匀变速扩展的圆盘形断层的远场辐射理论(二)

THE FAR-FIELD DISPLACEMENTS RADIATED FROM A CIRCULAR DISLOCATION EXPANDING WITH UNIFORMLY VARYING VELOCITY:PART Ⅱ

本文第一部分已讨论在以下两条假设下的匀变速扩展的圆盘形断层的远场辐射理论: 1.破裂是从中心开始的;2.均匀位错分布(n=0)。 本部分将讨论更一般的情形,第一部分中的结果可以从本部分的普遍公式中作为特殊情形过渡得到。

In this part the general problem about the far-field displacements radiated from a circular crack expanding with non-uniform velocity is studied. A general, closed analytical solution for the problem is obtained by means of Jacobi elliptic functions and the first, second and third kinds of Legenotre 's incomplete ellipitic integrals. The differences between our work and those having been done by other authors are as follows.(1) We assume that the rupture velocity is of the general form:V(t) = V0 + at (a=constant), with an acceleration term.(2) The cracking starts from R1. i.e. we consider that there is an initial crack, thereforewhere ξ(t) is the instantaneous radius of the crack front.(3) The general form of source function is assumed:where D0-maximum central dislocation, R2-terminal radius of the circular fault, g(t)-source time function.The case in which the spatial distribution of the dislocation is uniform will be obtained as n=0. As n=2 we obtain the case in which the source spatial functions are just equal to the first order of approximation of the static solution of the problem.As an example, the far-field displacements radiated from a complete fracture process (starting-accelerating-propagating with constant velocity-decelerating-stopping) are given.