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Dynamic behavior of a finite crack in functionally graded materials subjected to plane incident time-harmonic stress wave
Composite Structures 77 (2007) 10–17 www.elsevier.com/locate/compstruct
Dynamic behavior of a finite crack in functionally graded materials subjected to plane incident time-harmonic stress wave Chun-Hui Xia b
a,*
, Li Ma
b
a Department of Chemistry, Qiqihar Medical College, Qiqihar 161042, PR China Center for Composite Materials, Harbin Institute of Technology, Harbin 150001, PR China
Available online 21 June 2005
Abstract In the present paper, the dynamic behavior of a finite crack in functionally graded materials subjected to normally incident elastic time harmonic waves is investigated by means of the Schmidt method. By use of the Fourier transform and defining the jumps of the displacement across the crack surfaces as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement across the crack surfaces are expanded in a series. Numerical examples are provided to show the effect of the gradient parameter ba, the crack configuration on the dynamic intensity factors of cracked functionally graded materials. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Crack; Stress waves scattering; Functionally graded materials; Dual integral equations
1. Introduction In recent years, functionally graded materials (FGMs) have widely been applied in extremely high temperature environments. The major advantages of the graded material, especially in elevated temperature environments, stem from the tailoring capability to produce a gradual variation of its thermomechanical properties in the spatial domain [1]. In particular, the use of the graded material as interlayers in the bonded media is one of the highly effective and promising applications in eliminating various shortcoming resulting from stepwise property mismatch inherent in piecewise homogeneous composite media [2–4]. From the fracture mechanics viewpoint, the presence of a graded interlayer would play an important role in determining the crack driving forces and fracture
*
Corresponding author. Tel.: +86 452 6710544. E-mail address: [email protected] (C.-H. Xia).
0263-8223/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.05.012
resistance parameters. In an attempt to address the issues pertaining to the fracture analysis of bonded media with such transitional interfacial properties, a series of solutions to certain crack problems was obtained. Among them are a crack in the non-homogeneous interlayer bounded by dissimilar homogeneous media [5]; and a crack at the interface between homogeneous and non-homogeneous materials [6,7]. Similar problems of delamination or an interface crack between the functionally graded coating and the substrate were considered in [8–10]. Relatively fewer experimental and numerical investigations of the fracture behavior of FGMs have been conducted. Experimental investigations into the fracture of FGMs are limited due to the high cost and elaborate facilities required for processing FGMs [11– 13]. The finite element method has also been used to simulate the fracture behavior of cracked FGMs [14,15]. The crack problem in FGM layers under thermal stresses was studied by Edrogen and Wu [16]. They considered an unconstrained elastic layer under statically self-equilibrating thermal or residual stresses. The
C.-H. Xia, L. Ma / Composite Structures 77 (2007) 10–17
layer contained an embedded or surface crack perpendicular to its boundaries. After giving the distribution of thermal stresses, the stress intensity factors for the embedded and surface crack were presented along with the results of the crack/contact problem in a FGM layer that was under compression near and at the surface and tension in the interior region. The interface crack problem for a non-homogeneous coating bended to a homogeneous substrate was investigated in [17]. Existing studies which consider the fracture behavior of FGMs are mainly limited to quasi-static problems. These studies are suitable for cases where the scattering of the elastic wave by the Mode-I crack or impact load are not involved. In this case, inertia effects do not play a role and can be ignored. However, it should be mentioned that most FGMs will be used in critical situations, where significant dynamic loading may be involved. The dynamic fracture behavior of FGMs has received little attention from the scientific community. Examples of dynamic analysis include the study of the steady state dynamic crack propagation in an interphase with spatially varying elastic properties under antiplane loading conditions reported in [18], and the steady state dynamic fracture of FGMs under in-plane loading with the material properties being assumed to vary along the direction of crack propagation reported in [19]. Experimental studies of the dynamic fracture of a FGMs with discrete property variation using photoelasticity technique were also conducted in [20]. The dynamic crack propagation problems were studied in [21,22]. The dynamic crack problem for the non-homogeneous composite materials was considered in [23] but they considered the FGM layer as multi-layered homogeneous media. The transient internal crack problem for a functionally graded strip and eletromechanical impact of a cracked functionally graded piezoelectric medium was investigated in [24–26]. In spite of these efforts, the understanding of the dynamic fracture process of FGMs is still limited. It is therefore the objective of the current study to provide a theoretical analysis of the dynamic behavior of a finite crack in functionally graded materials subjected to normally incident time harmonic elastic waves is investigated by means of the Schmidt method. It is assumed that the elastic properties of FGMs spatially vary along to the direction of the crack. YoungÕs modulus and mass density of the model are assumed to vary exponentially while PoissonÕs ratio remains constant. The analytical study is based on the use of Fourier transform technique and a somewhat different approach, name as the Schmidt method [27–29]. To solve the dual integral equations, the jumps of the displacements across crack surfaces are expanded in a series. Numerical examples are given to show the effects of the material properties upon the dynamic stress intensity factors (DSIF).
11
2. Formulation of the problem Consider the plane problem of diffraction of normally incident longitudinal wave by a Griffith crack situated in a functionally graded plane. The crack is assumed to occupy the region jxj 6 a, y = 0, jzj < 1. For the present problem, it is convenient to divide the functionally graded plane into two regions, namely the material 1 in upper half plane and the material 2 located in lower half plane as show in Fig. 1. The elastic parameters l(x) and q(x) are approximated by ðl; qÞ ¼ ðl0 ebx ; q0 ebx Þ
ð1Þ
where l is the shear modulus, q is the mass density and b is the gradient parameter and is taken to be a constant. By denoting u(j)(x, y, t) and v(j)(x, y, t) as the displacement components in the x- and y-directions, and ðjÞ ðjÞ rðjÞ y ðx; y; tÞ, rx ðx; y; tÞ, sxy ðx; y; tÞ (the superscript j = 1, 2 correspond to the upper half plane and the lower half plane through in this paper) as the stress components, respectively. The constitutive relations for the FGMs are written as l0 ebx ouðjÞ ovðjÞ rðjÞ ð1 þ jÞ þ ð3 jÞ ðx; y; tÞ ¼ ; ðj ¼ 1; 2Þ x j1 ox oy ð2Þ rðjÞ y ðx; y; tÞ ¼
bx
ðjÞ
l0 e ovðjÞ ou ð1 þ jÞ þ ð3 jÞ ; ðj ¼ 1; 2Þ j1 oy ox ð3Þ
ðjÞ ovðjÞ ðjÞ bx ou þ sxy ðx; y; tÞ ¼ l0 e ; ðj ¼ 1; 2Þ oy ox
ð4Þ
where t is the PoissonÕs ratio, and j = 3 4t for the state of plane strain, j = (3 t)/(1 + t) for the state of generalized plane stress. The PoissonÕs ratio t is taken to be a constant, owing to the fact its variation within a practical range has the rather insignificant influence on the value of the near-tip driving for fracture mechanics [5–7]. b 5 0 for the functionally graded materials. When b = 0 it will return to the homogenous material case.
y
1
μ ( x) = μ 0 eβ x –a
x
a 2
Fig. 1. Geometry of a finite crack in the functionally graded materials.
12
C.-H. Xia, L. Ma / Composite Structures 77 (2007) 10–17
Let a plane time harmonic elastic wave originating at y = 1 be incident normally on the crack, and is defined by u¼0
ð5Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ v0 expfi½xy=ð ð1 þ jÞ=ðj 1Þct Þ xtg
ð6Þ
where 0 is a ffi constant, x is the circular frequency and pvffiffiffiffiffiffiffiffiffiffiffi ct ¼ l0 =q0 . Substituting these relations into Eqs. (2)– (4) results in n h .pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi io 3j r0 exp i xy rx ¼ ð1 þ jÞ=ðj 1Þct xt 1þk ð7Þ n h .pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi io ð1 þ jÞ=ðj 1Þct x ry ¼ r0 exp i xy ð8Þ rxy ¼ 0
ð9Þ
where r0 ðxÞ ¼
1þj ixl0 ebx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v0 j 1 ð1 þ jÞ=j 1Þct
ð10Þ
In the absence of body forces, the elastic behavior of the medium with the variable shear modulus and the variable density in (1) is governed by following equations: 2 ðjÞ
2 ðjÞ
2 ðjÞ
ou ou ov þ ðj 1Þ þ2 ð1 þ jÞ ox2 oy 2 oxoy ðjÞ ðjÞ ou ov þ b ð1 þ jÞ þ ð3 jÞ ox oy 2 ðj 1Þq0 x ðjÞ ¼ u ; ðj ¼ 1; 2Þ l0
ð11Þ
o2 vðjÞ o2 vðjÞ o2 uðjÞ þ ðj 1Þ þ ðj 1Þ þ 2 ð1 þ jÞ oy 2 ox2 oxoy ðjÞ ou ovðjÞ ðj 1Þq0 x2 ðjÞ b þ v ; ðj ¼ 1; 2Þ ¼ oy ox l0 ð12Þ Because it is a common factor in all equations, the timefactor exp(ixt) is dropped out hereafter. It is further supposed that the two surfaces of the crack do not contact during vibrations. If only the stress intensity factors are considered, the mixed boundary value problem shown in Fig. 1 must be solved under the following conditions: ð2Þ rð1Þ y ðx; 0Þ ¼ ry ðx; 0Þ ¼ r0 ;
ð1Þ ð2Þ sxy ðx; 0Þ ¼ sxy ðx; 0Þ ¼ 0;
jxj 6 a rð1Þ y ðx; 0Þ
ð13Þ ¼
ryð2Þ ðx; 0Þ;
sð1Þ xy ðx; 0Þ
¼
sð2Þ xy ðx; 0Þ;
jxj > a ð14Þ
ð1Þ
ð2Þ
u ðx; 0Þ ¼ u ðx; 0Þ;
ð1Þ
ð2Þ
v ðx; 0Þ ¼ v ðx; 0Þ;
jxj > a ð15Þ
3. Solution The system of above governing equations is solved using the Fourier integral transform technique to obtain the general expressions for the displacement components as 8 2 R1 P > ð1Þ > 1 > Aj ðsÞekj y eisx ds u ¼ 2p > 1 < j¼1 ð16Þ 2 > R P > 1 1 ð1Þ k y isx >v ¼ > mj ðsÞAj ðsÞe j e ds : 2p 1 j¼1 8 4 R1 P > > 1 > uð2Þ ¼ 2p Aj ðsÞekj y eisx ds > 1 < j¼3 ð17Þ 4 > > > vð2Þ ¼ 1 R 1 P mj ðsÞAj ðsÞekj y eisx ds > : 2p 1 j¼3
and from (2)–(4), the stress components are obtained as 8 2 R1 P > l0 ebx ð1Þ > r ðx; yÞ ¼ ½ðj þ 1Þmj ðsÞkj isð3 jÞ > y 2pðj1Þ 1 > > j¼1 > < Aj ðsÞekj y eisx ds > > > 2 R1 P > ð1Þ > l ebx > : sxy ðx; yÞ ¼ 02p 1 ½kj imj ðsÞsAj ðsÞekj y eisx ds j¼1
ð18Þ 8 4 R 1 P > l0 ebx > ½ðj þ 1Þmj ðsÞkj isð3 jÞ > ryð2Þ ðx; yÞ ¼ 2pðj1Þ 1 > > j¼3 > < Aj ðsÞekj y eisx ds > > > 4 > bx R 1 P > ð2Þ > ðx; yÞ ¼ l02pe 1 ½kj imj ðsÞsAj ðsÞekj y eisx ds : sxy j¼3
ð19Þ where s is the transform variable. A1, A2, A3 and A4 are unknowns, kj(s), j = 1, 2, 3 and 4 are the roots of the characteristic equation k4 D1 k2 þ D2 ¼ 0 where 3j 2 2j x2 b D1 ¼ 2s2 þ 2isb þ jþ1 j þ 1 c2t h i 2 ½sðs þ ibÞ c2t ð1 þ jÞsðs þ ibÞ ðj 1Þ xc2 t D2 ¼ ð1 þ jÞ Then, the root of Eq. (20) may be obtained as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D1 1 k1 ¼ D21 4D2 þ 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D1 1 D21 4D2 k2 ¼ 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D1 1 D21 4D2 þ k3 ¼ 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D1 1 D21 4D2 k4 ¼ 2 2
ð20Þ ð21Þ
ð22Þ
ð23Þ ð24Þ ð25Þ ð26Þ
C.-H. Xia, L. Ma / Composite Structures 77 (2007) 10–17
and mj(s), j = 1, 2, 3, and 4 are expressed for each root kj (s) as 2
mj ðsÞ ¼
ð1 þ jÞs2 þ ðj 1Þk2j isbð1 þ jÞ þ ðj 1Þ xc2 t
kj ½2is þ bð3 jÞ
Z
f1 ðxÞ ¼ uð1Þ ðx; 0Þ uð2Þ ðx; 0Þ
ð28Þ
f2 ðxÞ ¼ vð1Þ ðx; 0Þ vð2Þ ðx; 0Þ
ð29Þ
Applying the Fourier transforms and the boundary conditions (13)–(15), it can be obtained A1 ðsÞ A3 ðsÞ ½X 1 ¼ ½X 2 ð30Þ A2 ðsÞ A4 ðsÞ A1 ðsÞ A3 ðsÞ f 1 ðsÞ ½X 3 ½X 4 ¼ ð31Þ A2 ðsÞ A4 ðsÞ f 2 ðsÞ where " ½X 1 ¼ " ½X 1 ¼ ½X 3 ¼
l0 ½ðjþ1Þm1 ðsÞk1 isð3jÞ j1
l0 ½ðjþ1Þm2 ðsÞk2 isð3jÞ j1
l0 ½k1 im1 ðsÞs
l0 ½k2 im2 ðsÞs
l0 ½ðjþ1Þm3 ðsÞk3 isð3jÞ j1
l0 ½ðjþ1Þm4 ðsÞk4 isð3jÞ j1
l0 ½k3 im3 ðsÞs
l0 ½k4 im4 ðsÞs
1
1
m1 ðsÞ m2 ðsÞ 1 1 ½X 4 ¼ m3 ðsÞ m4 ðsÞ
#
#
½d 3 ðsÞf 1 ðsÞ þ d 4 ðsÞf 2 ðsÞeisx ds ¼ 0;
1
ð39Þ
f 2 ðsÞeisx ds ¼ 0;
jxj > a
ð40Þ
where d1(s), d2(s), d3(s) and d4(s) are known functions, and can be expressed as follows: ½X 5 ¼ ½X 3 ½X 4 ½X 2 1 ½X 1
d 1 ðsÞ
d 2 ðsÞ
d 3 ðSÞ
d 4 ðsÞ
¼ ½X 1 ½X 5
1
ð41Þ ð42Þ
To determine the unknown functions f 1 ðsÞ and f 2 ðsÞ, the above two pairs of dual integral Eqs. (37)–(40) must be solved.
4. Solution of the dual integral equation From the nature of the displacement along the crack line, it can be obtained that the jumps of the displacement across the crack surface are finite, differentiability and continuum functions. So the jumps of the displacement across the crack surface can be expanded by the following series [28]: f1 ðxÞ ¼
1 h x i 1X an sin 2nsin1 for 0 6 jxj 6 a p n¼1 a
ð43Þ
ð32Þ
ð44Þ f1 ðxÞ ¼ 0 jxj > a 1 h i X 1 x f2 ðxÞ ¼ bn sin ð2n 1Þsin1 0 6 jxj 6 a p n¼1 a
ð33Þ
ð45Þ
ð34Þ
ð35Þ
A superposed bar indicates the Fourier transform through the paper. And defined as follows: Z 1 Z 1 1 f ðxÞeisx dx; f ðxÞ ¼ f ðsÞeisx ds f ðsÞ ¼ 2p 1 1 ð36Þ By solving four Eqs. (24) and (25) with four unknown functions, substituting the solutions into Eq. (18) and applying the boundary conditions, it can be obtained Z ebx 1 rð1Þ ðx; 0Þ ¼ ½d 1 ðsÞf 1 ðsÞ þ d 2 ðsÞf 2 ðsÞeisx ds y 2p 1 ¼ r0 ðxÞ 0 6 jxj 6 a ð37Þ Z 1 ð38Þ f 1 ðsÞeisx ds ¼ 0; jxj > a 1
1
1
1
ð27Þ From Eqs. (16)–(19), it can be seen that there are four unknown constants (in Fourier space they are functions of s), i e., A1, A2, A3 and A4 which can be obtained from the boundary conditions. To solve the present problem, the jumps of the displacement across the crack surfaces can be defined as follows:
Z
ebx 2p 0 6 jxj 6 a
sð1Þ xy ðx; 0Þ ¼
13
f2 ðxÞ ¼ 0 jxj > a
ð46Þ
where an and bn are unknown coefficients. The Fourier transform of (43)–(46) is [29] f 1 ðsÞ ¼ i f 2 ðsÞ ¼
1 X
n¼1 1 X
an
bn
n¼1
2n J 2n ðsaÞ s
2n 1 J 2n1 ðsaÞ s
ð47Þ ð48Þ
Substituting Eqs. (47) and (48) into Eqs. (37)–(40), it can be shown that Eqs. (38) and (40) are automatically satisfied. After integration with respect to x in [a, x], Eqs. (37) and (39) reduce to 1 Z 1 1 X i ½i2nd 1 ðsÞan J 2n ðsaÞ 2p n¼1 1 s2 þ ð2n 1Þd 2 ðsÞbn J 2n1 ðsaÞðeisx eisa Þ ds Z x ¼ r0 ðxÞebx dx; 0 6 jxj 6 a a
ð49Þ
14
C.-H. Xia, L. Ma / Composite Structures 77 (2007) 10–17
1 Z 1 1 X i ½i2nd 3 ðsÞan J 2n ðsaÞ þ ð2n 1Þd 4 ðsÞ 2p n¼1 1 s2
bn J 2n1 ðsaÞðeisx eisa Þ ds ¼ 0; Z
0 6 jxj 6 a
ð50Þ
From the relationships [30] 1
1 J n ðsaÞ sinðbsÞ ds s 0 8 > sin½n sin1 ðb=aÞ > > ; a>b < n ¼ an sinðnp=2Þ > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ; b > a > : n½b þ b2 a2 Z 1 1 J n ðsaÞ cosðbsÞ ds s 0 8 cos½n sin1 ðb=aÞ > > ; a>b < n n ¼ a cosðnp=2Þ > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; b > a : n½b þ b2 a2 n
ð51Þ
where lims!±1 d1(s)/s = d1, lims!+1d2(s)/s = lims!1 d2(s)/s = d2, lims!+1d3(s)/s = lims!1d3(s)/s = d3, lims!!±1d4(s)/s = d4. These constants can be obtained by using the Mathematica program and are independent of the gradient parameter b. Also, these constants equal to the ones of the homogeneous material case. The semi-infinite integral in (51)–(56) can be evaluated directly, so Eqs. (49) and (50) can now be solved for the coefficients an and bn by the Schmidt method [26,27] as described in Appendix A.
5. Dynamic stress intensity factors
ð52Þ
The semi-infinite integral in Eqs. (49) and (50) can be modified as Z 1 h i d 1 ðsÞ 2id1 1 x isx isa sin 2nsin J ðsaÞðe e Þds ¼ 2n 2 a 2n 1 s Z 1 1 d 1 ðsÞ d1 J 2n ðsaÞðeisx eisa Þds þ ð53Þ s 1 s Z 1 d 2 ðsÞ J 2n1 ðsaÞðeisx eisa Þds 2 1 s h x i 2id2 n 2id2 ¼ sin ð2n 1Þsin1 þ ð1Þ a 2n 1 2n 1 Z 0 1 d 2 ðsÞ þ d2 J 2n1 ðsaÞðeisx eisa Þds þ s s 1 Z 1 1 d 2 ðsÞ d2 J 2n1 ðsaÞðeisx eisa Þds þ ð54Þ s s 0 Z 1 d 3 ðsÞ J 2n ðsaÞðeisx eisa Þds 2 1 s h x i 2d3 n 2d3 cos 2nsin1 þ ð1Þ ¼ a 2n 2n Z 0 1 d 3 ðsÞ þ d3 J 2n ðsaÞðeisx eisa Þds þ s 1 s Z 1 1 d 3 ðsÞ d3 J 2n ðsaÞðeisx eisa Þds þ ð55Þ s s 0 Z 1 d 4 ðsÞ J 2n1 ðsaÞðeisx eisa Þds 2 s 1 h x i 2d4 cos ð2n 1Þsin1 ¼ a 2n 1 Z 1 1 d 4 ðsÞ d4 J 2n1 ðsaÞðeisx eisa Þds ð56Þ þ s 1 s
The coefficients an and bn are known, so that the entire stress field can be obtained. However, in fracture mechanics, it is important to determine stress ryð1Þ in the vicinity of the crack tips. The stress components ryð1Þ along the crack line can be expressed as 1 Z 1 ebx X 2n ð1Þ ian d 1 ðsÞ J 2n ðsaÞ ry ðx; 0Þ ¼ s 2p n¼1 1 2n 1 þ bn d 2 ðsÞ J 2n1 ðsaÞ eisx ds s Z 1 1 ebx X d 1 ðsÞ d1 J 2n ðsaÞeisx ds ¼ ian 2n s 2p n¼1 1 Z 1 d 2 ðsÞ þ bn ð2n 1Þ d2 J 2n1 ðsaÞeisx ds s 0 Z 0 d 2 ðsÞ þ d2 J 2n1 ðsaÞeisx ds þ bn ð2n 1Þ s 1 Z 1 þ d1 an i2n J 2n ðsaÞeisx ds 1
þ d2 bn ð2n 1Þ
Z
1
J 2n1 ðsaÞeisx ds
0
d2 bn ð2n 1Þ
Z
0
J 2n1 ðsaÞeisx ds
ð57Þ
1
An examination of Eq. (57) shows that, the singular part of the stress field can be obtained from the relationships as follows [30]: Z
1
J n ðsaÞ cosðbsÞ ds
0
¼
8 cos½n sin1 ðb=aÞ > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; > > < a2 b2
a>b
> an sinðnp=2Þ > > p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiin ; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffih > > : b2 a 2 b þ b 2 a 2
ð58Þ b>a
C.-H. Xia, L. Ma / Composite Structures 77 (2007) 10–17 1
J n ðsaÞ sinðbsÞ ds 0 8 sin½n sin1 ðb=aÞ > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; a>b < 2 b2 a ¼ > an cosðnp=2Þ > > ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ; : pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b a2 ½b þ b2 a2 Z 1 J nþ1 ðsaÞeisx ds ¼ 0; jxj > a
ð59Þ b>a ð60Þ
1
For a < x, the singular part of the stress field can be expressed respectively as follows: Z 1 1 d2 ebx X r¼ 2bn ð2n 1Þ J 2n1 ðsaÞ cosðsxÞ ds 2p n¼1 0 1 d2 ebx X a2n1 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2n1 ¼ bn ð2n 1Þð1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi p n¼1 x2 a2 ½x þ x2 a2 ð61Þ For x < a, the singular part of the stress field can be expressed respectively as follows: Z 1 1 d2 ebx X 2bn ð2n 1Þ J 2n1 ðsaÞ cosðsxÞ ds r¼ 2p n¼1 0 1 d2 ebx X a2n1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ bn ð2n 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi p n¼1 x2 a2 ½jxj þ x2 a2 2n1 ð62Þ
namic stress intensity factor (DSIF) as K ¼ K I =K 0 , pffiffiffi where K 0 ¼ r0 a. To verify the validity of the present work, the variations of the normalized dynamic stress intensity factors with the normalized wave number xa/ct for a small gradient parameter ba, namely, ba = 0.0001 are displayed in Fig. 2. This case returned to the homogeneous isotropic case. These curve are very similar to those given by Sih in [33] for isotropic material. From this result, it can be seen that our analysis process is reasonable and effectively. The aim of the present paper is to give an approach to solve the dynamic behavior of a crack in the functionally graded materials subjected to the normally harmonic stress waves. In the present paper, the unknown variables of dual integral equations are the displacement across the crack surfaces. The solution can be returned to the same problem as the static problem for x = 0. The variations of the normalized DSIF with the normalized wave number xa/ct for different value of ba are
2.6 2.4 2.2
K/K0
Z
The stress intensity factors KI at the right tip of the crack can be given as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K IR ¼ limþ 2pðx aÞr d2 e ¼ pffiffiffiffiffiffi pa
1 X
n
ð1Þ bn ð2n 1Þ
x!a
KIL
1.8
KIR
1.6
1.2 1.0 0.8
ð63Þ
0.0
0.2
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 eba X 2pðjxj aÞr ¼ pffiffiffiffiffiffi bn ð2n 1Þ pa n¼0
0.4
0.6
0.8
1.0
ω a/ct
n¼1
The stress intensity factors KI at the left tip of the crack can be given as follows: K IL ¼ lim
2.0
1.4
x!a
ba
15
Fig. 2. Variations of normal stress intensity factor with normal wave number at ba = 0.0001.
ð64Þ 3.5
KIL
3.0
KIR
6. Numerical calculations and discussion K/K0
Numerical calculations are carried out for functionally graded materials. In the present work, we only consider the dynamic behavior of cracked functionally graded materials for plane stain condition. As discussed in the works [26,27,31,32], it can be seen that the Schmidt method is performed satisfactorily if the first ten terms of the infinite series to Eqs. (A.1) and (A.2) are retained.The behavior of the sum of the series stays steady with the increasing number of terms in (A.1) and (A.2). For convenience, we define the normalized dy-
2.5 2.0 1.5 1.0 0.5 0.0
0.5
1.0
1.5
2.0
ω a/ct
Fig. 3. Variations of normal stress intensity factor with normal wave number at ba = 0.1.
16
C.-H. Xia, L. Ma / Composite Structures 77 (2007) 10–17
the materials and frequency of the incident wave on dynamic stress intensity factors are investigated.
3.0 2.4
K/K0
KIL 1.8
Appendix A
KIR
For convenience, Eqs. (44) and (45) can be rewritten as 1 1 X X an En ðxÞ þ bn F n ðxÞ ¼ U 0 ðxÞ; a 6 x 6 a ðA:1Þ
1.2 0.6 0.0 0.0
0.3
0.6
0.9
1.2
1.5
1.8
ω a/ct
n¼1
n¼1
1 X
1 X
an Gn ðxÞ þ
n¼1
Fig. 4. Variations of normal stress intensity factor with normal wave number at ba = 0.5.
bn H n ðxÞ ¼ V 0 ðxÞ; a 6 x 6 a
A set of functions Pn(y) that satisfy the orthogonality condition Z a Z a P m ðxÞP n ðxÞ dx ¼ N n dmn N n ¼ P 2n ðxÞ dx ðA:3Þ a
K/K0
2.0
KIL KIR
1.0 0.5 0.0 0.0
0.2
0.4
0.6
0.8 1.0 ω a/ct
a
can be constructed from a given set of arbitrary functions, say Gn(x), such that n X M in Gi ðxÞ ðA:4Þ P n ðxÞ ¼ M nn i¼1
2.5
1.5
ðA:2Þ
n¼1
1.2
1.4
1.6
Fig. 5. Variations of normal stress intensity factor with normal wave number at ba = 1.5.
plotted in Figs. 3–5, where the broken straight lines indicate the corresponding static values. It is apparent from the figures that the predominant term of the dynamic stress intensity factor of crack tip in the stiffer side is greater than that in the softer side. And the peak values of DSIF are considerably larger than the corresponding static values.
7. Conclusions A theoretical and numerical treatment of a finite crack subjected to an in-plane incident harmonic stress wave in a functionally graded plane is presented. The analysis is based upon an integral transform technique. The Fredholm integral equation is solved by using the Schmidt method. The present method is applied to illustrate the fundamental behavior of a crack in FGMs under the dynamic loading. Furthermore, the effects of the geometry of the crack, the shear stress wave velocity of
where Min defined as d 11 d 21 Dn ¼ d n1
is the cofactor of element din of Dn, which is d 12 d 22
d 1n d 2n ; d nn
d in ¼
Z
a
Gi ðxÞGn ðxÞ dx a
ðA:5Þ Representing the second series in Eq. (A.2) by the orthogonal series Pn(x) with coefficient cn, the following relations can be given: 1 X
an Gn ðxÞ ¼
n¼1
1 X
cn P n ðxÞ ¼ V 0 ðxÞ
n¼1
1 X
bn H n ðxÞ
ðA:6Þ
n¼1
The second equality yields # Z a" 1 X 1 cn ¼ V 0 ðxÞ bi H i ðxÞ P n ðxÞ dx N n a i¼0
ðA:7Þ
The first equality becomes an ¼
1 X
cni bi þ dn
ðA:8Þ
i¼1
with Z a 1 X M nj H i ðxÞP j ðxÞ dx N j M jj a j¼n Z a 1 X M nj V 0 ðxÞP j ðxÞ dx dn ¼ N j M jj a j¼n
cni ¼
ðA:9Þ ðA:10Þ
C.-H. Xia, L. Ma / Composite Structures 77 (2007) 10–17
Substituting Eq. (A.8) into Eq. (A.1) reduce to 1 X
bn Y n ðxÞ ¼ W ðxÞ
ðA:11Þ
n¼1
with Y n ðxÞ ¼ F n ðxÞ þ W ðxÞ ¼ U 0 ðxÞ
1 X i¼1 1 X
cin Ei ðxÞ
ðA:12Þ
di Ei ðxÞ
ðA:13Þ
i¼1
Finally, coefficients bn can be determined by bn ¼
1 X j¼n
qj ¼
1 Kj
qj
Z
Lnj Ljj
ðA:14Þ 1 X Lij Y i ðxÞ Ljj i¼1
a
W ðxÞQj ðxÞ dx;
Qj ðxÞ ¼
a
ðA:15Þ where Ljn is the minant Dn e11 e12 e21 e22 Dn ¼ en1 en2
cofactor of the element ejn of the deter e1n e2n ; enn
ejn ¼
Z
a
Y j ðxÞY n ðxÞ dx
a
ðA:16Þ Coefficients an are calculated by using Eq. (A.8).
References [1] Koizumi M. The concept of FGM. In: Holt JB et al., editors. Ceramic transactions. Functionally graded materials, vol. 34. Ohio: American Ceramic Society; 1993. p. 3–10. [2] Lee YD, Erdogen F. Residual/thermal stress in FGM and laminated thermal barrier coating. Int J Fract 1994;69:145–65. [3] Suresh S, Mortensen A. Functionally graded metals and metal– ceramic composites. Part 2: Thermomechanical behaviour. Int Mater Rev 1997;29:306–12. [4] Choi HJ. The problem for bonded half-planes containing a crack at an arbitrary angle to the graded interfacial zone. Int J Solids Struct 2001;38:6559–88. [5] Delae F, Erdogen F. On the mechanical modeling of the interfacial region in bonded half-planes. ASME J Appl Mech 1988;55:317–24. [6] Chen YF. Interface crack in nonhomogeneous bonded materials of finite thickness. PhD Dissertation, Lehigh University, 1990. [7] Ozturk M, Edrogen F. Axisymmetric crack problem in bonded materials with a graded interfacial region. Int J Solids Struct 1996;33:193–219. [8] Jin ZH, Batra RC. Interface cracking between functionally graded coating and a substrate under antiplane shear. Int J Eng Sci 1996;34:1705–16. [9] Bao G, Cai H. Delamination cracking in functionally graded coating/metal substrate systems. ACTA Mater 1997;45: 1055–66.
17
[10] Shbeeb NI, Binienda WK. Analysis of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness. Eng Fract Mech 1999;64: 693–720. [11] Marur PR, Tippur HV. Evaluation of mechanical properties of functionally graded materials. J Testing Evaluat 1998;26:539–45. [12] Butcher RJ, Rousseau CE, Tippur HV. A functionally graded particulate composite:preparation, measurements and failure analysis. Acta Mater 1999;47:259–68. [13] Rousseau CE, Tippur HV. Compositionally graded materials with cracks normal to the elastic gradient. Acta Mater 2000;48: 4021–33. [14] Gu P, Dao M, Asaro RJ. A simplified method for calculating the crack tip field of functionally graded materials using the domain integral. ASME J Appl Mech 1999;66:101–8. [15] Anlas G, Santare MH, Lambros J. Numerical calculation of stress intensity factors in functionally graded materials. Int J Fract 2000; 104:131–43. [16] Edrogen F, Wu HB. Crack problems in FGM layer under thermal stress. J Thermal Stress 1996;19:237–65. [17] Chen YF, Erdogan F. The interface crack problem for a nonhomogeneous coating bended to a homogeneous substrate. J Mech Phys Solids 1996;44(5):771–87. [18] Wang XD, Meguid SA. On the dynamic crack propagation in an interface with spatially varying elastic properties. Int J Fract 1994; 69:87–99. [19] Parameswaran V, Shukla A. Crack-tip stress fields for dynamic fracture in functionally graded materials. Mech Mater 1999;33: 3303–11. [20] Parameswaran V, Shukla A. Dynamic fracture of a functionally graded material having discrete property variation. J Mater Sci 1998;31:1641–57. [21] Meguid SA, Wang XD, Jiang LY. On the dynamic propagation of a finite crack in functionally graded materials. Eng Fract Mech 2002;69:1753–68. [22] Jiang LY, Wang XD. On the dynamic propagation in an interphase with spatially varying elastic properties under inplane loading. Int J Fract 2002;114:225–44. [23] Wang BL, Han JC, Du SY. Crack problem for non-homogeneous composite materials subjected to dynamic loading. Int J Solids Struct 2000;37:1251–74. [24] Chen J, Liu Z, Zou Z. Transient internal crack problem for a nonhomogeneous orthotropic strip (mode I). Int J Eng Sci 2002; 40(15):1761–74. [25] Chen J, Liu Z, Zou Z. Dynamic response of a crack in a functionally graded interface of two dissimilar piezoelectric halfplanes. Arch Appl Mech 2003;72(9):686–96. [26] Chen J, Liu Z, Zou Z. Electromechanical impact of a crack in a functionally graded piezoelectric medium. Theoret Appl Fract Mech 2003;39:47–60. [27] Morse PM, Feshbach H, editors. Methods of theoretical physics, vol. 1. New York: McGraw-Hill; 1958. [28] Itou S. Three dimensional waves propagation in a cracked elastic solid. ASME J Appl Mech 1978;45:807–11. [29] Itou S. Dynamic stress intensity factors around two parallel cracks in an infinite-orthotropic plane subjected to incident harmonic stress waves. Int J Solids Struct 1997;34(9):1145–65. [30] Gradshteyn IS, Ryzhik IM. Table of integral, series and products. New York: Academic Press; 1980. [31] Zhou ZG, Bai YY, Zhang XW. Two collinear Griffith cracks subjected to uniform tension in infinitely long strip. Int J Solids Struct 1999;36:5597–609. [32] Zhou ZG, Shen YP. Investigation of the scattering of harmonic shear waves by two collinear cracks using the non-local theory. Acta Mech 1999;135:169–79. [33] Sih GC. Elastodynamic crack problems, mechanics of fracture 4. Leyden: Noordhoff International Publishing; 1977.
Dynamic behavior of a finite crack in functionally graded materials subjected to plane incident time-harmonic stress wave Chun-Hui Xia b
a,*
, Li Ma
b
a Department of Chemistry, Qiqihar Medical College, Qiqihar 161042, PR China Center for Composite Materials, Harbin Institute of Technology, Harbin 150001, PR China
Available online 21 June 2005
Abstract In the present paper, the dynamic behavior of a finite crack in functionally graded materials subjected to normally incident elastic time harmonic waves is investigated by means of the Schmidt method. By use of the Fourier transform and defining the jumps of the displacement across the crack surfaces as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement across the crack surfaces are expanded in a series. Numerical examples are provided to show the effect of the gradient parameter ba, the crack configuration on the dynamic intensity factors of cracked functionally graded materials. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Crack; Stress waves scattering; Functionally graded materials; Dual integral equations
1. Introduction In recent years, functionally graded materials (FGMs) have widely been applied in extremely high temperature environments. The major advantages of the graded material, especially in elevated temperature environments, stem from the tailoring capability to produce a gradual variation of its thermomechanical properties in the spatial domain [1]. In particular, the use of the graded material as interlayers in the bonded media is one of the highly effective and promising applications in eliminating various shortcoming resulting from stepwise property mismatch inherent in piecewise homogeneous composite media [2–4]. From the fracture mechanics viewpoint, the presence of a graded interlayer would play an important role in determining the crack driving forces and fracture
*
Corresponding author. Tel.: +86 452 6710544. E-mail address: [email protected] (C.-H. Xia).
0263-8223/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.05.012
resistance parameters. In an attempt to address the issues pertaining to the fracture analysis of bonded media with such transitional interfacial properties, a series of solutions to certain crack problems was obtained. Among them are a crack in the non-homogeneous interlayer bounded by dissimilar homogeneous media [5]; and a crack at the interface between homogeneous and non-homogeneous materials [6,7]. Similar problems of delamination or an interface crack between the functionally graded coating and the substrate were considered in [8–10]. Relatively fewer experimental and numerical investigations of the fracture behavior of FGMs have been conducted. Experimental investigations into the fracture of FGMs are limited due to the high cost and elaborate facilities required for processing FGMs [11– 13]. The finite element method has also been used to simulate the fracture behavior of cracked FGMs [14,15]. The crack problem in FGM layers under thermal stresses was studied by Edrogen and Wu [16]. They considered an unconstrained elastic layer under statically self-equilibrating thermal or residual stresses. The
C.-H. Xia, L. Ma / Composite Structures 77 (2007) 10–17
layer contained an embedded or surface crack perpendicular to its boundaries. After giving the distribution of thermal stresses, the stress intensity factors for the embedded and surface crack were presented along with the results of the crack/contact problem in a FGM layer that was under compression near and at the surface and tension in the interior region. The interface crack problem for a non-homogeneous coating bended to a homogeneous substrate was investigated in [17]. Existing studies which consider the fracture behavior of FGMs are mainly limited to quasi-static problems. These studies are suitable for cases where the scattering of the elastic wave by the Mode-I crack or impact load are not involved. In this case, inertia effects do not play a role and can be ignored. However, it should be mentioned that most FGMs will be used in critical situations, where significant dynamic loading may be involved. The dynamic fracture behavior of FGMs has received little attention from the scientific community. Examples of dynamic analysis include the study of the steady state dynamic crack propagation in an interphase with spatially varying elastic properties under antiplane loading conditions reported in [18], and the steady state dynamic fracture of FGMs under in-plane loading with the material properties being assumed to vary along the direction of crack propagation reported in [19]. Experimental studies of the dynamic fracture of a FGMs with discrete property variation using photoelasticity technique were also conducted in [20]. The dynamic crack propagation problems were studied in [21,22]. The dynamic crack problem for the non-homogeneous composite materials was considered in [23] but they considered the FGM layer as multi-layered homogeneous media. The transient internal crack problem for a functionally graded strip and eletromechanical impact of a cracked functionally graded piezoelectric medium was investigated in [24–26]. In spite of these efforts, the understanding of the dynamic fracture process of FGMs is still limited. It is therefore the objective of the current study to provide a theoretical analysis of the dynamic behavior of a finite crack in functionally graded materials subjected to normally incident time harmonic elastic waves is investigated by means of the Schmidt method. It is assumed that the elastic properties of FGMs spatially vary along to the direction of the crack. YoungÕs modulus and mass density of the model are assumed to vary exponentially while PoissonÕs ratio remains constant. The analytical study is based on the use of Fourier transform technique and a somewhat different approach, name as the Schmidt method [27–29]. To solve the dual integral equations, the jumps of the displacements across crack surfaces are expanded in a series. Numerical examples are given to show the effects of the material properties upon the dynamic stress intensity factors (DSIF).
11
2. Formulation of the problem Consider the plane problem of diffraction of normally incident longitudinal wave by a Griffith crack situated in a functionally graded plane. The crack is assumed to occupy the region jxj 6 a, y = 0, jzj < 1. For the present problem, it is convenient to divide the functionally graded plane into two regions, namely the material 1 in upper half plane and the material 2 located in lower half plane as show in Fig. 1. The elastic parameters l(x) and q(x) are approximated by ðl; qÞ ¼ ðl0 ebx ; q0 ebx Þ
ð1Þ
where l is the shear modulus, q is the mass density and b is the gradient parameter and is taken to be a constant. By denoting u(j)(x, y, t) and v(j)(x, y, t) as the displacement components in the x- and y-directions, and ðjÞ ðjÞ rðjÞ y ðx; y; tÞ, rx ðx; y; tÞ, sxy ðx; y; tÞ (the superscript j = 1, 2 correspond to the upper half plane and the lower half plane through in this paper) as the stress components, respectively. The constitutive relations for the FGMs are written as l0 ebx ouðjÞ ovðjÞ rðjÞ ð1 þ jÞ þ ð3 jÞ ðx; y; tÞ ¼ ; ðj ¼ 1; 2Þ x j1 ox oy ð2Þ rðjÞ y ðx; y; tÞ ¼
bx
ðjÞ
l0 e ovðjÞ ou ð1 þ jÞ þ ð3 jÞ ; ðj ¼ 1; 2Þ j1 oy ox ð3Þ
ðjÞ ovðjÞ ðjÞ bx ou þ sxy ðx; y; tÞ ¼ l0 e ; ðj ¼ 1; 2Þ oy ox
ð4Þ
where t is the PoissonÕs ratio, and j = 3 4t for the state of plane strain, j = (3 t)/(1 + t) for the state of generalized plane stress. The PoissonÕs ratio t is taken to be a constant, owing to the fact its variation within a practical range has the rather insignificant influence on the value of the near-tip driving for fracture mechanics [5–7]. b 5 0 for the functionally graded materials. When b = 0 it will return to the homogenous material case.
y
1
μ ( x) = μ 0 eβ x –a
x
a 2
Fig. 1. Geometry of a finite crack in the functionally graded materials.
12
C.-H. Xia, L. Ma / Composite Structures 77 (2007) 10–17
Let a plane time harmonic elastic wave originating at y = 1 be incident normally on the crack, and is defined by u¼0
ð5Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ v0 expfi½xy=ð ð1 þ jÞ=ðj 1Þct Þ xtg
ð6Þ
where 0 is a ffi constant, x is the circular frequency and pvffiffiffiffiffiffiffiffiffiffiffi ct ¼ l0 =q0 . Substituting these relations into Eqs. (2)– (4) results in n h .pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi io 3j r0 exp i xy rx ¼ ð1 þ jÞ=ðj 1Þct xt 1þk ð7Þ n h .pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi io ð1 þ jÞ=ðj 1Þct x ry ¼ r0 exp i xy ð8Þ rxy ¼ 0
ð9Þ
where r0 ðxÞ ¼
1þj ixl0 ebx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v0 j 1 ð1 þ jÞ=j 1Þct
ð10Þ
In the absence of body forces, the elastic behavior of the medium with the variable shear modulus and the variable density in (1) is governed by following equations: 2 ðjÞ
2 ðjÞ
2 ðjÞ
ou ou ov þ ðj 1Þ þ2 ð1 þ jÞ ox2 oy 2 oxoy ðjÞ ðjÞ ou ov þ b ð1 þ jÞ þ ð3 jÞ ox oy 2 ðj 1Þq0 x ðjÞ ¼ u ; ðj ¼ 1; 2Þ l0
ð11Þ
o2 vðjÞ o2 vðjÞ o2 uðjÞ þ ðj 1Þ þ ðj 1Þ þ 2 ð1 þ jÞ oy 2 ox2 oxoy ðjÞ ou ovðjÞ ðj 1Þq0 x2 ðjÞ b þ v ; ðj ¼ 1; 2Þ ¼ oy ox l0 ð12Þ Because it is a common factor in all equations, the timefactor exp(ixt) is dropped out hereafter. It is further supposed that the two surfaces of the crack do not contact during vibrations. If only the stress intensity factors are considered, the mixed boundary value problem shown in Fig. 1 must be solved under the following conditions: ð2Þ rð1Þ y ðx; 0Þ ¼ ry ðx; 0Þ ¼ r0 ;
ð1Þ ð2Þ sxy ðx; 0Þ ¼ sxy ðx; 0Þ ¼ 0;
jxj 6 a rð1Þ y ðx; 0Þ
ð13Þ ¼
ryð2Þ ðx; 0Þ;
sð1Þ xy ðx; 0Þ
¼
sð2Þ xy ðx; 0Þ;
jxj > a ð14Þ
ð1Þ
ð2Þ
u ðx; 0Þ ¼ u ðx; 0Þ;
ð1Þ
ð2Þ
v ðx; 0Þ ¼ v ðx; 0Þ;
jxj > a ð15Þ
3. Solution The system of above governing equations is solved using the Fourier integral transform technique to obtain the general expressions for the displacement components as 8 2 R1 P > ð1Þ > 1 > Aj ðsÞekj y eisx ds u ¼ 2p > 1 < j¼1 ð16Þ 2 > R P > 1 1 ð1Þ k y isx >v ¼ > mj ðsÞAj ðsÞe j e ds : 2p 1 j¼1 8 4 R1 P > > 1 > uð2Þ ¼ 2p Aj ðsÞekj y eisx ds > 1 < j¼3 ð17Þ 4 > > > vð2Þ ¼ 1 R 1 P mj ðsÞAj ðsÞekj y eisx ds > : 2p 1 j¼3
and from (2)–(4), the stress components are obtained as 8 2 R1 P > l0 ebx ð1Þ > r ðx; yÞ ¼ ½ðj þ 1Þmj ðsÞkj isð3 jÞ > y 2pðj1Þ 1 > > j¼1 > < Aj ðsÞekj y eisx ds > > > 2 R1 P > ð1Þ > l ebx > : sxy ðx; yÞ ¼ 02p 1 ½kj imj ðsÞsAj ðsÞekj y eisx ds j¼1
ð18Þ 8 4 R 1 P > l0 ebx > ½ðj þ 1Þmj ðsÞkj isð3 jÞ > ryð2Þ ðx; yÞ ¼ 2pðj1Þ 1 > > j¼3 > < Aj ðsÞekj y eisx ds > > > 4 > bx R 1 P > ð2Þ > ðx; yÞ ¼ l02pe 1 ½kj imj ðsÞsAj ðsÞekj y eisx ds : sxy j¼3
ð19Þ where s is the transform variable. A1, A2, A3 and A4 are unknowns, kj(s), j = 1, 2, 3 and 4 are the roots of the characteristic equation k4 D1 k2 þ D2 ¼ 0 where 3j 2 2j x2 b D1 ¼ 2s2 þ 2isb þ jþ1 j þ 1 c2t h i 2 ½sðs þ ibÞ c2t ð1 þ jÞsðs þ ibÞ ðj 1Þ xc2 t D2 ¼ ð1 þ jÞ Then, the root of Eq. (20) may be obtained as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D1 1 k1 ¼ D21 4D2 þ 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D1 1 D21 4D2 k2 ¼ 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D1 1 D21 4D2 þ k3 ¼ 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D1 1 D21 4D2 k4 ¼ 2 2
ð20Þ ð21Þ
ð22Þ
ð23Þ ð24Þ ð25Þ ð26Þ
C.-H. Xia, L. Ma / Composite Structures 77 (2007) 10–17
and mj(s), j = 1, 2, 3, and 4 are expressed for each root kj (s) as 2
mj ðsÞ ¼
ð1 þ jÞs2 þ ðj 1Þk2j isbð1 þ jÞ þ ðj 1Þ xc2 t
kj ½2is þ bð3 jÞ
Z
f1 ðxÞ ¼ uð1Þ ðx; 0Þ uð2Þ ðx; 0Þ
ð28Þ
f2 ðxÞ ¼ vð1Þ ðx; 0Þ vð2Þ ðx; 0Þ
ð29Þ
Applying the Fourier transforms and the boundary conditions (13)–(15), it can be obtained A1 ðsÞ A3 ðsÞ ½X 1 ¼ ½X 2 ð30Þ A2 ðsÞ A4 ðsÞ A1 ðsÞ A3 ðsÞ f 1 ðsÞ ½X 3 ½X 4 ¼ ð31Þ A2 ðsÞ A4 ðsÞ f 2 ðsÞ where " ½X 1 ¼ " ½X 1 ¼ ½X 3 ¼
l0 ½ðjþ1Þm1 ðsÞk1 isð3jÞ j1
l0 ½ðjþ1Þm2 ðsÞk2 isð3jÞ j1
l0 ½k1 im1 ðsÞs
l0 ½k2 im2 ðsÞs
l0 ½ðjþ1Þm3 ðsÞk3 isð3jÞ j1
l0 ½ðjþ1Þm4 ðsÞk4 isð3jÞ j1
l0 ½k3 im3 ðsÞs
l0 ½k4 im4 ðsÞs
1
1
m1 ðsÞ m2 ðsÞ 1 1 ½X 4 ¼ m3 ðsÞ m4 ðsÞ
#
#
½d 3 ðsÞf 1 ðsÞ þ d 4 ðsÞf 2 ðsÞeisx ds ¼ 0;
1
ð39Þ
f 2 ðsÞeisx ds ¼ 0;
jxj > a
ð40Þ
where d1(s), d2(s), d3(s) and d4(s) are known functions, and can be expressed as follows: ½X 5 ¼ ½X 3 ½X 4 ½X 2 1 ½X 1
d 1 ðsÞ
d 2 ðsÞ
d 3 ðSÞ
d 4 ðsÞ
¼ ½X 1 ½X 5
1
ð41Þ ð42Þ
To determine the unknown functions f 1 ðsÞ and f 2 ðsÞ, the above two pairs of dual integral Eqs. (37)–(40) must be solved.
4. Solution of the dual integral equation From the nature of the displacement along the crack line, it can be obtained that the jumps of the displacement across the crack surface are finite, differentiability and continuum functions. So the jumps of the displacement across the crack surface can be expanded by the following series [28]: f1 ðxÞ ¼
1 h x i 1X an sin 2nsin1 for 0 6 jxj 6 a p n¼1 a
ð43Þ
ð32Þ
ð44Þ f1 ðxÞ ¼ 0 jxj > a 1 h i X 1 x f2 ðxÞ ¼ bn sin ð2n 1Þsin1 0 6 jxj 6 a p n¼1 a
ð33Þ
ð45Þ
ð34Þ
ð35Þ
A superposed bar indicates the Fourier transform through the paper. And defined as follows: Z 1 Z 1 1 f ðxÞeisx dx; f ðxÞ ¼ f ðsÞeisx ds f ðsÞ ¼ 2p 1 1 ð36Þ By solving four Eqs. (24) and (25) with four unknown functions, substituting the solutions into Eq. (18) and applying the boundary conditions, it can be obtained Z ebx 1 rð1Þ ðx; 0Þ ¼ ½d 1 ðsÞf 1 ðsÞ þ d 2 ðsÞf 2 ðsÞeisx ds y 2p 1 ¼ r0 ðxÞ 0 6 jxj 6 a ð37Þ Z 1 ð38Þ f 1 ðsÞeisx ds ¼ 0; jxj > a 1
1
1
1
ð27Þ From Eqs. (16)–(19), it can be seen that there are four unknown constants (in Fourier space they are functions of s), i e., A1, A2, A3 and A4 which can be obtained from the boundary conditions. To solve the present problem, the jumps of the displacement across the crack surfaces can be defined as follows:
Z
ebx 2p 0 6 jxj 6 a
sð1Þ xy ðx; 0Þ ¼
13
f2 ðxÞ ¼ 0 jxj > a
ð46Þ
where an and bn are unknown coefficients. The Fourier transform of (43)–(46) is [29] f 1 ðsÞ ¼ i f 2 ðsÞ ¼
1 X
n¼1 1 X
an
bn
n¼1
2n J 2n ðsaÞ s
2n 1 J 2n1 ðsaÞ s
ð47Þ ð48Þ
Substituting Eqs. (47) and (48) into Eqs. (37)–(40), it can be shown that Eqs. (38) and (40) are automatically satisfied. After integration with respect to x in [a, x], Eqs. (37) and (39) reduce to 1 Z 1 1 X i ½i2nd 1 ðsÞan J 2n ðsaÞ 2p n¼1 1 s2 þ ð2n 1Þd 2 ðsÞbn J 2n1 ðsaÞðeisx eisa Þ ds Z x ¼ r0 ðxÞebx dx; 0 6 jxj 6 a a
ð49Þ
14
C.-H. Xia, L. Ma / Composite Structures 77 (2007) 10–17
1 Z 1 1 X i ½i2nd 3 ðsÞan J 2n ðsaÞ þ ð2n 1Þd 4 ðsÞ 2p n¼1 1 s2
bn J 2n1 ðsaÞðeisx eisa Þ ds ¼ 0; Z
0 6 jxj 6 a
ð50Þ
From the relationships [30] 1
1 J n ðsaÞ sinðbsÞ ds s 0 8 > sin½n sin1 ðb=aÞ > > ; a>b < n ¼ an sinðnp=2Þ > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ; b > a > : n½b þ b2 a2 Z 1 1 J n ðsaÞ cosðbsÞ ds s 0 8 cos½n sin1 ðb=aÞ > > ; a>b < n n ¼ a cosðnp=2Þ > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; b > a : n½b þ b2 a2 n
ð51Þ
where lims!±1 d1(s)/s = d1, lims!+1d2(s)/s = lims!1 d2(s)/s = d2, lims!+1d3(s)/s = lims!1d3(s)/s = d3, lims!!±1d4(s)/s = d4. These constants can be obtained by using the Mathematica program and are independent of the gradient parameter b. Also, these constants equal to the ones of the homogeneous material case. The semi-infinite integral in (51)–(56) can be evaluated directly, so Eqs. (49) and (50) can now be solved for the coefficients an and bn by the Schmidt method [26,27] as described in Appendix A.
5. Dynamic stress intensity factors
ð52Þ
The semi-infinite integral in Eqs. (49) and (50) can be modified as Z 1 h i d 1 ðsÞ 2id1 1 x isx isa sin 2nsin J ðsaÞðe e Þds ¼ 2n 2 a 2n 1 s Z 1 1 d 1 ðsÞ d1 J 2n ðsaÞðeisx eisa Þds þ ð53Þ s 1 s Z 1 d 2 ðsÞ J 2n1 ðsaÞðeisx eisa Þds 2 1 s h x i 2id2 n 2id2 ¼ sin ð2n 1Þsin1 þ ð1Þ a 2n 1 2n 1 Z 0 1 d 2 ðsÞ þ d2 J 2n1 ðsaÞðeisx eisa Þds þ s s 1 Z 1 1 d 2 ðsÞ d2 J 2n1 ðsaÞðeisx eisa Þds þ ð54Þ s s 0 Z 1 d 3 ðsÞ J 2n ðsaÞðeisx eisa Þds 2 1 s h x i 2d3 n 2d3 cos 2nsin1 þ ð1Þ ¼ a 2n 2n Z 0 1 d 3 ðsÞ þ d3 J 2n ðsaÞðeisx eisa Þds þ s 1 s Z 1 1 d 3 ðsÞ d3 J 2n ðsaÞðeisx eisa Þds þ ð55Þ s s 0 Z 1 d 4 ðsÞ J 2n1 ðsaÞðeisx eisa Þds 2 s 1 h x i 2d4 cos ð2n 1Þsin1 ¼ a 2n 1 Z 1 1 d 4 ðsÞ d4 J 2n1 ðsaÞðeisx eisa Þds ð56Þ þ s 1 s
The coefficients an and bn are known, so that the entire stress field can be obtained. However, in fracture mechanics, it is important to determine stress ryð1Þ in the vicinity of the crack tips. The stress components ryð1Þ along the crack line can be expressed as 1 Z 1 ebx X 2n ð1Þ ian d 1 ðsÞ J 2n ðsaÞ ry ðx; 0Þ ¼ s 2p n¼1 1 2n 1 þ bn d 2 ðsÞ J 2n1 ðsaÞ eisx ds s Z 1 1 ebx X d 1 ðsÞ d1 J 2n ðsaÞeisx ds ¼ ian 2n s 2p n¼1 1 Z 1 d 2 ðsÞ þ bn ð2n 1Þ d2 J 2n1 ðsaÞeisx ds s 0 Z 0 d 2 ðsÞ þ d2 J 2n1 ðsaÞeisx ds þ bn ð2n 1Þ s 1 Z 1 þ d1 an i2n J 2n ðsaÞeisx ds 1
þ d2 bn ð2n 1Þ
Z
1
J 2n1 ðsaÞeisx ds
0
d2 bn ð2n 1Þ
Z
0
J 2n1 ðsaÞeisx ds
ð57Þ
1
An examination of Eq. (57) shows that, the singular part of the stress field can be obtained from the relationships as follows [30]: Z
1
J n ðsaÞ cosðbsÞ ds
0
¼
8 cos½n sin1 ðb=aÞ > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; > > < a2 b2
a>b
> an sinðnp=2Þ > > p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiin ; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffih > > : b2 a 2 b þ b 2 a 2
ð58Þ b>a
C.-H. Xia, L. Ma / Composite Structures 77 (2007) 10–17 1
J n ðsaÞ sinðbsÞ ds 0 8 sin½n sin1 ðb=aÞ > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; a>b < 2 b2 a ¼ > an cosðnp=2Þ > > ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ; : pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b a2 ½b þ b2 a2 Z 1 J nþ1 ðsaÞeisx ds ¼ 0; jxj > a
ð59Þ b>a ð60Þ
1
For a < x, the singular part of the stress field can be expressed respectively as follows: Z 1 1 d2 ebx X r¼ 2bn ð2n 1Þ J 2n1 ðsaÞ cosðsxÞ ds 2p n¼1 0 1 d2 ebx X a2n1 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2n1 ¼ bn ð2n 1Þð1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi p n¼1 x2 a2 ½x þ x2 a2 ð61Þ For x < a, the singular part of the stress field can be expressed respectively as follows: Z 1 1 d2 ebx X 2bn ð2n 1Þ J 2n1 ðsaÞ cosðsxÞ ds r¼ 2p n¼1 0 1 d2 ebx X a2n1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ bn ð2n 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi p n¼1 x2 a2 ½jxj þ x2 a2 2n1 ð62Þ
namic stress intensity factor (DSIF) as K ¼ K I =K 0 , pffiffiffi where K 0 ¼ r0 a. To verify the validity of the present work, the variations of the normalized dynamic stress intensity factors with the normalized wave number xa/ct for a small gradient parameter ba, namely, ba = 0.0001 are displayed in Fig. 2. This case returned to the homogeneous isotropic case. These curve are very similar to those given by Sih in [33] for isotropic material. From this result, it can be seen that our analysis process is reasonable and effectively. The aim of the present paper is to give an approach to solve the dynamic behavior of a crack in the functionally graded materials subjected to the normally harmonic stress waves. In the present paper, the unknown variables of dual integral equations are the displacement across the crack surfaces. The solution can be returned to the same problem as the static problem for x = 0. The variations of the normalized DSIF with the normalized wave number xa/ct for different value of ba are
2.6 2.4 2.2
K/K0
Z
The stress intensity factors KI at the right tip of the crack can be given as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K IR ¼ limþ 2pðx aÞr d2 e ¼ pffiffiffiffiffiffi pa
1 X
n
ð1Þ bn ð2n 1Þ
x!a
KIL
1.8
KIR
1.6
1.2 1.0 0.8
ð63Þ
0.0
0.2
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 eba X 2pðjxj aÞr ¼ pffiffiffiffiffiffi bn ð2n 1Þ pa n¼0
0.4
0.6
0.8
1.0
ω a/ct
n¼1
The stress intensity factors KI at the left tip of the crack can be given as follows: K IL ¼ lim
2.0
1.4
x!a
ba
15
Fig. 2. Variations of normal stress intensity factor with normal wave number at ba = 0.0001.
ð64Þ 3.5
KIL
3.0
KIR
6. Numerical calculations and discussion K/K0
Numerical calculations are carried out for functionally graded materials. In the present work, we only consider the dynamic behavior of cracked functionally graded materials for plane stain condition. As discussed in the works [26,27,31,32], it can be seen that the Schmidt method is performed satisfactorily if the first ten terms of the infinite series to Eqs. (A.1) and (A.2) are retained.The behavior of the sum of the series stays steady with the increasing number of terms in (A.1) and (A.2). For convenience, we define the normalized dy-
2.5 2.0 1.5 1.0 0.5 0.0
0.5
1.0
1.5
2.0
ω a/ct
Fig. 3. Variations of normal stress intensity factor with normal wave number at ba = 0.1.
16
C.-H. Xia, L. Ma / Composite Structures 77 (2007) 10–17
the materials and frequency of the incident wave on dynamic stress intensity factors are investigated.
3.0 2.4
K/K0
KIL 1.8
Appendix A
KIR
For convenience, Eqs. (44) and (45) can be rewritten as 1 1 X X an En ðxÞ þ bn F n ðxÞ ¼ U 0 ðxÞ; a 6 x 6 a ðA:1Þ
1.2 0.6 0.0 0.0
0.3
0.6
0.9
1.2
1.5
1.8
ω a/ct
n¼1
n¼1
1 X
1 X
an Gn ðxÞ þ
n¼1
Fig. 4. Variations of normal stress intensity factor with normal wave number at ba = 0.5.
bn H n ðxÞ ¼ V 0 ðxÞ; a 6 x 6 a
A set of functions Pn(y) that satisfy the orthogonality condition Z a Z a P m ðxÞP n ðxÞ dx ¼ N n dmn N n ¼ P 2n ðxÞ dx ðA:3Þ a
K/K0
2.0
KIL KIR
1.0 0.5 0.0 0.0
0.2
0.4
0.6
0.8 1.0 ω a/ct
a
can be constructed from a given set of arbitrary functions, say Gn(x), such that n X M in Gi ðxÞ ðA:4Þ P n ðxÞ ¼ M nn i¼1
2.5
1.5
ðA:2Þ
n¼1
1.2
1.4
1.6
Fig. 5. Variations of normal stress intensity factor with normal wave number at ba = 1.5.
plotted in Figs. 3–5, where the broken straight lines indicate the corresponding static values. It is apparent from the figures that the predominant term of the dynamic stress intensity factor of crack tip in the stiffer side is greater than that in the softer side. And the peak values of DSIF are considerably larger than the corresponding static values.
7. Conclusions A theoretical and numerical treatment of a finite crack subjected to an in-plane incident harmonic stress wave in a functionally graded plane is presented. The analysis is based upon an integral transform technique. The Fredholm integral equation is solved by using the Schmidt method. The present method is applied to illustrate the fundamental behavior of a crack in FGMs under the dynamic loading. Furthermore, the effects of the geometry of the crack, the shear stress wave velocity of
where Min defined as d 11 d 21 Dn ¼ d n1
is the cofactor of element din of Dn, which is d 12 d 22
d 1n d 2n ; d nn
d in ¼
Z
a
Gi ðxÞGn ðxÞ dx a
ðA:5Þ Representing the second series in Eq. (A.2) by the orthogonal series Pn(x) with coefficient cn, the following relations can be given: 1 X
an Gn ðxÞ ¼
n¼1
1 X
cn P n ðxÞ ¼ V 0 ðxÞ
n¼1
1 X
bn H n ðxÞ
ðA:6Þ
n¼1
The second equality yields # Z a" 1 X 1 cn ¼ V 0 ðxÞ bi H i ðxÞ P n ðxÞ dx N n a i¼0
ðA:7Þ
The first equality becomes an ¼
1 X
cni bi þ dn
ðA:8Þ
i¼1
with Z a 1 X M nj H i ðxÞP j ðxÞ dx N j M jj a j¼n Z a 1 X M nj V 0 ðxÞP j ðxÞ dx dn ¼ N j M jj a j¼n
cni ¼
ðA:9Þ ðA:10Þ
C.-H. Xia, L. Ma / Composite Structures 77 (2007) 10–17
Substituting Eq. (A.8) into Eq. (A.1) reduce to 1 X
bn Y n ðxÞ ¼ W ðxÞ
ðA:11Þ
n¼1
with Y n ðxÞ ¼ F n ðxÞ þ W ðxÞ ¼ U 0 ðxÞ
1 X i¼1 1 X
cin Ei ðxÞ
ðA:12Þ
di Ei ðxÞ
ðA:13Þ
i¼1
Finally, coefficients bn can be determined by bn ¼
1 X j¼n
qj ¼
1 Kj
qj
Z
Lnj Ljj
ðA:14Þ 1 X Lij Y i ðxÞ Ljj i¼1
a
W ðxÞQj ðxÞ dx;
Qj ðxÞ ¼
a
ðA:15Þ where Ljn is the minant Dn e11 e12 e21 e22 Dn ¼ en1 en2
cofactor of the element ejn of the deter e1n e2n ; enn
ejn ¼
Z
a
Y j ðxÞY n ðxÞ dx
a
ðA:16Þ Coefficients an are calculated by using Eq. (A.8).
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