Stability boundaries for the Rayleigh-Taylor instability in accelerated elastic-plastic solid slabs

The linear theory of the incompressible Rayleigh-Taylor instability in elastic-plastic solid slabs is developed on the basis of the simplest constitutive model consisting in a linear elastic (Hookean) initial stage followed by a rigid-plastic phase. The slab is under the action of a constant acceleration, and it overlays a very thick ideal fluid. The boundaries of stability and plastic flow are obtained by assuming that the instability is dominated by the average growth of the perturbation amplitude and neglecting the effects of the higher oscillation frequencies during the stable elastic phase. The theory yields complete analytical expressions for such boundaries for arbitrary Atwood numbers and thickness of the solid slabs.

Figure 1

Schematic of the two-interfaces system formed by the elastic-plastic slab on the top of a semi-infinite ideal fluid.

Figure 2

Boundaries for stability (full lines) and for the elastic-plastic transition (dotted lines) for three values of the Atwood number and for different values of the parameter $\alpha$ indicated on the figures: (a) $A_T=1$, (b) $A_T=0.8$, and (c) $A_T=0.4$.

Figure 3

Stability boundaries for two particular cases of cylindrical implosions: (a) driven by magnetic pressure [22], (b) driven by heavy ion beams [60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70].

Figure 4

Stability boundaries for a spherical shell driven by high explosives [53]: (a) implosion phase, (b) explosion phase.

Figure 5

Comparison with two-dimensional numerical simulations reported in the literature for $A_T=1$: (a) stability and EP transition boundaries for W and Al thick slabs [31, 32], (b) stability boundaries for the Barnes experiments [3, 7], numerically determined in Ref. [6]. Full lines are the results of the present theory, and dots are from the simulations.