Nano-scale wave dispersion beyond the First Brillouin Zone simulated with inertia gradient continua

Nano-scale wave dispersion beyond the First Brillouin Zone is often observed as descending branches and inflection points when plotting frequency or phase velocity against the wave number. Modeling this with discrete chain models is hampered by their restricted resolution. A continuum model equipped with higher-order inertia gradients is here developed as a suitable and versatile alternative. This model can be derived from discrete chain models, thereby providing a lower-scale motivation for the higher-order gradient terms. The derived gradient model is without free parameters, as the material constants are calculated a priori by minimising the error with respect to the discrete chain response. Unlike asymptotic approximations that provide a best fit for vanishing wave numbers, the error is here minimised over the entire range of reduced wave numbers 0 to 1, which leads to a much improved accuracy beyond the First Brillouin Zone. The new gradient model has been validated against (i) phonon dispersion curves measured through neutron scattering experiments in bismuth, aluminum, and nickel and (ii) Molecular Dynamics (MD) flexural wave propagation simulations of carbon nanotubes. The model captures all qualitative aspects of the experimental and MD dispersion curves without requiring a bespoke curve fitting procedure. With the exception of one set of MD results, the accuracy of the gradient model is very good.Nano-scale wave dispersion beyond the First Brillouin Zone is often observed as descending branches and inflection points when plotting frequency or phase velocity against the wave number. Modeling this with discrete chain models is hampered by their restricted resolution. A continuum model equipped with higher-order inertia gradients is here developed as a suitable and versatile alternative. This model can be derived from discrete chain models, thereby providing a lower-scale motivation for the higher-order gradient terms. The derived gradient model is without free parameters, as the material constants are calculated a priori by minimising the error with respect to the discrete chain response. Unlike asymptotic approximations that provide a best fit for vanishing wave numbers, the error is here minimised over the entire range of reduced wave numbers 0 to 1, which leads to a much improved accuracy beyond the First Brillouin Zone. The new gradient model has been validated against (i) phonon dispersion curv...

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