Nonuniform motion of an edge dislocation in an anisotropic solid. I

Markenscoff, Xanthippi; Ni, Lu Qun

doi:10.1090/qam/724058

Authors: Xanthippi Markenscoff and Lu Qun Ni
Journal: Quart. Appl. Math. 41 (1984), 475-494
MSC: Primary 73S05; Secondary 73D99
DOI: https://doi.org/10.1090/qam/724058
MathSciNet review: 724058
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The two-dimensional problem of the nonuniform motion of an edge dislocation in an anisotropic solid (regular hyperbolic case) is solved by means of Laplace transforms with inversion according to the Cagniard-de Hoop technique. The solution is also evaluated asymptotically at the saddle points on the Cagniard-de Hoop contour which lies on a multi-sheet Riemann surface, the singular points of which are examined in detail also in connection to the slowness surface. The stress field is square root singular near the wavefront for a motion of constant velocity starting from rest, and 2/3 singular near the cusp-tips. For general nonuniform motion the stress at the wavefront is obtained as well and an example is given for a motion starting with constant acceleration.

A. W. Sáenz, J. Rat. Mech. Anal. 2 (1953), 83

R. Bullough and B. A. Bilby, Uniformly moving dislocations in anisotropic media, Proc. Phys. Soc. Sect. B. 67 (1954), 615–624. MR 0063272

L. J. Teutonico, Phys. Rev. 124 (1961), 1039

J. D. Eshelby, The equation of motion of a dislocation, Phys. Rev. (2) 90 (1953), 248–255. MR 66922

J. Weertman and J. R. Weertman, Dislocations in solids Vol. 3 (edited by F. R. N. Nabarro) North Holland, 1980

X. Markenscoff and R. J. Clifton, J. Mech. Phys. Soc. 29 (1981), 253

X. Markenscoff and R. J. Clifton, J. Appl. Mech. ASME 49 (1982), 792

G. F. D. Duff, The Cauchy problem for elastic waves in an anistropic medium, Philos. Trans. Roy. Soc. London Ser. A 252 (1960), 249–273. MR 111293, DOI https://doi.org/10.1098/rsta.1960.0006

L. J. Teutonico, Mathematical theory of dislocations (edited by T. Mura), ASME, New York, 1969, 49

J. D. Eshelby, Phil. Mag. 40 (1949), 903

J. D. Eshelby, W. T. Read and W. Shockley, Acta Met 1 (1953), 251

J. R. Willis, Phil. Mag., Sec. 8, 21 (1970), 931

R. Burridge, The directions in which Rayleigh waves may be propagated on crystals, Quart. J. Mech. Appl. Math. 23 (1970), 217–224. MR 266479, DOI https://doi.org/10.1093/qjmam/23.2.217

A. T. de Hoop, Appl. Sci. Res., B8 (1960), 349

R. Burridge, Lamb’s problem for an anisotropic half-space, Quart. J. Mech. Appl. Math. 24 (1971), 81–98. MR 299059, DOI https://doi.org/10.1093/qjmam/24.1.81

E. A. Kraut, Reviews of Geophysics 1 (1963), 401

M. J. P. Musgrave, Crystal acoustics, Halden-Day Inc. 1970

Xanthippi Markenscoff, The transient motion of a nonuniformly moving dislocation, J. Elasticity 10 (1980), no. 2, 193–201 (English, with French summary). MR 576167, DOI https://doi.org/10.1007/BF00044503
Leon Knopoff and Freeman Gilbert, First motion methods in theoretical seismology, J. Acoust. Soc. Amer. 31 (1959), 1161–1168. MR 108370, DOI https://doi.org/10.1121/1.1907845

N. Bleinstein and R. A. Handelman, Asymptotic expansions of integrals, Holt, Reinhart & Winston, New York, 1975

W. L. Ferrar, Higher algebra, Oxford University Press, 1948

L. B. Freund, Quart. Appl. Math. 30 (1972), 271

David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
L. Sirovich, Techniques of asymptotic analysis, Applied Mathematical Sciences, vol. 2, Springer-Verlag, New York-Berlin, 1971. MR 0275034

X. Markenscoff, Int. J. Engng. Sci. 20 (1982), 289