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Antiplane elastodynamic analysis of a strip weakened by multiple defects
Applied Mathematical Modelling 33 (2009) 3258–3268
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Antiplane elastodynamic analysis of a strip weakened by multiple defects M. Ayatollahi a, S.J. Fariborz b,* a b
Faculty of Engineering, Zanjan University, Zanjan 45195-313, Iran Department of Mechanical Engineering, Amirkabir University of Technology (Tehran Polytechnic) 424, Hafez Avenue, Tehran 158754413, Iran
a r t i c l e
i n f o
Article history: Received 11 November 2007 Received in revised form 18 October 2008 Accepted 22 October 2008 Available online 30 October 2008
Keywords: Antiplane Strip Time-harmonic Point load Multiple defects
a b s t r a c t The method of images is utilized to derive the solution of a screw dislocation under timeharmonic conditions for an elastic strip from the solution of infinite planes. The displacement and stress components are obtained for a strip under concentrated antiplane, timeharmonic traction. The dislocation solution is employed to formulate integral equation for a strip weakened by cracks and cavities. The effects of load frequency and crack orientation on the stress intensity factors are studied. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction The stress analysis in elastic regions, weakened by defects subject to dynamic loading, is imperative for evaluating the damage caused by fatigue of material. The analytical techniques for stress analysis are, however, only capable of handling regions with simple geometries. Nonetheless, the results may be used as a benchmark solution for the verification of numerical results. Apparently, the stress analysis in elastic strips weakened by cracks under dynamic loading, unlike the static case, has not attracted much attention. The earliest study dates back to an article by Sih and Chen [1], dealing with a semi-infinite central crack traveling with constant velocity under antiplane excitation. The problem was reduced to the Riemann–Hilbert problem which was solved in closed-form for stress field in a strip with two different cases of boundary conditions. Kuhn and Matczynski [2], investigated the in-plane stress distribution of a stationary central crack in a strip subjected to periodic loading by means of complex Fourier transform and the Weiner–Hopf technique. Two coplanar cracks in a strip under shear wave was studied by Srivastava et al. [3]. A strip weakened by stationary cracks composed of piezoelectric material under antiplane harmonic load on the cracks surfaces was analyzed by some researchers, see for instance Huang et al. [4]. A piezoelectric strip containing a crack parallel to the strip boundaries and traveling with constant velocity under the action of uniform anti-plane shear stress and electric field was investigated by Li [5]. By eliminating the electric field, the solution to an elastic strip with the same geometry and mechanical loading may be reached. In the present study, the solution of screw dislocation with harmonic motion in infinite planes obtained in [6] together with the method of images are employed to derive the solution of screw dislocation in an elastic strip. The elastodynamic problem of a flawless strip under anti-plane, self-equilibrium, time-harmonic point forces is solved. The solution may be viewed as the Green’s function for a strip under arbitrary self-equilibrating traction. The dislocation solutions are employed to formulate integral equations for a strip weakened by cracks and/or cavities. The integral equations are of Cauchy singular * Corresponding author. Tel.: +98 21 6454 3460; fax: +98 21 641 9736. E-mail address: [email protected] (S.J. Fariborz). 0307-904X/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2008.10.026
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Nomenclature Aij, Cij, Dij coefficients matrix A(n), B(n) unknown coefficients Burgers vector bz complex dislocation densities Bzj B1j, B2j real and imaginary parts of dislocation densities c shear wave velocity Ci(x), Si(x) sine and cosine integral functions regular terms of dislocation densities gij(p) h thickness of strip H(x) Heaviside step function Hankel function of the second kind of order p H2p ðxÞ Jp(x), Yp(x) Bessel functions of first and second kinds of order p stress intensity factors of left and right side of crack kLi, kRi stress intensity factor of a crack in strip k0 kij(s, p) kernel of integral equations kij1, kij2 real and imaginary parts of kernel of integral equations wave number kT M number of cavities N total number of defects number of embedded cracks N1 p time-harmonic concentrated load distance from right and left crack tips rLi, rRi w amplitude of displacement component x, y coordinates ai(s), bi(s) functions describing the geometry of defects d(n) Dirac delta function Kronecker delta dij crack orientation hi(s) l elastic shear modulus q mass density r1nz, r2nz real and imaginary parts of traction vector rxz, ryz out of plane stress components x angular frequency
type which are solved numerically for the dislocation density on the defects surfaces. Several examples of strips with cracks and cavities are solved to study the effects of excitation frequency on stress intensity factors and also the interaction between defects. 2. Dislocation solution The stress components in an infinite plane containing a time-harmonic screw dislocation with the amplitude of Burgers vector bz located at a point with coordinates (n, g) were derived in [6] and is recited here
Z
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
lbz ðy gÞ lbz ðy gÞ 1 lnðu þ u2 1ÞeikT ur rxz ¼ ½cosðk rÞ i sinðk rÞ du ½Y 1 ðkT rÞ þ iJ 1 ðkT rÞ þ T T 4r 2pr 2 pr2 ðu2 y2 =r 2 Þ2 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z lbz ðy gÞ3 1 u2 1eikT ur ilbz kT ðy gÞ 1 ðu u2 1 u2 þ y2 =r 2 lnðu þ u2 1ÞÞeikT ur þ du þ du 2 4 2pr pr u2 y2 =r 2 ðu2 y2 =r 2 Þ 1 1 lbz kT ðy gÞ
ryz ¼
lbz kT ðx nÞ 4r þ
"
½Y 1 ðkT rÞ þ iJ1 ðkT rÞ þ
ilbz kT ðx nÞ pr
Z 1
1
lbz k2T ðx nÞ ip CiðkT rÞ iSiðkT rÞ þ 2p 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u2 1eikT ur ðu2 y2 =r 2 Þ2
du;
Z 1
1
# pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu u2 1 u2 þ y2 =r2 ÞeikT ur du uðu2 y2 =r 2 Þ ð1Þ
where Ci(x) and Si(x) are, respectively, the sine and cosine integral functions, Abramowitz and Stegun [7], kT = x/c and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r ¼ ðx nÞ2 þ ðy gÞ2 . For a strip with thickness 0 6 y 6 h weakened by a screw dislocation with Bergers vector bz situated at a point with coordinates (n, g), the stress components may be obtained from the solution of an infinite plane by positioning dislocations with Bergers vector bz at points with coordinates x = n, y = g ± 2kh, k 2 {0,1,2,. . .} and dislocations with
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Bergers vector bz at points with coordinates x = n, y = 2h(1 ± k) g, k 2 {0,1,2,. . .}. Employing Eq. (1), and the principle of superposition, the stress components in the strip containing dislocation yield
rxz ¼ n!1 lim
ryz
n X
lbz
k¼n
kT ðy g 2khÞ kT ðy 2hð1 þ kÞ þ gÞ ½Y 1 ðkT r 3 Þ þ iJ1 ðkT r 3 Þ ½Y 1 ðkT r 4 Þ þ iJ1 ðkT r 4 Þ 4r 3 4r4
ðy g 2khÞ ðy 2hð1 þ kÞ þ gÞ þ ½cosðkT r 3 Þ i sinðkT r 3 Þ ½cosðkT r 4 Þ i sinðkT r 4 Þ 2pr 23 2pr 24 p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z ðy g 2khÞ 1 lnðu þ u2 1ÞeikT ur3 ðy 2hð1 þ kÞ þ gÞ 1 lnðu þ u2 1ÞeikT ur4 du þ du pr23 pr24 ðu2 ðy g 2khÞ2 =r 23 Þ2 ðu2 ðy 2hð1 þ kÞ þ gÞ2 =r 24 Þ2 1 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p p Z Z u2 1eikT ur3 u2 1eikT ur4 ðy g 2khÞ3 1 ðy 2hð1 þ kÞ þ gÞ3 1 du du þ 4 2 2 4 pr 3 pr 4 ðu2 ðy g 2khÞ =r23 Þ ðu2 ðy 2hð1 þ kÞ þ gÞ2 =r 24 Þ2 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ikT ðy g 2khÞ 1 ðu u2 1 u2 þ ðy g 2khÞ2 =r 23 lnðu þ u2 1ÞÞeikT ur3 du þ 2pr 3 u2 ðy g 2khÞ2 =r23 1 ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ikT ðy 2hð1 þ kÞ þ gÞ 1 ðu u2 1 u2 þ ðy 2hð1 þ kÞ þ gÞ2 =r 24 lnðu þ u2 1ÞÞeikT ur4 du 2pr 4 u2 ðy 2hð1 þ kÞ þ gÞ2 =r 24 1 ( n X ðx nÞ ðx nÞ ¼ lim lb z k T ½Y 1 ðkT r 3 Þ þ iJ1 ðkT r 3 Þ ½Y 1 ðkT r 4 Þ þ iJ1 ðkT r 4 Þ n!1 4r 4r4 3 k¼n " # ffi Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi kT ðx nÞ ip ðu u2 1 u2 þ ðy g 2khÞ2 =r 23 ÞeikT ur3 þ CiðkT r 3 Þ iSiðkT r3 Þ þ du 2p 2 uðu2 ðy g 2khÞ2 =r 23 Þ 1 " # p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z 1 kT ðx nÞ ip ðu u2 1 u2 þ ðy 2hð1 þ kÞ þ gÞ2 =r 24 ÞeikT ur1 CiðkT r 4 Þ iSiðkT r4 Þ þ du 2p 2 uðu2 ðy 2hð1 þ kÞ þ gÞ2 =r 24 Þ 1 ) p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z Z iðx nÞ 1 u u2 1eikT ur3 iðx nÞ 1 u u2 1eikT ur4 du du ; ð2Þ þ 2 2 pr3 1 ðu2 ðy g 2khÞ =r23 Þ pr4 1 ðu2 ðy 2hð1 þ kÞ þ gÞ2 =r24 Þ2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r 3 ¼ ðx nÞ2 þ ðy g 2khÞ2 ; r4 ¼ ðx nÞ2 þ ðy 2hð1 þ kÞ þ gÞ2 . It is worth mentioning that the integrands in stress components (2) are continuous functions in 1 6 u < 1 with a sufficiently fast rate of decay as u ? 1, making them susceptible to numerical evaluation. Nonetheless, the series in (2) behaves like 1/k, as k ? 1, requiring a large number of terms for convergence. 3. Strip under antiplane point-force The constitutive equations, for an elastic medium with elastic shear modulus l under anti-plane deformation W(x, y, t) are
oW ; ox oW ryz ðx; y; tÞ ¼ l oy
rxz ðx; y; tÞ ¼ l
ð3Þ
The equation of motion for an elastic body under anti-plane deformation is
o2 W o2 W 1 o2 W þ 2 ¼ 2 : ox2 oy c ot 2
ð4Þ
pffiffiffiffiffiffiffiffiffiffiffi In Eq. (4), the characteristic shear wave velocity is c ¼ l=q; where l is the elastic shear modulus and q is the mass density of material. We let the strip be under time-harmonic concentrated applied loads with magnitude p on the edges represented by
ryz ðx; 0; tÞ ¼ pdðxÞeixt ; ryz ðx; h; tÞ ¼ pdðxÞeixt
ð5Þ
where d(x) is the Dirac delta function. The solution to Eq. (4) is considered in separable form
Wðx; y; tÞ ¼ wðx; yÞeixt :
ð6Þ
The substitution of (6) into (4) results in
o2 w o2 w 2 þ 2 þ kT w ¼ 0; ox2 oy
ð7Þ
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where kT = x/c is the wave number. The complex Fourier transform is defined as
f ðnÞ ¼
Z
1
f ðxÞeinx dx;
ð8Þ
1
the inversion of Fourier transform yields
f ðxÞ ¼
1 2p
Z
1
f ðnÞeinx dn:
ð9Þ
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The application of Eqs. (8) to (7), in view of the requirement lim wðx; yÞ ¼ 0, where r ¼ x2 þ y2 , yields a second order orr!1 * dinary differential equation for w (n, y). The solution to the differential equation is readily known
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2 2 2 2 w ðn; yÞ ¼ AðnÞey n kT þ BðnÞey n kT :
ð10Þ
The coefficients in Eq. (10) may be obtained by the application of boundary condition (5) together with Fourier transform (8). By virtue of inverse transform (9) the displacement field becomes
2
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2 2 Z 1 2 2 einxy n kT einxþy n kT ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q dn dn pffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 1 1 n2 kT ð1 e2h n kT Þ n2 kT ð1 e2h n kT Þ 3 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi Z 1 inxðhyÞ n2 k2T inxþðhyÞ n2 k2T e e 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dn þ dn5: pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 2 2 1 2h n k 2h n k T T n kT ð1 e n kT ð1 e Þ Þ
p 6 wðx; yÞ ¼ 4 2pl
Z
1
1
Z
1
ð11Þ
To simplify the integrals in Eq. (11) the procedure outlined in Achenbach [8] is followed. The final result leads to
wðx; yÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i p h p Y 0 ðkT x2 þ y2 Þ þ iJ0 ðkT x2 þ y2 Þ Y 0 ðkT x2 þ ðh yÞ2 Þ þ iJ0 ðkT x2 þ ðh yÞ2 Þ 2l 2l Z Z 2ip 1 eir1 kT u A1 sin c1 2ip 1 eir2 kT u A2 sin c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ du du; 2 pl 1 pl 1 u2 1 1 þ A1 2A1 cos c1 u2 1 1 þ A22 2A2 cos c2
ð12Þ
where J0(kTr) and Y0(kTr) are the Bessel functions of zero order of first and second kinds, respectively, and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 ; h1 ¼ tan1 ðy=xÞ; c1 ¼ 2hkT u sin h1 ; A1 ¼ expð2hkT cos h1 u2 1Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 ¼ x2 þ ðh yÞ2 ; h2 ¼ tan1 ½ðh yÞ=x; c2 ¼ 2hkT u sin h2 ; A2 ¼ expð2hkT cos h2 u2 1Þ r1 ¼
ð13Þ
In the particular case of a half-plane under anti-plane point traction, we set h ? 1 in Eq. (12) carry out the integration and obtain the well-known solution
wðx; yÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i p h Y 0 ðkT x2 þ y2 Þ þ iJ0 ðkT x2 þ y2 Þ : 2l
ð14Þ
From Eqs. (3), (8), and (10), the Fourier transform of stress components become
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi " pffiffiffiffiffiffiffiffiffi # 2 2 2 2 2 2 2 2 ey n kT eðhyÞ n kT ey n kT eðhyÞ n kT pffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffi ; 2 2 2 2 2 1 e2h n kT 1 e2h n kT n2 kT pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi " pffiffiffiffiffiffiffiffiffi # 2 2 2 2 2 2 2 2 ey n kT þ eðhyÞ n kT ey n kT þ eðhyÞ n kT pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ¼p þ : 2 2 2 2 1 e2h n kT 1 e2h n kT ipn
rxz ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ryz
ð15Þ
Similar to the displacement field the Fourier inversion of Eq. (15) yield
rxz ¼
pkT cos h1 pk cos h2 ½Y 1 ðkT r 1 Þ þ iJ1 ðkT r 1 Þ þ T ½Y 1 ðkT r2 Þ þ iJ1 ðkT r2 Þ 2 2 Z Z pk cos h1 1 ueir1 kT u A1 sin c1 pkT cos h2 1 ueir2 kT u A2 sin c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ T du du 2 2 2 p p u 1ð1 þ A1 2A1 cos c1 Þ u 1ð1 þ A22 2A2 cos c2 Þ 1 1 pkT sin h1 pk sin h2 ½Y 1 ðkT r 1 Þ þ iJ1 ðkT r 1 Þ T ½Y 1 ðkT r 2 Þ þ iJ 1 ðkT r 2 Þ 2 2 Z Z pk sin h1 1 ueir1 kT u A1 sin c1 pkT sin h2 1 ueir2 kT u A2 sin c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ du: þ T 2 2 2 p p u 1ð1 þ A1 2A1 cos c1 Þ u 1ð1 þ A22 2A2 cos c2 Þ 1 1
ryz ¼
ð16Þ
At the points of application of load, i.e., r1 ? 0 and r2 ? 0, stress fields exhibit the familiar Cauchy type singularity, 1/r1 and 1/r2.
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4. Strip weekend by defects The dislocation solutions accomplished in the preceding section may be employed to analyze a strip with N arbitrarily oriented defects, curved cracks and cavities, with smooth surfaces. Cavities are considered as closed curved cracks without singularity [6]. The defects configuration may be described in parametric form as
xi ¼ ai ðsÞ yi ¼ bi ðsÞ
ð17Þ
1 6 s 6 1 i ¼ 1; 2; . . . ; N:
The moveable orthogonal t, n-coordinate system is chosen such that the origin may move on a defect while the t-axis remains tangent to the defect surface. The anti-plane traction on the surface of the ith defect in terms of stress components in Cartesian coordinates is
rnz ðxi ; yi Þ ¼ ryz cos hi ðsÞ rxz sin hi ðsÞ; 1
ð18Þ
ðb0i ðsÞ= 0i ðsÞÞ
where hi ðsÞ ¼ tan a is the angle between x- and t-axes on theq ith defect. Suppose dislocations with unknown comffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi plex density Bzj = B1j + iB2j are distributed on the infinitesimal segment ½a0j 2 þ ½b0j 2 dp at the surface of the jth defect where the parameter 1 6 p 6 1and prime denotes differentiation with respect to the relevant argument. The traction on the surface of ith defect due to the presence of dislocations on the surfaces of all N defects yields
rnz ðai ðsÞ; bi ðsÞÞ ¼
N Z X
1
1
j¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ij ðs; pÞ ½a0j ðpÞ2 þ ½b0j ðpÞ2 ½B1j ðpÞ þ iB2j ðpÞdp i ¼ 1; 2; . . . ; N:
ð19Þ
From Eqs. (16) and (18), the kernel of integral Eq. (19) becomes
K ij ðai ðsÞ; bi ðsÞ; aj ðpÞ; bj ðpÞÞ ¼ lim
n!1
n X k ða a Þ k ða a Þ l T i j ½Y 1 ðkT r3ij Þ þ iJ1 ðkT r3ij Þ T i j ½Y 1 ðkT r4ij Þ þ iJ1 ðkT r4ij Þ 4r 4r4ij 3ij k¼n
" # Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðu u2 1 u2 þ ðbi bj 2khÞ2 =r23ij ÞeikT ur3ij kT ðai aj Þ ip CiðkT r3ij Þ iSiðkT r 3ij Þ þ þ du 2 2p uðu2 ðbi bj 2khÞ2 =r23ij Þ 1
" # Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðu u2 1 u2 þ ðbi 2hð1 þ kÞ þ bj Þ2 =r 24ij ÞeikT ur4ij kT ðai aj Þ ip CiðkT r4ij Þ iSiðkT r 4ij Þ þ du 2 2p uðu2 ðbi 2hð1 þ kÞ þ bj Þ2 =r24ij Þ 1
þ
ikT ðai aj Þ pr3ij
Z
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u2 1eikT ur3ij
1
du ðu2 ðbi bj 2khÞ2 =r23ij Þ2
1
ikT ðai aj Þ pr4ij
Z
1 1
du cos hi ðsÞ ðu2 ðbi 2hð1 þ kÞ þ bj Þ2 =r24ij Þ2
kT ðbi bj 2khÞ kT ðbi 2hð1 þ kÞ þ bj Þ ½Y 1 ðkT r3ij Þ þ iJ1 ðkT r3ij Þ ½Y 1 ðkT r4ij Þ þ iJ 1 ðkT r4ij Þ 4r 3ij 4r4ij
þ
ðbi bj 2khÞ ðbi 2hð1 þ kÞ þ bj Þ ½cosðkT r 3ij Þ i sinðkT r3ij Þ ½cosðkT r 4ij Þ i sinðkT r4ij Þ 2pr 23ij 2pr24ij
ðbi bj 2khÞ pr23ij
Z
3
þ
)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u2 1eikT ur4ij
ðbi bj 2khÞ pr43ij
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 1ÞeikT ur3ij
du þ ðu2 ðbi bj 2khÞ2 =r23ij Þ2
1
Z
ikT ðbi bj 2khÞ þ 2pr3ij
lnðu þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 1eikT ur3ij
1
ðbi 2hð1 þ kÞ þ bj Þ pr24ij
du ðu2 ðbi bj 2khÞ2 =r23ij Þ2
1
Z
1
Z
ðbi 2hð1 þ kÞ þ bj Þ pr44ij
1
Z 1
u2 ðbi bj 2khÞ2 =r23ij
ikT ðbi 2hð1 þ kÞ þ bj Þ 2pr4ij
Z
1 1
ðu2 ðbi 2hð1 þ kÞ þ bj Þ2 =r 24ij Þ2
du
du
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðu u2 1 u2 þ ðbi 2hð1 þ kÞ þ bj Þ =r 24ij lnðu þ u2 1ÞÞeikT ur4ij u2 ðbi 2hð1 þ kÞ þ bj Þ2 =r24ij
du
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 1eikT ur4ij
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu u2 1 u2 þ ðbi bj 2khÞ2 =r23ij lnðu þ u2 1ÞÞeikT ur3ij
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 1ÞeikT ur4ij
ðu2 ðbi 2hð1 þ kÞ þ bj Þ2 =r24ij Þ2
1 3
lnðu þ
)
#
du sin hi ðsÞ ;
ð20Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r 3ij ¼ ðai aj Þ2 þ ðbi bj 2khÞ2 ; r4ij ¼ ðai aj Þ2 þ ðbi 2hð1 þ kÞ þ bj Þ2 . It is noteworthy to mention that in Eqs. (20), a and b with subscripts i and j are functions of s and p, respectively. It is easy to observe that the kernel of integral Eq. (19) exhibit Cauchy type singularity for i = j as r3ij ? 0 or r4ij ? 0.
M. Ayatollahi, S.J. Fariborz / Applied Mathematical Modelling 33 (2009) 3258–3268
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The anti-plane traction on the surface of the ith defect in view of Eqs. (16) and (18) becomes
(
bi b ½Y 1 ðkT r 1i Þ þ iJ1 ðkT r 1i Þ i ½Y 1 ðkT r 2i Þ þ iJ1 ðkT r 2i Þ: 2r 1i 2r 2i " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 2 2 kT bi 1 ir 1i kT u 2huai sin c1i þ ir 1i u 1 sin c1i 2hbi u 1 cos c1i þ A1 e 2 pr1i 1 ð1 þ A1 2A1 cos c1i Þr1i h i3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4A1 h sin c1i uai ðA1 cos c1i Þ bi u2 1 sin c1i 5du ð1 þ A21 2A1 cos c1i Þ2 r 1i " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 2 2 kT ðbi hÞ 1 ir2i kT u 2huai sin c2i þ ir 2i u 1 sin c2i 2hðh bi Þ u 1 cos c2i þ A2 e : 2 pr2i ð1 þ A2 2A2 cos ci Þr 2i 1 h i3 9 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 4A2 h sin c2i uai ðA2 cos c2i Þ ðh bi Þ u2 1 sin c2i 5du cos hi ðsÞ 2 2 ; ð1 þ A2 2A2 cos c2i Þ r2i ai ai þ pkT ½Y 1 ðkT r1i Þ þ iJ1 ðkT r1i Þ ½Y 1 ðkT r2i Þ þ iJ1 ðkT r 2i Þ 2r1i 2r2i " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 2 kT ai 2huai sin c1i þ ir 1i u2 1 sin c1i 2hbi u2 1 cos c1i A1 eir1i kT u pr1i 1 ð1 þ A21 2A1 cos c1i Þr1i h i3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4A1 h sin c1i uai ðA1 cos c1i Þ bi u2 1 sin c1i 5du ð1 þ A21 2A1 cos c1i Þ2 r 1i " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 2 2 kT ai 1 ir 2i kT u 2huai sin c2i þ ir 2i u 1 sin c2i 2hðh bi Þ u 1 cos c2i þ A2 e 2 pr2i 1 ð1 þ A 2A2 cos ci Þr 2i # ) pffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi 4A2 h sin c2i ½uai ðA2 cos c2i Þ ðh bi Þ u2 1 sin c2i du sin hi ðsÞ; ð1 þ A22 2A2 cos c2i Þ2 r 2i
rnz ¼ pkT
ð21Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r 1i ¼ a2i þ b2i ; r2i ¼ a2i þ ðh bi Þ2 and c1i = 2hkTubi/r1i, c2i = 2hkTubi/r2i. Employing the definition of dislocation density function, the anti-plane opening across the boundary of the jth curved crack is
wj ðsÞ wþj ðsÞ ¼
Z
s
1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½a0j ðpÞ2 þ ½b0j ðpÞ2 ½B1j ðpÞ þ iB2j ðpÞdp j ¼ 1; 2; . . . ; N:
ð22Þ
Let the strip be weakened by M, N1 and N2, cavities, embedded and edge cracks, respectively. The displacement field is singlevalued out of the surfaces of embedded defects. Consequently, the dislocation density functions are subjected to the following closure requirement:
Z
1
1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½a0j ðpÞ2 þ ½b0j ðpÞ2 Bij ðpÞdp ¼ 0 i ¼ 1; 2 j ¼ 1; 2; . . . ; M þ N 1 :
ð23Þ
5. Solution of integral equations To obtain dislocation density functions, the integral Eqs. (19) and (23) are to be solved simultaneously. This task is taken up by the methodology devised in Faal et al. [9]. Cavities are considered as closed curved cracks with bounded dislocation density at both ends leading to
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bij ðpÞ ¼ g ij ðpÞ 1 p2
16p61
i ¼ 1; 2 j ¼ 1; 2; . . . ; M:
ð24Þ
The stress fields for embedded cracks are singular at the crack tips with square root singularity. Therefore, the dislocation density functions for embedded cracks are represented as
g ij ðpÞ Bij ðpÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p2
1 6 p 6 1 i ¼ 1; 2 j ¼ M þ 1; M þ 2; . . . ; M þ N 1 :
ð25Þ
For the edge cracks, taking the embedded crack tip at p = 1, the following dislocation density function is considered
sffiffiffiffiffiffiffiffiffiffiffiffi 1p ; Bij ðpÞ ¼ g ij ðpÞ 1þp
1 6 p 6 1 i ¼ 1; 2 j ¼ M þ N1 þ 1; M þ N1 þ 2; . . . ; N:
ð26Þ
The functions gij(p) in Eqs. (24)–(26) are continuous in 1 6 p 6 1. Substituting Eqs. (24)–(26) into Eqs. (19) and (23) and discretizing the domain, 1 6 p 6 1 by m + 1 segments, the integral equations are reduced to the following system of N 2m linear algebraic equations
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2
A1N
32
g z1 ðpn Þ
3
2
q1 ðsr Þ
3
A11
A12
6A 6 21 6 . 6 . 4 . AN1
A22 .. . AN2
7 6 7 6 A2N 7 76 g z2 ðpn Þ 7 6 q2 ðsr Þ 7 7¼6 7 6 .. .. 7 . . 76 7 6 . 7; .. 5 4 . 5 . . 54 ANN g zN ðpn Þ qN ðsr Þ
ð27Þ
where the collocation points are
pr
r ¼ 1; 2; . . . ; m 1; pð2n 1Þ pn ¼ cos n ¼ 1; 2; . . . ; m: 2m
sr ¼ cos
m
ð28Þ
The components of matrix in Eq. (27) are
Aij ¼
Bij
C ij
C ij
Bij
ð29Þ
;
where
2
Hj1 K 1ij ðs1 ; p1 ÞDi ðp1 Þ
6 6 Hj1 K 1ij ðs2 ; p1 ÞDi ðp1 Þ 6 p6 6 .. Bij ¼ 6 . m6 6 6 Hj1 K 1ij ðsm1 ; p ÞDi ðp Þ 1 1 4 Hj1 Dij ðp1 ÞDi ðp1 Þ 2 Hj1 K 2ij ðs1 ; p1 ÞDi ðp1 Þ 6 6 Hj1 K 2ij ðs2 ; p1 ÞDi ðp1 Þ 6 p6 6 C ij ¼ 6 ... m6 6 6 Hj1 K 2ij ðsm1 ; p ÞDi ðp Þ 1 1 4 0
Hj2 K 1ij ðs1 ; p2 ÞDi ðp2 Þ
Hj2 K 1ij ðs2 ; p2 ÞDi ðp2 Þ
.. .
..
.
Hj2 K 1ij ðsm1 ; p2 ÞDi ðp2 Þ Hj2 Dij ðp1 ÞDi ðp2 Þ
Hj2 K 2ij ðs1 ; p2 ÞDi ðp2 Þ
Hj2 K 2ij ðs2 ; p2 ÞDi ðp2 Þ
.. .
..
.
Hj2 K 2ij ðsm1 ; p2 ÞDi ðp2 Þ 0
Hjn K 1ij ðs1 ; pm ÞDi ðpm Þ
3
7 Hjn K 1ij ðs2 ; pm ÞDi ðpm Þ 7 7 7 7 .. 7; . 7 7 Hjn K 1ij ðsm1 ; pm ÞDi ðpm Þ 7 5 Hjn Dij ðp1 ÞDi ðpm Þ 3 Hjn K 2ij ðs1 ; pm ÞDi ðpm Þ 7 Hjn K 2ij ðs2 ; pm ÞDi ðpm Þ 7 7 7 7 .. 7: . 7 7 Hjn K 2ij ðsm1 ; pm ÞDi ðpm Þ 7 5
ð30Þ
0
In matrices Bij and Cij, the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi coefficients K1ij and K2ij are the real and imaginary parts of Kij specified in Eq. (20), Di ðpj Þ ¼ ½a0i ðpj Þ2 þ ½b0i ðpj Þ2 , and Hjn and Dij(p) are as follows:
8 > 1 p2n Hjn ¼ 1 m> : 1 pn dij Dij ðpÞ ¼ K ij ð1; pÞ
p<
j ¼ 1; 2; . . . ; N1 j ¼ N1 þ 1; . . . ; M þ N1 ð31Þ
j ¼ M þ N1 þ 1; . . . ; N; i ¼ 1; . . . ; M þ N1 i ¼ M þ N 1 þ 1; . . . ; N:
The components of vectors in Eq. (27) are
T g zj ðpn Þ ¼ g 1j ðp1 Þ g 1j ðp2 Þ g 1j ðpm Þ g 2j ðp1 Þ g 2j ðp2 Þ . . . g 2j ðpm Þ ; qj ðsr Þ ¼
½r1nz ðxj ðs1 Þ; yj ðs1 ÞÞ r1nz ðxj ðs2 Þ; yj ðs2 ÞÞ . . . r1nz ðxj ðsm1 Þ; yj ðsm1 ÞÞ 0 : r2nz ðxj ðs1 Þ; yj ðs1 ÞÞ r2nz ðxj ðs2 Þ; yj ðs2 ÞÞ . . . r2nz ðxj ðsm1 Þ; yj ðsm1 ÞÞ 0T
ð32Þ
where r1nz and r2nz are, respectively, the real and imaginary parts of traction vector defined in Eq. (21) and superscript T stands for the transpose of a vector. The calculation of hoop stress on the surface of cavities are accomplished by employing the definition of complex dislocation density function, Bzi(s), and Hooke’s law, yielding
rzt ðai ðsÞ; bi ðsÞÞ ¼ lBzi ðsÞ 1 6 s 6 1;
i ¼ 1; 2; . . . ; M:
ð33Þ
The stress intensity factors for ith crack in terms of crack opening displacement are
pffiffiffi wþ ðsÞ w ðsÞ 2l kLi ¼ lim i pffiffiffiffiffi i ; r Li 4 rLi !0 pffiffiffi wþ ðsÞ w ðsÞ 2l ; kRi ¼ lim i pffiffiffiffiffiffi i r !0 r Ri 4 Ri
ð34Þ
where rLi and rRi designate distances, respectively, from the left and right crack tips to a point on the crack surface, determined as follows:
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M. Ayatollahi, S.J. Fariborz / Applied Mathematical Modelling 33 (2009) 3258–3268 1
r Li ¼ ½ðai ðsÞ ai ð1ÞÞ2 þ ðbi ðsÞ bi ð1ÞÞ2 Þ2 ;
ð35Þ
1 2
r Ri ¼ ½ðai ðsÞ ai ð1ÞÞ2 þ ðbi ðsÞ bi ð1ÞÞ2 Þ :
It is worth mentioning that in the limit as rLi ? 0 and rRi ? 0, the parameter s ? 1 and s ? 1, respectively. The substitution of (25) into (22), and the resultant equations and Eq. (35) into Eq. (34) in conjunction with the Taylor series expansion of functions ai(s) and bi(s) around the points s = ± 1 leads to the stress intensity factors for embedded cracks
kLi ¼ kRi ¼
l
1
2
½½a0i ð1Þ2 þ ½b0i ð1Þ2 4 ½g 1i ð1Þ þ ig 2i ð1Þ;
2
½½a0i ð1Þ2 þ ½b0i ð1Þ2 4 ½g 1i ð1Þ þ ig 2i ð1Þ:
l
ð36Þ
1
Analogously, for an edge crack the stress intensity factor reduces to 1
kLi ¼ l½½a0i ð1Þ2 þ ½b0i ð1Þ2 4 ½g 1i ð1Þ þ ig 2i ð1Þ:
ð37Þ
The solutions of system of Eq. (27) are plugged into Eqs. (33), (36) and (37) thereby obtaining the complex hoop stress for cavities and stress intensity factor for cracks.
3.5
3
k/k
0
2.5
2
1.5
1 0
0.05
0.1
c
0.15
0.2
0.25
Fig. 1. Stress intensity factor verses excitation frequency.
2.5 L1 R1 2
L
2
R
2
k/k
0
1.5
1
0.5
0
0
20
40
60
80 100 q (degrees)
120
140
Fig. 2. Stress intensity factors for stationary and rotating cracks.
160
180
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M. Ayatollahi, S.J. Fariborz / Applied Mathematical Modelling 33 (2009) 3258–3268
6. Numerical results We consider a strip with elastic shear wave velocity c = 3000 (m/s). In all examples, the strip is under concentrated antipffiffiffi plane point loads as specified in Eq. (5), the quantity for making stress intensity factors dimensionless is k0 ¼ p= a, where a half length of an embedded crack, and except for the first example, the frequency of excitation, x = 1000 (rad/s). Moreover, by stress intensity factor and hoop stress we mean the absolute value of the relevant complex quantities. The first example deals with a strip weakened by an embedded crack situated at two different locations yc = 0.5 and yc = 0.3. The problem is symmetric with respect to the y-axis. The non-dimensional stress intensity factor k/k0, versus the wave number, v = kT a, is shown in Fig. 1. Stress intensity factor of the off-center crack for all frequencies of excitation is higher than that of a central crack but exhibit the same trend of variation. The interaction between a stationary centerline crack and a rotating crack is shown in Fig. 2. At h = p/2, the problem becomes symmetric with respect to the center-line of the strip, hence the stress intensity factors at crack tips R2 and L2 are
5.5 5
L1
4.5
R1 L2
4 3.5
k/k
0
3 2.5 2 1.5 1 0.5 0 0
20
40
60
80
100
120
140
160
180
θ (degrees) Fig. 3. Interaction between the edge and embedded cracks.
4 R L
3.5 3
k/k0
2.5 2 1.5 1 0.5
0.1
0.15
0.2
0.25
0.3 6d/h
0.35
0.4
0.45
Fig. 4. Stress intensity factors of two centerline cracks adjacent to an elliptical cavity.
0.5
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M. Ayatollahi, S.J. Fariborz / Applied Mathematical Modelling 33 (2009) 3258–3268
identical. The range of variation of the stress intensity factor at tip R1 is wider than L1 which is due to the stronger interaction with the rotating crack. At crack angles h 78° and h 102°, the stress intensity factor at crack tips L2 and R2, respectively, vanishes. Fig. 3, depicts the stress intensity factors of a rotating embedded crack and a fixed edge crack. At h = p/2 the traction on the embedded crack vanishes and the interaction with the edge crack is weak resulting in a negligible stress intensity factor for the embedded crack. In the fourth example, the stress analysis is carried out in a strip weakened by an elliptical cavity and two adjacent centerline cracks with fixed dimensions. The cavity is treated as a closed curved crack without singularity. Stress intensity factors against non-dimensional distance 6d/h are shown in Fig. 4. The non-dimensional hoop stress on the cavity for three different values of 6d/h is illustrated in Fig. 5. At g = 0 and g = p the hoop stress is zero whereas it is a maximum at the vertices of the cavity. The interaction between defects enhances by diminishing 6d/h. In the last example, the strip is weakened by a circular cavity with radius h/6, an edge and an off-center embedded cracks, inset in Fig. 6. By increasing the length of cracks simultaneously, the distances between cracks tips and cavity decrease resulting in higher stress intensity factors for cracks, Fig. 6. The non-dimensional hoop stress for various d/h is illustrated in Fig. 7. As it was expected, the hoop stress is a minima at g = p.
12
6d/h=0.4 6d/h=0.3 6d/h=0.2
10
hp/τ 0
8 6 4 2 0 0
50
100
150
200 η (degrees)
250
300
350
Fig. 5. Hoop stress on the elliptical cavity located between two centerline cracks.
3.5 R
3
1
L1 L
2.5
2
k/k0
2 1.5 1 0.5 0
0.4
0.5
0.6
0.7
0.8
0.9
d/h Fig. 6. Stress intensity factors of an embedded crack and an edge crack adjacent to a circular cavity.
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M. Ayatollahi, S.J. Fariborz / Applied Mathematical Modelling 33 (2009) 3258–3268
10 9 8 7
h ση / τ0
6 5 4 3 2 1 0
50
100
150 200 η (degrees)
250
300
350
Fig. 7. Hoop stress on a circular cavity located between the edge and embedded cracks.
7. Conclusion The distributed dislocation technique is used to analyze the defects in the forms of cracks and cavities in a strip under time-harmonic anti-plane excitation. The analysis shows that centerline cracks experience lower stress intensity factor than identical off-center cracks parallel to strip. The steady of interaction of fixed and rotating cracks reveals that the variations of stress intensity factor at the tips of rotating cracks are much larger than those of stationary cracks. The interaction of a circular cavity with an edge and an embedded cracks result in large variation of hoop stress on the cavity surface. References [1] G.C. Sih, E.P. Chen, Moving cracks in finite strip under tearing action, J. Frank. Inst. 290 (1970) 25–35. [2] G. Kuhn, M. Matczynski, Elastic strip with a crack under periodic loading, Arch. Mech. 27 (1975) 459–472. [3] K.N. Srivastava, R.M. Palaiya, D.S. Karaulia, Interaction of shear waves with two coplanar Griffith cracks situated in an infinitely long elastic strip, Int. J. Fract. 23 (1983) 3–14. [4] H.M. Huang, H.J. Shi, Y.J. Yin, Multi-cracks problem for piezoelectric materials strip subjected to dynamic loading, Mech. Res. Commun. 29 (2002) 413– 424. [5] X.F. Li, Griffith crack moving in a piezoelectric strip, Arch. Appl. Mech. 72 (2003) 745–758. [6] M. Ayatollahi, S.J. Fariborz, M. Ahmadi Najafabadi, Antiplane elastodynamic analysis of planes with multiple defects, Appl. Math. Model. 33 (2009) 663– 676. [7] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, ninth ed., Dover, New York, 1972. [8] J.D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, Amsterdam, 1976. [9] R.T. Faal, S.J. Fariborz, H.R. Daghyani, Anti-plane deformation of orthotropic strips with multiple defects, J. Mech. Mater. Struct. 1 (2007) 1097–1113.
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Antiplane elastodynamic analysis of a strip weakened by multiple defects M. Ayatollahi a, S.J. Fariborz b,* a b
Faculty of Engineering, Zanjan University, Zanjan 45195-313, Iran Department of Mechanical Engineering, Amirkabir University of Technology (Tehran Polytechnic) 424, Hafez Avenue, Tehran 158754413, Iran
a r t i c l e
i n f o
Article history: Received 11 November 2007 Received in revised form 18 October 2008 Accepted 22 October 2008 Available online 30 October 2008
Keywords: Antiplane Strip Time-harmonic Point load Multiple defects
a b s t r a c t The method of images is utilized to derive the solution of a screw dislocation under timeharmonic conditions for an elastic strip from the solution of infinite planes. The displacement and stress components are obtained for a strip under concentrated antiplane, timeharmonic traction. The dislocation solution is employed to formulate integral equation for a strip weakened by cracks and cavities. The effects of load frequency and crack orientation on the stress intensity factors are studied. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction The stress analysis in elastic regions, weakened by defects subject to dynamic loading, is imperative for evaluating the damage caused by fatigue of material. The analytical techniques for stress analysis are, however, only capable of handling regions with simple geometries. Nonetheless, the results may be used as a benchmark solution for the verification of numerical results. Apparently, the stress analysis in elastic strips weakened by cracks under dynamic loading, unlike the static case, has not attracted much attention. The earliest study dates back to an article by Sih and Chen [1], dealing with a semi-infinite central crack traveling with constant velocity under antiplane excitation. The problem was reduced to the Riemann–Hilbert problem which was solved in closed-form for stress field in a strip with two different cases of boundary conditions. Kuhn and Matczynski [2], investigated the in-plane stress distribution of a stationary central crack in a strip subjected to periodic loading by means of complex Fourier transform and the Weiner–Hopf technique. Two coplanar cracks in a strip under shear wave was studied by Srivastava et al. [3]. A strip weakened by stationary cracks composed of piezoelectric material under antiplane harmonic load on the cracks surfaces was analyzed by some researchers, see for instance Huang et al. [4]. A piezoelectric strip containing a crack parallel to the strip boundaries and traveling with constant velocity under the action of uniform anti-plane shear stress and electric field was investigated by Li [5]. By eliminating the electric field, the solution to an elastic strip with the same geometry and mechanical loading may be reached. In the present study, the solution of screw dislocation with harmonic motion in infinite planes obtained in [6] together with the method of images are employed to derive the solution of screw dislocation in an elastic strip. The elastodynamic problem of a flawless strip under anti-plane, self-equilibrium, time-harmonic point forces is solved. The solution may be viewed as the Green’s function for a strip under arbitrary self-equilibrating traction. The dislocation solutions are employed to formulate integral equations for a strip weakened by cracks and/or cavities. The integral equations are of Cauchy singular * Corresponding author. Tel.: +98 21 6454 3460; fax: +98 21 641 9736. E-mail address: [email protected] (S.J. Fariborz). 0307-904X/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2008.10.026
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Nomenclature Aij, Cij, Dij coefficients matrix A(n), B(n) unknown coefficients Burgers vector bz complex dislocation densities Bzj B1j, B2j real and imaginary parts of dislocation densities c shear wave velocity Ci(x), Si(x) sine and cosine integral functions regular terms of dislocation densities gij(p) h thickness of strip H(x) Heaviside step function Hankel function of the second kind of order p H2p ðxÞ Jp(x), Yp(x) Bessel functions of first and second kinds of order p stress intensity factors of left and right side of crack kLi, kRi stress intensity factor of a crack in strip k0 kij(s, p) kernel of integral equations kij1, kij2 real and imaginary parts of kernel of integral equations wave number kT M number of cavities N total number of defects number of embedded cracks N1 p time-harmonic concentrated load distance from right and left crack tips rLi, rRi w amplitude of displacement component x, y coordinates ai(s), bi(s) functions describing the geometry of defects d(n) Dirac delta function Kronecker delta dij crack orientation hi(s) l elastic shear modulus q mass density r1nz, r2nz real and imaginary parts of traction vector rxz, ryz out of plane stress components x angular frequency
type which are solved numerically for the dislocation density on the defects surfaces. Several examples of strips with cracks and cavities are solved to study the effects of excitation frequency on stress intensity factors and also the interaction between defects. 2. Dislocation solution The stress components in an infinite plane containing a time-harmonic screw dislocation with the amplitude of Burgers vector bz located at a point with coordinates (n, g) were derived in [6] and is recited here
Z
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
lbz ðy gÞ lbz ðy gÞ 1 lnðu þ u2 1ÞeikT ur rxz ¼ ½cosðk rÞ i sinðk rÞ du ½Y 1 ðkT rÞ þ iJ 1 ðkT rÞ þ T T 4r 2pr 2 pr2 ðu2 y2 =r 2 Þ2 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z lbz ðy gÞ3 1 u2 1eikT ur ilbz kT ðy gÞ 1 ðu u2 1 u2 þ y2 =r 2 lnðu þ u2 1ÞÞeikT ur þ du þ du 2 4 2pr pr u2 y2 =r 2 ðu2 y2 =r 2 Þ 1 1 lbz kT ðy gÞ
ryz ¼
lbz kT ðx nÞ 4r þ
"
½Y 1 ðkT rÞ þ iJ1 ðkT rÞ þ
ilbz kT ðx nÞ pr
Z 1
1
lbz k2T ðx nÞ ip CiðkT rÞ iSiðkT rÞ þ 2p 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u2 1eikT ur ðu2 y2 =r 2 Þ2
du;
Z 1
1
# pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu u2 1 u2 þ y2 =r2 ÞeikT ur du uðu2 y2 =r 2 Þ ð1Þ
where Ci(x) and Si(x) are, respectively, the sine and cosine integral functions, Abramowitz and Stegun [7], kT = x/c and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r ¼ ðx nÞ2 þ ðy gÞ2 . For a strip with thickness 0 6 y 6 h weakened by a screw dislocation with Bergers vector bz situated at a point with coordinates (n, g), the stress components may be obtained from the solution of an infinite plane by positioning dislocations with Bergers vector bz at points with coordinates x = n, y = g ± 2kh, k 2 {0,1,2,. . .} and dislocations with
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M. Ayatollahi, S.J. Fariborz / Applied Mathematical Modelling 33 (2009) 3258–3268
Bergers vector bz at points with coordinates x = n, y = 2h(1 ± k) g, k 2 {0,1,2,. . .}. Employing Eq. (1), and the principle of superposition, the stress components in the strip containing dislocation yield
rxz ¼ n!1 lim
ryz
n X
lbz
k¼n
kT ðy g 2khÞ kT ðy 2hð1 þ kÞ þ gÞ ½Y 1 ðkT r 3 Þ þ iJ1 ðkT r 3 Þ ½Y 1 ðkT r 4 Þ þ iJ1 ðkT r 4 Þ 4r 3 4r4
ðy g 2khÞ ðy 2hð1 þ kÞ þ gÞ þ ½cosðkT r 3 Þ i sinðkT r 3 Þ ½cosðkT r 4 Þ i sinðkT r 4 Þ 2pr 23 2pr 24 p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z ðy g 2khÞ 1 lnðu þ u2 1ÞeikT ur3 ðy 2hð1 þ kÞ þ gÞ 1 lnðu þ u2 1ÞeikT ur4 du þ du pr23 pr24 ðu2 ðy g 2khÞ2 =r 23 Þ2 ðu2 ðy 2hð1 þ kÞ þ gÞ2 =r 24 Þ2 1 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p p Z Z u2 1eikT ur3 u2 1eikT ur4 ðy g 2khÞ3 1 ðy 2hð1 þ kÞ þ gÞ3 1 du du þ 4 2 2 4 pr 3 pr 4 ðu2 ðy g 2khÞ =r23 Þ ðu2 ðy 2hð1 þ kÞ þ gÞ2 =r 24 Þ2 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ikT ðy g 2khÞ 1 ðu u2 1 u2 þ ðy g 2khÞ2 =r 23 lnðu þ u2 1ÞÞeikT ur3 du þ 2pr 3 u2 ðy g 2khÞ2 =r23 1 ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ikT ðy 2hð1 þ kÞ þ gÞ 1 ðu u2 1 u2 þ ðy 2hð1 þ kÞ þ gÞ2 =r 24 lnðu þ u2 1ÞÞeikT ur4 du 2pr 4 u2 ðy 2hð1 þ kÞ þ gÞ2 =r 24 1 ( n X ðx nÞ ðx nÞ ¼ lim lb z k T ½Y 1 ðkT r 3 Þ þ iJ1 ðkT r 3 Þ ½Y 1 ðkT r 4 Þ þ iJ1 ðkT r 4 Þ n!1 4r 4r4 3 k¼n " # ffi Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi kT ðx nÞ ip ðu u2 1 u2 þ ðy g 2khÞ2 =r 23 ÞeikT ur3 þ CiðkT r 3 Þ iSiðkT r3 Þ þ du 2p 2 uðu2 ðy g 2khÞ2 =r 23 Þ 1 " # p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z 1 kT ðx nÞ ip ðu u2 1 u2 þ ðy 2hð1 þ kÞ þ gÞ2 =r 24 ÞeikT ur1 CiðkT r 4 Þ iSiðkT r4 Þ þ du 2p 2 uðu2 ðy 2hð1 þ kÞ þ gÞ2 =r 24 Þ 1 ) p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z Z iðx nÞ 1 u u2 1eikT ur3 iðx nÞ 1 u u2 1eikT ur4 du du ; ð2Þ þ 2 2 pr3 1 ðu2 ðy g 2khÞ =r23 Þ pr4 1 ðu2 ðy 2hð1 þ kÞ þ gÞ2 =r24 Þ2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r 3 ¼ ðx nÞ2 þ ðy g 2khÞ2 ; r4 ¼ ðx nÞ2 þ ðy 2hð1 þ kÞ þ gÞ2 . It is worth mentioning that the integrands in stress components (2) are continuous functions in 1 6 u < 1 with a sufficiently fast rate of decay as u ? 1, making them susceptible to numerical evaluation. Nonetheless, the series in (2) behaves like 1/k, as k ? 1, requiring a large number of terms for convergence. 3. Strip under antiplane point-force The constitutive equations, for an elastic medium with elastic shear modulus l under anti-plane deformation W(x, y, t) are
oW ; ox oW ryz ðx; y; tÞ ¼ l oy
rxz ðx; y; tÞ ¼ l
ð3Þ
The equation of motion for an elastic body under anti-plane deformation is
o2 W o2 W 1 o2 W þ 2 ¼ 2 : ox2 oy c ot 2
ð4Þ
pffiffiffiffiffiffiffiffiffiffiffi In Eq. (4), the characteristic shear wave velocity is c ¼ l=q; where l is the elastic shear modulus and q is the mass density of material. We let the strip be under time-harmonic concentrated applied loads with magnitude p on the edges represented by
ryz ðx; 0; tÞ ¼ pdðxÞeixt ; ryz ðx; h; tÞ ¼ pdðxÞeixt
ð5Þ
where d(x) is the Dirac delta function. The solution to Eq. (4) is considered in separable form
Wðx; y; tÞ ¼ wðx; yÞeixt :
ð6Þ
The substitution of (6) into (4) results in
o2 w o2 w 2 þ 2 þ kT w ¼ 0; ox2 oy
ð7Þ
M. Ayatollahi, S.J. Fariborz / Applied Mathematical Modelling 33 (2009) 3258–3268
3261
where kT = x/c is the wave number. The complex Fourier transform is defined as
f ðnÞ ¼
Z
1
f ðxÞeinx dx;
ð8Þ
1
the inversion of Fourier transform yields
f ðxÞ ¼
1 2p
Z
1
f ðnÞeinx dn:
ð9Þ
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The application of Eqs. (8) to (7), in view of the requirement lim wðx; yÞ ¼ 0, where r ¼ x2 þ y2 , yields a second order orr!1 * dinary differential equation for w (n, y). The solution to the differential equation is readily known
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2 2 2 2 w ðn; yÞ ¼ AðnÞey n kT þ BðnÞey n kT :
ð10Þ
The coefficients in Eq. (10) may be obtained by the application of boundary condition (5) together with Fourier transform (8). By virtue of inverse transform (9) the displacement field becomes
2
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2 2 Z 1 2 2 einxy n kT einxþy n kT ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q dn dn pffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 1 1 n2 kT ð1 e2h n kT Þ n2 kT ð1 e2h n kT Þ 3 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi Z 1 inxðhyÞ n2 k2T inxþðhyÞ n2 k2T e e 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dn þ dn5: pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 2 2 1 2h n k 2h n k T T n kT ð1 e n kT ð1 e Þ Þ
p 6 wðx; yÞ ¼ 4 2pl
Z
1
1
Z
1
ð11Þ
To simplify the integrals in Eq. (11) the procedure outlined in Achenbach [8] is followed. The final result leads to
wðx; yÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i p h p Y 0 ðkT x2 þ y2 Þ þ iJ0 ðkT x2 þ y2 Þ Y 0 ðkT x2 þ ðh yÞ2 Þ þ iJ0 ðkT x2 þ ðh yÞ2 Þ 2l 2l Z Z 2ip 1 eir1 kT u A1 sin c1 2ip 1 eir2 kT u A2 sin c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ du du; 2 pl 1 pl 1 u2 1 1 þ A1 2A1 cos c1 u2 1 1 þ A22 2A2 cos c2
ð12Þ
where J0(kTr) and Y0(kTr) are the Bessel functions of zero order of first and second kinds, respectively, and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 ; h1 ¼ tan1 ðy=xÞ; c1 ¼ 2hkT u sin h1 ; A1 ¼ expð2hkT cos h1 u2 1Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 ¼ x2 þ ðh yÞ2 ; h2 ¼ tan1 ½ðh yÞ=x; c2 ¼ 2hkT u sin h2 ; A2 ¼ expð2hkT cos h2 u2 1Þ r1 ¼
ð13Þ
In the particular case of a half-plane under anti-plane point traction, we set h ? 1 in Eq. (12) carry out the integration and obtain the well-known solution
wðx; yÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i p h Y 0 ðkT x2 þ y2 Þ þ iJ0 ðkT x2 þ y2 Þ : 2l
ð14Þ
From Eqs. (3), (8), and (10), the Fourier transform of stress components become
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi " pffiffiffiffiffiffiffiffiffi # 2 2 2 2 2 2 2 2 ey n kT eðhyÞ n kT ey n kT eðhyÞ n kT pffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffi ; 2 2 2 2 2 1 e2h n kT 1 e2h n kT n2 kT pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi " pffiffiffiffiffiffiffiffiffi # 2 2 2 2 2 2 2 2 ey n kT þ eðhyÞ n kT ey n kT þ eðhyÞ n kT pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ¼p þ : 2 2 2 2 1 e2h n kT 1 e2h n kT ipn
rxz ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ryz
ð15Þ
Similar to the displacement field the Fourier inversion of Eq. (15) yield
rxz ¼
pkT cos h1 pk cos h2 ½Y 1 ðkT r 1 Þ þ iJ1 ðkT r 1 Þ þ T ½Y 1 ðkT r2 Þ þ iJ1 ðkT r2 Þ 2 2 Z Z pk cos h1 1 ueir1 kT u A1 sin c1 pkT cos h2 1 ueir2 kT u A2 sin c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ T du du 2 2 2 p p u 1ð1 þ A1 2A1 cos c1 Þ u 1ð1 þ A22 2A2 cos c2 Þ 1 1 pkT sin h1 pk sin h2 ½Y 1 ðkT r 1 Þ þ iJ1 ðkT r 1 Þ T ½Y 1 ðkT r 2 Þ þ iJ 1 ðkT r 2 Þ 2 2 Z Z pk sin h1 1 ueir1 kT u A1 sin c1 pkT sin h2 1 ueir2 kT u A2 sin c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ du: þ T 2 2 2 p p u 1ð1 þ A1 2A1 cos c1 Þ u 1ð1 þ A22 2A2 cos c2 Þ 1 1
ryz ¼
ð16Þ
At the points of application of load, i.e., r1 ? 0 and r2 ? 0, stress fields exhibit the familiar Cauchy type singularity, 1/r1 and 1/r2.
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4. Strip weekend by defects The dislocation solutions accomplished in the preceding section may be employed to analyze a strip with N arbitrarily oriented defects, curved cracks and cavities, with smooth surfaces. Cavities are considered as closed curved cracks without singularity [6]. The defects configuration may be described in parametric form as
xi ¼ ai ðsÞ yi ¼ bi ðsÞ
ð17Þ
1 6 s 6 1 i ¼ 1; 2; . . . ; N:
The moveable orthogonal t, n-coordinate system is chosen such that the origin may move on a defect while the t-axis remains tangent to the defect surface. The anti-plane traction on the surface of the ith defect in terms of stress components in Cartesian coordinates is
rnz ðxi ; yi Þ ¼ ryz cos hi ðsÞ rxz sin hi ðsÞ; 1
ð18Þ
ðb0i ðsÞ= 0i ðsÞÞ
where hi ðsÞ ¼ tan a is the angle between x- and t-axes on theq ith defect. Suppose dislocations with unknown comffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi plex density Bzj = B1j + iB2j are distributed on the infinitesimal segment ½a0j 2 þ ½b0j 2 dp at the surface of the jth defect where the parameter 1 6 p 6 1and prime denotes differentiation with respect to the relevant argument. The traction on the surface of ith defect due to the presence of dislocations on the surfaces of all N defects yields
rnz ðai ðsÞ; bi ðsÞÞ ¼
N Z X
1
1
j¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ij ðs; pÞ ½a0j ðpÞ2 þ ½b0j ðpÞ2 ½B1j ðpÞ þ iB2j ðpÞdp i ¼ 1; 2; . . . ; N:
ð19Þ
From Eqs. (16) and (18), the kernel of integral Eq. (19) becomes
K ij ðai ðsÞ; bi ðsÞ; aj ðpÞ; bj ðpÞÞ ¼ lim
n!1
n X k ða a Þ k ða a Þ l T i j ½Y 1 ðkT r3ij Þ þ iJ1 ðkT r3ij Þ T i j ½Y 1 ðkT r4ij Þ þ iJ1 ðkT r4ij Þ 4r 4r4ij 3ij k¼n
" # Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðu u2 1 u2 þ ðbi bj 2khÞ2 =r23ij ÞeikT ur3ij kT ðai aj Þ ip CiðkT r3ij Þ iSiðkT r 3ij Þ þ þ du 2 2p uðu2 ðbi bj 2khÞ2 =r23ij Þ 1
" # Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðu u2 1 u2 þ ðbi 2hð1 þ kÞ þ bj Þ2 =r 24ij ÞeikT ur4ij kT ðai aj Þ ip CiðkT r4ij Þ iSiðkT r 4ij Þ þ du 2 2p uðu2 ðbi 2hð1 þ kÞ þ bj Þ2 =r24ij Þ 1
þ
ikT ðai aj Þ pr3ij
Z
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u2 1eikT ur3ij
1
du ðu2 ðbi bj 2khÞ2 =r23ij Þ2
1
ikT ðai aj Þ pr4ij
Z
1 1
du cos hi ðsÞ ðu2 ðbi 2hð1 þ kÞ þ bj Þ2 =r24ij Þ2
kT ðbi bj 2khÞ kT ðbi 2hð1 þ kÞ þ bj Þ ½Y 1 ðkT r3ij Þ þ iJ1 ðkT r3ij Þ ½Y 1 ðkT r4ij Þ þ iJ 1 ðkT r4ij Þ 4r 3ij 4r4ij
þ
ðbi bj 2khÞ ðbi 2hð1 þ kÞ þ bj Þ ½cosðkT r 3ij Þ i sinðkT r3ij Þ ½cosðkT r 4ij Þ i sinðkT r4ij Þ 2pr 23ij 2pr24ij
ðbi bj 2khÞ pr23ij
Z
3
þ
)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u2 1eikT ur4ij
ðbi bj 2khÞ pr43ij
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 1ÞeikT ur3ij
du þ ðu2 ðbi bj 2khÞ2 =r23ij Þ2
1
Z
ikT ðbi bj 2khÞ þ 2pr3ij
lnðu þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 1eikT ur3ij
1
ðbi 2hð1 þ kÞ þ bj Þ pr24ij
du ðu2 ðbi bj 2khÞ2 =r23ij Þ2
1
Z
1
Z
ðbi 2hð1 þ kÞ þ bj Þ pr44ij
1
Z 1
u2 ðbi bj 2khÞ2 =r23ij
ikT ðbi 2hð1 þ kÞ þ bj Þ 2pr4ij
Z
1 1
ðu2 ðbi 2hð1 þ kÞ þ bj Þ2 =r 24ij Þ2
du
du
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðu u2 1 u2 þ ðbi 2hð1 þ kÞ þ bj Þ =r 24ij lnðu þ u2 1ÞÞeikT ur4ij u2 ðbi 2hð1 þ kÞ þ bj Þ2 =r24ij
du
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 1eikT ur4ij
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu u2 1 u2 þ ðbi bj 2khÞ2 =r23ij lnðu þ u2 1ÞÞeikT ur3ij
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 1ÞeikT ur4ij
ðu2 ðbi 2hð1 þ kÞ þ bj Þ2 =r24ij Þ2
1 3
lnðu þ
)
#
du sin hi ðsÞ ;
ð20Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r 3ij ¼ ðai aj Þ2 þ ðbi bj 2khÞ2 ; r4ij ¼ ðai aj Þ2 þ ðbi 2hð1 þ kÞ þ bj Þ2 . It is noteworthy to mention that in Eqs. (20), a and b with subscripts i and j are functions of s and p, respectively. It is easy to observe that the kernel of integral Eq. (19) exhibit Cauchy type singularity for i = j as r3ij ? 0 or r4ij ? 0.
M. Ayatollahi, S.J. Fariborz / Applied Mathematical Modelling 33 (2009) 3258–3268
3263
The anti-plane traction on the surface of the ith defect in view of Eqs. (16) and (18) becomes
(
bi b ½Y 1 ðkT r 1i Þ þ iJ1 ðkT r 1i Þ i ½Y 1 ðkT r 2i Þ þ iJ1 ðkT r 2i Þ: 2r 1i 2r 2i " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 2 2 kT bi 1 ir 1i kT u 2huai sin c1i þ ir 1i u 1 sin c1i 2hbi u 1 cos c1i þ A1 e 2 pr1i 1 ð1 þ A1 2A1 cos c1i Þr1i h i3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4A1 h sin c1i uai ðA1 cos c1i Þ bi u2 1 sin c1i 5du ð1 þ A21 2A1 cos c1i Þ2 r 1i " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 2 2 kT ðbi hÞ 1 ir2i kT u 2huai sin c2i þ ir 2i u 1 sin c2i 2hðh bi Þ u 1 cos c2i þ A2 e : 2 pr2i ð1 þ A2 2A2 cos ci Þr 2i 1 h i3 9 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 4A2 h sin c2i uai ðA2 cos c2i Þ ðh bi Þ u2 1 sin c2i 5du cos hi ðsÞ 2 2 ; ð1 þ A2 2A2 cos c2i Þ r2i ai ai þ pkT ½Y 1 ðkT r1i Þ þ iJ1 ðkT r1i Þ ½Y 1 ðkT r2i Þ þ iJ1 ðkT r 2i Þ 2r1i 2r2i " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 2 kT ai 2huai sin c1i þ ir 1i u2 1 sin c1i 2hbi u2 1 cos c1i A1 eir1i kT u pr1i 1 ð1 þ A21 2A1 cos c1i Þr1i h i3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4A1 h sin c1i uai ðA1 cos c1i Þ bi u2 1 sin c1i 5du ð1 þ A21 2A1 cos c1i Þ2 r 1i " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 2 2 kT ai 1 ir 2i kT u 2huai sin c2i þ ir 2i u 1 sin c2i 2hðh bi Þ u 1 cos c2i þ A2 e 2 pr2i 1 ð1 þ A 2A2 cos ci Þr 2i # ) pffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi 4A2 h sin c2i ½uai ðA2 cos c2i Þ ðh bi Þ u2 1 sin c2i du sin hi ðsÞ; ð1 þ A22 2A2 cos c2i Þ2 r 2i
rnz ¼ pkT
ð21Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r 1i ¼ a2i þ b2i ; r2i ¼ a2i þ ðh bi Þ2 and c1i = 2hkTubi/r1i, c2i = 2hkTubi/r2i. Employing the definition of dislocation density function, the anti-plane opening across the boundary of the jth curved crack is
wj ðsÞ wþj ðsÞ ¼
Z
s
1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½a0j ðpÞ2 þ ½b0j ðpÞ2 ½B1j ðpÞ þ iB2j ðpÞdp j ¼ 1; 2; . . . ; N:
ð22Þ
Let the strip be weakened by M, N1 and N2, cavities, embedded and edge cracks, respectively. The displacement field is singlevalued out of the surfaces of embedded defects. Consequently, the dislocation density functions are subjected to the following closure requirement:
Z
1
1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½a0j ðpÞ2 þ ½b0j ðpÞ2 Bij ðpÞdp ¼ 0 i ¼ 1; 2 j ¼ 1; 2; . . . ; M þ N 1 :
ð23Þ
5. Solution of integral equations To obtain dislocation density functions, the integral Eqs. (19) and (23) are to be solved simultaneously. This task is taken up by the methodology devised in Faal et al. [9]. Cavities are considered as closed curved cracks with bounded dislocation density at both ends leading to
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bij ðpÞ ¼ g ij ðpÞ 1 p2
16p61
i ¼ 1; 2 j ¼ 1; 2; . . . ; M:
ð24Þ
The stress fields for embedded cracks are singular at the crack tips with square root singularity. Therefore, the dislocation density functions for embedded cracks are represented as
g ij ðpÞ Bij ðpÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p2
1 6 p 6 1 i ¼ 1; 2 j ¼ M þ 1; M þ 2; . . . ; M þ N 1 :
ð25Þ
For the edge cracks, taking the embedded crack tip at p = 1, the following dislocation density function is considered
sffiffiffiffiffiffiffiffiffiffiffiffi 1p ; Bij ðpÞ ¼ g ij ðpÞ 1þp
1 6 p 6 1 i ¼ 1; 2 j ¼ M þ N1 þ 1; M þ N1 þ 2; . . . ; N:
ð26Þ
The functions gij(p) in Eqs. (24)–(26) are continuous in 1 6 p 6 1. Substituting Eqs. (24)–(26) into Eqs. (19) and (23) and discretizing the domain, 1 6 p 6 1 by m + 1 segments, the integral equations are reduced to the following system of N 2m linear algebraic equations
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M. Ayatollahi, S.J. Fariborz / Applied Mathematical Modelling 33 (2009) 3258–3268
2
A1N
32
g z1 ðpn Þ
3
2
q1 ðsr Þ
3
A11
A12
6A 6 21 6 . 6 . 4 . AN1
A22 .. . AN2
7 6 7 6 A2N 7 76 g z2 ðpn Þ 7 6 q2 ðsr Þ 7 7¼6 7 6 .. .. 7 . . 76 7 6 . 7; .. 5 4 . 5 . . 54 ANN g zN ðpn Þ qN ðsr Þ
ð27Þ
where the collocation points are
pr
r ¼ 1; 2; . . . ; m 1; pð2n 1Þ pn ¼ cos n ¼ 1; 2; . . . ; m: 2m
sr ¼ cos
m
ð28Þ
The components of matrix in Eq. (27) are
Aij ¼
Bij
C ij
C ij
Bij
ð29Þ
;
where
2
Hj1 K 1ij ðs1 ; p1 ÞDi ðp1 Þ
6 6 Hj1 K 1ij ðs2 ; p1 ÞDi ðp1 Þ 6 p6 6 .. Bij ¼ 6 . m6 6 6 Hj1 K 1ij ðsm1 ; p ÞDi ðp Þ 1 1 4 Hj1 Dij ðp1 ÞDi ðp1 Þ 2 Hj1 K 2ij ðs1 ; p1 ÞDi ðp1 Þ 6 6 Hj1 K 2ij ðs2 ; p1 ÞDi ðp1 Þ 6 p6 6 C ij ¼ 6 ... m6 6 6 Hj1 K 2ij ðsm1 ; p ÞDi ðp Þ 1 1 4 0
Hj2 K 1ij ðs1 ; p2 ÞDi ðp2 Þ
Hj2 K 1ij ðs2 ; p2 ÞDi ðp2 Þ
.. .
..
.
Hj2 K 1ij ðsm1 ; p2 ÞDi ðp2 Þ Hj2 Dij ðp1 ÞDi ðp2 Þ
Hj2 K 2ij ðs1 ; p2 ÞDi ðp2 Þ
Hj2 K 2ij ðs2 ; p2 ÞDi ðp2 Þ
.. .
..
.
Hj2 K 2ij ðsm1 ; p2 ÞDi ðp2 Þ 0
Hjn K 1ij ðs1 ; pm ÞDi ðpm Þ
3
7 Hjn K 1ij ðs2 ; pm ÞDi ðpm Þ 7 7 7 7 .. 7; . 7 7 Hjn K 1ij ðsm1 ; pm ÞDi ðpm Þ 7 5 Hjn Dij ðp1 ÞDi ðpm Þ 3 Hjn K 2ij ðs1 ; pm ÞDi ðpm Þ 7 Hjn K 2ij ðs2 ; pm ÞDi ðpm Þ 7 7 7 7 .. 7: . 7 7 Hjn K 2ij ðsm1 ; pm ÞDi ðpm Þ 7 5
ð30Þ
0
In matrices Bij and Cij, the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi coefficients K1ij and K2ij are the real and imaginary parts of Kij specified in Eq. (20), Di ðpj Þ ¼ ½a0i ðpj Þ2 þ ½b0i ðpj Þ2 , and Hjn and Dij(p) are as follows:
8 > 1 p2n Hjn ¼ 1 m> : 1 pn dij Dij ðpÞ ¼ K ij ð1; pÞ
p<
j ¼ 1; 2; . . . ; N1 j ¼ N1 þ 1; . . . ; M þ N1 ð31Þ
j ¼ M þ N1 þ 1; . . . ; N; i ¼ 1; . . . ; M þ N1 i ¼ M þ N 1 þ 1; . . . ; N:
The components of vectors in Eq. (27) are
T g zj ðpn Þ ¼ g 1j ðp1 Þ g 1j ðp2 Þ g 1j ðpm Þ g 2j ðp1 Þ g 2j ðp2 Þ . . . g 2j ðpm Þ ; qj ðsr Þ ¼
½r1nz ðxj ðs1 Þ; yj ðs1 ÞÞ r1nz ðxj ðs2 Þ; yj ðs2 ÞÞ . . . r1nz ðxj ðsm1 Þ; yj ðsm1 ÞÞ 0 : r2nz ðxj ðs1 Þ; yj ðs1 ÞÞ r2nz ðxj ðs2 Þ; yj ðs2 ÞÞ . . . r2nz ðxj ðsm1 Þ; yj ðsm1 ÞÞ 0T
ð32Þ
where r1nz and r2nz are, respectively, the real and imaginary parts of traction vector defined in Eq. (21) and superscript T stands for the transpose of a vector. The calculation of hoop stress on the surface of cavities are accomplished by employing the definition of complex dislocation density function, Bzi(s), and Hooke’s law, yielding
rzt ðai ðsÞ; bi ðsÞÞ ¼ lBzi ðsÞ 1 6 s 6 1;
i ¼ 1; 2; . . . ; M:
ð33Þ
The stress intensity factors for ith crack in terms of crack opening displacement are
pffiffiffi wþ ðsÞ w ðsÞ 2l kLi ¼ lim i pffiffiffiffiffi i ; r Li 4 rLi !0 pffiffiffi wþ ðsÞ w ðsÞ 2l ; kRi ¼ lim i pffiffiffiffiffiffi i r !0 r Ri 4 Ri
ð34Þ
where rLi and rRi designate distances, respectively, from the left and right crack tips to a point on the crack surface, determined as follows:
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M. Ayatollahi, S.J. Fariborz / Applied Mathematical Modelling 33 (2009) 3258–3268 1
r Li ¼ ½ðai ðsÞ ai ð1ÞÞ2 þ ðbi ðsÞ bi ð1ÞÞ2 Þ2 ;
ð35Þ
1 2
r Ri ¼ ½ðai ðsÞ ai ð1ÞÞ2 þ ðbi ðsÞ bi ð1ÞÞ2 Þ :
It is worth mentioning that in the limit as rLi ? 0 and rRi ? 0, the parameter s ? 1 and s ? 1, respectively. The substitution of (25) into (22), and the resultant equations and Eq. (35) into Eq. (34) in conjunction with the Taylor series expansion of functions ai(s) and bi(s) around the points s = ± 1 leads to the stress intensity factors for embedded cracks
kLi ¼ kRi ¼
l
1
2
½½a0i ð1Þ2 þ ½b0i ð1Þ2 4 ½g 1i ð1Þ þ ig 2i ð1Þ;
2
½½a0i ð1Þ2 þ ½b0i ð1Þ2 4 ½g 1i ð1Þ þ ig 2i ð1Þ:
l
ð36Þ
1
Analogously, for an edge crack the stress intensity factor reduces to 1
kLi ¼ l½½a0i ð1Þ2 þ ½b0i ð1Þ2 4 ½g 1i ð1Þ þ ig 2i ð1Þ:
ð37Þ
The solutions of system of Eq. (27) are plugged into Eqs. (33), (36) and (37) thereby obtaining the complex hoop stress for cavities and stress intensity factor for cracks.
3.5
3
k/k
0
2.5
2
1.5
1 0
0.05
0.1
c
0.15
0.2
0.25
Fig. 1. Stress intensity factor verses excitation frequency.
2.5 L1 R1 2
L
2
R
2
k/k
0
1.5
1
0.5
0
0
20
40
60
80 100 q (degrees)
120
140
Fig. 2. Stress intensity factors for stationary and rotating cracks.
160
180
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M. Ayatollahi, S.J. Fariborz / Applied Mathematical Modelling 33 (2009) 3258–3268
6. Numerical results We consider a strip with elastic shear wave velocity c = 3000 (m/s). In all examples, the strip is under concentrated antipffiffiffi plane point loads as specified in Eq. (5), the quantity for making stress intensity factors dimensionless is k0 ¼ p= a, where a half length of an embedded crack, and except for the first example, the frequency of excitation, x = 1000 (rad/s). Moreover, by stress intensity factor and hoop stress we mean the absolute value of the relevant complex quantities. The first example deals with a strip weakened by an embedded crack situated at two different locations yc = 0.5 and yc = 0.3. The problem is symmetric with respect to the y-axis. The non-dimensional stress intensity factor k/k0, versus the wave number, v = kT a, is shown in Fig. 1. Stress intensity factor of the off-center crack for all frequencies of excitation is higher than that of a central crack but exhibit the same trend of variation. The interaction between a stationary centerline crack and a rotating crack is shown in Fig. 2. At h = p/2, the problem becomes symmetric with respect to the center-line of the strip, hence the stress intensity factors at crack tips R2 and L2 are
5.5 5
L1
4.5
R1 L2
4 3.5
k/k
0
3 2.5 2 1.5 1 0.5 0 0
20
40
60
80
100
120
140
160
180
θ (degrees) Fig. 3. Interaction between the edge and embedded cracks.
4 R L
3.5 3
k/k0
2.5 2 1.5 1 0.5
0.1
0.15
0.2
0.25
0.3 6d/h
0.35
0.4
0.45
Fig. 4. Stress intensity factors of two centerline cracks adjacent to an elliptical cavity.
0.5
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M. Ayatollahi, S.J. Fariborz / Applied Mathematical Modelling 33 (2009) 3258–3268
identical. The range of variation of the stress intensity factor at tip R1 is wider than L1 which is due to the stronger interaction with the rotating crack. At crack angles h 78° and h 102°, the stress intensity factor at crack tips L2 and R2, respectively, vanishes. Fig. 3, depicts the stress intensity factors of a rotating embedded crack and a fixed edge crack. At h = p/2 the traction on the embedded crack vanishes and the interaction with the edge crack is weak resulting in a negligible stress intensity factor for the embedded crack. In the fourth example, the stress analysis is carried out in a strip weakened by an elliptical cavity and two adjacent centerline cracks with fixed dimensions. The cavity is treated as a closed curved crack without singularity. Stress intensity factors against non-dimensional distance 6d/h are shown in Fig. 4. The non-dimensional hoop stress on the cavity for three different values of 6d/h is illustrated in Fig. 5. At g = 0 and g = p the hoop stress is zero whereas it is a maximum at the vertices of the cavity. The interaction between defects enhances by diminishing 6d/h. In the last example, the strip is weakened by a circular cavity with radius h/6, an edge and an off-center embedded cracks, inset in Fig. 6. By increasing the length of cracks simultaneously, the distances between cracks tips and cavity decrease resulting in higher stress intensity factors for cracks, Fig. 6. The non-dimensional hoop stress for various d/h is illustrated in Fig. 7. As it was expected, the hoop stress is a minima at g = p.
12
6d/h=0.4 6d/h=0.3 6d/h=0.2
10
hp/τ 0
8 6 4 2 0 0
50
100
150
200 η (degrees)
250
300
350
Fig. 5. Hoop stress on the elliptical cavity located between two centerline cracks.
3.5 R
3
1
L1 L
2.5
2
k/k0
2 1.5 1 0.5 0
0.4
0.5
0.6
0.7
0.8
0.9
d/h Fig. 6. Stress intensity factors of an embedded crack and an edge crack adjacent to a circular cavity.
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M. Ayatollahi, S.J. Fariborz / Applied Mathematical Modelling 33 (2009) 3258–3268
10 9 8 7
h ση / τ0
6 5 4 3 2 1 0
50
100
150 200 η (degrees)
250
300
350
Fig. 7. Hoop stress on a circular cavity located between the edge and embedded cracks.
7. Conclusion The distributed dislocation technique is used to analyze the defects in the forms of cracks and cavities in a strip under time-harmonic anti-plane excitation. The analysis shows that centerline cracks experience lower stress intensity factor than identical off-center cracks parallel to strip. The steady of interaction of fixed and rotating cracks reveals that the variations of stress intensity factor at the tips of rotating cracks are much larger than those of stationary cracks. The interaction of a circular cavity with an edge and an embedded cracks result in large variation of hoop stress on the cavity surface. References [1] G.C. Sih, E.P. Chen, Moving cracks in finite strip under tearing action, J. Frank. Inst. 290 (1970) 25–35. [2] G. Kuhn, M. Matczynski, Elastic strip with a crack under periodic loading, Arch. Mech. 27 (1975) 459–472. [3] K.N. Srivastava, R.M. Palaiya, D.S. Karaulia, Interaction of shear waves with two coplanar Griffith cracks situated in an infinitely long elastic strip, Int. J. Fract. 23 (1983) 3–14. [4] H.M. Huang, H.J. Shi, Y.J. Yin, Multi-cracks problem for piezoelectric materials strip subjected to dynamic loading, Mech. Res. Commun. 29 (2002) 413– 424. [5] X.F. Li, Griffith crack moving in a piezoelectric strip, Arch. Appl. Mech. 72 (2003) 745–758. [6] M. Ayatollahi, S.J. Fariborz, M. Ahmadi Najafabadi, Antiplane elastodynamic analysis of planes with multiple defects, Appl. Math. Model. 33 (2009) 663– 676. [7] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, ninth ed., Dover, New York, 1972. [8] J.D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, Amsterdam, 1976. [9] R.T. Faal, S.J. Fariborz, H.R. Daghyani, Anti-plane deformation of orthotropic strips with multiple defects, J. Mech. Mater. Struct. 1 (2007) 1097–1113.