In a nondimensionalized rectangular Cartesian coordinate system
(
x
1
,
x
2
)
\left ( {{x_1}, {x_2}} \right )
let
x
2
=
ε
Y
±
(
x
1
)
{x_2} = \varepsilon {Y_ \pm }\left ( {{x_1}} \right )
denote the upper and lower surfaces of a hole where
|
x
1
|
≤
1
\left | {{x_1}} \right | \le 1
and
ε
\varepsilon
is a small parameter. As
ε
\varepsilon
tends to zero, the hole degenerates into a crack of length 2. The functions
Y
±
{Y_ \pm }
, together with their derivatives, are continuous and
Y
+
−
Y
−
≥
0
{Y_ + } - {Y_ - } \ge 0
. For
ε
\varepsilon
not equal to zero, the hole is called a regularly (singularly) perturbed crack if
Y
+
′
(
±
1
)
=
Y
−
′
(
±
1
)
(
Y
+
′
(
±
1
)
≠
Y
−
′
(
±
1
)
)
{Y’_ + }\left ( { \pm 1} \right ) = {Y’_ - }\left ( { \pm 1} \right ) \left ( {Y’_ + } \left ( { \pm 1} \right ) \ne {Y’_ - }\left ( { \pm 1} \right ) \right )
. Regular perturbation procedures are applied to obtain the stress intensity factors existing at the tips of regularly perturbed cracks. It is shown that the second term of a two-term expansion is not always of the order of
ε
\varepsilon
. The notch-tip singularity associated with a singularly perturbed crack is obtained by the method of matched asymptotic expansions.