Xiao&Karihaloo(2002)

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International Journal of Fracture 118: 115, 2002.

2003Kluwer Academic Publishers. Printed in the Netherlands.

Coefficients of the crack tip asymptotic field for a standard

compact tension specimen

Q.Z. XIAO and B.L. KARIHALOODivision of Civil Engineering, School of Engineering, Cardiff University, Cardiff, CF24 0YF, UK;

e-mail: [email protected]

Received 12 March 2002; accepted 9 April 2002

Abstract. The coefficients of the crack tip asymptotic field of a standard compact tension (CT) specimen are

computed using a hybrid crack element (HCE). It allows the direct calculation (without post-processing) of not

only the stress intensity factor (SIF) but also the coefficients of higher order terms of the crack tip asymptotic field.

Approximate closed-form expressions for the first five terms for the CT specimen that are accurate for shallow

to very deep cracks are obtained by fitting the computed data. The SIF formula proposed by Brown and Srawley

(1966) is shown to be accurate when the crack length to depth ratio () ranges from 0.35 to 0.75. The formulaproposed by Newman (1974) and Srawley (1976) is accurate for 0.15. However, the accuracy of availableformulas for the second T-term in the literature is quite disappointing. Numerical results also show that, unlike

the notched three-point bend beam and the wedge splitting specimen, the second T-term of the CT specimen is

always positive.

Key words: Coefficients of crack tip asymptotic field, compact tension (CT) specimen, hybrid crack element,

stress intensity factor (SIF),T-stress.

1. Introduction

The standard compact tension (CT) specimen (Figure 1) is widely used for determining the

fracture toughness of metallic materials. If the crack with traction-free faces lies on the neg-ative x-axis, and the polar coordinates centred at the crack tip are designated r and ( is

measured counterclockwise from the positive x-axis) as in Figure 1, the displacement and

stress fields near the tip of the crack may be expressed in the so-called Williams expansion

(Williams, 1957; Owen and Fawkes, 1983; Karihaloo and Xiao, 2001ac):

u =

n=1

rn2

2an

+ n

2+ (1)n

cos

n

2 n

2cos(

n

2 2)

(1)

=

n=1r

n2

2an

n

2 (1)n

sin

n

2+ n

2sin(

n

2 2)

(2)

x=

n=1

n

2r

n21an

2 + n

2+ (1)n

cos(

n

2 1) ( n

2 1) cos( n

2 3)

(3)

y=

n=1

n

2r

n21an

2 n

2 (1)n

cos(

n

2 1)+ ( n

2 1) cos( n

2 3)

(4)


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2 Q.Z. Xiao and B.L. Karihaloo

Figure 1. A standard compact tension (CT) specimen. h= 0.6b, h1 = 0.275b, D = 0.25b, c= 0.25b, andthickness = b/2.Pis the load per unit thickness.

xy=

n=1

n

2r

n21an

(

n

2 1) sin( n

2 3)

n2

+ (1)n

sin(n

2 1)

(5)

where= E/(2(1 + )) is the shear modulus; the Kolosov constant = 3 4 for planestrain or = (3 )/(1 + ) for plane stress; E and are Youngs modulus and Poissonsratio, respectively.a1is related to the mode I stress intensity factor (SIF) KI as

a1=KI

2(6)

The SIF which controls the first (i.e. the singular) term of the crack tip asymptotic field

(15) has been used for years as the single controlling parameter for the initiation and prop-

agation of a crack in brittle materials and materials with limited ductility. However, recent

studies show higher order terms of the asymptotic field are of great relevance to predicting the

constraint of elasto-plastic crack tip fields (ODowd and Shih, 1991; Du and Hancock, 1991;

Karstensen et al., 1997; Nikishkov, 1998) and to interpreting the size effect of quasi-brittlematerials (Karihaloo, 1999; Karihaloo, et al., 2002).

The second term in (15) corresponds to a uniform in-plane stress

x= T= 4a2 (7)

acting near the crack tip in the direction parallel to the crack plane. This uniform stress is often

referred to as the elastic T-stress.

For the standard CT specimen shown in Figure 1, a SIF expression is given by Brown and

Srawley (1966) for the crack length to depth ratio 0.45 < = a/b 0.2 (see also,

Tada et al., 1985; Fett, 2002)

KI=2P (2 + )b(1 )3/2

(0.443 + 2.32 6.662 + 7.363 2.84) (9)


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Coefficients of the crack tip asymptotic field for a standard compact tension specimen 3

For the elastic T-stress of the standard CT specimen (Figure 1) with 0.2 0.7,the solutions of Cotterell (1970), of Leevers and Radon (1982), and of Kfouri (1986) can be

respectively fitted as (Sherry et al., 1995)

T

= 6.063 78.987 + 380.462 661.793 + 428.454; (10)

T

= 1.996 + 10.169 + 10.5462 (11)

and

T

= 2.616 + 8.019 + 16.4212 (12)

Sherry et al. (1995) showed that the above solutions are identical only for 0 .5 < < 0.6.

Recently, Fett (2002) gave the following expression for the biaxiality ratio B

B= T

a

KI= 0.2 + 2.0307 + 0.675

2 7.47563 + 6.3494 1.077251

(13)

As for the coefficients an of terms higher than order two (n > 2), to the authors best

knowledge, no reference solutions are available in the published literature.

Recently, the authors (Karihaloo and Xiao, 2001ac) extended the hybrid crack element

(HCE) originally introduced by Tong et al. (1973) for evaluating the SIF to calculate directly

(i.e., without the use of the energy related quantities like the J-integral, or other extra post-

processing) not only the SIF but also the coefficients of the higher order terms of the crack tip

asymptotic field. Extensive studies have proved the versatility of the element. In this paper, it

will be used to compute the coefficients anof the crack tip asymptotic field of the standard CT

specimen (Figure 1).

Approximate expressions for the SIF, as well as coefficients an of the higher order terms

that are accurate for shallow to very deep cracks will be given by fitting the computed results.

The computed SIFs and theT-stress will also be used to validate the known results mentionedabove.

2. Definition of the dimensionless shape functions

Consider the standard CT specimen shown in Figure 1, the SIF, theT-term and the coefficients

a3, a4and a5of the third to fifth order terms may be normalized to be dimensionless as follows:

k() = KI

2 b

= a1

b

(14)

t() =T

4 =a2

(15)

g3() =a3

/

b(16)

g4() =a4

/b(17)


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Figure 2. A typical n-node polygonal HCE. (a) For mixed mode crack; (b) For a pure mode I or mode II crack.

g5() =a5

/b1.5 (18)

With the use of (14), KI(8) can be normalized as

k() = 12

(29.61/2 185.53/2 + 655.75/2 1017.07/2 + 638.99/2), (19)

andKI(9) can be normalized as

k() = 2(2 + )2 (1 )3/2

(0.443 + 2.32 6.662 + 7.363 2.84) (20)

Formula (13) can be rewritten as

T

=

2k()

B. (21)

3. HCE for accurate determination of SIF and coefficients of higher order terms

In this section, for completeness, we will discuss briefly an n-node polygonal HCE with p-

adaptivity as shown in Figure 2. For details, refer to Karihaloo and Xiao (2001a). We ignore

the body force and assume that no element boundary displacements have been prescribed. The

simplified variational functional for formulating the HCE


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em=

Se

(1

2Ti Ti )ui ds

Ae Ti (

1

2uiui )ds (22a)

or in matrix form

em= SeuT(

1

2T T)ds

Ae(

1

2uT

uT)T ds (22b)

can be obtained from either the modified elemental Hellinger-Reissner functional or the po-

tential functional with relaxed continuity requirements after enforcing the equilibrium and

compatibility conditions within the element. The boundary of the element, Ae, is composed

of the segment Seon which the tractions Ti are prescribed and the interelement boundary,

Ae = Ae Se, common with the adjacent elements. Displacements ui and boundarytractionsT (= ijnj) are independent of the other elements, but boundary displacementsuihave to be the same for the two elements over their common boundaries Ae. ij denotestresses, andnjare the direction cosines of the unit outward normal to A

e. Subscripts i and

jtake the values 1 and 2, and the summation convention on repeated indices is used.

Note that the simplified functional (22) is only stationary with respect to variations in

stressesij

or boundary tractions Ti, or equivalently, in displacements u

iand boundary dis-

placementsui . A compatible displacement field is required only along the element boundariesinstead of the entire element, thus an n-node polygonal element with p-adaptivity can be

easily formulated.

In the formulation of elements using the simplified functional (22), the assumed element

displacement and stress fields, ui and ij, should meet the equations of equilibrium and the

stress-displacement relations. They should not include any rigid body or zero energy modes;

otherwise the inversion of matrix Hin (32)(34) in the following is impossible. If the trun-

catedNterms of the Mode I crack tip asymptotic field (15) are used in assuming the element

displacementsui and boundary tractions Ti , we have

u=

N

n=1 f1n(r,)

f2n(r,) an= U (23)

=N

n=1

f3n(r,)

f4n(r,)

f5n(r,)

an= P (24)where functions fin(r,), i= 1, . . . , 5, can be easily identified from (15). Obviously, thecomponents of vector are the coefficients an.

The boundary tractions are

T=

R (25)

withR=

n1 0 n20 n2 n1

P. The interelement boundary displacements ui may be written in a

matrix form as functions of the nodal displacements

u = Lq (26)


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The L matrix will be constructed in such a way thatu varies linearly between any two adjacentnodes in order to connect it compatibly with the respective standard linear elements. Consider

the boundary segment between two adjacent nodes, i and i+ 1, as shown in Figure 2. Thelinearly varying

ubetween nodesi and i + 1 can be written as

u = N1 0 N2 00 N1 0 N2

ui

i

ui+1

i+1

(27)whereN1= (1 )/2 andN2= (1 + )/2 are shape functions, and (ui , i ) the displacementsat node i . The local coordinate is = 1 at node i , and = 1 at node (i + 1). Thex andycoordinates within the segment are

x

y = N1 0 N2 0

0 N1 0 N2

xi

yi

xi+1

yi+1

(28)

where (xi , yi ) are the coordinates of node i. From (28), we have the determinant of the

Jacobian matrix J=

(xi+1 xi )2 + (yi+1 yi )2/2, which is half the length of the seg-ment. Substituting (23), (25) and (26) into functional (22), we can express the elemental

characteristic matricesH,G and Q as follows

H = 12

i,i+1Ae

(xi+1,yi+1)(xi ,yi )

(UTR + RTU)ds

i,i+1Se


(UTR + RTU)ds

= 12

i,i+1Ae 1

1(UTR + RTU)Jd

i,i+1Se 1

1(UTR + RTU)Jd (29)

G =

i,i+1Ae (xi+1,yi+1)

(xi ,yi )

LTRd s=

i,i+1Ae 1

1LTRJd (30)

Q =

i,i+1Se


UTT ds=

i,i+1Se

11

UTT J d (31)

Integrations on

Ae are performed only along the element outer boundary away from the crack

tip, thus avoiding errors due to the area integration of the singular integrand. Karihaloo andXiao (2001a) show that three-point Gauss integration ensures good accuracy. Integrations on

crack faces Semay be carried out analytically; if they are traction-free, as in most cases, the

integrals vanish. The element stiffness matrix

Ke = GH1GT, (32)

the consistent nodal force vector


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fe = GH1Q, (33)

and the coefficients of the crack tip asymptotic field can be obtained via

= H1GTq H1Q. (34)

The element must satisfy the following condition for stability

n nq nr (35)

where n and nq are the number of element stress and nodal displacement parameters em-

ployed in (24) and (26), respectively, and nr (= 3 for two-dimensional problems) the numberof independent rigid body modes. If (35) becomes an equality, the best parameter matching

condition is attained.

If we restrict our attention to pure mode I or mode II crack problems, the half polygonal

element with n nodes shown in Figure 2(b) may be used for constructing the HCE by exploit-

ing the symmetric (mode I) or asymmetric (mode II) conditions along the line of extension of

the crack. As

u2= 0 andT1= 0 in case of mode I,and

u1= 0 andT2= 0 in case of mode II,

we need only integrate along the outer boundary and avoid integrations along the line of

extension of the crack. For the case of pure mode II, the second term in the Williams expansion

should be dropped in the element formulation, otherwise it will result in a spurious zero energy

mode.

4. Finite element computations

Without loss of generality, the width b is chosen as 1. Only one half of the specimen needs

to be considered because of symmetry. The load Pis assumed to be 1 per unit thickness, and

applied on the specimen through three adjacent nodes with amplitudes of 0.25, 0.5 and 0.25,

as in Chan et al. (1970) who derived the SIFs by extrapolating the displacement to the crack

tip. The crack length to depth ratio is varied from 0.05 to 0.8.

Karihaloo and Xiao (2001a) showed that the accurate determination of the coefficients of

higher order terms requires a higher order HCE together with a finer discretisation of the

remainder of the body. In order to guarantee the accuracy of the HCE, finite elements for

modelling the remainder of the body should give results of the far field with high accuracy.

Thus a 21-node HCE (the first 39 terms in (15) are included in the element formulation)

together with a relatively fine discretisation of the remainder of the body will be used. Three-

point Gauss integration is used in the HCE. The PS element (Pian and Sumihara, 1984; Pianand Wu, 1988) is used in conjunction with the HCE, and the traction-free conditions on the

exterior boundary, as well as the loading hole are exactly satisfied using the special hybrid

stress boundary element HBE (Xiao et al., 1999). 2 2 and 3 3 Gauss quadratures areemployed for the formulation of PS and HBE, respectively. All the computations can be

carried out on an advanced PC. The finite element meshes used in the computations are shown

in Figure 3. The coordinate system used is the same as that shown in Figure 1. The HCE is


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Figure 3. Finite element meshes: (a) = 0.05, a HCE surrounded by 1450 four-node elements giving a total of1560 nodes; (b)

=0.1, a HCE surrounded by 1200 four-node elements giving a total of 1300 nodes; (c)

=0.2,

a HCE surrounded by 1050 four-node elements giving a total of 1140 nodes; (d) = 0.3, a HCE surrounded by750 four-node elements giving a total of 830 nodes; (e) = 0.4, a HCE surrounded by 850 four-node elementsgiving a total of 935 nodes; (f) = 0.5, a HCE surrounded by 750 four-node elements giving a total of 830nodes; (g) = 0.6, a HCE surrounded by 850 four-node elements giving a total of 935 nodes; (h) = 0.7, aHCE surrounded by 750 four-node elements giving a total of 830 nodes; (i)= 0.8, a HCE surrounded by 850four-node elements giving a total of 935 nodes.


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Figure 3. Continued.

symmetric about they -axis; it is a rectangular with its length (inx-direction) twice that of itsheight (in y -direction). The scaled coordinate axes are also included in Figure 3 to show the

location of the HCE relative to the rest of the mesh. The total number of elements and nodes

are indicated in the figure caption. Instead of the element combinations above, if we use only

general non-singular or singular elements together with the meshes in Figure 3, accurate SIFs

may be obtained via the J- or alternative energy related integrals. However, the coefficient of

each higher order term needs to be determined separately with special techniques. Generally,


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the accuracy deteriorates quickly with the increase of the order of the term. On the otherhand, the HCE determines the SIF as well as the higher order terms simultaneously with high

accuracy.

As the coefficients an in the asymptotic expansions (15) are independent of the material

constants, in the computations Youngs modulus Eis set at 1, and Poissons ratio at 0.25. The


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units of loading (Figure 1) are consistent with that ofE . A state of plane stress is consideredwith thickness assumed to be 1.


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Table 1. Results of the dimensionless shape functions for the first five

terms of a standard CT specimen.

= a/b k() t () g3() g4() g5()

0.05 1.2173 0.9173 4.2158 1.3749 18.63650.10 1.3060 0.7493 2.7925 3.3874 18.08190.20 1.7031 0.3717 1.4404 2.2262 5.39630.30 2.2390 0.4079 2.1290 0.0377 0.40760.40 2.8949 0.6950

3.4584 0.6544

0.6038

0.50 3.8343 1.0378 5.2621 0.6269 0.38670.60 5.4313 1.4325 8.6699 0.8078 2.40170.70 8.5914 2.1841 17.7671 2.7625 12.13560.80 16.3894 4.7180 52.0358 13.5101 64.6699

5. Results

For the standard CT specimen shown in Figure 1, the computed dimensionless shape functions

corresponding to coefficients an(1

n

5)are tabulated in Table 1. These results have been

fitted with the following closed-form expressions to a very high accuracy

k() = 373.085 567.334 + 321.473 73.1242 + 9.8345 + 0.8436 (36)

t() = 294.235 520.34 + 320.883 71.4182 + 2.3668 + 0.9443 (37)

g3() = 4774.16 + 9231.35 70834 + 2799.13 664.262 + 94.265 7.6254(38)


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Figure 4. Variation ofk()with.

Figure 5. Variation oft()with.

g4() = 4013.86 101915 + 106884 5797.43 + 1639.32 204.27 + 5.4205 (39)

g5() = 248196 + 604845 584354 + 277603 6422.22 + 563.07 + 3.4375 (40)

The computed k() are compared with formulas (19) of Brown and Srawley (1966) and

(20) of Newman (1974) and Srawley (1976) in Figure 4. It is clear that formula (19) results of

the SIF agree very well with the present finite element method when the crack length to depth

ratio is in the range 0.35 0.75. Formula (20) is identical to (36) when >0.15.


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The computed t ()are compared with formulas (10) of Cotterell (1970), (11) of Leevers

and Radon (1982), (12) of Kfouri (1986), and (21) of Fett (2002) in Figure 5. From this

comparison, it is obvious that the accuracy of formulas (1012) is quite disappointing since

they deviate from the finite element results. The accuracy of the latter has been well established

(Karihaloo and Xiao, 2001ac). Formula (21) of Fett (2002) gives reasonable results only for

0.45 <


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