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An integral equation approach to self-similar plane-elastodynamic crack problems. (English) Zbl 0731.73064

Summary: The elastodynamic problem of an expanding crack under homogeneous polynomial-form loading was reduced to the solution of a Cauchy singular integral equation. In this manner the solution of the original problem can be obtained by using well-known numerical treatments available for Cauchy SIEs. The procedure was accomplished by means of the Busemann- Chaplygin similarity technique and complex variable methods. The analysis has been restricted to the subsonic case.

MSC:

74R99 Fracture and damage
74S30 Other numerical methods in solid mechanics (MSC2010)
45E05 Integral equations with kernels of Cauchy type
74B99 Elastic materials
74H99 Dynamical problems in solid mechanics
65R20 Numerical methods for integral equations
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[1] V.I. Smirnov,A Course of Higher Mathematics. Vol. 3, New York: Pergamon Press (1964). · Zbl 0121.25904
[2] V.Z. Parton and P.I. Perlin,Mathematical Methods of the Theory of Elasticity. Moscow: Mir (1984). · Zbl 0626.73001
[3] B.V. Kostrov, The axisymmetric problem of propagation of a tension crack.PMM 28 (1964) 793-803. · Zbl 0139.20002
[4] B.V. Kostrov, Selfsimilar problems of propagation of shear cracks.PMM 28 (1964) 1077-1087. · Zbl 0139.20003
[5] E.F. Afanas’ev and G.P. Cherepanov, Some dynamic problems of the theory of elasticity.PMM 37 (1973) 584-606. · Zbl 0298.73033
[6] A.R. Robinson and J.C. Thompson, Transient stresses in an elastic half space resulting from the frictionless indentation of a rigid wedge-shaped die.ZAMM 54 (1974) 139-144. · Zbl 0286.73026
[7] A.R. Robinson and J.C. Thompson, Transient disturbances in a half-space during the first stage of frictionless indentation of a smooth rigid die of arbitrary shape.Q. Appl. Math. 33 (1975) 215-223. · Zbl 0334.73007
[8] F.R. Norwood, Similarity solutions in plane elastodynamics.Int. J. Solids Structures 9 (1973) 789-803. · Zbl 0269.73024 · doi:10.1016/0020-7683(73)90003-6
[9] F.R. Norwood, Correction and extension of Broberg’s results on brittle crack propagation.Int. J. Engng. Sci. 14 (1976) 477-488. · Zbl 0324.73066 · doi:10.1016/0020-7225(76)90039-2
[10] J.R. Willis, Self-similar problems in elastodynamics.Phil. Trans. Royal Soc. London 274 (1973) 435-471. · Zbl 0317.73062
[11] G.N. Ward,Linearized Theory of Steady High-Speed Flow, London: Cambridge University Press (1955). · Zbl 0064.43703
[12] J.D. Achenbach,Wave Propagation in Elastic Solids. Amsterdam: North-Holland (1973). · Zbl 0268.73005
[13] A.C. Eringen and E.S. Suhubi,Elastodynamics. New York: Academic Press (1975).
[14] J.W. Miles, Homogeneous solutions in elastic wave propagation.Q. Appl. Math. 18 (1960) 37-59. · Zbl 0094.21103
[15] J.W. Craggs, On two-dimensional waves in an elastic half-space.Proc. Camb. Soc. 56 (1960) 269-285. · Zbl 0094.38601 · doi:10.1017/S0305004100034551
[16] J.D. Achenbach, Bifurcation of a running crack in antiplane strain.Int. J. Sol. Structures 11 (1975) 1301-1314. · Zbl 0313.73083 · doi:10.1016/0020-7683(75)90059-1
[17] J.D. Achenbach, Wave propagation, elastodynamic stress singularities, and fracture, in:Theoretical and Applied Mechanics, W.T. Koiter (ed.), Amsterdam: North-Holland (1976). · Zbl 0355.73075
[18] L.M. Brock, Non-symmetric extension of a small flaw into a plane crack at a constant rate under polynomial-form loading.Int. J. Engng. Sci. 14 (1976) 181-190. · Zbl 0356.73075 · doi:10.1016/0020-7225(76)90087-2
[19] L.M. Brock, Two basic problems of plane crack extension?A unified treatment.Int. J. Engng. Sci. 15 (1977) 527-536. · Zbl 0364.73082 · doi:10.1016/0020-7225(77)90049-0
[20] L.M. Brock, Symmetrical frictionless indentation over a uniformly expanding contact region?I. Basic analysis.Int. J. Engng. Sci. 14 (1976) 191-199. · Zbl 0324.73017 · doi:10.1016/0020-7225(76)90088-4
[21] L.M. Brock, Dynamic rigid indentation induced by sliding frictionless contact.Int. J. Engng. Sci. 16 (1978) 545-550. · Zbl 0381.73100 · doi:10.1016/0020-7225(78)90018-6
[22] L.M. Brock, Symmetrical indentation over a uniformly expanding contact region?II. Perfect adhesion.Int. J. Engng. Sci. 15 (1977) 147-155. · Zbl 0354.73015 · doi:10.1016/0020-7225(77)90030-1
[23] L.M. Brock, Frictionless indentation by a rigid wedge: The effect of tangential displacements in the contact zone.Int. J. Engng. Sci. 17 (1979) 365-372. · Zbl 0393.73119 · doi:10.1016/0020-7225(79)90072-7
[24] N.I. Muskhelishvili,Singular Integral Equations. Groningen: Noordhoff (1953). · Zbl 0051.33203
[25] F.D. Gakhov,Boundary Value Problems, New York: Pergamon Press (1966). · Zbl 0141.08001
[26] E.C. Titchmarsh,Introduction to the Theory of Fourier Integrals, New York: Oxford (1937). · Zbl 0017.40404
[27] L.M. Jones,An Introduction to Mathematical Methods of Physics, Menlo Park: Benjamin/Cummings (1979).
[28] R.V. Churchill, J.W. Brown and R.F. Verhey,Complex Variables and Applications. New York: McGraw Hill (1974). · Zbl 0299.30003
[29] Y.C. Fung,Foundations of Solid Mechanics, Englewood Cliffs: Prentice Hall (1965).
[30] F. Erdogan, Approximate solutions of systems of singular integral equations.SIAM J. Appl. Math. 17 (1969) 1041-1059. · Zbl 0187.12404 · doi:10.1137/0117094
[31] H. Sekine, Crack problem for a semi-infinite solid with heated bounding surface.J. Appl. Mech. 44 (1977) 637-642. · Zbl 0369.73091 · doi:10.1115/1.3424149
[32] L. A. Galin and A.A. Shmatkova, Motion of a rigid stamp on the boundary of a viscoelastic half-plane.PMM 32 (1968) 446-455. · Zbl 0175.23406
[33] L.N. Karpenko, Approximate solution of a singular integral equation by means of Jacobi polynomials.PMM 30 (1966) 668-675. · Zbl 0308.45013
[34] F. Erdogan and G.D. Gupta, On the numerical solution of singular integral equations.Q. Appl. Math. 30 (1972) 525-534. · Zbl 0236.65083
[35] N.I. Ioakimidis,General Methods for the Solution of Crack Problems in the Theory of Plane Elasticity. Doctoral Thesis at the National Technical University of Athens (Univ. Micr. order No. 76-21, 056) Athens (1976). · Zbl 0351.73109
[36] S. Krenk, On the use of the interpolation polynomial for solutions of singular integral equations.Q. Appl. Math. 32 (1975) 479-484. · Zbl 0322.45023
[37] S. Krenk,Polynomial Solutions to Singular Integral Equations, with Applications to Elasticity Theory. Thesis at the Technical University of Denmark, Lyngby (1981).
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