Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Regularly and singularly perturbed cracks

Author: Chien H. Wu
Journal: Quart. Appl. Math. 52 (1994), 529-543
MSC: Primary 73M25; Secondary 73V35
DOI: https://doi.org/10.1090/qam/1292203
MathSciNet review: MR1292203
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Abstract: 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\vert {{x_1}} \right\vert \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.

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DOI: https://doi.org/10.1090/qam/1292203
Article copyright: © Copyright 1994 American Mathematical Society

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