Mixed mode axisymmetric cracks in transversely isotropic infinite solid cylinders

Accepted Manuscript

Mixed mode axisymmetric cracks in transversely isotropic infinite solid cylinders M. Pourseifi , R.T. Faal PII: DOI: Reference:

S0307-904X(17)30307-4 10.1016/j.apm.2017.04.035 APM 11743

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

1 December 2015 17 April 2017 19 April 2017

Please cite this article as: M. Pourseifi , R.T. Faal , Mixed mode axisymmetric cracks in transversely isotropic infinite solid cylinders, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.04.035

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ACCEPTED MANUSCRIPT Highlights New analytical dislocation solution in a transversely isotropic cylinder is developed.



Material anisotropy has an efficient effect on stress intensity factors.

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Interaction of cracks implies that need for mixed-mode analysis is essential.



Loading type and crack length have also a crucial effect on stress intensity factors.

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Mixed mode axisymmetric cracks in transversely isotropic infinite solid cylinders M. Pourseifi, R. T. Faal* Faculty of Engineering, University of Zanjan, P. O. Box 45195-313, Zanjan, Iran Abstract

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The present study examined mixed mode cracking in a transversely isotropic infinite cylinder. The solutions to axisymmetric Volterra climb and glide dislocations in an infinite circular cylinder of the transversely isotropic material are first obtained. The solutions are represented in terms of the biharmonic stress function. Next, the problem of a transversely isotropic infinite cylinder with a set of concentric axisymmetric penny-shaped, annular, and circumferential cracks is formulated using the distributed dislocation technique. Two types of loadings are considered: (i) the lateral cylinder is loaded by two self-equilibrating distributed shear stresses; (ii) the curved surface of the cylinder is under the action of a distributed normal stress. The resulting integral

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equations are solved by using a numerical scheme to compute the dislocation density on the borders of the cracks. The dislocation densities are employed to determine stress intensity factors for axisymmetric interacting cracks. Finally, a good amount of examples are solved to depict the effect of crack type and location on the stress intensity factors at crack tips and interaction between cracks. Numerical solutions for practical materials are presented and the effect of transverse isotropy on stress intensity factors is discussed.

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Keywords Infinite cylinder, Coaxial axisymmetric cracks, Edge dislocation, Cauchy singularity, Biharmonic Galerkin vector.

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1. Introduction One popular method used to obtain the elastic modulus and tensile strength of materials is uniaxial tension testing of cylindrical specimens. The uniaxial compression of such specimens is also of same importance. The first part of this review is allocated to an intact finite cylinder of transversely isotropic elastic material subjected to axial loading. The review is limited to two sample studies to preserve space. Non-uniform displacement and stress analysis of a finite cylinder composed of transversely isotropic elastic material has been carried out by Wei and Chau [1]. They applied compression loading to two ends of a cylinder using loading platens and the friction between the end surfaces of the cylinder and the platens was assessed. They also proposed the Lekhnitskii’s stress function in the form of a series which satisfies the governing equation and all end and lateral boundary conditions of the cylinder. Using a similar method, they resolved the problem of a finite cylinder of transversely isotropic material subjected to two axial point loads applied along the cylinder axis [2].

*

Corresponding author. Tel.: +98 24 3305 2600; Fax: +98 24 3228 3204 E-mail addresses: [email protected] (R.T. Faal), [email protected] (M.Pourseifi)

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Analysis of axisymmetric cracks in a cylinder is another important problem and studies on both isotropic and transversely isotropic cylinders are available. This review focuses on the solid cylinders of the transversely isotropic materials. Efforts have focused on other geometries of the transversely isotropic materials, such as infinite spaces and layers. In these investigations, the stress intensity factors at the crack tip, an important parameter of the fracture mechanics, were evaluated and the effects of anisotropy of the material constants on this parameter were studied. Dahan [3] analyzed an infinite solid of transversely isotropic material with a penny-shaped crack subjected to body forces parallel to the axis and perpendicular to the crack surface. A potential function of the Love type in integral form was introduced and the axisymmetric problem was reduced to an Abel integral equation by applying boundary conditions. Noda and Ashida [4] studied a similar solid weakened by an annular crack under heat absorption and exchange on the crack surface. By discretizing the time variable, the temperature field was calculated using the finite-difference method. The potential functions method was used to reduce the crack problem to a dual-integral equation. Numerically solving these equations obtained the transient thermal stress field and stress intensity factor of the crack tip. Danyluk et al. [5] studied a thick transversely isotropic layer weakened by a penny-shaped crack subjected to radial shear. The non-zero displacement and stress components were given in integral form, including the Bessel function. The Dugdale hypothesis and the Hankel transform theory were employed to study the effect of transverse isotropy on the size of the plastic zone. The problem was numerically solved by providing a Fredholm integral equation of the second kind. A similarly thick layer with an annular crack under uniform loading was investigated by Ataberk et al. [6]. The crack was located at the middle plane of the layer. The governing equation of the problem was derived in terms of a Love type stress function. Using the Hankel transform, the problem was reduced to a singular integral equation which was solved numerically using Gaussian quadrature. Li et al. [7] used Dugdale’s crack model to predict the size of the plastic zone of a pennyshaped crack tip embedded in a transversely isotropic medium. The axisymmetric crack was parallel to the isotropic plane of the medium and subjected to three types of loading. The first type was annular pressure applied to the upper and lower crack surfaces in the opposite directions. The second type was concentrated force applied to the upper and lower crack surfaces in the opposite directions. The third type was concentrated force applied at two symmetric opposing points outside of the crack on the axis normal to the crack surface passing from the crack center. The problem was analyzed based on potential theory and crack surface displacement was determined explicitly. Zhang [8] addressed the problem of a concentric penny-shaped crack at the interface between two transversely isotropic media. The two governing equations of the problem were differential equations exhibiting nonzero axial and radial displacements independent of the angle. By applying the Hankel transform, the axial and radial displacements were provided in the integral form and unknown coefficients which were determined using the boundary conditions. The problem was examined for normal and shear stress on the crack surfaces. Saxena and Dhaliwal [9] addressed the axisymmetric problem of an infinite fiber made of transversely isotropic elastic material which is perfectly bonded to a transversely isotropic elastic matrix of a different material. The fiber was surrounded by an annular crack subjected 3

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to a prescribed longitudinal tension. The solution to the displacement-based governing equations of the problem was given in terms of the integral form harmonic potential functions. The application of boundary conditions led to a singular integral equation. The use of Chebyshev polynomials transformed this integral equation into a system of algebraic equations which led to a stress intensity factor of the crack tips. Satapathy and Rourkela [10] analyzed the problem of an infinite cylinder engulfed by an infinite medium of different materials with a concentric penny-shaped crack under constant internal pressure. Both the materials were homogeneous and transversely isotropic. The displacements and stress components were introduced via two harmonic functions. The integral transform was used to reduce the problem to dual integral equations which were then converted to a Fredholm integral equation of the second kind. The ensuing integral equation was solved using the Gaussian quadrature formula. The strain-energy required to open the crack was also evaluated. Stress analysis of transversely isotropic cylinders weakened by cracks will be dealt with in the following part of the review. The stress intensity factor of the crack tip is generally evaluated. The problem of a penny-shaped crack in a transversely isotropic infinite cylinder with two boundary conditions was treated by Parhi and Atsumi [11]. The boundary conditions arose from a cylinder with a smooth rigid bore and a cylinder with a lateral stressfree surface. The problem progressed by stating the displacement and stress fields as integral forms of harmonic functions, which satisfied equilibrium equations. Applying the boundary conditions led to dual integral equations which were reduced to a Fredholm equation of the second kind, which was solved numerically. Quantities such as deformation of the crack surface, the strain energy required to open the crack, and the critical pressure required to spread the crack were calculated. Danyluk et al. [12] estimated the width of the plastic zone of a concentric penny-shaped crack in an elastic perfectly-plastic transversely isotropic cylinder. The crack opened under constant pressure, but the curved surface of the cylinder was free of stress. The Dugdale hypothesis states that the singularity of the normal stress at the leading edge of the crack is removed for linear elastic materials. The width of the plastic zone is obtained with a numerical procedure after reducing the problem to a Fredholm integral equation of the second kind. A singular stress field for a circumferential edge crack tip embedded in a transversely isotropic infinite cylinder was derived in closed form by Atsumi and Shindo [13]. The axisymmetric displacement components and stress components were first stated in terms of harmonic potential functions. Next, by taking the Hankel transform, a Cauchy singular integral equation of the first kind was obtained which was solved numerically. Fabrikant [14] assessed a transversely isotropic infinite circular cylinder with an crack of arbitrary shape embedded in it. This was considered for a crack under normal stress and one subjected to a tangential stress. The solution to the governing differential equations for the displacement components was expressed through three potential functions and the stress components were provided using these functions. The solution was presented as the sum of the solution of an infinite space with a flat crack and an integral term emanating from an integral transform. Applying the boundary conditions yielded the governing integral

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equations of the problem for unknown crack displacement discontinuities. For a pennyshaped crack, these equations were solved by the method of consecutive interactions. This portion of the review is related to dynamic fracture and begins with a study by Shindo and Nozak [15], who approached axisymmetric crack problems in an infinite cylinder. They analyzed the axisymmetric dynamic response of an eccentric penny-shaped crack in a transversely isotropic infinite cylinder under normal impact. The plane of the crack was assumed to coincide with one of the planes of the elastic symmetry of the material. For axially symmetric deformation field, the equations of motion were differential equations for axial and radial displacement. Laplace and Hankel transforms were applied to the equations of motion to reduce the problem to dual integral equations in the Laplace domain. The solution to these dual integral equations resulted in a Fredholm integral equation of the second kind. A numerical Laplace inversion was used and the solution was given over time. The dynamic stress intensity factor of crack tip was determined. The numerical solution of the axisymmetric crack problems in transversely isotropic solids has also been studied. A review of these methods is beyond of the scope of this issue, but the boundary element method is one option [16-17]. The methods of the problem solution utilized in the above-mentioned studies were based on the type of defects and loadings and their particular configuration. Most of them failed to investigate a domain being weakened by various types of arbitrarily located axisymmetric cracks. Among the methods of solution of the crack problems, the distributed dislocation technique is a well-organized way for treating multiple cracks. In this investigation, the distributed dislocation technique is presented which is independent of the defect type and location of the flaw. The distributed dislocation technique is a capable tool for analyzing multiple cracks in the domains with arbitrary forms. For instance/example, it is feasible to solve the problem of the cylinder containing arbitrary types of axisymmetric cracks with this method. The determination of stress fields caused by a single dislocation in the domain is by far one of the most momentous obstacles in this method. This task for cylinders containing two climb and glide edge dislocations is accomplished here. The only mentionable study in the literature which applies the distributed dislocation technique for isotopic infinite cylinder was done by Pourseifi et al. [20]. It was allocated to fracture analysis of infinite isotopic cylinders subjected to shear lateral loadings. In the present study, this analysis is extended for transversely isotropic cylinders subjected to both shear and normal lateral loadings. To solve the new problem, new constitutive and governing equations are used and consequently different solution forms are proposed. The present study examined mixed mode cracking in a transversely isotropic infinite cylinder. At first, the solutions to axisymmetric Volterra climb and glide dislocations in an infinite circular cylinder of the transversely isotropic material are obtained. Each of these dislocations is defined as cuts which its two surfaces are moved with respect to each other in a constant value. These surfaces are displaced in the axial and radial directions. Here, the cut of dislocation is in the radial direction that contains a circular area. To attain the stress fields in an infinite transversely isotropic domain caused by the presence of axisymmetric climb and glide edge dislocations, the Hankel transform is applied to the governing equations of biharmonic stress function. The solutions are represented in terms of the biharmonic stress function. The integral forms of displacement and stress fields contain 5

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several unknown coefficients which can be found by applying the boundary conditions. Next, the distributed dislocation technique is used to formulate the problem of a cracked transversely isotropic infinite cylinder. The cylinder contains a set of coaxial axisymmetric penny-shaped, annular, and circumferential cracks. Two types of loadings are considered: (i) the lateral cylinder being loaded by two self-equilibrating distributed shear stresses; (ii) the curved surface of the cylinder being under the action of a distributed normal stress. The distributed dislocation technique was utilized to formulate Cauchy-type integral equations for a set of concentric annular cracks in a cylinder subjected to axisymmetric tensile load. To obtain the dislocation densities on the borders of the cracks these equations were solved numerically. After determination of the dislocation densities, the Modes I and II stress intensity factors for axisymmetric interacting cracks can be found. The present paper presents analysis of the fundamental climb and glide edge dislocation solution for an infinite cylinder of transversely isotropic material and employs them to analyze related crack problems. There-examination of axisymmetric crack problems in cylinders is 2-fold: (1) to extend the literature to multiple interacting cracks instead of being restricted to a single crack; and (2) to solve the less-examined mixed mode crack problem. It is assumed that the cylinder is weakened by axisymmetric penny-shaped, annular, and circumferential edge cracks. The cylinder is subjected to distributed self-equilibrating shear or normal traction on its lateral surface. At first, stress and displacement fields on an infinite transversely isotropic cylinder with an axisymmetric Volterra climb/glide dislocation are obtained analytically. The solution of an intact cylinder under lateral shear or normal traction is presented in Section 2. Section 3 presents the distributed dislocation technique for these solutions to construct and solve the Cauchy-type singular integral equations for the cracked domain. Section 4 presents numerical examples to study the effect of loading, material anisotropy, and interaction of cracks on the resulting stress intensity factors at the crack tips. Section 5 offers concluding remarks. 2. Formulation

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The physical properties of a transversely isotropic material are symmetric about an axis that is normal to a plane of isotropy. Within this plane, the material properties are the same in all directions. Examples of transversely isotropic materials are the well-known on-axis unidirectional fiber composite lamina with the circular fibers in cross section and geological layers of rocks. A transversely isotropic material with a polar axis along the direction parallel to the axis is considered next. Hooke’s law for linear elastic transversely isotropic materials is (1)

where the planes of isotropy are planes parallel to the end surfaces of the cylinder and (2) (

)

(

)

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)

(

)

(

)(

)

(

)

(

)

(

(

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(

(

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(

(

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(

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(

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(4)

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(

(3)

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where (

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wherein, represents the longitudinal (polar) direction and represents the transverse direction.Also and are the transverse Young’s modulus and longitudinal Young’s modulus, respectively. Moreover, and are the shear moduli of the planes of isotropy and of planes perpendicular to the planes of isotropy, respectively. The Poisson’s ratios and describe transverse reductions in the plane of isotropy under tension in the same plane and under axial tension, respectively. The displacement and the stress components in the transversely isotropic materials can be expressed in terms of a stress function as follows [18]

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Substituting the stress components (3) into the first equation of the equilibrium results in )(

)

(

)

(5)

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(

It is easy to show that other equilibrium equations are satisfied identically. Solution to Eq. (5) is proposed in the integral form as follows (

)

∫

∫ 0 ( )

, ( ) ( √

) ( )

( ) ( √

1 (

))

where

7

( )

(6)

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[ [

√(

)

√(

)

(7) ] ]

wherein √ Also, ( ) and ( ) are the Bessel function and the modified Bessel function of the first kind of order 0, respectively. Substituting Eq. (6) into relations (3) the displacement and stress components are given by ) √

∫ , ( )

√

∫ * ( )[ ( )[ √

(

(

∫ * ( )[

, ( )

(

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(

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0 ( )

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where

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( ∫

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∫ , ( )

1 (

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( ) √

( )

( )

1 (

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(9)

,

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2.1. Dislocation solution 2.1.1. Climb edge dislocation Let us consider an infinite cylinder with radius (shown in Fig. 1). In the cylindrical coordinate a Volterra-type climb edge dislocation located at , is considered ) wherein the cut of dislocation in radial direction is a circular area ( The boundary conditions along the lateral surface read as ( (

) )

(10)

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Fig.1. Schematic view of an infinite cylinder with a climb/glide edge dislocation

A Volterra type climb ring dislocation is located at , the cut of dislocation in radial direction whose condition is (

)

(

)

(

)

in an infinite cylinder with (11)

(

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(

)

(

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where, and ( ) denote the dislocation Burgers vector and the Heaviside step-function, respectively. Besides, the traction vector is continuous on the cut of the dislocation. This implies that (

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(12)

(

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(

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(13)

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(

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Problem of the climb edge dislocation is symmetric with respect to the plane Therefore; the half-space is subjected to the following boundary conditions [19] )

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The zero order Hankel transform of the Heaviside step-function ( ) can be easily written as ( ) ∫ ( ) ( ) By virtue of this relation, the boundary conditions (13) are applied on Eqs. (8). This leads to

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( ) ( )

( )

(14)

( )

( )

Solution to the above equations is given by ( ) ( )

where

( (

(15)

) )

Applying the boundary conditions (10) to the stress field (8) results

in 9

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(

( )[

)

(

)

∫ *

√ √ (

(

(

) )

(

)

(

)]+ √

( ) (

1

(

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∫ (

)

√

√

0

, ( )

(16)

)]

√

0

)

∫

(

( √

1+ ( )

)-

( )

( ) (

)

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∫ * ( )[

Taking the inverse sine and cosine transforms, the above relation are simplified as (

)

( (

∫ *

√ √ (

(

)

, ( )

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)

(

)] √

)∫ (

)

∫

∫ (

√

√

∫ (

(

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( )

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⁄√

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∫

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( )[

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(

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(17)

)

)

)

(

)

(

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(18)

+

) ( ) ( )(

(

+ (

)

two equations in terms of coefficients

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By integrating of the above equations with respect to ( ) and ( ) are derived as ( )[

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( )[

) )

AC

The integrals with dummy variable can be evaluated using the formula given in Appendix A. Substituting them into Eqs. (18) yields ( )[

(

*

)

, .

( 0

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.

, ( )

(

)

(

/ / /

( )[

) (

(

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(

( ) .

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(

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.

/

(

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.

/

(

)-+

) .

(

/

10

(

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(

)]

(19)

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(

,

,

.

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,

. (

/

*

(

)

,

(

(

(

/

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(

) (

/

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.

(20)

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/

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(

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wherein (

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/

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(

/

(

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(

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( )

,

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(

) (

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)

(

) (

)-

(21)

Substituting the coefficients (20) into the equations of (8) which is linked to the stress components gives

∫ *

(

/

√

√

, ( )

∫

, ( )

∫ (

AC √

.

√

CE

∫ (

.

√

(

∫

)

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√ √ (

.

(

/

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( )[

.

For isotropic materials and then

. √

/

)

.

/]+

.

M

∫ * ( )[

)(

(

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) ( )

/ )

.

( ) (

/-

.

/

)

.

/-

√

(22)

) + ( )

( )

/ √

√

/]

)

(

.

/

) (

)

and Therefore, using Eqs. (2) and (4),we arrive at To verify the climb edge dislocation solution, we use the

⁄√ √ ∑ ( )( ) Taylor's expansion and evaluate the stress field by assuming and as In this case the stress field of the climb edge dislocation solution given in the reference [20] is derived which validates Eq. (22). The stress component ( )of the above dislocation solution exhibits the singularity as To show the strength of this singularity we use the following formula [21]

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( ) (

∫

)

{

( )

(23)

( ) ( )

where ( )

√ and ( ) ∫ √ are the complete ∫ elliptic integrals of the first and second kind, respectively.Existence of two kinds of singularities in the above integral that is, the Cauchy singularity

and the weak

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|has been reported by the reference [21]. It is worth mentioning that the singularity | first kind of the aforementioned singularity in the dislocation solution for an infinite isotropic domain containing a climb and glide edge dislocations is also reported by [19]. 2.1.2. Glide edge dislocation

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For want of space, in the case of the glide edge dislocation, the relation for and also the displacement and stress fields are not presented here again. For the glide dislocation, we apply some changes to Eqs. (8), including replacement of the sine functions by cosine functions and reversely. Furthermore, the sign of the first integral term of and are changed. The coefficients ( ) ( ) ( ) and ( )for glide edge dislocation are not the same as those for climb edge dislocation. But for the sake of brevity we do not introduce any new coefficients for them. The condition corresponding to a Volterra-type glide ring dislocation located at of an infinite cylinder in which the cut of dislocation to be in radial direction can be written as (24)

(

)

(

ED

where, is the dislocation Burgers vector. Also, the traction vector is continuous on the dislocation cut. This implies that )

(

)

(

)

(25)

(

)

(

(26)

)

CE

(

PT

For the glide edge dislocation, we deal with an antisymmetric problem with respect to the plane Therefore; the following boundary conditions are imposed to the half-space

)

AC

The above boundary conditions are applied similar to the case of the climb edge dislocation. To this end, the first order Hankel transform of the Heaviside step function is written as ( ) ∫ (∫ ( ) ) ( ) . Therefore we have ( ) ( )

(

∫ ∫

(27)

) (

)

( )( )( ) Using the relations (27) along with the Where modified version of Eqs. (8) for glide dislocation we will have

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( ),

(

∫

) )

(

)

)

(

(

(

∫

)

∫

)

(

) (

)

(

)-

( )

)

√

( )

(

)-

√

(

)

)+

√

(

∫

) (

( )

(

, ( )

∫

( ) √

√

, ( )

(28)

)-

√

√

∫

(

)-+

* (

)(

∫

)

(

(

∫

(

(

√

) (

( ) ) (

)

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∫ * ( ),

)

(

)

∫

(

)

, ( ) ∫

( )

)∫ (

(

)

(

(

*

(

)-

√

)

( ),

(

√

(

∫

, ( )

(

∫

)

(

√

)

(

)

)

(

)

AC

∫

(

(

*

(

CE

(

)

)

-

( ),

(

( )

)

(

(

(

)

)

(

)

-

(30)

) )

)( ) +

)

)(

( )

)

) ( (

)

)-

(

)

(29)

)-

yields

(

( ( )

(

+

√

(

)

)

(

∫

( )

)

)

)-

∫ (

PT

(

(

√

Integration of the above equations with respect to ( ),

)

√

)∫ (

( ) (

∫

)

M

∫

(

ED

( ),

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For the glide dislocation, the boundary conditions along the lateral surface of the cylinder are the same as those for the climb dislocation. By employing the inverse sine and cosine Fourier transform of Eqs. (28), the boundary condition (10) is expressed as below

) (

(

)

(

)

)(

)

We rewrite the above equations by use of the relations given in the Appendix A, as below ( ),

(

∫ ( ( )

)

( (

*,

) , ) ,

( )

(

( ∫

)

) ( ) .

)-

(

(

( ),

)

/

)

(

) (

(

(

)

) (

(

)

(

)-

)-+

) (

)

.

13

/

(

)-

(

)-

(31)

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(

)

, ( )

( )

Where

( ) ( )

∑

(

(32)

( ) ( )-

) (

( )

and

)

( )

∑

(

) (

)

are the modified Struve

functions. Equations (31) are simplified in view of the relations (32) as )

(

*

, (

, (

)

( )

(

)

)

)

)

(

)-

)

(

(

( ) )

(

(

(

*, ( , (

)

(

(

)

(

)-,

(

)-, (

)

)

(

) (

(

) (

)-

)-

( (

(

)

)

)

)

(

)

) (

(

(

( ),

)+

(33)

)-

CR IP T

(

(

)

)

(

(

)-+

)-

)

AN US

( ),

Solution to the above equations is given by , (

(

) ( , (

( )

))

* (

) (

(

)

)

(

)

, (

)

)-

(

(

)

(

,

( ) (

(

(

)

,

)

(

)

) (

)

(

(

(

)-*

(

)

)-, (

)-,

(

) (

, (

) (

)

) (

)

)-*

) (

) (

(

)

) (

(

,

) )

(

(34)

)

)-+ (

) (

(

)

(

)-+

(

)

)-+

(

)

(

)-+

PT

, (

)

M

*

ED

( )

CE

Similarly for isotropic materials, by assuming and as the above glide edge dislocation solution is simplified to the stress field given in reference [20] which verifies the solution. Analogous to the case of the climb dislocation, we can find the strength ( )as of the singularity of the stress component Using Mathematica software we the

AC

attain ∑

relation

( )

( ) ( )

(

∫

(

)in

which

is the Hypergeometric function and the symbol ( )

Pochhammer symbolwherein stress component

)

(

(

)

(

) ( )

is the

( )is the Gamma function.Substituting this relation into the

) leads to a term as ∫

(

) (

)

which it

can be integrated by the relation given in the following ( ∫

(

)

(

. /

. /)

) {

(

14

)

( )

(36)

ACCEPTED MANUSCRIPT (

Again we conclude that the stress component corresponding to the Cauchy singularity

) has two kinds of singularities

and also the weak singularity

|

|as

2.2. Solution of an uncracked cylinder under lateral shear or normal traction 2.2.1. An infinite cylinder subjected to two self-equilibrating shear stresses on the lateral surface

( (

) )

, (

)

(

CR IP T

Let us consider an intact infinite cylinder under two self-equilibrating shear stresses as ( ) , ( ) ( )- and ( ) , ( ) ( )as shown in Fig. 2a. A practical case is the tension test of the cylindrical specimens which experience the same kind of loadings. The problem is also symmetric with respect to the plane Satisfying the boundary conditions along the lateral surface of the cylinder for requires )-

(37)

(

)

, ( )

∫

,

(

) ( ( )

)-

(

)

)

(

)-

(

∫

( ),

(

)-

(

)

)

(

(

)

gives ) (

)-

( )

( )

AC

CE

PT

∫

(

)

ED

( ),

(

M

Heaviside step-function, i.e.,

AN US

The starting point for the solution is the solution form (6) and the resultant displacement and stress field (8). These equations are still valid but we eliminate the coefficients ( ) and ( ) because only two boundary conditions are required to be satisfied. The boundary conditions (37) are applied to Eqs. (8) by means of the Fourier sine transform of the

(a)

(b)

Fig.2. An infinite cylinder subjected to (a) two self-equilibrating shear stresses, (b) a normal stress

15

(38)

ACCEPTED MANUSCRIPT The above equations are rewritten to take the form ( ),

(

( )

) (

( )

) (

( )

)-

(

( ),

(

)

,

)

(

( )

) ( (

(39)

)-

)-

Solution of Eqs. (39) gives the coefficients of ( ) and ( ) which employing them and Eqs. (8), the following stress components are obtained

(

)

,

(

(

(

( (

)

(

)

(

)-,

(

)

) ( )-

)

(

)-

(

(

) (

(

(

)

(

)

(

) ( (

)

(

)

) ( )-,

)-

(

∫ *, ,

) (

) ∫ *,

,

(

)-

(

)

,

)+

(

)

)

) (

)-

(

)+

(

(

)-

(40)

)-+

CR IP T

,

)

(

)-

AN US

(

∫ *,

(

)

)

,

(

)

(

)-

(

)

2.2.2. An infinite cylinder under a normal stress on the lateral surface ( (

) )

, (

)

M

Conditions representing the loading shown in Fig. 2b, can be written as follows (

)-

0

(42)

PT

ED

Using the modified form of the displacement and the stress field (8) which has been used for analysis of the glide edge dislocation and ignoring the coefficients ( ) and ( ) the above boundary conditions are applied as ∫ * ( ), (

)

,

∫

(

(

(

(

) (

) (

)

)

AC

( )

)

CE

( ),

(

(

)-+

)-

( )

(43)

)( )

( )

(

)

Similar to the section 2.2.1 the above equations are first simplified and solved and the resulting stress field is given by ,

∫ * (

,

) ,

(

)

( (

(

) ( ) (

)-

∫ , (

( ) (

)

)

( (

) ( ) (

)-

)-+

) )

(

) (

16

)-

(44)

ACCEPTED MANUSCRIPT

,

(

)

(

)-

(

(

) (

)-

(

∫ , ,

(

)

(

) )

(

) (

)-

)

3. Axisymmetric crack formulation

(

) ( (

AN US

( ) ( ) ( )

CR IP T

Using the dislocation solutions (22) and (28), we are able to analyze axisymmetric crack problems in an infinite cylinder of transversely isotropic material. In particular, we intend here to investigate an infinite cylinder with various kinds of axisymmetric cracks. These cracks are including penny-shaped cracks, annular cracks and circumferential edge cracks which the cylinder is under different types of loading. We assume that the cylinder is weakened by penny-shaped cracks, annular cracks and circumferential edge cracks in which is number of all above-mentioned defects. Also these defects are located horizontally. The parametric form of the aforementioned cracks can be written as ) )

in which

(

(

{

(46)

M

(

) ) )

(45)

CE

PT

ED

Suppose ( ) and ( ) to be climb and glide edge dislocation densitieson the dimensionless length , respectively. By distributing these unknown dislocation densities on the infinitesimal segment at the surfaces of the j-th concentric crack, we can ( ( ) ) obtain the traction component on the surface of i-th crack. These cracks include penny-shaped, annular, circumferential edge cracks. Using Eqs. (22) and (28) and distributing of dislocations on all surfaces of the cracks, the traction components will be summarized here ( ( )

)

∑

(

∫ [

)

( )

(

)

( )]

(47)

AC

where the kernels of the integrals ( ) and ( ) are given in the Appendix B. The left-hand side of Eq. (47) can be found by use of the Bueckner’s principle (see reference [23]). This term is the negative value of the traction at the presumed borders of cracks caused by applying the shear or normal stress on the lateral surface of an intact cylinder, Fig. 2.The applied tractions on the intact cylinder are taken to be as Eqs. (37) and (42) and the resulting stress components are given by Eqs. (40) and (44) respectively. The crack opening displacements by virtue of the dislocation density functions are given by ( )

( )

∫

( )

( )

( )

∫

( )

(48)

17

ACCEPTED MANUSCRIPT Single-valuedness of the displacement field out of embedded cracks i.e. the annular cracks imply that their related dislocation densities to be imposed to the closure requirement as ( ) ( ) The Cauchy singular ∫ ∫ integral Eqs. (47) and these closure equations for embedded cracks are solved simultaneously to determine dislocation density functions. As expected, the stress components present a square-root singularity at the crack tips [16]. Hence, by choosing that to be representative for the singular embedded crack tips then the dislocation densities for each type of cracks are given by (49) ( )

( )

√ ( )√

( )

{

CR IP T

( )√

( )

AN US

By virtue of the relations (49), the integral equations (48) along with the closure equations for ( ) ( ) annular cracks i.e. ∫ are ∫ solved numerically. To this end, these integral equations were discretized in the collocation .

points

/

.

and

/

(see for more

)

(

)

√

AC

(

CE

PT

ED

M

details [24]). Because the number of discrete points is one less than the number of whole ensuing algebraic equations will be ( ) less than of the number of the unknowns ( ) Therefore we must discretize Eqs. (47) at ( ) new arbitrary distinct discrete points other than and that is, the arbitrary points and For the annular cracks, because of the existence of the closure equations the number of ensuing algebraic equations andunknowns are equal and we need no new distinct discrete points. At last, by considering the new arbitrary distinct collocation points, the integral equations are converted to a system of linear algebraic equations. The solution to these equations leads to the dislocation density functions on the crack tips. The crack opening for a penny-shaped crack with radius located in an infinite domain of transversely isotropic material subjected to an internal pressure is given by relation [16]

where Also

(

(50)

)

is the radial distance from the crack tip and the axis is normal to the crack surfaces. is the Mode I stress intensity factor of the singular crack tip and (

) and

(

)

Wherein

and

are the two

zeros of the quadratic equation of the material propertieswhich is given by the relation , ( ) In these relations the coefficient and are material constants defined by

18

ACCEPTED MANUSCRIPT (51) (

)

(

)

It is worth mentioning that the stress intensity factor can be written in terms of the stress component as ) ( ) which we didn't use √ (

CR IP T

where

it because Eq. (50) is easily linked to the dislocation density as follows. Substituting the first relation of Eqs. (49) into Eqs. (48) and substituting the ensuing relations into Eq. (50) yields )(

)

√ (

where ( )

)

(

√,

)

( )-

∫

( )√

(52)

is the parametric form of the radial coordinate on the

AN US

(

crack borders. With the help of L’Hospital’s rule the aforementioned relation will be (

expressed as

)(

)√

√ (

(

)

) Correspondingly, for all kinds of cracks, the

relations for stress intensity factors of crack tips are classified as follows

√ ( ( {

)√ )

)( √ (

)√ )

(

) (

)

(53)

M

)(

ED

(

AC

CE

PT

( ) with Mode II stress intensity factors of the crack tips are attained by replacing ( ). In summary, the stress intensity factors of the crack tips are computed according to the following flow chart:

19

ACCEPTED MANUSCRIPT Start

Specifying the geometry of the crack using Eqs. (45): ( )

(

)

( )

(

( )

(

) )

)

∑

( ( )

where

(

∫ [

)

( )

(

)

( )]

) are computed by use of Eqs. (47).

AN US

( ( )

CR IP T

( ) ( ) The kernel of the integrals, are computed using Eqs. (B.1) and (B.2) and ( ) are obtained by use of Eqs. (45). In Eqs. (B.1) and (B.2), ( ) and ( ) are obtained from the relations for ( )and ( ), i.e. Eqs. (20) and (34), respectively. Also and are eliminated and replaced by . Integral Equations:

( )

Closure equations for annular cracks:∫

( )

∫

Replacing the following equations into the integral and closure equations:

( )

√

( )√

PT

{

( )

ED

( )

M

( )√

( )

CE

Choosing .

and discretizing integral and closure equations with collocation points /

.

and

AC

factor in the last step

)( √ ( (

{

)√ )

)( √ (

)√ )

(

) (

, then, solving

( ) and calculating the stress intensity

the ensuing algebraic equations to find

(

/

)

End 20

ACCEPTED MANUSCRIPT 4. Numerical examples and discussions Cracks in a transversely isotropic long cylinder were analyzed to validate the proposed technique. The materials chosen were steel, magnesium, and cadmium at six compliances constants as shown in Table 1[25-26]. In the examples considered in this study, the material properties given by table 1 are used.

Steel

and corresponding elastic constants

-0.145

-0.785

-0.498

Cadmium

1.290

-0.150

-0.930

Steel Magnesium

2.1008 0.4545

0.3004 0.3568

2.1142 0.5076

Cadmium

0.7752

0.1163

0.473

1.240

1.97

6.10

3.690

6.400

0.3066 0.2528

0.8065 0.1639

AN US

-0.143

Magnesium

0.476 2.20

CR IP T

Table 1. Compliances material constants

0.2710

0.2520

0.1563

M

For these materials the anisotropy ratios are different. It is worth mentioning that the displacement and stress fields are given in terms of these parameters i.e. Moreover, for isotropic materials we have These materials have been used as a representative for transversely isotopic materials in the literature ([25, 26]).

ED

Example 1. An infinite transversely isotopic cylinder with a penny-shaped crack In this example, a uniform axial tension is applied on an infinite cylinder which is weakened by a penny-shaped crack with radius . Stress intensity factors are normalized dividing them

PT

by √ . The numerical results are compared with those of (Parhi and Atsumi) [11] to establish their accuracy and showed good agreement (see Table 2).

CE

Table 2.Normalized stress intensity factor for a penny-shaped crack (

Reference [11] 0.962 0.975

AC

Magnesium Cadmium

)

Present study 0.973 0.978

To compare predictions of the formulation given in this manuscript with finite element calculations, we have considered an embedded penny-shaped crack in an infinite isotropic cylinder. The nondimensional stress intensity factor of a penny-shaped crack with radius is presented in Table 3. The result is also compared with that of the reference [27] and good agreement is observed. The stress intensity factor is normalized by a divisor as The dimensionless stress intensity factors for this kind of crack are less than √ unity depicting that the lateral surface of cylinder plays a significant role on SIFs.

21

ACCEPTED MANUSCRIPT Table 3. Non-dimensional SIF values for a penny-shaped crack in an infinite isotropic cylinder (

Reference [27] 0.6406 0.6465 0.6850 0.8053 1.3180

0.1 0.3 0.5 0.7 0.9

)

Present study 0.6399 0.6492 0.6875 0.8114 1.3332

CR IP T

Example 2. An infinite transversely isotopic domain with a penny-shaped crack To compare evaluations of the formulation given in this manuscript with boundary element formulations, an infinite domain made of graphite-epoxy containing a penny-shaped crack with radius is considered. The crack surfaces are subjected to a uniform normal or a shear traction . Compliances material constants are chosen from Table 4.

Gr-Epoxy

9.6030

for the graphite-epoxy

AN US

Table 4. Compliances material constants

-4.6827

-0.1972

0.6381

14.144

0.0350

√

Table 5. Normalized

for a penny-shaped crack at transversely isotropic infinite domain

Reference [16] 0.634

ED

Analytic Solution [16] 0.6366

PT

Gr-Epoxy

√

Table 6. Normalized

CE

Gr-Epoxy

M

Modes I and II stress intensity factors are normalized by √ and √ respectively. The numerical results are compared with those of (Sfiez et. al) [16] to establish their accuracy. A good agreement is seen in the results (see Tables 5 and 6).

Present study 0.6372

for a penny-shaped crack at transversely isotropic infinite domain

Analytic Solution [16] 0.8115

Reference [16] 0.82

Present study 0.8137

AC

Example 3. An infinite transversely isotopic cylinder with a circumferential edge crack A circumferential edge crack with inner radius is considered. The dimensionless stress intensity factor of a singular crack tip is shown in Table 7. Comparison of the dimensionless stress intensity factor with that of (Atsumi and Shindo) [13] shows good agreement. The divisor for normalizing the stress intensity factor is set at Table 7. Normalized stress intensity factor for a circumferential edge crack (

Magnesium Cadmium

Reference [13] 1.923 1.931

22

Present study 1.9132 1.9210

√ )

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

Example 4. A cracked infinite cylinder containing a penny-shaped crack, an annular crack and a circumferential edge crack An infinite cylinder containing a penny-shaped crack, an annular crack and a circumferential edge crack is considered. These concentric interacting cracks (a penny-shaped crack, an annular crack, and a circumferential edge crack) are labeled by 1, 2, and 3, respectively. The cracks are assumed to be on the same plane( ) and have radii , and respectively. The penny-shaped and the circumferential-edge crack lengths are and ; the annular half-crack length is ( ) The cylinder is loaded by two self-equilibrating shear stresses represented symmetrically as , ( , ( ( ) ) ( )- and ( ) ) ( )- wherein The material of the cylinder is steel(transversely isotropic material). In the examples, the stress intensity factors are normalized as √ Figures 3 and 4 show the effects of dimensionless crack length on normalized stress intensity factors and for fixed values of the average radius of the annular crack ) ( ( ). In these figures, and denote the inner and outer edges of the annular crack, respectively. As crack length increases, Modes I and II stress intensity factors of the approaching crack tips of the penny-shaped and the annular cracks increase. Mode I stress intensity factors of the inner tip of the annular crack experience some local reduction in crack length which is representative of crack arrest. Mode I stress intensity factors for the outer tip of the annular crack increase slightly as the crack length increases and the outer tip approaches the curved surface of the cylinder. Mode II stress intensity factors for this tip decrease slightly with crack growth. Figures 3 and 4 show that the Modes I and II stress intensity factors for the circumferential edge crack tip are, in general, smaller than for other tips. The Mode I stress intensity factors increase slightly as this tip approaches the outer tip of the annular crack and the Mode II stress intensity factors remain constant. Three concentric non-planar cracks of the same type located on planes and are considered for the penny-shaped, annular, and circumferential edge cracks, respectively. Figures 5 and 6 show the graphs for and versus for the tips of the cracks.Variations in the stress intensity factors of the penny-shaped crack and the inner tip of the annular crack versus crack length are not similar to those in Figures 3 and 4. In this case, the stress intensity factors decrease for bigger crack lengths. This occurs because the interaction of the crack tips are dominant over the effect of crack length enlargement. Example 4 is reexamined with three coplanar concentric interacting cracks located on ( ) , ( ) when the cylinder is under a normal stress of ( )- on the lateral surface in which Figures 7 and 8 show graphs of the variation of and versus . The variation in the stress intensity factors of the tips of the cracks versus crack length are similar to those in Figures 3 and 4.

23

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

Fig. 3. Variation of the nondimensional Modes I stress intensity factors versus for three coplanar cracks under shear stresses on the cylinder lateral surface

24

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

Fig. 4. Variation of the nondimensional Modes II stress intensity factors versus for three coplanar cracks under shear stresses on the cylinder lateral surface

25

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

Fig. 5. Variation of the nondimensional Modes I stress intensity factors versus for three non-planar cracks under shear stresses on the cylinder lateral surface

26

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

Fig. 6. Variation of the nondimensional Modes II stress intensity factors versus for three non-planar cracks under shear stresses on the cylinder lateral surface

27

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

Fig. 7. Variation of the nondimensional Modes I stress intensity factors versus for three coplanar cracks under a normal stress on the lateral surface

28

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

CE

PT

Fig. 8. Variation of the nondimensional Modes II stress intensity factors versus for three coplanar cracks under a normal stress on the lateral surface

AC

Tables 8 through 11 show the effect of material anisotropy on the stress intensity factors at the crack tips of three interacting cracks. The isotropic materials have , , and Table 1 shows that magnesium is more similar to cadmium than is the isotropic material. Tables 8 and 9 assume that the cracks are located at and the loadings locations are specified by and Tables 10 and 11 are for cracks located at and the loading locations are specified by

29

ACCEPTED MANUSCRIPT Table 8. NormalizedMode I stress intensity factors for three coplanar interacting cracks under shear stresses on the cylinder lateral surface

Left tip of annular crack

Right tip of annular crack

Edge crack

0.7547 1.4511 1.8317

3.6675 4.7621 4.9013

1.5190 1.4728 0.7685

0.0589 0.0632 0.0956

0.2888 0.5483 1.2192

1.6459 1.9762 2.9495

0.9337 0.8389 0.8082

0.0482 0.0946 0.1724

Magnesium l/R=0.1 l/R=0.15 l/R=0.2 Cadmium l/R=0.1 l/R=0.15 l/R=0.2

CR IP T

Penny-shaped crack

AN US

Table 9. Normalized Mode II stress intensity factors for three coplanar interacting cracks under shear stresses on the cylinder lateral surface

Left tip of annular crack

Right tip of annular crack

0.0371 0.0453 0.0668

0.0486 0.0680 0.1071

0.0073 0.0041 0.0022

Magnesium l/R=0.1 l/R=0.15 l/R=0.2 Cadmium l/R=0.1 l/R=0.15 l/R=0.2

0.0088 0.0141 0.0243

4.7132 2.1017 6.5714

Edge crack

1.0058 1.8481 1.9474 3.0137 5.5727 7.5698

PT

ED

0.0061 0.0090 0.0149

M

Penny shaped crack

AC

CE

Table 10. Normalized Mode I stress intensity factors for three coplanar interacting cracks under a normal stress on the cylinder lateral surface

Magnesium l/R=0.1 l/R=0.15 l/R=0.2 Cadmium l/R=0.1 l/R=0.15 l/R=0.2

Penny shaped crack

Left tip of annular crack

Right tip of annular crack

Edge crack

0.3914 1.0145 0.8995

2.1166 3.5381 2.1705

1.1515 1.4439 0.7176

0.1216 0.0689 0.1154

0.4148 0.6699 1.2652

1.9230 2.1555 2.9118

0.7186 0.5273 0.4373

0.0144 0.0286 0.0609

30

ACCEPTED MANUSCRIPT Table 11. Normalized Mode II stress intensity factors for three coplanar interacting cracks under a normal stress on the cylinder lateral surface

Penny shaped

Left tip of

Right tip of

crack

annular crack

annular crack

Edge crack

Magnesium l/R=0.1

3.8

4.9

l/R=0.15

4.3

6.4

1

l/R=0.2

6.2

9.8

1

l/R=0.1

2.852

3.671

6.20

l/R=0.15

1.897

2.954

5.7

l/R=0.2

2.457

8

AN US

5.1

CR IP T

Cadmium

2

5. Concluding remarks

4.5 2.7 5

AC

CE

PT

ED

M

From assessment of the figures and tables it can be seen that: (1) In the case of the steel cylinder with three coplanar concentric cracks under shear stress on the lateral surface, Modes I and II SIFs of the approaching crack tips of the penny-shaped and the annular cracks increase as long as crack length increases. It is valid for magnesium and cadmium cylinders as well. For the inner and outer tips of the annular crack the following results are observed: a) There is a specific crack length value which is representative for crack arrest. This is because of the fact that Mode I stress intensity factors of the inner tip experience some local reduction for this crack length. b) Mode I stress intensity factors for the outer tip increase slightly as the crack length increases. On the contrary, Mode II stress intensity factors for this tip decrease slightly as long as crack length increases. For the circumferential edge crack it was also noted that: a) The Modes I and II SIFs for crack tip have, in general, smaller values than others. b) The Mode I stress intensity factors increase slightly as this tip approaches the outer tip of the annular crack and the Mode II stress intensity factors remain constant. (2) In the case of the steel cylinder with three non-planar concentric cracks under shear stress on the lateral surface, the stress intensity factors decrease for bigger crack lengths. (3) The magnesium cylinder with three coplanar concentric cracks under shear stress on the lateral surface shows greater SIF values for the penny-shaped and annular cracks than the cadmium cylinder. Similarly, for normal stress on the lateral surface of the Magnesium cylinder the results illustrate greater SIF values for the edge crack than the cadmium cylinder. 31

ACCEPTED MANUSCRIPT (4) The Mode II stress intensity factors of all cracks are greater for the magnesium cylinder than the cadmium cylinder with three coplanar concentric cracks under normal stresses on the lateral surface. (5) For magnesium and cadmium cylinders, the Mode I stress intensity factor depends on whether material anisotropy or crack length is dominant.

Appendix A

)( ) ( ) ( )( ( ) ( ) )( ( ) ( )

( (

∫

(

, (

∫

) ( (

, (

∫

)

(

)

(

( )( (

)

(

(

(

√

)

∫ (

)

)

(

)

(

)

)-

)

(

)-

) )

)(

(

)

(

)-

(

)

are (B.1)

( ) |

, ( ) |

)

)

)∫ ( ∫

)

(

)

(

(

)

)

CE

, ( )

AC

∫

)

( )

Climb dislocation: The kernel of the integrals, (

)

( | √

( | √

(A.1)

)

(

(

)

)

(

)(

)

(

(

( )

)

(

) )

(

PT

Appendix B

)

)

( )

)

(

)

(

( (

)

(

)

) (

)

)

(

(

(

(

(

)

,

) )

(

)

)

AN US

∫

(

ED

( ( ∫ (

) ( )

M

(

∫

CR IP T

The following relations can be easily written using the formula extracted from reference[22]

⁄ )

)-

(|

|

| √

( )

(

|

| √

| )

) ) (

( ⁄ )-

)

(

)

(|

) (

|

)

)

where ( ) and ( ) are obtained from the relations for ( ) and ( ), i.e. Eqs. (20), by eliminating from them and replacing by . All parameters that have the indices i and j are functions of and respectively.

32

ACCEPTED MANUSCRIPT (

Glide dislocation: The kernel of the integrals,

∫

, ( ) ∫

(

) (

∫

( ∫

(B.2)

)

)

|

(

, ( )

( ∫

(

(

)

∫

( )

)

(|

| √

|

)

|

| √

) (

)

)

) (

)-

( ) (

( |

)| √

(| |

| | √

)

CR IP T

(

) are

) (

)

AN US

where ( ) and ( ) are obtained from the relations for ( )and ( ), i.e. Eqs. (34), by eliminating from them and replacing by . All parameters that have the indices i and j are functions of and respectively.

References:

AC

CE

PT

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M

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