pÞ ¼
c120 p2 q0 ðc660 s2 þ p2 q0 Þ 2c110 c660 k21 k22 k23 k24
ðA:7cÞ
R21 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þsk21 ðd þ k21 Þ c110 c660 ðk21 k22 Þðk21 k23 Þðk21 k24 Þ
ðA:8aÞ
R22 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þsk22 ðd þ k22 Þ c110 c660 ðk21 k22 Þðk22 k23 Þðk22 k24 Þ
ðA:8bÞ
R23 ðs; u; pÞ ¼
c120 p2 dq0 s 2c110 c660 k21 k22 k23 k24
ðA:8cÞ
R31 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þs2 k23 c110 ðk21 k23 Þðk23 k22 Þðk23 k24 Þ
ðA:9aÞ
R32 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þs2 k24 c110 ðk21 k24 Þðk24 k22 Þðk24 k23 Þ
ðA:9bÞ
R33 ðs; u; pÞ ¼ R13 ðs; u; pÞ
ðA:9cÞ
R41 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þsk23 ðd þ k23 Þ c110 c660 ðk21 k23 Þðk23 k22 Þðk23 k24 Þ
ðA:10aÞ
R42 ðs; u; pÞ ¼
ðc2120 þ c110 c220 Þsk24 ðd þ k24 Þ c110 c660 ðk21 k24 Þðk24 k22 Þðk24 k23 Þ
ðA:10bÞ
R43 ðs; u; pÞ ¼ R23 ðs; u; pÞ pffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n11 C11 þ C12 4 pffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n C11 C12 p12 ¼ 4 12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n11 ¼ sign½Reð C11 þ C12 Þ p11 ¼
ðA:10cÞ ðA:11aÞ
ðA:11bÞ ðA:12aÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n12 ¼ sign½Reð C11 C12 Þ
ðA:12bÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where n11 and n12 give 1 or 1 depending on whether the real parts of C11 þ C12 and C11 C12 are negative, or positive. C11 ¼
c110 c220 c120 ðc120 þ 2c660 Þ c220 c660
ðA:13aÞ
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C12 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðc2120 c110 c220 Þ½ðc120 þ 2c660 Þ c110 c220 =ðc220 c660 Þ
ðA:13bÞ
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p21 ¼ n21 C21 þ C22 2
ðA:14aÞ
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p22 ¼ n22 C21 C22 2
ðA:14bÞ
p23 ¼ lim ðk21 p21 sÞ
ðA:14cÞ
p24 ¼ lim ðk22 p22 sÞ
ðA:14dÞ
s!1
s!1
n21 ¼ sign½Reð
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C21 þ C22 Þ
ðA:15aÞ
n22 ¼ sign½Reð
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C21 C22 Þ
ðA:15bÞ
c110 c220 c120 ðc120 þ 2c660 Þ c110 c660 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 C22 ¼ ðc2120 c110 c220 Þ½ðc120 þ 2c660 Þ c110 c220 =ðc110 c660 Þ
C21 ¼
Here n21 and n22 take 1 or 1, which depend on whether the real parts of negative or positive.
ðA:16aÞ ðA:16bÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C21 þ C22 and C21 C22 are
K21 ¼ ðc2120 c110 c220 Þp21 ðc220 þ c120 p221 Þfc220 ðc660 þ c110 p21 p22 Þ þ p22 ½c2120 p21 þ c120 c660 ðp21 p22 Þ þ c120 c660 p21 p222 g=K
ðA:17aÞ
K22 ¼ ðc2120 c110 c220 Þ½c220 ðc220 þ c120 p221 Þ þ p221 ðc2120 þ c110 c660 p221 Þp22 ðc220 þ c120 p222 Þ=K
ðA:17bÞ
K23 ¼ ðc2120 c110 c220 Þp21 ðc220 þ c120 p221 Þ½c220 ðc660 þ c110 p222 Þ p222 ðc2120 þ c110 c660 p222 Þ=K
ðA:17cÞ
K24 ¼ ðc2120 c110 c220 Þp22 ðc220 þ c120 p222 Þfc220 ðc660 þ c110 p21 p22 Þ þ p21 ½c2120 p22 þ c110 c660 p21 p22 þ c120 c660 ðp22 p21 Þg=K
ðA:17dÞ
2
K ¼ f2c110 ðp21 p22 Þ ðp21 þ p22 Þ½c3120 p21 p22 þ c2120 ðc220 c660 p21 p22 Þ þ c120 ðc110 c660 p221 p222 þ c220 ðc660 þ c110 p21 p22 ÞÞ þ c110 c220 ðc220 þ c660 ðp221 þ p21 p22 þ p222 ÞÞg
ðA:18Þ
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