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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In a nondimensionalized rectangular Cartesian coordinate system let denote the upper and lower surfaces of a hole where and is a small parameter. As tends to zero, the hole degenerates into a crack of length 2. The functions , together with their derivatives, are continuous and . For not equal to zero, the hole is called a regularly (singularly) perturbed crack if . 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 . The notch-tip singularity associated with a singularly perturbed crack is obtained by the method of matched asymptotic expansions.

**[1]**L. T. Berezhnitskii and V. M. Sadivskii,*Stress distribution near elastic inclusions with cuspidal points on the outline*, Soviet Material Sci.**12**, 261-265 (1976)**[2]**L. T. Berezhnitskii and V. M. Sadivskii,*Theory of sharp-pointed stress concentrations in anisotropic plates*, Soviet Material Sci.**13**, 303-311 (1977)**[3]**J. D. Cole,*Perturbation methods in applied mathematics*, Blaisdell, Waltham, MA, 1968**[4]**B. Cotterell and J. Rice,*Slightly curved or kinked cracks*, Internat. J. Fracture**16**, 155-169 (1980)**[5]**A. H. England,*Complex Variable Methods in Elasticity*, Wiley-Interscience, New York, 1971**[6]**R. V. Goldstein and R. L. Salganik,*Brittle fracture of solids with arbitrary cracks*, Internat. J. Fracture**10**, 507-423 (1974)**[7]**H. K. Moffatt and B. R. Duffy,*Local similarity solutions and their limitations*, J. Fluid Mech.**96**, 299-313 (1980)**[8]**N. I. Muskhelishvili,*Some Basic Problems of the Mathematical Theory of Elasticity*, translated by J. R. M. Redok, Noordhoff, Groningen, 1953**[9]**V. V. Panasynk and L. T. Berezhnitskii,*Limit equilibrium of plates with sharp stress raisers*, Soviet Material Sci.**1**, 293-301 (1965)**[10]**T. C. T. Ting,*The wedge subjected to tractions*:*A paradox re-examined*, J. Elasticity**14**, 235-247 (1984)**[11]**T. C. T. Ting,*Asymptotic solution near the apex of an elastic wedge with curved boundaries*, Quart. Appl. Math.**42**, 467-476 (1985)**[12]**M. Van Dyke,*Perturbation Methods in Fluid Mechanics*, Parabolic Press, Stanford, CA, 1975**[13]**M. L. Williams,*Stress singularities resulting from various boundary conditions in angular corners of plates in extension*, J. Appl. Mech.**19**, 526-528 (1952)**[14]**R. A. Westman,*Pressurized star crack*, J. Math. Phys.**43**, 191-198 (1964)**[15]**C. H. Wu,*Unconventional internal cracks Part*1:*Symmetric variations of a straight crack*, J. Appl. Mech.**40**, 62-68 (1982)**[16]**C. H. Wu,*Unconventional internal cracks Part*2:*Method of generating simple cracks*, J. Appl. Mech.**49**, 383-388 (1982)**[17]**C. H. Wu,*Stress and notch-stress concentration induced by slight depressions and protrusions*, J. Appl. Mech.**60**, 992-997 (1993)

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
73M25,
73V35

Retrieve articles in all journals with MSC: 73M25, 73V35

Additional Information

DOI:
https://doi.org/10.1090/qam/1292203

Article copyright:
© Copyright 1994
American Mathematical Society