Selection of hexagonal buckling patterns by the elastic Rayleigh-Taylor instability

Abstract

We investigate the non-linear buckling patterns produced by the elastic Rayleigh-Taylor instability in a hyper-elastic slab hanging below a rigid horizontal plane, using a combination of experiments, weakly nonlinear expansions and numerical simulations. Our experiments reveal the formation of hexagonal patterns through a discontinuous transition. As the unbuckled state is transversely isotropic, a continuum of linear modes become critical at the first bifurcation load: the critical wavevectors form a circle contained in a horizontal plane. Using a weakly non-linear post-bifurcation expansion, we investigate how these linear modes cooperate to produce buckling patterns: by a mechanism documented in other transversely isotropic structures, three-modes coupling make the unbuckled configuration unstable with respect to hexagonal patterns by a transcritical bifurcation. Stripe and square patterns are solutions of the post-bifurcation expansion as well but they are unstable near the threshold. These analytical results are confirmed and complemented by numerical simulations.

A meniscus fingering instability in viscoelastic fluids

“…The most spectacular occurrences of this analogy arise through mechanical instabilities. A rope falling onto a solid surface typically forms a series of regular coils, reminiscent of viscous coiling [6,7]; soft cylinders can develop Rayleigh-Plateau pearling instabilities [8,9]; the free surface of a soft elastic solid facing downwards is unstable with respect to the Rayleigh-Taylor instability [10][11][12] and wrinkles may appear at the free surface of an elastomer film subjected to an electric field [13], a phenomenon first observed with liquid films [14]. More recently, the free surface of a confined elastic layer subjected to a hydrostatic pressure has been shown to undergo a bulk fingering instability [15] reminiscent of the Saffman-Taylor instability [16].…”

Buckling of a compliant hollow cylinder attached to a rotating rigid shaft

“…In the limit of thin hollow cylinders this instability is closely related to the elastic Rayleigh-Taylor instability [18,[25][26][27]. Our control parameter α can be rewritten as…”

“…The first-order correction t 1 is the linear mode from Section III. The asymptotic post-buckling expansion method [12][13][14][15][16][17][18] proceeds by inserting the expansion in Eqs.19-20 into the non linear equilibrium written earlier in Eq.5 as…”

“…where δt(r, θ) is the set of virtual functions δR, δΘ, δq that represent infinitesimal increments satisfying the kinematic boundary conditions. Equation 21 is then expanded order by order in ε [4,[18][19][20]. At order ε 2 , we obtain an equation for t 2 , ∀δt k.a.,…”

Buckling of a spinning elastic cylinder: linear, weakly nonlinear and post-buckling analyses

“…The Koiter post-bifurcation expansion is a perturbation method that has already been used in various elastic systems [15][16][17][18][19][20][21]. In our recent paper dealing with the post-buckling analysis in the elastic Rayleigh-Taylor instability [22], Koiter's method was rederived in a self-consistent way; the present work makes use of the same notations and of the same solution strategy as in this previous paper.…”