brief notes

On the Griffith-lrwin Fracture Theory

J. L. SANDERS, JR. 1
In the usual case a crack length longer than that determined by
THE purpose of the present paper is to give a new mathematical (4) would extend and lead to failure of the structure. However,
formulation of the criterion for crack extension according to the in other cases the solution of equation (4) may determine the
Griffith-lrwin theory in terms of a path-independent line integral. conditions under which a running crack would be arrested (neg-
The derivation will be made for the case of a crack in a plate lecting inertia effects) and so the proper physical interpretation
where the plane-stress theory applies, but it may be generalized of the solution to (4) must be left to depend on the case at hand.
easily to other cases (three-dimensional elasticity, shells, etc.). The most useful feature of the present form of the criterion (4)
An account of the Griffith-lrwin theory as such may be found in is that the integral I is independent of the path C (by the manner
[1 ]2 and in the papers referred to in that paper. of its derivation). One obvious advantage is that there will be
no difficulties connected with calculating the strain energy in in-
Analysis infinite regions. A simple modification of (4) is that C need not
Suppose we have a plate with arbitrary boundaries on which necessarily be a closed path but may begin and end on a free
we prescribe arbitrary boundary conditions (possibly time-de- boundary in such a way as to enclose the tip of the crack.
pendent). The plate is supposed to contain a crack of length L(a) Since the integral I is independent of the path C, one might
where a is some parameter which increases when L increases. expect it to take a simple form if the solution of the plane-stress
The edges of the crack itself are assumed stress-free and all inertia problem is known in terms of the two analytic functions tfi and >p
effects are neglected. We will assume that the (quasistatic) of the Muskhelishvili formulation of plane-stress theoiy [2],
solution of this plane-stress problem is known, that is, <r;,-, 6,,, and This is indeed the case. According to Muskhelishvili,
ut are known functions of x, y, t, and a. Let C be some simple
closed curve surrounding the crack. According to the Griffith- E(u - iv) = (3 - v)$ - (1 + v)z<t>' - (1 + (5)
lrwin theoiy, the following energy balance must hold during a
(virtual) extension of the crack: where Ui = u, ih = v, and a bar over a quantity denotes the com-
The rate at which work is being done by forces acting across plex conjugate. Let Tids = dX, Tzds = dY, and let a subscript
C equals the rate of increases of strain energy stored in the a denote differentiation with respect to a. The integral I may
material inside C plus the rate at which energy is dissipated be written
by the growing crack.
The last term in this equality is assumed to be proportional to
1 = I Re I fc [(M" + iVa)d(X ~ iY) ~ + iv)d(Xa ~ }
dL dL da
= L'a = ^ Im j f c [ ( l + v)z$a' - (3 - v)<t> „
dt da dt
+ (1 + v)j;a]d(z<t>' + 4> + i)
In symbols the equality reads:

r dui , 1 d r - [(1 + v)z§' - (3 - vW + (1 + v)t]d{z<t>*' + fa + ia) j

Jc'r<MdS = 2 ^JcTiUids + GL'a (1)

where Ti are the boundary tractions, and G is a constant which where Re or Im means the real or imaginary part of, and where
depends on the power required to extend the crack at a given d(X — iY) = id{z<j>' + $ + \f/). After some manipulation (6)
rate. Equation (1) may be transformed into becomes:
1 bu
•f ( T ' ^ A j I / =
E
Im j / C ( < M > ' + i'4>a)dz\ (7)

If the path C begins at point A and ends at point B on a free

(2) boundary where the boundary condition is z(f>' + <j> + ip = 0.
^ fc(T<,i' ~ = GL'&
then equation (7) must be modified to read:
where

diij diij I = - | Im \[z4>'4>a]AB + f f Wat' + <P'4>a)dz\(8)

— = Ui -f a
dt ' da
Applications
and similarly for dTi/dt. Now the second integral on the left is
The following two examples are not new results but are pre-
easily shown to vanish, and so we have the result:
sented to illustrate the application of the criterion (4).
Consider the problem of a cracked rectangular plate loaded on
two opposite edges which are constrained to remain straight and
free on the other two edges. On the first loaded edge let Ui = Ui
Ordinarily equation (3) has no solution other than a = 0 and we = 0. Let C be the boundary of the plate; then the only con-
have here a kind of eigenvalue problem. For certain values of tribution to I comes from the second loaded edge. We have:
the crack length (or load level) which may be called critical the
1 bU_
following equation holds: / = r f Trfds
2 da I 2 da
Gordon M c K a y Lecturer on Structural Mechanics, Division of
1

Engineering and Applied Physics, Harvard University, Cambridge.

Mass.
2 Numbers in brackets indicate References at end of Note.

Manuscript received by A S M E Applied Mechanics Division, 1 ™ F _ l u * Z (9)

October 2, 1959. 2 da 2 da

352 / j u n e 19 6 0 Transactions of the ASME

Copyright © 1960 by ASME

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 brief notes

where F = J^'l'tds, Ui = 0, and u* = U on the second loaded pressed in the form of a contour integral. The path independence
edge. For the present purpose, all we need to know about the of the integral makes it clear that the amount of strain energy
solution of the plane-stress problem is that, for a fixed value of a, available in the cracked structure is irrelevant. The quantity
there exists a linear relation between the resultant, load F and that does matter is the strength of the square-root singularity at
the displacement U, namely the tip of the crack. This strongly suggests that the Griffith-Irwin
approach to fracture mechanics via energy concepts is equivalent
k(a)U (10) to an approach via stress-concentration factors (which reinforces
a conclusion reached in Ref. [4]).
From the foregoing we can show that the critical crack length for
a fixed grip test ( b U / d a = 0) is the same as for a dead load test References
(dF/da = 0). In the first case we have from (9) and (10)
1 G. R. Irwin, "Analysis of Stresses and Strains Near the End of
a Crack Traversing a Plate," JOURNAL OF APPLIED MECHANICS, vol.
24, TRANS. A S M E , vol. 79,1957, pp. 361-364.
2 da - 2 U V (ID 2 N . I. Muskhelishvili, "Some Basic Problems of the Mathemati-
cal Theory of Elasticity," P. Noordhoff Ltd., Groningen, 1953.
3 H. M. Westergaard, "Bearing Pressures and Cracks," JOURNAL
In the second case from (9) and (10)
OF APPLIED MECHANICS, v o l . 6, TRANS. A S M E , v o l . 61, 1939, pp.
A-49-A-53.
= V — = - 1 4 G. R. Irwin, J. A. Kies, and H. L. Smith, "Fracture Strengths
k' = - - Vk' (12) Relative to Onset and Arrest of Crack Propagation," Proceedings of
2 dot 2 \ k ) 2
the A S T M , vol. 58, 1958, pp. 640-660.

The value of I is the same in both cases and hence so is the critical
crack length. This result has been noted before by several
authors.
For another example, consider the problem of an infinite plate
On the Flexure of Plastic Plates1
with a periodic array of cracks of length 2a along the .r-axis with a S. LERNER 2 a n d W . PRAGER 3
distance I between the centers of the cracks. The plate is sub-
jected to a uniform stress <ry = a at infinity. The problem of I N THEIR pioneering work on bending of rigid, perfectly plastic
determining the distribution in this case was solved by Wester- plates, Hopkins and Prager4 used the yield condition of Tresca
gaard [3]. In terms of the Muskhelishvili functions the solution and the associated flow rule. Hopkins and Wang5 later extended
the analysis to arbitrary yield conditions and the associated flow
rules. On account of the mathematical difficulties presented by
other shapes, work in this area has been almost exclusively con-
2)7 , cerned with circular plates under rotationally symmetric loading.
4> — — r z i cosh - 1 (13) Attempts at experimental verification of the theory have likewise
7T
been restricted to circular plates under rotationally symmetric
loading.6 Such tests can at best provide indirect verification of
the theory because they deal with the over-all deformation of the
\p = -Z<t>' + 0 + -<TZ (14) circular plate under a nonuniform distribution of radial and cir-
cumferential bending moments. The purpose of the present note
is to draw attention to a simple test that provides a direct check
Let C enclose the crack which extends from z = —a to z = a.
on the fundamental assumptions of the theory, when the two
By using (14) we have
principal bending moments have opposite signs.

J = - \ Im jfc(M' + rta)dz Analysis

Fig. 1 illustrates the proposed test: A horizontal rhomboid
plate of the uniform thickness h and semidiagonals of the lengths
= - E I m
a and b is subjected to four vertical loads of the same intensity P,
two of wliich act downward at the end points of one diagonal,
7TZ 7TZ while the remaining two act upward at the end points of the other
sin — cos — diagonal. As was first pointed out by Kelvin and Tait, this type
tan — Im •dz (15) of loading is compatible with a distribution of bending and twist-
E I f ^
In TTZ
J C
m s ! I— I 1 The results presented in this Note were obtained in the course of

research sponsored by the Office of Naval Research under Contract

Nonr 562(10).
The path C encloses the two simple poles at z = a and z = —a ! Associate Professor of Engineering, Brown University, Provi-
so the integral may be evaluated easily. After the calculations dence, R . I.
have been performed the criterion reads ' L. Herbert Ballou University Professor, Brown University, Provi-
dence, R. I.
4 I I . G. Hopkins and W. Prager, " T h e Load-Carrying Capacity
2 <rH TCI _
— tan j = 2G (10) of Circular Plates," Journal of the Mechanics and Physics of Solids,
vol. 2, 1952, pp. 1-13.
' H. G. Hopkins and A. J. Wang, "Load-Carrying Capacities for
and hence the critical stress for a crack length L = 2a is Circular Plates of Perfectly Plastic Material With Arbitrary Yield
Conditions," Journal of the Mechanics and Physics of Solids, vol. 3,
1954, pp. 117-129.
TeS ttoT/.
=
LTcofcyJ
6 R . M . Haythornthwaite and E. T . Onat, " T h e Load-Carrying
(17)
Capacity of Initially Flat Circular Steel Plates Under Reversed
Loading," Journal of the Aeronautical Sciences, vol. 22, 1955, pp. 8 6 7 -
869.
Conclusions
Manuscript rec:ived by A S M E Applied Mechanics Division, Sept.
The Griffith-Irwin criterion for crack extension has been ex- 21, 1959.

Journal of Applied Mechanics J U N E 1 9 6 0 / 3 5 3

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