Hussein M I, Khajehtourian R
Ann and H.J. Smead Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO 80309-0429, USA.
Proc Math Phys Eng Sci. 2018 Sep;474(2217):20180173. doi: 10.1098/rspa.2018.0173. Epub 2018 Sep 5.
The introduction of nonlinearity alters the dispersion of elastic waves in solid media. In this paper, we present an analytical formulation for the treatment of finite-strain Bloch waves in one-dimensional phononic crystals consisting of layers with alternating material properties. Considering longitudinal waves and ignoring lateral effects, the exact nonlinear dispersion relation in each homogeneous layer is first obtained and subsequently used within the transfer matrix method to derive an approximate nonlinear dispersion relation for the overall periodic medium. The result is an amplitude-dependent elastic band structure that upon verification by numerical simulations is accurate for up to an amplitude-to-unit-cell length ratio of one-eighth. The derived dispersion relation allows us to interpret the formation of spatial invariance in the wave profile as a balance between hardening and softening effects in the dispersion that emerge due to the nonlinearity and the periodicity, respectively. For example, for a wave amplitude of the order of one-eighth of the unit-cell size in a demonstrative structure, the two effects are practically in balance for wavelengths as small as roughly three times the unit-cell size.
非线性的引入改变了固体介质中弹性波的色散。在本文中，我们提出了一种解析公式，用于处理由具有交替材料特性的层组成的一维声子晶体中的有限应变布洛赫波。考虑纵向波并忽略横向效应，首先在每个均匀层中获得精确的非线性色散关系，随后在转移矩阵方法中使用该关系来推导整个周期性介质的近似非线性色散关系。结果是一种与振幅相关的弹性带结构，经数值模拟验证，对于高达八分之一的振幅与单胞长度比是准确的。所推导的色散关系使我们能够将波剖面中空间不变性的形成解释为分别由于非线性和周期性而在色散中出现的硬化和软化效应之间的平衡。例如，对于一个演示结构中波振幅为单胞尺寸的八分之一左右的情况，对于波长小至约三倍单胞尺寸的情况，这两种效应实际上是平衡的。
