Saint-Venant torsion of orthotropic bars with rectangular cross section weakened by cracks

International Journal of Solids and Structures 52 (2015) 165–179

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Saint-Venant torsion of orthotropic bars with rectangular cross section weakened by cracks A.R. Hassani, R.T. Faal ⇑ Faculty of Engineering, University of Zanjan, P.O. Box 45195-313, Zanjan, Iran

a r t i c l e

i n f o

Article history: Received 29 April 2014 Received in revised form 8 September 2014 Available online 15 October 2014 Keywords: Saint-Venant torsion Orthotropic bar Rectangular cross section Stress intensity factor Dislocation density Torsional rigidity

a b s t r a c t The solution to problem of a Volterra-type screw dislocation in an orthotropic bar with rectangular cross section is first obtained by means of a finite Fourier cosine transform. The bar is under axial torque which is governed by the Saint-Venant torsion theory. The series solution is then derived for displacement and stress fields in the bar cross section. The dislocation solution is employed to derive a set of Cauchy singular integral equations for the analysis of curved cracks. The solution to these equations is used to determine the torsional rigidity of bar and the stress intensity factors (SIFs) for the tips of the cracks. Several examples of a single straight crack and an arc-crack are solved. Furthermore, the interaction between two cracks is studied. Finally, the stress components around an inclined edge crack tip are used to define the boundary of the plastic region employing von Mises yield criterion. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Shafts or bars in torsion are often subjected to cracking during their service life. Though the torsion problem of bars is a rather old one in the theory of elasticity, the torsion problem of bars with multiple curved cracks has not been adequately developed. The problems of an elastic bar in torsion have been considered by a number of investigators. These investigations may be grouped into two major categories: those primarily dealing with bars with no defects, and those studying bars with single or multiple defects. Within the first category, there are numerous investigations in the literature (Ecsedi, 2009; Rongqiao et al., 2010).The bars contained with multiple defects have been the subject of other earlier investigations. First, bars with rectangular cross section are reviewed. Torsion problem of a rectangular bar containing one or two edge cracks perpendicular to the cross section sides was developed by Chen (1980). The succeeding Dirichlet problem of the Laplace equation was solved by subdividing the cross section into several rectangular regions and using the Duhem theorem (a theorem of continuation of harmonic function). The torsional rigidity and also the compliance coefficient were obtained and lastly, the stress intensity factors of crack tips were calculated. The applied method was restricted only to the edge cracks, which were normal to the sides of cross section. ⇑ Corresponding author. Tel.: +98 241 515 2600; fax: +98 241 515 2762. E-mail address: [email protected] (R.T. Faal). http://dx.doi.org/10.1016/j.ijsolstr.2014.10.002 0020-7683/Ó 2014 Elsevier Ltd. All rights reserved.

Chen et al. (1997) gave a solution of the torsion problem for an orthotropic rectangular bar containing an edge crack, bisecting and normal to one side of the cross section. The solution of governing equation was accomplished by mapping a rectangular plane with cut to another one. Indeed, by applying this mapping, the problem was reduced to the Laplace equation. After solving this Laplace equation, the compliance coefficient as an inverse of torsion rigidity coefficient was evaluated. Considering the fact that energy release rate can be obtained in terms of applied torque and the compliance coefficient and also as a function of stress intensity factor of crack tip, a formula for the stress intensity factor was presented. The flexural and torsional problem of bars with circular, elliptical and semi-elliptical cross sections weakened by some edge and embedded cracks was subject of study by Sih (1963). The problem was formulated based on three complex flexure functions including the classical torsion function and the non-trivial stresses were calculated in terms of the aforementioned complex functions. By choosing the appropriate complex flexure function, which satisfies the necessary boundary conditions of outer boundary of the bar and also those corresponding to crack, the singular stress field near the crack tips was found. Then, the closed form relations for stress intensity factors of crack tips were given. The analytical solutions for the torsion rigidities of a circular bar weakened by different length radial cracks were achieved by Lebedev et al. (1979). Also, the problem of the twisting of a rod of elliptical cross section with two edge cracks extending to its foci was analyzed and the torsional rigidity was calculated.

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Chen and Chen (1983) conducted the solution of a bar with a cracked ring section under torsion. The cross section weakened by three edge cracks emanating from outer boundary of section and all the cracks were equally spaced. By partitioning the plan of cross section and applying conformal mapping, the problem was analyzed in the rectangular region and by employing the harmonic function continuation technique the final solution was derived. Finally, torsional rigidity and stress intensity factors of tips of cracks were achieved. Duality between the problem of the circular plane with multiple cracks in anti-plane elasticity and problem of multiple cracks of the round bar under torsion was proposed by Chen (1984). Namely, one should interprets the warping function of bar as the displacement function in anti-plane elasticity. The method of complex functions was used for analyzing the problem. Using the Hankel transform and Fourier series, the torsion problem of a finite cylinder containing a concentric penny-shaped crack was solved by Chang (1985). The governing equation of the problem was in terms of the only non-zero displacement component, namely the circumferential component of displacement. The problem was reduced to a Fredholm integral equation of the second kind which was solved by an iterative scheme and the Mode III stress intensity factor of the crack tip was evaluated. Torsion of a finite orthotropic cylinder with a concentric pennyshaped crack was treated by Zhang (1988). Making use of the Hankel transform and Fourier series, the problem was reduced to a Fredholm integral equation of the second kind. The numerical solution of this equation led to stress intensity factor of the crack tip. Analytical expression of the torsion function for a cylinder containing a screw dislocation was first given in the work accomplished by Xiao-chun and Ren-ji (1988). This solution may also be found by the method of images. Using the corresponding dislocation solution and via dislocation distribution technique, the problem was reduced to solve a system of singular integral equations for the unknown dislocation density functions. Numerical solution of the ensuing singular integral equations was led to the torsional rigidity of cylinder and also stress intensity factors of crack tips for a cylinder with several cracks was obtained. Renji and Yulan (1992) presented a set of solutions for torsional problems of a cylinder with a rectangular hole and a rectangular cylinder containing a crack. The torsion problem of circular cylinder containing several cracks solved by dislocation distribution technique was split into two above-mentioned problems. Then, torsional rigidities and the stress intensity factor at the crack tip were determined. Also for the circular cylinder with a rectangular hole the expressions for the singular stresses around the concave corner points were derived and the stress concentration factors were achieved. Yi-Zhou and Yi-Heng (1983) focused on the study of a bar with sector shape cross section under torsion, weakened by radial and circumferential edge cracks. Using the conformal mapping the cross section was transformed to a rectangular cross section with edge cracks normal to the sides of cross section. Subsequently, analogous to reference Chen (1999), a general solution of the Dirichlet problem for the Laplace equation in the bar with rectangular cross section was obtained by using the harmonic function continuation technique as well as the compliance method. Finally, stress intensity factors of crack tips were evaluated. Fang-ming and Ren-ji (1993) analyzed Saint-Venant torsion of a circular cylinder with internal crack that reinforced by an eccentric rod with different material of cylinder. Using Muskhelishvili single-layer potential function solution and single crack solution for the torsion problem of a cylinder, the problem was reduced to a set of mixed-type integral equation with generalized Cauchykernel. These integral equations were a mixture of Fredholm

integrals of single-layer potential density functions and also Cauchy-type singular integrals of unknown dislocation density function. By numerical solution of these integral equations, torsional rigidity and stress intensity factors were obtained. There is a subgroup of torsion problems dealing with bars weakened by circumferential and interfacial defects which are reviewed in the following. Problem of torsion and extension of an infinite solid cylinder with an edge annular crack as a slit on the outer surface of cylinder was treated by Kudriavtsev and Parton (1973). The stress field was determined in the vicinity of slit which its surfaces were normal to the axis of cylinder and the crack was symmetric with respect to the cylinder axis. In the case of pure torsion and extension along the cylinder axis, they found stress intensity factors of the crack tips via the stress field. Torsion of a long bar with circular cross-section and uniform surface coating layer subjected to external torque at two ends was investigated by Wu et al. (2008). In their study a periodic of circumferential cracks in the coating layer governed by pure shearing was considered. To determine the progressive cracking density in the coating layers, the free-edge stresses near crack tip and the shear stresses in the cross-section of bar were obtained based on the variational principle of complementary strain energy. Assuming strain energy conservation, a criterion for progressive cracking in the coating layer was established and the crack density was estimated. Problem of torsion of a cylindrical interface crack in a bi-layered tubular composite of finite thickness was investigated by Li et al. (2013). Two elastic cylinders were perfectly bonded to each other on the cylindrical surfaces, except for an interface cylindrical crack. The problem was assumed to be axisymmetric and by applying cosine integral transform to the governing equation, the problem was led to a Cauchy singular integral equation which was solved numerically to obtain stress intensity factor of crack tips. The simultaneous effect of geometrical and physical parameters on the interfacial fracture behavior was discussed. Lazzarin et al. (2007), were derived the closed form stress fields of a semi-elliptic circumferential notch and peripheral crack in an axisymmetric shaft under torsional loading. The problem was formulated by using complex potential functions and choosing a proper elliptic coordinate system having the origin located at the center of the notch. Solution of governing Laplace’s equation was written as the real part of an analytical (holomorphic) function and stress field was attained in terms of complex potential function. By choosing a suitable complex potential function, the boundary conditions of the problem were satisfied and the stress field was acquired. Relations for the Mode III notch stress intensity factors, NSIFs, for circumferentially-sharply-notched rounded bars under torsion were presented by Zappalorto et al. (2009). A closed-form solution for the NSIFs was obtained for deep notches; subsequently the solution was extended also to finite notched components. In the last part of this review, we cite investigations with emphasis on numerical methods. The torsion problem of a rectangular cross section bar with inner crack as two Dirichlet problems with the definite boundary values was studied by Chen (1998). The corresponding Dirichlet problems are solved using the finite difference method. The torsion rigidity coefficient and the subsequent compliance coefficient as an inverse of the torsion rigidity coefficient was reached by numerical integration. At the end, the stress intensity factor for an embedded straight crack was evaluated. Also, the evaluation was carried out for four inner branch cracks, which are perpendicular to each other. In another study by Chen (1999), the torsion problem of thinwalled cylinder weakened by four inner edge crack normal to inner

A.R. Hassani, R.T. Faal / International Journal of Solids and Structures 52 (2015) 165–179

surface of cylinder was investigated. Using a method similar to the above-mentioned (Chen, 1998), the problem was reduced to a Dirichlet problem of the Laplace equation and solved via the finite difference method. The torsional rigidity and the stress intensity factor of the crack tips were also evaluated. Torsion problem of a thick-walled cylinder with a radial external crack was investigated by Yuanhan (1990). An expansion for the stress function was introduced so that the ensuing stresses at the crack tip possess the inverse square root singularity. The boundary collocation method was used to obtain the unknown coefficients of expansion. At the end, torsional rigidity of the cylinder and Mode III stress intensity factor of crack tip were calculated. Using the boundary collocation method, as a numerical method, torsional rigidity and stress intensity factor for bars with circular, elliptical and rectangular cross sections containing two edge cracks was investigated by Cheung and Wang (1991). The Saint-Venant torsion problems of a cylinder with curvilinear cracks were the subject of study by Wang and Lu (2005). Using the boundary element method, the problem was reduced to the solution of boundary integral equations only on cracks borders which are solved numerically. Torsion rigidity and stress intensity factors of the crack tips were given for a straight, kinked or eccentric circular-arc crack. Among the numerical techniques, the finite element method is an important means to analyze the crack problems. In finite element analysis, it is conventional to model the crack tip singularity by using elements in which midside nodes are moved to quarter points. There is a widely used method of calculating of stress intensity factors which substitute the displacements, or the stresses, obtained from the finite element calculations, into standard field equations in the vicinity of crack tips. (For example, see (Pook, 2007)). One of the reasons of considering this reference here is the concept of Volterra distorsioni introduced there which is another feature of dislocation distribution technique. Pook, considered a pair of infinitesimal elements which are situated on the upper and lower surface of an unloaded crack respectively. These elements are connected by a ring element of infinitesimal width having a square cross section and located around the crack tip. He introduced the three of modes of dislocations of the six Volterra distorsioni (distortions), correspond to the three modes of crack tip surface displacement. The remaining three Volterra distorsioni involve rotation of the above-mentioned elements which are usually called three modes of disclinations. He explained that the modes of crack surface displacements are a combination of some modes of dislocation and disclinations. In fact, the six Volterra distorsioni are useful in the description of crack displacements under load, and may also be used to describe narrow notch surface displacements as a model of crack. An interesting matter mentioned in this reference and also (Pook, 2003) which may be the main reason of considering these references here is about that different modes of displacements cannot exist in isolation. It can be seen in some applications such as, a cracked square plate and a cracked long bar in Mode III loading, but it is only for a special positions of the cracks. In particular, a cracked long bar under torsion may be regarded as a Mode III loading which is also the subject of preset work. We will reveal the validity of the existence of such a single mode for a long bar in torsion. According to the above reviews, the torsion problem of cracked bars is a challenging problem which has been taken into consideration by several researchers. Nevertheless, to the authors’ knowledge, no dislocation solution has been reported previously on the stress analysis of a torsion bar with rectangular cross-section and multiple curved cracks. In the sections to follow, series solution of stress and displacement fields will be first calculated on an orthotropic bar with rectangular cross section weakened by one screw dislocation (Section 2). Dislocation solutions are then

167

employed to formulate and solve the Cauchy type singular integral equations for the orthotropic bar weakened by several cracks. Formula for evaluation of torsional rigidity of cracked bars is presented (Section 3). Numerical examples will be presented to demonstrate the effectiveness of the proposed solutions, as well as to understand the effect of moment, material properties and cracks orientation, geometry and interaction on the ensuing stress intensity factors at crack tips (Section 4). Finally, concluding remarks are included in Section 5. 2. Problem formulation 2.1. Dislocation solution We consider a long bar with rectangular cross section with a finite length a and width h in the x and y directions, respectively, (as shown in Fig. 1). The x-axis is located at the distance g top the lower edge of the rectangular. Similarly, the y-axis is situated at the distance n from the left edge of the rectangular plane. As such, the cross section consists of two orthotropic sub-rectangular regions with finite widths h  g and g which have been attached together along the x-axis. For a long prismatic rod twisted by end torques each cross-section rotates as a rigid body (No ‘‘distortion’’ of cross-section shape in x, y) and we have a constant rate of twist. Also, the cross-sections are free to warp in the z-direction but the warping is the same for all cross-sections. Under these assumptions, the Saint-Venant torsion theory is valid and consequently we have the Mode III of fracture only. For portions of bar far from the ends, the Saint-Venant torsion theory is valid and then the Mode III is in isolation as it has been reported in the reference Pook (2003). Also in the following references, the decoupling of Mode III and other modes of fracture for the Saint-Venant torsion problem of the orthotropic bars was reported by Chen et al. (1997). Moreover, the validity of using the Saint-Venant torsion theory for the orthotropic bars was also reported (see: Benveniste and Chen, 2003; Gracia and Doblare, 1988; Rongqiao et al., 2010; Santoro, 2010). As said by Saint-Venant torsion theory, the displacement components read as

u ¼ hzy;

v ¼ hzx;

w ¼ huðx; yÞ

Fig. 1. Rectangular cross section of bar with screw dislocation.

ð1Þ

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in which h is the twist angle of unit length, and uðx; yÞ represents the warping function. The out of plane displacement component wðx; yÞ under Saint-Venant torsion is of interest. Consequently, the two nonzero stress components in the z direction are

  @ uðx; yÞ rzx ðx; yÞ ¼ Gzx h y @x

ð2Þ

  @ uðx; yÞ rzy ðx; yÞ ¼ Gzy h þx @y

ð3Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where G ¼ Gzx =Gzy . Appling the finite Fourier cosine transRa form,Uðn; yÞ ¼ 0 uðX; yÞcosðnpX=aÞdX to above equation, in which X = x + n, (see Fig. 1) leads to 2

 ðGnjÞ2 Ul ðn;yÞ þ G2 ð1Þn

@ ul ða;yÞ @ ul ð0; yÞ  G2 ¼ 0; l ¼ 1; 2 @X @X ð4Þ

where j = p/a and the subscripts l = 1, 2 is used to refer to the upper and lower rectangular regions, respectively. The traction-free condition on the lateral surface of the bar requires that

a; l ¼ 1; 2

By virtue of Eq. (4) and the first equation of (5), we easily arrive at

   ðGnjÞ2 Ul ðn; yÞ ¼ G2 1  ð1Þn y:

dy

Using

the

inverse P1

ul ðX; yÞ ¼ Ul ða0;yÞ þ 2a

n¼1

Ul ð0; yÞ a

þ

ð9Þ

P1

n¼1

ð1Þn 1 ½cosðn n2

finite

Fourier

cosine

transform,

Ul ðn; yÞcosðnjX Þ, the warping function

" 1 2X Aln sinhðGnjyÞ þ Bln coshðGnjyÞ a n¼1

  # 1  ð1Þn y cosðnjX Þ;  ðnjÞ2

B20 ¼ B10 ¼ na 

a2 2

ð10Þ

B1n ¼ A1n cothðGnjðh  gÞÞ þ Kn cschðGnjðh  gÞÞ B2n ¼ A2n cothðGnjgÞ  Kn cschðGnjgÞ   where Kn ¼ 2 1  ð1Þn =GðnjÞ3 . Making use of the expansion P 1 HðX  nÞ ¼ 1  an  p2 1 n¼1 n sinðnjnÞcosðnjX Þ; the conditions (9) are applied to Eqs. (6), i.e.

bz ða  nÞ h bz ¼ A1n ¼ Cn sinðnjnÞsinhðGnjgÞsinhðGnjðh  gÞÞ jh

þnKn ½sinhðGnjgÞ þ sinhðGnjðh  gÞÞ

ð11Þ

where Cn ¼ 1=ðnsinhðGnjhÞÞ. Substitution of determinate coefficients in Eqs. (11) into (10) leads to the coefficients B1n and B2n. Next, with substituting the coefficients Aln, Bln, l = 1, 2 into Eqs. (6), we may state the warping functions as A20 bz ða  nÞ þ þ ðX þ n  aÞy ah a 1 2bz X C sinhðGnjgÞsinðnjnÞcoshðGnjðy  ðh  gÞÞÞcosðnjX Þ  ph n¼1 n    1 2X h þ Xn sinh Gnj y þ g  cosðnjX Þ; 0  y  h  g a n¼1 2 ð12Þ

A20 þ ðX þ n  aÞy a 1 2bz X C sinhðGnjðh  gÞÞsinðnjnÞcoshðGnjðy þ ph n¼1 n

uðX; yÞ ¼ l ¼ 1; 2

ð6Þ

where in light of Eqs. (4) and (5) we have Ul ð0; yÞ ¼ Al0 þ Bl0 y; l ¼ 1; 2. The solution of the warping function can also be represented in the form of exponential functions but in order to easy compare the resulting stress field in the special case i.e. G = 1 with those in the literature we state the solution in terms of hyperbolic functions. A Volterra dislocation of screw type with Burgers vector bz is situated in the domain under consideration with straight cut y = 0, x > 0. First, we suppose that bar is under pure torsion ðh–0Þ and then we make a screw dislocation in the cross section. On the other hand, the Eqs. (1) and (2) hold. The boundary condition representing this dislocation under anti-plane deformation is:

  w x; 0þ  wðx; 0 Þ ¼ bz HðxÞ

jX Þ

uðX; yÞ ¼

takes the form

ul ðX; yÞ ¼

bz HðX  nÞ h

cosðnjnÞ, the second and third conditions (5) are applied to Eq. (6), leading to

A2n

@ u2 ðX; yÞ þ X  n ¼ 0 at y ¼ g @y

2



@ u1 ðX; 0Þ @ u2 ðX; 0Þ ¼ @y @y

ð5Þ

@ u1 ðX; yÞ þ X  n ¼ 0 at y ¼ h  g @y

2

In view of Eqs. (1) and (2), The boundary and continuity conditions (7) and (8) can be rewritten in the form

A10 ¼ A20 þ

@ ul ðX; yÞ  y ¼ 0 at X ¼ 0; @X

d Ul ðn; yÞ

ð8Þ

With the help of the expansion X  n ¼ p2a2

@ 2 uðx; yÞ @ 2 uðx; yÞ G þ ¼0 @x2 @y2

dy





2

2



rzy X; 0þ ¼ rzy ðX; 0 Þ

u1 X; 0þ  u2 ðX; 0 Þ ¼

where Gzx, Gzy are the orthotropic shear moduli. The equilibrium equations in the absence of body forces rij,j = 0, in view of Eq. (1) reduce to

d Ul ðn; yÞ

where HðÞ is the Heaviside step function. The continuity condition (self-equilibrium of stress) in the rectangular plane containing the dislocation is as follows

ð7Þ

þgÞÞcosðnjX Þ    1 2X h cosðnjX Þ; g 6 y  0 Xn sinh Gnj y þ g  þ a n¼1 2 where Xn ¼ Kn sechðGnjh=2Þ: According to the Eqs. (2), the stress field takes the form 1 2Gzx h X bz n  Cn sinðnjnÞsinhðGnjgÞcoshðGnjðy a n¼1 h   

h sinðnjX Þ; ðh  gÞÞÞ þ jXn sinh Gnj y þ g  2

rzx ðX; yÞ ¼ 

06y6hg

ð13Þ

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( 1 2Gzy h bz G X rzy ðX; yÞ ¼ Gzy hð2X  aÞ þ ½Em ðX þ n; y þ hÞ a 8h m¼1

1 2G h X bz rzx ðX; yÞ ¼  zx n Cn sinðnjnÞsinhðGnjðh a n¼1 h

gÞÞcoshðGnjðy þ gÞÞ   

h þjXn sinh Gnj y þ g  sinðnjX Þ; 2

Em ðX  n; y þ hÞ þ Em ðX þ n; ðy þ hÞÞ Em ðX  n; ðy þ hÞÞ  Em ðX þ n; y  ðh  2gÞÞ

g 6 y 6 0

þEm ðX  n; y  ðh  2gÞÞ  Em ðX þ n; y þ ðh  2gÞÞ þEm ðX  n; y þ ðh  2gÞÞ )    1 X h cosðnjX Þ ; þGj nXn cosh Gnj y þ g  2 n¼1

rzy ðX; yÞ ¼ Gzy hð2X  aÞ

(

   1 2Gzy h X h þ n GjXn cosh Gnj y þ g  a n¼1 2 

g6y60

bz G Cn sinðnjnÞsinhðGnjgÞsinhðGnjðy h )

ðh  gÞÞÞ cosðnjX Þ;

where

06y6hg

   1 2Gzy h X h n GjXn cosh Gnj y þ g  a n¼1 2

bz G þ Cn sinðnjnÞsinhðGnjðh  gÞÞsinhðGnjðy þ gÞÞ cosðnjX Þ; h

rzy ðX;yÞ ¼ Gzy hð2X  aÞ þ

g6y60

Making use of the series of the form nCn ¼ cschðGnjhÞ ¼ P ð2m1ÞGnjh 2 1 and expanding sinh, cosh, sin and cos functions m¼1 e in terms of exponential functions, stress field due to a screw dislocation can be recast by means of the formulas given in Appendix A of the report by Faal et al. (2004) as

rzx ðX; yÞ ¼ 

( 1 2Gzx h bz X  ½F m ðX þ n; y  ðh  2gÞÞ a 8h m¼1

F m ðX  n; y  ðh  2gÞÞ  F m ðX þ n; y þðh  2gÞÞ þ F m ðX  n; y þ ðh  2gÞÞ

Em ðx; yÞ ¼

sinðjxÞ coshðGj½y  ð2m  1ÞhÞ  cosðjxÞ

F m ðx; yÞ ¼

sinhðGj½y  ð2m  1ÞhÞ coshðGj½y  ð2m  1ÞhÞ  cosðjxÞ

The stress field of (14) takes into account both singular and non-singular terms. It is worth mentioning that for bz = 0, the stress field (14) reduces to the stress field of an intact bar under torque which there is no defect or dislocation in the cross section. In this case, for isotropic bar (G = 1) the stress field (14) reduces to identical results of reference Sadd (2009). Moreover, the series including pre-defined functions Em and Fm rapidly converge with increasing index m. One can conclude that the stress component rzy(x, 0) has Cauchy singularity which was also previously reported, e.g., by Faal et al. (2011) for rectangular plane under anti-plane elasticity. The proof is given in the Appendix A. Integration of the stress component rzy ðX; yÞ with respect to y in light of Eq. (2) yields uðx; yÞ and the out of plane displacement component wðx; yÞ: That is

F m ðX þ n; y  hÞ þ F m ðX  n; y  hÞ

(

þF m ðX þ n; h  yÞ  F m ðX  n; h  yÞ )    1 X h sinðnjX Þ ; þj nXn sinh Gnj y þ g  2 n¼1 06y6hg

wðx; yÞ ¼ hðXy þ ny  ayÞ þ

ð14Þ

F m ðX þ n; ðy þ hÞÞ þ F m ðX  n; ðy þ hÞÞ F m ðX þ n; y  ðh  2gÞÞ þ F m ðX  n; y  ðh  2gÞÞ þF m ðX þ n; y þ ðh  2gÞÞ  F m ðX  n; y þ ðh  2gÞÞ )    1 X h sinðnjX Þ ; þj nXn sinh Gnj y þ g  2 n¼1

rzy ðX; yÞ ¼ Gzy hð2X  aÞ

( 1 2Gzy h bz G X  ½Em ðX þ n; y  hÞ a 8h m¼1



1 bz X ½Gm ðX þ n; y  hÞ 2j m¼1

Gm ðX  n; y  hÞ  Gm ðX þ n; h  yÞ

( 1 2Gzx h bz X rzx ðX; yÞ ¼  ½F m ðX þ n; y þ hÞ  F m ðX  n; y þ hÞ a 8h m¼1

g6y60

þGm ðX  n; h  yÞ  Gm ðX þ n; y  ðh  2gÞÞ þGm ðX  n; y  ðh  2gÞÞ þ Gm ðX þ n; y þ ðh  2gÞÞ Gm ðX  n; y þ ðh  2gÞÞ )    1 2h X h þ cosðnjX Þ ; Xn sinh Gnj y þ g  a n¼1 2 06y6hg

ð16Þ (

wðx; yÞ ¼ hðXy þ ny  ayÞ þ

1 bz X ½Gm ðX  n; y þ hÞ 2j m¼1

Gm ðX þ n; y þ hÞ þ Gm ðX þ n; ðy þ hÞÞ Gm ðX  n; ðy þ hÞÞ þ Gm ðX þ n; ðy  ðh  2gÞÞÞ Gm ðX  n; ðy  ðh  2gÞÞÞ  Gm ðX þ n; y þ ðh  2gÞÞ

þEm ðX  n; h  yÞ þ Em ðX þ n; y  ðh  2gÞÞ

þGm ðX  n; y þ ðh  2gÞÞ )    1 2h X h cosðnjX Þ ; Xn cosh Gnj y þ g  þ a n¼1 2

Em ðX  n; y  ðh  2gÞÞ þ Em ðX þ n; y

g6y60

þ

þEm ðX  n; y  hÞ  Em ðX þ n; h  yÞ

þðh  2gÞÞ  Em ðX  n; y þ ðh  2gÞÞ )    1 X h cosðnjX Þ ; þGj nXn cosh Gnj y þ g  2 n¼1 06y6hg

ð15Þ

where Gm ðx; yÞ ¼ tan1 ½cothðGk ðy  ð2m  1ÞhÞÞtanðkx=2Þ. 2 We consider a system of coordinates ðX; Y Þ attached to lower and left edges of bar cross-section, Fig. 2, which is compatible with the former change of variable i.e. X = x + n. Consequently, we also have Y = y + g. Substituting the stress relations (2) into

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"

Ra Rh

M ¼ 0 0 ðX rzY ðX; Y Þ  Y rzX ðX; Y ÞÞdXdY ¼ Dh (This equation has been used even for warped cross section in many of the references, for example (Chen, 1980, 1998; Fang-ming and Ren-ji, 1993; Renji and Yulan, 1992; Rongqiao et al., 2010), torsional rigidity of bar can be easily calculated by using the following formula D ¼ Gzy

Z

a 0

h

Z

XðX  nÞ þ G2 YðY  gÞ þ X

0

sY i z ðX i ; Y i Þ ¼ Gzy hcosui 2X i  a #    1 2Gj X h þ nXn cosh Gnj Y i  cosðnjX i Þ 2 a n¼1    1 X 2jGzx h h þ sinðnjX i Þ sinui nXn sinh Gnj Y i  a 2 n¼1 ( 1 X    2Gzy G þ Em X i þ X j ; Y i  Y j  h cosui  Bzj dkj 8 a m¼1     þEm X i  X j ; Y i  Y j  h  Em X i þ X j ; Y i þ Y j þ h     þEm X i  X j ; Y i þ Y j þ h þ Em X i þ X j ; Y i þ Y j  h     Em X i  X j ; Y i þ Y j  h þ Em X i þ X j ; Y i  Y j þ h )   Em X i  X j ; Y i  Y j þ h

 @ uðX;Y Þ @ uðX;Y Þ  YG2 dXdY @Y @X ð17Þ

where M is the applied torque. Considering wðX; Y Þ ¼ huðX; Y Þ and also substituting Eqs. (12) into above equations we arrive at

D ¼ D0 

1 16bz a2 Gzy X 1 sechðnjGh=2ÞsinðnjnÞsinhðGnjðh 3 p h n¼1;3;... n3

gÞ=2ÞsinhðGnjg=2Þ h

where D0 ¼ Gzy a3 h=3  ð64a4 =Gp5 Þ

ð18Þ P1

1 n¼1;3;... n5

i

( " 1 X   2Gzx 1 þ sinui  Bzj dkj Fm Xi þ Xj; Y i þ Y j  h 8 a m¼1     F m X i  X j ; Y i þ Y j  h  F m X i þ X j ; Y i  Y j þ h     þF m X i  X j ; Y i  Y j þ h  F m X i þ X j ; Y i  Y j  h     þF m X i  X j ; Y i  Y j  h þ F m X i þ X j ; Y i þ Y j þ h #)   ; Yj 6 Yi 6 h F m X i  X j ; Y i þ Y j þ h ð20Þ

tanhðnjGh=2Þ is the

torsional rigidity of intact bar. Comparing D0 for isotropic bar to that provided by Sadd (2009) shows identical result. 3. Analyses with multiple cracks In what follows, we implement the dislocation solutions accomplished in the prior section to analyze bars of rectangular cross section with several cracks. The anti-plane stress components on the local coordinate ðX i ; Y i Þ; Fig. 2, located on the surface of the ith crack become

sY i z ðX; Y Þ ¼ szy ðX; Y Þcosui  szx ðX; Y Þsinui

ð19Þ

sY i z ðX i ; Y i Þ ¼ Gzy hcosui ½2X i  a  1 2j X

#   h cosðnjX i Þ þ nXn cosh Gnj Y i  2 a n¼1    1 X 2jGzx h h þ sinðnjX i Þ sinui nXn sinh Gnj Y i  a 2 n¼1 ( 1 X    2Gzy h G þ cosui Bzj dkj Em X i þ X j ; Y i  Y j þ h a 8h  m¼1    Em X i  X j ; Y i  Y j þ h þ Em X i þ X j ;  Y i  Y j þ h      Em X i  X j ;  Y i  Y j þ h  Em X i þ X j ; Y i þ Y j  h      þEm X i  X j ; Y i þ Y j  h  Em X i þ X j ;  Y i þ Y j  h    þEm X i  X j ;  Y i þ Y j  h 1 X    2Gzx h 1 þ Bzj dkj sinui Fm Xi þ Xj; Y i  Y j þ h a 8h m¼1     F m X i  X j ; Y i  Y j þ h  F m X i þ X j ; Y i þ Y j  h     þF m X i  X j ; Y i þ Y j  h  F m X i þ X j ; Y i þ Y j  h     þF m X i  X j ; Y i þ Y j  h þ F m X i þ X j ; Y i  Y j þ h   F m X i  X j ; Y i  Y j þ h ; 0 6 Y i 6 Y j

sXi z ðX; Y Þ ¼ szy ðX; Y Þsinui þ szx ðX; Y Þcosui where ui denotes angle between Xi (local) and X (global) axes. Suppose dislocations with unknown density Bzj are distributed on the infinitesimal segment dkj located at a point with coordinates   X j ; Y j on the jth crack surface. First, we find the traction on the surface of the ith crack due to the presence of above distribution of dislocations. Utilizing Eqs. (14) and (19), the anti-plane stress components become

We have to emphasize that for evaluating D, the term bz in the Eq. (18) is replaced by Bzj dkj and also n and g is replaced by Xj and Yj, respectively. Next, Eq. (18) is integrated on the crack surfaces. The integration of Eq. (18) can be assisted by describing crack configurations in a parametric form X j ¼ X j ðtÞ; Y j ¼ Y j ðtÞ; i ¼ 1; 2; . . . ; N; where 1 6 t 6 1. We recall that h = M/D is a constant which depends on geometry of bar cross section and cracks. Finally the torsional rigidity is given by (

  N Z 1 1   16a2 Gzy X 1 njGh X sech sin njX j ðt Þ 3 3 2 p M n¼1;3;... n 1 j¼1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )     h i2 h i2  Gnj  Gnj  sinh h  Y j ðt Þ sinh Y j ðt Þ bzj ðt Þ X 0j ðt Þ þ Y 0j ðt Þ dt 2 2

D ¼ D0 = 1 þ

ð21Þ

Fig. 2. Rectangular cross section of bar with a smooth curved crack.

where, bzj ðt Þ is the dislocation density along the dimensionless length 1 6 t 6 1. Analogously, the integration of Eq. (20), leads to the resultant tractions on the crack surfaces.

171

A.R. Hassani, R.T. Faal / International Journal of Solids and Structures 52 (2015) 165–179

Substituting h ¼ M into Eqs. (20) gives the final form of the tracD tion on the surface of the ith crack. As we expect, the crack surfaces are stress free, therefore the left side of the equations made by integration of Eq. (20) are vanished. Consequently, the terms of Eqs. (20) which are not multiplied by bzj ðtÞ are moved to the left side of these equations. Finally we have

Q i ðX i ðsÞ; Y i ðsÞÞ ¼

N Z X j¼1

1

kij ðs;t Þbzj ðt Þdt; 1 6 s 6 1; i ¼ 1;2;. .. ; N 1

ð22Þ where the kernels kij ðs; t Þ and the left side of Eqs. (22), i.e. Q i ðX i ðsÞ; Y i ðsÞÞ are given in Appendix A. By virtue of Bueckner’s principle (see, e.g., Hills et al. (1996)), the left-hand side of Eq. (22) after

changing, the sign is the traction caused by the external torque on the uncracked bar at the presumed surfaces of cracks. In fact, we implicitly use the Bueckner’s principle, to derive Eqs. (22). As a result of single-valuedness of displacement field away from the surfaces of embedded cracks, the dislocation density functions are subjected to the following closure requirement (Faal et al., 2006)

Z

1

1

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i2 h i2 X 0j ðt Þ þ Y 0j ðt Þ bzj ðt Þdt ¼ 0

ð23Þ

The Cauchy singular integral Eqs. (22) and (23) should to be solved simultaneously to determine dislocation density functions. To this end, the original numerical procedure proposed by

Table 1 Values of 104D/la4 for isotropic bar with an edge crack, bisecting and normal to one side of cross section. Study source

l a

l a

¼0

l a

¼ 0:2

l a

¼ 0:4

¼ 0:6

l a

¼ 0:8

l a

¼ 0:95

¼ 0:1

Present work

23.305

19.892

15.894

11.901

8.066

6.306

h 2a

¼ 0:2

Chen (1980) Present work

23.301 159.594

19.800 136.726

15.830 105.535

11.855 75.729

8.035 52.904

6.303 46.725

h 2a

¼ 0:25

Chen (1980) Present work

159.590 285.852

136.110 245.265

105.060 186.870

75.426 133.173

52.797 96.403

46.718 87.890

h 2a

¼ 0:5

Chen (1980) Present work

285.850 1405.770

244.040 1227.265

185.840 938.954

132.580 707.611

96.207 590.770

87.880 571.921

h 2a

¼1

Chen (1980) Present work

1405.700 4573.634

1221.100 4198.614

933.840 3557.023

704.920 3062.952

590.070 2842.055

571.890 2811.832

h 2a

¼ 2:5

Chen (1980) Present work

4573.700 14565.838

4179.800 14123.802

5540.100 13354.734

3054.800 12762.401

2839.900 12502.843

2811.800 12468.571

h 2a

¼5

Chen (1980) Present work

14567 31232.504

14077 30789.833

13322 30019.537

12740 29426.247

12498 29166.316

12469 29132.011

Chen (1980)

31245

30702

29943

29390

29164

29137

h 2a

Table 2 Values of kIIIa2.5/M for isotropic bar with an edge crack, bisecting and normal to one side of cross section. Study source

l a

l a

¼ 0:05

l a

¼ 0:2

l a

¼ 0:4

l a

¼ 0:6

l a

¼ 0:8

¼ 0:9

h 2a

¼ 0:1

Present work

17.765

22.469

28.144

37.483

51.701

47.274

h 2a

¼ 0:2

Chen (1980) Present work

17.355 6.123

22.483 9.058

28.236 11.805

37.651 15.568

51.316 16.764

45.199 11.115

h 2a

¼ 0:25

Chen (1980) Present work

6.094 4.415

9.091 6.884

11.847 9.088

15.572 11.614

16.485 11.117

11.017 6.783

h 2a

¼ 0:5

Chen (1980) Present work

4.435 1.754

6.918 3.036

9.108 3.950

11.597 4.221

10.942 2.870

6.758 1.466

h 2a

¼1

Chen (1980) Present work

1.795 0.757

3.053 1.313

3.951 1.552

4.190 1.384

2.835 0.777

1.487 0.366

h 2a

¼ 2:5

Chen (1980) Present work

0.789 0.256

1.327 0.426

1.546 0.453

1.369 0.362

0.764 0.189

0.368 0.087

h 2a

¼5

Chen (1980) Present work

0.283 0.119

0.429 0.196

0.449 0.202

0.353 0.158

0.180 0.083

0.084 0.039

Chen (1980)

0.146

0.198

0.120

0.149

0.074

0.034

Table 3 Values of 104D/la4 for isotropic bar with an edge crack, non-bisecting and normal to one side of cross section. h a

¼1

Study source

d a

¼ 0:1

Present work

1402.043

1363.562

1281.325

1198.849

1146.694

1136.094

d a

¼ 0:2

Chen (1980) Present work

1401.300 1397.141

1361.800 1310.734

1279.400 1147.014

1197.600 1000.156

1146.300 917.455

1136.100 902.761

d a

¼ 0:3

Chen (1980) Present work

1395.500 1393.214

1301.200 1265.932

1143.600 1035.630

998.130 842.360

916.890 740.361

902.740 723.278

d a

¼ 0:4

Chen (1980) Present work

1390.900 1390.771

1260.700 1237.116

1031.300 963.683

839.780 741.972

739.920 628.825

723.250 610.409

d a

¼ 0:5

Chen (1980) Present work

1388.000 1389.948

1231.100 1227.265

958.610 938.954

739.270 707.611

628.110 590.770

610.380 571.921

Chen (1980)

1387.100

1221.100

933.840

704.920

590.070

571.890

l a

¼ 0:05

l a

¼ 0:2

l a

¼ 0:4

l a

¼ 0:6

l a

¼ 0:8

l a

¼ 0:95

172

A.R. Hassani, R.T. Faal / International Journal of Solids and Structures 52 (2015) 165–179

Erdogan et al. (1973) may not be directly applicable, since it is not able to simultaneously consider all types of defects; i.e. embedded and edge cracks. In the study by Faal et al. (2006), a minor generalization of this numerical procedure was introduced to overcome this issue. For want of space, we did not repeat it here but we use this generalized method to determine dislocation density functions. Stress fields for embedded cracks in orthotropic media are singular at crack tips with square root singularity (Delale, 1984), thus for embedded cracks the dislocation density functions are represented by Faal et al. (2006)

g zj ðt Þ bzj ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1  t2

1 6 t 6 1

ð24Þ

For edge cracks, taking the embedded crack tip at t = 1, we let (Faal et al., 2006)

rffiffiffiffiffiffiffiffiffiffiffi 1t ; 1þt

bzj ðtÞ ¼ g zj ðt Þ

1 6 t 6 1

ð25Þ

The stress intensity factors for the embedded and edge cracks are (Faal et al., 2006)

8 h i2 h i2 14 > > GGzy 0 0 > > k ¼ X ð 1 Þ þ Y ð 1 Þ g zj ð1Þ IIILi < j j 2 for embedded crack h i2 h i2 14 > > > GGzy 0 0 > X j ð 1Þ þ Y j ð 1Þ g zj ð1Þ : kIIIRi ¼  2 ð26Þ kIIILi ¼ GGzy

h i2 h i2 14 X 0j ð1Þ þ Y 0j ð1Þ g zj ð1Þfor edge crack

In order to determine the torsional rigidity of cracked bar the integral included in Eq. (21) can be evaluated by use of discretiza

Þ tion of domain of integration at the points t k ¼ cos pð2k1 ; 2n0 k ¼ 1; 2; . . . ; n0 where n0 denotes number of discrete points. This method was explained completely in Faal et al. (2006). Consequently the torsional rigidity emerge as

Table 4 Values of kIIIa2.5/M for isotropic bar with an edge crack, bisecting and normal to one side of cross section. h a

¼1

Study source

l a

d a

¼ 0:1

Present work

0.833

1.372

1.639

1.581

1.070

0.846

d a

¼ 0:2

Chen (1980) Present work

0.848 1.280

1.378 2.086

1.635 2.508

1.557 2.455

1.052 1.610

0.833 1.250

d a

¼ 0:3

Chen (1980) Present work

1.303 1.555

2.094 2.606

2.502 3.238

2.434 3.284

1.588 2.171

1.235 1.677

d a

¼ 0:4

Chen (1980) Present work

1.587 1.706

2.619 2.927

3.233 3.759

3.259 3.955

2.141 2.666

1.660 2.058

d a

¼ 0:5

Chen (1980) Present work

1.746 1.754

2.942 3.036

3.757 3.951

3.926 4.223

2.628 2.875

2.037 2.219

Chen (1980)

1.795

3.053

3.951

4.190

2.835

2.198

l a

¼ 0:05

l a

¼ 0:2

l a

¼ 0:4

l a

¼ 0:6

l a

¼ 0:8

¼ 0:9

Table 5 Values of 104D/la4 for isotropic bar with two edge cracks bisecting and normal to opposite sides. Study source

l a

¼ 0:05

l a

l a

¼ 0:2

l a

¼ 0:4

¼ 0:6

l a

¼ 0:8

l a

¼ 0:95

¼ 0:1

Present work

22.974

20.436

16.480

12.486

8.589

6.438

h 2a

¼ 0:25

Chen (1980) Present work

22.871 283.453

20.283 257.575

16.361 205.413

12.405 150.766

8.544 106.083

6.427 88.969

h 2a

¼ 0:5

Chen (1980) Present work

282.730 1397.362

255.790 1294.504

203.730 1062.103

149.610 818.208

105.550 637.996

88.858 575.944

h 2a

¼1

Chen (1980) Present work

1394.400 4557.420

1286.100 4350.277

1052.800 3860.535

811.560 3336.841

634.760 2951.814

575.270 2820.483

h 2a

¼ 2:5

Chen (1980) Present work

4588.600 14547.060

4325.900 14305.003

3831.100 13725.848

3413.400 13100.265

2939.600 12637.395

2818.400 12479.044

h 2a

¼5

Chen (1980) Present work

14523 31213.702

14240 30971.315

13649 30391.305

13035.000 29764.735

12606 29301.102

12476 29142.484

Chen (1980)

31179

30845

30239

29648.000

29256

29143

h 2a

Table 6 Values of kIIIa2.5/M for isotropic bar with two edge cracks bisecting and normal to opposite sides. Study source

l a

¼ 0:05

l a

¼ 0:2

l a

¼ 0:4

l a

¼ 0:6

l a

¼ 0:8

l a

¼ 0:9

¼ 0:1

Present work

14.805

21.496

27.128

35.774

49.916

52.112

h 2a

¼ 0:25

(Chen, 1980) Present work

15.167 3.360

21.599 5.883

27.228 8.109

35.919 10.696

49.649 12.392

50.420 10.518

h 2a

¼ 0:5

Chen (1980) Present work

3.543 1.292

5.954 2.407

8.153 3.348

10.714 4.078

12.238 3.989

10.226 3.118

h 2a

¼1

Chen (1980) Present work

I.3858 0.552

2.441 1.028

3.367 1.347

4.073 1.465

3.927 1.256

3.016 0.937

h 2a

¼ 2:5

Chen (1980) Present work

0.613 0.186

1.049 0.339

1.354 0.413

1.457 0.409

1.226 0.322

0.887 0.234

h 2a

¼5

Chen (1980) Present work

0.235 0.087

0.351 0.157

0.414 0.187

0.400 0.180

0.302 0.139

0.207 0.100

Chen (1980)

0.131

0.168

0.186

0.171

0.122

0.083

h 2a

A.R. Hassani, R.T. Faal / International Journal of Solids and Structures 52 (2015) 165–179

Fig. 3. Bar cross section with an edge crack bisecting and normal to its side.

173

Fig. 5. Bar cross section with two edge cracks bisecting and normal to opposite sides.

4. Numerical examples and discussions

Fig. 4. Bar cross section with an edge crack non bisecting and normal to its side.

( D ¼ D0 = 1 þ

  1 16a2 Gzy X 1 njGh sech 2 p2 n0 M n¼1 n3

ð27Þ

    n0 N X X    Gnj  Gnj sin njX j ðt k Þ sinh h  Y j ðt k Þ sinh Y j ðt k Þ 2 2 j¼1 k¼1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) h i2 h i2 g zj ðt k ÞDðtk Þ X 0j ðtk Þ þ Y 0j ðt k Þ where

Dðt k Þ ¼



1

for embedded cracks

1  tk

for edge cracks

ð28Þ

The analysis framework, developed in the preceding section, allowed the consideration of an orthotropic bar with rectangular cross section containing an arbitrary number of defects. These defects may includes smooth embedded and edge cracks with different orientations. Let us first validate our solution with the published results for isotropic cases (Tables 1–6). To check the correctness and validation of the present method, the results of this paper will be compared with those in the literature during the following examples. To verify the validity of formulation for isotropic bar, three examples solved by Chen (1980) concerning an edge crack bisecting/non bisecting one side of cross section and two edge cracks bisecting two opposite sides of the cross section, are re-examined. Good agreement is observed for torsional rigidities and stress intensity factors, Tables 1–6 of the foregoing reference. In the following, another example of reference Chen et al. (1997) is examined and the results are validated for orthotropic case. Because of huge number of data in the aforementioned references, we present only some data as a comparative selective set. Adopting the notations of the above-mentioned references according to this study we provide some tables to compare the results as follows. 4.1. Example 1. An edge crack, bisecting and normal to one side of cross section In this example, the crack configuration can be seen in Fig. 3. Results for dimensionless torsional rigidities 104D/la4, where l is the shear modulus of isotropic bar, can be found in Table 1. Values of non-dimensional stress intensity factors of singular crack tip kIIIa2.5/M for isotropic bar with an edge crack, bisecting and normal to one side of cross section were also listed in Table 2. There is one pffiffiffiffi more coefficient p in definition of stress intensity factor in the works done by Chen et al. (1997) and Chen (1980) in comparison with those in Eq. (26) which is well-known difference in the literpffiffiffiffi ature. From now on, we consider the coefficient as p; which is applied to all examples.

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A.R. Hassani, R.T. Faal / International Journal of Solids and Structures 52 (2015) 165–179

Table 7 Values of kIIIa2.5/M for orthotropic bar with an edge crack, bisecting and normal to one side of cross section and h/2a = 0.5. h 2a

¼ 0:5

2

G = 0.2 G2 = 0.5 G2 = 1 G2 = 2 G2 = 5

Study source

l a

Present work Chen et al. (1997) Present work Chen et al. (1997) Present work Chen et al. (1997) Present work Chen et al. (1997) Present work Chen et al. (1997)

2.322 2.366 1.934 1.978 1.754 1.798 1.624 1.670 1.480 1.532

¼ 0:05

l a

¼ 0:2

3.528 3.535 3.223 3.233 3.036 3.048 2.842 2.854 2.556 2.569

l a

¼ 0:4

4.631 4.634 4.279 4.282 3.951 3.953 3.542 3.542 2.969 2.966

l a

¼ 0:6

l a

¼ 0:8

l a

¼ 0:95

6.026 6.022 5.055 5.047 4.223 4.212 3.430 3.416 2.594 2.576

6.129 6.098 4.030 4.000 2.875 2.847 2.078 2.051 1.433 1.405

2.005 1.954 1.066 1.030 0.680 0.651 0.455 0.430 0.297 0.275

l a

l a

l a

Table 8 Values of kIIIa2.5/M for orthotropic bar with an edge crack, bisecting and normal to one side of cross section and h/2a = 1. h 2a

¼1

2

G = 0.2 G2 = 0.5 G2 = 1 G2 = 2 G2 = 5

Study source

l a

Present work Chen et al. (1997) Present work Chen et al. (1997) Present work Chen et al. (1997) Present work Chen et al. (1997) Present work Chen et al. (1997)

0.902 0.924 0.812 0.835 0.757 0.782 0.706 0.734 0.649 0.687

¼ 0:05

l a

¼ 0:2

1.548 1.553 1.421 1.427 1.313 1.320 1.205 1.212 1.087 1.095

l a

¼ 0:4

2.033 2.034 1.771 1.771 1.552 1.551 1.355 1.352 1.166 1.162

¼ 0:6

¼ 0:8

¼ 0:95

2.246 2.241 1.715 1.708 1.385 1.377 1.141 1.131 0.939 0.926

1.602 1.588 1.039 1.025 0.778 0.764 0.614 0.600 0.493 0.476

0.391 0.376 0.227 0.215 0.163 0.151 0.126 0.115 0.100 0.089

l a

l a

l a

Table 9 Values of kIIIa2.5/M for orthotropic bar with an edge crack, bisecting and normal to one side of cross section and h/2a = 2. h 2a

¼2

G2 = 0.2 G2 = 0.5 G2 = 1 G2 = 2 G2 = 5

Study source

l a

Present work Chen et al. (1997) Present work Chen et al. (1997) Present work Chen et al. (1997) Present work Chen et al. (1997) Present work Chen et al. (1997)

0.387 0.399 0.353 0.367 0.331 0.348 0.314 0.337 0.300 0.335

¼ 0:05

l a

¼ 0:2

0.674 0.678 0.602 0.606 0.556 0.560 0.522 0.526 0.494 0.499

4.2. Example 2. An edge crack, non-bisecting and perpendicular to one side of cross section In the second example, we consider an edge crack, nonbisecting and perpendicular to one side of the cross section, Fig. 4. Values of dimensionless torsional rigidities 104D/la4 and dimensionless stress intensity factors of singular crack tip kIIIa2.5/M can be seen in Tables 3 and 4 respectively. 4.3. Example 3. Two edge cracks bisecting two opposite sides of the cross section In the third example, we consider two edge cracks bisecting two opposite sides of the cross section, Fig. 5. Values of dimensionless torsional rigidities 104D/la4 and dimensionless stress intensity factors of singular crack tip kIIIa2.5/M can be seen in Tables 5 and 6 respectively. To verify the validity of formulation for the orthotropic bar, an example solved by Chen et al. (1997) concerning an edge crack bisecting one side of the cross section (Fig. 3), is re-examined and good agreement is observed for the stress intensity factors, Tables 7–9 of the foregoing reference. Adopting the notations of the above-mentioned reference according to this study we provide some tables to compare the results as follows.

l a

¼ 0:4

0.811 0.811 0.677 0.676 0.602 0.600 0.551 0.548 0.511 0.506

¼ 0:6

0.741 0.737 0.570 0.565 0.489 0.483 0.438 0.430 0.400 0.389

¼ 0:8

0.425 0.418 0.307 0.300 0.257 0.249 0.228 0.219 0.207 0.194

¼ 0:95

0.090 0.084 0.063 0.057 0.052 0.047 0.046 0.040 0.042 0.035

The capabilities of procedure are also demonstrated by solving five more examples, as follows. In all examples, we consider the case that a/h = 2. The problem were solved for a orthotropic sector with two different orthotropy ratios, G = 0.8811 and G = 1.1349. 4.4. Example 4. A rotating edge crack emanating from midpoint of one side of cross section In the first non-comparative example we consider a rotating edge crack emanating from midpoint of one side of the cross section, Fig. 6a. The crack length is l/h = 0.1 and the crack orientation angle h is measured with respect to horizontal side of the cross section. Variation of 104D/Gzxa4 with respect to the orientation angle h can be seen in Fig. 6b. As seen from the figure, we observe a slow increasing for torsional rigidity of cracked cross section. Variation of dimensionless stress intensity factor of singular crack tip kIIIa2.5/M with respect to rotation angle h is depicted in Fig. 7. We see an increasing for the dimensionless stress intensity factor of the singular crack tip while the crack is rotated and being close to the right side of section. In the orthotropic sector (G = 0.8811), the weaker material stiffness in the longitudinal direction compared to that of the transversal direction increased the stress intensity factor.

175

A.R. Hassani, R.T. Faal / International Journal of Solids and Structures 52 (2015) 165–179 300 Isotropic, G=1 Orthotropic, G=0.8811 Orthotropic, G=1.1349

290 280

104 D G zx a 4

270 260 250 240 230 220

(a)

210 0.05

305

0.1

0.15

0.2

300

104 D G zx a 4

0.25

0.3

0.35

0.4

Fig. 8. Variation of 104D/Gzxa4 versus dimensionless crack length l/a for an inclined fixed edge crack.

295 Isotropic, G=1 Orthotropic, G=0.8811 Orthotropic, G=1.1349

290

l a

285

6

280

5.5

275

265 260

0

10

20

30

40

50

60

70

80

θ

K I I I a 2.5 M

270

5

4.5

(b) 4

Fig. 6. Variation of 104D/Gzxa4 with respect to orientation angle h.

3.5 0.05

5 Isotropic, G=1 Orthotropic, G=0.8811 Orthotropic, G=1.1349

4.5

Isotropic, G=1 Orthotropic, G=0.8811 Orthotropic, G=1.1349 0.1

0.15

0.2

0.25

0.3

0.35

0.4

l a Fig. 9. Variation of dimensionless stress intensity factor kIIIa2.5/M versus dimensionless crack length l/a for an inclined fixed edge crack.

4

K I I I a 2.5 M

3.5 3 2.5 2 1.5 1

0

10

20

30

40

θ

50

60

70

80

Fig. 7. Variation of dimensionless stress intensity factor kIIIa2.5/M with respect to orientation angle h and l/h = 0.1.

4.5. Example 5. An oblique fixed edge crack, emanating from midpoint of one side of cross section In the previous example, we considered a fixed oblique edge crack, h = tan1(h/a). In this case, plot of 104D/Gzxa4 versus nondimensional crack length l/a is illustrated in Fig. 8. As expected the torsional rigidity of cross section reduced by increasing l/a.

Variation of dimensionless stress intensity factor of singular crack tip kIIIa2.5/M versus non-dimensional crack length l/a was depicted in Fig. 9. The dimensionless stress intensity factor of singular crack tip is firstly increased with crack grows as we expect, but by approaching this tip to the free surface of adjacent side kIIIa2.5/M is reduced. Similar trend of previous example for variations of torsional rigidity and stress intensity factor for two different orthotropy ratios can be realized. In the fracture mechanics, the size of plastic region at a crack tip is a factor of extreme importance. Therefore, we specify the boundary of plastic region around the singular crack tip of this example by means of von Mises yield criterion. Under the assumption of small scale yielding and invoking von Mises yield criterion, at the boundary of plastic zone in anti-plane deformation, the relation 0:75½r2zx ðx; yÞ þ r2zy ðx; yÞ ¼ s2y holds where sy is the shear yield stress of the material. By using the stress field of (14), the boundary of plastic region is assigned which is depicted in Fig. 10 for isotropic as well as orthotropic bar with two different orthotropy ratios. This analysis takes into account both singular and non-singular terms of the stress fields. The advantage of the dislocation distribution technique is considering of all singular and regular terms of

176

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4 0.01

3.5 Crack Line

3

0

K I I I a 2.5 M

y a 3τy 2 ( ) l M

0.005

−0.005

−0.01

2.5

2

1.5 Isotropic, G=1 Orthotropic, G=0.8811 Orthotropic, G=1.1349

−0.015 −0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

Isotropic, G=1 Orthotropic, G=0.8811 Orthotropic, G=1.1349

1

0.03

x a 3 τy 2 ( ) l M

0.5

Fig. 10. Plastic region around the singular tip of an inclined fixed edge crack for M ¼ 0:02. a3 s y

0

0.2

0.4

0.6

0.8

1

Fig. 12. Variation of dimensionless stress intensity factor kIIIa2.5/M versus dimensionless crack length l/a for a circular edge crack.

(a)

(a)

305

305 300

300 Isotropic, G=1 Orthotropic, G=0.8811 Orthotropic, G=1.1349

295

295

Isotropic, G=1 Orthotropic, G=0.8811 Orthotropic, G=1.1349

290

104 D G zx a 4

290

104 D G zx a 4

l a

285

285 280

280

275 275

270

270 265

265

0

0.2

0.4

0.6

0.8

1

260

0.05

0.1

l a

l a

(b)

(b)

Fig. 11. Variation of 104D/Gzxa4 versus dimensionless crack length l/a for a circular edge crack.

the stress field. As we know from fracture mechanics, in anti-plane deformation of the isotropic materials, the plastic zone for only singular term is a circle around the crack tip which in this paper it is not exactly circle because of the nonsingular terms effect.

0.15

0.2

Fig. 13. Values of 104D/Gzxa4 versus non-dimensional crack length l/a for an embedded circular crack.

4.6. Example 6. A circular edge crack, emanating from one side of cross section As of another example, we consider an arc edge crack emanating from the right side of the cross section, in which the crack

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In Fig. 12, variation of dimensionless stress intensity factor of singular crack tip kIIIa2.5/M versus the dimensionless crack length l/a can be observed. By enlarging the crack length, the dimensionless stress intensity factor is noticeable enhanced because the crack is going to eliminate a semicircle portion of cross section.

1.8 1.6 1.4

K I I I a 2.5 M

1.2

4.7. Example 7. An embedded circular crack 1 0.8 0.6 0.4 Isotropic, G=1 Orthotropic, G=0.8811 Orthotropic, G=1.1349

0.2 0

0.05

0.1

l a

0.15

0.2

Fig. 14. Variation of kIIIa2.5/M versus dimensionless crack length l/a for an embedded circular crack.

As an alternative example, we considered a concentric embedded circular crack with crack radius R = 0.15a, Fig. 13a. We suppose the crack length increases equally from its two tips. Values of 104D/Gzxa4 versus non-dimensional crack length l/a where l = R(p  2a) are displayed in Fig. 13b. As expected, the dimensionless torsional rigidities tend to be less as the crack length grows up. Variation of the dimensionless stress intensity factor of crack tips kIIIa2.5/M versus dimensionless crack length l/a is presented graphically in Fig. 14. As it can be seen, increasing the crack length give rise to the dimensionless stress intensity factors of the crack tips. This trend is continued for l/a  0.17 but after that we see a decreasing of dimensionless stress intensity factors while the crack length is increased. 4.8. Example 8. An embedded crack and an edge crack

(a) 300 Isotropic, G=1 Orthotropic, G=0.8811 Orthotropic, G=1.1349

290

270

18

260

16

250

14

240

12

R1 L

1

L

2

K I I I a 2.5 M

104 D G zx a 4

280

In a new example, let us consider bar with a rectangular cross section weakened by a central embedded crack and an edge crack bisecting the right side of cross section. Fig. 15a. The center of embedded crack is in the distance d = 2a/3 from the left side of the cross section. The plot of non-dimensional torsional rigidity 104D/Gzxa4 versus non-dimensional crack length l/a is displayed in Fig. 15b. As expected, the existence of cracks makes the dimensionless torsional rigidities to be less when the crack length rises. Variation of dimensionless stress intensity factors of singular tips of cracks kIIIa2.5/M versus dimensionless crack length l/a are also presented in Fig. 16 for isotropic materials. As can be seen from this figure, stress intensity factors, increases rapidly while the distance between the singular tips of cracks (R1, L2) decreases. The formation of regions with high stress levels is because of the interaction of geometric singularities. For orthotropic material, the graphs of stress intensity factors are slightly different from those for isotropic case that because of space limitation, we didn’t bring it here.

230 220 0.05

0.1

l a

10 8

0.15

(b) Fig. 15. Values of 104D/Gzxa4 versus non-dimensional crack length l/a for an embedded crack and an edge crack.

center coincides to the midpoint of the right side of the cross section, Fig. 11a. The crack radius is considered to be R = 0.05a. Plot of 104D/Gzxa4 versus crack non-dimensional crack length l/a can be seen in Fig. 11b. As we expect, the torsional rigidity of cross section is reduced by increasing l/a which makes a weaker section.

6 4 2 0 0.05

0.1

l a

0.15

Fig. 16. Variation of kIIIa2.5/M versus dimensionless crack length l/a for an embedded crack and an edge crack in the isotropic case.

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The arc edge crack with length l emanating from the right side of the cross section, in which the crack center coincides to the midpoint of the right side of the cross section, Fig. 17a. Centers of embedded cracks with lengths 2l are located on the horizontal center-line of the cross section in the distances 0.2a and 0.4a from the right side of the cross section. Variation of kIIIa2.5/M for all singular tips of cracks versus dimensionless crack length l/a was depicted in Fig. 17b. Rapidly increasing of stress intensity factors, while the distance between the singular tips of cracks i.e. (R1, L2) and (R2, L3) decreases is noticeable. This is because of interaction of approaching crack tips. Graph of values of 104D/Gzxa4 versus non-dimensional crack length 1/a is also plotted in Fig. 18 which shows reduction of dimensionless torsional rigidity of cracked cross section with growing the lengths of cracks as one expected.

(a) 15 R1 L1

5. Concluding remarks

L

Analytical solution to the torsion problem of orthotropic bars weakened by Volterra-type dislocation was obtained in terms of two predefined functions. Consequently, stress field in the cross section under axial torsion was given in a series form, by which the singularity of stress field was also shown. The unknown dislocation density on cracks surfaces became available by solving a set of integral equations of Cauchy singular type. At last, the distributed dislocation technique was used to solve example problems with multiple cracks and smooth geometries. Briefly, results in particular suggested that:

R2 2 3

K I I I a 2.5 M

10

L

5

0 0.07

0.08

0.09

0.1

0.11

l a

0.12

0.13

0.14

0.15

(b) Fig. 17. Variation of kIIIa2.5/M versus dimensionless crack length l/a for two embedded cracks and an edge crack in the orthotropic case, G = 0.8811.

285

1. Dimensionless stress intensity factor of crack tip is increased with crack grows but for an embedded central circular crack there is a critical length which for crack length bigger than it, the dimensionless stress intensity factor is reduced. In other words, we see the crack growth is limited. 2. The plastic region around the crack tip is a circle if only the singular term of the stress field to be considered. Considering the effects of the both singular and non singular terms of the stress field changes the shape of plastic zone and it is no longer a circle. Future work will have to include different geometry of cross section and material, e.g., considering elliptical cross section and bars made of different material types such as FGMs.

280 275 270

Appendix A. Proof of Cauchy singularity for stress field of a dislocation

104 D G zx a 4

265 260

From the stress field (14) we find

255

( 1 2Gzy h bz G X rzy ðX; 0Þ ¼ Gzy hð2X  aÞ þ  ½Em ðX þ n; hÞ a 8h m¼1

250 245

þEm ðX  n; hÞ  Em ðX þ n; hÞ þ Em ðX  n; hÞ

240 235 0.07

þEm ðX þ n; 2g  hÞ  Em ðX  n; 2g  hÞ 0.08

4

0.09

0.1

0.11

l a

0.12

0.13

0.14

0.15

4

Fig. 18. Values of 10 D/Gzxa versus non-dimensional crack length l/a for two embedded cracks and an edge crack in the orthotropic case, G = 0.8811.

4.9. Example 9. Two embedded cracks and an edge crack In last example, we consider an orthotropic bar (G = 0.8811) with rectangular cross section weakened by an edge arc crack and two embedded circular cracks with identical radii R = 0.1a.

þEm ðX þ n; h  2gÞ  Em ðX  n; h  2gÞ )    1 X h cosðnjX Þ þGj nXn cosh Gnj g  2 n¼1

ðA1Þ

  It is easy to show that E1 ðx; hÞ ¼ cot j2x  j2x as x ! 0: Therefore the above stress components rzy on the dislocation cut y = 0 consist of the term

: It means that by approaching to disloEm ðX  n; hÞ ¼ cot jðXnÞ 2 cation point i.e. x ? 0 or X ? n, the stress components rzy on the 2 ¼ j2x. This kind dislocation cut is singular and behaves like jðXnÞ of singularity is called Cauchy-type singularity.

A.R. Hassani, R.T. Faal / International Journal of Solids and Structures 52 (2015) 165–179

Kernels of Integral equations are:

References

kij ðs; tÞ ¼ Pij ðX i ðsÞ; Y i ðsÞÞQ ij ðX j ðtÞ; Y j ðtÞÞ " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi( 1 X 1 h 0 i2 h 0 i2 X j ðtÞ þ Y j ðtÞ GGzy cosui Em ðX i ðsÞ  4a m¼1 þX j ðtÞ; Y i ðsÞ  Y j ðtÞ  hÞ þ Em ðX i ðsÞ  X j ðtÞ; Y i ðsÞ Y j ðtÞ  hÞ  Em ðX i ðsÞ þ X j ðtÞ; ðY i ðsÞ  Y j ðtÞ  hÞÞ þEm ðX i ðsÞ  X j ðtÞ; ðY i ðsÞ  Y j ðtÞ  hÞÞ þ Em ðX i ðsÞ þX j ðtÞ; Y i ðsÞ þ Y i ðtÞ  hÞ  Em ðX i ðsÞ  X j ðtÞ; Y i ðsÞ þY j ðtÞ  hÞ þ Em ðX i ðsÞ þ X j ðtÞ; ðY i ðsÞ þ Y j ðtÞ  hÞÞ # Em ðX i ðsÞ  X j ðtÞ; ðY i ðsÞ þ Y j ðtÞ  hÞÞ " 1 X þ Gzx sinui F m ðX i ðsÞ þ X j ðtÞ; Y i ðsÞ þ Y j ðtÞ  hÞ m¼1   F m X i ðsÞ  X j ðt Þ; Y i ðsÞ þ Y j ðt Þ  h    F m X i ðsÞ þ X j ðt Þ;  Y i ðsÞ þ Y j ðt Þ  h    þF m X i ðsÞ  X j ðt Þ;  Y i ðsÞ þ Y j ðt Þ  h   F m X i ðsÞ þ X j ðt Þ; Y i ðsÞ  Y j ðt Þ  h   þF m X i ðsÞ  X j ðt Þ; Y i ðsÞ  Y j ðt Þ  h    þF m X i ðsÞ þ X j ðt Þ;  Y i ðsÞ  Y j ðt Þ  h #)   Y j ðt Þ F m X i ðsÞ  X j ðtÞ; ðY i ðsÞ  Y j ðtÞ  hÞ

 Y i ðsÞ  hkij ðs; tÞ ¼ Pij ðX i ðsÞ; Y i ðsÞÞQ i;j ðX j ðtÞ; Y j ðtÞÞ " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi( 1 X 1 h 0 i2 h 0 i2 þ Em ðX i ðsÞ X j ðtÞ þ Y j ðtÞ GGzy cosui 4 m¼1 þX j ðtÞ; Y i ðsÞ  Y j ðtÞ þ hÞ  Em ðX i ðsÞ þ X j ðtÞ; Y i ðsÞ Y j ðtÞ þ hÞ þ Em ðX i ðsÞ þ X j ðtÞ; ðY i ðsÞ  Y j ðtÞ þ hÞÞ Em ðX i ðsÞ þ X j ðtÞ; ðY i ðsÞ  Y j ðtÞ þ hÞÞ  Em ðX i ðsÞ þX j ðtÞ; ðY i ðsÞ þ Y j ðtÞ  hÞ þ Em ðX i ðsÞ  X j ðtÞ; Y i ðsÞ þY j ðtÞ þ hÞ  Em ðX i ðsÞ þ X j ðtÞ; ðY i ðsÞ þ Y j ðtÞ  hÞÞ # þEm ðX i ðsÞ  X j ðtÞ; ðY i ðsÞ þ Y j ðtÞ  hÞÞ " 1 X F m ðX i ðsÞ þ X j ðtÞ; Y i ðsÞ  Y j ðtÞ þ hÞ þ Gzx sinui m¼1   F m X i ðsÞ  X j ðt Þ; Y i ðsÞ  Y j ðt Þ þ h  F m ðX i ðsÞ    þX j ðt Þ;  Y i ðsÞ  Y j ðt Þ þ h þ F m X i ðsÞ  X j ðt Þ;     Y i ðsÞ  Y j ðtÞ þ h  F m X i ðsÞ þ X j ðt Þ; Y i ðsÞ    þY j ðt Þ  h þ F m X i ðsÞ  X j ðt Þ; Y i ðsÞ þ Y j ðt Þ  h    þF m X i ðsÞ þ X j ðt Þ;  Y i ðsÞ þ Y j ðt Þ  h  F m ðX i ðsÞ #)   ; 0 6 Y i ðsÞ 6 Y j ðt Þ X j ðt Þ;  Y i ðsÞ þ Y j ðt Þ  h

Q i ðX i ðsÞ; Y i ðsÞÞ ¼ 

ðA2Þ

M Pij ðX i ðsÞ; Y i ðsÞÞ D0

where

Pij ðX i ðsÞ; Y i ðsÞÞ ¼ Gzy cosui ðsÞ½2X i ðsÞ  a #    1 2Gj X h cosðnjX i ðsÞÞ nXn cosh Gnj Y i ðsÞ  þ 2 a n¼1 1 X 2jGzx sinui ðsÞ nXn sinhðGnjðY i ðsÞ a n¼1  h sinðnjX i ðsÞÞ  2   1   16Gzy a2 X   1 njGh sin njX j ðtÞ Q ij X j ðtÞ; Y j ðt Þ ¼ sech 3 3 2 p D0 n¼1;3;... n

þ

 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   h i2 h i2  Gnj  Gnj X 0j ðt Þ þ Y 0j ðt Þ sinh h  Y j ðt Þ sinh Y j ðt Þ 2 2

179

ðA3Þ

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