弹性纵波垂直通过非线性结构面的傅里叶级数分析法

利用傅里叶级数计算任意函数形式的弹性纵波垂直通过非线性结构面的透射波和反射波的速度波形。采用双曲模型描述结构面的变形特性，基于位移不连续模型，结合波振面处动量守恒定律，推导了弹性纵波垂直通过非线性结构面的基本方程。假设应力波在含结构面岩体中传播时，结构面的存在不改变应力波波形函数的最小正周期，运用傅里叶级数理论和周期延拓方法，得到了任意函数形式的弹性纵波垂直入射时透射波和反射波速度波形的傅里叶级数解，并验证了傅里叶级数解是合理的。利用傅里叶级数解，分析了单一频率的正弦谐波入射至结构面时，透射波中各阶谐波的振幅和相位与谐波阶数的关系。研究结果表明，各阶谐波的振幅与阶数呈负指数关系衰减，前7阶谐波振幅的衰减指数为谐波阶数的二次函数，当谐波阶数大于7时，衰减指数为谐波阶数的一次函数；各阶谐波的相位与谐波阶数呈线性关系。

Fourier series analysis of a elastic longitudinal wave vertically propagating through a nonlinear joint

To calculate the transmitted and reflected velocity waveform of elastic longitudinal wave with arbitrary function form propagating through nonlinear joint by Fourier series theory. Based on the discontinuous displacement model and the momentum balance equation at the wave front, the governing equation for the propagation of an elastic longitudinal wave through the nonlinear joint is developed, in which the joint deformation is described by the Barton-Bandis model. With assuming that the minimal positive period of stress waveform function remains unchanged as stress wave propagates across the nonlinear joint, the arbitrary Fourier series solutions of transmitted and reflected waves are obtained using the Fourier series method and the periodic extension method, and the Fourier series solutions are validated. Based on Fourier series solutions, the dependence of the amplitude and phase on the order number of the harmonic waves generated by the transmitted wave at the joint is analyzed. It is shown that the relationship between the amplitude and the order number of the harmonic waves follows a negative exponential law. The attenuation index of the amplitude is a quadratic function of the order number when the order number is less than 7, whereas the attenuation index of the amplitude is a linear function of the order number when the order number is less than 7. A linear relationship exists between the phase and the order number for harmonic waves of different orders.