Stress intensity factors for internal circular cracks in fibers under tensile loading

Engineering Fracture Mechanics 71 (2004) 365–377

www.elsevier.com/locate/engfracmech

Stress intensity factors for internal circular cracksin fibers under tensile loading

G.V. Guinea *, F.J. Rojo, M. Elices

Departamento de Ciencia de Materiales, Universidad Polit�eecnica de Madrid ETSI Caminos, Canales y Puertos,

Ciudad Universitaria s/n, 28040 Madrid, Spain

Received 7 March 2002; received in revised form 7 October 2002; accepted 14 March 2003

Abstract

Stress intensity factors for inner circular cracks placed eccentrically in a fiber with round cross section were com-

puted and are presented in this paper in both analytical and graphical form. The crack plane was perpendicular to the

fiber axis and remote tensile loading was assumed. The stress intensity factors were numerically computed using the

finite element method. Mesh objectivity and some other aspects of computational precision are considered. The as-

ymptotic behaviour when the crack size and the ligament depth vanish were considered in order to formulate accurate

interpolation expressions.

� 2003 Elsevier Ltd. All rights reserved.

Keywords: Fiber; Internal circular crack; Stress intensity factor; Tensile load

1. Introduction

Brittle failure of fibers is often initiated by cracks located on their surfaces or in their interior. In both

cases, the crack configuration is three-dimensional in nature, and the determination of the stress intensity

factors has to resort to numerical analysis.A number of works, experimental as well as numerical, have been devoted to the analysis of fibers with

surface cracks, mainly those with circular shape [1–5]. Particularly, the results due to Levan and Royer [5]

are a good compendium of stress intensity factors for this kind of geometry. In contrast, fibers with internal

cracks have attracted less attention from researchers. Practical reasons, like the fact that fatigue cracks

ordinarily originate from the surface or that many fiber manufacturing processes are intrinsically associated

with certain surface defects, may be at the root of this lack of interest, along with other difficulties inherent

in the stress analysis required, specially when the internal crack approaches the fiber surface. To the au-

thors� knowledge, the available solutions for the stress intensity factor of internal cracks are scarce, andmost of them are limited to infinite or semi-infinite solids or symmetrical configurations where the crack is

* Corresponding author. Tel.: +34-91-336-6679; fax: +34-91-336-6680.

E-mail address: [email protected] (G.V. Guinea).

0013-7944/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0013-7944(03)00115-2

mail to: [email protected]

366 G.V. Guinea et al. / Engineering Fracture Mechanics 71 (2004) 365–377

centered with the fiber axis [6–8]. Only Mori et al. [9] have considered the case of elliptical cracks in a bar

under tension and bending, although their numerical analysis is restricted to small cracks with sizes up to

16% of fiber radius.

The purpose of this paper is to provide expressions, general enough, for the stress intensity factor for aninternal circular crack, arbitrarily located in a plane normal to the fiber axis, in a fiber under tensile loading.

Polynomial expressions are given which allow accurate computations of the stress intensity factor for a

wide range of crack geometries.

This paper is structured in three parts: The first part describes the geometry of the problem and the

numerical analysis performed to compute the stress intensity factor KI. The results for KI are graphically

presented as a function of the crack size and the ligament depth. The second part of the paper gives accurate

interpolation expressions for KI which take into account the asymptotic behaviour for vanishingly small

cracks and ligament depths. Finally, in the third an last part the KI values obtained from the interpolationexpressions are compared with some available results for symmetrical crack configurations and other

limiting cases. Mesh objectivity and some other aspects of computational precision are analyzed, and an

upper bound of the interpolation error is estimated.

2. Numerical analysis

A planar circular crack perpendicular to the fiber axis is the simplest idealization of many internal flaws

in fibers, usually flat pore-like defects [10]. This kind of crack configuration is three-dimensional in essence,and cannot be simulated by means of a two-dimensional crack. For this reason, and excluding certain

simple geometries such as those of infinite or semi-infinite solids, the computation of the stress intensity

factor along the crack front must be based on numerical methods. Well over other numerical techniques,

the finite element method has become the most used tool of analysis, and very accurate KI values can be

obtained with a proper design of the mesh, as was shown in a recent paper by the authors [11]. The fol-

lowing subsections present the specimen geometry, the mesh, and the numerical technique applied to ex-

tract KI factors. The computed values of the stress intensity factor and their dependence on crack size and

ligament depth are given at the end of the section.

2.1. Mesh geometry

The specimen geometry is shown in Fig. 1. A planar circular crack of radius r is placed normal to theapplied tensile stress at a distance b from the surface of the fiber. The fiber radius is R, and a is the distance

between the fiber axis and the crack center. The four geometrical parameters r, b, a and R are related by the

equation:

aRþ bRþ rR¼ 1 ð1Þ

A boundary condition of remote tensile stress was simulated by taking the fiber length, 2L as equal to six

diameters––12R––and by applying a uniform tensile stress r at the two extremes. No displacements or

rotations were imposed at the ends of the fiber, and no body load was considered.

Six relative ligament depths were investigated by taking b=R as equal to 0.005, 0.01, 0.05, 0.1, 0.3 and 0.5.

These values were combined with five different relative crack radii, r=R, equal to 0.05, 0.1, 0.2, 0.4 and 0.6.

To obtain smooth enough interpolation curves, other intermediate values (b=R, r=R) were analyzed where

appropriate. As a result, a total of 48 different specimen geometries were computed.

Fig. 2 shows the meshes used for r=R ¼ 0:1 b=R ¼ 0:5 and r=R ¼ 0:1 b=R ¼ 0:01 geometries. Finiteelement computations were carried out using the commercial finite element code ANSYS. Given the

L

L

σ

σ

r b

2 R

a

Fig. 1. Geometry and notation used for numerical computations.

G.V. Guinea et al. / Engineering Fracture Mechanics 71 (2004) 365–377 367

symmetry of the problem, only a quarter of the specimen was modeled. The mesh structure was the samefor all the geometries, and was based on a semi-torus with the crack front as the line of revolution, as shown

in Fig. 3. The radius of the torus was always equal to a tenth of the crack radius r or the ligament depth b,whichever was smaller.

A good angular discretization around the crack front was achieved in all the cases by placing at least six

15-node wedge elements surrounding the crack line––12 for the whole fiber if we take the symmetry into

account––(Fig. 3). This is a necessary condition when the stress intensity factor is determined by extra-

polation of the displacement field [11]. The r�1=2 stress singularity at the crack front was simulated by

shifting by a quarter the midside nodes of all surrounding elements, as illustrated in Fig. 3 [12]. ThePoisson�s ratio was taken as equal to 0.3.

(a)

(b)

Fig. 2. Some of the finite element meshes used for numerical computation: (a) r=R ¼ 0:1 b=R ¼ 0:5, (b) r=R ¼ 0:1 b=R ¼ 0:01.


The fiber volume outside the torus was meshed with 10-node tetrahedral solid elements whose size in-

creased uniformly from the crack front. On average, 60.000 nodes (180.000 degrees of freedom) were used

for each crack geometry, the meshes with b=R ¼ 0:005 being the most dense, with up to 190.000 nodes(580.000 degrees of freedom).

2.2. KI evaluation technique

Because of their technical significance, the stress analysis of three-dimensional cracks––often modeled as

elliptical cracks––have received detailed attention in the literature. The local stress field near the crack front

of an elliptical crack can be expressed––in a form analogous to the two-dimensional case––as a combi-

nation of the three plane modes of crack deformation: opening (I), sliding (II) and tearing (III). When the

coordinate system shown in Fig. 4 is considered, the stresses in the normal plane (n; b) in the proximity ofan elliptical crack are given by [13]:

Fig. 3. Detail of the finite element mesh around the crack tip.

b

d

nt

θ

,

,σtt utunσnn,

ubσbb

Fig. 4. Stresses and displacements in the plane normal to the elliptical crack front.


rnn ¼KIffiffiffiffiffiffiffiffi2pd

p cosh2

1

�� sin

h2sin

3h2

�� KIIffiffiffiffiffiffiffiffi

2pdp sin

h2

2

�þ cos

h2cos

3h2

�ð2aÞ

rbb ¼KIffiffiffiffiffiffiffiffi2pd

p cosh2

1

�þ sin

h2sin

3h2

�þ KIIffiffiffiffiffiffiffiffi

2pdp sin

h2cos

h2cos

3h2

ð2bÞ


rtt ¼ 2mKIffiffiffiffiffiffiffiffi2pd

p cosh2

�� KIIffiffiffiffiffiffiffiffi

2pdp sin

h2

�ð2cÞ

rnb ¼KIffiffiffiffiffiffiffiffi2pd

p sinh2cos

h2cos

3h2þ KIIffiffiffiffiffiffiffiffi

2pdp cos

h2

1

�� sin

h2sin

3h2

�ð2dÞ

rnt ¼ � KIIIffiffiffiffiffiffiffiffi2pd

p sinh2

ð2eÞ

rbt ¼KIIIffiffiffiffiffiffiffiffi2pd

p cosh2

ð2fÞ

where m is the Poisson�s ratio and KI, KII and KIII are the opening, sliding and tearing mode stress intensity

factors. The displacement field close to the crack front takes the form (Fig. 4) [14]:ffiffiffiffiffiffir ffiffiffiffiffiffir
un ¼
KI

2l2dpcos

h2

ð1�

� 2mÞ þ sin2 h2

�þ KII

2l2dpsin

h2

2ð1�

� mÞ þ cos2h2

�ð3aÞ

ub ¼KI

2l

ffiffiffiffiffiffi2dp

rsin

h2

2ð1�

� mÞ � cos2h2

�� KII

2l


rcos

h2

ð1�

� 2mÞ þ sin2 h2

�ð3bÞ

ut ¼KIII

2l


rsin

h2

ð3cÞ

where l ¼ E=2ð1þ mÞ is the shear modulus.

The stresses and displacements given by Eqs. (2) and (3) have a structure coincident with that of a two-

dimensional state of plane strain. In particular, from Eqs. (2) it is readily confirmed that rtt ¼ mðrnn þ rbbÞ.This fact allows the determination of the three-dimensional stress intensity factors by the same field ex-

trapolation methods as those used in two-dimensional problems.In this work the stress intensity factor in opening mode, KI, is evaluated by the displacement extrapo-

lation technique following the recommendations of the authors in [11]. For each given point O of the crack

front (Fig. 5), KI is determined from the displacements of the nodes of the singular element at the upper face

b

n

t

θ= πB

A3h/4

h/4

ubB

ubA

O

Fig. 5. Nodal displacements in a plane normal to the crack front.


of the crack––A and B––placed in the normal plane. The stress intensity factor has been computed from the

equation:

KI ¼E0

4

ffiffiffiffiffiffi2ph

rð4uAb � uBb Þ ð4Þ

where h is the length OB of the singular element, and uAb and uBb the nodal displacements. A detailed jus-

tification of Eq. (4) can be found in [11].

2.3. KI values for internal circular cracks

Figs. 6 and 7 plot in non-dimensional form the maximum stress intensity factor for the crack geometries

analyzed. The stress intensity factor was computed at the point of the crack front closest to the fiber

1.0

2.0

3.0

0.001 0.01 0.1 1

r/R=0.4

r/R=0.2

r/R=0.05

r/R=0.1

σ

σ

r/R=0.6

b2 R

a r

NO

ND

IMEN

SIO

NAL

STR

ESS

INTE

NSI

TY F

ACTO

R,

KI /

[ (2/

π) σ

( πr)1/

2 ]

RELATIVE LIGAMENT DEPTH, b/R

Fig. 6. Non-dimensional KI factor versus relative ligament depth.

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0 0.2 0.4 0.6 0.8 1RELATIVE CRACK RADIUS, r/R

NO

ND

IMEN

SIO

NAL

STR

ESS

INTE

NSI

TY F

ACTO

R,

KI /

[ (2/

π) σ

( πr)1/

2 ]

b/R=0.05

b/R=0.1b/R=0.3

σ

s

b/R=0.01

b/R=0.005

b/R=0.5

b2 R

a r

Fig. 7. Non-dimensional KI factor versus relative crack radius.


surface, where it reaches its maximum value. The dimensionless KI has been obtained by dividing it by the

reference value:

K0 ¼2

pr

ffiffiffiffiffipr

pð5Þ

which corresponds to the stress intensity factor for a circular crack of radius r in an infinite medium

subjected to remote tension r (penny-shape crack) [8].

Fig. 6 shows the dependence of KI on the relative ligament depth b=R for different crack radii r=R. Thestress intensity factor diverges when b=R tends to zero and has its minimum when the crack is centered with

the fiber. Curves in Fig. 6 are stopped at this point (a=R ¼ 0), that corresponds––recalling Eq. (1)––to aligament depth equal to 1� r=R.

-4.0

-2.0

0.0

2.0

4.0

1 104 2 104 3 104 4 104 5 104

NUMBER OF NODES

PER

CEN

T D

IFFE

REN

CE

IN K

Ir/R=0.1 b/R=0.01

r/R=0.1 b/R=0.5

Fig. 8. Variation of KI with the number of nodes for two meshes with different ligament. The values of the most dense meshes are taken

as reference.


It is worth noting that from a practical point of view KI differs from the value for the penny-shape crack

K0 (Eq. (5)) only for very small values of b=R. When the ligament is greater than a tenth of the fiber radius(b=R > 0:1) KI exceeds K0 by no more than 20% for most practical crack sizes (r=R < 0:3).

The effect of the relative crack radius in KI for a given ligament depth is illustrated in Fig. 7. The plot of

these curves stops at the point where the crack is centered with the fiber, r=R ¼ 1� b=R. As stated above,

for small values of the crack radius KI approaches the value given by Eq. (5) for a circular crack in an

infinite medium.

To check the mesh sensitivity and convergence, some selected cases were analyzed with meshes of in-

creasing density. The results are shown in Fig. 8, where the relative differences between the computed values

of KI are plotted as a function of the number of nodes used in the mesh. The reference values were thosecorresponding to the two meshes with the highest number of nodes. As seen in this figure, a good con-

vergence in KI is achieved when the number of nodes is high enough, typically above 50.000. As a practical

convergence criterion, no further refinement in the mesh was considered necessary when the differences

between two consecutive meshes were less than 0.6%.

3. Interpolation expressions for K I

The KI values obtained in the preceding section were numerically fitted to polynomial expressions to

simplify their use in practical situations. To help the fitting procedure, the asymptotic behaviour of small

cracks and ligament depths is taken into account. In addition, some known solutions for a crack centered

with the fiber axis are considered. The details are presented in the following subsections.


3.1. Asymptotic behaviour

As mentioned previously, the stress intensity factor reaches the maximum value at the point of the crack

front closest to the fiber surface, and may be written, without loss of generality, as:

KI ¼ K0f ðr=R; b=RÞ ð6Þ

where K0 is the stress intensity factor for a circular crack of radius r in a infinite medium loaded with a

tensile stress r, given by Eq. (5), and f is a shape function of the relative crack radius r=R and ligament

depth b=R.To compute an approximate expression for f , with a range of validity as wide as possible, it is worth

considering the limit behaviour of Eq. (6) when r=R or b=R tends to zero, and when r=Rþ b=R ¼ 1.For very small cracks; i.e. when the crack radius goes to zero while remaining constant the size of the

ligament, the KI values along the crack front tend towards the penny-shape crack solution K0. The shape

function f is, in the limit, equal to unity:

f ðr=R ! 0; b=RÞ ¼ 1 ð7Þ

When a crack with constant radius approaches the surface of the fiber, the stress distribution at the liga-

ment is similar, at first approximation, to the two-dimensional problem shown in Fig. 9. The semi-infinite

crack is subjected to remote tension, and no rotation is allowed at the loading points. The stress intensityfactor for this geometry can be found in Ref. [8], and its value is given by:

KI ¼pffiffiffiffiffiffiffiffiffiffiffiffiffi

p2 � 4p 2Pffiffiffiffiffiffi

pbp ð8Þ

where P is the load per unit thickness. The equivalence to the three-dimensional problem is obtained by

setting P equal to the total load on the fiber, rpR2, divided by the perimeter 2pR. The stress intensity factor

is then equal to:

b

P

P

Fig. 9. Two-dimensional semi-infinite crack subjected to tensile loading.

Table

Coeffic

i

1

2

3

4

5


KI ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi

pp2 � 4

rrRffiffiffib

p ð9Þ

and the shape function, in the limit, behaves as:

f ðr=R; b=R ! 0Þ ¼ p

2ffiffiffiffiffiffiffiffiffiffiffiffiffip2 � 4

p 1ffiffiffiffiffiffiffiffib=R

p 1ffiffiffiffiffiffiffiffir=R

p ð10Þ

A third limiting case is when the crack is aligned with the axis of the fiber, and a=R is equal to zero. From

Eq. (1) r=Rþ b=R ¼ 1 for this case, and due to the symmetry, KI is constant along the crack front. A closed-

form equation for KI, valid for any value of r=R, is found in [6,8]:

KI ¼ K0

1� 0:5r=Rþ 0:148ðr=RÞ3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� r=R

p ð11Þ

with K0 given again by Eq. (5). This expression for KI is claimed to have an accuracy better than 0.5% for

any r=R. The shape function in this case is readily obtained as:

f ðr=Rþ b=R ¼ 1Þ ¼ 1� 0:5r=Rþ 0:148ðr=RÞ3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� r=R

p ð12Þ

which is concordant with the limits given in Eqs. (7) and (10): when r=R goes to zero f is equal to 1, and

when r=R approaches unity f diverges with the ligament b=R ¼ 1� r=R as p=2ffiffiffiffiffiffiffiffiffiffiffiffiffip2 � 4

pðb=RÞ�1=2

� 0:648ðb=RÞ�1=2.

In the next subsection, Eqs. (7), (10) and (12) which give the behaviour of the shape function f in the

three limiting cases; r=R ! 0, b=R ! 0, and r=Rþ b=R ! 1, are used to seek an accurate expression that

interpolates KI in a wide range of values (r=R, b=R).

3.2. Polynomial interpolation of KI

The numerical data for the 48 computed geometries are fitted by the least square method to the ex-pression given in Eq. (6). To fulfill automatically the conditions settled by Eqs. (7) and (10), an appropriate

structure for the shape function has been devised by adopting for f the following polynomial form:

f ðr=R; b=RÞ ¼ 1þX5

i¼1

Ci0ðr=RÞð2iþ1Þ=2 þX3

i¼1

Ln½1þ ðr=RÞ2i� Ci1Ln2½ðb=RÞðr=RÞ�(

þ Ci2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr=RÞðb=RÞ

p)

ð13Þ

1

ients Cij

Ci0 Ci1 Ci2

+1.242� 10�2 )3.097� 10�1 +1.185� 10þ0

)6.388� 10þ0 +1.547� 10þ0 )3.723� 10þ0

+1.689� 10þ1 )8.769� 10�1 +2.628� 10þ0

)9.838� 10þ0

)1.228� 10þ0


where the coefficients Cij are given in Table 1. The condition prescribed by Eq. (12) for centered cracks is

indirectly introduced in (13) by forcing the numerical fit to satisfy the values obtained with (12) for a

discrete set of points from r=R ¼ 0:05 to 0.9. The mean quadratic errors resulting from the fitting are under

1%, a figure comparable to the accepted accuracy of the finite element method.

4. Error estimation and final comments

Like other numerical methods, the displacement extrapolation technique used in this paper does not give

directly an upper bound of the estimation error for KI. As an approximate figure, the evaluation error can

be well under 1% when some good-practice recommendations are taken into account [11], but a detailed

assessment requires an independent estimation based on other theoretical or numerical solutions. Unfor-

tunately, to the authors� knowledge no other results––apart from the limit solution for a centered crack

given by Eq. (11)––are available for the problem under consideration when not too small cracks and very

small ligaments are considered, for geometries where KI differs significantly (more than a few percent) fromthe penny shape solution K0.

Based on these comments, the symmetric crack configuration is taken as a reference to check the per-

formance of the meshing procedure and to estimate the numerical error for KI. To this end, some selected

meshes with inner centered cracks of r=R ¼ 0:1 to 0.9 were analyzed, and their results compared with those

predicted by Eq. (11). Mesh structure, size and quality were kept similar to that used for eccentric cracks.

Fig. 10 shows the difference in percentage between the computed values of KI and the reference solution

given by (11). From these results it can be noticed that the numerical error is well under 0.5% for all the

cases, and although this value cannot be extrapolated directly to other crack configurations, it furnishes afigure of reference for the computations presented in this paper.

-0.4

-0.2

0.0

0.2

0.4

0 0.2 0.4 0.6 0.8 1

RELATIVE CRACK RADIUS, r/RPER

CEN

T D

IFFE

REN

CE

IN K

I, 10

0 x

(KI,C

OM

PU

TE

D -

KI,

Eq.

11)

/ K

I, E

q.11

Fig. 10. Fiber with a centered crack (a=R ¼ 0): Deviation from the reference value given by Eq. (11).


Acknowledgements

The authors gratefully acknowledge financial support for this research provided by the Comision In-

terministerial de Ciencia y Tecnolog�ııa (Spain) under grants MAT2000-1334 and MAT 2000-1355.

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Documents

Stress intensity factors for internal circular cracks in fibers under tensile loading