International Journal of Fracture 104: 317–327, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
Perturbed cracks in two dimensions: An integral-equation
approach
P. A. MARTIN
Department of Mathematical & Computer Sciences Colorado School of Mines, Golden, CO 80401-1887, U.S.A.
(e-mail: pamartin@mines.edu)
Received 19 October 1999; accepted in revised form 18 February 2000
Abstract. A nominally straight crack of finite length is subjected to plane-strain loadings. A perturbation method
is developed for calculating the stress-intensity factors, based on an asymptotic analysis of the governing hypersingular boundary integral equation for the crack-opening displacement. Known exact results for a shallow
circular-arc crack are recovered, correct to second order in the small geometrical parameter. The method will
extend to three-dimensional problems as it does not make essential use of two-dimensional techniques.
1. Introduction
Slightly curved or kinked cracks is the title of a well known paper by Cotterell and Rice
(1980), in which a nominally straight crack is subject to plane-strain loading. The problem is
to calculate the stress-intensity factors, correct to first order in ε, where the perturbed crack is
defined by
y = ε f (x),
−a ≤ x ≤ a,
(1)
with
f (a) = f (−a) = 0;
(2)
here, x and y are Cartesian coordinates, f is a given function and ε is a small parameter.
Cotterell and Rice (1980) found expressions for the stress-intensity factors, using the complexvariable techniques of Muskhelishvili (1953). Expressions correct to second order in ε were
subsequently obtained by Wu (1994), using similar methods.
Wu (1994) claimed that the perturbation expansion in ε is regular, in the sense that the
approximations obtained are uniformly valid throughout the solid; this was implicitly assumed
by Cotterell and Rice (1980). Why is this so? The reason is (2): in two dimensions, one can
always arrange that the two crack tips are fixed, independently of ε, implying that the places
where the stresses are singular do not move as the crack is perturbed from the straight (ε = 0)
reference position. In general, we do not have this luxury in three dimensions, where the edge
of a non-planar crack (a simple closed curve) need not lie in a plane.
The discussion above motivates the present work. We reconsider the plane-strain problem
for a slightly curved crack, using integral-equation methods. We begin by reformulating the
boundary-value problem as a boundary integral equation; we choose to use a hypersingular
integral equation for the crack-opening displacement (COD). Next, we parametrise the curve
defining the crack, leading to a one-dimensional hypersingular integral equation on a finite
318 P. A. Martin
interval. At this stage, the analysis is exact. Then, we introduce (1), leading to a sequence of
hypersingular integral equations for each term in the regular expansion of the COD in powers
of ε. We verify that the method yields approximations in agreement with Muskhelishvili’s
(1953) exact solution for a circular-arc crack under constant loads, correct to second order in
ε.
Our method does not require that f (x) satisfies (2). We obtain regular perturbation expansions because we work only on the crack faces, not within the solid. (Displacements and
stresses off the crack can be obtained, if desired, by inserting the approximations to the COD
into the integral representation (5).) Moreover, our method does not make any essential use of
two-dimensional techniques (such as those based on functions of a complex variable). Thus,
it is perhaps not surprising that our method extends to perturbed cracks in three dimensions.
We have obtained analogous results for some scalar problems (potential flow past wrinkled
discs) (Martin, 1998a, b), and will describe our results on three-dimensional crack problems
elsewhere.
We note that Dreilich and Gross (1985) have briefly considered perturbed cracks, using a
singular integral equation for the derivative of the COD. Numerical solutions of this integral
equation were presented by Chen et al. (1991). In fact, there are many other papers on curved
cracks in two dimensions (see, for example, Lin’kov and Mogilevskaya, 1990, 1994; Sur and
Altiero, 1988; Zang and Gudmundson, 1988), but we are mainly interested here in asymptotic
results.
Finally, let us make some remarks on the asymptotic method developed by Movchan, Gao
and Willis (1998) for two-dimensional (and three-dimensional) crack perturbations. Their
theory uses local expansions near the crack tips and Bueckner weight functions for the unperturbed reference crack. It is a form of singular-perturbation theory, with a boundary layer
near each crack tip; this ‘occurs due to relocation of the coordinate system from the actual
crack tip to the end of the reference crack’(Movchan et al., 1998, Section 2.1). These technical
difficulties do not arise when the problem is first reformulated as a boundary integral equation,
as we do below.
2. An integral equation
In this section, we derive an exact hypersingular integral equation for the COD when a
curved crack is loaded under plane-strain conditions. The basic ingredient is the well-known
fundamental solution
1
∂R ∂R
1
Gij (P , Q) =
.
(3 − 4ν) δij log +
8π µ(1 − ν)
R ∂xi ∂xj
Here, i, j = 1, 2, µ is the shear modulus, ν is Poisson’s ratio, the points P and Q have
Cartesian coordinates (x1 , x2 ) and (x1′ , x2′ ), respectively, δij is the Kronecker delta, and
1/2
R = (x1 − x1′ )2 + (x2 − x2′ )2
(3)
is the distance between P and Q (see, for example Rizzo, 1967).
Consider a cavity in an otherwise unbounded homogeneous isotropic elastic solid. Assume
that the surface of the cavity, S (a simple closed curve) is loaded in such a way that the
displacement components ui = o(1) and the stress components τij = o(R−1 ) as R → ∞,
where R is distance from some fixed point in the vicinity of the cavity. Then, a familiar
calculation yields the integral representation
Perturbed cracks in two dimensions 319
ui (P ) =
Z
S
uj (q) Tij (P , q) − tj (q) Gij (P , q) dsq ,
i = 1, 2,
(4)
q
q
q
where summation over repeated subscripts is implied, tj (q) = nk τj k , n(q) = (n1 , n2 ) is the
unit normal at q ∈ S pointing into the solid,
q
q
Tij = [4π(1 − ν)R]−1 {(1 − 2ν)[Rj ni − Ri nj ]
q
−Rn [(1 − 2ν)δij + 2Ri Rj ]},
Ri =
xi − xi′
∂R
=
∂xi
R
Rnq ≡
and
∂R
q
q ∂R
= ni ′ = −ni Ri .
∂nq
∂xi
The formula (4) is the basis for boundary-element methods in two-dimensional elastostatics.
Next, let the cavity S shrink to a crack Ŵ. Denote the two sides of the crack by Ŵ + and Ŵ − ,
and let q + and q − be corresponding points on Ŵ + and Ŵ − , respectively. Define n(q) = n(q + )
so that n(q − ) = −n(q). Also, define the crack-opening displacement (COD) by
[ui (q)] = ui (q + ) − ui (q − ).
We assume that the imposed stresses are continuous across Ŵ. Then, we find that (4) reduces
to
Z
ui (P ) = [uj (q)] Tij (P , q) s.q , i = 1, 2,
(5)
Ŵ
an integral representation for the displacement components at any point P in the solid.
We can compute the tractions ti (p) on Ŵ corresponding to ui , using Hooke’s law. The result
is
Z
1
ti (p) = × [uj (q)] Sij (p, q) s.q , i = 1, 2, p ∈ Ŵ.
(6)
µ
Ŵ
In this formula, the cross on the integral sign means that the integral is to be interpreted as
a Hadamard finite-part integral; for references, see, for example, Martin and Rizzo (1989) or
Martin et al. (1998). The kernel Sij is given by
p q
q p
π Sij = R −2 {AN δij + Ani nj + (1 − 3A)ni nj + (1 − 2A)N Ri Rj −
q
p
p
p
q
q
−Rn [2Ani Rj + (1 − 2A)Ri nj ] + Rn [(1 − 2A)ni Rj + 2ARi nj ]
p
(7)
q
+Rn Rn [8(1 − A)Ri Rj − (1 − 2A)δij ]}
q
p q
p
p q
= (1 − A)R −2 {ni nj + N Ri Rj − Rn Ri nj + Rn ni Rj +
p
q
+Rn Rn (8Ri Rj − δij )},
p q
where N = ni ni ,
A=
1 − 2ν
2(1 − ν)
and
Rnp ≡
∂R
p ∂R
p
= ni
= n i Ri .
∂np
∂xi
(8)
320 P. A. Martin
p
All the terms inside the curly brackets are bounded as p → q; in fact, in this limit, Rn →
q
0, Rn → 0 and N → 1, assuming that Ŵ is a twice-differentiable curve. Thus, (8) exhibits the
expected non-integrable R −2 singularity that is typical of one-dimensional finite-part integrals.
From (8), one can also show that Sij = Sj i .
Once the traction components ti (p) are prescribed on Ŵ, (6) becomes a hypersingular
boundary integral equation for the components of the COD, [uj ]. It is to be solved subject
to the natural end-point conditions,
[ui (q] = 0, i = 1, 2, when q is an end-point of Ŵ.
(9)
(Note that (6) can also be used if Ŵ represents several disjoint cracks.)
3. Projection
Our basic integral equation, (6), is exact and it holds on the curve Ŵ. It is more convenient to
write (6) on a fixed reference curve, which we take to be a straight line segment of length 2. We
do this projection by simply parametrising the curve Ŵ. Thus, we suppose that the integration
point q = (x1′ , x2′ ) is specified by
x1′ = ax,
x2′ = aF (x),
−1 ≤ x ≤ 1,
where a is a length-scale and the (dimensionless) function F specifies the shape of Ŵ. Similarly, the point p = (x1 , x2 ) is specified by
x1 = ax0 ,
x2 = aF (x0 ),
−1 ≤ x0 ≤ 1.
For the unit normals n(q) and n(p), we have n(q) = N(q)/N(q) where
p
N(q) = (−F ′ (x), 1) and N(q) = |N(q)| = 1 + [F ′ (x)]2 ,
with a similar expression for n(p). √
From (3), we have R = a|x − x0 | 1 + 32 , where
3=
F (x) − F (x0 )
.
x − x0
Next, we have
R1 = √

1+
N(q) Rnq =
32
,
R2 = √
−(3 − F ′ )
,
√
1 + 32
3
1 + 32
,
(3 − F0′ )
N(p) Rnp = √
1 + 32
and N(p) N(q) N = 1 + F ′ F0′ , where F ′ ≡ F ′ (x), F0′ ≡ F ′ (x0 ) and
 = (x0 − x)/|x0 − x|
so that 2 = 1. These expressions allow us to write the kernel Sij in terms of x, x0 and F .
Thus, we define a new kernel matrix S̃ij by
2π(1 − ν)a 2 (x − x0 )2 N(p) N(q) Sij = S̃ij (x0 , x);
(10)
Perturbed cracks in two dimensions 321
it is given explicitly as follows:
S̃11 = (1 + 32 )−3 {1 − 632 + 34 − (3 − 632 − 34 )F ′ F0′ +
+23(3 − 32 )(F ′ + F0′ )},
S̃22 = (1 + 32 )−3 {1 + 632 − 334 + (1 − 632 + 34 )F ′ F0′ −
−23(1 − 332 )(F ′ + F0′ )},
S̃12 = S̃21 = (1 + 32 )−3 {23(1 − 332 ) − 23(3 − 32 )F ′ F0′ −
−(1 − 632 + 34 )(F ′ + F0′ )}.
Finally, if we multiply the integral equation (6) by 2(1 − ν)N(p), and note that dsq =
aN(q) dx, we obtain
Z
dx
1 1
× ũj (x) S̃ij (x0 , x)
= t˜i (x0 ), i = 1, 2,
(11)
π −1
(x − x0 )2
for −1 < x0 < 1, where
ũi (x) = a −1 [ui (q)]
and
t˜i (x0 ) = 2(1 − ν)µ−1 N(p) ti (p).
(12)
Equation (11) is our basic hypersingular integral equation for a curved crack Ŵ under plainstrain loading. In fact, it is a pair of coupled scalar integral equations for ũ1 and ũ2 . This pair
is to be solved subject to the end-point conditions (9), which become
ũi (1) = ũi (−1) = 0,
i = 1, 2,
after projection. The (vector) integral equation (11) is exact, and it can be solved numerically.
However, our focus will be on slightly curved cracks, where Ŵ is approximately straight.
4. The straight crack
For a straight crack, we have
F (x) = mx + c,
where m and c are constants. Hence, F ′ = F0′ = 3 = m. Elementary calculations then give
S̃11 = S̃22 = 1
and
S̃12 = S̃21 = 0,
so that the 2 × 2 system of scalar integral equations decouples into
(H ũi ) (x0 ) = t˜i (x0 ),
i = 1, 2,
where the operator H is defined by
Z
1 1 u(x)
dx,
(Hu) (x0 ) = ×
π −1 (x − x0 )2
−1 < x0 < 1,
−1 < x0 < 1.
(13)
322 P. A. Martin
The hypersingular integral equation Hu = b can be solved explicitly. When the end-point
conditions u(1) = u(−1) = 0 are imposed, the solution is given by


Z
|x
−
x
|
1 1
0
 dx
q
u(x0 ) =
b(x) log 
π −1
2
2
1 − xx0 + (1 − x )(1 − x0 )
for −1 < x0 < 1, provided that b satisfies certain conditions (Martin, 1992). (For example,
it is sufficient that b be continuous with integrable end-point singularities.) In practice, it is
often simpler to use the following formula (Kaya and Erdogan, 1987; Martin, 1992):
Z √
1 1 1 − x 2 Un (x)
×
dx = −(n + 1) Un (x0 ), n = 0, 1, 2, . . . .
π −1 (x − x0 )2
Here, Un (x) is a Chebyshev polynomial of the second kind, defined by Un (cos θ) =
[sin (n + 1)θ]/ sin θ. Thus, if
X
b(x) =
bn Un (x),
n=0
then
p
X
u(x) = − 1 − x 2
(n + 1)−1 bn Un (x)
n=0
is the unique solution of H u = b, subject to u(1) = u(−1)
√ = 0. Note that, as the Chebyshev
polynomials are orthogonal with respect to the weight 1 − x 2 , we have
Z
2 1p
bn =
1 − x 2 Un (x) b(x) dx,
π −1
so that the coefficients bn are known in terms of the given function b.
5. Slightly curved cracks
Suppose that
F (x) = ε f (x),
where ε is a small dimensionless parameter and f is independent of ε. Setting
3 = ελ
with λ = {f (x) − f (x0 )}/(x − x0 ),
we find that
2
S̃11 = 1 + ε 2 S11
+ O(ε 4 ),
2
S̃22 = 1 + ε 2 S22
+ O(ε 4 ),
1
S̃12 = S̃21 = εS12
+ O(ε 3 )
as ε → 0, where
2
S22
= 3λ2 + f ′ f0′ − 2λ(f ′ + f0′ ),
Perturbed cracks in two dimensions 323
1
S12
= 2λ − f ′ − f0′ ,
2
2
S11
= −3S22
,
1
2
f ′ ≡ f ′ (x) and f0′ ≡ f ′ (x0 ). Note that S12
(x0 , x) and S22
(x0 , x) are both O((x − x0 )2 ) as
|x − x0 | → 0. These kernels are also symmetric in x and x0 .
We expand t˜i similarly. Suppose that the prescribed tractions are defined in terms of a stress
field τij (x1 , x2 ), so that
t˜i (x0 ) = B{τi2 (ax0 , aF0 ) − F0′ τi1 (ax0 , aF0 )},
exactly, where F0 ≡ F (x0 ) and B = 2(1 − ν)/µ. From Taylor’s theorem, we have
τij (ax0 , εaf0 ) = τij(0) (x0 ) + ετij(1) (x0 ) + ε 2 τij(2) (x0 ) + · · · ,
where f0 ≡ f (x0 ) and
(af0 )n ∂ n
(n)
τij (x0 ) =
τij (ax0 , y) .
n
n!
∂y
y=0
Hence, we deduce that
t˜i (x0 ) = ti0 + εti1 + ε 2 ti2 + · · · ,
(14)
where
ti0 = Bτi2(0)
and
tin = B τi2(n) − f0′ τi1(n−1)
(15)
for n = 1, 2, . . . . In particular, if the given stress field is constant, then
t˜i (x0 ) = B(τi2 − εf0′ τi1 )
exactly.
Next, we expand the crack-opening displacement ũ = (ũ1 , ũ2 ), writing
ũi = u0i + εu1i + ε 2 u2i + · · · .
j
Substituting in (11) then gives integral equations for ui . These are all uncoupled scalar equations of the form
j
j
Hui = bi ,
i = 1, 2,
j = 0, 1, 2, . . . ,
j
where H is defined by (13) and bi is known. For j = 0, we obtain
bi0 = ti0 ,
i = 1, 2,
so that the leading-order approximation is obtained by solving the integral equations for a
straight crack. Having found u0i , the first-order approximation is obtained using
Z
dx
1 1 0
1
1
1
u (x) S12
b1 (x0 ) = t1 (x0 ) −
π −1 2
(x − x0 )2
and
b21 (x0 )
=
t21 (x0 )
1
−
π
Z
1
−1
1
u01 (x) S12
dx
.
(x − x0 )2
324 P. A. Martin
At second order, we have
1
π
Z
1
−
π
Z
b12 (x0 ) = t12 (x0 ) +
1
−1
0
2
1
3u1 (x) S22
− u12 (x) S12
dx
(x − x0 )2
and
b22 (x0 )
=
t22 (x0 )
1
−1
0
2
1
u2 (x) S22
+ u11 (x) S12
dx
.
(x − x0 )2
6. An example: quadratic cracks
As a simple example, let us take a quadratic crack,
f (x) = a0 + a1 x + a2 x 2 ,
under uniform loading; here, a0 , a1 and a2 are constants. Hence f ′ = 2a2 x + a1 , λ = a2 (x +
1
2
x0 ) + a1 , S12
= 0 and S22
= −a22 (x − x0 )2 . From (15), we have
ti0 = Bτi2 ,
ti1 (x) = −B(2a2 x + a1 )τi1
and
ti2 = 0.
Hence
p
u0i (x) = −Bτi2 1 − x 2
and
p
u1i (x) = B(a2 x + a1 )τi1 1 − x 2 ,
√
where we have noted that the solution of (H u)(x0 ) = 2ax0 + b is u(x) = −(ax + b) 1 − x 2 .
For the second-order solution, we have
Z
a22 1 0
2
b2 (x0 ) =
u (x) dx = − 12 Ba22 τ22
π −1 2
and b12 = 23 Ba22 τ12 , whence u2i is proportional to u0i . So, combining all these results, we find
that
p
u1 (x) = −B τ12 − ε(a2 x + a1 )τ11 + 23 ε 2 a22 τ12
1 − x2
and
p
u2 (x) = −B τ22 − ε(a2 x + a1 )τ12 − 12 ε 2 a22 τ22
1 − x2,
correct to second order in ε.
The quantities of most interest are the stress-intensity factors, K1 and K2 . We define these
by
p
p
(16)
un (x) ∼ −BK1 2ρ and ut (x) ∼ −BK2 2ρ as ρ → 0,
where
ρ=
p
(1 − x)2 + ε 2 [f (1) − f (x)]2
is distance from the edge at x = 1, and un and ut are the normal and tangential components,
respectively, of the crack-opening displacement, ũ. The slope of the crack near x = 1 is
εf ′ (1), so that if θ is the inclination of the crack to the x-axis at x = 1, then
Perturbed cracks in two dimensions 325
sin θ = εf ′ (1)
and
cos θ = 1 − 12 ε 2 [f ′ (1)]2 ,
with an error of O(ε 3 ) as ε → 0. Thus, near x = 1,
un = u2 cos θ − u1 sin θ
∼ u02 + ε{u12 − u01 f ′ (1)} + ε 2 {u22 − 12 u02 [f ′ (1)]2 − u11 f ′ (1)}
√
∼ −B 2(1 − x){τ22 − ε(3a2 + 2a1 )τ12
+ε 2 [(a2 + a1 )(2a2 + a1 )τ11 − 21 (5a22 + 4a2 a1 + a12 )τ22 ]}
and
ut = u1 cos θ + u2 sin θ
∼ u01 + ε{u11 + u02 f ′ (1)} + ε 2 {u21 − 12 u01 [f ′ (1)]2 + u12 f ′ (1)}
√
∼ −B 2(1 − x){τ12 + ε[(2a2 + a1 )τ22 − (a2 + a1 )τ11 ]
−ε 2 τ12 [ 52 a22 + 5a2 a1 + 23 a12 ]}.
We now compare these expressions with (16), noting that
√
1/4
√
ρ = 1 − x 1 + ε 2 [a2 (1 + x) + a1 ]2
∼
√
1 − x 1 + 14 ε 2 (2a2 + a1 )2 .
Hence, we find that
K1 = τ22 − ε(3a2 + 2a1 )τ12
and
+ε 2 (a2 + a1 )(2a2 + a1 )τ11 −
7 2
a
2 2
+ 3a2 a1 + 34 a12 τ22
K2 = τ12 + ε {(2a2 + a1 )τ22 − (a2 + a1 )τ11 }
−ε 2 τ12
7
a2
2 2
+ 6a2 a1 + 74 a12 .
A special case of this geometry is a shallow circular arc, given by
q
x2 = aF (x1 ) = c − c2 − x12 , −a < x1 < a,
where c is the radius of the arc. The arc subtends an angle of 2α at the centre of the circle,
where sin α = a/c. For a shallow arc, we suppose that c/a is large, with a fixed. Hence, we
see that
f (x) = 12 x 2
with
ε = a/c = sin α,
326 P. A. Martin
where α = ε + O(ε 3 ) as ε → 0. Thus, the previous calculation, with a2 =
gives
K1 = τ22 − 23 ετ12 + ε 2 21 τ11 − 78 τ22
1
2
and a1 = 0,
and
K2 = τ12 + ε(τ22 − 21 τ11 ) − 78 ε 2 τ12 ,
with an error of O(ε 3 ) as ε → 0. This agrees with Muskhelishvili’s exact solution (Muskhelishvili, 1953, Section 124a), as re-expressed by Cotterell and Rice (1980, p. 158].
7. Discussion
We have presented a perturbation method for calculating the crack-opening displacement
for a crack that is nominally straight and subjected to plane-strain loading. The method is
general and takes proper account of the edge behaviour; we do not have to assume that the
two crack tips lie on the x-axis. At each perturbation order, we have to solve two uncoupled
scalar integral equations of the form H u = b, where H is the basic one-dimensional hypersingular operator; such equations can be solved exactly. For any given crack shape f ,
j
the main difficulty comes from the evaluation of the forcing terms (bi ) for each integral
equation. Nevertheless, the method does generalise to three-dimensional problems, such as
non-planar perturbations of a penny-shaped crack under mixed-mode loadings; this work will
be described elsewhere.
Acknowledgements
I am grateful to an anonymous referee for pointing out that (7) could be simplified to (8),
although the expression (10) was originally obtained from (7).
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