262
THE OPENING OF A GRIFFITH CRACK UNDER
INTERNAL PRESSURE*
BY
I. N. SNEDDON {University of Glasgow) and H. A. ELLIOTT (University of Bristol)
1. The determination of the distribution of stress in the neighbourhood of a
crack in an elastic body is of importance in the discussion of certain properties of the
solid state. The theory of cracks in a two-dimensional elastic medium was first de-
veloped by Griffith1 who succeeded in solving the equations of elastic equilibrium in
two dimensions for a space bounded by two concentric coaxial ellipses; by considering
the inner ellipse to be of zero eccentricity and by assuming that the major axis of the
outer ellipse was very large Griffith then derived the solution corresponding to a very
thin crack in the interor of an infinite elastic solid. Because of the nature of the co-
ordinate system employed by Griffith the expressions he derives for the components
of stress in the vicinity of the crack do not lend themselves easily to computation.
An alternative method of determining the distribution of stress in the neighbourhood
of a Griffith crack was given recently by one of us2 making use of a complex stress-
function stated by Westergaard.3 This method suffers from the disadvantage that the
Westergaard stress-function refers only "to the case in which the Griffith crack is
opened under the action of a uniform internal pressure; the stress-function correspond-
ing to a variable internal pressure does not appear to be known.
In the present note we discuss the distribution of stress in the neighbourhood of a
Griffith crack which is subject to an internal pressure, which may vary along the
length of the crack, by considering the corresponding boundary value problem for a
semi-infinite two-dimensional medium. The analysis is the exact analogue of that for
the three-dimensional "circular" cracks developed in the previous paper2 except that
now we employ a Fourier cosine transform method in place of the Hankel transform
method used there. A method is given for determining the shape of the crack resulting
from the application of a variable internal pressure to a very thin crevice in the in-
terior of an elastic solid, and for determining the distribution of stress throughout the
solid. The converse problem of determining the distribution of pressure necessary to
open a crevice to a crack of prescribed shape is also considered. As an example of the
use of the method the expressions for the components of stress, due to the opening
of a crack under a uniform pressure, are derived and are found to be in agreement
with those found in the earlier paper.2
2. We consider the distribution of stress in the interior of an infinite two-dimen-
sional elastic medium when a very thin internal crack — c^y^c, * = 0 is opened un-
der the action of a pressure which may be considered to vary in magnitude along the
length of the crack. For simplicity we shall consider the symmetrical case in which the
applied pressure is a function of |y| but the analysis may easily be extended to the
* Received March 12, 1946.
* A. A. Griffith, Phil. Trans. (A) 221, 163 (1921).
* I. N. Sneddon, Proc. Roy. Soc. (A) (in the press).
® H. M. Westergaard, J. Appl. Mech. 6, A49 (1939).
THE OPENING OF A GRIFFITH CRACK 263
more general case in which there is no such symmetry. The stress in such a medium
may be described by three components of stress <rx, cr„ and rxu; the corresponding
components of the displacement vector will be denoted by ux and uv. The differential
equations determining the stress-components are4
do" x xy dT d(X«
— + ~L = 0- (!) ^+T^ = 0- (2)
dx dy dx dy
The boundary conditions to be satisfied are that all the components of stress and of
the displacement vector must tend to zero as x1+y- tends to infinity, and that
TXy ~ 0, <Tx p(y), (3)
when x = 0 and — cgygc.
It is obvious from the symmetry about the axis x = 0 that the problem of determin-
ing the distribution of stress in the neighbourhood of the crevice is equivalent to that
of determining the stress in the semi-infinite elastic medium *^0 when the boundary
x = 0 is subjected to the following conditions:
(i) rxy = 0, for all values of y,
(ii) <rx = —p(y), jy| gc,
uz = 0 iy \ ^c.
From the symmetry about the second axis y = 0 we may take as solutions of the
elastic equations (1) and (2) the expressions:5
2 r"
= — I <?(p)( 1 + px)e fX cos pydp, (4)
X J 0
2 r
<Jy = — I
7T J o
$(p)(l — px)e 'x cos pydp, (5)
2 x r"
Txy = — ) sin pydp. (6)
7T J o
These expressions satisfy the equations of equilibrium and the boundary condition (i)
above; the function #(p) is determined from the set of conditions (ii). The compo-
nents of the displacement vector are similarly found to be
2(1 + o) C°° , > cos py
ux = — I $(p)e~pI{ 2(1 — a) + px) dp, (7)
tE J o p
2(1 + a) r °° . , sin py
uy = —— I <£(p)e_,,I{(l — 2tr) — px) dp. (8)
irE J o P
When x = 0, equations (4) and (7) reduce to
22 r
= - *(#»)
IT J 0
cos pydp, (9)
4(1 — a2) rx cos py
ux = ~ *(p) dp. (10)
7rE J o
4 A. E. H. Love, The mathematical theory of elasticity, 4th ed., Cambridge, 1934, p. 208.
6 I. N. Sneddon, Proc. Cambridge Phil. Soc. 40, 229 (1944).
264 i. N. SNEDDON AND H. A. ELLIOTT [Vol. IV, No. 3
If we insert the boundary conditions (ii) into Eqs. (9) and (10) and make the substitu-
tions
p = £/<". y = v, g(v) = — P(vc), = £1/2^"(£). (}1)
we obtain a pair of "dual" integral equations
1
f iF(£)J-ut(frl)dZ = g(v), 0 < v < 1
J 0
r" i
I'WJ-yMdt = 0, v > 1
J 0 )
(12)
for the determination of the function /*'(£). Once F(£) has been found, <£(p) can be
written down and the components of stress calculated by means of Eqs. (4), (5)
and (6).
3. The dual integral equations (12) are a special case of a pair of equations con-
sidered by Busbridge;6 the solution may be obtained by substituting a = l, v= —1/2
in the general solution given in the paper.6 In this we obtain
F(l) = i"2[-/„(*) J'y"*(l -
+ £ j" m1/2(1 — u-)ir-du J" ^//Vitt,)^]. (13)
Thus if the pressure p(y) is given by a Taylor series of the form
P(y) = Po H o„ (—\ , (14)
„_o \c /
convergent for — c^y^c, then the corresponding expression for #(p) is readily found
to be
" F('w _)_ 1) r 1
4>(p) = - W'-t1/2p£ —TT <J„<7o(cp) + CP I y"+Ui(cpy)c'y> . (15)
«=o r(i» + 2) i Jo '
Substituting for $(p) from Eq. (15) into Eq. (10) and making use of the results7
r" l
I Jo(cp) cos p\dp = ,—- - 0 < v < c
J u " ^ c- - v2
r
I pJ i (cp) cos pydp = —————' 0 < y < c
J 0
c
{c-
we find tliat the normal component of the displacement along the crack is given by w,
where
2(1 - <72)p0r " r(i» + a)
w -
\'tc E
£ r<*» + «, / +(LX" f "• 1. (i6)
f-, r(s» + 2] V</ J, («•-!)"■ I
I. W. Busbridge, Proc. London Malh. Soc. (2) 44, 115 (1938).
G. N. Watson, The theory of Bessel functions, 2nd ed., Cambridge, 1944, p. 405.
1946] THE OPENING OF A GRIFFITH CRACK 265
For the case of a uniform pressure p» we take ««=!, = ^ 1 and find
2(1 — (T-)po
w = — \V — y-. (17)
E
If we write
b = 2(1 - <72)/>oc/E,
Eq. (17) reduces to the form
1 = 1
c2 b-
which shows that the effect of the uniform pressure is to widen the crevice into an
elliptic crack.
4. It is also of interest to determine what distribution of pressure will produce a
crack of prescribed shape. In this case we assume that the value of the normal dis-
placement ux is known all along the y-axis; we have
ux =
lw(y), y ^ | c|, x = 0,
I 0, y ^ | c |, x = 0.
Inverting Eq. (10) by the Fourier cosine rule and substituting this value for ux we
have
E
E rc
<?(p) = ~ — — P I w(y) cos pydy.
2(1 — a1) J o
(18)
2(1 - „*)
With this value of <£(p) in Eqs. (4), (5) and (6) we obtain expressions for the compo-
nents of stress in the interior of the elastic solid.
For example if we take
-•(>-■£)•
then, from Eq. (18)
Et /sin cp \
$(p) = - — IT ( cos CP). (19)
(1 — C )cp \ Cp )
Substituting from (19) into Eq. (9) we obtain for the normal component of the stress
along * = 0,
yu
2 Et
ir(l — cr2)c
Now
i f
_ C J 0
sin u sin
c
du
u
(20)
/.
cos qx — cos px p-
dx = | log —
so that Eq. (20) reduces to
2 Et
°x= ~ T(1 - <r2)c
ty c + y~\
1 - — log , 0 < y < c,
2c c - yj
(21)
266 I. N. SNEDDON AND H. A. ELLIOTT [Vol. IV, No. 3
giving the normal component of stress along the crack. This stress is negative when
v = 0 but becomes positive for a value of y between 0 and c, so that if a crack of this
shape is to be maintained the applied stress must be tensile (and very large) near the
edges y= +c of the crack.
5. Expressions for the potential functions w(z), 12(z) of Stevenson corresponding
to this problem can easily be deduced from the analysis of Section 3. It was shown by
Stevenson,8 that if we write
0 — j! | (Tj/, 'I' = (7 f IT y | 2 IT xy, D — U x I lily
then the components of the stress and the displacement can be expressed in terms of
two "potential" functions «(z), £2(z) by means of the equations
1 —<T . _ .
D = £{(3 - 4(7)12(3) - zi2'(z) - w'(z)|
4
20 = V.'(z) + l2'(z)
— 2<i> = z 12" (z) + J>"(z)
(22)
in the absence of body forces.
It follows from Eqs. (4) to (8) that the stresses and the components of the dis-
placement vector may be derived from the potential functions
4 f *(p) 4 r m
U(z) = I e-"'dp, u (z) = — I (1 + pz)e~**dp;
T J 0 p T J 0 P
(23)
where #(p) is given by Eq. (15) in the case where the applied internal pressure is
given by Eq. (14).
6. We now consider the distribution of stress in the solid when the crevice
— c^y^c, x = 0 is opened up by the action of a uniform pressure pa- Taking ao=l,
an = 0, n>0, in Eq. (15) we obtain for $(p) the expression
<£(p) = — jirp0c2p 0(cp) + — J zl/i(z)</zj .
Now,
f z2Ji(z)dz = c2p-Ji{cp)
J o
and, by a well-known recurrence relation,
2
Jo(cp) + J 2 (cp) = J l (fp)
cp
so that
<?(p) = — h^pocJ i(cp). (24)
Substituting from Eq. (24) into (4), (5) and (6) we obtain the equations
5(^1+ <rv) = — poc I e~"x cos pyJi(cp)dp, (25)
J 0
8 A. C. Stevenson, Phil. Mag., (7) 34, 766 (1943).