An estimation of local stress intensity factors for semi-elliptical surface cracks

Engineering Fracture Mechanics Vol. 34. No. 4,

pp.883-890, 1989

Printed in Great Britain.

0013-7944/89 $3.00 + 0.00 CJ 1989 Pergamon Press plc.

AN ESTIMATION OF LOCAL STRESS INTENSITY FACTORS FOR SEMI-ELLIPTICAL SURFACE CRACKS T. FE’IT Kemforschungszentrum

Karlsruhe, Institut fur Material- und Festkijrperforschung Postfach 3640, Karlsruhe 7500, F.R.G.

IV,

Abstract-A simple method is described which allows the estimation of local stress intensity factors of two-dimensional cracks by use of the weight function basic relation. The method is applied especially to semi-elliptical surface cracks. Two examples are considered. The half-penny shaped crack under bending load is treated based on the tension reference case and the results are compared with literature data. In a second example, the semi-elliptical crack is loaded with a quadratic stress distribution. The result allows an extension of the Newman-Raju formulae which provide stress intensity factors for u _ x0 and u ... x’.

1. INTRODUCTION LOCAL stress intensity factors for semi-elliptical surface cracks in finite bodies are known only for the simplest load cases such as pure tension and pure bending. The best-known solutions were given by Newman and Raju[ 11. For many problems, e.g. cracks under thermal stresses in thermal fatigue experiments or cracks in residual stress fields, much more complex stress distributions are of interest. One possibility of deriving stress intensity factors is given by the weight function method where so-called “averaged weighted stress intensity factors K” are determined[2]. Since this method uses mean values of K only, details of the K-distribution along the crack front may be lost. To allow an estimative determination of local stress intensity factors, the weight function method can be used in a modified application.

2. PROCEDURE OF ESTIMATION The procedure is based on the fact that stress intensity factors for elliptical cracks in an infinite body are available for arbitrary normal stress distributions. The procedure is outlined in detail for stresses which depend only on the depth coordinate. The problem of semi-elliptical surface cracks is reduced to the elliptical crack problem by cutting along the y-axis of the elliptical crack (Fig. 1) and by introducing a correction function which considers the free surface and finite thickness of the real structure. In this model, only stresses symmetrical to the cutting line C-C in Fig. 1 are of interest for the description of the surface crack loading state. The stress intensity factor for the semi-elliptical surface crack problem can be written

where the subscript “s” denotes the semi-elliptical surface crack and “e” stands for the stress intensity factor of an embedded elliptical crack. Index “I” denotes the case of constant stress and “2” stands for an arbitrary load case. The function g considers the free surface and the finite thickness. EFM W4-3

883

T. FE’l-f

884

al

Fig. 1. (a) Semi-elliptical surface crack in a plate under stress distribution a(x). (b) Elliptical crack in an infinite solid under identic stresses.

If the geometric function Y, defined by K=aYJ;;

(2)

is used, eq. (1) reads (3)

where

&

Y,, =-

E(k)

a2

1 114

K> c

cos2 r$ + sin2 C$

as given in [3], with the elliptical integral of second kind E(k) and the modulus k = ,/w. A very rough but frequently used estimation can be obtained by setting g = 1[4]. By expanding the unknown function g in a Fourier series one obtains (4) where the coefficients D, depend on a/c and a/t and, evidently, on the type of stress distribution. Two of the coefficients D, can be determined easily by the weight function method. As shown by Rice[5], it holds for any virtual crack extension AS

a

s

HZG u,tsdS==

’

KK,d(AS) s

where I(, is the crack opening displacement in a reference loading case ai-_often O, = const-and K, is the reference stress intensity factor.

(9 chosen to be

~~~liipti~l

surface cracks

Applying the two-degree of freedom virtual crack extensions, as proposed Besuner[6], the following system of equations results

885

by Cruse and

~~~~~~d~=~~~~sin2~d~

(6)

&.&Sy.dS=j;KK~cosz$d&

(7)

and for a one-degree of freedom crack increment keeping a/c constant it reads ~~~~~~d~=~~~~~d~.

(8)

Two of the relations (6)-(s) allow the coefficients D,, and D, to be determined. If eqs (6) and (8) are chosen,

--- i a63

Y$ (Ye2/Y,,)cos(2# - n)sin2 4 d4

A aa

(9)

Y~,(~~~~,~sin' Q,dd, ' A aa Ialc=const s ia@ --Y:, ( Ye2 I Y,, 1 d4 A da I C=COnSt s

L) =‘!?

where

Then, the estimative solution is found. 3. APPLICATIONS 3.1. The half-penny shaped crack Analytical solutions for penny-shaped cracks in an infinite body are available in the literature. In [3] the relation

X=asin#~

x=psin4

and

f&)=+)/a(a)

(10)

is reported to evaluate any stress distribution dependent on x only. As a first example of the estimation procedure the local stress intensity factors of a half-penny shaped surface crack under bending is treated. The related stress distribution is

T. FETT

886

From [3] one obtains Y = -?-. 1 - 2 4 6 + #sin #)3’z(Jm& & i

(12)

- .JZJ]j

No exact solution is known for the reference displacement field of a half-penny shaped surface crack. Therefore, an approximative crack opening displacement field u,, based on the reference stress gl, is used as proposed in [7& The reference crack opening displacements in the reference loading case, e’r= const, are described by

%(P,6) = i cvt4w - Ple+1’2*

(13)

v=5

The coefficients are functions of the crack geometry (a/c, u/t) and the angle Qtand can be expressed by W%J)=A,+~*G(~)

v =O, 192

(14)

where A5=0

B,= 1

A t = -B, G(O)

B, = (y ‘P - C,(O) - 4Q)/(G - C,(O))

A2 = ($ + Be)C~(O) B2 = -(B, i- 1)

(1%

1 -l/4

C,(4) = fi

$ Y(a/c, a/t, #)a

cos2 i$ + sin’ #

69 y=:“_.!L.

477Hacs

Y*a’ dS(a’)

(17)

The reference stress intensity factors were taken from the Finite-Element calculations by Newman and Raj@l] where the geometric function is expressed by

J;; h=~(k)F.

(19)

The function F is defined by F = (M, + M2a2 + M,cw4)f& where M, = 1.13 -0.0s~ M*=

0.89 p -o*54+o.2+B

jt4,=Oe5

a = a/t,

#3= a/c

g = 1 + (0.1 I- 0,35aZ)(l - sin 4)’

1 - + 14(1 - #I)% f4 = (J2 co2 f#~+ sin* $)“4. 0.65+B

It is quite obvious that this numerical solution will somewhat deviate from the exact solution. The possibilities of errors should be kept in mind. For not completely exact solutions of stress intensity

Semi-elliptical surface cracks

887

factors-this is always the case for semi-elliptical surface crack problems-the function Y is dependent on the chosen integration path[7’J. To minimize additional errors, it is recommended to use the same shape of integration increments for evaluating eq. (17) as used in eqs (6)-(s). It results from eq. (17)

c=

const

a=const

Y =A!-

a/c=const

Y =-

Y2 sin2 4a’ da’ db,

for eq. (6)

Y2cos2 4a’ da’ d4

for eq. (7)

Y2a” da’ d&.

for eq. (8)

Figure 2 shows the estimated solution for the local stress intensity factor for bending load a a function of the angle 4 = 0 and several values of a/t. Here only a/t ratios are included which show positive K-values. The estimated geometric functions are compared with Finite-Element results given by Newman and Raju[l, lo] for c/b = 0.2, where b is the width of the plate. In Fig. 3 the estimated geometric functions are additionally compared with other literature data[8,9] obtained for the special case of the “wide plate” (c/b+O). Both figures show a good agreement. 3.2. Semi-elliptical surface cracks with quadratic stress distribution In many practical crack problems involving semi-elliptical surface cracks the acting stresses can be approximated by a linear stress distribution. Such cases can easily be treated by an appropriate superposition of Newman-Raju’s tensile and bending solution. In more general applications such as thermal shock and thermal fatigue experiments or for cracks in residual stress fields, non-linear stress distributions have to be expected. Such stress distributions can be better fitted by higher polynomials. In this investigation an estimative solution for a quadratic stress term shall be provided.

e b

A

1.2 h

0

HI

q

eqs.(1,4,12)

a/c =l

8

0.8

a/t=0.6

8

Q A

0.4

0

a

q this investigation

2 P

0

Au0

181

A 191

0

0 I101

9

y‘

@A

I

I

0

20/n Fig. 2. Stress intensity factors in bending computed with the presented method and compared with FE-results from Newman and Raju(1, lo] for c/b = 0.2.

.o

A

0.5

I

1

3 Fig. 3. Stress intensity factors in bending computed with the presented method and compared with literature data for the “wide plate” (i.e. c/b +O).

888

T. FE’lT

The fully elliptical crack in an infinite solid was investigated by Kassir and Sih[ll] who provided a number of solutions for polynomial loading. If a stress distribution x2 (20) 05 is given where c is a characte~stic length one can conclude from Kassir and Sih’s results the geometric function a=q)

with

F2 = J2, + SJ,

where sn(t), u%(t) are the Jacobian elliptic functions and K(k) is the complete elliptic integral of the first kind with modulus k = 1 - (a/c)‘. A great number of integrals J,, expressed by simpler relations is given in [I I]. The in~~als in eq. (21) are listed in the Appendix. using Y,, as Y,, in

1

Fig. 4. Geometric function Y for a semi-elliptical surface crack under a quadratic stress d~~butio~ u(x).

Fig. 5. Coefficients DO and D, for a surface crack under quadratic stress ~s~bution.

Semi-elliptical surface cracks

eq. (3) and taking the geometric function of Newman-Raju eqs (3), (6) and (8) can be evaluated. In Fig. 3 the resulting geometric function Y = Ylw(YKslY,,)[&l+

889

Y,, for the constant stress case as Y,,

D, cos (24 -

n)l

(22)

is shown as a function of the angle r/~for several values of a/c and for a/t = 0.6. Figure 4(b) represents the values of Y for the deepest point of the ellipse as a function of a/t with a/c = l/2. These data are plotted in Fig. 5. In Table 1 the coefficients D,, and D, are given as a function of the variables a/c, a/t.

Table 1. Coefficients D, and D, for a semi-elliptical surface crack loaded by a quadratic stress distribution ale

ah

D,

D,

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.01 0.10 0.20 0.30 0.40 0.50 0.60 0.70

0.9585 0.9582 0.9574 0.9562 0.9549 0.9538 0.9533 0.9538

0.0549 0.0545 0.0489 0.0400 0.0281 0.0139 -0.0020 -0.0188

213 213 213 213

0.01 0.10 0.20 0.30

0.0111

213 213 213

8% 0.60 0.70

0.9730 0.9713 0.9710 0.9707 0.9704 0.9704 0.9708 0.9719

0.0103 -0.0010 -0.0184 -0.0397 -0.0627 -0.0849 -0.1037

112

112

0.01 0.10 0.20 0.30 0.40 0.50 0.60 0.70

0.9798 0.9816 0.9823 0.9834 0.9850 0.9870 0.9892 0.9916

-0.0085 -0.0166 - 0.0330 -0.0575 - d.0868 -0.1171 -0.1449 -0.1670

113 113

0.01 0.10

l/3 l/3

0.20 0.30

0.9952 0.9952 0.9989 1.0044

l/3 l/3 l/3 l/3

0.40 0.50 0.60 0.70

1.0109 1.0178 1.0243 1.0299

-0.0403 -0.0473 -0.0734 -0.1113 -0.1546 -0.1974 -0.2349 -0.2634

l/4 l/4 l/4 l/4 114 114 l/4 114

0.01 0.10 0.20 0.30 0.40 0.50 0.60 0.70

1.0020 1.0032 1.0104 1.0207 1.0321 1.0430 1.0524 1.0594

-0.0537 -0.0637 -0.0991 -0.1488 -0.2035 -0.2553 -0.2984 -0.3294

115

0.01 0.10

-0.0582 -0.0741 -0.1176 -0.1775 -0.2411 -0.2987 -0.3442 -0.3740

l/2

112 112 112 l/2 l/2

l/5

0.60

1.0045 1.0083 1.0190 1.0336 1.0490 1.0626 1.0729

l/5

0.70

1.0790

115 115 l/5 l/5 115

0.20 0.30 it:

890

T. FETT

Since an approximation for stress intensity factors under quadratically distributed stresses is known, any stress distribution described by a polynomial of the type o

0 ;

2

=A+B;+C;

0

(23)

can be evaluated by K = J;;[AY(O) + BY(l) + CY(2)]

(24)

where Y(0) = YNR(tension) Y( 1) = g Y,,(tension)

- YNR(bending))

Y(2) = Ycalculated by eq. (21) using 5 = t. 4. SUMMARY A simple procedure is proposed to estimate local stress intensity factors in arbitrarily loaded structures for two-dimensional crack problems. The necessary formulae are developed and the application is demonstrated for surface cracks in plates of finite widths. As a result, the geometric function is derived for a stress distribution of the type cr N x2. REFERENCES [l] J. C. Newman and I. S. Raju, Engng Fracture Mech. 15, 185-192 (1981). [2] C. Mattheck, P. Morawietz and D. Munz, Znr. J. Fracture 23, 201-212 (1983). [3] H. Tada, The Stress Analysis of Cracks Handbook. Del Research Corporation (1986). [4] H. J. Underwood, ASTM STP 513, 59 (1972). [5] J. R. Rice, Znr. J. Solidr Struct. 8, 751 (1972). [6] T. A. Cruse and P. M. Resuner, J. Aircraji 12, 369-375 (1975). [7] T. Fett, Znt. J. Fracture 36, 55-69 (1988). [8] A. S. Kobayashi, N. Polvanich, A. F. Emery and W. J. Love, Proc. 12th Annual Meeting o/the Society of Engineering Science, pp. 343-352 (October 1975). [9] K. Kathiresan, Ph.D. Thesis, Georgia Inst. of Tech. (1976) (quoted in [IO]). [lo] J. C. Newman and I. S. Raju, NASA Technical Paper 1578 (1979) [l l] M. K. Kassir and G. C. Sih, Three Dimensional Crack Problems. Noordhoff, Leyden (1975).

APPENDIX Integrals occurring in eq. (20) Jm = fk -‘[(2 + k*)K(k) - 2(1 + k*)E(k)] Jo2 = i(k’k)-“[k’*(2 J,, = k-‘[--K(k) J,* = fk+‘k’-*[(8 J2, = tk-6[-(8

- 3k*)K(k) + 2(2k2 - l)E(k)] + (1 + k’-2)E(k)] - 9k*)K(k) - (10 - 3k2 - 2k’-*)E(k)]

+ k*)K(k) + (5 + 2k2 + 3k’-2)E(k)]

&, = $k -6[8 + 3k2 + 4k4)K(k) - (8 + 7k2 + 8k4)E(k)] J,,, = iLjk-6k’-4[(8k’-2 k’2=

- 23k*)E(k)

- (8 - 19k2 + 15k4)K(k)].

1 -k’ (Received 20 October 1988)