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Dynamic stress intensity factors of multiple cracks in an orthotropic strip with FGM coating
Engineering Fracture Mechanics 109 (2013) 45–57
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Dynamic stress intensity factors of multiple cracks in an orthotropic strip with FGM coating M.M. Monfared, M. Ayatollahi ⇑ Faculty of Engineering, University of Zanjan, P.O. Box 45195-313, Zanjan, Iran
a r t i c l e
i n f o
Article history: Received 16 April 2012 Received in revised form 23 May 2013 Accepted 2 July 2013 Available online 12 July 2013 Keywords: Dynamic stress intensity factors Functionally graded coating Orthotropic strip Multiple cracks
a b s t r a c t The problem of determining the dynamic stress intensity factors (DSIFs) in an orthotropic strip with functionally graded materials coating weakened by multiple arbitrary cracks under time-harmonic excitation are investigated. Using Fourier transform, the problem is to the solution of integral equations of the Cauchy singular type. The integral equations were solved by Erdogan’s collocation method and dynamic stress intensity factors were obtained. The proposed method of solution permits an examination of the multiple curved cracks problem with arbitrary patterns. Several examples are solved to obtain the dynamic stress intensity factors with various angular frequency, crack lengths and material properties. A knowledge of dynamic stress intensity factor is essential to the clear understanding of the propagation of cracks through structural components undergoing oscillation. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction With the application of functionally graded materials in engineering, the fracture analysis concerning inter facial zone has absorbed the attention of more and more researchers. Functionally graded materials (FGMs) are favorable coatings for structures exposed to harsh environments and severe thermal shocks. Reliable design of mechanical components necessitates their accurate stress analysis, avoiding unrealistic physical assumptions such as perfect bonding in bodies comprised of dissimilar materials. Most of the current researches on the fracture analysis of the FGMs interface have been devoted to the FGMs interlayer [1–3] and the interface between the FGM coating and the homogeneous substrate [4–8]. The dynamic problem for a finite crack was first investigated by Loeber and Sih [9]. They obtained the stress intensity factor around a crack during the passage of a time-harmonic anti-plane shear wave. Keer and Luong [10] studied the diffraction of anti-plane shear waves by a crack in a layered composite. The problem of a crack perpendicular to the direction of interface of two half-plane of different FGMs was obtained by Erdogan [11]. Li and Tai [12] presented a dynamic fracture analysis of two isotropic dissimilar strips between isotropic dissimilar half-spaces with an interfacial crack subjected to the anti-plane shear impact loading. Two bonded isotropic half-planes with non-homogeneous interface, was analyzed by Erdogan et al. [13]. Dynamic anti-plane analysis of a medium composed of a FGM coating and an orthotropic half-plane substrate where the FGM layer was weakened by a periodic array of cracks perpendicular to the boundary was carried out by Wang and Mai [14]. The stress intensity factor and strain energy release rate for an interface crack between an elastic halfplane and a non-homogenous strip whose other edge was bonded to another elastic half-plane were obtained by Ozturk and Erdogan [15]. Daros [16] solved the dynamic problem of time harmonic SH waves in a special class of anisotropic inhomogeneous materials. The influences of the inhomogeneities on the stress intensity factor of crack were presented. Babaei and Lukasiewicz [17] obtained the dynamic stress intensity factor (DSIF) for the crack in a non-homogeneous layer between ⇑ Corresponding author. Tel.: +98 241 515 2488; fax: +98 241 515 2762. E-mail address: [email protected] (M. Ayatollahi). 0013-7944/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2013.07.002
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Nomenclature a, b lengths of major and minor semi-axes of elliptical crack bz Burgers vector Bzj complex dislocation densities A(s), B(s), C1(s), D1(s), C2(s), D2(s) unknowns coefficients c, cx shears wave velocity gij(p) regular terms of dislocation densities h1, h2 thickness orthotropic and FGM layers, respectively H(x) Heaviside step function Hij coefficients matrix kIII Li, kIII Ri stress intensity factors of left and right side of crack k0 stress intensity factor of a crack in infinite plane kij(s, p) kernel of integral equations kT, kP wave numbers l half lengths of straight crack N total number of cracks rLi, rRi distance from right and left crack tips W out of plane displacement component w amplitude of displacement component x, y coordinates ai(s), ci(s) functions describing the geometry of cracks g, f dislocation coordinates k FGM exponent d(k) Dirac delta function dij Kronecker delta l shear moduli in FGM medium hi(s) crack orientation Gzx, Gzy orthotropic shear moduli of elasticity of material q1, q2 mass densities in orthotropic and FGM region, respectively rnz traction vector rzx, rzy out of plane stress components x angular frequency Acronyms DSIF FGM
dynamic stress intensity factors functionally graded materials
isotropic half-spaces under the anti-plane shear impact load. The scattering problem of elastic wave by two collinear cracks in a functionally graded interlayer bonded to dissimilar half-plane under anti-plane shear were studied by Ma et al. [18]. The stress analysis in a non-homogeneous interfacial zone between two dissimilar elastic solids was accomplished by Wang et al. [19]. Li [20] carried out an analysis for interfacial crack between two bonded dissimilar semi-infinite orthotropic strips where the crack surface was under anti-plane traction. Closed form stress intensity factors were obtained for a strip with either clamped or traction-free boundaries. The problem of cracked multi-layered and FGM coatings of a coating-substrate composite under normal traction on the crack surface was investigated by Chi and Chung [21]. A multi-layered model to the crack problem of functionally graded materials under plane stress state deformation with arbitrarily varying Young’s modulus were investigated by Huang et al. [22]. The problem of arbitrarily oriented crack in a FGM layer bonded to a homogeneous half-plane was solved by Long and Delale [23]. The effect of crack length, crack orientation and the non-homogeneity parameter of the strip on the fracture behavior of the FGM layer were presented. A crack at the interface of a FGM and a rigid half-plane was analyzed by Erdogan and Ozturk [24]. Anti-plane problem of periodic cracks in a functionally graded coatingsubstrate structure was studied by Chen [25]. The dynamic stress intensity factor in an orthotropic strip weakened by multiple cracks under point forces loading were investigated by Monfared et al. [26]. Lekesiz et al. [27] studied effective spring stiffness for a planer periodic array of collinear cracks at interface between two dissimilar materials. The effects of elastic dissimilarity, crack density and crack interaction on the effective spring stiffness was presented. In the present paper, the aim is to find the dynamic stress intensity factors of multiple cracks with arbitrary patterns in an orthotropic strip reinforced with FGM subjected to oscillating anti-plane loading. The technique necessities the solution of screw Volterra dislocation in the region. The complex Fourier transform is employed to obtain transformed displacement and stress fields. The dislocation solution and Buckner’s principle are employed to derive integral equations for dislocation density function in an orthotropic strip with FGM coating weakened by multiple arbitrary cracks. The integral equations
M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
47
are of Cauchy singular types which are solved numerically for the dislocation density on the cracks faces. Several examples of cracks are solved to illustrate the applicability of the procedure. 2. Dislocation solution A layer of FGM with thickness h1 is used to reinforce an orthotropic homogenous layer with thickness h2 as depicted in Fig. 1. In crack problems a so called ‘‘distributed dislocation technique’’ is often used in treating cracks with smooth geometries Hills et al. [28]. We must first calculate the stress fields due to a single dislocation in the region. We now take up this task for an orthotropic strip with functionally graded materials coating containing a screw dislocation that located in the orthotropic layer under time-harmonic excitation. For a medium under anti-plane deformation, the only nonzero displacement component is the out of plane component W(x, y, t). Hence, of all the stress components, only rzx and r zy are different from zero
rzx ðx; y; tÞ ¼ lðyÞ @W ; rzy ðx; y; tÞ ¼ lðyÞ @W 0 6 y 6 h1 ; @x @y @W rzx ðx; y; tÞ ¼ Gzx @W ; r ðx; y; tÞ ¼ G ; h zy zy @y 2 6 y 6 0; @x
ð1Þ
where l(y) is the shear modulus in the FGM layer, Gzx and Gzy are the shear moduli of elasticity of the orthotropic strip in the x- and y-directions, respectively. Substituting Eq. (1) into the equilibrium equation, we obtain the expressions 2
2
2
lðyÞ @@yW2 þ l0 ðyÞ @W þ lðyÞ @@xW2 ¼ qðyÞ @@tW 0 6 y 6 h1 ; 2 @y 2
2
2
Gzy @@yW2 þ Gzx @@xW2 ¼ q1 @@tW 2
h2 6 y 6 0;
ð2Þ
where q1 and q(y) are the materials mass density in the orthotropic and FGM layers, respectively. Since the problem has steady-state time dependence, the displacement and stresses may be taken to vary harmonically in time as
Wðx; y; tÞ ¼ wðx; yÞeixt :
ð3Þ
pffiffiffiffiffiffiffi In Eq. (3) i ¼ 1; x is the angular frequency of motion and w(x, y) is an unknown function. In terms of the function w(x, y), Eqs. (2) become @2 w @y2
0
ðyÞ þ llðyÞ
Gzy @ 2 w Gzx @y2
@w @y
2
x þ @@xw2 þ qðyÞ lðyÞ w ¼ 0 0 6 y 6 h1 ; 2
2
þ @@xw2 þ xGzxq1 w ¼ 0 2
h2 6 y 6 0:
ð4Þ
For the sake of convenience, the time factor exp(ixt) will be suppressed and only w(x, y) will be considered. To facilitate the solution of the above differential equation, we will focus on special class of FGM in which the variations of material properties have the same material gradient parameter which based on the reasonable and generally accepted assumption. Therefore we assume
lðyÞ ¼ le2ky ; qðyÞ ¼ q2 e2ky ;
ð5Þ
where l, q2 and k are material constants of FGM. Substituting Eqs. (5) into Eq. (4), the equation of motion can be rewritten as @2 w @y2
2
2
þ 2k @w þ @@xw2 þ xc2 w ¼ 0 0 6 y 6 h1 ; @y
1 @2 w f 2 @y2
2
2
þ @@xw2 þ xc2 w ¼ 0 x
h2 6 y 6 0:
ð6Þ
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In the above equation f ¼ Gzx =Gzy is the ratio of shear moduli of the elasticity of the orthotropic medium. c ¼ l=q2 and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cx ¼ Gzx =q1 are the represents the velocity of the anti-plane shear wave propagating along FGM and orthotropic layers, respectively. The traction-free condition on the strip boundaries implies that
rzy ðx; h1 Þ ¼ 0; rzy ðx; h2 Þ ¼ 0; jxj < 1:
Fig. 1. Schematic of the medium with screw dislocation.
ð7Þ
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M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
Let a Volterra type screw dislocation with Burgers vector bz be situated in the medium at a point with coordinates (g, f). The dislocation line chosen in the orthotropic layer with the positive direction of x-axis depicted in Fig. 1. The conditions representing the dislocation under anti-plane deformation are
wðx; fþ Þ wðx; f Þ ¼ bz Hðx gÞ;
ð8Þ
rzy ðx; fþ Þ ¼ rzy ðx; f Þ; jxj < 1;
where H( ) is the Heaviside step function. The first of Eqs. (8) enforces the multivaluedness of displacement while the second implies the continuity of traction along the dislocation line (self-equilibrium of stress). The continuity conditions at the interface of two materials imply that
wðx; 0þ Þ ¼ wðx; 0 Þ;
ð9Þ
rzy ðx; 0þ Þ ¼ rzy ðx; 0 Þ; jxj < 1:
The solution to the current dynamic problem is obtained by means of the following complex Fourier transformation defined by
F ðsÞ ¼
Z
1
f ðxÞeisx dx:
ð10Þ
1
The inversion of complex Fourier transformation yields
f ðxÞ ¼
1 2p
Z
1
F ðsÞeisx ds:
ð11Þ
1
The application of Eq. (10) to Eqs. (6) with the aid of integrations by parts and in conjunction with the property of decaying behavior of displacement and stress components as jxj ? 1 leads to a second order ordinary differential equation, in each region as
2 þ 2k dw s2 kP w ¼ 0; 0 6 y 6 h1 ; dy 2 d2 w s2 kT w ¼ 0; h2 6 y 6 0: dy2
d2 w dy2 1 f2
ð12Þ
The wave number kP is equal to the circular frequency x divided by the shear-wave velocity c, i.e., kP = x/c and kT = x/cx are the wave numbers. The solution to Eqs. (12) yields
w ðs; yÞ ¼ AðsÞeðbkÞy þ BðsÞeðbþkÞy ;
0 6 y 6 h1 ;
w ðs; yÞ ¼ C 1 ðsÞ sinhðf ayÞ þ D1 ðsÞ coshðf ayÞ; f 6 y 6 0;
ð13Þ
w ðs; yÞ ¼ C 2 ðsÞ sinhðf ayÞ þ D2 ðsÞ coshðf ayÞ; h2 6 y 6 f: A, B, C1, D1, C2 and D2 are undetermined function of the transform variable s, while the function a and b are given by
1=2 2 ¼ i kT s2 ; 1=2 1=2 2 2 ¼ i kP s2 k2 : b ¼ s2 þ k2 kP
a ¼ s2 k2T
1=2
ð14Þ
The Fourier transform of (7)–(9) by virtue of (10) yields dw ðs;h1 Þ dy
¼ 0;
dw ðs;h2 Þ dy þ
¼ 0;
w ðs; f Þ w ðs; f Þ ¼ bz ðpdðsÞ i=sÞeisg ; ðs; fþ Þ ¼ dw ðs; f Þ; dy
dw dy
ð15Þ
w ðs; 0þ Þ ¼ w ðs; 0 Þ;
l dw ðs; 0þ Þ ¼ Gzy dw ðs; 0þ Þ; dy dy where d(s) is the Dirac delta function. Application conditions (15) to Eqs. (13) yields the algebraic relation ( f aGzy bz ðpdðsÞ i=sÞeisg sinhðf aðf þ h2 ÞÞ ðb þ kÞebh1 ; 2EðsÞ ðb kÞebh1 ; ( 2 C 1 ðsÞ bz ðpdðsÞ i=sÞ sinhðf aðf þ h2 ÞÞeisg l s2 kP sinhðbh1 Þ; ¼ 2EðsÞ D1 ðsÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ; h i 2 2 sinhðbh b ð p dðsÞ i=sÞ l s k Þ coshðf afÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf afÞ eisg sinhðf ah2 Þ; z 1 P C 2 ðsÞ ¼ EðsÞ D2 ðsÞ coshðf ah2 Þ;
AðsÞ BðsÞ
¼
ð16Þ
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M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
where
2 EðsÞ ¼ l s2 kP sinhðbh1 Þ coshðf ah2 Þ þ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf ah2 Þ.
Substituting
Eq.
(16)
into
Eq. (13), and taking the inverse of Fourier transform (11), the displacements fields yields
wðx; yÞ ¼
sinðfkT ðf þ h2 ÞÞ½c coshðcðy h1 ÞÞ þ k sinhðcðy h1 ÞÞ lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z ifGzy bz eky 1 a sinhðf aðf þ h2 ÞÞ ½b coshðbðy h1 ÞÞ þ k sinhðbðy h1 ÞÞeisðxgÞ ds; 0 6 y 6 h1 ; sEðsÞ 2p 1 bz fkT Gzy eky 2
h i 2 bz sinðfkT ðf þ h2 ÞÞ lkP sinhðch1 Þ sinðfykT Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ cosðfykT Þ wðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z 1 i ibz sinhðf aðf þ h2 ÞÞ h 2 þ l s k2P sinhðbh1 Þ sinhðf ayÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ coshðf ayÞ eisðxgÞ ds; f 6 y 6 0; sEðsÞ 2p 1 h i 2 bz cosðfkT ðy þ h2 ÞÞ lkP sinhðch1 Þ cosðf fkT Þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðf fkT Þ wðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z i ibz 1 coshðf aðy þ h2 ÞÞ h 2 þ l s k2P sinhðbh1 Þ coshðf afÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf afÞ eisðxgÞ ds; sEðsÞ 2p 1 h2 6 y 6 f;
2
where c ¼ k2 kP
1=2
ð17Þ
1=2 2 ¼ i kP k2 . By splitting the integrals in (17) into odd and even parts with respect to the param-
eter s and the displacement fields are written as
wðx; yÞ ¼
bz fkT Gzy eky 2 þ
fGzy bz eky
sinðfkT ðf þ h2 ÞÞ½c coshðcðy h1 ÞÞ þ k sinhðcðy h1 ÞÞ 2 kP 1
l sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ a sinhðf aðf þ h2 ÞÞ
Z
p
0
sEðsÞ
½b coshðbðy h1 ÞÞ þ k sinhðbðy h1 ÞÞ sinðsðx gÞÞds;
0 6 y 6 h1 ;
h i 2 bz sinðfkT ðf þ h2 ÞÞ lkP sinhðch1 Þ sinðfykT Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ cosðfykT Þ wðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z 1 i bz sinhðf aðf þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ sinhðf ayÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ coshðf ayÞ sinðsðx gÞÞds; sEðsÞ p 0 f 6 y 6 0; h i 2 bz cosðfkT ðy þ h2 ÞÞ lkP sinhðch1 Þ cosðf fkT Þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðf fkT Þ wðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z 1 i bz coshðf aðy þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ coshðf afÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf afÞ sinðsðx gÞÞds; sEðsÞ p 0 h2 6 y 6 f:
ð18Þ
The stress components by virtue of Eq. (1), first Eqs. (5) and (18), are expressed as
bz lfGzy eky
rzx ðx; yÞ ¼ rzy ðx; yÞ ¼
p
Z 0
1
a sinhðf aðf þ h2 ÞÞ EðsÞ
½b coshðbðy h1 ÞÞ þ k sinhðbðy h1 ÞÞ cosðsðx gÞÞds;
bz clfkT Gzy eky sinðfkT ðf þ h2 ÞÞ½c sinhðcðy h1 ÞÞ þ k coshðcðy h1 ÞÞ 2 2 lkP sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z bz lfGzy eky 1 a sinhðf aðf þ h2 ÞÞ 2 2 þ s kP sinhðbðy h1 ÞÞ sinðsðx gÞÞds; 0 6 y 6 h1 ; sEðsÞ p 0
rzx ðx; yÞ ¼
bz Gzx
p
Z
1 0
i sinhðf aðf þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ sinhðf ayÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ coshðf ayÞ cosðsðx gÞÞds; EðsÞ
h i 2 bz fkT Gzy sinðfkT ðf þ h2 ÞÞ lkP sinhðch1 Þ cosðfykT Þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfykT Þ rzy ðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z i bz fGzy 1 a sinhðf aðf þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ coshðf ayÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf ayÞ sinðsðx gÞÞds; sEðsÞ p 0 f 6 y 6 0;
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M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
rzx ðx; yÞ ¼
bz Gzx
p
Z
1
0
i coshðf aðy þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ coshðf afÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf afÞ cosðsðx gÞÞds; EðsÞ
h i 2 bz fkT sinðfkT ðy þ h2 ÞÞ lkP sinhðch1 Þ cosðf fkT Þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðf fkT Þ rzy ðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z 1 i bz fGzy a sinhðf aðy þ h2 ÞÞ h 2 2 l s kP sinhðbh1 Þ coshðf afÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf afÞ sinðsðx gÞÞds; sEðsÞ p 0 ð19Þ h2 6 y 6 f:
In the region 0 6 y 6 h1, the integrands of Eqs. (19) decay sufficiently rapidly as s ? 1. Therefore, the integrals can be evaluated numerically. However, in the region h2 6 y 6 0, integrals diverge for points in the vicinity of dislocation. The singularity must occur as s tends to infinity. The asymptotic behavior of stress components at dislocation location may be studied by employing the following integrals
Z
1
x ; y < 0; x2 þ y 2 Z 1 y esy cosðsxÞds ¼ 2 ; y < 0: x þ y2 0 esy sinðsxÞds ¼
0
ð20Þ
The character of the solution of Eq. (19) depends on whether s > kT or s < kT. This leads to the necessity of distinguishing between the fundamental above-mention cases. Stress components in the region h2 6 y 6 0 may be recast to more appropriate forms
rzx ðx; yÞ ¼
bz Gzx 2p
(
f ðy fÞ
ðx gÞ2 þ f 2 ðy fÞ2 Z kT i 2 sinðf a1 ðf þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ sinðf a1 yÞ f a1 Gzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ cosðf a1 yÞ þ efsðyfÞ cosðsðx gÞÞds E1 ðsÞ 0 Z 1 i 2 sinhðf aðf þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ sinhðf ayÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ coshðf ayÞ þ efsðyfÞ cosðsðx gÞÞds EðsÞ kT
h i 2 bz fkT Gzy sinðfkT ðf þ h2 ÞÞ lkP sinhðch1 Þ cosðfykT Þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfykT Þ rzy ðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ ( Z kT bz fGzy xg 2a1 sinðf a1 ðf þ h2 ÞÞ h 2 þ lðs k2P Þ sinhðbh1 Þ cosðf a1 yÞ 2 2 2 sE1 ðsÞ 2p ðx gÞ þ f ðy fÞ 0 o þf a1 Gzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinðf a1 yÞ efsðyfÞ sinðsðx gÞÞds Z 1 i 2a sinhðf aðf þ h2 ÞÞ h 2 þ l s k2P sinhðbh1 Þ coshðf ayÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf ayÞ efsðyfÞ sEðsÞ kT sinðsðx gÞÞds ; f 6 y 6 0; (
2 cosðf a1 ðy þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ cosðf a1 fÞ E ðsÞ 1 0 o þf a1 Gzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinðf a1 fÞ efsðfyÞ cosðsðx gÞÞds Z 1 i 2 coshðf aðy þ h2 ÞÞ h 2 þ l s k2P sinhðbh1 Þ coshðf afÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf afÞ efsðfyÞ EðsÞ kT cosðsðx gÞÞds ;
rzx ðx; yÞ ¼
bz Gzx 2p
f ðf yÞ
þ ðx gÞ2 þ f 2 ðy fÞ2
Z
kT
h i 2 bz fkT sinðfkT ðy þ h2 ÞÞ lkP sinhðch1 Þ cosðf fkT Þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðf fkT Þ rzy ðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ ( Z kT bz fGzy xg 2a1 sinðf a1 ðy þ h2 ÞÞ h 2 þ l s k2P sinhðbh1 Þ cosðf a1 fÞ 2 2 sE1 ðsÞ 2p ðx gÞ þ f 2 ðy fÞ 0 o þf a1 Gzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinðf a1 fÞ efsðfyÞ sinðsðx gÞÞds Z 1 i 2a sinhðf aðy þ h2 ÞÞ h 2 þ l s k2P sinhðbh1 Þ coshðf afÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf afÞ efsðfyÞ sEðsÞ kT sinðsðx gÞÞds ; h2 6 y 6 f:
ð21Þ
M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
51
In the above equation, we use the relation
2 E1 ðsÞ ¼ l s2 kP sinhðbh1 Þ cosðf a1 h2 Þ f a1 Gzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinðf a1 h2 Þ;
ð22Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where a1 ¼ kT s2 . For x < xcr = ck we have that
1 1 2 2 2 2 b ¼ s2 þ k2 kP s > k2 kP ; 12 12 a ¼ s2 k2T ¼ i k2T s2 :
1 2 2 s < k2 kP ;
And for x > xcr = ck we arrive at
1 1 2 2 2 2 b ¼ s2 þ k2 kP s > k2 kP
12 12 2 2 and b ¼ i kP s2 k2 s < kP k2 :
The response is thus made up of a propagating and a non-propagating term. The solution given by Eq. (21) show clearly the non-wave nature and simple vibration both occur simultaneously. In term of energy, we may interpret the one term as continuously radiated energy and the other as local stored energy. This phenomenon will occur also in more complex problems [29]. All integrals in Eq. (21) decay reasonably fast as s ? 1. Consequently, integrals are regular and stress fields exhibit the familiar Cauchy type singularity at dislocation location. 3. Strip under anti-plane point-force In this section we will consider the layer, in the absence of defects, be subjected to a pair of anti-plane point forces with magnitude s0 applied at x = 0 on the boundaries. Consequently, the boundary conditions become
rzy ðx; h1 Þ ¼ s0 dðxÞ; rzy ðx; h2 Þ ¼ s0 dðxÞ; jxj < 1:
ð23Þ
Eq. (6) should be solved subjected to the boundary condition (9) and (23). A procedure analogous to that of dislocation solution leads to the following stress components
(
s ekðyh1 Þ x rzx ðx; yÞ ¼ 0 þ 2 p x2 þ ðh1 yÞ
Z
1
s ½bl coshðf ah2 Þ coshðbyÞ þ ðkl coshðf ah2 Þ EðsÞ þf aGzy sinhðf ah2 ÞÞ sinhðbyÞ lðb coshðbðy h1 ÞÞ þ k sinhðbðy h1 ÞÞÞ esðh1 yÞ sinðsxÞds ; (
rzy ðx; yÞ ¼
Z
0
1 2 ½f abGzy sinhðf ah2 Þ coshðbyÞ þ l s2 kP coshðf ah2 Þ kf aGzy sinhðf ah2 Þ EðsÞ 0 2 2 sinhðbyÞ l s kP sinhðbðy h1 ÞÞ esðh1 yÞ cosðsxÞds ; 0 6 y 6 h1 ;
s0 ekðyh1 Þ h1 y þ 2 p x2 þ ðh1 yÞ
f s0
(
Z
1
s ½f abGzy ekh1 coshðf aðy þ h2 ÞÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ coshðf ayÞ aEðsÞ 0 x2 þ f 2 ðy þ h2 Þ 2 þl s2 kP sinhðbh1 Þ sinhðf ayÞ þ efsðyþh2 Þ sinðsxÞds ;
rzx ðx; yÞ ¼
p
(
x
1
2
s0 f ðy þ h2 Þ þ p x2 þ f 2 ðy þ h2 Þ2
Z
1
1 ½f abGzy ekh1 sinhðf aðy þ h2 ÞÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf ayÞ EðsÞ 2 þl s2 kP sinhðbh1 Þ coshðf ayÞ efsðyþh2 Þ cosðsxÞds ; h2 6 y 6 0:
rzy ðx; yÞ ¼
0
ð24Þ
The stress components (24) are Cauchy singular at the points of application of loads. Nonetheless, the integrands in Eqs. (24) decay reasonably fast as s ? 1. However integrands in Eqs. (24) are regular and can be evaluated numerically. 4. Solution of the integral equations The dislocation solutions accomplished in the preceding section may be employed to analyze an orthotropic strip reinforced by functionally graded materials with N arbitrarily curved cracks, with smooth surfaces. The stress components caused by a screw dislocation located at (g, f) with dislocation line parallel to the x-axis may be written as
8 1 k ðx; y; g; fÞ; 0 6 y 6 h1 > > < zj rzj ðx; yÞ ¼ bz k2zj ðx; y; g; fÞ; f 6 y 6 0 > > : 3 kzj ðx; y; g; fÞ; h2 6 y 6 f
j ¼ x; y;
ð25Þ
52
M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57 l
where kzj ðx; y; g; fÞ; l ¼ 1; 2; 3 j ¼ x; y are the coefficients of bz and may be deduced from (21) and first Eq. (19) for the three different formulations. The curved crack configuration may be described in parametric form as
xi ¼ ai ðsÞ; yi ¼ bi ðsÞ;
i ¼ 1; 2; . . . N;
ð26Þ
1 6 s 6 1:
The moveable orthogonal t, n coordinate system is chosen such that the origin may move on the crack while the t-axis remains tangent to the crack face. The anti-plane traction on the face of ith crack in the terms of the stress components in Cartesian coordinates become
rnz ðxi ; yi Þ ¼ rzy cos hi ðsÞ rzx sin hi ðsÞ; i ¼ 1; 2; . . . ; N;
ð27Þ
where hi ðsÞ ¼ tan1 b0i ðsÞ=a0i ðsÞ is the angle between x and t-axis and prime denotes differentiation with respect to the argument. Suppose screw dislocations with unknown density Bzj(p) are distributed on the infinitesimal segment rhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 h i2 a0j ðpÞ þ b0j ðpÞ dp at the face of the jth crack where the parameter 1 6 p 6 1. The anti-plane traction on the face of ith crack due to the presence of above-mentioned distribution dislocations on the faces of all N cracks yield
rnz ðai ðsÞ; bi ðsÞÞ ¼
N Z X
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i2 h i2 a0j ðpÞ þ b0j ðpÞ Bzj ðpÞdp;
1
K ij ðs; pÞ 1
j¼1
1 6 s 6 1;
i ¼ 1; 2; . . . N;
ð28Þ
where
8 1 1 k ða ; b ; a ; b Þ cos hi kzx ðai ; bi ; aj ; bj Þ sin hi ; 0 6 bi 6 h1 ; > > < zy i i j j K ij ðs; pÞ ¼ k2zy ðai ; bi ; aj ; bj Þ cos hi k2zx ðai ; bi ; aj ; bj Þ sin hi ; bj 6 bi 6 0; > > : 3 3 kzy ðai ; bi ; aj ; bj Þ cos hi kzx ðai ; bi ; aj ; bj Þ sin hi ; h2 6 bi 6 bj :
ð29Þ
l
The function kzj ðai ; bi ; aj ; bj Þ; l ¼ 1; 2; 3 j ¼ x; y are introduced in (25), we should point out that in (29) quantities with subscript i are functions of s whereas those with subscript j are functions of p. By virtue of the Buckner’s principal [28], the elasticity problem of a strip in the absence of cracks under external loading should be solved which yields the traction on the crack faces with opposite sign. Therefore, the left-hand side of Eq. (28) may be specified. As was discussed previously, the dominant singularity of stress fields for dislocation as r ? 0 is of Cauchy type. Consequently, Eq. (28) is Cauchy singular integral equations for the dislocation densities. Employing the definition of dislocation density function, the equation for the crack opening displacement across the jth crack become
wj ðsÞ wþj ðsÞ ¼
Z
s
1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i2 h i2 a0j ðpÞ þ b0j ðpÞ Bzj ðpÞdp;
j ¼ 1; 2; . . . N:
ð30Þ
The displacement field is single-value out of crack face. Thus, dislocation density for an embedded crack is subjected to the following closure requirement
Z
1
1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i2 h i2 a0j ðpÞ þ b0j ðpÞ Bzj ðpÞdp ¼ 0;
j ¼ 1; 2; . . . N;
ð31Þ
where j is the crack number. To obtain the dislocation density for embedded cracks, the integral Eqs. (28) and (31) are to be solved simultaneously. This task is taken up by the methodology devised in Erdogan et al. [30]. The stress fields for the cracks are singular at the crack tips with square root singularity. Therefore, the dislocation densities for embedded cracks are taken as
g zj ðpÞ Bzj ðpÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 p2
1 6 p 6 1;
j ¼ 1; 2; . . . ; N:
ð32Þ
The function gzj(p) in above equation are continuous in 1 6 p 6 1. Substituting Eq. (32) into (28) and (31) and discretizing the domain, 1 6 p 6 1, by m + 1 segments, we arrive at the following system of N m algebraic equations
2
32
g z1 ðpn Þ
3
2
q1 ðsr Þ
3
H11
H12
H1N
6H 6 21 6 . 6 . 4 .
H22 .. .
.. .
7 6 7 6 H1N 7 76 g z2 ðpn Þ 7 6 q2 ðsr Þ 7 7¼6 7 6 .. 7 .. 76 7 6 .. 7; 5 4 . 5 . 54 .
HN1
HN2
HNN
g zN ðpn Þ
ð33Þ
qN ðsr Þ
where the collocation points are
pr ; r ¼ 1; 2; . . . ; m 1;
m pð2n 1Þ pn ¼ cos ; n ¼ 1; 2; . . . ; m: 2m
sr ¼ cos
ð34Þ
M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
53
The components of matrices in (33) are
2
K ij ðs1 ; p1 ÞDj ðp1 Þ
K ij ðs1 ; p2 ÞDj ðp2 Þ
K ij ðs1 ; pm ÞDj ðpm Þ
3
6 K ðs ; p ÞD ðp Þ K ij ðs2 ; p2 ÞDj ðp2 Þ K ij ðs2 ; pm ÞDj ðpm Þ 7 7 6 ij 2 1 j 1 7 7 . . . .. .. .. .. Hij ¼ 6 7: . 7 m6 7 6 4 K ij ðsm1 ; p1 ÞDj ðp1 Þ K ij ðsm1 ; p2 ÞDj ðp2 Þ K ij ðsm1 ; pm ÞDj ðpm Þ 5
p6 6
dij Dj ðp1 Þ
dij Dj ðp2 Þ
ð35Þ
dij Dj ðpm Þ
In (35), dij in the last row of Hij designates the Kronecker delta, Dj ðpÞ ¼ (33) are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 0 2ffi ai ðpÞ þ bi ðpÞ . The components of vectors in
g zj ðpn Þ ¼ ½g j ðp1 Þ g j ðp2 Þ g j ðpm ÞT ; qj ðsr Þ ¼ ½rnz ðxj ðs1 Þ; yj ðs1 ÞÞ
rnz ðxj ðs2 Þ; yj ðs2 ÞÞ rnz ðxj ðsm1 Þ; yj ðsm1 ÞÞ 0T ;
ð36Þ
the superscript T stands for the transpose of a vector. The stress intensity factor at the tip of ith crack in terms of the crack opening displacement is [11].
kIII kIII
Li
Ri
pffiffiffi w ðsÞ wþ ðsÞ 2 lðyLi Þ lim j pffiffiffiffiffi j ; r Li !0 4 rLi pffiffiffi w ðsÞ wþ ðsÞ 2 ¼ lðyRi Þ lim j pffiffiffiffiffiffi j ; r Ri !0 4 r Ri
¼
ð37Þ
where the subscripts L and R designate the left and right crack tips, respectively, and the geometry of a crack implies 1
r Li ¼ ½ðai ðsÞ ai ð1ÞÞ2 þ ðbi ðsÞ bi ð1ÞÞ2 2 ; 1
ð38Þ
r Ri ¼ ½ðai ðsÞ ai ð1ÞÞ2 þ ðbi ðsÞ bi ð1ÞÞ2 2 : In order to take the limits for rLi ? 0 and rR i ? 0, we should let, in Eq. (38), the parameter s ? 1 and s ? 1, respectively. The substitution of (32) into (30) and the resultant equations into Eq. (37) in conjunction with the Taylor series expansion of functions ai(s) and bi(s) around the points s = ±1 yield
kIII kIII
Li
Ri
14 f lðyLi Þ 0 ai ð1Þ 2 þ b0i ð1Þ 2 g zi ð1Þ; 2 f lðyLi Þ 0 2 0 2 14 ¼ ai ð1Þ þ bi ð1Þ g zi ð1Þ; i ¼ 1; . . . ; N: 2
¼
ð39Þ
The solution of Eq. (33) is plugged into (39) to calculate stress intensity factors. 5. Numerical results All the proceeding examples, thickness of the FGM coating is h1/h2 = 0.1. The analysis developed in preceding section, allows the consideration of an orthotropic strip with functionally graded materials coating weakened by multiple curved cracks subjected to a pair of time-harmonic concentrated point forces. Variations of angular frequency, crack length and materials properties investigated on the dynamic stress intensity factors. Several examples are solved to demonstrate the applicability of the distributed dislocation technique. The quantity for making the dynamic stress intensity factors dimenpffi sionless is K 0 ¼ s0 = l where l is the half length of crack. The first and second examples deal with an orthotropic strip weakened by an embedded crack situated at 2l/h2 = 1.0 with different ratios of shear moduli and FGM constants, respectively. In these examples, the effects of material properties and circular frequency on the dynamic stress intensity factors are investigated. The dimensionless stress intensity factors K/K0, versus v = lkT are depicted in Figs. 2 and 3. At xl/cx = 0.94, 1.005, 1.05, the maximum values of stress intensity factor occur and the increase of ratio of shear moduli only lower these values while having negligible effect on the other values of dynamic stress intensity factor. The variation of K/K0, versus the dimensionless crack length is also included in Fig. (4). The stress intensity factors of crack tips for both x > xcr and x < xcr are increased by an increase of the crack length. In the next example, two collinear cracks with two different ratios of moduli for k = 5 are considered. The problem is symmetric with respect to the y-axis. The plots of dynamic stress intensity factors against v are shown in Fig. 5. The distances between cracks centers are 2b/h2 = 1.2. The next example deals with the interaction of a stationary curved crack and growing straight crack with fixed center, the parametric representations of straight and curved cracks are, respectively
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M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
1.8
f=1.2, λ=5 f=1.0, λ=5 f=0.9, λ=5
1.6 1.4
K/K0
1.2 1 0.8 0.6 0.4 0.2 0
0
0.2
0.4
0.6
0.8
1
1.2
χ Fig. 2. Variation of the dynamic stress intensity factor verses angular frequency for multiple ratios of muduli.
a1 ðtÞ ¼ a þ lt; c1 ðtÞ ¼ b; hp i a2 ðtÞ ¼ a cos ð1 tÞ ; 4 hp i c2 ðtÞ ¼ b sin ð1 tÞ 1 6 t 6 1; 4
where a/h2 = 0.3 and b/h2 = 0.2. Fig. 6 shows the variation of dimensionless stress intensity factors versus the length of the straight crack in strip. As it may be observed, k/k0 changes rapidly for two approaching crack tips. In contrast the variations of k/k0, for crack tips L1 and R2 are not that pronounced. In the last example, we consider two identical curved cracks which are portions of the circumference of an ellipse, Fig. 7. The lengths of major and minor semi-axes of ellipse are a and b, respectively. The cracks may be represented in the following parametric forms
1 ð1 ð1Þi tÞw ; 2
1 1 6 t 6 1; bi ðtÞ ¼ yc þ ð1Þi b sin ð1 ð1Þi tÞw 2
ai ðtÞ ¼ xc þ ð1Þi a cos
i ¼ 1; 2;
where the angle:
a tan1 ðcot g/Þ; b 2.5
f=1.2, λ=5.0 f=1.2, λ=2.5 f=1.2, λ=0.0
2
1.5
K/K0
w¼
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
χ Fig. 3. Variation of the dynamic stress intensity factor verses angular frequency for three FGM constant.
55
M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
0.8 f=1.2, ω/ω =0.5 cr
f=1.2, ω/ω =1.0
0.7
cr
f=1.2, ω/ω =2.0 cr
0.6
K/K0
0.5 0.4 0.3 0.2 0.1 0 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
l/h1 Fig. 4. Variation of the dynamic stress intensity factor with changing l/h1.
3
f=1.2, λ=5, L f=1.2, λ=5, R f=1.0, λ=5, L f=1.0, λ=5, R
2.5
K/K0
2
1.5
1
0.5
0 0
0.2
0.4
0.6
0.8
1
1.2
χ Fig. 5. Variation of the dynamic stress intensity factors of two approaching cracks.
1.4
f=0.9, λ=5, R1 f=0.9, λ=5, L1
1.2
f=0.9, λ=5, R2
K/K0
1
f=0.9, λ=5, L
2
0.8 0.6 0.4 0.2 0
0
0.5
1
1.5
2
2.5
3
2l/b Fig. 6. Variation of the dynamic stress intensity factors of a curved crack and a straight interacting cracks.
56
M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
3.5
f=1, λ=5.0, R f=1, λ=5.0, L f=1, λ=2.5, R f=1, λ=2.5, L
3
K/K0
2.5 2 1.5 1 0.5 0
0
0.2
0.4
0.6
0.8
1
1.2
χ Fig. 7. Variation of the dynamic stress intensity factors of two curved crack.
and (xc, yc) are the coordinates of the center of the ellipse. The problem is symmetric with respect to the y-axis. The dimensionless stress intensity factors versus angular frequency for two FGM constants are plotted. 6. Conclusion The anti-plane stress analysis of a medium composed of an orthotropic substrate weakened by multiple cracks and reinforced with a FGM coating under time-harmonic excitation by means of Fourier transformation is carried out in this article. The stress fields exhibit the familiar Cauchy-type singularity at dislocation and point force position. The dislocation method is applied to illustrate the fundamental behavior of multiple arbitrary cracks in an orthotropic strip with functionally graded materials coating under the anti-plane deformation. The effect of excitation frequency on the stress intensity factors of cracks and interaction of cracks are studied. It is observed that under point-load traction, the stress intensity factor increases with increasing load frequency reaching a maximum value then decreases with a faster rate. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Dynamic stress intensity factors of multiple cracks in an orthotropic strip with FGM coating M.M. Monfared, M. Ayatollahi ⇑ Faculty of Engineering, University of Zanjan, P.O. Box 45195-313, Zanjan, Iran
a r t i c l e
i n f o
Article history: Received 16 April 2012 Received in revised form 23 May 2013 Accepted 2 July 2013 Available online 12 July 2013 Keywords: Dynamic stress intensity factors Functionally graded coating Orthotropic strip Multiple cracks
a b s t r a c t The problem of determining the dynamic stress intensity factors (DSIFs) in an orthotropic strip with functionally graded materials coating weakened by multiple arbitrary cracks under time-harmonic excitation are investigated. Using Fourier transform, the problem is to the solution of integral equations of the Cauchy singular type. The integral equations were solved by Erdogan’s collocation method and dynamic stress intensity factors were obtained. The proposed method of solution permits an examination of the multiple curved cracks problem with arbitrary patterns. Several examples are solved to obtain the dynamic stress intensity factors with various angular frequency, crack lengths and material properties. A knowledge of dynamic stress intensity factor is essential to the clear understanding of the propagation of cracks through structural components undergoing oscillation. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction With the application of functionally graded materials in engineering, the fracture analysis concerning inter facial zone has absorbed the attention of more and more researchers. Functionally graded materials (FGMs) are favorable coatings for structures exposed to harsh environments and severe thermal shocks. Reliable design of mechanical components necessitates their accurate stress analysis, avoiding unrealistic physical assumptions such as perfect bonding in bodies comprised of dissimilar materials. Most of the current researches on the fracture analysis of the FGMs interface have been devoted to the FGMs interlayer [1–3] and the interface between the FGM coating and the homogeneous substrate [4–8]. The dynamic problem for a finite crack was first investigated by Loeber and Sih [9]. They obtained the stress intensity factor around a crack during the passage of a time-harmonic anti-plane shear wave. Keer and Luong [10] studied the diffraction of anti-plane shear waves by a crack in a layered composite. The problem of a crack perpendicular to the direction of interface of two half-plane of different FGMs was obtained by Erdogan [11]. Li and Tai [12] presented a dynamic fracture analysis of two isotropic dissimilar strips between isotropic dissimilar half-spaces with an interfacial crack subjected to the anti-plane shear impact loading. Two bonded isotropic half-planes with non-homogeneous interface, was analyzed by Erdogan et al. [13]. Dynamic anti-plane analysis of a medium composed of a FGM coating and an orthotropic half-plane substrate where the FGM layer was weakened by a periodic array of cracks perpendicular to the boundary was carried out by Wang and Mai [14]. The stress intensity factor and strain energy release rate for an interface crack between an elastic halfplane and a non-homogenous strip whose other edge was bonded to another elastic half-plane were obtained by Ozturk and Erdogan [15]. Daros [16] solved the dynamic problem of time harmonic SH waves in a special class of anisotropic inhomogeneous materials. The influences of the inhomogeneities on the stress intensity factor of crack were presented. Babaei and Lukasiewicz [17] obtained the dynamic stress intensity factor (DSIF) for the crack in a non-homogeneous layer between ⇑ Corresponding author. Tel.: +98 241 515 2488; fax: +98 241 515 2762. E-mail address: [email protected] (M. Ayatollahi). 0013-7944/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2013.07.002
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Nomenclature a, b lengths of major and minor semi-axes of elliptical crack bz Burgers vector Bzj complex dislocation densities A(s), B(s), C1(s), D1(s), C2(s), D2(s) unknowns coefficients c, cx shears wave velocity gij(p) regular terms of dislocation densities h1, h2 thickness orthotropic and FGM layers, respectively H(x) Heaviside step function Hij coefficients matrix kIII Li, kIII Ri stress intensity factors of left and right side of crack k0 stress intensity factor of a crack in infinite plane kij(s, p) kernel of integral equations kT, kP wave numbers l half lengths of straight crack N total number of cracks rLi, rRi distance from right and left crack tips W out of plane displacement component w amplitude of displacement component x, y coordinates ai(s), ci(s) functions describing the geometry of cracks g, f dislocation coordinates k FGM exponent d(k) Dirac delta function dij Kronecker delta l shear moduli in FGM medium hi(s) crack orientation Gzx, Gzy orthotropic shear moduli of elasticity of material q1, q2 mass densities in orthotropic and FGM region, respectively rnz traction vector rzx, rzy out of plane stress components x angular frequency Acronyms DSIF FGM
dynamic stress intensity factors functionally graded materials
isotropic half-spaces under the anti-plane shear impact load. The scattering problem of elastic wave by two collinear cracks in a functionally graded interlayer bonded to dissimilar half-plane under anti-plane shear were studied by Ma et al. [18]. The stress analysis in a non-homogeneous interfacial zone between two dissimilar elastic solids was accomplished by Wang et al. [19]. Li [20] carried out an analysis for interfacial crack between two bonded dissimilar semi-infinite orthotropic strips where the crack surface was under anti-plane traction. Closed form stress intensity factors were obtained for a strip with either clamped or traction-free boundaries. The problem of cracked multi-layered and FGM coatings of a coating-substrate composite under normal traction on the crack surface was investigated by Chi and Chung [21]. A multi-layered model to the crack problem of functionally graded materials under plane stress state deformation with arbitrarily varying Young’s modulus were investigated by Huang et al. [22]. The problem of arbitrarily oriented crack in a FGM layer bonded to a homogeneous half-plane was solved by Long and Delale [23]. The effect of crack length, crack orientation and the non-homogeneity parameter of the strip on the fracture behavior of the FGM layer were presented. A crack at the interface of a FGM and a rigid half-plane was analyzed by Erdogan and Ozturk [24]. Anti-plane problem of periodic cracks in a functionally graded coatingsubstrate structure was studied by Chen [25]. The dynamic stress intensity factor in an orthotropic strip weakened by multiple cracks under point forces loading were investigated by Monfared et al. [26]. Lekesiz et al. [27] studied effective spring stiffness for a planer periodic array of collinear cracks at interface between two dissimilar materials. The effects of elastic dissimilarity, crack density and crack interaction on the effective spring stiffness was presented. In the present paper, the aim is to find the dynamic stress intensity factors of multiple cracks with arbitrary patterns in an orthotropic strip reinforced with FGM subjected to oscillating anti-plane loading. The technique necessities the solution of screw Volterra dislocation in the region. The complex Fourier transform is employed to obtain transformed displacement and stress fields. The dislocation solution and Buckner’s principle are employed to derive integral equations for dislocation density function in an orthotropic strip with FGM coating weakened by multiple arbitrary cracks. The integral equations
M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
47
are of Cauchy singular types which are solved numerically for the dislocation density on the cracks faces. Several examples of cracks are solved to illustrate the applicability of the procedure. 2. Dislocation solution A layer of FGM with thickness h1 is used to reinforce an orthotropic homogenous layer with thickness h2 as depicted in Fig. 1. In crack problems a so called ‘‘distributed dislocation technique’’ is often used in treating cracks with smooth geometries Hills et al. [28]. We must first calculate the stress fields due to a single dislocation in the region. We now take up this task for an orthotropic strip with functionally graded materials coating containing a screw dislocation that located in the orthotropic layer under time-harmonic excitation. For a medium under anti-plane deformation, the only nonzero displacement component is the out of plane component W(x, y, t). Hence, of all the stress components, only rzx and r zy are different from zero
rzx ðx; y; tÞ ¼ lðyÞ @W ; rzy ðx; y; tÞ ¼ lðyÞ @W 0 6 y 6 h1 ; @x @y @W rzx ðx; y; tÞ ¼ Gzx @W ; r ðx; y; tÞ ¼ G ; h zy zy @y 2 6 y 6 0; @x
ð1Þ
where l(y) is the shear modulus in the FGM layer, Gzx and Gzy are the shear moduli of elasticity of the orthotropic strip in the x- and y-directions, respectively. Substituting Eq. (1) into the equilibrium equation, we obtain the expressions 2
2
2
lðyÞ @@yW2 þ l0 ðyÞ @W þ lðyÞ @@xW2 ¼ qðyÞ @@tW 0 6 y 6 h1 ; 2 @y 2
2
2
Gzy @@yW2 þ Gzx @@xW2 ¼ q1 @@tW 2
h2 6 y 6 0;
ð2Þ
where q1 and q(y) are the materials mass density in the orthotropic and FGM layers, respectively. Since the problem has steady-state time dependence, the displacement and stresses may be taken to vary harmonically in time as
Wðx; y; tÞ ¼ wðx; yÞeixt :
ð3Þ
pffiffiffiffiffiffiffi In Eq. (3) i ¼ 1; x is the angular frequency of motion and w(x, y) is an unknown function. In terms of the function w(x, y), Eqs. (2) become @2 w @y2
0
ðyÞ þ llðyÞ
Gzy @ 2 w Gzx @y2
@w @y
2
x þ @@xw2 þ qðyÞ lðyÞ w ¼ 0 0 6 y 6 h1 ; 2
2
þ @@xw2 þ xGzxq1 w ¼ 0 2
h2 6 y 6 0:
ð4Þ
For the sake of convenience, the time factor exp(ixt) will be suppressed and only w(x, y) will be considered. To facilitate the solution of the above differential equation, we will focus on special class of FGM in which the variations of material properties have the same material gradient parameter which based on the reasonable and generally accepted assumption. Therefore we assume
lðyÞ ¼ le2ky ; qðyÞ ¼ q2 e2ky ;
ð5Þ
where l, q2 and k are material constants of FGM. Substituting Eqs. (5) into Eq. (4), the equation of motion can be rewritten as @2 w @y2
2
2
þ 2k @w þ @@xw2 þ xc2 w ¼ 0 0 6 y 6 h1 ; @y
1 @2 w f 2 @y2
2
2
þ @@xw2 þ xc2 w ¼ 0 x
h2 6 y 6 0:
ð6Þ
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In the above equation f ¼ Gzx =Gzy is the ratio of shear moduli of the elasticity of the orthotropic medium. c ¼ l=q2 and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cx ¼ Gzx =q1 are the represents the velocity of the anti-plane shear wave propagating along FGM and orthotropic layers, respectively. The traction-free condition on the strip boundaries implies that
rzy ðx; h1 Þ ¼ 0; rzy ðx; h2 Þ ¼ 0; jxj < 1:
Fig. 1. Schematic of the medium with screw dislocation.
ð7Þ
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M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
Let a Volterra type screw dislocation with Burgers vector bz be situated in the medium at a point with coordinates (g, f). The dislocation line chosen in the orthotropic layer with the positive direction of x-axis depicted in Fig. 1. The conditions representing the dislocation under anti-plane deformation are
wðx; fþ Þ wðx; f Þ ¼ bz Hðx gÞ;
ð8Þ
rzy ðx; fþ Þ ¼ rzy ðx; f Þ; jxj < 1;
where H( ) is the Heaviside step function. The first of Eqs. (8) enforces the multivaluedness of displacement while the second implies the continuity of traction along the dislocation line (self-equilibrium of stress). The continuity conditions at the interface of two materials imply that
wðx; 0þ Þ ¼ wðx; 0 Þ;
ð9Þ
rzy ðx; 0þ Þ ¼ rzy ðx; 0 Þ; jxj < 1:
The solution to the current dynamic problem is obtained by means of the following complex Fourier transformation defined by
F ðsÞ ¼
Z
1
f ðxÞeisx dx:
ð10Þ
1
The inversion of complex Fourier transformation yields
f ðxÞ ¼
1 2p
Z
1
F ðsÞeisx ds:
ð11Þ
1
The application of Eq. (10) to Eqs. (6) with the aid of integrations by parts and in conjunction with the property of decaying behavior of displacement and stress components as jxj ? 1 leads to a second order ordinary differential equation, in each region as
2 þ 2k dw s2 kP w ¼ 0; 0 6 y 6 h1 ; dy 2 d2 w s2 kT w ¼ 0; h2 6 y 6 0: dy2
d2 w dy2 1 f2
ð12Þ
The wave number kP is equal to the circular frequency x divided by the shear-wave velocity c, i.e., kP = x/c and kT = x/cx are the wave numbers. The solution to Eqs. (12) yields
w ðs; yÞ ¼ AðsÞeðbkÞy þ BðsÞeðbþkÞy ;
0 6 y 6 h1 ;
w ðs; yÞ ¼ C 1 ðsÞ sinhðf ayÞ þ D1 ðsÞ coshðf ayÞ; f 6 y 6 0;
ð13Þ
w ðs; yÞ ¼ C 2 ðsÞ sinhðf ayÞ þ D2 ðsÞ coshðf ayÞ; h2 6 y 6 f: A, B, C1, D1, C2 and D2 are undetermined function of the transform variable s, while the function a and b are given by
1=2 2 ¼ i kT s2 ; 1=2 1=2 2 2 ¼ i kP s2 k2 : b ¼ s2 þ k2 kP
a ¼ s2 k2T
1=2
ð14Þ
The Fourier transform of (7)–(9) by virtue of (10) yields dw ðs;h1 Þ dy
¼ 0;
dw ðs;h2 Þ dy þ
¼ 0;
w ðs; f Þ w ðs; f Þ ¼ bz ðpdðsÞ i=sÞeisg ; ðs; fþ Þ ¼ dw ðs; f Þ; dy
dw dy
ð15Þ
w ðs; 0þ Þ ¼ w ðs; 0 Þ;
l dw ðs; 0þ Þ ¼ Gzy dw ðs; 0þ Þ; dy dy where d(s) is the Dirac delta function. Application conditions (15) to Eqs. (13) yields the algebraic relation ( f aGzy bz ðpdðsÞ i=sÞeisg sinhðf aðf þ h2 ÞÞ ðb þ kÞebh1 ; 2EðsÞ ðb kÞebh1 ; ( 2 C 1 ðsÞ bz ðpdðsÞ i=sÞ sinhðf aðf þ h2 ÞÞeisg l s2 kP sinhðbh1 Þ; ¼ 2EðsÞ D1 ðsÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ; h i 2 2 sinhðbh b ð p dðsÞ i=sÞ l s k Þ coshðf afÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf afÞ eisg sinhðf ah2 Þ; z 1 P C 2 ðsÞ ¼ EðsÞ D2 ðsÞ coshðf ah2 Þ;
AðsÞ BðsÞ
¼
ð16Þ
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M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
where
2 EðsÞ ¼ l s2 kP sinhðbh1 Þ coshðf ah2 Þ þ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf ah2 Þ.
Substituting
Eq.
(16)
into
Eq. (13), and taking the inverse of Fourier transform (11), the displacements fields yields
wðx; yÞ ¼
sinðfkT ðf þ h2 ÞÞ½c coshðcðy h1 ÞÞ þ k sinhðcðy h1 ÞÞ lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z ifGzy bz eky 1 a sinhðf aðf þ h2 ÞÞ ½b coshðbðy h1 ÞÞ þ k sinhðbðy h1 ÞÞeisðxgÞ ds; 0 6 y 6 h1 ; sEðsÞ 2p 1 bz fkT Gzy eky 2
h i 2 bz sinðfkT ðf þ h2 ÞÞ lkP sinhðch1 Þ sinðfykT Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ cosðfykT Þ wðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z 1 i ibz sinhðf aðf þ h2 ÞÞ h 2 þ l s k2P sinhðbh1 Þ sinhðf ayÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ coshðf ayÞ eisðxgÞ ds; f 6 y 6 0; sEðsÞ 2p 1 h i 2 bz cosðfkT ðy þ h2 ÞÞ lkP sinhðch1 Þ cosðf fkT Þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðf fkT Þ wðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z i ibz 1 coshðf aðy þ h2 ÞÞ h 2 þ l s k2P sinhðbh1 Þ coshðf afÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf afÞ eisðxgÞ ds; sEðsÞ 2p 1 h2 6 y 6 f;
2
where c ¼ k2 kP
1=2
ð17Þ
1=2 2 ¼ i kP k2 . By splitting the integrals in (17) into odd and even parts with respect to the param-
eter s and the displacement fields are written as
wðx; yÞ ¼
bz fkT Gzy eky 2 þ
fGzy bz eky
sinðfkT ðf þ h2 ÞÞ½c coshðcðy h1 ÞÞ þ k sinhðcðy h1 ÞÞ 2 kP 1
l sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ a sinhðf aðf þ h2 ÞÞ
Z
p
0
sEðsÞ
½b coshðbðy h1 ÞÞ þ k sinhðbðy h1 ÞÞ sinðsðx gÞÞds;
0 6 y 6 h1 ;
h i 2 bz sinðfkT ðf þ h2 ÞÞ lkP sinhðch1 Þ sinðfykT Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ cosðfykT Þ wðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z 1 i bz sinhðf aðf þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ sinhðf ayÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ coshðf ayÞ sinðsðx gÞÞds; sEðsÞ p 0 f 6 y 6 0; h i 2 bz cosðfkT ðy þ h2 ÞÞ lkP sinhðch1 Þ cosðf fkT Þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðf fkT Þ wðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z 1 i bz coshðf aðy þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ coshðf afÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf afÞ sinðsðx gÞÞds; sEðsÞ p 0 h2 6 y 6 f:
ð18Þ
The stress components by virtue of Eq. (1), first Eqs. (5) and (18), are expressed as
bz lfGzy eky
rzx ðx; yÞ ¼ rzy ðx; yÞ ¼
p
Z 0
1
a sinhðf aðf þ h2 ÞÞ EðsÞ
½b coshðbðy h1 ÞÞ þ k sinhðbðy h1 ÞÞ cosðsðx gÞÞds;
bz clfkT Gzy eky sinðfkT ðf þ h2 ÞÞ½c sinhðcðy h1 ÞÞ þ k coshðcðy h1 ÞÞ 2 2 lkP sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z bz lfGzy eky 1 a sinhðf aðf þ h2 ÞÞ 2 2 þ s kP sinhðbðy h1 ÞÞ sinðsðx gÞÞds; 0 6 y 6 h1 ; sEðsÞ p 0
rzx ðx; yÞ ¼
bz Gzx
p
Z
1 0
i sinhðf aðf þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ sinhðf ayÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ coshðf ayÞ cosðsðx gÞÞds; EðsÞ
h i 2 bz fkT Gzy sinðfkT ðf þ h2 ÞÞ lkP sinhðch1 Þ cosðfykT Þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfykT Þ rzy ðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z i bz fGzy 1 a sinhðf aðf þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ coshðf ayÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf ayÞ sinðsðx gÞÞds; sEðsÞ p 0 f 6 y 6 0;
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M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
rzx ðx; yÞ ¼
bz Gzx
p
Z
1
0
i coshðf aðy þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ coshðf afÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf afÞ cosðsðx gÞÞds; EðsÞ
h i 2 bz fkT sinðfkT ðy þ h2 ÞÞ lkP sinhðch1 Þ cosðf fkT Þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðf fkT Þ rzy ðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ Z 1 i bz fGzy a sinhðf aðy þ h2 ÞÞ h 2 2 l s kP sinhðbh1 Þ coshðf afÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf afÞ sinðsðx gÞÞds; sEðsÞ p 0 ð19Þ h2 6 y 6 f:
In the region 0 6 y 6 h1, the integrands of Eqs. (19) decay sufficiently rapidly as s ? 1. Therefore, the integrals can be evaluated numerically. However, in the region h2 6 y 6 0, integrals diverge for points in the vicinity of dislocation. The singularity must occur as s tends to infinity. The asymptotic behavior of stress components at dislocation location may be studied by employing the following integrals
Z
1
x ; y < 0; x2 þ y 2 Z 1 y esy cosðsxÞds ¼ 2 ; y < 0: x þ y2 0 esy sinðsxÞds ¼
0
ð20Þ
The character of the solution of Eq. (19) depends on whether s > kT or s < kT. This leads to the necessity of distinguishing between the fundamental above-mention cases. Stress components in the region h2 6 y 6 0 may be recast to more appropriate forms
rzx ðx; yÞ ¼
bz Gzx 2p
(
f ðy fÞ
ðx gÞ2 þ f 2 ðy fÞ2 Z kT i 2 sinðf a1 ðf þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ sinðf a1 yÞ f a1 Gzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ cosðf a1 yÞ þ efsðyfÞ cosðsðx gÞÞds E1 ðsÞ 0 Z 1 i 2 sinhðf aðf þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ sinhðf ayÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ coshðf ayÞ þ efsðyfÞ cosðsðx gÞÞds EðsÞ kT
h i 2 bz fkT Gzy sinðfkT ðf þ h2 ÞÞ lkP sinhðch1 Þ cosðfykT Þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfykT Þ rzy ðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ ( Z kT bz fGzy xg 2a1 sinðf a1 ðf þ h2 ÞÞ h 2 þ lðs k2P Þ sinhðbh1 Þ cosðf a1 yÞ 2 2 2 sE1 ðsÞ 2p ðx gÞ þ f ðy fÞ 0 o þf a1 Gzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinðf a1 yÞ efsðyfÞ sinðsðx gÞÞds Z 1 i 2a sinhðf aðf þ h2 ÞÞ h 2 þ l s k2P sinhðbh1 Þ coshðf ayÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf ayÞ efsðyfÞ sEðsÞ kT sinðsðx gÞÞds ; f 6 y 6 0; (
2 cosðf a1 ðy þ h2 ÞÞ h 2 l s k2P sinhðbh1 Þ cosðf a1 fÞ E ðsÞ 1 0 o þf a1 Gzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinðf a1 fÞ efsðfyÞ cosðsðx gÞÞds Z 1 i 2 coshðf aðy þ h2 ÞÞ h 2 þ l s k2P sinhðbh1 Þ coshðf afÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf afÞ efsðfyÞ EðsÞ kT cosðsðx gÞÞds ;
rzx ðx; yÞ ¼
bz Gzx 2p
f ðf yÞ
þ ðx gÞ2 þ f 2 ðy fÞ2
Z
kT
h i 2 bz fkT sinðfkT ðy þ h2 ÞÞ lkP sinhðch1 Þ cosðf fkT Þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðf fkT Þ rzy ðx; yÞ ¼ 2 lk2P sinhðch1 Þ cosðfkT h2 Þ þ fkT Gzy ðc coshðch1 Þ k sinhðch1 ÞÞ sinðfkT h2 Þ ( Z kT bz fGzy xg 2a1 sinðf a1 ðy þ h2 ÞÞ h 2 þ l s k2P sinhðbh1 Þ cosðf a1 fÞ 2 2 sE1 ðsÞ 2p ðx gÞ þ f 2 ðy fÞ 0 o þf a1 Gzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinðf a1 fÞ efsðfyÞ sinðsðx gÞÞds Z 1 i 2a sinhðf aðy þ h2 ÞÞ h 2 þ l s k2P sinhðbh1 Þ coshðf afÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf afÞ efsðfyÞ sEðsÞ kT sinðsðx gÞÞds ; h2 6 y 6 f:
ð21Þ
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51
In the above equation, we use the relation
2 E1 ðsÞ ¼ l s2 kP sinhðbh1 Þ cosðf a1 h2 Þ f a1 Gzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinðf a1 h2 Þ;
ð22Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where a1 ¼ kT s2 . For x < xcr = ck we have that
1 1 2 2 2 2 b ¼ s2 þ k2 kP s > k2 kP ; 12 12 a ¼ s2 k2T ¼ i k2T s2 :
1 2 2 s < k2 kP ;
And for x > xcr = ck we arrive at
1 1 2 2 2 2 b ¼ s2 þ k2 kP s > k2 kP
12 12 2 2 and b ¼ i kP s2 k2 s < kP k2 :
The response is thus made up of a propagating and a non-propagating term. The solution given by Eq. (21) show clearly the non-wave nature and simple vibration both occur simultaneously. In term of energy, we may interpret the one term as continuously radiated energy and the other as local stored energy. This phenomenon will occur also in more complex problems [29]. All integrals in Eq. (21) decay reasonably fast as s ? 1. Consequently, integrals are regular and stress fields exhibit the familiar Cauchy type singularity at dislocation location. 3. Strip under anti-plane point-force In this section we will consider the layer, in the absence of defects, be subjected to a pair of anti-plane point forces with magnitude s0 applied at x = 0 on the boundaries. Consequently, the boundary conditions become
rzy ðx; h1 Þ ¼ s0 dðxÞ; rzy ðx; h2 Þ ¼ s0 dðxÞ; jxj < 1:
ð23Þ
Eq. (6) should be solved subjected to the boundary condition (9) and (23). A procedure analogous to that of dislocation solution leads to the following stress components
(
s ekðyh1 Þ x rzx ðx; yÞ ¼ 0 þ 2 p x2 þ ðh1 yÞ
Z
1
s ½bl coshðf ah2 Þ coshðbyÞ þ ðkl coshðf ah2 Þ EðsÞ þf aGzy sinhðf ah2 ÞÞ sinhðbyÞ lðb coshðbðy h1 ÞÞ þ k sinhðbðy h1 ÞÞÞ esðh1 yÞ sinðsxÞds ; (
rzy ðx; yÞ ¼
Z
0
1 2 ½f abGzy sinhðf ah2 Þ coshðbyÞ þ l s2 kP coshðf ah2 Þ kf aGzy sinhðf ah2 Þ EðsÞ 0 2 2 sinhðbyÞ l s kP sinhðbðy h1 ÞÞ esðh1 yÞ cosðsxÞds ; 0 6 y 6 h1 ;
s0 ekðyh1 Þ h1 y þ 2 p x2 þ ðh1 yÞ
f s0
(
Z
1
s ½f abGzy ekh1 coshðf aðy þ h2 ÞÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ coshðf ayÞ aEðsÞ 0 x2 þ f 2 ðy þ h2 Þ 2 þl s2 kP sinhðbh1 Þ sinhðf ayÞ þ efsðyþh2 Þ sinðsxÞds ;
rzx ðx; yÞ ¼
p
(
x
1
2
s0 f ðy þ h2 Þ þ p x2 þ f 2 ðy þ h2 Þ2
Z
1
1 ½f abGzy ekh1 sinhðf aðy þ h2 ÞÞ f aGzy ðb coshðbh1 Þ k sinhðbh1 ÞÞ sinhðf ayÞ EðsÞ 2 þl s2 kP sinhðbh1 Þ coshðf ayÞ efsðyþh2 Þ cosðsxÞds ; h2 6 y 6 0:
rzy ðx; yÞ ¼
0
ð24Þ
The stress components (24) are Cauchy singular at the points of application of loads. Nonetheless, the integrands in Eqs. (24) decay reasonably fast as s ? 1. However integrands in Eqs. (24) are regular and can be evaluated numerically. 4. Solution of the integral equations The dislocation solutions accomplished in the preceding section may be employed to analyze an orthotropic strip reinforced by functionally graded materials with N arbitrarily curved cracks, with smooth surfaces. The stress components caused by a screw dislocation located at (g, f) with dislocation line parallel to the x-axis may be written as
8 1 k ðx; y; g; fÞ; 0 6 y 6 h1 > > < zj rzj ðx; yÞ ¼ bz k2zj ðx; y; g; fÞ; f 6 y 6 0 > > : 3 kzj ðx; y; g; fÞ; h2 6 y 6 f
j ¼ x; y;
ð25Þ
52
M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57 l
where kzj ðx; y; g; fÞ; l ¼ 1; 2; 3 j ¼ x; y are the coefficients of bz and may be deduced from (21) and first Eq. (19) for the three different formulations. The curved crack configuration may be described in parametric form as
xi ¼ ai ðsÞ; yi ¼ bi ðsÞ;
i ¼ 1; 2; . . . N;
ð26Þ
1 6 s 6 1:
The moveable orthogonal t, n coordinate system is chosen such that the origin may move on the crack while the t-axis remains tangent to the crack face. The anti-plane traction on the face of ith crack in the terms of the stress components in Cartesian coordinates become
rnz ðxi ; yi Þ ¼ rzy cos hi ðsÞ rzx sin hi ðsÞ; i ¼ 1; 2; . . . ; N;
ð27Þ
where hi ðsÞ ¼ tan1 b0i ðsÞ=a0i ðsÞ is the angle between x and t-axis and prime denotes differentiation with respect to the argument. Suppose screw dislocations with unknown density Bzj(p) are distributed on the infinitesimal segment rhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 h i2 a0j ðpÞ þ b0j ðpÞ dp at the face of the jth crack where the parameter 1 6 p 6 1. The anti-plane traction on the face of ith crack due to the presence of above-mentioned distribution dislocations on the faces of all N cracks yield
rnz ðai ðsÞ; bi ðsÞÞ ¼
N Z X
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i2 h i2 a0j ðpÞ þ b0j ðpÞ Bzj ðpÞdp;
1
K ij ðs; pÞ 1
j¼1
1 6 s 6 1;
i ¼ 1; 2; . . . N;
ð28Þ
where
8 1 1 k ða ; b ; a ; b Þ cos hi kzx ðai ; bi ; aj ; bj Þ sin hi ; 0 6 bi 6 h1 ; > > < zy i i j j K ij ðs; pÞ ¼ k2zy ðai ; bi ; aj ; bj Þ cos hi k2zx ðai ; bi ; aj ; bj Þ sin hi ; bj 6 bi 6 0; > > : 3 3 kzy ðai ; bi ; aj ; bj Þ cos hi kzx ðai ; bi ; aj ; bj Þ sin hi ; h2 6 bi 6 bj :
ð29Þ
l
The function kzj ðai ; bi ; aj ; bj Þ; l ¼ 1; 2; 3 j ¼ x; y are introduced in (25), we should point out that in (29) quantities with subscript i are functions of s whereas those with subscript j are functions of p. By virtue of the Buckner’s principal [28], the elasticity problem of a strip in the absence of cracks under external loading should be solved which yields the traction on the crack faces with opposite sign. Therefore, the left-hand side of Eq. (28) may be specified. As was discussed previously, the dominant singularity of stress fields for dislocation as r ? 0 is of Cauchy type. Consequently, Eq. (28) is Cauchy singular integral equations for the dislocation densities. Employing the definition of dislocation density function, the equation for the crack opening displacement across the jth crack become
wj ðsÞ wþj ðsÞ ¼
Z
s
1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i2 h i2 a0j ðpÞ þ b0j ðpÞ Bzj ðpÞdp;
j ¼ 1; 2; . . . N:
ð30Þ
The displacement field is single-value out of crack face. Thus, dislocation density for an embedded crack is subjected to the following closure requirement
Z
1
1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i2 h i2 a0j ðpÞ þ b0j ðpÞ Bzj ðpÞdp ¼ 0;
j ¼ 1; 2; . . . N;
ð31Þ
where j is the crack number. To obtain the dislocation density for embedded cracks, the integral Eqs. (28) and (31) are to be solved simultaneously. This task is taken up by the methodology devised in Erdogan et al. [30]. The stress fields for the cracks are singular at the crack tips with square root singularity. Therefore, the dislocation densities for embedded cracks are taken as
g zj ðpÞ Bzj ðpÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 p2
1 6 p 6 1;
j ¼ 1; 2; . . . ; N:
ð32Þ
The function gzj(p) in above equation are continuous in 1 6 p 6 1. Substituting Eq. (32) into (28) and (31) and discretizing the domain, 1 6 p 6 1, by m + 1 segments, we arrive at the following system of N m algebraic equations
2
32
g z1 ðpn Þ
3
2
q1 ðsr Þ
3
H11
H12
H1N
6H 6 21 6 . 6 . 4 .
H22 .. .
.. .
7 6 7 6 H1N 7 76 g z2 ðpn Þ 7 6 q2 ðsr Þ 7 7¼6 7 6 .. 7 .. 76 7 6 .. 7; 5 4 . 5 . 54 .
HN1
HN2
HNN
g zN ðpn Þ
ð33Þ
qN ðsr Þ
where the collocation points are
pr ; r ¼ 1; 2; . . . ; m 1;
m pð2n 1Þ pn ¼ cos ; n ¼ 1; 2; . . . ; m: 2m
sr ¼ cos
ð34Þ
M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
53
The components of matrices in (33) are
2
K ij ðs1 ; p1 ÞDj ðp1 Þ
K ij ðs1 ; p2 ÞDj ðp2 Þ
K ij ðs1 ; pm ÞDj ðpm Þ
3
6 K ðs ; p ÞD ðp Þ K ij ðs2 ; p2 ÞDj ðp2 Þ K ij ðs2 ; pm ÞDj ðpm Þ 7 7 6 ij 2 1 j 1 7 7 . . . .. .. .. .. Hij ¼ 6 7: . 7 m6 7 6 4 K ij ðsm1 ; p1 ÞDj ðp1 Þ K ij ðsm1 ; p2 ÞDj ðp2 Þ K ij ðsm1 ; pm ÞDj ðpm Þ 5
p6 6
dij Dj ðp1 Þ
dij Dj ðp2 Þ
ð35Þ
dij Dj ðpm Þ
In (35), dij in the last row of Hij designates the Kronecker delta, Dj ðpÞ ¼ (33) are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 0 2ffi ai ðpÞ þ bi ðpÞ . The components of vectors in
g zj ðpn Þ ¼ ½g j ðp1 Þ g j ðp2 Þ g j ðpm ÞT ; qj ðsr Þ ¼ ½rnz ðxj ðs1 Þ; yj ðs1 ÞÞ
rnz ðxj ðs2 Þ; yj ðs2 ÞÞ rnz ðxj ðsm1 Þ; yj ðsm1 ÞÞ 0T ;
ð36Þ
the superscript T stands for the transpose of a vector. The stress intensity factor at the tip of ith crack in terms of the crack opening displacement is [11].
kIII kIII
Li
Ri
pffiffiffi w ðsÞ wþ ðsÞ 2 lðyLi Þ lim j pffiffiffiffiffi j ; r Li !0 4 rLi pffiffiffi w ðsÞ wþ ðsÞ 2 ¼ lðyRi Þ lim j pffiffiffiffiffiffi j ; r Ri !0 4 r Ri
¼
ð37Þ
where the subscripts L and R designate the left and right crack tips, respectively, and the geometry of a crack implies 1
r Li ¼ ½ðai ðsÞ ai ð1ÞÞ2 þ ðbi ðsÞ bi ð1ÞÞ2 2 ; 1
ð38Þ
r Ri ¼ ½ðai ðsÞ ai ð1ÞÞ2 þ ðbi ðsÞ bi ð1ÞÞ2 2 : In order to take the limits for rLi ? 0 and rR i ? 0, we should let, in Eq. (38), the parameter s ? 1 and s ? 1, respectively. The substitution of (32) into (30) and the resultant equations into Eq. (37) in conjunction with the Taylor series expansion of functions ai(s) and bi(s) around the points s = ±1 yield
kIII kIII
Li
Ri
14 f lðyLi Þ 0 ai ð1Þ 2 þ b0i ð1Þ 2 g zi ð1Þ; 2 f lðyLi Þ 0 2 0 2 14 ¼ ai ð1Þ þ bi ð1Þ g zi ð1Þ; i ¼ 1; . . . ; N: 2
¼
ð39Þ
The solution of Eq. (33) is plugged into (39) to calculate stress intensity factors. 5. Numerical results All the proceeding examples, thickness of the FGM coating is h1/h2 = 0.1. The analysis developed in preceding section, allows the consideration of an orthotropic strip with functionally graded materials coating weakened by multiple curved cracks subjected to a pair of time-harmonic concentrated point forces. Variations of angular frequency, crack length and materials properties investigated on the dynamic stress intensity factors. Several examples are solved to demonstrate the applicability of the distributed dislocation technique. The quantity for making the dynamic stress intensity factors dimenpffi sionless is K 0 ¼ s0 = l where l is the half length of crack. The first and second examples deal with an orthotropic strip weakened by an embedded crack situated at 2l/h2 = 1.0 with different ratios of shear moduli and FGM constants, respectively. In these examples, the effects of material properties and circular frequency on the dynamic stress intensity factors are investigated. The dimensionless stress intensity factors K/K0, versus v = lkT are depicted in Figs. 2 and 3. At xl/cx = 0.94, 1.005, 1.05, the maximum values of stress intensity factor occur and the increase of ratio of shear moduli only lower these values while having negligible effect on the other values of dynamic stress intensity factor. The variation of K/K0, versus the dimensionless crack length is also included in Fig. (4). The stress intensity factors of crack tips for both x > xcr and x < xcr are increased by an increase of the crack length. In the next example, two collinear cracks with two different ratios of moduli for k = 5 are considered. The problem is symmetric with respect to the y-axis. The plots of dynamic stress intensity factors against v are shown in Fig. 5. The distances between cracks centers are 2b/h2 = 1.2. The next example deals with the interaction of a stationary curved crack and growing straight crack with fixed center, the parametric representations of straight and curved cracks are, respectively
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M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
1.8
f=1.2, λ=5 f=1.0, λ=5 f=0.9, λ=5
1.6 1.4
K/K0
1.2 1 0.8 0.6 0.4 0.2 0
0
0.2
0.4
0.6
0.8
1
1.2
χ Fig. 2. Variation of the dynamic stress intensity factor verses angular frequency for multiple ratios of muduli.
a1 ðtÞ ¼ a þ lt; c1 ðtÞ ¼ b; hp i a2 ðtÞ ¼ a cos ð1 tÞ ; 4 hp i c2 ðtÞ ¼ b sin ð1 tÞ 1 6 t 6 1; 4
where a/h2 = 0.3 and b/h2 = 0.2. Fig. 6 shows the variation of dimensionless stress intensity factors versus the length of the straight crack in strip. As it may be observed, k/k0 changes rapidly for two approaching crack tips. In contrast the variations of k/k0, for crack tips L1 and R2 are not that pronounced. In the last example, we consider two identical curved cracks which are portions of the circumference of an ellipse, Fig. 7. The lengths of major and minor semi-axes of ellipse are a and b, respectively. The cracks may be represented in the following parametric forms
1 ð1 ð1Þi tÞw ; 2
1 1 6 t 6 1; bi ðtÞ ¼ yc þ ð1Þi b sin ð1 ð1Þi tÞw 2
ai ðtÞ ¼ xc þ ð1Þi a cos
i ¼ 1; 2;
where the angle:
a tan1 ðcot g/Þ; b 2.5
f=1.2, λ=5.0 f=1.2, λ=2.5 f=1.2, λ=0.0
2
1.5
K/K0
w¼
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
χ Fig. 3. Variation of the dynamic stress intensity factor verses angular frequency for three FGM constant.
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M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
0.8 f=1.2, ω/ω =0.5 cr
f=1.2, ω/ω =1.0
0.7
cr
f=1.2, ω/ω =2.0 cr
0.6
K/K0
0.5 0.4 0.3 0.2 0.1 0 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
l/h1 Fig. 4. Variation of the dynamic stress intensity factor with changing l/h1.
3
f=1.2, λ=5, L f=1.2, λ=5, R f=1.0, λ=5, L f=1.0, λ=5, R
2.5
K/K0
2
1.5
1
0.5
0 0
0.2
0.4
0.6
0.8
1
1.2
χ Fig. 5. Variation of the dynamic stress intensity factors of two approaching cracks.
1.4
f=0.9, λ=5, R1 f=0.9, λ=5, L1
1.2
f=0.9, λ=5, R2
K/K0
1
f=0.9, λ=5, L
2
0.8 0.6 0.4 0.2 0
0
0.5
1
1.5
2
2.5
3
2l/b Fig. 6. Variation of the dynamic stress intensity factors of a curved crack and a straight interacting cracks.
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M.M. Monfared, M. Ayatollahi / Engineering Fracture Mechanics 109 (2013) 45–57
3.5
f=1, λ=5.0, R f=1, λ=5.0, L f=1, λ=2.5, R f=1, λ=2.5, L
3
K/K0
2.5 2 1.5 1 0.5 0
0
0.2
0.4
0.6
0.8
1
1.2
χ Fig. 7. Variation of the dynamic stress intensity factors of two curved crack.
and (xc, yc) are the coordinates of the center of the ellipse. The problem is symmetric with respect to the y-axis. The dimensionless stress intensity factors versus angular frequency for two FGM constants are plotted. 6. Conclusion The anti-plane stress analysis of a medium composed of an orthotropic substrate weakened by multiple cracks and reinforced with a FGM coating under time-harmonic excitation by means of Fourier transformation is carried out in this article. The stress fields exhibit the familiar Cauchy-type singularity at dislocation and point force position. The dislocation method is applied to illustrate the fundamental behavior of multiple arbitrary cracks in an orthotropic strip with functionally graded materials coating under the anti-plane deformation. The effect of excitation frequency on the stress intensity factors of cracks and interaction of cracks are studied. It is observed that under point-load traction, the stress intensity factor increases with increasing load frequency reaching a maximum value then decreases with a faster rate. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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