Curvature-induced stiffness and the spatial variation of wavelength in wrinkled sheets.

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.

Communication: Dominance of extreme statistics in a prototype many-body Brownian ratchet.

Many forms of cell motility rely on Brownian ratchet mechanisms that involve multiple stochastic processes. We present a computational and theoretical study of the nonequilibrium statistical dynamics of such a many-body ratchet, in the specific form of a growing polymer gel that pushes a diffusing obstacle. We find that oft-neglected correlations among constituent filaments impact steady-state kinetics and significantly deplete the gel's density within molecular distances of its leading edge. These behaviors are captured quantitatively by a self-consistent theory for extreme fluctuations in filaments' spatial distribution.

Coexistence of scale-invariant states in incompressible elastomers.

Cavitation and sulcification of soft elastomers are two examples of thresholdless, nonlinear instabilities that evade detection by linearization. I show that the onset of such instabilities can be understood as a kind of phase coexistence between multiple scale-invariant states, and I constructively enumerate the possible scale-invariant states of incompressible rubber in two dimensions. Whereas true phases (like the affine deformations of rubber) are homogeneous, the alternatives are inhomogeneous. In terms of the thermodynamics of solids, both classes of states must generally be given equal consideration.

Contractile equilibration of single cells to step changes in extracellular stiffness.

Extracellular stiffness has been shown to alter long timescale cell behaviors such as growth and differentiation, but the cellular response to changes in stiffness on short timescales is poorly understood. By studying the contractile response of cells to dynamic stiffness conditions using an atomic force microscope, we observe a seconds-timescale response to a step change in extracellular stiffness. Specifically, we observe acceleration in contraction velocity (µm/min) and force rate (nN/min) upon a step decrease in stiffness and deceleration upon a step increase in stiffness. Interestingly, this seconds-timescale response to a change in extracellular stiffness is not altered by inhibiting focal adhesion signaling or stretch-activated ion channels and is independent of cell height and contraction force. Rather, the response timescale is altered only by disrupting cytoskeletal mechanics and is well described by a simple mechanical model of a constant velocity actuator pulling against an internal cellular viscoelastic network. Consistent with the predictions of this model, we find that an osmotically expanding hydrogel responds to step changes in extracellular stiffness in a similar manner to cells. We therefore propose that an initial event in stiffness sensing is establishment of a mechanical equilibrium that balances contraction of the viscoelastic cytoskeleton with deformation of the extracellular matrix.

Scale and nature of sulcification patterns.

Sulci are surface folds commonly seen in strained soft elastomers and form via a strongly subcritical, yet scale-free, instability. Treating the threshold for nonlinear instability as a nonlinear critical point, we explain the nature of sulcus patterns in terms of the scale and translation symmetries which are broken by the formation of an isolated, small sulcus. Our perturbative theory and simulations show that sulcus formation in a thick, compressed slab can arise either as a supercritical or as a weakly subcritical bifurcation relative to this nonlinear critical point, depending on the boundary conditions. An infinite number of competing, equilibrium patterns simultaneously emerge at this critical point, but the one selected has the lowest energy. We give a simple, physical explanation for the formation of these sulcification patterns using an analogy to a solid-solid phase transition with a finite energy of transformation.

Unfolding the sulcus.

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, subcritical 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 quasistatically shrinks to a point with a lower critical strain, explained by breaking of scale symmetry. At intermediate strains the system is linearly stable but nonlinearly unstable with no energy barrier. Simple experiments confirm the existence of these two critical strains.

Sheet on a deformable sphere: wrinkle patterns suppress curvature-induced delamination.

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.