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微结构固体中的孤立波及其存在条件
Solitary waves and their existence conditions in microstructured solids
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摘要: 根据Mindlin理论，考虑宏观尺度非线性效应、二次和三次微尺度非线性效应以及微尺度频散效应，建立了描述一维微结构固体中纵波传播的一种新模型。用动力系统定性分析理论，分析了微结构固体中孤立波的存在条件及其几何特征，证明了当介质参数和孤立波传播速度满足适当条件时，在二次微尺度非线性效应的影响下微结构固体中可以形成一种非对称孤立波，在三次微尺度非线性效应的影响下微结构固体中可以形成一种对称孤立波。最后，用数值方法进一步验证了上述结论。Abstract: By taking into account the macroscale nonlinear effect, quadratic and cubic microscale nonlinear effects, and microscale dispersion effect, a new model capable to describe the longitudinal wave propagation in one-dimensional microstructured solid is established based on the Mindlin theory. Using the qualitative analysis theory in the planar dynamical systems, we have analyzed the existence conditions and geometrical characteristics of solitary waves, and proved that the faces under the influence of quadratic microscale nonlinear effect, can form an asymmetric solitary wave in the microstructured solid; and under the influence of cubic microscale nonlinear effect, they can form a symmetric solitary wave in the microstructured solid, when the medium parameters and the propagation speeds of solitary waves satisfy certain appropriate conditions. Finally, the above results are further verified using a numerical method.
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