International Journal of Fracture 3 4 : R 2 3 - R 2 6 (1987) R23 © Martinus Nijhoff Publishers, Dordrecht - Printed in the Netherlands

A PHYSICAL INTERPRETATION OF THE M-INTEGRAL FOR A GRIFFITH CRACK

John S~ort Department of Mechanical and Industrial Engineering, University of Utah Salt Lake City, Utah 84112 USA tel: (801) 581-8981

The relationship between the mode I energy release rate G_ and the mode i

I stress intensity factor K I is well known. Eshelby [i], for example, de- rived the result

G I = KI2/E' (i)

using a simple traction work calculation. As usual, E' in this expression takes the value E for plane stress and E/(I-~ 2) for plane strain loading conditions.

In recent years a number of authors [2-4] have discussed another energy release rate M, which can be interpreted [3] as the energy release rate associated with self-similar expansion within a crack tip bounding contour. Freund [4] has shown that the so-called M integral is given by

M = YiJi (2)

where y. is a characteristic dimension of the contour in the i direction as measure~ from the crack tip and J. is the translational energy release rate vector [3]. I

Based on (I) and (2) and the equivalence of GTTand J , the M integral for crack tip contours in linear elastic bodies s~bjeete@ to mode I loading should have the value

M = YI(KI2/E ') (3)

If (3) is correct, then an identical expression should result from an evalu- ation, similar to Eshelby's [i], of differences in traction work associated with self-similar expansion within a contour at the tip of a crack subjected to linear elastic mode I loading. Such an evaluation, useful in developing an appreciation of the M-integral's physical significance, is presented be- low.

Consider a Griffith crack [5] with a circular contour of diameter p at its tip. Let the crack be slit over the distance p and then reclosed by applying normal tractions which have the finalvalue

Oy = Ki//(2~r) (4)

Int Journ of Fracture 34 (1987)

R24

after the crack has been reclosed. Here r measures distance in the crack plane ahead of the crack tip. Now, imagine the contour to be extended to diameter p+AO in a self-similar manner as shown in Fig. i. With this figure in mind , the expansional energy release rate M may be expressed as the total increase in traction work AW required to close the crack tip from r = 0 to r = p, per unit contour expansion Ae

p

M = S (~W/Ae)dr (5) O

Defining Ae = Ap/p [3] and AW=~W/~o)Ap, (5) becomes

P

S (~W/~o)H r (6) M=P O

For tractions varying from 0 to ~ in a linear elastic fashion the trac- tion work W, for both crack faces, is y

W = 2(~y~/2) (7)

Noting [6] that for p<<athe crack opening displacement v is approximately

= 2Kl/[2(p-r) ]//~E' (8)

(7) becomes

W = 2Kl2/[(p-r)/r]/~E' (9)

Differentiating W with respect to p and substituting into (6)

P

M = 2K12 p I i//(pr_r2)dr/~E , O

The integral above is easily shown to equal~/2 so that

M = P(Kl2/E' ) (i0)

as expected.

This result was of course anticipated. However, the preceding develop- ment is nonethel~ss informative, being based as it is on a physical inter- pretation of the expansional energy release rate M.

REFERENCES

[i] J.D. Eshelby, in Fracture Toughness, Iron and Steel Institute Publication

Int Journ of Fracture 34 (1987)

R25

121 (1968( 12-48.

[2] J.K. Knowles and E. Sternberg, Archive for Rational Mechanics and Analysis 44 (1972) 187-211.

[3] B. Budiansky and J.R. Rice, Journal of Applied Mechanics 37 (1970) 201-203.

[4] L.B. Freund, International Journal of Solids and Structures 14 (1978) 241-250.

[5] A.A. Griffith, Philosophical Transactions of the Royal Society of London A221 (1921) 163-197.

[6] D. Broek, Elementary Engineering Fracture Mechanics, Sijthoff and Nootdhoff, Alphen aan den Rijn, The Netherlands (1978)

17 March 1987

Int Journ of Fracture 34 (1987)

R26

0

ttttttt

W

I I

P

By analogy with [1], the total AW (shaded area) is equal to the energy released during an increment of self-similar expansion.

m.~r v

p+~p

Fig. i. Illustration of the change in traction work, AW, due to self-similar expansion.

I~t Journ of Fracture 34 (1987)