APDEs Seminar

Recent experiments have observed the emergence of standing waves at the free surface of elastic bodies attached to a rigid oscillating substrate and subjected to critical values of forcing frequency and amplitude [4, 3]. This phenomenon, known as Faraday instability, is now well understood for viscous fluids [2] but surprisingly eluded any theoretical explanation for soft solids.

In this talk, we characterize Faraday waves in soft incompressible slabs using the Floquet theory to study the onset of harmonic and subharmonic resonance eigenmodes. We consider a ground state corresponding to a finite homogeneous deformation of the elastic slab. We transform the incremental boundary value problem into an algebraic eigenvalue problem characterized by the three dimensionless parameters, that characterize the interplay of gravity, capillary and elastic waves. Remarkably, we found that Faraday instability in soft solids is characterized by a harmonic resonance in the physical range of the material parameters. This seminal result [1] is in contrast to the subharmonic resonance that is known to characterize viscous fluids, and opens the path for using Faraday waves for a precise and robust experimental method that is able to distinguish solid-like from fluid-like responses of soft matter at different scales.

[1] G. Bevilacqua, X. Shao, J. R. Saylor, J. B. Bostwick, and P. Ciarletta, Faraday waves in soft elastic solids, Proceedings of the Royal Society A 476 (2020), p. 20200129.

[2] K. Kumar, Linear theory of Faraday instability in viscous liquids, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452(1948)(1996), 1113–1126.

[3] X. Shao, J. R. Saylor, and J, B. Bostwick, JB, Extracting the surface tension of soft gels from elastocapillary wave behavior, Soft Matter 14(36) (2018), 7347–7353.

[4] X. Shao, G. Bevilacqua, P. Ciarletta, J. R. Saylor, and J, B. Bostwick, JB, Experimental observation of Faraday waves in soft gels, Physical Review E 102(6) (2020), p. 060602.