Nadkarni Neel, Daraio Chiara, Kochmann Dennis M
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, California 91125, USA.
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, California 91125, USA and Department of Mechanical and Process Engineering, ETH Zurich, Zurich, Switzerland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Aug;90(2):023204. doi: 10.1103/PhysRevE.90.023204. Epub 2014 Aug 22.
We investigate the nonlinear dynamics of a periodic chain of bistable elements consisting of masses connected by elastic springs whose constraint arrangement gives rise to a large-deformation snap-through instability. We show that the resulting negative-stiffness effect produces three different regimes of (linear and nonlinear) wave propagation in the periodic medium, depending on the wave amplitude. At small amplitudes, linear elastic waves experience dispersion that is controllable by the geometry and by the level of precompression. At moderate to large amplitudes, solitary waves arise in the weakly and strongly nonlinear regime. For each case, we present closed-form analytical solutions and we confirm our theoretical findings by specific numerical examples. The precompression reveals a class of wave propagation for a partially positive and negative potential. The presented results highlight opportunities in the design of mechanical metamaterials based on negative-stiffness elements, which go beyond current concepts primarily based on linear elastic wave propagation. Our findings shed light on the rich effective dynamics achievable by nonlinear small-scale instabilities in solids and structures.
我们研究了由质量块通过弹性弹簧连接而成的双稳态元件周期性链的非线性动力学，其约束结构会引发大变形 snap-through 不稳定性。我们表明，由此产生的负刚度效应在周期性介质中产生了三种不同的（线性和非线性）波传播模式，这取决于波幅。在小振幅下，线性弹性波会经历色散，其可通过几何形状和预压缩水平来控制。在中等到大振幅下，孤波会出现在弱非线性和强非线性模式中。对于每种情况，我们都给出了封闭形式的解析解，并通过具体的数值示例证实了我们的理论发现。预压缩揭示了一类部分正负势的波传播。所呈现的结果突出了基于负刚度元件的机械超材料设计中的机会，这超越了目前主要基于线性弹性波传播的概念。我们的发现揭示了固体和结构中非线性小尺度不稳定性可实现的丰富有效动力学。
