Contact-based and spheroidal vibrational modes of a hexagonal monolayer of microspheres on a substrate

• Published in: Wave motion 2018-01, Vol.76 (C), p.122-133
• Main Authors: Vega-Flick, A., Duncan, R.A., Wallen, S.P., Boechler, N., Stelling, C., Retsch, M., Alvarado-Gil, J.J., Nelson, K.A., Maznev, A.A.
• Format: Article
• Language: English
• Subjects: 2D granular crystals; Acoustics; Dispersion; Displacements (lattice); Hertzian contact; Hexagonal lattice; Lattice vibration; Mechanics; Microspheres; Physics; Propagation; Shear stress; Spheroidal vibrations; Studies; Substrates; Waves in granular media
• ISSN: 0165-2125; 1878-433X
• DOI: 10.1016/j.wavemoti.2017.10.010

We analytically study acoustic modes of a close-packed hexagonal lattice of spheres adhered to a substrate, propagating along a high-symmetry direction. The model, accounting for both normal and shear coupling between the spheres and between the spheres and the substrate, yields three contact-based vibrational modes involving both translational and rotational motion of the spheres. In addition to contact-based modes, we also study the effect of sphere–substrate and sphere–sphere contacts on spheroidal vibrational modes of the spheres using a perturbative approach. The sphere–substrate interaction results in a frequency upshift for the modes having a non-zero displacement at the contact point with the substrate as well as mode-splitting for some of the degenerate modes of the free sphere. Sphere–sphere interactions result in dispersion of spheroidal modes. Analytical dispersion relations for both contact-based and spheroidal modes are presented and compared with results obtained for a square lattice. •We analyze acoustic waves in a close-packed 2D lattice of spheres on a substrate.•Spheres interact with each other and the substrate via Hertzian contacts.•Dispersion of translational and rotational contact-based modes is calculated.•Sphere contact perturbation causes frequency shifts, mode-splitting and dispersion.