用非线性梯度弹性模型预测局部平稳地震波
Localized stationary seismic waves predicted using a nonlinear gradient elasticity model
摘要：
本文旨在研究局部化驻波在浅层地下的本构行为是由双曲线模型，这意味着非多项式非线性和应变相关的剪切模量的存在。为此，我们推导出一个新的运动方程的非线性梯度弹性模型，其中高阶梯度项捕获的小尺度土壤异质性/微观结构的影响。我们还提出了一种新的有限差分格式来解决空间和时间的非线性运动方程。任意初始脉冲的传播的模拟清楚地揭示了非线性的影响：应变相关的速度一般，因此，锐化的脉冲。通过引入运动参考系和平稳性假设，得到了运动方程的平稳解。通过分析相图中获得的常微分方程，并沿不同的轨迹积分，发现周期性（有和没有下降趋势）以及局部驻波。定域驻波实际上是一种扭折波，可以通过沿同宿轨道积分得到。一般来说，轨道越接近同宿轨道，相应的周期性驻波的边缘越尖锐，其周期越大。最后，我们发现扭结波实际上不是一个真正的孤子，因为两个碰撞的扭结波在相互作用后不能恢复原来的形状。然而，它可能具有高振幅并到达表面，这取决于阻尼机制（尚未考虑）。因此，地震现场响应分析不应事先排除这种局部驻波的存在。
英文摘要：
This paper aims at investigating the existence of localized stationary waves in the shallow subsurface whose constitutive behaviour is governed by the hyperbolic model, implying non-polynomial nonlinearity and strain-dependent shear modulus. To this end, we derive a novel equation of motion for a nonlinear gradient elasticity model, where the higher-order gradient terms capture the effect of small-scale soil heterogeneity/micro-structure. We also present a novel finite-difference scheme to solve the nonlinear equation of motion in space and time. Simulations of the propagation of arbitrary initial pulses clearly reveal the influence of the nonlinearity: strain-dependent speed in general and, as a result, sharpening of the pulses. Stationary solutions of the equation of motion are obtained by introducing the moving reference frame together with the stationarity assumption. Periodic (with and without a descending trend), as well as localized stationary waves, are found by analyzing the obtained ordinary differential equation in the phase portrait, and integrating it along the different trajectories. The localized stationary wave is in fact a kink wave and is obtained by integration along a homoclinic orbit. In general, the closer the trajectory lies to a homoclinic orbit, the sharper the edges of the corresponding periodic stationary wave and the larger its period. Finally, we find that the kink wave is in fact not a true soliton as the original shapes of two colliding kink waves are not recovered after interaction. However, it may have high amplitude and reach the surface depending on the damping mechanisms (which have not been considered). Therefore, seismic site response analyses should not a priori exclude the presence of such localized stationary waves.
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