Sulci are localized furrows on the surface of soft materials that form by a compression-induced instability. We unfold this instability by breaking its natural scale and translation invariance, and compute a limiting bifurcation diagram for sulcfication showing that it is a scale-free, sub-critical nonlinear instability. In contrast with classical nucleation, sulcification is continuous, occurs in purely elastic continua and is structurally stable in the limit of vanishing surface energy. During loading, a sulcus nucleates at a point with an upper critical strain and an essential singularity in the linearized spectrum. On unloading, it quasi-statically shrinks to a point with a lower critical strain, Sulci are usually seen in combination with complementary protrusions known as gyri on the surface of the primate brain, but are also seen on the palm of our hand, in our elbows and knees, in swollen cellular foams (such as bread) and gels, and in geological strata; a few representative examples are shown in Fig. 1. While observations of sulci are ancient, their systematic study is fairly recent; an early reference is the reticulation patterns in photographic gelatin [1], and there has been a small interest in these objects both experimentally [2][3][4][5][6] and theoretically [3,[7][8][9], starting with the pioneering work of Biot [7] over the past 50 years. Despite this, there is no careful analysis of the fundamental instability and bifurcation that leads to sulci. Here we study the formation of a sulcus in a bent slab of soft elastomer, e.g. PDMS: as the slab is bent strongly, it pops while forming a sulcus that is visible in the lower right panel of Fig. 1; releasing the bend causes the sulcus to vanish continuously, in sharp contrast with familiar hysteretic instabilities that pop in both directions. We find that sulcification is a fundamentally new kind of nonlinear subcritical surface instability with no scale and a strongly topological character, yet has no energetic barrier relative to an entire manifold of linearly stable solutions. We also argue that sulcification instabilities are relevant to the stability of soft interfaces generally, and provide one of the first physical examples of the consequences of violating the complementing condition [12] (during loading) and quasiconvexity at the boundary [6] (during unloading), keystones in the mathematical theory of elliptic partial differential equations and the calculus of variations.To understand the unusual nature of sulcification, we first recall Biot's calculation for the linear instability of the surface of an infinite half-space of an incompressible elastomer that is uniformly stressed laterally. Because the the free surface is the softest part of the system, and there is no characteristic length scale in the equations of elasticity or in the boundary conditions, instability first arises when the Rayleigh surface wave speed vanishes. For incompressible rubber, Biot [7] showed that this threshold is reached when the compressive strain exceeds ...

Wrinkle patterns in compressed thin sheets are ubiquitous in nature and technology, from the furrows on our foreheads to crinkly plant leaves, from ripples on plastic-wrapped objects to the protein film on milk. The current understanding of an elementary descriptor of wrinkles-their wavelength-is restricted to deformations that are parallel, spatially uniform, and nearly planar. However, most naturally occurring wrinkles do not satisfy these stipulations. Here we present a scheme that quantitatively explains the wrinkle wavelength beyond such idealized situations. We propose a local law that incorporates both mechanical and geometrical effects on the spatial variation of wrinkle wavelength. Our experiments on thin polymer films provide strong evidence for its validity. Understanding how wavelength depends on the properties of the sheet and the underlying liquid or elastic subphase is crucial for applications where wrinkles are used to sculpt surface topography, to measure properties of the sheet, or to infer forces applied to a film.elastic sheets | wrinkles | curved topography W rinkles emerge in response to confinement, allowing a thin sheet to avoid the high energy cost associated with compressing a fraction e Δ of its length ( Fig. 1) (1-7). The wavelength, λ, of wrinkles reflects a balance between two competing effects: the bending resistance, which favors large wavelengths, and a restoring force that favors small amplitudes of deviation from the flat, unwrinkled state. Two such restoring forces are those due to the stiffness of a solid foundation or the hydrostatic pressure of a liquid subphase (Fig. 1A). Cerda and Mahadevan (1) realized that a tension in the sheet can give rise to a qualitatively similar effect ( Fig. 1B) and thereby proposed a universal law that applies in situations where the wrinkled sheet is nearly planar and subjected to uniaxial loading:Here the bending modulus B = Et 3 =½12ð1 − Λ 2 Þ (with E the Young's modulus, t the sheet's thickness, and Λ the Poisson ratio), whereas out-of-plane deformation is resisted by an effective stiffness, K eff , which can originate from a fluid or elastic substrate, an applied tension, or both. Eq. 1 is appealing in its simplicity, but it applies only for patterns that are effectively one-dimensional. In particular, it does not apply when the stress varies spatially or when there is significant curvature along the wrinkles. Here, we study two experimental settings in which these limitations are crucial: (i) indentation of a thin polymer sheet floating on a liquid, which leads to a horn-shaped surface with negative Gaussian curvature, and (ii) a circular sheet attached to a curved liquid meniscus with positive Gaussian curvature. In both cases, wrinkle patterns live on a curved surface, show spatially varying wavelengths, and are limited in spatial extent. The extent of finite wrinkle patterns in a variety of such 2D situations has recently been addressed (6,(8)(9)(10)(11) and was found to depend largely on external forces and boundary conditions. Howeve...

It is proposed that the event horizon of a black hole is a quantum phase transition of the vacuum of space-time analogous to the liquid-vapor critical point of a bose fluid. The equations of classical general relativity remain valid arbitrarily close to the horizon yet fail there through the divergence of a characteristic coherence length ξ. The integrity of global time, required for conventional quantum mechanics to be defined, is maintained. The metric inside the event horizon is different from that predicted by classical general relativity and may be de Sitter space. The deviations from classical behavior lead to distinct spectroscopic and bolometric signatures that can, in principle, be observed at large distances from the black hole.

The adhesion of a stiff film onto a curved substrate often generates elastic stresses in the film that eventually give rise to its delamination. Here we predict that delamination of very thin films can be dramatically suppressed through tiny, smooth deformations of the substrate, dubbed here "wrinklogami," that barely affect the macro-scale topography. This "prolamination" effect reflects a surprising capability of smooth wrinkles to suppress compression in elastic films even when spherical or other doubly curved topography is imposed, in a similar fashion to origami folds that enable construction of curved structures from an unstretchable paper. We show that the emergence of a wrinklogami pattern signals a nontrivial isometry of the sheet to its planar, undeformed state, in the doubly asymptotic limit of small thickness and weak tensile load exerted by the adhesive substrate. We explain how such an "asymptotic isometry" concept broadens the standard usage of isometries for describing the response of elastic sheets to geometric constraints and mechanical loads.