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Produced by the NASA Center for Aerospace Information (CASI)

https://ntrs.nasa.gov/search.jsp?R=19840003445 2018-05-11T19:30:45+00:00Z
NASA Contractor Report 168280

(NASA-Ct-118180) EFFECT OF CRACK CURVATURE 984-1 15 13

011 5111333 INTIMSITY FACTORS 102 ASTA

STANDARD COU ACT TENSION SPECININS Final

Report (Casa Nestor& Reserve Univ.) 19 P Unclas

UC E02.,/Ut 301 CSCL 20K G3/39 42464

EFFECT OF CRACK CURVATURE ON STRESS INTENSITY FACTOkS FOR ASTM

STANDARD COMPACT TENSION SPECIMENS

s

Javed Alam and Alexander Mendelsen

Case Western Reserve University

Cleveland, Ohio

NOV 103

RECEIVED

WM 911 sm FNSM

Mo.

It

October 1383

Prepared for

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

Lewis Research Center

Under Grant NSG3-251
INTRODUCTION

The use of linear elastic fracture mechanics in the design of structures

requires the knowledge of the fracture toughness of the material. For this

purpose the ASTM has established s standard test method, (E399) for obtaining

the plane-strain fracture toughness of a given material, [1]. One of the

standard specimens used in such a test is the compact tension specimen (CTS)

shown in figure 1.

Very rigid standards have been imposed in reference [1] to insure that

valid fracture toughness values will be obtained in a given test. One of the

requirements is the production of a fatigue crack. However, the fatigue crack

will usually not grow uniformly across + .e specimen thickness, i.e. the initially

straigth crack front will become curved. The crack length thus varies across

the specimen thickness and some average value must be used to calculate the

fracture toughness. A standard measure is provided in reference [1] such that

if the crack front curvature, as measured after the specimen is broken,is greater than this standard measure, the test is invalid.

Recently, however, suggestions have been made,[2] that the above standard

is too rigid and can be relaxed since the effect of the curved crack front may

be less than originally anticipated. These suggestions, however, are based

on rather weak evidence, [3] and it is not yet clear whether the E 399 standard

should be relaxed or not in this respect. What is needed is a clear definitive

analysis of the effect of the crack front curvature on the stress intensity

factor. The objective of this investigation is to perform this analysis.

Several three-dimensional finite element analyses have been made of the

compact tension specimen, e.g. [4,5]. These analyses showed that for the CTS,

the stress intensity factor is a maximum at the center of the specimen and a

1
minimum at the surface of the specimen. When crack curvature is introduced,

the maximum stress intensity factor moves to the surface of the specimen and the

minimum occurs at the center. The dimensions of the specimens treated in these

analyses differed from those of the standard specimen. A few attempts have also

been made to determine experimentally the effect of crack tip curvature using

a 3-D stress freezing technique [6], and scattered light speckle interferometry

[7]. Both these investigations showed qualitative agreement with the above

results.

The present investigation uses the method of lines (MOL) to obtain an

accurate three-dimensional solution for the standard CTS with ^urved crack

fronts. The MOL is described fully in references [8] and [9], where it is also

shown that accurate solutions of crack problems can be obtained. An improved

version of the method and the application to the standard CTS with a straight

crack front has been presented in a previous paper,[10]. The method will be

only briefly reviewed here for completeness.

BACKGROUND OF THE METHOD OF LINES

Within the framework of linearized elasticity theory, the equations of

elastic equilibrium in terms of displacements are

G Q2 ui + (a + G) 9 'i = 0 , i = 1, 2, 3 (1)

9-Ui1i

where X and G are Lame's constants, u are the displacement and the usual tensor

notation is used.

For a finite geometry solid we construct three sets of parallel lines,

Fig. 2. Each set of lines is parallel to one of the coordinate axes and thus

perpendicular to the corresponding coordinate plane. An approximate solution

2

^,R

no , - -
3

in refe _ence [9].

of Eq. (1) can then be obtained by developing solutions of ordinary differential

equations along ti ► e x-directional lines. As seen in the figure, there are a

total of Z = NY x NZ such lines where NY is the number of lines along the

y-direction and NZ is the number of lines alung the z-direction in a given

plane, respectively. We define the displacements along these lines as u l , u2,

u C The derivatives of the y-directional displacements on these lines

with respect to y are defined as v'1 10 v'1 2 , . . ., v1 1V and the derivatives

of the z-directional displacements with respect to z are defined as w') , w'I

1 2

. ., w'I p . These displacements and derivatives can then be regarded as

functions of x only since they are variables on x-directional lines. When these

definitions are used, the ordinary differential equation along a generic lire

ij using a 5-paint difference formula, may be written as:

2

2 2(1 -2 [ v) 12h 2 ( ui-2,j - 16u 30u i-l'j + ij - 16ui+l,j

dx y

+ u

i+2,j ) + 12h z 2 ( ui,j-2 - 16ui^j-1 + 30u ij - 16u

i1j+1 +

f (x)

ui.j +2 )I + dx 2(j( ) = 0 (2)

where

fij(x) v' + w'; v' _ dy and w'

dz (3)

Similar differential equations are obtained along the other x-directional lines.

Since each equation has the terms of the displacement on the surrounding lines,

These equations constitute a system of ordinary differential equations for the

displacements u

i 2

, u . . . , u V It is to be noted that for lines lying on or

near a boundary appropriate noncentral difference formulas must be used. Also,

the shear boundary conditions are incorporated in these equations as described
The set of R second order differential equations represented by Eq. (2)

can be reduced to a set of 2R, first order differential equations by treating

the derivatives of the u's as an additional set of k unknowns.

du dug

u Q+l s dx ' 

u

R+2 dx etc. (4)

The resulting 2k equations can now be written as a single first order matrix

differential equation

dU m A U + dR(x) (5)

dx 1 dx

where U and R are column matrices of 2R elements each and A l is a 2k x 29, matrix

of the constant coefficients appearing in Eqs. (2) and (4).

In a similar manner, to solve Eq. (1) for i = 2 and 3 ordinary differential equations are constructed along the y- and z-directional lines, respectively.

These equations are also expressed in an analogous form to Eq. (5); they are

d_V = A V + dS^ (6)

dy 2 dy

dW

A W + dT(z) (7) aZ 3 d z

Equations (5) to (7) are linear first-order ordinary matrix differential

equations. They are, however, not independent, but are coupled through the

vectors R, S, and T whose components are given by equations similar to Eq. (3).

They must therefore be solved iteratively correcting R, S and T after each

iteration. The elements of the coefficient matrices A , A , and A are all

1 2 3

constants, being functions of the coordinate increments and Poisson's ratio

only.

To solve equations (5) - (7) a recurrence relation algorithm was set up

as described in detail in reference (10). This algorithm is particularly useful

4
for solving two-point boundary value problems, requiring one sweep through from

f one boundary to the opposite one to obtain the missing boundary conditions and

a second sweep through to obtain the complete solution everywhere.

OF POOR QUALI-f'Y

CALCULATION OF STRESS INTENSITY FACTOR v

The stress intensity factor K is defined as follows:

K I = kim ay(27R)n

R-►O

where R is measured from the crack tip and is normal to the crack front. n is

the singularity. It was found, however, that due to the coarseness of the grid

used, the usual plotting and extrapolating techniques gave results that were

erratic and of questionable accuracy. This was compounded by the fact that the

precise crack tip location is not really known except that it is approximately

midway between two lines, one of which has zero stress specified. It was found,

however, that by using two terms in the stress and displacement series expansions

around the crack tip, good re.olts could be obtained even with the coarse grid

used. Furthermore, this also permitted the determination of the actual crack

tip location from the computed results. The method utilized is as follows, [11],

we take

L

v1 y=0 = aKI 27rr + KI (R —+r ) (8)

I

L

cy l ys0 KI [ 1 + KI

(R - r ] (9)

2n(^ I

where a is a function of Poisson's ratio, n was assumed to be ', and r is the

crack edge position corr•ction measured from the originally assumed midpoint

position. Using displacement data from three adjacent nodes to the crack edge

5 
OF POOR QUALITY

in Eq. (8), values of a K I LI /KI and r are calculated for etch value of z,

with R also measured from the halfway point between nodes specifying boundary

stresses and displacements, respectively. Note that a would be equal to 3.56

for the plane strain case and 4.0 for the plane stress case. In the case of

curved crack fronts a normal is drawn at the point where K is determined. The

displacement value p at the points where the normal meets the grid lines are extrapolated using the nodal displacemer.t values. These extrapolated values of -

placement are used to obtain the local K values for the curved crE_k fror

RESULTS AND DISCUSSION

The MOL was first applied to a compact tension specimen with tensile loading. The specimen dimensions were W/a - 2.0, B/a — 1.0 and L/a - 2.4. Where

a, B, W and L are the crack length, specimen thickness, width and length respectively. Crack length to width ratio was maintained close to 0.5. Figures 2

and 3 show the coordinate system and line number designation.

Figure 4 shows a plot of the local stress intensity factor variation

through the thickness of the specimen with tensile loading. Curve 1 is for a

straight crack front. For this case the maximum SIF occurs at the center of

the specimen. It drops by 13 percent on the surface. This trend is in accordance with the previously reported results of Raju (12).

The remaining curves in the figure are for increasing crack tunnel depth.

Crack tunnel depth is defined as the difference between the crack lengths at

the center and the surface of the specimen. Since approximate parabolic curves

were used in the study, it was difficult to assign a single crack curvature for

the crack front. An increasing crack tunnel depth reflects an increasing crack

curvature. For crack tunnel depth of 0.1 we see that SIF decreases at ti;e center

b y 9.5 percent, while at the surface its value increases by 20.5 percent. Simi-

lar type of variation is reported by Pereira [4]. However, a direct comparison ui

6
ORIGINAL PAC; JS

Of POOR QUALI'1Y

the results is not possible because a different crack and specimen size were

used in that study. Herein, the dimensions pre€cribed in ASTM standard [1]

are used.

For higher crack tunnel depths the maximum SIF value no longer occurs at

the surface, but shifts to the adjacent interior point of the grid. This

trend is not reported in reference [4]. There could be two possible reasons

for this differece, 1) the crack length a/W used in reference [4] is 0.25

which is smaller than the one used in the present study, 2) three layers of

elements were taken in tie thickness direction, possibly causing the maximum

SIF to be missed.

Figure 5 shows the Variation of the non-dimensional SIF for the same

specimen with a parabolic shear load along the load line. The results are

similar to those of fig, i,e 4 for the tensile loading.

Figure: 6 shows the variation of center (c), surface (S) and thickness

average (av) stress intensity factors with increasing crack tunnel depth for a

F t_andard compact tension specimen under shear loading. The stress intensity

factors are non-dimensionalized with the respective values of SIF occuring for a straight crack front (SC). We find that non-dimensional SIF value at

the center of the specimen keeps on decreasing as the crack tunnel depth increases. However, the value of SIF at the surface increases to a maximum and

then decreases. The thickness average SIF constantly decreases with increasing

crack tunnel depth. Similar trend for the thickness average SIF was observed

by McGowan [6] in his study on single edge notch specimen. This figure shows

clearly that the presence of a curved crack front can significantly change the

value of the stress intensity factor.

In figure 7 the change in the thickness average SIF is plotted with a nondimensional factor n. The value of n corresponds to the difference between the

average crack length (a ) and the surface trace of the crack divided by the

av

7 
average crack length (a av ). The calculation of a av is done by computing an

average of the three c'-ack measurements taken at the center of the crack front

and midway between the center and the end of the crack front on each side, which

is the definition used fn [1). Reference [1) restricts the length of either

surface trace of the crack to greater than 90 percent of the average crack length

(aav). Based on this criterion, the ASTM region -!s marked on the figure 7. Any

test in which the specimen has a surface trace within the marked region is considered a valid tent. For the worst case when the surface trace is 90 percent

of a av in the specimen, there is a 7 percent lowering in the thickness average

stress intensity factor, which indicates that the use of plane strain formula

-( given in reference (1) will result in an overestimation of the fracture toughness

of a material by 7 iercent.

On the basis of the experimental results and three-dimensional analysis of

Neale and Pereira [4,5], an amendment has recently been incorporated, [.3) to

extend the limit on the surface trace to 85 percent of at-rage crack length.

This figure corresponds to a n value of 15 percent. From the figure 7 we find

a lowering of 12.5 percent in the thickness average SIF for this value of n.

This implies that the use of two-dimensional plane strain formula will result

in an overestimation of the fracture toughness of a material by 12.5 percent.

These figures thus show the effect of crack curvature on the overestimation of

the fracture toughness for the compact tension specimen.

CONCLUSIONS AND RECOMMENDATIONS

1) The results from the numerical analysis show that the stress intensity factor

decreases at the center of the compact specimen due to crack curvature. For

the low values of crack tunnel depth the maximum stress intensity factor Is

found to be at the surface of the specimen. If the crack tunnel depth is

8
Increased, the maximum value of stress intensity factor no longer remains

at the surface, but shifts to an interior point of the specimen, still

remaining close to the surface.

2) The average value of stress intensity factor through the thickness keeps on

a

decreasing with increasing crack tunnel depth.

3) The ASTM test method L11 allows either surface trace of the crack to be less

than 90 percent of the average crack length. Based on an average SIF calculation, this will overestimate the fracture toughness by 7 percent. It is

further estimated that the new amendment changing the 90 percent to 85 percent will lead to an overestimation of fracture toughness by 12.5 percent.

The plots for stress intensity factor through the thickness of compact

tension specimen for two different types of loading predict that the crack growth

should start at the surface of the specimen first. This conclusion from analytical study is contrary to the experimentally observed crack propagation. It is

anticipated that plastic flow can significantly change the stress conditions

existing at the surface, which is, of course, not accounted for in an elastic

analysis. Neale [14] developed an elasto-plastic model to fix this discrepancy,

but it is an approximate analysis. To fully study the problem, we need to

analyze a full three-dimensional elastic-plastic case. The results of such a

study will be presented in a subsequent publication.
RZFERENCES

1. Standard Test Method for Plane-Strain Fracture Toughness of Metallic

Materials, ANSI/ASTM E399-78.

2. Prt., ate cosimunication from J.G. Kaufman. Chairman of ASTM Committee E-24

oh Fracture Testing to W.F. Brown, Jr., NASA Lewis Research Center,

Cleveland, Ohio.

3. Private Communication from W.F. Brown, Jr., to J.G. Kaufmann.

4. Pereira, M.F.S., Head, J.L. and Turner. C.E., "Three Dimensional Crack

Analysis Using Standard Isoparamatrie Elements", Report of the Imperial

College of Science and Technology, London.

S. Neale, B.K., "The Influence of Crack Shape on Fracture Toughness Testing",

Int. J. of Fracture. Vol. 12 (1976), p. 499-502.

6. McGowan, J.J., "A Photoelastic Investigation of the Effect of Crack Front

Curvature on the Stress Intensity Factor Distribution in a Single Edge

Notch Specimen". BER 242-97, September. 1977.

7. Fourney, M.E., "Experimental Determination of the Effect of Crack Front

Curvature in an ASTM Compact Tension Specimen", Proceedings of the Fourth

Brazilian Congress of Mechanical Engineering, 1977, p. 13-26.

8. Gyekeneesi, J.P., "Solution of Some Mixed Boundary Value Problems of

Three-Dimensional Elasticity by The Method of Lines", Ph.D. Thesis.Michigan

State University, 1972.

9. Gyekenyesi, J.P. and Mendelson, A., "Three Dimensional Elastic Stress

Analysis of Finite Geometry Solids Containing Cracks", Int. J. of Fracture.

Vol. 11, No. 3, June, 1975.

10. Mendelson, A. and Alam, J., "The Use of the Method of Lines in 3-D Fracture

Mechanics Analysis with Applications to Compact Tension Specimens", Submitted to Int. J. of Fracture.

11. Gyekenyesi. J.P. and Mendelson, A., "Stress Analysis and Stress Intensity

Factors for Finite Geometry Solids Containing Rectangular Surface Cracks",

J. of Appl. Mach., Trans. ASNE, Vol. 44, Series E, No. 3, Sept. 1977.

12. Raju, I.S. and Newman, J.C., Jr., "Three-Dimensional Finite Element

Analysis of Finite-Thickness Fracture Specimens". NASA TND-8+414, 1977.

13. Stand,.:d Test Method for Plane-Strain Fracture Toughness of Metallic Materials,

ASTM E399-81.

14. Neale, B.K., "The Stress Intensity Factors Associated with Curved r.rack Fronts

in a Compact Tension Specimen", Proc. of the First International Conference

on Numerical Methods in Fracture Mechanics. Swansea. 1978, Ed. A.R. Luxmeore

and D.R.J. Owen, p. 218-224.

10
m

An

It

J

I

esML•wrR

N

^

f

WM••H•aAwr

33

IZ

rI
NY-1

Y

X

t

NX

NX-1

hAlYxNX lines

j.

NY-1 NY Y

1

2

1

Z

1 =NiXNZ lines m=NZXNX lilies

f--=^-• NZ

• i i

i 41 NZ-1 bX i

i C - e—i i • i

r

i 41 41 0 i i i

X= i i • i i i

1 40

NZ-1 NZ Z NX-1 NX X

>igere 2. ?Mroe sets of lines parallel to X-, Y-, and Z-coordinates and perpendicular

to corresponding coordinate planes.

12
Y Ni

hx

NX

X

hz

Z

Figure 3. Coordinate system and line numbering for

single edge notch and compact tension

specimens.

13
d 4.

3

.s

e

r

w

w

r

b

C

1f

r

C

h

h

Sf i

rN

h 

h 2. r

C

O

W

C

O

E

0

ORKMAL PAGE III

OR POOR QUALM

u 11 0.2 0.3 0.4 0.0

Distance Along the Thickness, z/B

figure.4. Uriation of dimensionless Slf with thickness for different

c rack tunnel depths for compact tension specimen under

tensile loading.

14
0.2 0.3 0.4

Distance Along the Thickness. z/B

Figure 5. Variation of dimensionless SIF through the thickness for

different crack tunnel depths, for compact tension

specimen under parabolically applied shear load.
w

H

\ 1

iC

O

r

HA

r

0

C

O

r

C

h

1A

11 rrn 0

Nh

_O

C

O

h

C

O

c 0

,

0.1 0.2 0.3 0.4

Dimension'ess Crack Tunnel Depth, Al/B

Figure S. Variation of center, surface and thickness 

average stress Intensity factor with

increasing crack tunnel depth.

16
OF POM QUALM

.

ASTM REGION

(E399-81)

ASTM

10 REGION

(E399-78)

0 10

—10

0

c —20

20 30 40

Curvature Parameter n,%

—30

^ y —40

1

[i

fr

i

Figure 7. Percent variation in average stress intensity

factors between straight crack and curved

crack fronts.

W, .

17