Discrete Phononic Crystal Dynamics Modeling And Nonlinear Dispersion Analysis

This paper mainly studies the elastic wave propagation and analysis methods of discrete nonlinear periodic structures.The periodic arrangement of materials with high acoustic impedance contrast can be used to control wave propagation.These systems,often referred to as phononic crystals or metamaterials,introduce non-linear elements that not only make their bandgap adjustable but also bring new amplitude-dependent wave propagation characteristics.The design of these systems relies on the calculation and analysis of the structure of the dispersion relationship that contains information about the velocity and direction of the wave propagation.In this paper,we consider the chromatic dispersion related to amplitude by incorporating the Bloch-wave theory and the perturbation method for nonlinear element systems including periodic elements.It provides a new choice for the adjustable band structure.Several important one-dimensional and two-dimensional discrete periodic structures are provided: monoatomic chain,diatomic chain,monoatomic lattice,diatomic lattice,etc.1.Several typical one-dimensional mass-cubic nonlinear springs models are analyzed,Approximate closed-form,first-order dispersion relations capture the effect of nonlinearities on harmonic wave propagation.Nonlinear stiffness changes the cut-off frequency and its dispersion relationship not only depend on the frequency of the input harmonic but also depend on the input wave amplitude.Numerical simulations verify the dispersion relations obtained from the perturbation analysis.2.In the two-dimensional dispersion analysis of two-dimensional lattice structure,considering several typical nonlinear two-dimensional lattice,the relationship between the frequence dispersion relations and the nonlinear coefficients in different directions is futher explored,and the relationship between the group velocity and the nonlinear coefficients is analysis in detail.The lattices with isotropic linear stiffness and anisotropic nonlinear stiffness can show the wave-direction propagation related to the amplitude;As the harmonic amplitude increases,the nonlinear stiffness will affect the overall lattice isotropy.There is a similar optical Of caustics;and correspondingly affect the pattern of energy flowing in the lattice.