On the EshelbyKostrov property for the wave equation in the plane
Authors:
M. A. Herrero, G. E. Oleaga and J. J. L. Velázquez
Journal:
Trans. Amer. Math. Soc. 358 (2006), 36733695
MSC (2000):
Primary 74R05, 74B05, 74G70
Published electronically:
March 28, 2006
MathSciNet review:
2218994
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: This work deals with the linear wave equation considered in the whole plane except for a rectilinear moving slit, represented by a curve with Along either homogeneous Dirichlet or Neumann boundary conditions are imposed. We discuss existence and uniqueness for these problems, and derive explicit representation formulae for solutions. The latter have a simple geometrical interpretation, and in particular allow us to derive precise asymptotic expansions for solutions near the tip of the curve. In the Neumann case, we thus recover a classical result in fracture dynamics, namely the form of the stress intensity factor in crack propagation under antiplane shear conditions.
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 J.D.Eshelby: The elastic field of a crack extending nonuniformly under general antiplane loading. Journal of the Mechanics and Physics of Solids 17 (1969), 177199.
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 L.B. Freund: Dynamic Fracture Mechanics. Cambridge University Press (1998). MR 1054375 (92f:73044)
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 A. Friedman, B. Hu and J.J.L. Velázquez: The evolution of stress intensity factors and the propagation of cracks in elastic media. Arch. Rat. Mech. Anal. 152 (2000), 103139. MR 1760415 (2001h:74026)
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 A. Friedman, B. Hu and J.J.L. Velázquez: The evolution of stress intensity factors in the propagation of two dimensional cracks. Euro. Jnl. of Appl. Math. 11 (2000), 453471. MR 1799921 (2001h:74027)
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 J. Herrmann and J.R. Walton: A new method for solving dynamically accelerating crack problems. Part I: The case of a semiinfinite mode III crack in elastic material revisited, Quart. Appl. Math. 50, No. 2 (1992), 373387. MR 1162281 (93c:73086)
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 M.A.Herrero, G.E.Oleaga and J.J. Velázquez: A note on planar cracks running along piecewise linear paths. Proceedings of the Royal Society of London A 460 (2004), 581601. MR 2034657 (2004j:74116)
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 F. John: Partial Differential Equations. SpringerVerlag, New York (1978). MR 0514404 (80f:35001)
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 B.V.Kostrov: On the crack propagation with variable velocity. International Journal of Fracture 11 (1975), 4756.
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 J.B. Leblond: Crack paths in plane situations  I. General forms of the expansion of the stress intensity factors. Int. J. Solid Structures 25 (1981), 13111325. MR 1138834
 12.
 T.L. Leise and J.R. Walton, A general method for solving dynamically accelerating multiple colinear cracks, Int. J. Frac. 111 (2001), 116.
 13.
 B. Noble: Methods based on the WienerHopf Technique. Pergamon, New York (1958).
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 V. Saraikin and L. Slepyan: Plane problem of the dynamics of a crack in an elastic solid, Mechanics of Solids 14 (1979), 4662. MR 0572877 (81c:73052)
 15.
 J.R. Willis, Accelerating cracks and related problems, in Elasticity: Mathematical Methods and Applications, Ed. G. Eason and R. W. Ogden, Ellis Horwood, Chichester (1990), 397409.
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Additional Information
M. A. Herrero
Affiliation:
Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, Madrid 28040, Spain
G. E. Oleaga
Affiliation:
Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, Madrid 28040, Spain
J. J. L. Velázquez
Affiliation:
Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, Madrid 28040, Spain
DOI:
http://dx.doi.org/10.1090/S000299470603995X
PII:
S 00029947(06)03995X
Received by editor(s):
September 15, 2003
Received by editor(s) in revised form:
September 12, 2004
Published electronically:
March 28, 2006
Article copyright:
© Copyright 2006
American Mathematical Society
