Regularly and singularly perturbed cracks

In a nondimensionalized rectangular Cartesian coordinate system $\left ( {{x_1}, {x_2}} \right )$ let ${x_2} = \varepsilon {Y_ \pm }\left ( {{x_1}} \right )$ denote the upper and lower surfaces of a hole where $\left | {{x_1}} \right | \le 1$ and $\varepsilon$ is a small parameter. As $\varepsilon$ tends to zero, the hole degenerates into a crack of length 2. The functions ${Y_ \pm }$, together with their derivatives, are continuous and ${Y_ + } - {Y_ - } \ge 0$ . For $\varepsilon$ not equal to zero, the hole is called a regularly (singularly) perturbed crack if ${Y’_ + }\left ( { \pm 1} \right ) = {Y’_ - }\left ( { \pm 1} \right ) \left ( {Y’_ + } \left ( { \pm 1} \right ) \ne {Y’_ - }\left ( { \pm 1} \right ) \right )$. 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 $\varepsilon$ . The notch-tip singularity associated with a singularly perturbed crack is obtained by the method of matched asymptotic expansions.