QUARTERLY OF APPLIED MATHEMATICS 123
JULY, 1974
SKEW CRACK PROPAGATION AT THE DIFFRACTION
OF A TRANSIENT STRESS WAVE*
By
J. D. ACHENBACH and V. K. VARATHARAJULU
Northwestern University
1. Introduction. When a stress wave strikes a crack, a stress field is generated which
contains square-root singularities at the tips of the crack. The question of fracture
under the influence of such a stress field was investigated in [1] and [2] for anti-plane
and in-plane motions, respectively. In both [1] and [2] a semi-infinite crack in a two-
dimensional geometry was considered, and it was assumed that the crack propagates
in its own plane. The principal task of the analysis of [1] and [2] consisted in solving an
unusual transient diffraction problem, namely, the diffraction of an elastic wave by a
semi-infinite crack which starts to extend after the crack tip has been struck.
In this paper we also consider transient diffraction of an elastic wave by an extending
crack, but with the distinction that the crack may propagate under an arbitrary angle
with its own plane. The incident wave is a plane horizontally-polarized wave. It is
assumed that crack propagation at a constant velocity is generated at the instant that
the tip of a semi-infinite crack is struck. An expression is derived for the stress intensity
factor in terms of the speed of crack propagation and the angle of crack propagation.
For a step-stress incident wave a solution for the particlc velocity is sought which
shows dynamic similarity inside the circular region of the diffracted wave. A crucial
step in the analysis is the use of Chaplygin's transformation, which reduces the problem
to the solution of Laplace's equation in a semi-infinite strip containing a slit. Other
applications in elastodynamics of dynamic similarity and Chaplygin's transformation
can be found in [3] and [4]. The Schwarz -Chris toff el transformation is subsequently
employed to map the semi-infinite strip on a half-plane. The appropriate analytic
function in the half-plane is obtained by a method which was discussed in great detail
by Muskhelishvili [5].
Expressions for the shear stresses and the particle velocity in the vicinity of the
crack tip were obtained by means of a limiting process which is analogous to the one
worked out by Sih [6] for problems of static equilibrium. The interesting aspect of the
results is that the dependence of the near-tip fields on the angle of crack propagation
separates from the dependence on a polar angle centered at the moving crack tip.
2. The diffraction problem. Horizontally polarized wave motion in a homogeneous,
isotropic, linearly elastic solid is governed by the two-dimensional wave equation
* Received January 20, 1973; revised version received February 20, 1973. This work was supported
by the Advanced Research Projects Agency of the U.S. Department of Defense through the
Northwestern University Materials Research Center.
124 J. D. ACHENBACH AND V. K. VARATHARAJULU
d2w/dx + d2w/dy2 = (1 /cT2)d2w/dt2 (2.1)
where w(x, y, t) is the displacement normal to the x-y plane and cT is the velocity of
transverse waves,
cT = (m/p)1/2, (2.2)
where n and p are the shear modulus and the mass density, respectively.
At time t = 0 a plane incident wave of the form
Wine 0, y, t) = /(r) (2.3)
where
r = t + (x/cT) sin a — (y/cT) cos a (2.4)
and /(r) = 0 for r < 0, strikes the tip of a semi-infinite crack. The position of the wave-
front prior to time t = 0 is shown in Fig. la. It is assumed that the crack starts to propa-
gate at a constant velocity v, where v/cT < 1, at the instant that the crack tip is struck.
In contradistinction to earlier work, the crack is, however, not assumed to propagate
in its own plane, but rather in a plane which makes an angle kit with the plane of the
crack (see Fig. lb). Thus, at time t > 0 the crack tip is located at point D which is
0
UNDISTURBED / DISTURBED
(a) t<0
(b) t>0
Fig. 1. Pattern of wavefronts and position of the crack tip.
SKEW CRACK PROPAGATION 125
defined by r = vt, 6 = kit. At time t the crack has generated a plane reflected wave and
a cylindrical diffracted wave. The pattern of wavefronts and the position of the crack
tip is shown in Fig. lb. Note that P and Q denote points which coincide with the origin 0
but which are located on the lower and upper surfaces of the crack, respectively.
The reflection and diffraction of the incident wave involve horizontally polarized
motions only which are governed by Eq. (2.1). To investigate the cylindrical wave it
is convenient to express the Laplacian in polar coordinates, with fixed origin at 0. Eq.
(2.1) is thus rewritten as
ld_(dw
r dr V dr.
, 1 = J_ djw
r dr V dr) r 3d' cT2 dt2 ^ '
The relevant shear stresses are
tTz — ndw/dr, Tez = (ji/r)dw/dQ. (2.6a, b)
In this paper we will consider the case of a step-stress incident wave. The function
/(r) in Eq. (2.3) then is of the form
/(r) = WutH(t) (2.7)
where H(r) is the Heaviside step function. The method of solution of Eq. (2.5) is based
on the premise that certain field variables should show dynamic similarity, since there
is no characteristic length in this diffraction problem. This implies that these field
variables depend on the independent variables r, 9 and t only through dependence
on d and on the ratio r/t. For an incident wave of the form (2.7) the particle velocity
W = dw/dt (2.8)
shows this property, i.e., W(r, 8, t) = IF(s, 0), where
s = r/t. (2.9)
To solve for W{s, 6) it is convenient to introduce s = r/t as a new variable. The
equation governing W(s, d) follows from Eq. (2.1) as
s2\d2W , L 2s2\dW d2W ,„in,
s I1 " c"V Tf + 'I1 " 17 + W " °' <2'I0)
In terms of s and d the boundary conditions on W(s, d) take the form (see Fig. la, b):
6 = —7r, s < cT , dW/dd = 0, (2.11)
—7T < 6 < —(a + x/2), s = cT , W = 2W0 , (2-12)
— (a + 7r/2) < 6 < a + 7r/2, s = cT , W = W0 , (2.13)
a -f" 7r/2 < 6 < 7r, s = cT , W = 0, (2.14)
d = nr, s <cT , dW/dd = 0, (2.15)
6 = kit ± e, s < v, dW/dd = 0. (2.16)
For s < cT , Eq. (2.10) is elliptic. By means of Chaplygin's transformation
/3 = arcosh (cT/s), (2.17)
126 J. D. ACHENBACH AND V. K. VARATHARAJULU
Eq. (2.10) can then be simplified to Laplace's equation
d2W/d/32 + d2W/dd2 = 0. (2.18)
The real transformation (2.17) maps the circular domain s < cT , ~w < d < ir, into a
semi-infinite strip containing a slit in the 6 — 0 plane (see Fig. 2). Inside this domain
W(0, 6) satisfies Eq. (2.18). The boundary conditions are indicated in Fig. 2.
Within the semi-infinite strip in the 6 — 0 plane the solution of Laplace's equation
may be written as the real part of an analytic function G(y)
where
Eq. (2.19) implies
where
W = Re G(y) (2.19)
y = 0 + id. (2.20)
Tvz — iT#z = nG'(y) exp (if) (2.21)
Tpz = dT„./dt, T+z = dT+Jdt. (2.22a, b)
Here p and ^ are polar coordinates centered at the point D in the 6 — 0 plane (see Fig. 2)
and tvz and t+z are defined analogously to Eqs. (2.6a, b). A prime denotes a derivative
with respect to the argument of the function.
The function G(y) can, in principle, be obtained by the use of conformal mapping
techniques. The region in the 7-plane can be mapped on the upper half of the f-plane,
where f = £ + it], by the following Schwarz-Christoffel transformation:
7
- »«•> - c- jf+ + c. <2-23>
a+^ir
(a+^7r)
w = 0
E
-C
0
■Q
*P
—=0
w = w0
-B
>=2w0
Fig. 2. Domain in the 6-0 plane.
SKEW CRACK PROPAGATION 127
V
B
D Q
-£ -i -i K i i z
P Q E
Fig. 3. Mapping on f-plane.
where C\ and C2 are complex constants. Schwarz-Christoffel transformations are dis-
cussed in some detail in [7]. The three arbitrary choices in the mapping are that the
points A, F and D are mapped into the points f = — 1, f = +1 and f = k, respectively.
The f-plane is shown in Fig. 3.
The integral in Eq. (2.23) can be evaluated to yield
7 = n t 2V/2 ^ ~~ ^2)V2(1 — f2)V2 + £"!/> + 1] — In [f + %p]\
CP + ?q U ~ CP )
~tTi, (i-Ly/2 {ln [(1 ~ *°2)V2(1 ~f2)1/2 ~ +1] ~ln [f ~?o]1
+ C2 . (2.24)
The mapping of the point F gives C2 = iir. By examining the changes of the imaginary
parts at P and Q we obtain
1 + k = -&> + «)C,/ft, + £0)(1 - ?p2)1/2, (2.25)
1 -*=-(£« - icJC./ft, + f„)(l - £Q2)1/2, (2.26)
respectively. Elimination of C\ from (2.25) and (2.26) yields
(1 + «)/(1 - *) = + K)(l - |«2)l/2/($0 - <0(1 - £p2)'/2. (2.27)
In view of Eqs. (2.25) and (2.26), the mapping can be rewritten as
7 = (1 + K){ln [(1 — £pV/2( 1 — f2)172 + f£p + 1] — In [f + £P]j
+ (1 - «)jln [(1 - £02)1/2(1 - f2)1/2 - ft, + i] _ in [f - {<,]} + iT. (2.28)
Finally, comparing the coordinates of D in the y and f-planes, we obtain the following
relation:
ln [cT/v + {(cT/vf - 1]1/21
= (1 +k) ln {[(1 - £/)1/2(l - *2)1/2 + kZp + 1 ]/({, + k)\
+ (1 - k) In {[(1 - £a2),/2(l - k2)1/2 - + !]/&, - «)}• (2.29)