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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)=V_0+at (a=常数)其中V_0是初始破裂速度,V_0=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R_1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)=R_1+V_0t+1/2at~2.R_1=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)=D_0[1-(ζ/R_2)~n]g(t). (n=0,1,2,……)其中,D_0是圆盘中心最终错距,R_2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R_1=0,n=0的情形,其他内容将在第二部分中讨论。
In this paper, we study the far-field displacements of disc-shaped faults extended by uniform velocity expansion. By using special functions such as the first, the second and the third nonholonomic integrals of the Jacobi elliptic function and the Legendre paradigm, a general closed form analytical solution of the problem is given. The issues discussed in this article are the general case. Compared with the previous work has the following differences: 1 set the rupture speed V (t) V (t) = V_0 + at (a = constant) where V_0 is the initial rupture speed, V_0 = 0 that the initial velocity is zero A is the crack acceleration; a> 0, a = 0 and a <0 correspond to the special case of accelerating crack, uniform crack and decelerating crack respectively. 2. Rupture starts from radius R_1. That is, there can be initial cracks. So that the instantaneous radius of expansion ζ (t) is ζ (t) = R_1 + V_0t + ½at ~ 2.R_1 = 0, corresponding to the case of expansion starting from the center. 3. The hypocenter hypothesis has the following form: S (ζ, t) = D_0 [1- (ζ / R_2) ~ n] g (t) (n = 0,1,2, ...) where D_0 is Final center of disk discordance, R 2 is the final rupture radius, g (t) is the source time function. When n = 0, the source space function is obtained as a uniform distribution. When n = 2, the first order approximation of the static solution to the crack problem is obtained. Finally, as an example, the far-field displacement equation caused by the entire rupture process (initial-acceleration-uniform-deceleration-stop) is given. The first part of this article only discusses the case of R_1 = 0, n = 0, the rest will be discussed in the second part.