非线性阻尼作用时Ⅲ型断裂动力学的解析解

ANALYTICAL SOLUTION FOR MODE Ⅲ DYNAMIC RUPTURE SUBJECTED TO NONLINEAR DAMPING

本文用非线性Ｒａｙｌｅｉｇｈ阻尼描述破裂初期有激发使破裂加速至最大破裂速度后变为衰减至运动停止的Ⅲ型断裂动力学，得出问题的解析解，采用Ｇａｌｉｌｅｏ变换把固定坐标变为动坐标，用动坐标的Ｆｏｕｒｉｅｒ级数把控制方程的非线性偏微分方程约化为非线性常微分方程组，再用逐次逼近法。

In this paper, the nonlinear damping which depicts the exciting process on the initial stage is adopted. After the rupture velocity reaches its maxium value, decaying process succeeds.We obtain the analytical solution for the mode Ⅲ dynamic rupture subjected to the nonlinear Rayleigh damping. First, the Galileo transformation is adopted to transform the fixed coordinates into moving coordinates.Then the unknown function is assumed to be Fourier series in moving coordinates with coefficient as functions of time variable only. Applying the orthognality of Fourier series, the nonlinear PDE of the governing equation is reduced into two infinite systems of nonlinear ODEs. By successive approximation method, the analytical solution is obtained.