Effects of curvature on the propagation of undulatory waves in lower dimensional elastic materials
Effects of curvature on the propagation of undulatory waves in lower dimensional elastic materials
复制标题
DOI：
10.1103/physreve.103.013002
复制
发表时间：
2021-01-11
影响因子：
2.4
通讯作者：
Levine, Alex J.
中科院分区：
文献类型：
--
作者：
Kernes, Jonathan;Levine, Alex J.
低维弹性结构的力学很大程度上取决于其无应力状态的几何形状。弹性变形分为面内拉伸和低能面外弯曲变形。对于具有弯曲无应力状态的弹性结构，这两种弹性模式在线性弹性内耦合。我们研究了曲率引起的耦合对低维弹性结构中波传播的影响，重点关注最简单的例子——二维弯曲弹性杆。我们仅关注弯曲和纵向（面内）应变之间的几何引起的耦合，这对于二维杆和弹性壳来说是常见的。我们发现，在有限曲率存在的情况下，波的色散关系变得有间隙；低于与杆的曲率成比例的频率时不存在弯曲模式。通过研究波状波离开均匀曲率区域的散射，我们发现具有与弯曲区域相关的间隙中的频率的波状波通过转换成压缩波穿过该弯曲区域。这些结果应该直接适用于许多弯曲棒状弹性固体（包括碳纳米管和生物聚合物丝）中声子模式的频谱和空间分布。
The mechanics of lower dimensional elastic structures depends strongly on the geometry of their stress-free state. Elastic deformations separate into in-plane stretching and lower energy out-of-plane bending deformations. For elastic structures with a curved stress-free state, these two elastic modes are coupled within linear elasticity. We investigate the effect of that curvature-induced coupling on wave propagation in lower dimensional elastic structures, focusing on the simplest example-a curved elastic rod in two dimensions. We focus only on the geometry-induced coupling between bending and longitudinal (in-plane) strain that is common to both rods in two dimensions and to elastic shells. We find that the dispersion relation of the waves becomes gapped in the presence of finite curvature; bending modes are absent below a frequency proportional to the curvature of the rod. By studying the scattering of undulatory waves off regions of uniform curvature, we find that undulatory waves with frequencies in the gap associated with the curved region tunnel through that curved region via conversion into compression waves. These results should be directly applicable to the spectrum and spatial distribution of phonon modes in a number of curved rod-like elastic solids, including carbon nanotubes and biopolymer filaments.