In-plane analysis of a cracked orthotropic half-plane

International Journal of Solids and Structures 44 (2007) 1608–1627 www.elsevier.com/locate/ijsolstr

In-plane analysis of a cracked orthotropic half-plane A.R. Fotuhi, R.T. Faal, S.J. Fariborz

*

Department of Mechanical Engineering, Amirkabir University of Technology (Tehran Polytechnic), 424, Hafez Avenue, Tehran 158754413, Iran Received 24 April 2006; received in revised form 12 June 2006 Available online 5 July 2006

Abstract The stress fields in an orthotropic half-plane containing Volterra type climb and glide edge dislocations under plane stress condition are derived. The dislocation solutions are utilized to formulate integral equations for dislocation density functions on the surface of smooth cracks embedded in the half-plane under in-plane loads. The integral equations are of Cauchy singular type which are solved numerically. The dislocation density functions are employed to evaluate modes I and II stress intensity factors for multiple cracks with different configurations.  2006 Elsevier Ltd. All rights reserved. Keywords: Orthotropic half-plane; Volterra dislocation; Dislocation density; Cauchy singular; Curved crack

1. Introduction Orthotropic materials are vulnerable to cracking. The defects may initiate during the manufacturing process or in the regions subjected to steep stress gradient in the course of service life of a mechanical component. Multiple cracks with any shape and direction may exist in the material making the analytical stress analysis of a body intractable. Therefore, only restricted geometries may be tackled without resorting to approximation methods. Various attempts have been made to analyze half-plane of composite materials weakened by cracks. The solution of displacement discontinuity in an anisotropic half-plane was obtained and adopted by Wen (1989) for the analysis of crack problems. Sung and Liou (1995a), used the basic solution obtained by Suo (1990) and Ting (1992) to formulate the integral equations for dislocation density on a straight embedded crack in an anisotropic half-plane. The solution to integral equations was accomplished and the numerical values of stress intensity factor were determined in a half-plane where an axis of material orthotropy was parallel to the boundary of half-plane. The article also contains the solution of a cracked half-plane made up of socalled degenerate material. In another article, Sung and Liou (1995b), determined the stress intensity factor for a straight crack embedded in a composite half-plane with clamped boundary. The boundary element formu*

Corresponding author. Tel.: +98 21 6454 3460; fax: +98 21 641 9736. E-mail address: [email protected] (S.J. Fariborz).

0020-7683/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2006.06.041

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1609

lation for a cracked anisotropic elastic half-plane was accomplished by Pan et al. (1997). They also studied the effects of material anisotropy on the magnitude of stress intensity factors. Huang and Kardomateas (2004), analyzed a crack in an anisotropic half-plane by means of dislocation-based boundary element method. In this article, the stress fields in an orthotropic half-plane caused by climb and glide Volterra type dislocations are obtained in closed forms. The half-plane is such that an axis of orthotropy is parallel with the boundary of half-plane. For an isotropic half-plane the dislocation solutions recover the well-known results in literature. The stress fields due to dislocations are then used to derive singular integral equations for a half-plane with multiple cracks under tensile and shear tractions. The case of crack closure, however, is not studied. Consequently, the crack geometry and applied traction should be such as to prevent the possibility of crack closure. To demonstrate the applicability of the procedure, four examples of straight and curved cracks are solved and cracks interaction is investigated. 2. Formulation of the problem The dislocation solution for the anisotropic half-plane under plane-strain conditions was given by Pande and Chou (1971). They applied Eshelby et al. (1953) formulation and made use of analogy with the solution of isotropic half-plane to derive the stress components. Nevertheless, a systematic procedure is taken up here to obtain dislocation solution in the orthotropic half-plane for the plane-stress case. The procedure is applicable to plane-strain situation as well. The Hooke’s law in plane-stress elasticity for orthotropic materials, taking the coordinate system as the axes of principal orthotropy, are 1 mxy rx  ry Ex Ex 1 mxy ey ¼ ry  rx Ey Ex 1 cxy ¼ rxy Gxy

ex ¼

ð1Þ

The equilibrium equations are satisfied by expressing the stress components in terms of Airy stress function as rx ¼

o2 / ; oy 2

ry ¼

o2 / ; ox2

rxy ¼ 

o2 / oxoy

ð2Þ

The only equation of compatibility which should be satisfied is 2 o2 exx o2 eyy o cxy ¼0 þ  oy 2 ox2 oxoy

Substituting (1) and (2) into (3), we arrive at   4 E x o4 / Ex o/ o4 / þ  2m þ ¼0 xy ox2 oy 2 oy 4 Ey ox4 Gxy

ð3Þ

ð4Þ

We consider the half-plane y > h and situate a climb and a glide dislocation with Burgers vectors By, and Bx, respectively, at the origin. To facilitate the solution of Eq. (4), the dislocation line is chosen to be the positive part of the x-axis. Therefore, the conditions representing the dislocation are vðx; 0 Þ  vðx; 0þ Þ ¼ By H ðxÞ uðx; 0 Þ  uðx; 0þ Þ ¼ Bx H ðxÞ

ð5Þ

where H(x) is the Heaviside step function. Moreover, for both types of dislocations the continuity of stress components along the x-axis should be satisfied. Consequently ry ðx; 0þ Þ ¼ ry ðx; 0 Þ rxy ðx; 0þ Þ ¼ rxy ðx; 0 Þ

ð6Þ

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The traction-free conditions on the boundary of half-plane yields ry ðx; hÞ ¼ 0 rxy ðx; hÞ ¼ 0

ð7Þ

The solution to Eq. (4) may be accomplished by means of the Fourier transform. The complex Fourier transform of Airy stress function in the x-direction is defined as Uðx; yÞ ¼

Z

1

/ðx; yÞeixx dx

ð8Þ

1

where i ¼

pffiffiffiffiffiffiffi 1: The inversion of (8) is

/ðx; yÞ ¼

1 2p

Z

1

Uðx; yÞeixx dx

ð9Þ

1

The application of Eq. (8) to Eq. (4) with the aid of regularity condition lim /ðx; yÞ ¼ 0, leads to two fourth jxj!1

order ordinary differential equations for U(x, y), in the strip, h < y < 0, and half-plane, y > 0, regions. The equations are solved and the Fourier inversion formula (9) is applied to obtain the Airy stress function as Z 1 1 /¼ ½A1 ðxÞer1 yx þ B1 ðxÞer2 yx þ C 1 ðxÞer1 yx þ D1 ðxÞer2 yx eixx dx; 2p 1 Z 1 1 /¼ ½A2 ðxÞer1 yjxj þ B2 ðxÞer2 yjxj eixx dx; y P 0 2p 1

h 6 y 6 0 ð10Þ

where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u  s 2 u E E E t x x x r1 ¼  mxy þ  mxy  2Gxy 2Gxy Ey vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u  s 2 u E Ex Ex t x  mxy   mxy  r2 ¼ 2Gxy 2Gxy Ey

ð11Þ

The constants r1 and r2 for orthotropic materials are real and positive, Lekhnitskii (1963). The unknown coefficients in Eq. (10) are determined by utilizing the Fourier transforms of Eqs. (5)–(7), for both dislocations. The expressions for the first two coefficients in Eq. (10) are obtained as A1 ðxÞ ¼

Ex Bx ½ðr1 þ r2 Þð1 þ sgnðxÞÞe2r2 hx  2r2 ð1 þ sgnðxÞÞeðr1 þr2 Þhx þ ðr1  r2 Þð1  sgnðxÞÞ ½1 þ pixdðxÞ 2 x2 2ðr1  r2 Þ ðr1 þ r2 Þ½ð1 þ sgnðxÞÞe2ðr1 þr2 Þhx þ 1  sgnðxÞ



B1 ðxÞ ¼

Ex By ½ðr1 þ r2 Þð1 þ sgnðxÞÞe2r2 hx  2r1 ð1 þ sgnðxÞÞeðr1 þr2 Þhx  ðr1  r2 Þð1  sgnðxÞÞ ½i  pxdðxÞ x2 2r1 ðr1  r2 Þ2 ðr1 þ r2 Þ½ð1 þ sgnðxÞÞe2ðr1 þr2 Þhx þ 1  sgnðxÞ

Ex Bx ½ðr1 þ r2 Þð1 þ sgnðxÞÞe2r1 hx  2r1 ð1 þ sgnðxÞÞeðr1 þr2 Þhx  ðr1  r2 Þð1  sgnðxÞÞ ½1 þ pixdðxÞ 2 x2 2ðr1  r2 Þ ðr1 þ r2 Þ½ð1 þ sgnðxÞÞe2ðr1 þr2 Þhx þ 1  sgnðxÞ



Ex By ½ðr1 þ r2 Þð1 þ sgnðxÞÞe2r1 hx  2r2 ð1 þ sgnðxÞÞeðr1 þr2 Þhx þ ðr1  r2 Þð1  sgnðxÞÞ ½i  pxdðxÞ 2 x2 2r2 ðr1  r2 Þ ðr1 þ r2 Þ½ð1 þ sgnðxÞÞe2ðr1 þr2 Þhx þ 1  sgnðxÞ ð12Þ

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where sgn(x) and d(x) are the sign and the Dirac delta functions, respectively. For the sake of brevity, the other coefficients in Eq. (10) are given in terms of A1(x) and B1(x) 1 ½ðr1 þ r2 ÞA1 ðxÞe2r1 hx þ 2r2 B1 ðxÞeðr1 þr2 Þhx  ðr1  r2 Þ 1 ½2r1 A1 ðxÞeðr1 þr2 Þhx þ ðr1 þ r2 ÞB1 ðxÞe2r2 hx  D1 ðxÞ ¼  ðr1  r2 Þ 1 ½ðr1 þ r2 Þðr1 sgnðxÞ þ r2 Þe2r1 hx þ 2r1 r2 ð1 þ sgnðxÞÞeðr1 þr2 Þhx þ ðr1  r2 Þðr1 sgnðxÞ  r2 ÞA1 ðxÞ A2 ðxÞ ¼ ðr1  r2 Þ2 r2 þ ½ðr1 þ r2 Þð1 þ sgnðxÞÞe2r2 hx  2ðr1 sgnðxÞ þ r2 Þeðr1 þr2 Þhx  ðr1  r2 Þð1  sgnðxÞÞB1 ðxÞ ðr1  r2 Þ2 r1 ½ðr1 þ r2 Þð1 þ sgnðxÞÞe2r1 hx  2ðr1 þ r2 sgnðxÞÞeðr1 þr2 Þhx þ ðr1  r2 Þð1  sgnðxÞÞA1 ðxÞ B2 ðxÞ ¼ ðr1  r2 Þ2 1 ½ðr1 þ r2 Þðr1 þ r2 sgnðxÞÞe2r2 hx þ 2r1 r2 ð1 þ sgnðxÞÞeðr1 þr2 Þhx þ ðr1  r2 Þðr1  r2 sgnðxÞÞB1 ðxÞ þ ðr1  r2 Þ2 C 1 ðxÞ ¼

ð13Þ

Plugging Eq. (10) into Eq. (2) and carrying out the integrations, the stress fields result in ( E x Bx r32 ðr1  r2 Þy r31 ðr1  r2 Þy r31 ðr1 þ r2 Þð2h þ yÞ r32 ðr1 þ r2 Þð2h þ yÞ rx ðx; yÞ ¼  2 2 þ 2 þ 2 r 1 y þ x2 2pðr1  r2 Þ2 ðr1 þ r2 Þ r22 y 2 þ x2 r1 ð2h þ yÞ2 þ x2 r2 ð2h þ yÞ2 þ x2 ) ( 2r21 r2 ðr1 h þ r2 h þ r1 yÞ 2r1 r22 ðr1 h þ r2 h þ r2 yÞ E x By r2 ðr1  r2 Þx r1 ðr1  r2 Þx    2 2  2 2 2 2 2 r 1 y þ x2 2pðr1  r2 Þ ðr1 þ r2 Þ r22 y 2 þ x2 ðr1 h þ r2 h þ r1 yÞ þ x ðr1 h þ r2 h þ r2 yÞ þ x ) r1 ðr1 þ r2 Þx r2 ðr1 þ r2 Þx 2r21 x 2r22 x  2  þ þ r1 ð2h þ yÞ2 þ x2 r22 ð2h þ yÞ2 þ x2 ðr1 h þ r2 h þ r1 yÞ2 þ x2 ðr1 h þ r2 h þ r2 yÞ2 þ x2 ( Ex Bx r2 ðr1  r2 Þy r1 ðr1  r2 Þy r1 ðr1 þ r2 Þð2h þ yÞ r2 ðr1 þ r2 Þð2h þ yÞ  2 2 þ 2 þ 2 ry ðx; yÞ ¼  2 r 1 y þ x2 2pðr1  r2 Þ ðr1 þ r2 Þ r22 y 2 þ x2 r1 ð2h þ yÞ2 þ x2 r2 ð2h þ yÞ2 þ x2 ) ( 2r2 ðr1 h þ r2 h þ r1 yÞ 2r1 ðr1 h þ r2 h þ r2 yÞ Ex By r1 ðr1  r2 Þx r2 ðr1  r2 Þx þ   2 2  2 2 2 2 2 r 1 y þ x2 2pr1 r2 ðr1  r2 Þ ðr1 þ r2 Þ r22 y 2 þ x2 ðr1 h þ r2 h þ r1 yÞ þ x ðr1 h þ r2 h þ r2 yÞ þ x ) r2 ðr1 þ r2 Þx r1 ðr1 þ r2 Þx 2r1 r2 x 2r1 r2 x  2  þ þ r1 ð2h þ yÞ2 þ x2 r22 ð2h þ yÞ2 þ x2 ðr1 h þ r2 h þ r1 yÞ2 þ x2 ðr1 h þ r2 h þ r2 yÞ2 þ x2 ( Ex Bx r2 ðr1  r2 Þx r1 ðr1  r2 Þx r1 ðr1 þ r2 Þx r2 ðr1 þ r2 Þx  2 2 þ 2 þ rxy ðx; yÞ ¼  r 1 y þ x2 2pðr1  r2 Þ2 ðr1 þ r2 Þ r22 y 2 þ x2 r1 ð2h þ yÞ2 þ x2 r22 ð2h þ yÞ2 þ x2 ) ( 2r1 r2 x 2r1 r2 x E x By r2 ðr1  r2 Þy r1 ðr1  r2 Þy     2 2 r 1 y þ x2 2pðr1  r2 Þ2 ðr1 þ r2 Þ r22 y 2 þ x2 ðr1 h þ r2 h þ r1 yÞ2 þ x2 ðr1 h þ r2 h þ r2 yÞ2 þ x2 ) r1 ðr1 þ r2 Þð2h þ yÞ r2 ðr1 þ r2 Þð2h þ yÞ 2r1 ðr1 h þ r2 h þ r1 yÞ 2r2 ðr1 h þ r2 h þ r2 yÞ  2  2 þ þ r1 ð2h þ yÞ2 þ x2 r2 ð2h þ yÞ2 þ x2 ðr1 h þ r2 h þ r1 yÞ2 þ x2 ðr1 h þ r2 h þ r2 yÞ2 þ x2 ð14Þ

It is worth mentioning that only the first and second terms in each bracket of stress components (14) are Cauchy singular at dislocation location. In the particular case of an isotropic medium in two-dimensional elasticity, the material properties simplify to 8l 1þj 3j vxy ¼ 1þj Gxy ¼ l Ex ¼ Ey ¼

ð15Þ

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where the Kolosov constant for plane strain and plane stress situations are j = 3  4m and j = (3  m)/(1 + m), respectively. By virtue of (15) we have r1 = r2 = 1, and the stress components (14) reduce to 8 9 9 93 8 8 2 8 r yð3x2 þ y 2 Þ > xðx2  y 2 Þ > > > > > < = = = < x> < < 2l 1 6 7 ry ¼  By xð3y 2 þ x2 Þ 5 Bx yðy 2  x2 Þ 4 2 2 2 > > > > > > > pð1 þ jÞ :ðx þ y Þ ; ; ; : : : rxy xðy 2  x2 Þ yðx2  y 2 Þ 9 93 9 8 2 8 Mx > Nx > > > > = = = < < 1 6 7 M N þ þ B B 4 5 y y x y 2 4 > > > > ðx2 þ ð2h  yÞ Þ ; ; ; > : : M xy N xy

ð16Þ

The expressions for Ms and Ns are lengthy and are presented in the Appendix. The dislocation solution in infinite isotropic plane, cited for instance in Hills et al. (1996), may readily be recovered by letting h ! 1 in Eq. (16). Substituting (14) into (1), using the strain–displacement relationships, eij = (ui,j + uj,i)/2, i, j = x, y, integrating the resultant equations and ignoring rigid body motion we arrive at the displacement components u and v as      Bx x x 2 1 2 1  r Þðr þ m Þ tan  r Þðr þ m Þ tan ðr uðx; yÞ ¼ þ ðr 1 2 xy 1 2 xy 1 2 2 r1 y r2 y 2pðr1  r2 Þ ðr1 þ r2 Þ     x x þ ðr1 þ r2 Þðr21 þ mxy Þ tan1 þ ðr1 þ r2 Þðr22 þ mxy Þ tan1 r1 ð2h þ yÞ r2 ð2h þ yÞ     x x 2 1 2 1 2r2 ðr1 þ mxy Þ tan  2r1 ðr2 þ mxy Þ tan r1 h þ r 2 h þ r1 y r1 h þ r2 h þ r2 y þ

By 2

4pr1 r2 ðr1  r2 Þ ðr1 þ r2 Þ

fr2 ðr1  r2 Þðr21 þ mxy Þ ln½r21 y 2 þ x2   r1 ðr1  r2 Þðr22 þ mxy Þ ln½r22 y 2 þ x2  2

2

þ r2 ðr1 þ r2 Þðr21 þ mxy Þ ln½r21 ð2h þ yÞ þ x2  þ r1 ðr1 þ r2 Þðr22 þ mxy Þ ln½r22 ð2h þ yÞ þ x2  2

2

 2r1 r2 ðr21 þ mxy Þ ln½ðr1 h þ r2 h þ r1 yÞ þ x2   2r1 r2 ðr22 þ mxy Þ ln½ðr1 h þ r2 h þ r2 yÞ þ x2 g vðx; yÞ ¼

Bx 2

4pðr1  r2 Þ ðr1 þ r2 Þ

fr1 ðr1  r2 Þðr22 þ mxy Þ ln½r21 y 2 þ x2   r2 ðr1  r2 Þðr21 þ mxy Þ ln½r22 y 2 þ x2  2

2

 r1 ðr1 þ r2 Þðr22 þ mxy Þ ln½r21 ð2h þ yÞ þ x2   r2 ðr1 þ r2 Þðr21 þ mxy Þ ln½r22 ð2h þ yÞ þ x2  2

2

þ 2r1 r2 ðr22 þ mxy Þ ln½ðr1 h þ r2 h þ r1 yÞ þ x2  þ 2r1 r2 ðr21 þ mxy Þ ln½ðr1 h þ r2 h þ r2 yÞ þ x2 g      By 2 1 r1 y 2 1 r2 y  r Þðr þ m Þ tan  r Þðr þ m Þ tan ðr þ þ ðr 1 2 xy 1 2 xy 2 1 2 x x 2pðr1  r2 Þ ðr1 þ r2 Þ     r1 ð2h þ yÞ r2 ð2h þ yÞ  ðr1 þ r2 Þðr22 þ mxy Þ tan1  ðr1 þ r2 Þðr21 þ mxy Þ tan1 x x     r1 h þ r 2 h þ r1 y r1 h þ r2 h þ r2 y þ2r1 ðr22 þ mxy Þ tan1 þ 2r2 ðr21 þ mxy Þ tan1 x x ð17Þ By choosing the proper branch of multiple-valued function tan1( ), in Eq. (17) we may observe that the boundary data (5) are readily satisfied. To derive the integral equations for the crack problem, the distributed dislocation technique described in Hills et al. (1996) is employed. Let the climb and glide dislocations with densities bx and by, respectively, be distributed on a line in the half-plane performing a crack. The stress fields caused at a point by the

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1613

Fig. 1. Schematic view of a curved crack in a half-plane.

above-mentioned distribution of dislocations employing Eq. (14) while changing the coordinate system such that the x-axis coincides with the free boundary of half-planes, Fig. 1, are Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi( Ex r3 ðr1  r2 Þðy  bÞ r3 ðr1  r2 Þðy  bÞ rx ðx; yÞ ¼ þ 22 ½a0 ðtÞ2 þ ½b0 ðtÞ2  2 1 2 2 2 2pðr1  r2 Þ ðr1 þ r2 Þ 1 r1 ðy  bÞ þ ðx  aÞ r2 ðy  bÞ2 þ ðx  aÞ2 ) r31 ðr1 þ r2 Þðy þ bÞ r32 ðr1 þ r2 Þðy þ bÞ 2r21 r2 ðr2 b þ r1 yÞ 2r1 r22 ðr1 b þ r2 yÞ þ 2 bx ðtÞdt þ   r1 ðy þ bÞ2 þ ðx  aÞ2 r22 ðy þ bÞ2 þ ðx  aÞ2 ðr2 b þ r1 yÞ2 þ ðx  aÞ2 ðr1 b þ r2 yÞ2 þ ðx  aÞ2 Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi( Ex r1 ðr1  r2 Þðx  aÞ r2 ðr1  r2 Þðx  aÞ  2 ½a0 ðtÞ2 þ ½b0 ðtÞ2 2 þ 2 2 2 r1 ðy  bÞ þ ðx  aÞ r2 ðy  bÞ2 þ ðx  aÞ2 2pðr1  r2 Þ ðr1 þ r2 Þ 1 ) r1 ðr1 þ r2 Þðx  aÞ r2 ðr1 þ r2 Þðx  aÞ 2r21 ðx  aÞ 2r22 ðx  aÞ þ 2 þ   by ðtÞdt r1 ðy þ bÞ2 þ ðx  aÞ2 r22 ðy þ bÞ2 þ ðx  aÞ2 ðr2 b þ r1 yÞ2 þ ðx  aÞ2 ðr1 b þ r2 yÞ2 þ ðx  aÞ2 Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi( Ex r1 ðr1  r2 Þðy  bÞ r2 ðr1  r2 Þðy  bÞ  ½a0 ðtÞ2 þ ½b0 ðtÞ2 2 ry ðx; yÞ ¼ r1 ðy  bÞ2 þ ðx  aÞ2 r22 ðy  bÞ2 þ ðx  aÞ2 2pðr1  r2 Þ2 ðr1 þ r2 Þ 1 ) r1 ðr1 þ r2 Þðy þ bÞ r2 ðr1 þ r2 Þðy þ bÞ 2r2 ðr2 b þ r1 yÞ 2r1 ðr1 b þ r2 yÞ  2 bx ðtÞdt  þ þ r1 ðy þ bÞ2 þ ðx  aÞ2 r22 ðy þ bÞ2 þ ðx  aÞ2 ðr2 b þ r1 yÞ2 þ ðx  aÞ2 ðr1 b þ r2 yÞ2 þ ðx  aÞ2 Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi( Ex r2 ðr1  r2 Þðx  aÞ r1 ðr1  r2 Þðx  aÞ þ 2 ½a0 ðtÞ2 þ ½b0 ðtÞ2  2 þ 2 2 2 r1 ðy  bÞ þ ðx  aÞ r2 ðy  bÞ2 þ ðx  aÞ2 2pr1 r2 ðr1  r2 Þ ðr1 þ r2 Þ 1 ) r2 ðr1 þ r2 Þðx  aÞ r1 ðr1 þ r2 Þðx  aÞ 2r1 r2 ðx  aÞ 2r1 r2 ðx  aÞ  2 by ðtÞdt  þ þ r1 ðy þ bÞ2 þ ðx  aÞ2 r22 ðy þ bÞ2 þ ðx  aÞ2 ðr2 b þ r1 yÞ2 þ ðx  aÞ2 ðr1 b þ r2 yÞ2 þ ðx  aÞ2 Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi( Ex r1 ðr1  r2 Þðx  aÞ r2 ðr1  r2 Þðx  aÞ rxy ðx; yÞ ¼  2 ½a0 ðtÞ2 þ ½b0 ðtÞ2 2 2 2 2 r1 ðy  bÞ þ ðx  aÞ r2 ðy  bÞ2 þ ðx  aÞ2 2pðr1  r2 Þ ðr1 þ r2 Þ 1 ) r1 ðr1 þ r2 Þðx  aÞ r2 ðr1 þ r2 Þðx  aÞ 2r1 r2 ðx  aÞ 2r1 r2 ðx  aÞ  2  þ þ bx ðtÞdt r1 ðy þ bÞ2 þ ðx  aÞ2 r22 ðy þ bÞ2 þ ðx  aÞ2 ðr2 b þ r1 yÞ2 þ ðx  aÞ2 ðr1 b þ r2 yÞ2 þ ðx  aÞ2 Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi( Ex r1 ðr1  r2 Þðy  bÞ r2 ðr1  r2 Þðy  bÞ  ½a0 ðtÞ2 þ ½b0 ðtÞ2 2 þ r1 ðy  bÞ2 þ ðx  aÞ2 r22 ðy  bÞ2 þ ðx  aÞ2 2pðr1  r2 Þ2 ðr1 þ r2 Þ 1 ) r1 ðr1 þ r2 Þðy þ bÞ r2 ðr1 þ r2 Þðy þ bÞ 2r1 ðr2 b þ r1 yÞ 2r2 ðr1 b þ r2 yÞ þ 2 þ   by ðtÞdt r1 ðy þ bÞ2 þ ðx  aÞ2 r22 ðy þ bÞ2 þ ðx  aÞ2 ðr2 b þ r1 yÞ2 þ ðx  aÞ2 ðr1 b þ r2 yÞ2 þ ðx  aÞ2

ð18Þ

1614

A.R. Fotuhi et al. / International Journal of Solids and Structures 44 (2007) 1608–1627

In Eq. (18), x = a(t), y = b(t), where 1 6 t 6 1, specify the geometry of the crack with respect to the coordinate system (x, y) and prime denotes differentiation with respect to the relevant argument. In order to analyze curved cracks, the moveable orthogonal coordinates (s, n) are chosen such that the origin may move on the crack while s-axis remains tangent to the crack surface, Fig. 1. The stress components (18) and the dislocation densities bx and by are transformed to (s, n) coordinates. Employing the principle of superposition, the components of traction vector at a point with coordinates aj(g), bj(g), where parameter 1 6 g 6 1, on the surface of jth crack for a half-plane weakened by N cracks become rn ðaj ðgÞ; bj ðgÞÞ ¼

N Z X i¼1

rns ðaj ðgÞ; bj ðgÞÞ ¼

1

k 11ij ðg; tÞbsi ðtÞ dt þ

1

N Z X i¼1

N Z X i¼1

1

k 21ij ðg; tÞbsi ðtÞ dt þ

1

1

k 12ij ðg; tÞbni ðtÞ dt 1

N Z X i¼1

ð19Þ

1

k 22ij ðg; tÞbni ðtÞ dt;

j ¼ 1; 2; . . . ; N

1

where the kernels in Eq. (19) are k 11ij ðg; tÞ ¼ K 11ij ðg; tÞ cos wi ðtÞ þ K 12ij ðg; tÞ sin wi ðtÞ k 12ij ðg; tÞ ¼ K 12ij ðg; tÞ cos wi ðtÞ  K 11ij ðg; tÞ sin wi ðtÞ

ð20Þ

k 21ij ðg; tÞ ¼ K 21ij ðg; tÞ cos wi ðtÞ þ K 22ij ðg; tÞ sin wi ðtÞ k 22ij ðg; tÞ ¼ K 22ij ðg; tÞ cos wi ðtÞ  K 21ij ðg; tÞ sin wi ðtÞ

2

k I / k0 k II / k0

1.8

1.6

R 1.4

θ

2l

1.2

k/k

0

y 1

L

x

0.8

0.6

0.4

0.2

0

0

10

20

30

40

50

60

θ (degrees)

Fig. 2. Stress intensity factor for a rotating crack.

70

80

90

A.R. Fotuhi et al. / International Journal of Solids and Structures 44 (2007) 1608–1627

1615

In the above equalities wi ðtÞ ¼ tan1 ðb0i ðtÞ=a0i ðtÞÞ is the angle between s and x axes and the functions K 11ij ðg; tÞ ¼ A11ij ðg; tÞ þ A21ij ðg; tÞ þ ½A21ij ðg; tÞ  A11ij ðg; tÞ cos 2wj ðgÞ  A31ij ðg; tÞ sin 2wj ðgÞ K 12ij ðg; tÞ ¼ A22ij ðg; tÞ þ A12ij ðg; tÞ þ ½A22ij ðg; tÞ  A12ij ðg; tÞ cos 2wj ðgÞ  A32ij ðg; tÞ sin 2wj ðgÞ

ð21Þ

K 21ij ðg; tÞ ¼ ½A21ij ðg; tÞ  A11ij ðg; tÞ sin 2wj ðgÞ þ A31ij ðg; tÞ cos 2wj ðgÞ K 22ij ðg; tÞ ¼ ½A22ij ðg; tÞ  A12ij ðg; tÞ sin 2wj ðgÞ þ A32ij ðg; tÞ cos 2wj ðgÞ

The coefficients in Eq. (21) are given in the Appendix. The kernels in Eq. (21) exhibit Cauchy type singularity for i = j as t ! g and may be represented as K 11jj ðg; tÞ ¼

1 a11;1j X m þ a11;mj ðg  tÞ gt m¼0

K 12jj ðg; tÞ ¼

1 a12;1j X m þ a12;mj ðg  tÞ gt m¼0

ð22Þ

1 a21;1j X þ K 21jj ðg; tÞ ¼ a21;mj ðg  tÞm gt m¼0

K 22jj ðg; tÞ ¼

1 a22;1j X m þ a22;mj ðg  tÞ gt m¼0

15

L, Orthotropic L, Isotropic R, Orthotropic R, Isotropic Ashbaugh

R 2

10

L 0.1

k I/k0

α

5

0

0

10

20

30

40

θ (degrees)

50

60

70

Fig. 3. Mode I stress intensity factor for a crack rotating around tip L.

80

90

1616

A.R. Fotuhi et al. / International Journal of Solids and Structures 44 (2007) 1608–1627

The coefficients of singular terms are obtained via Taylor series expansion of ai(t) and bi(t) in the vicinity of g. These coefficients are given in the Appendix. By virtue of the Bueckner’s superposition theorem the left-hand side of Eq. (19), after changing the sign, is the traction caused by external loading on the uncracked half-plane at the presumed surfaces of cracks. Let the half-plane on y = 0 be under concentrated applied force, Fig. 1, represented by ry ðx; 0Þ ¼ r0 dðx  x0 Þ

ð23Þ

sxy ðx; 0Þ ¼ s0 dðx  x0 Þ

Utilizing complex Fourier transform in the x-direction, Eq. (3) is solved subject to boundary data (23) and the requirement that stress fields vanish at the far-field. The stress components in the half-plane leads to " # r1 r2 r0 y  s0 ðx  x0 Þ r21 r22 rx ¼  pðr1  r2 Þ r21 y 2 þ ðx  x0 Þ2 r22 y 2 þ ðx  x0 Þ2 " # r1 r2 r0 y  s0 ðx  x0 Þ 1 1 ry ¼  2 ð24Þ 2 2 pðr1  r2 Þ r22 y 2 þ ðx  x0 Þ r1 y 2 þ ðx  x0 Þ " ! !# 1 r1 r0 ðx  x0 Þ þ r2 s0 y r2 r0 ðx  x0 Þ þ r1 s0 y rxy ¼ r2  r1 2 2 pðr1  r2 Þ r22 y 2 þ ðx  x0 Þ r21 y 2 þ ðx  x0 Þ

10

L, Orthotropic L, Isotropic R, Orthotropic R, Isotropic Ashbaugh

8

6

4

k II/k0

2

0

-2

R 2

-4

-6

L -8

-10

0.1

α

0

10

20

30

40

50

60

70

θ (degrees)

Fig. 4. Mode II stress intensity factor for a crack rotating around tip L.

80

90

A.R. Fotuhi et al. / International Journal of Solids and Structures 44 (2007) 1608–1627

1617

It is worth mentioning that above solution is the Green’s function for the derivation of stress field in a halfplane under general applied traction. Utilizing Eq. (24), the components of normal and shear traction vector, rnj and rnsj, respectively, on the surface of jth crack should be expressed as ( ½r1 r2 r0 y  s0 ðx  x0 Þ½ð1  r21 Þ þ ð1 þ r21 Þ cosð2wj Þ  2r1 ½r2 r0 ðx  x0 Þ þ r1 s0 y sinð2wj Þ 1 rnj ¼ 2pðr1  r2 Þ r21 y 2 þ ðx  x0 Þ2 ) ½r1 r2 r0 y  s0 ðx  x0 Þ½ð1  r22 Þ þ ð1 þ r22 Þ cosð2wj Þ  2r2 ½r1 r0 ðx  x0 Þ þ r2 s0 y sinð2wj Þ  r22 y 2 þ ðx  x0 Þ2 ( ðr21 þ 1Þ½r1 r2 r0 y  s0 ðx  x0 Þ sinð2wj Þ þ 2r1 ½r2 r0 ðx  x0 Þ þ r1 s0 y cosð2wj Þ 1 rnsj ¼ 2pðr1  r2 Þ r21 y 2 þ ðx  x0 Þ2 ) ðr22 þ 1Þ½r1 r2 r0 y  s0 ðx  x0 Þ sinð2wj Þ þ 2r2 ½r1 r0 ðx  x0 Þ þ r2 s0 y cosð2wj Þ  ; j ¼ 1; 2; . . . ; N r22 y 2 þ ðx  x0 Þ2 ð25Þ Employing the definition of dislocation density function, the equations for the crack opening displacement across the ith crack are Z g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2  uþ ðgÞ  u ðgÞ ¼ ½a0i ðtÞ þ ½b0i ðtÞ ½cosðwi ðgÞ  wi ðtÞÞbsi ðtÞ þ sinðwi ðgÞ  wi ðtÞÞbni ðtÞdt si si 1 Z g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 þ  uni ðgÞ  uni ðgÞ ¼ ½a0i ðtÞ þ ½b0i ðtÞ ½cosðwi ðgÞ  wi ðtÞÞbni ðtÞ  sinðwi ðgÞ  wi ðtÞÞbsi ðtÞdt; i ¼ 1; 2;... ;N 1

ð26Þ

0.8

L, Orthotropic L, Isotropic R, Orthotropic R, Isotropic

R

0.7

θ

2l 0.6

y 0.5

2l

L

τ0

0.4

x

k I/k0

σ0 0.3

0.2

0.1

0

-0.1

0

20

40

60

80

100

120

140

θ (degrees)

Fig. 5. Mode I stress intensity factor for a crack rotating around the center.

160

180

1618

A.R. Fotuhi et al. / International Journal of Solids and Structures 44 (2007) 1608–1627

For embedded cracks, the displacement field is single-valued out of crack surfaces. Thus, the dislocation densities are subjected to the following closure requirements: Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½a0i ðtÞ2 þ ½b0i ðtÞ2 ½cosðwi ð1Þ  wi ðtÞÞbsi ðtÞ þ sinðwi ð1Þ  wi ðtÞÞbni ðtÞ dt ¼ 0 ð27Þ Z11 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ½a0i ðtÞ þ ½b0i ðtÞ ½cosðwi ð1Þ  wi ðtÞÞbni ðtÞ  sinðwi ð1Þ  wi ðtÞÞbsi ðtÞ dt ¼ 0; i ¼ 1; 2; . . . ; N 1

To evaluate the dislocation density, the Cauchy singular integral equations (19) and (27) ought to be solved simultaneously. This is accomplished by means of the Gauss–Chebyshev quadrature scheme developed by Erdogan et al. (1973). As was mentioned in Liebowitz (1968), for the p embedded cracks in an orthotropic medffiffi ium, the stress fields in the neighborhood of crack tips behave like 1= r where r is the distance from the crack tip Fig. 1. Therefore, the dislocation densities are taken in the following forms: gsi ðtÞ bsi ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 1  t2 ð28Þ gni ðtÞ bni ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ; 1 < t < 1; i ¼ 1; 2; . . . ; N 1  t2 Substituting Eq. (28) into Eqs. (19) and (27) and discretizing the domain, 1 < t < 1 by m + 1 segments, we arrive at the following system of 2N · m algebraic equations 3 2 3 32 2 g1 ðtp Þ q1 ðgr Þ A11 A12    A1N 7 6 7 76 6A 6 21 A22    A2N 76 g2 ðtp Þ 7 6 q2 ðgr Þ 7 7 7 6 6 6 . ð29Þ .. 76 .. 7 ¼ 6 .. 7 .. .. 7 6 . 4 . . 54 . 5 4 . 5 . . AN 1

AN 2



gN ðtp Þ

ANN

qN ðgr Þ

2 L, Orthotropic L, Isotropic R, Orthotropic R, Isotropic

1.5

1

k II/k0

0.5

0

R -0.5

θ

2l

-1

y

2l

L -1.5

τ0 -2

x

σ0 0

20

40

60

80

100

120

140

θ (degrees)

Fig. 6. Mode II stress intensity factor for a crack rotating around the center.

160

180

A.R. Fotuhi et al. / International Journal of Solids and Structures 44 (2007) 1608–1627

1619

where the collocation points are chosen as pr gr ¼ cos ; r ¼ 1; . . . ; m  1 m  pð2p  1Þ ; p ¼ 1; . . . ; m tp ¼ cos 2m

ð30Þ

The components of matrix and vectors in Eq. (29) are 2 k 11ij ðg1 ;t1 Þ k 11ij ðg1 ; t2 Þ   k 11ij ðg1 ; tm Þ k 12ij ðg1 ;t1 Þ k 12ij ðg1 ;t2 Þ 6 k ðg ;t Þ k ðg ; t Þ   k ðg ; t Þ k ðg ;t Þ k 12ij ðg2 ;t2 Þ 6 11ij 2 1 11ij 2 2 11ij 2 m 12ij 2 1 6 6 .. .. .. .. .. .. 6 . . . . . . 6 6 6 k 11ij ðgm1 ; t1 Þ k 11ij ðgm1 ;t2 Þ   k 11ij ðgm1 ;tm Þ k 12ij ðgm1 ;t1 Þ k 12ij ðgm1 ; t2 Þ 6 dij Di ðt2 Þ . .. dij Di ðtm Þ 0 0 p6 6 dij Di ðt1 Þ Aij ¼ 6 m 6 k 21ij ðg1 ;t1 Þ k 21ij ðg1 ; t2 Þ   k 21ij ðg1 ; tm Þ k 22ij ðg1 ;t1 Þ k 22ij ðg1 ;t2 Þ 6 6 k ðg ;t Þ k 21ij ðg2 ; t2 Þ   k 21ij ðg2 ; tm Þ k 22ij ðg2 ;t1 Þ k 22ij ðg2 ;t2 Þ 6 21ij 2 1 6 .. .. .. .. 6 .. .. 6 . . . . . . 6 6 4 k 21ij ðgm1 ; t1 Þ k 21ij ðgm1 ;t2 Þ   k 21ij ðgm1 ;tm Þ k 22ij ðgm1 ;t1 Þ k 22ij ðgm1 ; t2 Þ 0

0

. ..

dij Di ðt1 Þ

0

gj ðtp Þ ¼ ½gsj ðt1 Þ gsj ðt2 Þ    gsj ðtm Þ gnj ðt1 Þ gnj ðt2 Þ  gnj ðtm Þ

dij Di ðt2 Þ

    .. .   .. .     .. .   .. .

k 12ij ðg1 ;tm Þ

3

7 7 7 7 7 7 7 k 12ij ðgm1 ; tm Þ 7 7 7 0 7 7 k 22ij ðg1 ;tm Þ 7 7 k 22ij ðg2 ;tm Þ 7 7 7 .. 7 7 . 7 7 k 22ij ðgm1 ; tm Þ 5 dij Di ðtm Þ k 12ij ðg2 ;tm Þ .. .

T

qj ðgr Þ ¼ ½rnj ðg1 Þrnj ðg2 Þ.. .rnj ðgm1 Þ0rsj ðg1 Þrsj ðg2 Þ. ..rsj ðgm1 Þ0

T

ð31Þ

L1 L2 R1 R2

y 0.6a

0.6a R2

L2

0.6 2l

L1

a

R1 2l 0.75a

0.5

τ0

x

kI/k0

σ0

0.4

0.3

0.2

0.1

0

0.2

0.4

0.6

0.8

1

l/a

Fig. 7. Mode I stress intensity factor for two parallel cracks in orthotropic half-plane.

1.2

1620

A.R. Fotuhi et al. / International Journal of Solids and Structures 44 (2007) 1608–1627

L1 L2 R1 R2

0.3

0.2

y

0.1

kII/k0

0.6a

0.6a

R2

L2 0

2l

L1

a

R1 2l 0.75a

-0.1

τ0

x

σ0 -0.2

0

0.2

0.4

0.6

0.8

1

1.2

l/a

Fig. 8. Mode II stress intensity factor for two parallel cracks in orthotropic half-plane.

L1 L2 R1 R2

y 0.6a R2

L2

1

0.6a

2l

L1

a

R1 2l 0.75a

0.8

τ0

x

kI/k0

σ0 0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

l/a

Fig. 9. Mode I stress intensity factor for two parallel cracks in isotropic half-plane.

1.2

A.R. Fotuhi et al. / International Journal of Solids and Structures 44 (2007) 1608–1627

1621

L1 L2 R1 R2

0.5

0.4

0.3

0.2

kII/k0

0.1

y 0.6a

0.6a

0

R2

L2 -0.1

2l

L1

a

-0.2

R1 2l 0.75a

-0.3

τ0 σ0

-0.4

-0.5

x

0

0.2

0.4

0.6

0.8

1

1.2

l/a

Fig. 10. Mode II stress intensity factor for two parallel cracks in isotropic half-plane.

0.5

L1 L2 R1 R2

0.45

y φ

R1

0.4

φ

L2 b

0.35

c

L1

R2

b

a

kI/k0

0.3

x

τ0

0.25

σ0

0.2

0.15

0.1

0.05

0

10

20

30

40

θ (degrees)

50

60

70

Fig. 11. Mode I stress intensity factor for two curved cracks in orthotropic half-plane.

80

90

1622

A.R. Fotuhi et al. / International Journal of Solids and Structures 44 (2007) 1608–1627

where dij in matrix Aij is the Kronecker delta, superscript T stands for the transpose of vectors, klmij(g, t) are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi defined in Eq. (20), and Di ðtÞ ¼ ½a0i ðtÞ2 þ ½b0i ðtÞ2 . The mode I and II stress intensity factors derived by Faal and Fariborz (in press) are utilized. These are 



 2 2 1 Ex ð½a0i ð1Þ þ ½b0i ð1Þ Þ4 gni ð1Þ=r1 r2 2ðr1 þ r2 Þ gsi ð1Þ   1   2 2 0 k IR Ex ð½a0i ð1Þ þ ½bi ð1Þ Þ4 gni ð1Þ=r1 r2 ¼ 2ðr1 þ r2 Þ k IIR gsi ð1Þ k IL k IIL

¼

ð32Þ

The solution of Eq. (29) should be plugged into Eq. (32) to obtain stress intensity factors. 3. Results The analysis developed in the preceding section, allows the consideration of an orthotropic half-plane with multiple curved cracks subjected to normal and shear tractions. In what follows, the ratios of moduli of elasticity of the orthotropic material are taken as Ey/Ex = 0.04, Gxy/Ex = 0.02 and the Poisson’s ratio mxy = 0.25 which are representative of those for high-modulus graphite/epoxy composite. pffiffi To render the results dimensionless, unless otherwise stated, stress intensity factors are divided by k 0 ¼ r0 l, where l is the half length of a straight crack. To verify the validity of formulation, the problem of an infinite plane under constant far-field applied traction weakened by a straight crack rotating around its center is examined. A rotating crack is situated at the far distance from the boundary of half-plane. A uniform normal traction r0 is applied on the crack surface. The stress intensity factors at crack tips are identical. Fig. 2, shows modes I and II stress intensity factors.

0.8 L1 L2 R1 R2

y R1

φ

φ

L2

0.6

b c

L1

R2

b

a

0.4

x

τ0 σ0

kII/k0

0.2

0

-0.2

-0.4

-0.6

10

20

30

40

φ (degrees)

50

60

70

Fig. 12. Mode II stress intensity factor for two curved cracks in orthotropic half-plane.

80

90

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1623

The results are in excellent agreement with the analytical solutions obtained in Faal and Fariborz (in press). As another verification of results, the problem of an embedded crack in the isotropic and orthotropic half-planes where the crack is under constant normal traction r0 and rotates around point L, inset in Fig. 3, is solved. The values of kI/k0 and kII/k0, Figs. 3 and 4, for isotropic half-plane closely match those obtained for the same problem by Ashbaugh (1975). The capabilities of procedure are demonstrated by solving several examples. In all examples, the half-plane is under normal and shear point forces represented by Eq. (23). The opening of cracks is ensured by taking r0 = 2s0. 3.1. A rotating crack A crack with length 2l = 1 cm is rotating around its center. The plots of stress intensity factors versus crack orientation in isotropic and orthotropic half-planes are shown in Figs. 5 and 6. At h = p/2, the traction on the crack surface, Eq. (25), vanishes. Therefore, the stress intensity factors are zero. The comparison of stress intensity factors of isotropic and orthotropic half-planes reveals that material orthotropy enhances mode II, but attenuates mode I stress intensity factors. 3.2. Two parallel cracks We consider two equal-length cracks parallel to the boundary of half-plane, inset in Fig. 7. The centers of cracks remain fixed while the crack lengths are changing with the same rate. The variation of dimensionless stress intensity factors kI/k0 and kII/k0, where a = 1 cm, are depicted in Figs. 7 and 8. As it was expected the highest kI/k0 occurs where the distance between the interacting crack tips L1 and R2 is minimal, i.e., l/ a = 0.6. For larger values of l/a, crack tip L1 passes crack tip R2 reducing kI/k0 at L1 and R2 while enhancing

0.45 L1 L2 R1 R2

0.4

2l L2 0.35

R1

L1

b

y a

0.3

a

R2

b/2 0.25

x

kI/k0

τ0 σ0

0.2

0.15

0.1

0.05

0 0.5

1

1.5

2

2.5

2l/b

Fig. 13. Mode I stress intensity factor for a curved and a straight cracks in orthotropic half-plane.

3

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it at R1 and L2. Moreover, in comparison with crack tips L1 and R2, a slower variation of stress intensity factors for L2 and R1 is observed which may be attributed to a weaker interaction at these tips. The same problem is analyzed for isotropic half-planes, Figs. 9 and 10. These plots show a similar trend as of those for orthotropic material but with larger range of variation of stress intensity factors. In the orthotropic half-plane the stiffer material property in the crack direction attenuates mode I stress intensity factors. 3.3. Two curved cracks We consider two identical curved cracks which are portions of the circumference of an ellipse, Figs. 11 and 12. The lengths of major and minor semi-axes of ellipse are a = 3 cm and b = 1 cm, respectively. The cracks may be represented in the following parametric forms:  

1 i 1 a ai ðtÞ ¼ ð1Þ a cos ð1  ð1Þ tÞ tan cot u 2 b  

1 i 1 a bi ðtÞ ¼ b þ b sin ð1  ð1Þ tÞ tan cot u  1 6 t 6 1; i ¼ 1; 2 2 b i



ð33Þ

The stress intensity factors can be calculated by plugging Eq. (33) into Eq. (32). In Figs. 11 and 12 the dimensionless stress intensity factors for various cracks lengths for orthotropic half-plane are plotted. For this case, pffiffiffi we take k 0 ¼ r0 a. Comparing stress intensity factors at different crack tips shows that the mode I fracture is dominant at R1 and L2, whereas at L1 and R2, mode II dominants. In isotropic half-plane, under the abovementioned applied load, crack closing occurs for u > 37. Thus solution is not valid for larger values of angle u and plots of stress intensity factors are not included.

1

L1 L2 R1 R2

2l L2 R1

L1

b

y 0.8

a

a

R2

b/2 τ0

0.6

x

kII/k0

σ0

0.4

0.2

0

0.5

1

1.5

2

2.5

2l/b

Fig. 14. Mode II stress intensity factor for a curved and a straight cracks in orthotropic half-plane.

3

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1625

3.4. A curved and a straight crack As the last example, we consider a stationary curved crack and a growing straight crack with fixed center, inset in Fig. 13. The parametric representations of straight and curved cracks are, respectively, a1 ðtÞ ¼ a þ lt 3b b1 ðtÞ ¼ 2   p a2 ðtÞ ¼ a cos ð1  tÞ 4 p  b b2 ðtÞ ¼ þ b sin ð1  tÞ 2 4

ð34Þ 16t 61

where a = 3 cm and b = 2 cm. Figs. 13 and 14 show the variation of dimensionless stress intensity factors versus the length of the straight crack in orthotropic material. As it may be observed kI/k0 for the two approaching crack tips changes rapidly. In contrast the variations of kI/k0for crack tips L1 and R2 are not that pronounced. Another analysis was carried out for an isotropic half-plane with identical cracks. The comparison of results revealed that kII/k0 at R2 in orthotropic half-plane was an order of magnitude higher than that for isotropic material. On the contrary, for the other cracks tips the effect of orthotropy was not that significant. 4. Conclusion The solution of edge dislocation in an orthotropic half-plane is obtained and the verification of solution is carried out by showing the satisfaction of boundary conditions. The closed-form elasticity solution of orthotropic half-planes under normal and shear concentrated applied tractions is achieved via Fourier transform technique. The dislocation solution is utilized as the Green’s function to derive Cauchy singular integral equations for the stress analyzes of an orthotropic half-plane weakened by several crack patterns. The interaction between two cracks in various examples is studied. Appendix The functions appeared in Eq. (16) are M x ¼ 128h7  512h6 y þ 64h5 ð13y 2 þ x2 Þ  16h4 ð45y 2 þ 11x2 Þ þ 8h3 ð45y 4 þ 26x2 y 2 þ 5x4 Þ  8h2 ð13y 4 þ 16x2 y 2 þ 7x4 Þy þ 8hðx6 þ 2y 6 þ 5x2 y 4 þ 4x4 y 2 Þ  yð3x6 þ y 6 þ 7x4 y 2 þ 5x2 y 4 Þ N x ¼ 128h5 xy þ 16h4 xð5x2  17y 2 Þ þ 32h3 ð7y 2 þ 3x2 Þ þ 8h2 xð11y 4 þ 4x2 y 2 þ 3x4 Þ þ 16hxyðy 4  x4 Þ  xy 2 ðx4 þ y 4 þ x2 y 2 Þ M y ¼ 128h7  384h6 y þ 64h5 ð7y 2  x2 Þ þ 16h4 yð15y 2 þ 11x2 Þ  8h3 ð5y 4 þ 22x2 y 2 þ 3x4 Þ þ 16h2 ðy 4 þ 5x2 y 2 þ 2x4 Þy  8hy 2 ðx2 þ y 2 Þ2  yðx6  y 6 þ x4 y 2  x2 y 4 Þ N y ¼ 384h6 x þ 1152h5 xy  48h4 xð3x2 þ 87y 2 Þ þ 288h3 xyð3y 2 þ x2 Þ  16h2 xð18y 4 þ 13x2 y 2 þ x4 Þ þ 2hxyð24y 4 þ 8x4 þ 3x2 y 2 Þ  x3 ðx4 þ 7y 4 þ 5x2 y 2 Þ M xy ¼ 128xh6  384xyh5 þ 16h4 xð27y 2 þ x2 Þ  32h3 xyð7y 2 þ x2 Þ þ 16h2 xy 2 ð3y 2 þ x2 Þ þ xðx6  y 6 þ x4 y 2  x2 y 4 Þ N xy ¼ 128h7  512h6 y þ 64h5 ð13y 2  x2 Þ þ 8h3 ð45y 4  14x2 y 2  3x4 Þ þ 8h2 ð13y 4 þ 4x2 y 2 þ 5x4 Þy þ 16hy 2 ðy 4  x4 Þ þ yðx6  y 6 þ x4 y 2  x2 y 4 Þ

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The coefficients of Eq. (21) are as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi( r3 ðr1  r2 Þðbj  bi Þ r3 ðr1  r2 Þðbj  bi Þ 2 2 A11ij ðg;tÞ ¼ þ 2 2 ½a0i ðtÞ þ ½b0i ðtÞ  2 1 2 2 2 4pðr1  r2 Þ ðr1 þ r2 Þ r1 ðbj  bi Þ þ ðaj  ai Þ r2 ðbj  bi Þ2 þ ðaj  ai Þ2 Ex

þ

r31 ðr1 þ r2 Þðbj þ bi Þ

r32 ðr1 þ r2 Þðbj þ bi Þ

2r21 r2 ðr2 bi þ r1 bj Þ

2r1 r22 ðr1 bi þ r2 bj Þ

r1 ðr1 þ r2 Þðaj  ai Þ

r2 ðr1 þ r2 Þðaj  ai Þ

2r21 ðaj  ai Þ

2r22 ðaj  ai Þ

r1 ðr1 þ r2 Þðbj þ bi Þ

r2 ðr1 þ r2 Þðbj þ bi Þ

2r2 ðr2 bi þ r1 bj Þ

2r1 ðr1 bi þ r2 bj Þ

r2 ðr1 þ r2 Þðaj  ai Þ

r1 ðr1 þ r2 Þðaj  ai Þ

2r1 r2 ðaj  ai Þ

2r1 r2 ðaj  ai Þ

r1 ðr1 þ r2 Þðaj  ai Þ

r2 ðr1 þ r2 Þðaj  ai Þ

2r1 r2 ðaj  ai Þ

2r1 r2 ðaj  ai Þ

r1 ðr1 þ r2 Þðbj þ bi Þ

r2 ðr1 þ r2 Þðbj þ bi Þ

2r1 ðr2 bi þ r1 bj Þ

2r2 ðr1 bi þ r2 bj Þ

þ   r21 ðbj þ bi Þ2 þ ðaj  ai Þ2 r22 ðbj þ bi Þ2 þ ðaj  ai Þ2 ðr2 bi þ r1 bj Þ2 þ ðaj  ai Þ2 ðr1 bi þ r2 bj Þ2 þ ðaj  ai Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi( Ex r1 ðr1  r2 Þðaj  ai Þ r2 ðr1  r2 Þðaj  ai Þ A12ij ðg;tÞ ¼  2 ½a0i ðtÞ2 þ ½b0i ðtÞ2 2 2 2 2 r1 ðbj  bi Þ þ ðaj  ai Þ r2 ðbj  bi Þ2 þ ðaj  ai Þ2 4pðr1  r2 Þ ðr1 þ r2 Þ þ

þ   2 2 2 2 2 2 2 2 r21 ðbj þ bi Þ þ ðaj  ai Þ r22 ðbj þ bi Þ þ ðaj  ai Þ ðr2 bi þ r1 bj Þ þ ðaj  ai Þ ðr1 bi þ r2 bj Þ þ ðaj  ai Þ ( qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r1 ðr1  r2 Þðbj  bi Þ r2 ðr1  r2 Þðbj  bi Þ Ex A21ij ðg;tÞ ¼  ½a0i ðtÞ2 þ ½b0i ðtÞ2 2 2 2 2 2 2 r1 ðbj  bi Þ þ ðaj  ai Þ r22 ðbj  bi Þ þ ðaj  ai Þ 4pðr1  r2 Þ ðr1 þ r2 Þ 

 þ þ 2 2 2 2 2 2 2 2 r21 ðbj þ bi Þ þ ðaj  ai Þ r22 ðbj þ bi Þ þ ðaj  ai Þ ðr2 bi þ r1 bj Þ þ ðaj  ai Þ ðr1 bi þ r2 bj Þ þ ðaj  ai Þ ( qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ex r2 ðr1  r2 Þðaj  ai Þ r1 ðr1  r2 Þðaj  ai Þ 2 2 A22ij ðg;tÞ ¼ þ 2 ½a0i ðtÞ þ ½b0i ðtÞ  2 2 2 2 r1 ðbj  bi Þ þ ðaj  ai Þ r2 ðbj  bi Þ2 þ ðaj  ai Þ2 4pr1 r2 ðr1  r2 Þ ðr1 þ r2 Þ 

 þ þ r21 ðbj þ bi Þ2 þ ðaj  ai Þ2 r22 ðbj þ bi Þ2 þ ðaj  ai Þ2 ðr2 bi þ r1 bj Þ2 þ ðaj  ai Þ2 ðr1 bi þ r2 bj Þ2 þ ðaj  ai Þ2 ( qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ex r1 ðr1  r2 Þðaj  ai Þ r2 ðr1  r2 Þðaj  ai Þ A31ij ðg;tÞ ¼  ½a0i ðtÞ2 þ ½b0i ðtÞ2 2 r1 ðbj  bi Þ2 þ ðaj  ai Þ2 r22 ðbj  bi Þ2 þ ðaj  ai Þ2 2pðr1  r2 Þ2 ðr1 þ r2 Þ 

 þ þ 2 2 2 2 2 2 2 2 r21 ðbj þ bi Þ þ ðaj  ai Þ r22 ðbj þ bi Þ þ ðaj  ai Þ ðr2 bi þ r1 bj Þ þ ðaj  ai Þ ðr1 bi þ r2 bj Þ þ ðaj  ai Þ ( qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r1 ðr1  r2 Þðbj  bi Þ r2 ðr1  r2 Þðbj  bi Þ Ex 2 2 A32ij ðg;tÞ ¼  2 ½a0i ðtÞ þ ½b0i ðtÞ 2 2 2 2 2 2 r1 ðbj  bi Þ þ ðaj  ai Þ r2 ðbj  bi Þ þ ðaj  ai Þ 2pðr1  r2 Þ ðr1 þ r2 Þ þ

r21 ðbj þ bi Þ2 þ ðaj  ai Þ2

þ

  r22 ðbj þ bi Þ2 þ ðaj  ai Þ2 ðr2 bi þ r1 bj Þ2 þ ðaj  ai Þ2 ðr1 bi þ r2 bj Þ2 þ ðaj  ai Þ2

The coefficients of singular terms in Eq. (22) are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 a11;1j ¼ Ex ða0j Þ þ ðb0j Þ fb0j ½ðr21 þ r22 þ r1 r2  1Þða0j Þ þ r1 r2 ðr1 r2 þ 1Þðb0j Þ  2

2

þ b0j ½ðr21 þ r22 þ r1 r2 þ 1Þða0j Þ þ r1 r2 ðr1 r2  1Þðb0j Þ  cos 2wj þ 2a0j ðr1 r2 ðb0j Þ 2

a12;1j

2

2

2

2

2

 ða0j Þ Þ sin 2wj g=4pðr1 þ r2 Þ½ðr1 ðb0j ÞÞ þ ða0j Þ ½ðr2 ðb0j ÞÞ þ ða0j Þ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 ¼ Ex ða0j Þ þ ðb0j Þ fa0j ½ðr21 þ r22 þ r1 r2 ð1  r1 r2 ÞÞðb0j Þ þ ð1 þ r1 r2 Þða0j Þ  2

2

þ a0j ½ðr21 þ r22 þ r1 r2 ð1 þ r1 r2 ÞÞðb0j Þ þ ð1  r1 r2 Þða0j Þ  cos 2wj þ 2r1 r2 ðr1 r2 ðb0j Þ 2

2

2

2

2

2

a21;1j

 ða0j Þ Þb0j sinð2wj Þg=4pr1 r2 ðr1 þ r2 Þ½ðr1 ðb0j ÞÞ þ ða0j Þ ½ðr2 ðb0j ÞÞ þ ða0j Þ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 ¼ Ex ða0j Þ þ ðb0j Þ f2a0j ðr1 r2 ðb0j Þ  ða0j Þ Þ cos 2wj þ b0j ½ðr21 þ r22 þ r1 r2 þ 1Þða0j Þ

a22;1j

þ r1 r2 ðr1 r2  1Þðb0j Þ  sin 2wj g=4pðr1 þ r2 Þ½ðr1 ðb0j ÞÞ þ ða0j Þ ½ðr2 ðb0j ÞÞ þ ða0j Þ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ex ða0j Þ2 þ ðb0j Þ2 f2r1 r2 b0j ðr1 r2 ðb0j Þ2  ða0j Þ2 Þ cos 2wj þ a0j ½ðr21 þ r22 þ r1 r2 ð1 þ r1 r2 ÞÞðb0j Þ2

2

2

2

2

2

þ ða0j Þ2 ð1  r1 r2 Þ sin 2wj g=4pr1 r2 ðr1 þ r2 Þ½ðr1 ðb0j ÞÞ2 þ ða0j Þ2 ½ðr2 ðb0j ÞÞ2 þ ða0j Þ2  where wj ¼ tan1 ðb0j ðgÞ=a0j ðgÞÞ.

)

)

)

)

)

)

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References Ashbaugh, N., 1975. Stress solution for a crack at an arbitrary angle to an interface. International Journal of Fracture 11 (2), 205–219. Erdogan, F., Gupta, G.D., Cook, T.S., 1973. Numerical solution of integral equations. In: Sih, G.C. (Ed.), Methods of Analysis and Solution of Crack Problems. Noordhoof, Leyden, Holland. Eshelby, J.D., Read, W.T., Shockley, W., 1953. Anisotropic elasticity with applications to dislocation theory. Acta Metallurgica 1, 251– 259. Faal, R.T., Fariborz, S.J., in press. In-plane stress analysis of an orthotropic plane with multiple smooth cracks. Applied Mathematical Modelling. Hills, D.A., Kelly, P.A., Dai, D.N., Korsunsky, A.M., 1996. Solution of Crack Problems: The Distributed Dislocation Technique. Kluwer Academic Publishers, Netherlands. Huang, H., Kardomateas, G.A., 2004. Dislocation-based boundary-element method for crack problems in anisotropic half-planes. AIAA Journal 42 (3), 650–657. Lekhnitskii, S.G., 1963. Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day, San Francisco. Liebowitz, H., 1968. Fracture Mechanics. Academic Press, New York. Pan, E., Chen, C.-S., Amadei, B., 1997. A BEM formulation for anisotropic half-plane problems. Engineering Analysis with Boundary Elements 20, 185–195. Pande, C.S., Chou, Y.T., 1971. Edge dislocation in semi-infinite anisotropic media. Physica Status Solidi (a) 6, 499–503. Sung, J.C., Liou, J.Y., 1995a. Analysis of a crack embedded in a linear elastic half-plane solid. Journal of Applied Mechanics, Transactions of the ASME 62 (1), 78–85. Sung, J.C., Liou, J.Y., 1995b. An internal crack in a half-plane solid with clamped boundary. Computer Methods in Applied Mechanics and Engineering 121, 361–372. Suo, Z., 1990. Singularities, interface and cracks in dissimilar anisotropic media. Proceedings of the Royal Society of London A 427, 331–358. Ting, T.C.T., 1992. Image singularities of Green’s functions for anisotropic elastic half-spaces and bimaterials. The Quarterly Journal of Mechanics and Applied Mathematics 22, 119–139. Wen, P., 1989. The solution of a displacement discontinuity for an anisotropic half-plane and its applications to fracture mechanics. Engineering Fracture Mechanics 34, 1145–1154.