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Edge cracks in a transversely isotropic hollow cylinder
Engineering Fracture Mechanics 72 (2005) 2159–2173 www.elsevier.com/locate/engfracmech
Note
Edge cracks in a transversely isotropic hollow cylinder F. Suat Kadiog˘lu
*
Department of Mechanical Engineering, Middle East Technical University, 06531 Ankara, Turkey Received 22 September 2004; received in revised form 15 December 2004; accepted 24 January 2005 Available online 3 May 2005
Abstract The analytical solution for the linear elastic, axisymmetric problem of inner and outer edge cracks in a transversely isotropic infinitely long hollow cylinder is considered. The z = 0 plane on which the crack lies is a plane of symmetry. The loading is uniform crack surface pressure. The mixed boundary value problem is reduced to a singular integral equation where the unknown is the derivative of the crack surface displacement. An asymptotic analysis is done to derive the generalized Cauchy kernel associated with edge cracks. It is shown that the stress intensity factor is a function of three material parameters. The singular integral equation is solved numerically. Stress intensity factors are presented for various values of material and geometric parameters. 2005 Elsevier Ltd. All rights reserved. Keywords: Axisymmetric edge crack; Transversely isotropic hollow cylinder; Stress intensity factor
1. Introduction Because of its technical significance, many authors have addressed crack problems in hollow cylinders. For example, Erdol and Erdogan [1] considered a thick walled cylinder with an axisymmetric internal or edge crack under axisymmetric loading. It is stated that this problem approximates and provides a limiting case for the more realistic problem of a part-through circumferential crack lying in a plane perpendicular to the axis when the radial dimension of the flaw is relatively constant and the circumferential dimension is large compared to the wall thickness. Later Nied and Erdogan [2] considered the same geometry for the case of non-axisymmetric loads. Nied [3] also considered the problem of thermal shock in a circumferentially cracked hollow cylinder with cladding. In that study, formulation was such that cladding may have different thermal properties than the base material but the structure is elastically homogeneous. In all of the *
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above mentioned studies stress intensity factors were given. Varfolomeyev [4,5] presented weight functions which allows for calculation of mode I stress intensity factors for external and internal circumferential cracks in hollow cylinders subjected to axisymmetric loadings. Meshii and Watanabe [6] also presented a method to calculate mode I stress intensity factors for an inner circumferential crack in a finite length cylinder. Altinel et al. [7] introduced an extension in a different direction, by considering a transversely isotropic thick-walled cylinder rather than an isotropic one. Their formulation is axisymmetric. In that study, however, only internal cracks have been considered. A survey of literature reveals that the case of edge cracks in a transversely isotropic hollow cylinder has not been addressed since then. (On the other hand a solid cylinder with circumferential edge crack has been considered by Atsumi and Shindo [8] and Uyaner et al. [9].) The purpose of the current study is to complement the work of Altinel et al. [7] by extending the formulation to the case of inner and outer edge cracks. The formulation and the method of solution adopted here, follows steps very similar to those in [10]. By adopting a smeared approach, crack problems in fiber reinforced composite cylinders may be addressed by this model.
2. Formulation of the problem 2.1. Derivation of singular integral equation Geometry of the problem is given in Fig. 1. It was shown in Altinel et al. [7] that the solution of axisymmetric elasticity problems for transversely isotropic media can be reduced to the determination of a potential function of the Love type. The governing partial differential equation for this stress function is given as follows: o4 u 2 o3 u 1 o2 u 1 ou o4 u ða þ cÞ o3 u ou4 þ ða þ cÞ 2 2 þ þ d 4 ¼ 0. þ 2 2 þ 3 4 3 2 or r or r or r or or oz r oz or oz
2R 2Rc
z r
2Ri 2Ro Fig. 1. Geometry of the problem.
ð1Þ
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The stresses of interest and the displacement in z-direction can be expressed in terms of this stress function as o o2 u b ou o2 u o o2 u 1 ou o2 u þ a þ a rrr ¼ þ ¼ þ ; s ; ð2Þ rz oz or2 r or oz2 or or2 r or oz2 2 o o u c ou o2 u c 2 þ þd 2 ; rzz ¼ oz or r or oz
2 o u 1 ou o2 u þ d 2a aÞ ; þ ða w ¼ a44 33 13 or2 r or oz2
ð3Þ
where a¼
a13 ða11 a12 Þ ; a11 a33 a213
c¼
a13 ða11 a12 Þ þ a11 a44 ; a11 a33 a213
b¼
a13 ða13 þ a44 Þ a12 a33 ; a11 a33 a213 d¼
a211 a212 . a11 a33 a213
ð4Þ
In (4), a11, a12, a13, a33 and a44 are the compliances of the transversely isotropic material which are given in terms of the engineering constants as follows: 3 2 1 mTT mLT 0 7 6 ET ET EL 7 6 7 6 m 1 mLT 7 6 TT 0 7 6 7 6 ET ET EL 7. ð5Þ ½aij ¼ 6 7 6 m mTL 1 TL 7 6 0 7 6 7 6 ET EL ET 7 6 4 1 5 0 0 0 GTL Note that the matrix in (5) is symmetric, hence mLT mTL ¼ . EL ET
ð6Þ
In order to facilitate the investigation of the influence of material parameters on stress intensity factors, one can extend the definitions of effective material parameters given by [11] (for plane orthotropy) to the case of transversely isotropic materials: pffiffiffiffiffiffiffiffiffiffiffi E ¼ ET E L ; ð7Þ m¼
pffiffiffiffiffiffiffiffiffiffiffiffi mTL mLT ;
ET mTL ¼ ; EL mLT 1 pffiffiffiffiffiffiffiffiffiffiffi 1 mTL mLT j¼ ET EL ; 2 GTL ET EL
d4 ¼
ð8Þ ð9Þ ð10Þ
where E is the effective stiffness, m the effective poisson ratio, d the stiffness ratio and j the shear parameter. By using (6) along with the definitions (7)–(10) one can rewrite (5) in the following form:
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2
d2
6 2 16 6 mTT d ½aij ¼ 6 E 6 m 4
3
mTT d2
m
0
d2
m
0
d2
0
0
2ðj þ mÞ
m
0
0
7 7 7 7. 7 5
ð11Þ
Note that if the 2nd row and 2nd column of the 4 · 4 matrix in (11) is discarded, the remaining matrix is identical to that in Eq. (2.8) of [11]. At this point, it is possible to express material functions a, b, c and d given in (4), in terms of the newly defined effective parameters. Then one obtains, a¼
mð1 þ mTT Þ ; d2 ð1 m2 Þ
b¼
2jm þ m2 mTT ; 1 m2
c¼
2j þ m mmTT ; d2 ð1 m2 Þ
d¼
1 m2TT . d ð1 m2 Þ 4
ð12Þ
From (12) one can observe that a, b, c and d are all independent of effective stiffness E. Then from (1) to (3) one can conclude that for a transversely isotropic elastic domain, the stress field will be independent of E provided that there are only stress boundary conditions. The boundary conditions of the problem under consideration are given by the following equations: rrr ðR; zÞ ¼ 0; rrr ðRc ; zÞ ¼ 0;
srz ðR; zÞ ¼ 0; srz ðRc ; zÞ ¼ 0;
srz ðr; 0Þ ¼ 0; wðr; 0Þ ¼ 0;
ð13Þ ð14Þ ð15Þ
Rc < r < Ri and R > r > Ro ;
rzz ðr; 0Þ ¼ pðrÞ;
Ri < r < Ro .
ð16Þ ð17Þ
Note that (13) and (14) are the stress free boundary conditions on the inner and outer surfaces, (15) is a symmetry condition, (16) and (17) are the mixed boundary conditions. Then the Love stress function which satisfy the governing equation (1), and has the capacity to satisfy the boundary conditions (13)–(17) can be written as Z 1 2 uðr; zÞ ¼ kðm2 er2 z þ m4 er4 z ÞJ 0 ðkrÞ dk þ p 0 Z 1 1
½A I ðc arÞ þ A K ðc arÞ þ A I ðc arÞ þ A K ðc arÞ sinðazÞ da; ð18Þ 1 0 1 2 0 1 3 0 2 4 0 2 ðc21 c22 Þa2 0 for the hollow cylinder containing the crack. In this expression, J0 is the Bessel function, I0 and K0 are the modified Bessel functions of the first and the second kind; m2(k), m4(k), A1(a), A2(a), A3 (a), and A4(a) are functions to be determined by using the boundary conditions. Other functions, r2, r4, c1, c2 are given as 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v þ 1= d v 1= dA r2;4 ¼ k v v2 1=d ¼ k@ ; ð19Þ 2 2 where v¼
a þ c d2 ðj mmTT Þ ¼ 2d 1 m2TT
ð20Þ
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and pffiffiffi d pffiffiffi c2 ¼ ¼ d^r2 c1
1 c1 ¼ ; ^r2
where ^rk ¼ rk =k ðk ¼ 2; 4Þ. ð21Þ pffiffiffi In general v can be greater than, equal to or less than 1= d (hence ^r2 and ^r4 can have two distinct real values,porffiffiffi ^r2 ¼ ^r4 , or ^r2 and ^r4 are complex conjugates). In this study attention is limited to the case where v > 1= d . Not only the sample materials considered here, but many other transversely isotropic materials such as Al2O3, Cadmium, SiC, etc. fall into this category. pffiffiffi By using Eqs. (12) and (20) the condition v > 1= d can also be expressed in terms of the effective parameters as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j > mmTT þ ð1 m2 Þð1 m2TT Þ. ð22Þ Note that another physical limitation on the effective parameter j is that, considering (5) and (11), and observing that GTL > 0; j þ m > 0.
ð23Þ
By using Eqs. (2b), (3b) and (18), and observing that the second integrals that would arise due to (18) in (2b) and (3b) would vanish at z = 0 due to the sin(az) term, one can obtain Z 1 k2 J 1 ðkrÞ½k2 ðm2 þ m4 Þ aðm2 r22 þ m4 r24 Þ dk; ð24Þ srz ðr; 0Þ ¼ 0
ow ðr; 0Þ ¼ or
Z
1
k2 J 1 ðkrÞ½k2 a44 ðm2 þ m4 Þ ða33 d 2a13 aÞðm2 r22 þ m4 r24 Þ dk.
ð25Þ
0
Then by introducing the auxiliary function ow ðr; 0Þ ¼ GðrÞ ð26Þ or and observing that G(r) = 0 outside the interval r 2 (Ri, Ro) one can express m2 and m4 in terms of the auxiliary function G(r) by using Eqs. (15), (24) and (25) by taking inverse Hankel Transforms. Then, Z Ro 1 mk ¼ X k 3 qGðqÞJ 1 ðkqÞ dq; k ¼ 2; 4; ð27Þ k Ri where X2 ¼
ð1 a^r24 Þ ; ð^r22 ^r24 Þða33 d 2a13 a a44 aÞ
X4 ¼
ð1 a^r22 Þ . ð^r22 ^r24 Þða33 d 2a13 a a44 aÞ
ð28Þ
By substituting (27) and (28) first into (18) and then substituting the stress function in terms of G(r) into (2) and (3) the expressions for displacements and stresses in a hollow cylinder which contains a crack on z = 0 at r 2 (Ri, Ro) plane can be obtained. Then by using the homogeneous boundary conditions (13) and (14) one can express the unknown functions A1, A2, A3, A4 in terms of G(r) as well. (The details of this procedure is very similar to that followed in [10] and the interested reader may refer to that article.) Once A1, A2, A3, A4 are expressed in terms of G(r), one can substitute these functions into (17) to obtain the singular integral equation of this problem as follows: Z Ro c mðr; qÞ 1 mðr; qÞ þc þ GðqÞ ð29Þ þ 2qk 2 ðr; qÞ dq ¼ ppðrÞ; Ri 6 r 6 Ro . qr qr qþr Ri
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The terms in Eq. (29) are given as follows: c ¼ X 2^r2 ðc d^r22 Þ þ X 4^r4 ðc d^r24 Þ;
ð30Þ
8 r > > < E q ; q > r; ð31Þ mðr; qÞ ¼ q q2 r2 > q r > :E þK ; q < r; r q r rq Z 1 2 ðcc1 dÞ 0 0 0 2 2 0 k 2 ðr; qÞ ¼ ½A a I ðc arÞ þ A K ðc arÞ þ ðcc dÞa ½A I ðc arÞ þ A K ðc arÞ da; 0 1 2 0 1 3 0 2 4 0 2 2 ðc21 c22 Þ 1 0 ð32Þ where K and E are the complete elliptic integrals of the first and the second kind, respectively. and A04 are given in Appendix A. In (29), the first term gives the well known Cauchy type singularity, associated with the square root stress singularities at the tips of internal cracks. The second term contains a logarithmic singularity at r = q. This singularity can be treated as was done in [10]. k2(r, q) is a Fredholm kernel. A01 ,
A02 ,
A03
2.2. Dependence of integral equation on material parameters In problems, such as the one considered in this study, it is possible that the quantities of interest such as the stress intensity factors would be dependent only on some of the material constants. The remaining constants may enter the analysis as multiplicative factors or they may not have any influence at all. For example, in the isotropic counterpart of this study (given in [2]), it is seen that the stress intensity factors depend only on the poisson ratio and they are independent of the modulus of rigidity. Therefore it is worthwhile to take a closer look at the material constant combinations appearing in the integral equation (29). At this point by using (12), (20) and (21), one can easily show that, c1 ðj; m; mTT Þ c ðj; m; mTT Þ ; c2 ¼ 2 . ð33Þ d d In (33) ^r2 , ^r4 , c1 and c2 are functions of j, m and mTT only, that is d (or 1/d) appears as a multiplicative factor in the definitions of ^r2 , ^r4 , c1 and c2. From their definitions it is clear that ^r2 , ^r4 , c1 and c2 are independent of E. Then by substituting the definitions of aij given by (11) along with (12) and (33) into (28), after some algebraic manipulations, it is possible to show that, ^r2 ¼ d^r2 ðj; m; mTT Þ;
X 2 ¼ EX 2 ðj; m; mTT Þ;
^r4 ¼ d^r4 ðj; m; mTT Þ;
c1 ¼
X 4 ¼ EX 4 ðj; m; mTT Þ.
ð34Þ
In other words, X k ðk ¼ 2; 4Þ are functions of j, m and mTT only, E appears as a multiplicative factor in the definitions of Xk, and Xk are independent of d. Finally, by using (12), (30), (33) and (34) one can show that c¼
E c ðj; m; mTT Þ. d
ð35Þ
Note that the quantities with a superscript Ô*Õ appearing in Eqs. (33)–(35) can be easily obtained through substitution and factorization, hence they are not explicitly given in this article. Proceeding in a similar manner, one can also show that the material constant combinations appearing in the Fredholm kernel can be given in the following form: K1 ¼
cc21 d K1 ðj; m; mTT Þ ¼ ; c21 c22 d2
K2 ¼ cc22 d ¼
K2 ðj; m; mTT Þ . d4
ð36Þ
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By using (33) and (36), and defining a new dummy variable of integration, a b¼ ; d the Fredholm kernel given in (32) can be written as follows: Z 1 0 k 2 ðr; qÞ ¼ bK1 A1 I 0 ðc1 brÞ þ A02 K 0 ðc1 brÞ þ b3 K2 ½A03 I 0 ðc2 brÞ þ A04 K 0 ðc2 brÞ db.
2165
ð37Þ
ð38Þ
0
Now by using the definitions given in Appendix A and following the procedure outlined in Appendix B, it could be shown that Ed can be factored out of A01 , A02 , A03 and A04 as well. Hence the Fredholm kernel in (38) and the integral equation in (29) can be written respectively as k 2 ðr; qÞ ¼ Z
Ro
GðqÞ Ri
Z
E d
1 3 0 0 0 fbK1 ½A0 1 I 0 ðc1 brÞ þ A2 K 0 ðc1 brÞ þ b K2 ½A3 I 0 ðc2 brÞ þ A4 K 0 ðc2 brÞg db;
ð39Þ
0
c mðr; qÞ 1 mðr; qÞ d þ þ c þ 2qk 2 ðr; qÞ dq ¼ p pðrÞ; qr qþr E qr
Ri 6 r 6 Ro .
ð40Þ
In this form, the left hand side of (40) contains neither E nor d. 2.3. Extraction of singularities The Fredholm kernel k2(r, q) is bounded as long as the inner and outer crack tips remain in the cylinder, i.e. Rc < Ri and Ro < R. However when the crack is terminating at one of the free surfaces (i.e. Rc = Ri or Ro = R), the integral in (32) becomes unbounded. For this case an asymptotic analysis for a ! 1 is required. Letting, Kðr; q; aÞ ¼
ðcc21 dÞ 0 ½A I 0 ðc1 arÞ þ A02 K 0 ðc1 arÞa þ ðcc22 dÞa3 ½A03 I 0 ðc2 arÞ þ A04 K 0 ðc2 arÞ; ðc21 c22 Þ 1
ð41Þ
one can write, k 2 ðr; qÞ ¼
Z
1
½Kðr; q; aÞ K 1 ðr; q; aÞ da þ 0
Z
1
K 1 ðr; q; aÞ da;
ð42Þ
0
where K1(r, q, a) gives the asymptotic behavior of K(r, q, a) as a ! 1. After a lengthy analysis K1(r, q, a) can be obtained as follows: 1 K 1 ðr; q; aÞ ¼ pffiffiffiffiffi ^r2 j5 eaððqRÞ=^r2 þc1 ðrRÞÞ þ ^r4 j6 eaððqRÞ=^r4 þc1 ðrRÞÞ 2 rq 1 þ ^r2 j7 eaððqRÞ=^r2 þc2 ðrRÞÞ þ ^r4 j8 eaððqRÞ=^r4 þc2 ðrRÞÞ þ pffiffiffiffiffi ^r2 j5 eaððqRc Þ=^r2 þc1 ðrRc ÞÞ 2 rq þ ^r4 j6 eaððqRc Þ=^r4 þc1 ðrRc ÞÞ þ ^r2 j7 eaððqRc Þ=^r2 þc2 ðrRc ÞÞ þ ^r4 j8 eaððqRc Þ=^r4 þc2 ðrRc ÞÞ .
ð43Þ
In (43), the first square bracket contains the terms related to the case of outer edge crack and the second square bracket contains the terms related to the case of inner edge crack. The material parameters j5, j6, j7 and j8 are given in Appendix A. It can be shown that j6 = j7. Having known K1(r, q, a), the second integral in (42) can be evaluated in closed form, giving;
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Z
1 0
^r22 j5 ^r24 j6 ^r22 j7 1 K 1 ðr; q; aÞ da ¼ pffiffiffiffiffi 2 rq R q þ c1^r2 ðR rÞ R q þ c1^r4 ðR rÞ R q þ c2^r2 ðR rÞ ^r24 j8 ^r22 j5 ^r24 j6 þ þ R q þ c2^r4 ðR rÞ q Rc þ c1^r2 ðr Rc Þ q Rc þ c1^r4 ðr Rc Þ ^r22 j7 ^r24 j8 þ þ . q Rc þ c2^r2 ðr Rc Þ q Rc þ c2^r4 ðr Rc Þ
ð44Þ
Multiplying (44) with 2q gives the generalized Cauchy kernel. The singularity extraction has been done starting from Eq. (32), but it is obvious that it could have been done starting from Eq. (39), that is after factoring out E/d. Hence one can see that E/d can also be factored out of K1(r, q, a) given in (43) and its integrated form given in (44), which indeed is the case. 2.4. Stress intensity factors The quantities of interest in this study are the stress intensity factors for inner and outer edge cracks. Noting that gðrÞ GðrÞ ¼ R i < r < Ro ; ð45Þ x g; ðRo rÞ ðr Ri Þ where g(r) is a bounded function, the stress intensity factors at the crack tips can be expressed as follows: (a) Inner edge crack: (Ri = Rc, Ro < R, x = 1/2, g = 0) pffiffiffi pffiffiffi kðRo Þ ¼ limþ 2ðr Ro Þ1=2 rzz ðr; 0Þ ¼ c 2gðRo Þ;
ð46Þ
(b) Outer edge crack: (Ri > Rc, Ro = R, x = 0, g = 1/2) pffiffiffi pffiffiffi kðRi Þ ¼ lim 2ðRi rÞ1=2 rzz ðr; 0Þ ¼ c 2gðRi Þ.
ð47Þ
r!Ro
r!Ri
But from (35), one can also write; E pffiffiffi E pffiffiffi kðRo Þ ¼ c 2gðRo Þ; kðRi Þ ¼ c 2gðRi Þ. d d
ð48Þ
On the other hand, from (40) and (45) one can observe that g(r) is directly proportional to d/E, that is d g ðrÞ; ð49Þ E hence, one can easily see that the stress intensity factors are independent of E and d. The singular integral equation (29) is solved numerically by using the method outlined in [10] for various values of the material parameters j, m and mTT; sample material pairs and for various values of geometric parameters. The loading has been taken to be uniform, i.e. p(r) = r0. The results obtained are presented and discussed in the next section. gðrÞ ¼
3. Results and discussion For the problem under consideration there are three distinct crack geometries, namely, embedded, inner edge, and outer edge cracks. The geometric parameters that describe these configurations are the hole ra-
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dius (Rc), the coordinates of the inner and outer crack tips (Ri, Ro), and the outer radius R. In addition to these geometric parameters, there are three non-dimensional material parameters, namely j, m and mTT. Two sets of results are obtained in this study. In the first set of results a particular crack and cylinder geometry is selected and the influence of material parameters on stress intensity factors (SIF) is examined. In the second set of results, three particular materials, namely, magnesium (j = 1.226, m = 0.239, mTT = 0.357), a graphite-epoxy composite (j = 4.314, m = 0.084, mTT = 0.4) and steel (j = 1.001, m = 0.306, mTT = 0.300) are considered and the variation of SIFÕs with respect to the geometric parameters are presented. Only edge crack configurations are considered. Of these materials the former two are transversely isotropic where as steel is isotropic. At this point it should be noted that the formulation is made for transversely isotropic materials and it becomes degenerate for isotropic materials (j = 1, m = mTT, d = 1). This problem is overcome by introducing small perturbations in the compliance values of the isotropic material, thereby artificially rendering it ÔslightlyÕ anisotropic [8]. The compliances of the sample materials are also given in Table 1. The number of geometric parameters is reduced to two by normalizing them using outer radius, R, of the cylinder. In the tables, wall thickness of the cylinder, h = R Rc and crack length, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a = Ro Ri are used in presenting the results. Stress intensity factors are also normalized by r0 Ro Ri . In order to establish the validity and accuracy of the numerical results comparisons are made with [2] (bold values in tables). The agreement is found to be quite good. Fig. 2 shows the variation of SIFÕs for an inner edge crack as a function of effective poisson ratio, for selected values of shear parameter and inplane poisson ratio.
Table 1 Compliances of sample materials · 1011 [m2/N] Material
a11
a12
a13
a33
a44
Steel Magnesium Gr-Epoxy
0.476 2.20 8.55
0.143 0.785 3.42
0.145 0.498 0.210
0.473 1.97 0.725
1.240 6.10 21.9
1.58 1.54
κ =1.05
1.50 k
σ 0a1/2
κ =3.00 1.46 1.42
κ =5.00
1.38 1.34 0.0
0.1
0.2
0.3
0.4
0.5
ν Fig. 2. Stress intensity factors for an inner edge crack as a function of m(Rc/R = 0.9, a/h = 0.4). (Groups of curves correspond to j = 1.05, 3.00, and 5.00 in top to bottom order, within each group, curves correspond to mTT = 0.4, 0.25 and 0.10 in top to bottom order.)
2168
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It is observed that the dependence of SIF on both Poisson ratios is rather weak (note the scale of the graph) and this dependence could be neglected for practical purposes. A similar trend is observed for outer edge cracks in Fig. 3. On the other hand, Fig. 4 shows that significant reduction in SIF can be obtained by increasing shear parameter beyond 1.0 (which is the value for isotropic material). If j is further increased, SIF tends to a constant value. Table 2 shows some sample results for inner surface cracks. Basic trends regarding the variation of geometric parameters are the same for all the materials. SIF for a given crack size increase as the wall thickness
1.64
κ =1.05
1.60 1.56
κ =3.00 k 1.52 σ 0a1/2 1.48
κ =5.00
1.44 1.40 0.0
0.1
0.2
0.3
0.4
0.5
ν Fig. 3. Stress intensity factors for an outer edge crack as a function of m(Rc/R = 0.9, a/h = 0.4). (Groups of curves correspond to j = 1.05, 3.00, and 5.00 in top to bottom order, within each group, curves correspond to mTT = 0.4, 0.25 and 0.10 in top to bottom order.)
1.64 1.56 1.48 1.40 k σ 0a1/2 1.32 1.24 1.16
κ =1.05 1.08 0
10 20 30 40 50 60 70 80 90 100 ν
Fig. 4. Stress intensity factors for inner (bottom) and outer (top) edge cracks as a function of j. (mTT = 0.25, m = 0.25, Rc/R = 0.9, a/h = 0.4.)
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Table 2 Normalized stress intensity factors for inner surface cracks (a = Ro Ri, h = R Rc, Ri = Rc) Rc/R
a/h 0.2
0.4
0.6
Mg
0.1 0.3 0.5 0.7 0.9
0.7699 0.9354 1.0473 1.1409 1.2436
0.7470 0.9097 1.0711 1.2569 1.5464
0.8080 0.9837 1.1856 1.4533 1.9788
Graphite-Epoxy
0.1 0.3 0.5 0.7 0.9
0.7623 0.9036 0.9956 1.0762 1.1788
0.7369 0.8743 0.9985 1.1387 1.3866
0.7831 0.9303 1.0849 1.2788 1.6750
Steel
0.1
0.7707 0.775 0.9388 0.942 1.0531 1.055 1.1486 1.150 1.2521 1.253
0.7492 0.754 0.9150 0.920 1.0807 1.085 1.2712 1.275 1.5645 1.568
0.8139 0.820 0.9936 1.000 1.2014 1.208 1.4774 1.484 2.0160 2.025
0.3 0.5 0.7 0.9
Table 3 Normalized stress intensity factors for outer surface cracks (a = Ro Ri, h = R Rc, Ro = R) Rc/R
a/h 0.2
0.4
0.6
Mg
0.1 0.3 0.5 0.7 0.9
1.2343 1.2198 1.2204 1.2347 1.2739
1.5109 1.4290 1.4129 1.4559 1.6178
2.1744 1.8657 1.7595 1.7917 2.1083
Graphite-Epoxy
0.1 0.3 0.5 0.7 0.9
1.1842 1.1599 1.1513 1.1603 1.2058
1.4762 1.3632 1.3102 1.3145 1.4481
2.1451 1.7801 1.6157 1.5777 1.7834
Steel
0.1
1.2460 1.244 1.2320 1.231 1.2328 1.232 1.2465 1.247 1.2839 1.285
1.5197 1.513 1.4416 1.437 1.4294 1.427 1.4758 1.475 1.6387 1.641
2.1803 2.159 1.8782 1.866 1.7792 1.773 1.8199 1.818 2.1485 2.153
0.3 0.5 0.7 0.9
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decreases. For relatively thick walled cylinders SIF first decrease and then starts to increase as the crack length increases. This indicates a possibility of crack arrest. It can be observed that the influence of transverse isotropy on SIF is minimal for large wall thicknesses. However for the highly anisotropic graphite epoxy composite, significantly lower SIF can be achieved. Table 3 shows some sample results for outer surface cracks. SIFÕs for outer cracks are in general larger than those of inner cracks. Again, basic trends regarding the variation of geometric parameters are the same for all the materials. Over a wide range of wall thickness ratios, SIFÕs either decrease or remain pretty much constant with the decreasing wall thickness. On the other hand, as the crack lengths increase, so does the SIF for all the wall thicknesses considered. It can again be observed that for relatively thick walled cylinders such as Rc/R = 0.1, the influence of transverse isotropy on SIF is minimal for all the crack sizes considered. This is actualy consistent with the findings of [8]. For relatively thin walled cylinders, however, significant reduction in SIF can be achieved for highly anisotropic material.
Appendix A. Terms used in the calculation of kernels 2
A01
3
2
f11
f12
f13
6 07 6 6 A2 7 6 f21 f22 f23 6 7¼6 6 A0 7 6 f 4 3 5 4 31 f32 f33 f41 f42 f43 A04 R Ro note that A1 ¼ Ri qGðqÞA01 dq, l1 ¼
f14
7 f24 7 7 f34 7 5 f44
L1
3
6 7 6 L2 7 6 7 6L 7 4 35 L4
etc.
c21 a ¼ l1 ðj; m; mTT Þ; c21 c22
l3 ¼ c22 a ¼
31 2
1 l3 ðj; m; mTT Þ; d2
l2 ¼
c1 ðb 1Þ ¼ dl2 ðj; m; mTT Þ; c21 c22
1 l4 ¼ c2 ðb 1Þ ¼ l4 ðj; m; mTT Þ; d
c1 ðc21 aÞ 1 1 ¼ l5 ðj; m; mTT Þ; l6 ¼ c2 ðc22 aÞ ¼ 3 l6 ðj; m; mTT Þ; d c21 c22 d ! ! 1 a^r22 E 1 a^r24 E W 2 ¼ X2 ¼ 2 W 2 ðj; m; mTT Þ; W 4 ¼ X 4 ¼ 2 W 4 ðj; m; mTT Þ; ^r22 ^r24 d d
l5 ¼
Y2 ¼ X2
2 a^r2 1 E ¼ Y 2 ðj; m; mTT Þ; ^r2 d
Z 2 ¼ X 2 ðb 1Þ ¼ EZ 2 ðj; m; mTT Þ; f11 ¼ l1 aI 0 ðc1 aRc Þ þ
l2 I 1 ðc1 aRc Þ; Rc
Y4 ¼ X4
2 a^r4 1 E ¼ Y 4 ðj; m; mTT Þ; ^r4 d
Z 4 ¼ X 4 ðb 1Þ ¼ EZ 4 ðj; m; mTT Þ; f 12 ¼ l1 aK 0 ðc1 aRc Þ
f13 ¼ l3 a3 I 0 ðc2 aRc Þ þ
l4 2 a I 1 ðc2 aRc Þ; Rc
f21 ¼ l5 I 1 ðc1 aRc Þ;
f 22 ¼ l5 K 1 ðc1 aRc Þ;
l2 K 1 ðc1 aRc Þ; Rc
f 14 ¼ l3 a3 K 0 ðc2 aRc Þ
l4 2 a K 1 ðc2 aRc Þ; Rc
f 23 ¼ l6 a2 I 1 ðc2 aRc Þ;
f 24 ¼ l6 a2 K 1 ðc2 aRc Þ;
F. Suat Kadiog˘lu / Engineering Fracture Mechanics 72 (2005) 2159–2173
f31 ¼ l1 aI 0 ðc1 aRÞ þ
l2 I 1 ðc1 aRÞ; R
f33 ¼ l3 a3 I 0 ðc2 aRÞ þ
f 32 ¼ l1 aK 0 ðc1 aRÞ
l4 2 a I 1 ðc2 aRÞ; R
2171
l2 K 1 ðc1 aRÞ; R
f 34 ¼ l3 a3 K 0 ðc2 aRÞ
l4 2 a K 1 ðc2 aRÞ; R
f41 ¼ l5 I 1 ðc1 aRÞ; f 42 ¼ l5 K 1 ðc1 aRÞ; f 43 ¼ l6 a2 I 1 ðc2 aRÞ; f 44 ¼ l6 a2 K 1 ðc2 aRÞ; Rc a qa aRc aq 1 Rc a qa Rc a qa L1 ¼ I 0 I1 K1 Y 2a þ I 0 K1 Y 4a K1 Z2 þ I 1 K1 Z4 ; ^r2 ^r2 ^r4 ^r2 ^r2 ^r4 ^r4 ^r4 Rc Rc a qa Rc a qa L2 ¼ I 1 K1 W 2 þ I1 K1 W 4; ^r2 ^r2 ^r4 ^r4 qa Ra qa Ra 1 a Ra a Ra I1 q L3 ¼ I 1 K0 Y 2a I 1 K0 Y 4a K1 Z2 þ I 1 q K1 Z4 ; ^r2 ^r2 ^r4 ^r4 ^r2 ^r2 ^r4 ^r4 R qa Ra qa Ra L4 ¼ I 1 K1 W 2 þ I1 K1 W4 ; ^r2 ^r2 ^r4 ^r4 j5 ¼
ðd cc21 Þð1 þ c2^r2 Þð1 þ a^r22 ÞX 2 ; ða c21 Þðc2 c1 Þ^r22
j6 ¼
ðd cc21 Þð1 þ c2^r4 Þð1 þ a^r24 ÞX 4 ; ða c21 Þðc2 c1 Þ^r24
j7 ¼
ðcc22 dÞð1 þ c1^r2 Þð1 þ a^r22 ÞX 2 ; ða c22 Þðc2 c1 Þ^r22
j8 ¼
ðcc22 dÞð1 þ c1^r4 Þð1 þ a^r24 ÞX 4 . ða c22 Þðc2 c1 Þ^r24
Appendix B. Factorization of A0i ði ¼ 1; . . . ; 4Þ From Appendix A, one can observe that 4 X
fij A0j ¼ Li
ði ¼ 1; . . . ; 4Þ;
ðA:1Þ
j¼1
and from (37) a ¼ db.
ðA:2Þ
Then using the definitions of li(i = 1, . . ., 6) along with (33), one can write for example, f11 as f11 ¼ l1 dbI 0 ðc1 bRc Þ þ
l2 d I 1 ðc1 bRc Þ ¼ df11 . Rc
ðA:3Þ
Similarly one can show that f1j ¼ df1j ;
ðj ¼ 1; . . . ; 4Þ;
ðA:4Þ
1 f2j ¼ f2j d
ðj ¼ 1; . . . ; 4Þ;
ðA:5Þ
F. Suat Kadiog˘lu / Engineering Fracture Mechanics 72 (2005) 2159–2173
2172
f3j ¼ df3j
ðj ¼ 1; . . . ; 4Þ;
ðA:6Þ
1 f4j ¼ f4j d
ðj ¼ 1; . . . ; 4Þ;
ðA:7Þ
where fij ði; j ¼ 1; . . . ; 4Þ do not contain the material parameters E and d. Also one can express, for example L1, as Rc db qdb E dbRc dbq E Y db þ I 0 Y db L1 ¼ I 0 K1 K1 d^r2 d^r2 d 2 d^r4 d 4 d^r4 1 Rc db qdb Rc db qdb I1 K1 Z2E þ I 1 K1 Z4E ; Rc d^r2 d^r2 d^r4 d^r4 Rc b qb bRc bq L1 ¼ E I 0 K 1 Y 2b þ I 0 K 1 Y 4 b ^r2 ^r2 ^r4 ^r4 1 Rc b qb Rc b qb I1 K 1 Z 2 þ I 1 K 1 Z 4 ; ^r2 ^r2 ^r4 ^r4 Rc L1 ¼ EL1 ;
ðA:8Þ
ðA:9Þ ðA:10Þ
and proceeding similarly, L2 ¼
E L2 ; d2
ðA:11Þ
L3 ¼ EL3 ; L4 ¼
ðA:12Þ
E L4 ; d2
ðA:13Þ
where Li ði ¼ 1; . . . ; 4Þ is independent of E and d. Then from (A.1), (A.4)–(A.7) and (A.10)–(A.13) it is obvious that 4 X
fij A0j ¼
j¼1
E L d i
ði ¼ 1; . . . ; 4Þ.
ðA:14Þ
(A.14) implies that the functions A0j ðbÞ, which are supposed to be used in (38) are directly proportional to E/d.
References [1] Erdol R, Erdogan F. A thick-walled cylinder with an axisymmetric internal or edge crack. J Appl Mech 1978;45:281–6. [2] Nied HF, Erdogan F. The elasticity problem for a thick-walled cylinder containing a circumferential crack. Int J Fract 1983;22:277–301. [3] Nied HF. Thermal shock in a circumferentially cracked hollow cylinder with cladding. Engng Fract Mech 1984;20:113–37. [4] Varfolomeyev IV. Weight function for external circumferential cracks in hollow cylinders subjected to axisymmetric opening mode loading. Engng Fract Mech 1998;60:333–9. [5] Varfolomeyev IV. Stress intensity factors for internal circumferential cracks in thin- and thick-walled cylinders. Engng Fract Mech 1998;60:491–500. [6] Meshii T, Watanabe K. Stress intensity factor for a circumferential crack in a finite length thin to thick-walled cylinder under an arbitrary biquadratic stress distribution on the crack surfaces. Engng Fract Mech 2001;68:975–86.
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[7] Altinel T, Fildisß H, Yahsßi OS. The stress intensity factors for an infinitely long transversely isotropic, thick walled cylinder which contains a ring-shaped crack. Int J Fract 1996;78:211–25. [8] Atsumi A, Shindo Y. Singular stresses in a transversely isotropic circular cylinder with circumferential edge crack. Int J Engng Sci 1979;17:1229–36. [9] Uyaner M, Akdemir A, Erim S, Avci A. Plastic zones in a transversely isotropic solid cylinder containing a ring-shaped crack. Int J Fract 2000;106:161–75. [10] Kadioglu FS. Axisymmetric crack terminating at the interface of transversely isotropic, dissimilar media. Int J Fract 2002;116:51–79. [11] Krenk S. On the elastic constants of plane orthotropy. J Compos Mater 1979:108–16.
Note
Edge cracks in a transversely isotropic hollow cylinder F. Suat Kadiog˘lu
*
Department of Mechanical Engineering, Middle East Technical University, 06531 Ankara, Turkey Received 22 September 2004; received in revised form 15 December 2004; accepted 24 January 2005 Available online 3 May 2005
Abstract The analytical solution for the linear elastic, axisymmetric problem of inner and outer edge cracks in a transversely isotropic infinitely long hollow cylinder is considered. The z = 0 plane on which the crack lies is a plane of symmetry. The loading is uniform crack surface pressure. The mixed boundary value problem is reduced to a singular integral equation where the unknown is the derivative of the crack surface displacement. An asymptotic analysis is done to derive the generalized Cauchy kernel associated with edge cracks. It is shown that the stress intensity factor is a function of three material parameters. The singular integral equation is solved numerically. Stress intensity factors are presented for various values of material and geometric parameters. 2005 Elsevier Ltd. All rights reserved. Keywords: Axisymmetric edge crack; Transversely isotropic hollow cylinder; Stress intensity factor
1. Introduction Because of its technical significance, many authors have addressed crack problems in hollow cylinders. For example, Erdol and Erdogan [1] considered a thick walled cylinder with an axisymmetric internal or edge crack under axisymmetric loading. It is stated that this problem approximates and provides a limiting case for the more realistic problem of a part-through circumferential crack lying in a plane perpendicular to the axis when the radial dimension of the flaw is relatively constant and the circumferential dimension is large compared to the wall thickness. Later Nied and Erdogan [2] considered the same geometry for the case of non-axisymmetric loads. Nied [3] also considered the problem of thermal shock in a circumferentially cracked hollow cylinder with cladding. In that study, formulation was such that cladding may have different thermal properties than the base material but the structure is elastically homogeneous. In all of the *
Tel.: +90 312 210 52 94; fax: +90 312 210 12 66. E-mail address: [email protected]
0013-7944/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2005.01.011
2160
F. Suat Kadiog˘lu / Engineering Fracture Mechanics 72 (2005) 2159–2173
above mentioned studies stress intensity factors were given. Varfolomeyev [4,5] presented weight functions which allows for calculation of mode I stress intensity factors for external and internal circumferential cracks in hollow cylinders subjected to axisymmetric loadings. Meshii and Watanabe [6] also presented a method to calculate mode I stress intensity factors for an inner circumferential crack in a finite length cylinder. Altinel et al. [7] introduced an extension in a different direction, by considering a transversely isotropic thick-walled cylinder rather than an isotropic one. Their formulation is axisymmetric. In that study, however, only internal cracks have been considered. A survey of literature reveals that the case of edge cracks in a transversely isotropic hollow cylinder has not been addressed since then. (On the other hand a solid cylinder with circumferential edge crack has been considered by Atsumi and Shindo [8] and Uyaner et al. [9].) The purpose of the current study is to complement the work of Altinel et al. [7] by extending the formulation to the case of inner and outer edge cracks. The formulation and the method of solution adopted here, follows steps very similar to those in [10]. By adopting a smeared approach, crack problems in fiber reinforced composite cylinders may be addressed by this model.
2. Formulation of the problem 2.1. Derivation of singular integral equation Geometry of the problem is given in Fig. 1. It was shown in Altinel et al. [7] that the solution of axisymmetric elasticity problems for transversely isotropic media can be reduced to the determination of a potential function of the Love type. The governing partial differential equation for this stress function is given as follows: o4 u 2 o3 u 1 o2 u 1 ou o4 u ða þ cÞ o3 u ou4 þ ða þ cÞ 2 2 þ þ d 4 ¼ 0. þ 2 2 þ 3 4 3 2 or r or r or r or or oz r oz or oz
2R 2Rc
z r
2Ri 2Ro Fig. 1. Geometry of the problem.
ð1Þ
F. Suat Kadiog˘lu / Engineering Fracture Mechanics 72 (2005) 2159–2173
2161
The stresses of interest and the displacement in z-direction can be expressed in terms of this stress function as o o2 u b ou o2 u o o2 u 1 ou o2 u þ a þ a rrr ¼ þ ¼ þ ; s ; ð2Þ rz oz or2 r or oz2 or or2 r or oz2 2 o o u c ou o2 u c 2 þ þd 2 ; rzz ¼ oz or r or oz
2 o u 1 ou o2 u þ d 2a aÞ ; þ ða w ¼ a44 33 13 or2 r or oz2
ð3Þ
where a¼
a13 ða11 a12 Þ ; a11 a33 a213
c¼
a13 ða11 a12 Þ þ a11 a44 ; a11 a33 a213
b¼
a13 ða13 þ a44 Þ a12 a33 ; a11 a33 a213 d¼
a211 a212 . a11 a33 a213
ð4Þ
In (4), a11, a12, a13, a33 and a44 are the compliances of the transversely isotropic material which are given in terms of the engineering constants as follows: 3 2 1 mTT mLT 0 7 6 ET ET EL 7 6 7 6 m 1 mLT 7 6 TT 0 7 6 7 6 ET ET EL 7. ð5Þ ½aij ¼ 6 7 6 m mTL 1 TL 7 6 0 7 6 7 6 ET EL ET 7 6 4 1 5 0 0 0 GTL Note that the matrix in (5) is symmetric, hence mLT mTL ¼ . EL ET
ð6Þ
In order to facilitate the investigation of the influence of material parameters on stress intensity factors, one can extend the definitions of effective material parameters given by [11] (for plane orthotropy) to the case of transversely isotropic materials: pffiffiffiffiffiffiffiffiffiffiffi E ¼ ET E L ; ð7Þ m¼
pffiffiffiffiffiffiffiffiffiffiffiffi mTL mLT ;
ET mTL ¼ ; EL mLT 1 pffiffiffiffiffiffiffiffiffiffiffi 1 mTL mLT j¼ ET EL ; 2 GTL ET EL
d4 ¼
ð8Þ ð9Þ ð10Þ
where E is the effective stiffness, m the effective poisson ratio, d the stiffness ratio and j the shear parameter. By using (6) along with the definitions (7)–(10) one can rewrite (5) in the following form:
F. Suat Kadiog˘lu / Engineering Fracture Mechanics 72 (2005) 2159–2173
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2
d2
6 2 16 6 mTT d ½aij ¼ 6 E 6 m 4
3
mTT d2
m
0
d2
m
0
d2
0
0
2ðj þ mÞ
m
0
0
7 7 7 7. 7 5
ð11Þ
Note that if the 2nd row and 2nd column of the 4 · 4 matrix in (11) is discarded, the remaining matrix is identical to that in Eq. (2.8) of [11]. At this point, it is possible to express material functions a, b, c and d given in (4), in terms of the newly defined effective parameters. Then one obtains, a¼
mð1 þ mTT Þ ; d2 ð1 m2 Þ
b¼
2jm þ m2 mTT ; 1 m2
c¼
2j þ m mmTT ; d2 ð1 m2 Þ
d¼
1 m2TT . d ð1 m2 Þ 4
ð12Þ
From (12) one can observe that a, b, c and d are all independent of effective stiffness E. Then from (1) to (3) one can conclude that for a transversely isotropic elastic domain, the stress field will be independent of E provided that there are only stress boundary conditions. The boundary conditions of the problem under consideration are given by the following equations: rrr ðR; zÞ ¼ 0; rrr ðRc ; zÞ ¼ 0;
srz ðR; zÞ ¼ 0; srz ðRc ; zÞ ¼ 0;
srz ðr; 0Þ ¼ 0; wðr; 0Þ ¼ 0;
ð13Þ ð14Þ ð15Þ
Rc < r < Ri and R > r > Ro ;
rzz ðr; 0Þ ¼ pðrÞ;
Ri < r < Ro .
ð16Þ ð17Þ
Note that (13) and (14) are the stress free boundary conditions on the inner and outer surfaces, (15) is a symmetry condition, (16) and (17) are the mixed boundary conditions. Then the Love stress function which satisfy the governing equation (1), and has the capacity to satisfy the boundary conditions (13)–(17) can be written as Z 1 2 uðr; zÞ ¼ kðm2 er2 z þ m4 er4 z ÞJ 0 ðkrÞ dk þ p 0 Z 1 1
½A I ðc arÞ þ A K ðc arÞ þ A I ðc arÞ þ A K ðc arÞ sinðazÞ da; ð18Þ 1 0 1 2 0 1 3 0 2 4 0 2 ðc21 c22 Þa2 0 for the hollow cylinder containing the crack. In this expression, J0 is the Bessel function, I0 and K0 are the modified Bessel functions of the first and the second kind; m2(k), m4(k), A1(a), A2(a), A3 (a), and A4(a) are functions to be determined by using the boundary conditions. Other functions, r2, r4, c1, c2 are given as 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v þ 1= d v 1= dA r2;4 ¼ k v v2 1=d ¼ k@ ; ð19Þ 2 2 where v¼
a þ c d2 ðj mmTT Þ ¼ 2d 1 m2TT
ð20Þ
F. Suat Kadiog˘lu / Engineering Fracture Mechanics 72 (2005) 2159–2173
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and pffiffiffi d pffiffiffi c2 ¼ ¼ d^r2 c1
1 c1 ¼ ; ^r2
where ^rk ¼ rk =k ðk ¼ 2; 4Þ. ð21Þ pffiffiffi In general v can be greater than, equal to or less than 1= d (hence ^r2 and ^r4 can have two distinct real values,porffiffiffi ^r2 ¼ ^r4 , or ^r2 and ^r4 are complex conjugates). In this study attention is limited to the case where v > 1= d . Not only the sample materials considered here, but many other transversely isotropic materials such as Al2O3, Cadmium, SiC, etc. fall into this category. pffiffiffi By using Eqs. (12) and (20) the condition v > 1= d can also be expressed in terms of the effective parameters as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j > mmTT þ ð1 m2 Þð1 m2TT Þ. ð22Þ Note that another physical limitation on the effective parameter j is that, considering (5) and (11), and observing that GTL > 0; j þ m > 0.
ð23Þ
By using Eqs. (2b), (3b) and (18), and observing that the second integrals that would arise due to (18) in (2b) and (3b) would vanish at z = 0 due to the sin(az) term, one can obtain Z 1 k2 J 1 ðkrÞ½k2 ðm2 þ m4 Þ aðm2 r22 þ m4 r24 Þ dk; ð24Þ srz ðr; 0Þ ¼ 0
ow ðr; 0Þ ¼ or
Z
1
k2 J 1 ðkrÞ½k2 a44 ðm2 þ m4 Þ ða33 d 2a13 aÞðm2 r22 þ m4 r24 Þ dk.
ð25Þ
0
Then by introducing the auxiliary function ow ðr; 0Þ ¼ GðrÞ ð26Þ or and observing that G(r) = 0 outside the interval r 2 (Ri, Ro) one can express m2 and m4 in terms of the auxiliary function G(r) by using Eqs. (15), (24) and (25) by taking inverse Hankel Transforms. Then, Z Ro 1 mk ¼ X k 3 qGðqÞJ 1 ðkqÞ dq; k ¼ 2; 4; ð27Þ k Ri where X2 ¼
ð1 a^r24 Þ ; ð^r22 ^r24 Þða33 d 2a13 a a44 aÞ
X4 ¼
ð1 a^r22 Þ . ð^r22 ^r24 Þða33 d 2a13 a a44 aÞ
ð28Þ
By substituting (27) and (28) first into (18) and then substituting the stress function in terms of G(r) into (2) and (3) the expressions for displacements and stresses in a hollow cylinder which contains a crack on z = 0 at r 2 (Ri, Ro) plane can be obtained. Then by using the homogeneous boundary conditions (13) and (14) one can express the unknown functions A1, A2, A3, A4 in terms of G(r) as well. (The details of this procedure is very similar to that followed in [10] and the interested reader may refer to that article.) Once A1, A2, A3, A4 are expressed in terms of G(r), one can substitute these functions into (17) to obtain the singular integral equation of this problem as follows: Z Ro c mðr; qÞ 1 mðr; qÞ þc þ GðqÞ ð29Þ þ 2qk 2 ðr; qÞ dq ¼ ppðrÞ; Ri 6 r 6 Ro . qr qr qþr Ri
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The terms in Eq. (29) are given as follows: c ¼ X 2^r2 ðc d^r22 Þ þ X 4^r4 ðc d^r24 Þ;
ð30Þ
8 r > > < E q ; q > r; ð31Þ mðr; qÞ ¼ q q2 r2 > q r > :E þK ; q < r; r q r rq Z 1 2 ðcc1 dÞ 0 0 0 2 2 0 k 2 ðr; qÞ ¼ ½A a I ðc arÞ þ A K ðc arÞ þ ðcc dÞa ½A I ðc arÞ þ A K ðc arÞ da; 0 1 2 0 1 3 0 2 4 0 2 2 ðc21 c22 Þ 1 0 ð32Þ where K and E are the complete elliptic integrals of the first and the second kind, respectively. and A04 are given in Appendix A. In (29), the first term gives the well known Cauchy type singularity, associated with the square root stress singularities at the tips of internal cracks. The second term contains a logarithmic singularity at r = q. This singularity can be treated as was done in [10]. k2(r, q) is a Fredholm kernel. A01 ,
A02 ,
A03
2.2. Dependence of integral equation on material parameters In problems, such as the one considered in this study, it is possible that the quantities of interest such as the stress intensity factors would be dependent only on some of the material constants. The remaining constants may enter the analysis as multiplicative factors or they may not have any influence at all. For example, in the isotropic counterpart of this study (given in [2]), it is seen that the stress intensity factors depend only on the poisson ratio and they are independent of the modulus of rigidity. Therefore it is worthwhile to take a closer look at the material constant combinations appearing in the integral equation (29). At this point by using (12), (20) and (21), one can easily show that, c1 ðj; m; mTT Þ c ðj; m; mTT Þ ; c2 ¼ 2 . ð33Þ d d In (33) ^r2 , ^r4 , c1 and c2 are functions of j, m and mTT only, that is d (or 1/d) appears as a multiplicative factor in the definitions of ^r2 , ^r4 , c1 and c2. From their definitions it is clear that ^r2 , ^r4 , c1 and c2 are independent of E. Then by substituting the definitions of aij given by (11) along with (12) and (33) into (28), after some algebraic manipulations, it is possible to show that, ^r2 ¼ d^r2 ðj; m; mTT Þ;
X 2 ¼ EX 2 ðj; m; mTT Þ;
^r4 ¼ d^r4 ðj; m; mTT Þ;
c1 ¼
X 4 ¼ EX 4 ðj; m; mTT Þ.
ð34Þ
In other words, X k ðk ¼ 2; 4Þ are functions of j, m and mTT only, E appears as a multiplicative factor in the definitions of Xk, and Xk are independent of d. Finally, by using (12), (30), (33) and (34) one can show that c¼
E c ðj; m; mTT Þ. d
ð35Þ
Note that the quantities with a superscript Ô*Õ appearing in Eqs. (33)–(35) can be easily obtained through substitution and factorization, hence they are not explicitly given in this article. Proceeding in a similar manner, one can also show that the material constant combinations appearing in the Fredholm kernel can be given in the following form: K1 ¼
cc21 d K1 ðj; m; mTT Þ ¼ ; c21 c22 d2
K2 ¼ cc22 d ¼
K2 ðj; m; mTT Þ . d4
ð36Þ
F. Suat Kadiog˘lu / Engineering Fracture Mechanics 72 (2005) 2159–2173
By using (33) and (36), and defining a new dummy variable of integration, a b¼ ; d the Fredholm kernel given in (32) can be written as follows: Z 1 0 k 2 ðr; qÞ ¼ bK1 A1 I 0 ðc1 brÞ þ A02 K 0 ðc1 brÞ þ b3 K2 ½A03 I 0 ðc2 brÞ þ A04 K 0 ðc2 brÞ db.
2165
ð37Þ
ð38Þ
0
Now by using the definitions given in Appendix A and following the procedure outlined in Appendix B, it could be shown that Ed can be factored out of A01 , A02 , A03 and A04 as well. Hence the Fredholm kernel in (38) and the integral equation in (29) can be written respectively as k 2 ðr; qÞ ¼ Z
Ro
GðqÞ Ri
Z
E d
1 3 0 0 0 fbK1 ½A0 1 I 0 ðc1 brÞ þ A2 K 0 ðc1 brÞ þ b K2 ½A3 I 0 ðc2 brÞ þ A4 K 0 ðc2 brÞg db;
ð39Þ
0
c mðr; qÞ 1 mðr; qÞ d þ þ c þ 2qk 2 ðr; qÞ dq ¼ p pðrÞ; qr qþr E qr
Ri 6 r 6 Ro .
ð40Þ
In this form, the left hand side of (40) contains neither E nor d. 2.3. Extraction of singularities The Fredholm kernel k2(r, q) is bounded as long as the inner and outer crack tips remain in the cylinder, i.e. Rc < Ri and Ro < R. However when the crack is terminating at one of the free surfaces (i.e. Rc = Ri or Ro = R), the integral in (32) becomes unbounded. For this case an asymptotic analysis for a ! 1 is required. Letting, Kðr; q; aÞ ¼
ðcc21 dÞ 0 ½A I 0 ðc1 arÞ þ A02 K 0 ðc1 arÞa þ ðcc22 dÞa3 ½A03 I 0 ðc2 arÞ þ A04 K 0 ðc2 arÞ; ðc21 c22 Þ 1
ð41Þ
one can write, k 2 ðr; qÞ ¼
Z
1
½Kðr; q; aÞ K 1 ðr; q; aÞ da þ 0
Z
1
K 1 ðr; q; aÞ da;
ð42Þ
0
where K1(r, q, a) gives the asymptotic behavior of K(r, q, a) as a ! 1. After a lengthy analysis K1(r, q, a) can be obtained as follows: 1 K 1 ðr; q; aÞ ¼ pffiffiffiffiffi ^r2 j5 eaððqRÞ=^r2 þc1 ðrRÞÞ þ ^r4 j6 eaððqRÞ=^r4 þc1 ðrRÞÞ 2 rq 1 þ ^r2 j7 eaððqRÞ=^r2 þc2 ðrRÞÞ þ ^r4 j8 eaððqRÞ=^r4 þc2 ðrRÞÞ þ pffiffiffiffiffi ^r2 j5 eaððqRc Þ=^r2 þc1 ðrRc ÞÞ 2 rq þ ^r4 j6 eaððqRc Þ=^r4 þc1 ðrRc ÞÞ þ ^r2 j7 eaððqRc Þ=^r2 þc2 ðrRc ÞÞ þ ^r4 j8 eaððqRc Þ=^r4 þc2 ðrRc ÞÞ .
ð43Þ
In (43), the first square bracket contains the terms related to the case of outer edge crack and the second square bracket contains the terms related to the case of inner edge crack. The material parameters j5, j6, j7 and j8 are given in Appendix A. It can be shown that j6 = j7. Having known K1(r, q, a), the second integral in (42) can be evaluated in closed form, giving;
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Z
1 0
^r22 j5 ^r24 j6 ^r22 j7 1 K 1 ðr; q; aÞ da ¼ pffiffiffiffiffi 2 rq R q þ c1^r2 ðR rÞ R q þ c1^r4 ðR rÞ R q þ c2^r2 ðR rÞ ^r24 j8 ^r22 j5 ^r24 j6 þ þ R q þ c2^r4 ðR rÞ q Rc þ c1^r2 ðr Rc Þ q Rc þ c1^r4 ðr Rc Þ ^r22 j7 ^r24 j8 þ þ . q Rc þ c2^r2 ðr Rc Þ q Rc þ c2^r4 ðr Rc Þ
ð44Þ
Multiplying (44) with 2q gives the generalized Cauchy kernel. The singularity extraction has been done starting from Eq. (32), but it is obvious that it could have been done starting from Eq. (39), that is after factoring out E/d. Hence one can see that E/d can also be factored out of K1(r, q, a) given in (43) and its integrated form given in (44), which indeed is the case. 2.4. Stress intensity factors The quantities of interest in this study are the stress intensity factors for inner and outer edge cracks. Noting that gðrÞ GðrÞ ¼ R i < r < Ro ; ð45Þ x g; ðRo rÞ ðr Ri Þ where g(r) is a bounded function, the stress intensity factors at the crack tips can be expressed as follows: (a) Inner edge crack: (Ri = Rc, Ro < R, x = 1/2, g = 0) pffiffiffi pffiffiffi kðRo Þ ¼ limþ 2ðr Ro Þ1=2 rzz ðr; 0Þ ¼ c 2gðRo Þ;
ð46Þ
(b) Outer edge crack: (Ri > Rc, Ro = R, x = 0, g = 1/2) pffiffiffi pffiffiffi kðRi Þ ¼ lim 2ðRi rÞ1=2 rzz ðr; 0Þ ¼ c 2gðRi Þ.
ð47Þ
r!Ro
r!Ri
But from (35), one can also write; E pffiffiffi E pffiffiffi kðRo Þ ¼ c 2gðRo Þ; kðRi Þ ¼ c 2gðRi Þ. d d
ð48Þ
On the other hand, from (40) and (45) one can observe that g(r) is directly proportional to d/E, that is d g ðrÞ; ð49Þ E hence, one can easily see that the stress intensity factors are independent of E and d. The singular integral equation (29) is solved numerically by using the method outlined in [10] for various values of the material parameters j, m and mTT; sample material pairs and for various values of geometric parameters. The loading has been taken to be uniform, i.e. p(r) = r0. The results obtained are presented and discussed in the next section. gðrÞ ¼
3. Results and discussion For the problem under consideration there are three distinct crack geometries, namely, embedded, inner edge, and outer edge cracks. The geometric parameters that describe these configurations are the hole ra-
F. Suat Kadiog˘lu / Engineering Fracture Mechanics 72 (2005) 2159–2173
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dius (Rc), the coordinates of the inner and outer crack tips (Ri, Ro), and the outer radius R. In addition to these geometric parameters, there are three non-dimensional material parameters, namely j, m and mTT. Two sets of results are obtained in this study. In the first set of results a particular crack and cylinder geometry is selected and the influence of material parameters on stress intensity factors (SIF) is examined. In the second set of results, three particular materials, namely, magnesium (j = 1.226, m = 0.239, mTT = 0.357), a graphite-epoxy composite (j = 4.314, m = 0.084, mTT = 0.4) and steel (j = 1.001, m = 0.306, mTT = 0.300) are considered and the variation of SIFÕs with respect to the geometric parameters are presented. Only edge crack configurations are considered. Of these materials the former two are transversely isotropic where as steel is isotropic. At this point it should be noted that the formulation is made for transversely isotropic materials and it becomes degenerate for isotropic materials (j = 1, m = mTT, d = 1). This problem is overcome by introducing small perturbations in the compliance values of the isotropic material, thereby artificially rendering it ÔslightlyÕ anisotropic [8]. The compliances of the sample materials are also given in Table 1. The number of geometric parameters is reduced to two by normalizing them using outer radius, R, of the cylinder. In the tables, wall thickness of the cylinder, h = R Rc and crack length, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a = Ro Ri are used in presenting the results. Stress intensity factors are also normalized by r0 Ro Ri . In order to establish the validity and accuracy of the numerical results comparisons are made with [2] (bold values in tables). The agreement is found to be quite good. Fig. 2 shows the variation of SIFÕs for an inner edge crack as a function of effective poisson ratio, for selected values of shear parameter and inplane poisson ratio.
Table 1 Compliances of sample materials · 1011 [m2/N] Material
a11
a12
a13
a33
a44
Steel Magnesium Gr-Epoxy
0.476 2.20 8.55
0.143 0.785 3.42
0.145 0.498 0.210
0.473 1.97 0.725
1.240 6.10 21.9
1.58 1.54
κ =1.05
1.50 k
σ 0a1/2
κ =3.00 1.46 1.42
κ =5.00
1.38 1.34 0.0
0.1
0.2
0.3
0.4
0.5
ν Fig. 2. Stress intensity factors for an inner edge crack as a function of m(Rc/R = 0.9, a/h = 0.4). (Groups of curves correspond to j = 1.05, 3.00, and 5.00 in top to bottom order, within each group, curves correspond to mTT = 0.4, 0.25 and 0.10 in top to bottom order.)
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It is observed that the dependence of SIF on both Poisson ratios is rather weak (note the scale of the graph) and this dependence could be neglected for practical purposes. A similar trend is observed for outer edge cracks in Fig. 3. On the other hand, Fig. 4 shows that significant reduction in SIF can be obtained by increasing shear parameter beyond 1.0 (which is the value for isotropic material). If j is further increased, SIF tends to a constant value. Table 2 shows some sample results for inner surface cracks. Basic trends regarding the variation of geometric parameters are the same for all the materials. SIF for a given crack size increase as the wall thickness
1.64
κ =1.05
1.60 1.56
κ =3.00 k 1.52 σ 0a1/2 1.48
κ =5.00
1.44 1.40 0.0
0.1
0.2
0.3
0.4
0.5
ν Fig. 3. Stress intensity factors for an outer edge crack as a function of m(Rc/R = 0.9, a/h = 0.4). (Groups of curves correspond to j = 1.05, 3.00, and 5.00 in top to bottom order, within each group, curves correspond to mTT = 0.4, 0.25 and 0.10 in top to bottom order.)
1.64 1.56 1.48 1.40 k σ 0a1/2 1.32 1.24 1.16
κ =1.05 1.08 0
10 20 30 40 50 60 70 80 90 100 ν
Fig. 4. Stress intensity factors for inner (bottom) and outer (top) edge cracks as a function of j. (mTT = 0.25, m = 0.25, Rc/R = 0.9, a/h = 0.4.)
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Table 2 Normalized stress intensity factors for inner surface cracks (a = Ro Ri, h = R Rc, Ri = Rc) Rc/R
a/h 0.2
0.4
0.6
Mg
0.1 0.3 0.5 0.7 0.9
0.7699 0.9354 1.0473 1.1409 1.2436
0.7470 0.9097 1.0711 1.2569 1.5464
0.8080 0.9837 1.1856 1.4533 1.9788
Graphite-Epoxy
0.1 0.3 0.5 0.7 0.9
0.7623 0.9036 0.9956 1.0762 1.1788
0.7369 0.8743 0.9985 1.1387 1.3866
0.7831 0.9303 1.0849 1.2788 1.6750
Steel
0.1
0.7707 0.775 0.9388 0.942 1.0531 1.055 1.1486 1.150 1.2521 1.253
0.7492 0.754 0.9150 0.920 1.0807 1.085 1.2712 1.275 1.5645 1.568
0.8139 0.820 0.9936 1.000 1.2014 1.208 1.4774 1.484 2.0160 2.025
0.3 0.5 0.7 0.9
Table 3 Normalized stress intensity factors for outer surface cracks (a = Ro Ri, h = R Rc, Ro = R) Rc/R
a/h 0.2
0.4
0.6
Mg
0.1 0.3 0.5 0.7 0.9
1.2343 1.2198 1.2204 1.2347 1.2739
1.5109 1.4290 1.4129 1.4559 1.6178
2.1744 1.8657 1.7595 1.7917 2.1083
Graphite-Epoxy
0.1 0.3 0.5 0.7 0.9
1.1842 1.1599 1.1513 1.1603 1.2058
1.4762 1.3632 1.3102 1.3145 1.4481
2.1451 1.7801 1.6157 1.5777 1.7834
Steel
0.1
1.2460 1.244 1.2320 1.231 1.2328 1.232 1.2465 1.247 1.2839 1.285
1.5197 1.513 1.4416 1.437 1.4294 1.427 1.4758 1.475 1.6387 1.641
2.1803 2.159 1.8782 1.866 1.7792 1.773 1.8199 1.818 2.1485 2.153
0.3 0.5 0.7 0.9
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decreases. For relatively thick walled cylinders SIF first decrease and then starts to increase as the crack length increases. This indicates a possibility of crack arrest. It can be observed that the influence of transverse isotropy on SIF is minimal for large wall thicknesses. However for the highly anisotropic graphite epoxy composite, significantly lower SIF can be achieved. Table 3 shows some sample results for outer surface cracks. SIFÕs for outer cracks are in general larger than those of inner cracks. Again, basic trends regarding the variation of geometric parameters are the same for all the materials. Over a wide range of wall thickness ratios, SIFÕs either decrease or remain pretty much constant with the decreasing wall thickness. On the other hand, as the crack lengths increase, so does the SIF for all the wall thicknesses considered. It can again be observed that for relatively thick walled cylinders such as Rc/R = 0.1, the influence of transverse isotropy on SIF is minimal for all the crack sizes considered. This is actualy consistent with the findings of [8]. For relatively thin walled cylinders, however, significant reduction in SIF can be achieved for highly anisotropic material.
Appendix A. Terms used in the calculation of kernels 2
A01
3
2
f11
f12
f13
6 07 6 6 A2 7 6 f21 f22 f23 6 7¼6 6 A0 7 6 f 4 3 5 4 31 f32 f33 f41 f42 f43 A04 R Ro note that A1 ¼ Ri qGðqÞA01 dq, l1 ¼
f14
7 f24 7 7 f34 7 5 f44
L1
3
6 7 6 L2 7 6 7 6L 7 4 35 L4
etc.
c21 a ¼ l1 ðj; m; mTT Þ; c21 c22
l3 ¼ c22 a ¼
31 2
1 l3 ðj; m; mTT Þ; d2
l2 ¼
c1 ðb 1Þ ¼ dl2 ðj; m; mTT Þ; c21 c22
1 l4 ¼ c2 ðb 1Þ ¼ l4 ðj; m; mTT Þ; d
c1 ðc21 aÞ 1 1 ¼ l5 ðj; m; mTT Þ; l6 ¼ c2 ðc22 aÞ ¼ 3 l6 ðj; m; mTT Þ; d c21 c22 d ! ! 1 a^r22 E 1 a^r24 E W 2 ¼ X2 ¼ 2 W 2 ðj; m; mTT Þ; W 4 ¼ X 4 ¼ 2 W 4 ðj; m; mTT Þ; ^r22 ^r24 d d
l5 ¼
Y2 ¼ X2
2 a^r2 1 E ¼ Y 2 ðj; m; mTT Þ; ^r2 d
Z 2 ¼ X 2 ðb 1Þ ¼ EZ 2 ðj; m; mTT Þ; f11 ¼ l1 aI 0 ðc1 aRc Þ þ
l2 I 1 ðc1 aRc Þ; Rc
Y4 ¼ X4
2 a^r4 1 E ¼ Y 4 ðj; m; mTT Þ; ^r4 d
Z 4 ¼ X 4 ðb 1Þ ¼ EZ 4 ðj; m; mTT Þ; f 12 ¼ l1 aK 0 ðc1 aRc Þ
f13 ¼ l3 a3 I 0 ðc2 aRc Þ þ
l4 2 a I 1 ðc2 aRc Þ; Rc
f21 ¼ l5 I 1 ðc1 aRc Þ;
f 22 ¼ l5 K 1 ðc1 aRc Þ;
l2 K 1 ðc1 aRc Þ; Rc
f 14 ¼ l3 a3 K 0 ðc2 aRc Þ
l4 2 a K 1 ðc2 aRc Þ; Rc
f 23 ¼ l6 a2 I 1 ðc2 aRc Þ;
f 24 ¼ l6 a2 K 1 ðc2 aRc Þ;
F. Suat Kadiog˘lu / Engineering Fracture Mechanics 72 (2005) 2159–2173
f31 ¼ l1 aI 0 ðc1 aRÞ þ
l2 I 1 ðc1 aRÞ; R
f33 ¼ l3 a3 I 0 ðc2 aRÞ þ
f 32 ¼ l1 aK 0 ðc1 aRÞ
l4 2 a I 1 ðc2 aRÞ; R
2171
l2 K 1 ðc1 aRÞ; R
f 34 ¼ l3 a3 K 0 ðc2 aRÞ
l4 2 a K 1 ðc2 aRÞ; R
f41 ¼ l5 I 1 ðc1 aRÞ; f 42 ¼ l5 K 1 ðc1 aRÞ; f 43 ¼ l6 a2 I 1 ðc2 aRÞ; f 44 ¼ l6 a2 K 1 ðc2 aRÞ; Rc a qa aRc aq 1 Rc a qa Rc a qa L1 ¼ I 0 I1 K1 Y 2a þ I 0 K1 Y 4a K1 Z2 þ I 1 K1 Z4 ; ^r2 ^r2 ^r4 ^r2 ^r2 ^r4 ^r4 ^r4 Rc Rc a qa Rc a qa L2 ¼ I 1 K1 W 2 þ I1 K1 W 4; ^r2 ^r2 ^r4 ^r4 qa Ra qa Ra 1 a Ra a Ra I1 q L3 ¼ I 1 K0 Y 2a I 1 K0 Y 4a K1 Z2 þ I 1 q K1 Z4 ; ^r2 ^r2 ^r4 ^r4 ^r2 ^r2 ^r4 ^r4 R qa Ra qa Ra L4 ¼ I 1 K1 W 2 þ I1 K1 W4 ; ^r2 ^r2 ^r4 ^r4 j5 ¼
ðd cc21 Þð1 þ c2^r2 Þð1 þ a^r22 ÞX 2 ; ða c21 Þðc2 c1 Þ^r22
j6 ¼
ðd cc21 Þð1 þ c2^r4 Þð1 þ a^r24 ÞX 4 ; ða c21 Þðc2 c1 Þ^r24
j7 ¼
ðcc22 dÞð1 þ c1^r2 Þð1 þ a^r22 ÞX 2 ; ða c22 Þðc2 c1 Þ^r22
j8 ¼
ðcc22 dÞð1 þ c1^r4 Þð1 þ a^r24 ÞX 4 . ða c22 Þðc2 c1 Þ^r24
Appendix B. Factorization of A0i ði ¼ 1; . . . ; 4Þ From Appendix A, one can observe that 4 X
fij A0j ¼ Li
ði ¼ 1; . . . ; 4Þ;
ðA:1Þ
j¼1
and from (37) a ¼ db.
ðA:2Þ
Then using the definitions of li(i = 1, . . ., 6) along with (33), one can write for example, f11 as f11 ¼ l1 dbI 0 ðc1 bRc Þ þ
l2 d I 1 ðc1 bRc Þ ¼ df11 . Rc
ðA:3Þ
Similarly one can show that f1j ¼ df1j ;
ðj ¼ 1; . . . ; 4Þ;
ðA:4Þ
1 f2j ¼ f2j d
ðj ¼ 1; . . . ; 4Þ;
ðA:5Þ
F. Suat Kadiog˘lu / Engineering Fracture Mechanics 72 (2005) 2159–2173
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f3j ¼ df3j
ðj ¼ 1; . . . ; 4Þ;
ðA:6Þ
1 f4j ¼ f4j d
ðj ¼ 1; . . . ; 4Þ;
ðA:7Þ
where fij ði; j ¼ 1; . . . ; 4Þ do not contain the material parameters E and d. Also one can express, for example L1, as Rc db qdb E dbRc dbq E Y db þ I 0 Y db L1 ¼ I 0 K1 K1 d^r2 d^r2 d 2 d^r4 d 4 d^r4 1 Rc db qdb Rc db qdb I1 K1 Z2E þ I 1 K1 Z4E ; Rc d^r2 d^r2 d^r4 d^r4 Rc b qb bRc bq L1 ¼ E I 0 K 1 Y 2b þ I 0 K 1 Y 4 b ^r2 ^r2 ^r4 ^r4 1 Rc b qb Rc b qb I1 K 1 Z 2 þ I 1 K 1 Z 4 ; ^r2 ^r2 ^r4 ^r4 Rc L1 ¼ EL1 ;
ðA:8Þ
ðA:9Þ ðA:10Þ
and proceeding similarly, L2 ¼
E L2 ; d2
ðA:11Þ
L3 ¼ EL3 ; L4 ¼
ðA:12Þ
E L4 ; d2
ðA:13Þ
where Li ði ¼ 1; . . . ; 4Þ is independent of E and d. Then from (A.1), (A.4)–(A.7) and (A.10)–(A.13) it is obvious that 4 X
fij A0j ¼
j¼1
E L d i
ði ¼ 1; . . . ; 4Þ.
ðA:14Þ
(A.14) implies that the functions A0j ðbÞ, which are supposed to be used in (38) are directly proportional to E/d.
References [1] Erdol R, Erdogan F. A thick-walled cylinder with an axisymmetric internal or edge crack. J Appl Mech 1978;45:281–6. [2] Nied HF, Erdogan F. The elasticity problem for a thick-walled cylinder containing a circumferential crack. Int J Fract 1983;22:277–301. [3] Nied HF. Thermal shock in a circumferentially cracked hollow cylinder with cladding. Engng Fract Mech 1984;20:113–37. [4] Varfolomeyev IV. Weight function for external circumferential cracks in hollow cylinders subjected to axisymmetric opening mode loading. Engng Fract Mech 1998;60:333–9. [5] Varfolomeyev IV. Stress intensity factors for internal circumferential cracks in thin- and thick-walled cylinders. Engng Fract Mech 1998;60:491–500. [6] Meshii T, Watanabe K. Stress intensity factor for a circumferential crack in a finite length thin to thick-walled cylinder under an arbitrary biquadratic stress distribution on the crack surfaces. Engng Fract Mech 2001;68:975–86.
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[7] Altinel T, Fildisß H, Yahsßi OS. The stress intensity factors for an infinitely long transversely isotropic, thick walled cylinder which contains a ring-shaped crack. Int J Fract 1996;78:211–25. [8] Atsumi A, Shindo Y. Singular stresses in a transversely isotropic circular cylinder with circumferential edge crack. Int J Engng Sci 1979;17:1229–36. [9] Uyaner M, Akdemir A, Erim S, Avci A. Plastic zones in a transversely isotropic solid cylinder containing a ring-shaped crack. Int J Fract 2000;106:161–75. [10] Kadioglu FS. Axisymmetric crack terminating at the interface of transversely isotropic, dissimilar media. Int J Fract 2002;116:51–79. [11] Krenk S. On the elastic constants of plane orthotropy. J Compos Mater 1979:108–16.