On the Eshelby-Kostrov property for the wave equation in the plane
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- by M. A. Herrero, G. E. Oleaga and J. J. L. Velázquez PDF
- Trans. Amer. Math. Soc. 358 (2006), 3673-3695 Request permission
Abstract:
This work deals with the linear wave equation considered in the whole plane $\mathbb {R}^{2}$ except for a rectilinear moving slit, represented by a curve $\Gamma \left ( t\right ) =\left \{ \left ( x_{1},0\right ) :-\infty <x_{1}<\lambda \left ( t\right ) \right \}$ with $t\geq 0.$ Along $\Gamma \left ( t\right ) ,$ 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.References
- M. Amestoy and J.-B. Leblond, Crack paths in plane situations. II. Detailed form of the expansion of the stress intensity factors, Internat. J. Solids Structures 29 (1992), no. 4, 465–501. MR 1138337, DOI 10.1016/0020-7683(92)90210-K
- 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), 177-199.
- Gilles A. Francfort and Christopher J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math. 56 (2003), no. 10, 1465–1500. MR 1988896, DOI 10.1002/cpa.3039
- L. B. Freund, Dynamic fracture mechanics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, Cambridge, 1990. MR 1054375, DOI 10.1017/CBO9780511546761
- Avner Friedman, Bei Hu, and Juan J. L. Velazquez, The evolution of stress intensity factors and the propagation of cracks in elastic media, Arch. Ration. Mech. Anal. 152 (2000), no. 2, 103–139. MR 1760415, DOI 10.1007/s002050000072
- Avner Friedman, Bei Hu, and Juan J. L. Velazquez, The evolution of stress intensity factors in the propagation of two dimensional cracks, European J. Appl. Math. 11 (2000), no. 5, 453–471. MR 1799921, DOI 10.1017/S0956792500004265
- J. R. Walton and J. M. Herrmann, A new method for solving dynamically accelerating crack problems. I. The case of a semi-infinite mode $\textrm {III}$ crack in elastic material revisited, Quart. Appl. Math. 50 (1992), no. 2, 373–387. MR 1162281, DOI 10.1090/qam/1162281
- M. A. Herrero, G. E. Oleaga, and J. J. L. Velázquez, Planar cracks running along piecewise linear paths, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), no. 2042, 581–601. MR 2034657, DOI 10.1098/rspa.2003.1173
- Fritz John, Partial differential equations, 3rd ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York-Berlin, 1978. MR 514404, DOI 10.1007/978-1-4684-0059-5
- B. V. Kostrov: On the crack propagation with variable velocity. International Journal of Fracture 11 (1975), 47-56.
- J.-B. Leblond, Crack paths in plane situations. I. General form of the expansion of the stress intensity factors, Internat. J. Solids Structures 25 (1989), no. 11, 1311–1325. MR 1138834, DOI 10.1016/0020-7683(89)90094-2
- T.L. Leise and J.R. Walton, A general method for solving dynamically accelerating multiple co-linear cracks, Int. J. Frac. 111 (2001), 1-16.
- B. Noble: Methods based on the Wiener-Hopf Technique. Pergamon, New York (1958).
- V. A. Saraĭkin and L. I. Slepyan, Plane problem of the dynamics of a crack in an elastic solid; Russian transl., Mech. Solids 14 (1979), no. 4, 54–73. MR 572877
- 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), 397-409.
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
- MR Author ID: 289301
- Received by editor(s): September 15, 2003
- Received by editor(s) in revised form: September 12, 2004
- Published electronically: March 28, 2006
- © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 3673-3695
- MSC (2000): Primary 74R05, 74B05, 74G70
- DOI: https://doi.org/10.1090/S0002-9947-06-03995-X
- MathSciNet review: 2218994