Rayleigh-Taylor Instability in Elastoplastic Solids: A Local Catastrophic Process
Résumé
We show that the Rayleigh-Taylor instability in elastoplastic solids takes the form of local perturbations penetrating the material independently of the interface size, in contrast with the theory for simple elastic materials. Then, even just beyond the stable domain, the instability abruptly develops as bursts rapidly moving through the other medium. We show that this is due to the resistance to penetration of a finger which is minimal for a specific finger size and drops to a much lower value beyond a small depth (a few millimeters). The Rayleigh-Taylor instability (RTI) is a well-known instability which occurs when a denser fluid rests on top of a lighter one [1]. As it develops, the two fluids penetrate one another, in the form of fingers. Instability is driven by the density difference and the acceleration to which the fluids are submitted, while surface tension provides a stabilizing effect. In contrast, RTI in solids is much less studied and understood, even though it relates to many application fields and can cause irreversible damage to structures. Examples include metal plates submitted to strong pressure or acceleration in high-energy density physics experiments [2], magnetic implosion of impactor liners [3,4], assessment of solid strength under high strain rate [5], slowly accreting neutron stars [6]. Other applications are found in geology: volcanic island formation [7], salt dome formation [8], and more generally, magmatic diapirism in Earth's mantle and continental crust [9,10], correspond to situations where a liquid opens its way through a layer of denser solid material above it. In most approaches to this problem [7–9,11], the upper material was considered as a highly viscous fluid, which allowed simple simulations of the process, but could also be misleading. Another situation concerns oil well cementing operations, in which yield stress fluids of different densities (drilling muds and cement, e.g.), which behave as solids at rest, may be pumped into the well in an ill-favored density order [12]. The basic approach to RTI for solids assumes linear elastic materials. The problem appears similar to that for simple fluids, except that the role of surface tension effects, neglected for solids, is played by elasticity. For a single solid above a liquid with a (positive) density difference Δρ, the instability criterion (A) is given by gΔρ > 4απG=L, where G and L are the shear modulus and length of the sample, respectively, and g denotes the gravitational acceleration. Depending on boundary conditions, factor α was found to be 1 [3,13], 1.6 [14], or 2 [15]. A couple of experiments on metal plates [16] and with a yogurt [17] provided some support to this theory. From a more complete study [18] using soft elastic solids, the overall validity of this approach was proved but the wavelength was shown to be smaller than expected from theory and dependent on uncontrollable, slight disturbances of the surface [19]. RTI for solids is further complicated by the fact that yielding may occur beyond a critical deformation. So far, this aspect has been considered separately, leading to the conclusion that instability results from a sufficiently large initial perturbation amplitude ε 0 (penetration depth). The instability criterion (B) then reads gΔρ > βτ c =ε 0 , where τ c denotes the material's yield stress (in simple shear), and where 0.5 ≤ β ≤ 2 depending on the sample aspect ratio [13–15,18,24,25]. Some tests with a single material were apparently in agreement with this criterion [17] but the plastic regime for this material was not so well-defined [19]. Finally, it was suggested [2] that elastic and plastic stability criteria should be taken into account successively, and deep theoretical analysis [26] predicted that for plastic materials, once the threshold is reached somewhere, the perturbation grows unlimitedly. These approaches have the advantage of considering independently the elasticity and the yielding effects. However, one cannot exclude that the interplay of both mechanisms could play a crucial role in the early stage of the perturbation growth. Here we aim at clarifying this problem through experiments on well-characterized materials, linearly elastic below a critical deformation and elastoplastic beyond this deformation. We show that the RTI in solids does not develop as predicted by the theory for simple elastic materials, but results from the ability of local perturbations to penetrate the material by involving, from the start, both elastic and plastic effects. At some point during the process, resistance to penetration drops, causing an abrupt
Domaines
Matière Condensée [cond-mat]
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