Journal
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
Volume 474, Issue 2217, Pages -Publisher
ROYAL SOC
DOI: 10.1098/rspa.2018.0173
Keywords
nonlinear dispersion; finite-strain waves; Green-Lagrange strain; periodic media; phononic crystals; solitary waves
Categories
Funding
- National Science Foundation CAREER [1254931]
- Div Of Civil, Mechanical, & Manufact Inn
- Directorate For Engineering [1254931] Funding Source: National Science Foundation
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The introduction of non linearity 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.
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