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On boundary value problems in fracture of elastic Composites

Published online by Cambridge University Press:  26 September 2008

Gennady S. Mishuris  and
Zbigniew S. Olesiak
Gennady S. Mishuris
Affiliation:
Department of Applied Mathematics and Mechanics, University of Warsaw, Poland
Zbigniew S. Olesiak
Affiliation:
Department of Applied Mathematics and Mechanics, University of Warsaw, Poland
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Abstract

We present a method of solution of a class of fracture problems in the theory of elasticity. The method can be applied to any problem reducible to Poisson's equation, e.g. heat conduction and mass diffusion in solids, theory of consolidation and the like. The novelty of the paper is that we address regions of layered composites with notches, or, in a particular case, with a crack. Within the framework of classical analysis, we apply Fourier and Mellin transforms, 'fit' them together, and reduce the problem to solving a singular integral equation with fixed singularities on a semi-axis. We show the existence and uniqueness of solutions of the equations under consideration, and justify the asymptotics necessary for applications. We show the practical usefulness of the method on the examples of an antiplane problem of fracture mechanics. From our solution, we are able to find the stress intensity factor in the case when a crack tip penetrates a layered composite consisting of 60 layers, and show the limits of applicability of the anisotropic model of such composites.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 6 , December 1995 , pp. 591 - 610
Copyright
Copyright © Cambridge University Press 1995

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References

[1] Banerjee, P. K. & Butterfield, R. 1977 Boundary Element Method in Geomechcmics. Wiley.Google Scholar
[2] Kolchin, G. V. & Faverman, A. A. 1972 Theory of Elasticity of Non-homogeneous Bodies. Sticenci, Kishinev (in Russian).Google Scholar
[3] Sneddon, I. N. 1977 The Use of Integral Transforms. McGraw-Hill.Google Scholar
[4] Uflyand, Ya. S. 1967 Integral transforms in problems of the theory of elasticity. Translated from Russian edition: Izd. Ak. Nauk SSSR, Moscow-Leningrad.Google Scholar
[5] Linkov, A. & Filippov, N. 1991 Difference equations approach to the analysis of layered systems. Mecchanica 26, 195–209.CrossRefGoogle Scholar
[6] Hilton, P. D. & Sih, G. S. 1971 Laminate composite with a crack normal to the interface. Int. J. Solids Struct. 7, 913–930.CrossRefGoogle Scholar
[7] Srivastava, S. N. & Dhavan, G. K. 1977 Stress distribution due to a Griffith crack at the interface of an elastic layer bonded to a half plane. Indian J. Pure Appl. Math. 8, 1354–1369.Google Scholar
[8] Duduchava, R. V. 1979 Integral equations of convolution with discontinuous symbols, singular integral equations with fixed singularities and their applications to problems of mechanics. Tr. Tbilis. Mat. Inst, A. N. Gruz. SR, 10 (in Russian).Google Scholar
[9] Soldatov, A. P. 1974 Solution of a boundary value problem of the theory of functions with shift. Diff. Uravn., X, 143–152 (in Russian).Google Scholar
[10] Tranter, C. J. 1951 Integral Transforms in Mathematical Physics. Methuen.Google Scholar
[11] Mishuris, G. S. & Paukshto, M. V. 1984 On an antiplane displacement of a wedge leaned on a layered medium. In: Dynamics and Stability of the Mechanical Systems, Izdat. Len. University (in Russian).Google Scholar
[12] Mishuris, G. S. 1985 Applications of the integral transforms in problems of elastic solids with notches. PhD thesis, stored at V1NITI, Nr 0485004348, 174 pp. (in Russian).Google Scholar
[13] Mishuris, G. S. 1984 On a class matrix integral equations. VINITI, Nr 637384 (in Russian).Google Scholar
[14] Nowacki, W. & Olesiak, Z. S. 1991 Thermodiffusion in solid bodies. Polish Scientific Publishers (PWN), Warsaw (in Polish).Google Scholar
[15] Prudnikov, A. P., Brychkov, Yu. A. & Marichev, O. U. 1981 Integrals and Series. Elementary Functions, Tables. Nauka, Moscow (in Russian).Google Scholar
[16] Nazarov, S. A. 1983 Introduction to asymptotic methods of the theory of elasticity. Leningrad, Izdat. Len. University (in Russian).Google Scholar
[17] Khoroshun, L. P. 1966 Stress-strain relationship in layered continua. Prikl. Mekh. 2, 14–19 (in Russian).Google Scholar
[18] Gokhberg, I. U. & Feldman, N. A. 1971 Equations in Convolutions and Projectional Methods of their Solutions. Moscow, Nauka (in Russian).Google Scholar
[19] Gokhberg, I. U. & Krein, M. G. 1958 Systems of Integral Equations on Semi-axis with Kernels Dependent on the Difference of the Arguments. Usp. Mat. Nauk. XIII, 3–72 (in Russian).Google Scholar
[20] Ma, C. C. & Hour, B. L. 1990 Antiplane problems in composite materials with an inclined crack terminating at bimaterial interface. Int. J. Solids Struct. 26, 1387–1400.Google Scholar
[21] Schovanec, L. 1988 A mode-Ill crack problem in a bonded composite material. Eng. Tract. Mech. 31, 437–449.CrossRefGoogle Scholar