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

Abstract

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 singular asymptotic expansion for the field near a moving dislocation loop

“…The stress fields radiated from a dislocation moving in an elastic solid have been studied since 1949 by Frank [7] and Eshelby [5] and subsequently by other investigators who studied steady state and accelerating motions of screw and edge dislocations in isotropic and anisotropic media. Some selected references include Eshelby [6], Kuisalaas and Mura [9,10], Markenscoff and Clifton [13], and Markenscoff and Ni [15,16]. The motion of dislocation loops has also been studied.…”

The Moving Plane Inhomogeneity Boundary with Transformation Strain

“…The radiated fields obtained by Markenscoff and Ni [4], as given by (25) to (27), can be used in order to obtain the radiated fields from the inhomogeneous (of different elastic constants C * ijkl ) moving boundary. Eshelby's [3] equivalent inclusion method will be extended here to dynamics for a moving plane boundary of a half-space expanding inhomogeneity with transformation strain.…”

“…In the case of the dynamically expanding plane boundary of an inhomogeneity with eigenstrain under applied loading, the integrations over the expanding inclusion domain can be performed for an isotropic material, and one can use the results of Markenscoff and Ni [4] of (25) to (27). The point-wise equivalent stresses are given by the same equation…”

“…and is the time when a wavelet emitted by the moving front l(τ ) has the time to reach the field point (x, t), and to contribute to the strain at that point, now related to the equivalent eigenstrain by the solution for the strain of the radiated fields of the moving inclusion plane boundary (with general eigenstrain), obtained above through (25), (26), (27), as a function of the eigenstrain and the boundary velocity. With the total equivalent eigenstrain being determined from (37), the radiated fields for the moving boundary of an inhomogeneity with eigenstrain can be evaluated from (1):…”

“…1), then the opposite loading may be applied to cancel the tractions, but these fields will not affect the fields in the bulk (see also, Dundurs and Markenscoff [24]), so that the driving force in the bulk will remain the same. It may be also noted that, while the elastodynamic solution for the moving plane inclusion boundary has been calculated analytically for an isotropic material, the solution for some classes of anisotropic materials may possibly be obtained partially analytically, as in the analysis of a generally moving dislocation in a cubic/hexagonal crystal by Markenscoff and Ni [27,28], where the solution was obtained by Laplace transforms and the analysis of the Riemann surfaces corresponding to the four roots of the determinantal equation. Thus, the applicability to physical phenomena could be expanded.…”