Mapping flat cracks onto penny-shaped cracks, with application to somewhat circular tensile cracks
Author:
P. A. Martin
Journal:
Quart. Appl. Math. 54 (1996), 663-675
MSC:
Primary 73M25; Secondary 73B50
DOI:
https://doi.org/10.1090/qam/1417230
MathSciNet review:
MR1417230
Full-text PDF Free Access
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Abstract: Consider a three-dimensional homogeneous isotropic elastic solid containing a flat pressurized crack, $\Omega$. The problem of finding the resulting stress distribution can be reduced to a hypersingular integral equation over $\Omega$ for the crack-opening displacement. Here, this equation is transformed into a similar equation over a circular region $D$, using a conformal mapping between $\Omega$ and $D$. This new equation is then regularized analytically by using an appropriate expansion method (Fourier series in the azimuthal direction and series of orthogonal polynomials in the radial direction). Analytical results for regions that are approximately circular are also obtained. The method will generalize to other scalar problems and to vector problems (such as shear loading of the crack).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 1965
L. V. Ahlfors, Complex Analysis, 2nd edition, McGraw-Hill, New York, 1966
H. F. Bueckner, Field singularities and related integral representations, in Mechanics of Fracture 1, Methods of Analysis and Solutions of Crack Problems (ed. G. C. Sih), Noordhoff, Leyden, 1973, pp. 239–314
H. D. Bui, Application des potentiels élastiques à l’étude des fissures planes de forme arbitraire en milieu tridimensionnel, C. R. Acad. Sci. Paris, Série A 280, 1157–1160 (1975)
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, vol. 2, McGraw-Hill, New York, 1953
V. J. Ervin and E. P. Stephan, Collocation with Chebyshev polynomials for a hypersingular integral equation on an interval, J. Comput. Appl. Math. 43, 221–229 (1992)
V. I. Fabrikant, Flat crack of arbitrary shape in an elastic body: analytical approach, Philos. Mag. A. 56, 175–189 (1987)
A. Frenkel, A Chebyshev expansion of singular integrodifferential equations with a ${\partial ^2} ln \left | {s - t} \right |/\partial s\partial t$ kernel, J. Comput. Phys. 51, 335–342 (1983)
H. Gao and J. R. Rice, Somewhat circular tensile cracks, Internat. J. Fracture 33, 155–174 (1987)
M. A. Golberg, The convergence of several algorithms for solving integral equations with finite-part integrals, J. Integral Equations 5, 329–340 (1983)
M. A. Golberg, The convergence of several algorithms for solving integral equations with finite-part integrals. II, J. Integral Equations 9, 267–275 (1985)
J. T. Guidera and R. W. Lardner, Penny-shaped cracks, J. Elasticity 5, 59–73 (1975)
N. I. Ioakimidis, Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity, Acta Mech. 45, 31–47 (1982)
A. C. Kaya and F. Erdogan, On the solution of integral equations with strongly singular kernels, Quart. Appl. Math. 45, 105–122 (1987)
H. Kober, Dictionary of Conformal Representations, Dover, New York, 1957
E. Kossecka, Defects as surface distributions of double forces, Arch. Mech. 23, 481–494 (1971)
S. Krenk, A circular crack under asymmetric loads and some related integral equations, J. Appl. Mech. 46, 821–826 (1979)
G. Krishnasamy, L. W. Schmerr, T. J. Rudolphi, and F. J. Rizzo, Hypersingular boundary integral equations: some applications in acoustic and elastic wave scattering, J. Appl. Mech. 57, 404–414 (1990)
P. A. Martin, The discontinuity in the elastostatic displacement vector across a penny-shaped crack under arbitrary loads, J. Elasticity 12, 201–218 (1982)
P. A. Martin, Orthogonal polynomial solutions for pressurized elliptical cracks, Quart. J. Mech. Appl. Math. 39, 269–287 (1986)
P. A. Martin, End-point behaviour of solutions to hypersingular integral equations, Proc. Roy. Soc. London Ser. A 432, 301–320 (1991)
P. A. Martin and F. J. Rizzo, On boundary integral equations for crack problems, Proc. Roy. Soc. London Ser. A 421, 341–355 (1989)
V. V. Panasyuk, Limiting Equilibrium of Brittle Solids with Fractures, Management Information Services, Detroit, 1970. [Lib. of Congr. No. 70-135093] (Translation from Russian edition, Naukova Dumka, Kiev, 1968)
V. V. Panasyuk, A. E. Andrejkiv, and M. M. Stadnik, Three-dimensional static crack problems solution (a review), Engrg. Fract. Mech. 14, 245–260 (1981)
N. F. Parsons and P. A. Martin, Scattering of water waves by submerged plates using hypersingular integral equations, Appl. Ocean Res. 14, 313–321 (1992)
E. Z. Polch, T. A. Cruse, and C.-J. Huang, Traction BIE solutions for flat cracks, Computational Mech. 2, 253–267 (1987)
J. R. Rice, Weight function theory for three-dimensional elastic crack analysis, in Fracture Mechanics: Perspectives and Directions (Twentieth Symposium) (ed. R. P. Wei and R. P. Gangloff), American Society for Testing and Materials, Philadelphia, 1989, pp. 29–57
K. Takakuda, T. Koizumi, and T. Shibuya, On integral equation methods for crack problems, Bull. Japan. Soc. Mech. Engrs. 28, 217–224 (1985)
J. Weaver, Three-dimensional crack analysis, Internat. J. Solids and Structures 13, 321–330 (1977)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 1965
L. V. Ahlfors, Complex Analysis, 2nd edition, McGraw-Hill, New York, 1966
H. F. Bueckner, Field singularities and related integral representations, in Mechanics of Fracture 1, Methods of Analysis and Solutions of Crack Problems (ed. G. C. Sih), Noordhoff, Leyden, 1973, pp. 239–314
H. D. Bui, Application des potentiels élastiques à l’étude des fissures planes de forme arbitraire en milieu tridimensionnel, C. R. Acad. Sci. Paris, Série A 280, 1157–1160 (1975)
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, vol. 2, McGraw-Hill, New York, 1953
V. J. Ervin and E. P. Stephan, Collocation with Chebyshev polynomials for a hypersingular integral equation on an interval, J. Comput. Appl. Math. 43, 221–229 (1992)
V. I. Fabrikant, Flat crack of arbitrary shape in an elastic body: analytical approach, Philos. Mag. A. 56, 175–189 (1987)
A. Frenkel, A Chebyshev expansion of singular integrodifferential equations with a ${\partial ^2} ln \left | {s - t} \right |/\partial s\partial t$ kernel, J. Comput. Phys. 51, 335–342 (1983)
H. Gao and J. R. Rice, Somewhat circular tensile cracks, Internat. J. Fracture 33, 155–174 (1987)
M. A. Golberg, The convergence of several algorithms for solving integral equations with finite-part integrals, J. Integral Equations 5, 329–340 (1983)
M. A. Golberg, The convergence of several algorithms for solving integral equations with finite-part integrals. II, J. Integral Equations 9, 267–275 (1985)
J. T. Guidera and R. W. Lardner, Penny-shaped cracks, J. Elasticity 5, 59–73 (1975)
N. I. Ioakimidis, Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity, Acta Mech. 45, 31–47 (1982)
A. C. Kaya and F. Erdogan, On the solution of integral equations with strongly singular kernels, Quart. Appl. Math. 45, 105–122 (1987)
H. Kober, Dictionary of Conformal Representations, Dover, New York, 1957
E. Kossecka, Defects as surface distributions of double forces, Arch. Mech. 23, 481–494 (1971)
S. Krenk, A circular crack under asymmetric loads and some related integral equations, J. Appl. Mech. 46, 821–826 (1979)
G. Krishnasamy, L. W. Schmerr, T. J. Rudolphi, and F. J. Rizzo, Hypersingular boundary integral equations: some applications in acoustic and elastic wave scattering, J. Appl. Mech. 57, 404–414 (1990)
P. A. Martin, The discontinuity in the elastostatic displacement vector across a penny-shaped crack under arbitrary loads, J. Elasticity 12, 201–218 (1982)
P. A. Martin, Orthogonal polynomial solutions for pressurized elliptical cracks, Quart. J. Mech. Appl. Math. 39, 269–287 (1986)
P. A. Martin, End-point behaviour of solutions to hypersingular integral equations, Proc. Roy. Soc. London Ser. A 432, 301–320 (1991)
P. A. Martin and F. J. Rizzo, On boundary integral equations for crack problems, Proc. Roy. Soc. London Ser. A 421, 341–355 (1989)
V. V. Panasyuk, Limiting Equilibrium of Brittle Solids with Fractures, Management Information Services, Detroit, 1970. [Lib. of Congr. No. 70-135093] (Translation from Russian edition, Naukova Dumka, Kiev, 1968)
V. V. Panasyuk, A. E. Andrejkiv, and M. M. Stadnik, Three-dimensional static crack problems solution (a review), Engrg. Fract. Mech. 14, 245–260 (1981)
N. F. Parsons and P. A. Martin, Scattering of water waves by submerged plates using hypersingular integral equations, Appl. Ocean Res. 14, 313–321 (1992)
E. Z. Polch, T. A. Cruse, and C.-J. Huang, Traction BIE solutions for flat cracks, Computational Mech. 2, 253–267 (1987)
J. R. Rice, Weight function theory for three-dimensional elastic crack analysis, in Fracture Mechanics: Perspectives and Directions (Twentieth Symposium) (ed. R. P. Wei and R. P. Gangloff), American Society for Testing and Materials, Philadelphia, 1989, pp. 29–57
K. Takakuda, T. Koizumi, and T. Shibuya, On integral equation methods for crack problems, Bull. Japan. Soc. Mech. Engrs. 28, 217–224 (1985)
J. Weaver, Three-dimensional crack analysis, Internat. J. Solids and Structures 13, 321–330 (1977)
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Article copyright:
© Copyright 1996
American Mathematical Society
