匀变速扩展的圆盘形断层的远场辐射理论（一）

本文研究了由匀变速扩展的圆盘形断层所辐射的远场位移。通过Jacobi椭圆函数和Legendre范式的第一、第二、第三种非完全椭圆积分等特殊函数,给出了该问题的普遍形式的闭合解析解。 本文所讨论的问题是普遍情形。与已往工作比较具有以下不同之处: 1.设破裂速度为V（t） V（t）=V₀+at （a=常数）其中V₀是初始破裂速度,V₀=0即初速度为零的特殊情形;a是破裂加速度;a>0、a=0及a<0分别对应于加速破裂、匀速破裂及减速破裂的特殊情形。 2.破裂是从半径R₁开始的。即可以有初始裂纹存在。从而扩展的瞬时半径ζ（t）为 ζ（t）=R₁+V₀t+1/2at².R₁=0,相应于从中心开始扩展的情形。 3.震源函数假设具有下述形式: S（ζ,t）=D₀[1-（ζ/R₂）ⁿ]g（t）. （n=0,1,2,……）其中,D₀是圆盘中心最终错距,R₂是最终破裂半径,g（t）是震源时间函数。n=0时得到震源空间函数为均匀分布情形。n=2时得到该裂纹问题静态解的一级近似的情形。 最后,作为例子,给出了整个破裂过程（起始-加速-匀速-减速-停止）所引起的远场位移公式。 本文第一部分只讨论R₁=0,n=0的情形,其他内容将在第二部分中讨论。

FAR-FIELD THEORY RADIATED FROM A CIRCULAR DISLOCATION EXPANDING WITH UNIFORMLY VARYING VELOCITY Part （Ⅰ）

The far-field displacements radiated from a circular dislocation expanding with uniformly varying velocity are studied in this paper. A closed analytical solution to the general form of this problem is obtained. The problems discussed in this paper are general and differ from previous works as follows:1. Assume that the rupture velocity V（t） is given by V（t）=V₀+αt （α=constant） where V₀ is the initial rupture velocity, and a is the rupture acceleration.2. Cracking starts from an initial radius of R¹ which means that a preexisting crack is allowed. The instantaneous radius ξ（t） is ξ（t） = R¹+V₀t+1/2αt². R₁= 0 corresponds to the case when cracking initiates from the centre.3. The source function is assumed to take the formwhere D₀ is the final dislocation at the centre, R² is the final fracture radius, and g（t） is the source time function, n = 0 corresponds to a source function that is uniform spatially, whereas n = 2 corresponds to the first order approximation of the static crack problem.Lastly, an example is given for the expression of the far-field displacements due to a complete fracture process （initiation-acceleration-constant velocity-deceleration-arrest）.This paper is divided into two parts. In the first part, only the case R₁=0 and n=0 is discussed. All other cases are included in the second part.