Torsion analysis of a hollow cylinder with an orthotropic coating weakened by multiple cracks

Torsion analysis of a hollow cylinder with an orthotropic coating weakened by multiple cracks

Mostafa Karimi

^a,b

, Amir Atrian

^a,b

, Aazam Ghassemi

^b,⇑

, Meisam Vahabi

^a,b

aModern manufacturing technologies research center, Najafabad Branch, Islamic Azad University, Najafabad, Iran

bDepartment of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran

a r t i c l e i n f o

Article history:

Received 21 January 2017 Revised 22 February 2017 Accepted 8 March 2017 Available online 10 March 2017

Keywords:

Saint-Venant torsion Hollow cylinder Coating

Stress intensity factor

Distribution dislocation technique Torsional rigidity

a b s t r a c t

This paper presents an analytical solution for an infinite hollow cylinder with an orthotropic coating con- taining multiple cracks under Saint-Venant torsion by means of the distribution dislocation technique.

First, the solution of a hollow cylinder with an orthotropic coating weakened by a Volterra-type screw dislocation is achieved with the aid of the finite Fourier sine transform. Next, the problem is reduced to a set of singular integral equations with a Cauchy type kernel in the hollow cylinder by using distri- bution dislocation technique. The singular integral equations are then solved numerically and the stress intensity factors of the crack tips and torsional rigidity in the whole domain are also obtained. Finally, several examples are presented to show the accuracy and efficiency of the dislocation technique in Saint-Venant torsion problems.

Ó2017 Elsevier Ltd. All rights reserved.

1. Introduction

The study of the torsion of shafts is important in the design of various structures. Due to high rigidity per unit weight and sim- plicity of manufacturing, the hollow cylinders are particularly important. Hollow cylinders are often subjected to torsional load- ing in the process of working. In the study of the fracture mechan- ics of shafts, the structure of the coating seems to play an important role in a problem with multiple cracks. One important challenge in material design is reduction of the stress intensity fac- tor in the cracked bars. To overcome this drawback, an efficient method is to introduce an effective coating layer. Coating is often applied to the surfaces of polymeric, metallic or composite struc- tures. Coating layers are used for many reasons such as protecting, decorating, serving as a barrier, or providing unique surface prop- erties. An appropriate coating can improve efficiency, component durability and fuel economy. In this paper, we use an orthotropic coating layer for reducing the stress intensity factor. The orthotro- pic materials, with properties that differ along three mutually- orthogonal twofold axes of rotational symmetry, are increasingly used as a coating of conventional materials in aerospace engineer- ing as well as automobile and ship vehicles. This can be attributed

to their high strength/density and stiffness/density ratios, high resistance to wear and heat penetration, low coefficient of friction and relatively low cost. On the other hand, the use of orthotropic materials as a coating for isotropic materials may be suggested for structural purposes, such as the reduction of stress intensity factors at the crack tips. Though the torsion problem of a hollow cylinder is a rather old one in the theory of elasticity, the effect of coating structure on stress intensity factors in a hollow cylinder with multiple cracks has not yet been adequately investigated.

The problems of elastic cylindrical shafts under torsional load- ing have been investigated by numerous researchers. In order to review torsion problems, it is convenient to categorize them into two major groups: those primarily dealing with domains without any crack, and those studying shafts containing single or several cracks. Within the first category, a number of researchers have studied torsion problems in the intact bars[1–4].

There are other investigations studying the shafts with single or several cracks, but the shafts with multiple, arbitrarily oriented curved cracks have not been developed sufficiently. The defects in the following papers were assumed to be extended throughout the shaft axis. At first, we review the shafts with circular cross section.

The complete analysis of the torsional rigidity of a solid cylinder with radial cracks was carried out by Lebedev et al.[5]. The authors investigated the problem of the twisting of an elliptical cross section containing two edge cracks extended to its foci.

http://dx.doi.org/10.1016/j.tafmec.2017.03.005 0167-8442/Ó2017 Elsevier Ltd. All rights reserved.

⇑Corresponding author: Tel.: +98 314 229 2883; Fax: +98 314 229 1016.

E-mail addresses: Mk_darani@yahoo.com(M. Karimi), amiratrian@gmail.com (A. Atrian), a_ghassemi@pmc.iaun.ac.ir (A. Ghassemi), m.vahabi@pmc.iaun.ac.ir (M. Vahabi).

Contents lists available atScienceDirect

Theoretical and Applied Fracture Mechanics

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m/ l o c a t e / t a f m e c

Xiao-chun and Ren-ji[6]presented an analytical solution for a solid cylinder weakened by a screw dislocation. The problem was reduced to solve a singular integral equation for the unknown dislocation density with the aid of the dislocation distribution technique. The stress intensity factor and torsional rigidity were calculated by solving the ensuing singular integral equation numerically.

Analysis of a solid cylinder with curvilinear cracks subjected to Saint-Venant torsion was done by Wang and Lu[7]. With the aid of the boundary element method, the authors evaluated boundary integral equations only on the cracks surfaces. Also, the stress intensity factor of the crack tip and torsional rigidity were deter- mined for a straight, kinked or eccentric circular-arc crack.

The analysis of the hollow cylinder with four edge cracks nor- mal to the inner boundary of the cylinder under torsion was the subject of a study done by Chen[8]. The author made use of a method similar to that used in Ref.[7]. The problem was reduced to a Dirichlet problem of the Laplace equation and evaluated with the help of the finite difference method. Finally, the stress intensity factors of the crack tips and the torsional rigidity were calculated numerically.

Tweed and Rooke[9]analyzed the Saint-Venant torsion prob- lem of a circular cross section containing a symmetric array of edge cracks. By symmetry, the problem reduced an integral equation to that of finding the warping functions in some sectors. Finally, the stress intensity factor and crack energy were computed by solving the ensuing integral equation.

Yuanhan [10]studied the problem of a thick-walled cylinder with a radial edge crack under torsional loading. An expansion for the stress function was introduced so that the ensuing stresses at the crack tip possess the inverse square root singularity. The unknown coefficients of the expansion were calculated by the boundary collocation method. At the end, the torsional rigidity of the thick-walled cylinder and the stress intensity factor of the crack tip were achieved.

Chen et al.[11]analyzed a circular cross section bar weakened by a straight edge crack under Saint-Venant torsion with the aid of the dual boundary element method. The authors indicated that the dual boundary element method provided excellent accuracy and simplified the modeling. The dual boundary element method involved modeling only on the boundary without considering the artificial boundary as the multi-zone method. The domain cell was not discretized, since the domain of the integral for the calcu- lation of the torsion rigidity was divided into two boundary inte- grals by means of the Green’s second identity and Gauss theorem.

The problem of an orthotropic bar with circular cross section under Saint-Venant torsion was treated by Hassani and Faal[12].

The solution of a Volterra-type screw dislocation was first obtained with the aid of a finite Fourier cosine transform. Next, the disloca- tion solution was employed to derive a set of Cauchy singular inte- gral equations for analysis of the bar with multiple cracks. The solution to these equations was used to determine the torsional rigidity of the shaft and the stress intensity factors at the crack tips.

Yi-Zhou [13] studied a hollow bar with outer or inner keys under torsion by the harmonic function continuation technique and conformal mapping. The domain under consideration was sep- arated into several sub-regions; then these regions were mapped into some rectangular domains. By using the conformal mapping, the harmonic continuation conditions along the dividing lines were presented. For the convenience of solution, the problem was divided into two Dirichlet problems by defining a new func- tion which was conjugated to the warping function and a constant.

Finally, the torsional rigidity of cross section was calculated.

Fang-ming and Ren-ji[14]addressed the torsion problem of a circular bar weakened by an internal crack reinforced by a ring of rod made of different material of the cylinder. By employing

the Muskhelishvili single-layer potential function solution and the single crack solution for the problem of a cylinder under Saint-Venant torsion, the problem was reduced to a set of mixed- type generalized Cauchy singular integral equations. The ensuring integral equations were a combination of Fredholm integrals of single-layer potential density functions with Cauchy-type singular integrals of dislocation density. Finally, the torsional rigidity and the stress intensity factors were evaluated numerically.

The analysis for the flexure and torsion of cylindrical bars con- taining some edge and embedded cracks was done by Sih[15]. The problem was evaluated based on three complex flexure functions including the classical torsion function. By choosing the appropri- ate complex flexure function, which satisfied the necessary bound- ary conditions on the outer boundary of the cross section and boundary conditions on the crack surfaces, the closed form rela- tions for the stress intensity factors at the crack tips were determined.

Renji and Yulan[16]presented solutions for torsion problems of a circular bar with a rectangular hole and a rectangular bar weak- ened by an embedded crack. The torsion problem of the circular bar weakened by multiple cracks was evaluated by dislocation technique and divided into the two aforementioned problems.

Also, the torsional rigidity and the stress intensity factor of crack tip were found. Then, for the circular bar with a rectangular hole, the relations for the singular stresses around the concave corner points were computed and the generalized stress intensity factors were achieved.

Li et al.[17]addressed the problem of a circular bar containing a polygonal opening and an embedded crack. The formulation was founded on degenerating a system of the connecting line cracks that extended them making the polygonal opening. In the follow- ing, singular integral equations were found to model the torsion problem of a circular cylinder with a polygonal opening and an embedded crack. The stress intensity factors of the crack tips were derived to illustrate the effect of the crack size and its position. The singular behavior of the stresses near a rectangular corner was, showing to be different from that of the crack tip.

Jiang and Henshall[18]presented a finite element model for the torsional analysis of prismatic bars by considering only a slice of the cross section. A set of coupling equations independent of the situation of the axis of rotation were formulated. The conventional three dimensional solid brick elements were used for the structural discretization. The non-linear material behavior and the large deformation effects could also be included by using an incremental loading process. Also, the authors presented the analyses for an orthotropic elastic square cross section bar; also, an elastoplastic circular cross section weakened by a radial crack was employed for the geometrically nonlinear deformation of a thin-walled I- section beam.

The torsion problem of a bar containing a cracked ring section was the subject of a study done by Chen and Chen[19]. The cross section of the bar weakened by three edge cracks emanating from the outer boundary of the bar and all cracks were considered equally spaced. The problem was analyzed in the rectangular region. The solution was divided by following three steps: (1) the partitioning plan and conformal mapping technique, (2) the har- monic function continuation technique, and (3) the compliance method. At the end, the torsional rigidity and the stress intensity factor at the crack tip were computed.

This part of the review is related to the torsion problems of bars with rectangular cross section. We begin with a study done by Chen[20], who analyzed a rectangular bar with one or two edge cracks perpendicular to the cross section sides. The ensuring Dirichlet problem of the Laplace equation was solved by dividing the cross section into several rectangular sub-regions and using the Duhem theorem. The torsional rigidity and also, the

M. Karimi et al. / Theoretical and Applied Fracture Mechanics 90 (2017) 110–121 111

compliance coefficient were calculated and finally, the stress inten- sity factors of crack tips were obtained. It should be noted that the solution was restricted to the edge cracks, which were perpendic- ular to the boundaries of the cross section.

Chen et al. [21] presented a solution for the problem of an orthotropic rectangular cross section with an edge crack bisecting and perpendicular to one boundary of the cross section. The prob- lem was reduced to the Laplace equation by mapping a rectangular plane with cut to another one. The compliance coefficient as an inverse of torsional rigidity was computed by solving the Laplace equation. The energy release rate as a function of stress intensity factor was obtained in terms of the applied moment and the com- pliance coefficient. After finding energy release, a formula for the stress intensity factor at the crack tip was presented.

Recently, Hassani and Faal[22]focused on study an orthotropic bar with rectangular cross section with the aid of the distribution dislocation technique. The bar was under Saint-Venant torsion.

The solution of displacement and stress field was achieved by means of a finite Fourier transform in terms of the dislocation den- sity. The problem was reduced to a set of Cauchy singular integral equations for the analysis of multiple cracks. The stress intensity factors and torsional rigidity were presented by solving the ensu- ing singular integral equations numerically.

According to the above review, the fracture problem of the shafts under torsion is an interesting problem. It is worth noting that all of the above works were limited to the shafts with a par- ticular orientation and geometry. Also, no work has been pub- lished concerning the effect of the coating on the stress intensity factor of the crack tips in the hollow cylinder subjected to tor- sional loading. Also, to authors’ knowledge, no analytical solution has been presented on the Saint-Venant torsion of a hollow cylin- der with multiple cracks by considering the effect of the coating.

In this paper, the closed form solution of the stress fields and warping functions are achieved for a hollow cylinder with an orthotropic coating containing a Volterra-type screw dislocation (Section2.1). The torsional rigidity of the cracked shaft with coat- ing is evaluated in terms of the dislocation density (Section2.2).

The problem is reduced to the solution of a Cauchy singular inte- gral equation (Section3). The numerical examples, as presented in Section 4, and results are validated by employing the available results from the literature. Finally, Section5offers the concluding remarks.

2. General formulation 2.1. Dislocation solution

Consider a prismatic hollow cylinder with an orthotropic coat- ing as shown inFig. 1.R1andR2refer to the inner and outer radius of the cylinder, respectively and thickness of the coating is assumed to beR₃R₂. By considering cylindrical coordinate sys- tem, it is assumed that the origin of cylindrical coordinate is located at Oand z-axis is coincided with the axis of the hollow cylinder. The coating is made of an orthotropic material, where GrzandG_hzare the shear moduli inrzandhzplanes. A Volterra type screw dislocation having the Burgers vectorbz is located at r¼a; with the line of dislocations in the radial direction (h¼0, a6r6R2). We divide the whole domain into three regions:

R16r<a,a6r6R2andR26r6R3.

When the shaft is subjected to Saint-Venant torsional loading, components of displacement in the directions ofx;y and z axes denoted asu;

v

^andw, respectively, which are given in terms of the angle of twist per unit length of the bar

a

; also the warping function

u

^ðx;yÞis presented as[23]

u¼

a

^zy

v

^¼

a

^zx

w¼

au

^ðx;yÞ

ð1Þ

It is convenient to treat this problem in the cylindrical coordi- nate system; therefore, the cylindrical transformation is applied to Eqs.(1). Thus, we have

ur¼0 u_h¼

a

^rz

w¼

au

^ðr;hÞ

ð2Þ

The non-vanishing stress components in terms of warping func- tion can be expressed as

s

^rz¼

la

^@

u

^ðr;hÞ

@r ; R1<r<R2

s

hz¼

la

¹_r^@

u

ðr;hÞ

@h þr

; R1<r<R2

s

rz¼Grz

a

^@

u

ðr;hÞ

@r ; R2<r<R3

s

hz¼G_hz

a

¹_r^@

u

ðr;hÞ

@h þr

; R2<r<R3

ð3Þ

where

l

denotes shear modulus in the bar. These stress compo- nents must satisfy the equilibrium equation^@_@r^srzþ¹_r^@_@h^shzþ¹_r

s

^rz¼0.

We obtain the governing equation of the coating as follows

r²@²

u

^ðr;hÞ

@r² þr@

u

^ðr;hÞ

@r þG²@²

u

^ðr;hÞ

@h² ¼0 ð4Þ

whereG¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G_hz=Grz

p is defined as the orthotropic ratio in the coat- ing. The governing equation for the hollow cylinder can be easily derived by settingG¼1 in Eq.(4). The above partial differential equation is solved by means of the finite Fourier sine transform for a regular functionfðr;hÞas

Fsðr;nÞ ¼ Z _p

0

fðr;hÞsinðnhÞdh ð5Þ

The inverse of the finite Fourier sin transform is expressed as

fðr;hÞ ¼2

p

X¹

n¼1

Fsðr;nÞsinðnhÞ ð6Þ

a

Fig. 1.Cross section of a hollow cylinder with an orthotropic coating weakened by a screw dislocation.

It should be mentioned that the hollow cylinder and its coating are twisted by an applied momentMand then the dislocation cut is made in the cross section of the hollow cylinder.

The boundary condition representing a Volterra-type screw dis- location is

u

ðr;0^þÞ

u

ðr;0Þ ¼bz

a

^{½HðraÞ HðrR2Þ ð7Þ}

whereHðÞis the Heaviside step function. The continuity of stress components along the dislocation cut requires that

@

u

ðr;0^þÞ

@h ¼@

u

ðr;0Þ

@h ð8Þ

The problem is anti-symmetric with respect to the diameter of the cross section containing the dislocation line; therefore we consider a dislocation solution for the region 06h6

p

^{and the}

boundary conditions(7) and (8)are expressed by

u

^{ðr;0Þ ¼}₂^b

a

^{z½HðraÞ HðrR2Þ}

u

ðr;

p

^{Þ ¼0 ð9Þ}

By using the integral transform(6), the partial differential equation (4)can be reduced to the form

r²@²U^sðr;nÞ

@r² þr@U^sðr;nÞ

@r n²U^sðr;nÞ

¼ bzn

2

a

^{½HðraÞ HðrR2Þ ð10Þ}

The general solution to Eq.(10)can be expressed as U^sðr;nÞ ¼A1nrⁿþB1nrⁿ forR16r6a

U^sðr;nÞ ¼A2nrⁿþB2nrⁿþ_2n^bz_a fora6r6R2

U^sðr;nÞ ¼A3nr^GnþB3nr^Gn forR26r6R3

ð11Þ

According to Eq.(6), the warping function in the whole domain is written as

u

ðr;hÞ ¼_p²X¹

n¼1

ðA1nrⁿþB1nrⁿÞsinðnhÞ forR16r6a

u

^{ðr;hÞ ¼}_p^2X1

n¼1

ðA2nrⁿþB2nrⁿþ_2n^bz_aÞsinðnhÞ fora6r6R2

u

^{ðr;hÞ ¼}_p^2X1

n¼1

ðA3nr^GnþB3nr^GnÞsinðnhÞ forR26r6R3

ð12Þ Upon substituting the above relations into Eq. (6), the warping functions in the whole domain are obtained. Also, the stresses are obtained from Eq.(3)as

s

hzðr;hÞ ¼2

a l p

^r

X¹

n¼1

nðA1nrⁿþB1nrⁿÞcosðnhÞ þr

a l

; R16r6a

s

hzðr;hÞ ¼ b_z

l

2

p

^rþ2

a l p

^r

X¹

n¼1

nðA2nrⁿþB2nrⁿÞcosðnhÞ þr

a l

; a6r6R2

s

hzðr;hÞ ¼2

a

^Ghz

p

^r

X¹

n¼1

nðA3nrⁿþB3nrⁿÞcosðnhÞ þr

a l

; R26r6R3

s

^rzðr;hÞ ¼2

a l p

^r

X¹

n¼1

nðA1nrⁿB1nrⁿÞsinðnhÞ; R16r6a

s

^rzðr;hÞ ¼2

a l p

^r

X¹

n¼1

nðA2nrⁿB2nrⁿÞsinðnhÞ; a6r6R2

s

^rzðr;hÞ ¼2

a

^Grz

p

^r

X¹

n¼1

nðA3nrⁿB3nrⁿÞsinðnhÞ; R26r6R3

ð13Þ

whereAln;Bln;l¼1;2;3 are unknown coefficients which are deter- mined by following boundary and continuity conditions

s

^rzðR1;hÞ ¼0

s

^rzða;hÞ ¼

s

^rzðaþ;hÞ

u

^ða;hÞ ¼

u

^ðaþ;hÞ

u

^ðR2;hÞ ¼

u

^ðRþ2;hÞ

s

^rzðR2;hÞ ¼

s

^rzðRþ2;hÞ

s

rzðR3;hÞ ¼0

ð14Þ

Application of the conditions(14)to Eqs.(13) and (3)leads to A1nRⁿ₁B1nRⁿ₁ ¼0

A1naⁿB1naⁿ¼A2naⁿB2naⁿ A1naⁿþB1naⁿ¼A2naⁿþB2naⁿþ bz

2n

a

A2nRⁿ₂þB2nRⁿ₂ þ bz

2n

a

^{¼A3nRGn2 þB3nRGn2}

l

^ðA2nRn2B2nRⁿ₂ Þ ¼GrzGðA3nR^Gn₂ B3nR^Gn₂ Þ A3nR^Gn₃ B3nR^Gn₃ ¼0

ð15Þ

The solution of Eqs.(15)gives

A1n¼ bz

4n

a

^{Cn a}

j

²³

R²₂

!n

þ ðCeq1Þðð

j

²³=R2ÞⁿRⁿ₂ Þ

"

þCeqðða=R²2Þⁿ ð

j

²³=aÞⁿÞ aⁿi B1n¼ bz

4n

a

^Cnh^{ðCeq1ÞððR21}

j

²³=R2Þⁿ ðR²₁=R2ÞⁿÞ þCeqðða

j

^{12Þn ðR2}1

j

^{23=aÞnÞ þ ða}

j

¹²

j

^{23Þn ðR2}1=aÞⁿi A2n¼ bz

4n

a

^Cnh^ða

j

^23=R22Þⁿ ð

j

¹²

j

^{23=aÞnþCeqðða=R2}2Þⁿ ð

j

^{12=aÞnÞ þ ð1CeqÞðRn}2 ð

j

^23=R2ÞnÞ

B2n¼ bz

4n

a

^Cnh^ða

j

¹²

j

^{23Þnþ ðCeq1ÞððR2}1

j

²³=R2Þⁿ

ðR²₁=R2ÞⁿÞ ðR²₁=aÞⁿþCeqðða

j

^{12Þn ðR2}1

j

^23=aÞnÞi_4n^bz

a

^an

A3n¼ bzCn

a

^{nð1CeqÞ ðaRG12 Þ}

nþ ð

j

^12RG2ÞⁿR^Gn₂ ðR^G1₂ R²₁=aÞⁿ

B3n¼ bzCⁿ

a

^{nð1CeqÞ ðaRG12 R2G3 Þ}

nþ ð

j

^12RG2R^2G₃Þⁿ ðR^G₂ R^2G₃Þⁿ

ðR^G1₂ R²₁R^2G₃=aÞⁿ

ð16Þ in which

Ceq¼

l

^GGrz

l

^þGGrz

C^n¼₁

j

ⁿ12

j

ⁿ23þ¹Ceqð

j

ⁿ23

j

ⁿ12Þ

j

^{12¼ R}_R¹

2 2

j

^{23¼ R}_R²

3 2G

ð17Þ

Thus, the warping functions are determined and the stress fields can be obtained from Eq.(3). It is noteworthy that we do not need the stress field in region 3 because the cracks are located in the hol- low cylinder, not in the coating.

M. Karimi et al. / Theoretical and Applied Fracture Mechanics 90 (2017) 110–121 113

s^rz¼bzl

2p^r

X¹

n¼1

Cnfðraj²³=R²2Þⁿþ ðr=aÞⁿþ ðaj¹²j²³=rÞⁿ ðR²₁=ðraÞÞⁿ

þCeqððrj23=aÞⁿ ðra=R²2Þⁿþ ðaj12=rÞⁿ ðR²₁j23=ðraÞÞⁿÞ þ ðCeq1ÞððR²₁j²³=ðrR2ÞÞⁿ ðR²₁=ðrR2ÞÞⁿ ðrj²³=R2Þⁿ ðr=R2ÞⁿÞgsinðnhÞ R16r6a

shz¼ bzl

2pr X¹

n¼1

Cnfðraj23=R²₂Þⁿþ ðaj12j23=rÞⁿ ðr=aÞⁿ ðR²1=raÞⁿ

þ ðCeq1Þððrj²³=R2Þⁿ ðr=R2Þⁿþ ðR²₁j²³=ðrR2ÞÞⁿ ðR²₁=rR2ÞⁿÞ þCeqððaj12=rÞⁿ ðR²1j23=ðraÞÞⁿþ ðra=R²₂Þⁿ ðrj23=aÞⁿÞgcosðnhÞ þral ^R16r6a

s^rz¼bzl

2p^r

X¹

n¼1

Cnfðrj¹²j²³=aÞⁿ ðraj²³=R²2Þⁿþ ðaj¹²j²³=rÞⁿ ðR²₁=raÞⁿ

þ ðCeq1ÞððR²1j23=rR2Þⁿ ðR²1=rR2Þⁿþ ðr=R2Þⁿ ðrj23=R2ÞⁿÞ þ ða=rÞⁿ=Cn

þCeqððaj¹²=rÞⁿ ðra=R²2Þⁿþ ðrj¹²=aÞⁿ ðR²₁j²³=ðraÞÞⁿÞgsinðnhÞ a6r6R2

shz¼ bzl

2p^rbzl

2p^r

X¹

n¼1

Cnfðraj23=R²₂Þⁿ ðrj12j23=aÞⁿþ ðaj12j23=rÞⁿ ðR²1=raÞⁿ

þ ðCeq1ÞððR²₁j²³=ðrR2ÞÞⁿþ ðrj²³=R2Þⁿ ðr=R2Þⁿ ðR²₁=ðrR2ÞÞⁿÞ þ ða=rÞⁿ=Cn

þCeqððra=R²₂Þⁿ ðrj12=aÞⁿþ ðR²1j23=ðraÞÞⁿþ ðaj12=rÞⁿÞgcosðnhÞ þral ^a6r6R2

ð18Þ With the aid of following expansion of theCn

C^n¼X1

m¼0

X^m

i¼0

X¹

j¼0

C^m_eqð1Þ^iþmC^ðjþmþ1Þ

C^ðiþ1ÞC^ðjþ1ÞC^{ðiþmþ1Þð}

j

^miþj23

j

^jþi12Þⁿ ð19Þ the stress components(18)can be summed over the whole domain, leading to

srzðr;hÞ ¼bzl 4p^r

X¹

m¼0

X^m

i¼0

X¹

j¼0

K^mij w^mij

r a;h

w^mij

raj²³ R²₂ ;h

! þw^mij

aj¹²j¹³ r ;h

"

w^mij

R²₁ ra;h

! þCeq w^mij

rj23

a ;h

w^mij

ra R²₂;h

! þw^mij

aj12

r ;h

w^mij

R²₁j23

ra ;h

!!

þðCeq1Þ w^mij

R²₁j23

rR2 ;h

! w^mij

R²₁ rR2;h

! þw^mij

r R2;h

w^mij

R²₁j23

ra ;h

! !#

; R1<r<a

srzðr;hÞ ¼bzl 4p^r

X¹

m¼0

X^m

i¼0

X¹

j¼0

K^mij w^mij

rj12j23

a ;h

w^mij

raj23

R²₂ ;h

! þw^mij

aj12j13

r ;h

"

w^mij

R²₁ ra;h

! þCeq w^mij

rj¹² a ;h

w^mij

ra R²₂;h

! þw^mij

aj¹² r ;h

w^mij

R²₁j²³ ra ;h

!

þðCeq1Þ w^mij

R²₁j23

rR2 ;h

! w^mij

R²₁ rR2;h

! þw^mij

r R2;h

w^mij

rj23

R2;h

!#

þbzl 4p^{rw000 ar;h}

; a<r<R2

shzðr;hÞ ¼bzl 4p^r

X¹

m¼0

X^m

i¼0

X¹

j¼0

K^mij /^mij

raj²³ R²₂ ;h

! /^mij

r a;h

þ/^mij

aj¹²j¹³ r ;h

"

/^mij

R²₁ ra;h

! þCeq /^mij

ra R²₂;h

! /^mij

rj23

a ;h

þ/^mij

aj12

r ;h

/^mij

R²₁j23

ra ;h

!!

þðCeq1Þ /^mij

R²₁j²³ rR2 ;h

! /^mij

R²₁ rR2;h

! þ/^mij

rj²³ R2 ;h

/^mij

r R2;h

!#

þral; R1<r<a

shzðr;hÞ ¼ bzl 4p^rþbzl

4p^r X¹

m¼0

X^m

i¼0

X¹

j¼0

K^mij /^mij

raj23

R²₂ ;h

! /^mij

rj12j23

a ;h

"

/^mij

R²₁ ra;h

! þ/^mij

aj¹²j²³ r ;h

þCeq /^mij

ra R²₂;h

! /^mij

rj¹² a ;h

þ/^mij

aj¹² r ;h

/^mij

R²₁j23

ra ;h !!

þ ðCeq1Þ /^mij

R²₁j23

rR2 ;h

! /^mij

R²₁ rR2;h

! þ/^mij

rj23

R2 ;h

/^mij

r R2;h

þbzl 4p^r/000

a r;h

þral; a<r<R2

ð20Þ

whereK^mij ¼ C^m_eqð1Þ^iþmC^ðjþmþ1Þ C^ðiþ1ÞC^ðjþ1ÞC^{ðiþmþ1Þand} w^mijðx;hÞ ¼ sinðhÞ

coshðlnðx

j

^miþj23

j

^jþi12ÞÞ cosðhÞ /^mijðx;hÞ ¼ sinhðlnðx

j

^miþj23

j

^jþi12ÞÞ

coshðlnðx

j

^miþj23

j

^jþi12ÞÞ cosðhÞ

ð21Þ

It is obvious that the warping function and stress fields satisfy the equilibrium equation and all of the specified boundary and continuity conditions. It should be noted that we rewrote stress fields(18)in the form of Eq.(20), since stress fields are converged quickly by introducing the functionsw^mijðx;hÞand/^mijðx;hÞ. In a spe- cial case, the hollow cylinder with an orthotropic coating can be simplified as a solid shaft without coating by settingR1¼0 and R2¼R3. The functionsw^mijðx;hÞ and /^mijðx;hÞ are vanished for all i;j;m–0. In this case, Eq.(20)is reduced to

s

rzðr;hÞ ¼bz

l

4

p

^{r w000 ar;h}

w⁰00

ra R²₂;h

!

" #

; R16r6a

s

hzðr;hÞ ¼bz

l

4

p

^{r /000 ra}R²₂;h

! /⁰00

r a;h

" #

þ

a

^Ghzr; R16r6a

s

^rzðr;hÞ ¼bz

l

4

p

^{r w000 ar;h}

w⁰00

ra R²₂;h

!

" #

; R16r6R2

s

hzðr;hÞ ¼bz

l

4

p

^{r /000 ra}R²₂;h

! þ/⁰00

a r;h

" #

þ

a

^Ghzr; R16r6R2

ð22Þ which is exactly the same as that given by Hassani and Faal[12], proving the correctness of the derivation of the stress fields in Eq.

(20). Also, for a homogenous hollow cylinder (Ceq¼0;G¼1), the functionsw^mijðx;hÞand/^mijðx;hÞare vanished for alli;j–0. Therefore, in this special case, the stress field(20)can be simplified, such that only the functionsw^m00ðx;hÞand/^m00ðx;hÞremain nonzero.

Referring toFig 1, we define a local coordinate systemðr⁰;h⁰Þto investigate the behavior of the stress singularity. The local coordi- nateðr⁰;h⁰Þis related to global coordinateðr;hÞthrough relations

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a²þ ðr⁰Þ²þ2ar⁰cosh⁰ q

h¼sin¹ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir⁰sinh⁰ a²þ ðr⁰Þ²þ2ar⁰cosh⁰ q

0 B@

1

CA; 06h⁰62

p

^ð23Þ

Substituting Eq.(23)into Eq.(20)and employing some manip- ulations will yield the following results

w^mijðr=a;hÞ 1

r⁰ asr⁰!0 /^mijðr=a;hÞ 1

r⁰ asr⁰!0

ð24Þ

The functionsw^mijðx;hÞand/^mijðx;hÞhave a Cauchy type singular- ity at the dislocation location. Therefore, stress components(20) are Cauchy singular at this point. It should be noted that the Cau- chy type singularity has been reported by Weertman and Weert- man [24] for the two-dimensional body media weakened by a Volterra-type screw dislocation.

2.2. Calculation of torsional rigidity

In this section, torsional rigidity for a hollow cylinder with an orthotropic coating weakened by a Volterra-type screw dislocation

has been calculated. The torsional rigidity, in cylindrical coordi- nate, is determined by the following form[25]

M¼D

a

^{¼Z 2p}

0

Z R2 R1

r²

s

hzðr;hÞdrdh ð25Þ

in which

s

hzðr;hÞrefers to the stress components given in Eq.(20) andMis the twisting moment in the entire domain. Substituting Eq.(20)into Eq(25), the torsional rigidity can be written as

D¼D0bz

2

a l

^{ðR22a2Þ ð26Þ}

in which D0¼

p

2

a

^ð

l

^ðR42R⁴₁Þ þG_hzðR⁴₃R⁴₂ÞÞ ð27Þ It is clear from the aforementioned equation that the torsional rigidity is dependent onbz; andD0 denotes torsional rigidity in the intact hollow cylinder with the orthotropic coating.

3. Analyses with multiple cracks

At this stage, we define a local coordinate systemðn;tÞattached to the surface of thei-th crack (seeFig. 2). The local stress fields on the crack surfaces can be expressed by the following relations

s

tzðri;hiÞ ¼

s

hzðri;hiÞsin

u

iþ

s

rzðri;hiÞcos

u

i

s

nzðri;hiÞ ¼

s

hzðri;hiÞcos

u

i

s

rzðri;hiÞsin

u

i

ð28Þ Note that

u

iis the angle between the tangent to thei-th crack and radial direction. As an application of the derived dislocation solu- tion, we analyze the problem under consideration as weakened by multiple cracks. Let dislocations with unknown densities to be dis- tributed on the surface of thei-th crack. By substituting Eq.(20)into Eq.(28), stress components at a point with the coordinateðri;hiÞ could be achieved. Since the dislocation cut is situated ath¼0,hi

is replaced byhihjin the local stress fields.

s^nzðri;hÞ ¼bzl

4p^ri cosð/iÞX¹

m¼0

X^m

i¼0

X¹

j¼0

K^mij /^mij

rirjj23

R²₂ ;hihj

! /^mij

ri

rj;hihj

"

(

þ/^mij

rjj12j13

ri ;hihj

/^mij

R²₁ rirj;hihj

! þCeq /^mij

rirj

R²₂;hihj

!

/^mij

rij23

rj ;hihj

þ/^mij

rjj12

ri ;hihj

/^mij

R²₁j23

ria ;hihj

!

þðCeq1Þ /^mij

R²₁j23

riR2;hihj

! /^mij

R²₁ riR2;hihj

! þ/^mij

r_ij23

R2 ;hihj

/^mij

ri

R2;hihj

sinð/iÞX¹

m¼0

X^m

i¼0

X¹

j¼0

K^mij w^mij

ri

rj;hihj

w^mij

rirjj²³ R²₂ ;hihj

!

"

þw^mij

aj¹²j¹³ ri ;hihj

w^mij

R²₁ rirj;hihj

! þCeq w^mij

rij²³ rj ;hihj

w^mij

rirj

R²₂;hihj

! þw^mij

rjj12

ri ;hihj

w^mij

R²₁j23

ria ;hihj

!!

þðCeq1Þ w^mij

R²₁j23

riR2;hihj

! w^mij

R²₁ riR2;hihj

! þw^mij

ri

R2;hihj

w^mij

R²₁j23

rirj ;hihj

!!#)

;R1<ri<rj

snzðri;hiÞ ¼b_zl 4p^ri

cosð/iÞX¹

m¼0

X^m

i¼0

X¹

j¼0

K^mij /^mij

r_ir_jj23

R²₂ ;hihj

! /^mij

R²₁ r_ir_j;hihj

!

"

(

þ/^mij

rjj¹²j²³ ri ;hihj

/^mij

rij¹²j²³ rj ;hihj

þCeq /^mij

rirj

R²₂;hihj

!

/^mij

rij¹² rj ;hihj

þ/^mij

rjj¹² ri ;hihj

/^mij

R²₁j²³ rirj ;hihj

!!

þðCeq1Þ /^mij

R²₁j²³ riR2;hihj

! /^mij

R²₁ riR2;hihj

! þ/^mij

rij²³ R2 ;hihj

/^mij

ri

R2;hihj

þ /⁰00

rj

ri;hihj

1

cosð/iÞ

sinð/iÞX¹

m¼0

X^m

i¼0

X¹

j¼0

K^mij