Gravity Driven Instability in Elastic Solid Layers

We demonstrate the instability of the free surface of a soft elastic solid facing downwards. Experiments are carried out using a gel of constant density $\rho$, shear modulus $\mu$, put in a rigid cylindrical dish of depth $h$. When turned upside down, the free surface of the gel undergoes a normal outgoing acceleration $g$. It remains perfectly flat for $\rho gh/\mu <{\alpha }^{*}$ with ${\alpha }^{*}\simeq 6$, whereas a steady pattern spontaneously appears in the opposite case. This phenomenon results from the interplay between the gravitational energy and the elastic energy of deformation, which reduces the Rayleigh waves celerity and vanishes it at the threshold.

Figure 1

Scheme of a sinusoidal disturbance of a downward facing and initially flat surface of a heavy elastic slab. The other surface is fixed on the rigid substrate. The energy change due to this disturbance can be positive or negative depending on the value of the dimensionless ratio $\rho gh/\mu$ with $\rho$ the mass density, $g$ the (gravitational) acceleration, $h$ the thickness, and $\mu$ the elastic modulus.

Figure 2

Views of the downward facing free surface of gels with different shear moduli. (a) $\mu =78\pm$ 0.5 Pa, (b) $44\pm 0.5\mathrm{Pa}$, (c) $43.3\pm 0.5\mathrm{Pa}$, (d) $42.8\pm 0.5\mathrm{Pa}$, (e) $41.0\pm 0.5\mathrm{Pa}$, (f),(g),(h) $40.0\pm 0.5\mathrm{Pa}$. The cylindrical dish is 18 cm diameter and 2.75 cm deep. (a)–(g) Steady patterns obtained just after reversal. (h) The sample temperature was 50° when reversed. The snapshot is taken after the sample has cooled to room temperature.

Figure 3

Quantitative analysis of the surface distortion. (a) Experimental setup. The distance mirror-projector sample is 1.5 m while the dish diameter is 18 cm. (b) Full circles: intersections of the distorted lines of the grid observed on a sample ($\mu =41.5\pm 0.5\mathrm{Pa}$). Solid lines defined the grid that best fits the observed one. (c) Square root of the mean squared error with the grid that best fits the observed one as a function of $\mu$ with $h=2.75\mathrm{cm}$ (full circles). The solid line is the best fit with the power law $a+b({\mu }^{*}-\mu {)}^c$. We find ${\mu }^{*}=44.6\pm 1.8\mathrm{Pa}$.

Figure 4

Predictions for waves propagating at the surface of a heavy elastic material. Left: dimensionless frequency squared ${\omega }^2{h}^2\mu /\rho$ as a function of the dimensionless wave number $kh$ for different values of $\alpha$. For $\alpha <{\alpha }^{*}$, the flat surface is stable and $\omega$ is the Rayleigh frequency. For $\alpha >{\alpha }^{*}$ the flat surface is unstable and the growth rate of the most unstable mode is $\mathrm{\Omega }=\sqrt{-{\omega }_{\min }^2}$, ${\omega }_{\min }$ being the minimum of $\omega$. Right: Dimensionless growth rate of the mode $k$ with the maximum growth rate obtained from Eq. (1) as a function of the gap with the instability threshold (solid line).