Rayleigh-Taylor instability in elastic-plastic solid slabs bounded by a rigid wall

The linear evolution of the incompressible Rayleigh-Taylor instability for the interface between an elastic-plastic slab medium and a lighter semi-infinite ideal fluid beneath the slab is developed for the case in which slab is attached to a rigid wall at the top surface. The theory yields the maps for the stability in the space determined by the initial perturbation amplitude and wavelength, as well as for the transition boundary from the elastic to the plastic regimes for arbitrary thicknesses of the slab and density contrasts between the media. In particular, an approximate but very accurate scaling law is found for the minimum initial perturbation amplitude required for instability and for the corresponding perturbation wavelength at which it occurs. These results allows for an interpretation of the recent experiments by Maimouni et al. [Phys. Rev. Lett. 116, 154502 (2016)].

Figure 1

Schematic of an elastic-plastic slab bounded by a rigid wall on the top ($y=-h$), and by an unstable interface with a semi-infinite ideal fluid on the bottom ($y=0$).

Figure 2

Dimensionless oscillation frequency $\mathrm{\Omega }$ for the stable regimes as a function of the dimensionless perturbation wavenumber $\kappa$, for $A_T=1$ (a), and for $A_T=0.4$ (b), and for different values of the dimensionless slab thickness $\alpha$.

Figure 3

Normalized dimensionless slab thickness $2A_T\alpha /(1+A_T)$ as a function of the cutoff wave number $k_{c}h$.

Figure 4

Boundaries for stability ${\xi }^{*}$ (full lines) and for the elastic-plastic transition ${\xi }_{\mathrm{EP}}^{*}$ (dotted lines) for $A_T=1$ (a), and for $A_T=0.4$ (b), and for different values of the dimensionless slab thickness $\alpha$.

Figure 5

Scaling laws for the normalized dimensionless minimum initial perturbation amplitude $2A_T{\xi }_m^{*}/(1+A_T)$ (a), and for the corresponding perturbation wavelength $2A_T{\lambda }_m^{*}/(1+A_T)$ (b), in terms of the normalized dimensionless slab thickness $2A_T\alpha /(1+A_T)$.