Wrinkling Hierarchy in Constrained Thin Sheets from Suspended Graphene to Curtains

Abstract

We show that thin sheets under boundary confinement spontaneously generate a universal self-similar hierarchy of wrinkles. From simple geometry arguments and energy scalings, we develop a formalism based on wrinklons, the localized transition zone in the merging of two wrinkles, as building blocks of the global pattern. Contrary to the case of crumpled paper where elastic energy is focused, this transition is described as smooth in agreement with a recent numerical work [R. D. Schroll, E. Katifori, and B. Davidovitch, Phys. Rev. Lett. 106, 074301 (2011)]. This formalism is validated from hundreds of nanometers for graphene sheets to meters for ordinary curtains, which shows the universality of our description. We finally describe the effect of an external tension to the distribution of the wrinkles. The drive towards miniaturization in technology is demanding for increasingly thinner components, raising new mechanical challenges [1]. Thin films are, however, unstable to boundary or substrate-induced compressive loads: moderate compression results in regular wrinkling [2][3][4][5][6] while further confinement can lead to crumpling [7,8]. Regions of stress focusing can be a hindrance, acting as nucleation points for mechanical failure. Conversely, these deformations can be exploited constructively for tunable thin structures. For example, singular points of deformation dramatically affect the electronic properties of graphene [9].Here, we show that thin sheets under boundary confinement spontaneously generate a universal self-similar hierarchy of wrinkles, from strained suspended graphene to ordinary hanging curtains. We develop a formalism based on wrinklons, a localized transition zone in the merging of two wrinkles, as building blocks to describe these wrinkled patterns.To illustrate this hierarchical pattern, in Fig. 1(a), we show a wrinkled graphene sheet along with an ordinary hanged curtain. These patterns are also similar to the selfsimilar circular patterns first reported by Argon et al. for the blistering of thin films adhering on a thick substrate [10]. The diversity and complexity of those systems, characterized by various chemical and physical conditions, could suggest, a priori, that the underlying mechanisms governing the formation of these patterns are unrelated. However, these systems can be depicted, independently from the details of the experiments, as a thin sheet constrained at one edge while the others are free to adapt their morphology. These constraints can take the form of an imposed wavelength at one edge or just the requirement that it should remain flat.

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

“…Thus, H 1 encapsulates bending and splaying of wrinkles beyond those prescribed by the asymptotic stress field through aðxÞ. The specific form of H 1 remains unknown, despite some recent works that addressed the energetic cost associated with smooth and sharp transformations of n wrinkles to n + m wrinkles (17,18,25). However, the unexpectedly good agreement obtained between the simple local λ law and experiments suggests that, in circumstances that remain to be understood, the effect of H 1 may safely be neglected.…”

“…4C presents data from large sheets; here the edge fell outside the illuminated region, so the edge cascade was not visible). The local effect of the liquid meniscus or other boundary forces (18,(28)(29)(30) are not accounted for in Eqs. 1 and 2.…”

Prototypical model for tensional wrinkling in thin sheets

“…Furthermore, in both cases, the subdominant energies consist of a mixture of stretching and bending terms that determine small-scale features of the pattern: wavelength (wrinkling) and the size of stress-focusing zones (crumpling). Recent experiments (23,24) have shown that this mixture of subdominant energies leads to the complex shapes of curtains: multiscale, hierarchical patterns that interpolate between crumpling-and wrinkling-like shapes.…”

Wrinkle to fold transition: influence of the substrate response

“…In order to study carefully the properties of the emerging structures, we focus on bilayer systems compressed uniaxially in the plane of the rigid membrane that have been intensively studied. [48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63] Notice, however, that more complex geometries have already been studied by considering biaxial/radial stresses (with or without substrate) or curved substrates [64][65][66][67][68][69][70][71] where wrinkle to fold transitions can also be observed for large enough deformations of the systems.…”