Stamping and Wrinkling of Elastic Plates

Abstract

We study the peculiar wrinkling pattern of an elastic plate stamped into a spherical mold. We show that the wavelength of the wrinkles decreases with their amplitude, but reaches a maximum when the amplitude is of the order of the thickness of the plate. The force required for compressing the wrinkled plate presents a maximum independent of the thickness. A model is derived and verified experimentally for a simple one-dimensional case. This model is extended to the initial situation through an effective Young modulus representing the mechanical behavior of wrinkled state. The theoretical predictions are shown to be in good agreement with the experiments. This approach provides a complement to the "tension field theory" developed for wrinkles with unconstrained amplitude.Wrinkling patterns are observed when thin plates are put under compression, spanning scales from geological patterns [1], skin wrinkles resulting from aging processes or scars [2,3] to cells locomotion generating strains on substrates [4]. They have important applications in micro-engineering such as the formation of controlled patterns [5], or the estimation of mechanical properties from the number and the extent of the wrinkles [6][7][8][9]. While most studies have focused on near threshold patterns, recent contributions have pushed further the description of finely wrinkled plates, i.e., well above the initial buckling threshold [10,11]. These studies consider plates submitted to strong in-plane tension on the boundaries, which prevents stress focusing commonly observed in crumpled paper [12]. In these descriptions, wrinkles are assumed to totally relax compressive stresses, as in traditional tension field theory [13,14]. In addition, both amplitude and wavelength of the wrinkles vanish with the thickness of the plate. We propose to study a conceptually different wrinkling regime where the boundaries are free from tension but the amplitude of the wrinkles is highly constrained. We focus on the simplest example, an elastic plate compressed in a spherical mold with a confinement defined by a gap δ (Fig. 1). This stamping configuration is common in industrial processes where metal plates are plastically embossed, the mismatch in Gaussian curvature generally leading to regular wrinkles [15][16][17]. In the elastic case, crumpling singularities first appear as the mold is progressively closed down (Fig. 1c) and evolve into a pattern of apparently smooth radial wrinkles for high confinement (Fig. 1d-e). We show that constraining the amplitude does not lead to the collapse of compressive stress. Instead, the wrinkling pattern derives from a nontrivial balance between compression and bending stresses, with surprising consequences : the wavelength of the wrinkles does not vanish and the constraining force reaches a maximum independent of the thickness of the plate.One-dimensional problem. We start by considering the simpler problem of a plate of length L, thickness h and unit width. In-plane displacements are imposed at both ends u x (±L/2) = ±∆/2 (Fig...

Mesoscale structure of wrinkle patterns and defect-proliferated liquid crystalline phases

“…metrically-favored) Gaussian curvature G tar is forced to reside near a "substrate" whose shape has a curvature G sub > G tar (e.g. a naturally-planar sheet attached to a liquid drop [19] or a rigid sphere [2,18], Fig. 1c).…”

Geometrically incompatible confinement of solids

“…In the article, we aim to bridge the gap between the bending-dominated mechanics of Euler and straindominated geometry of Gauss through the "Gauss-Euler [6]); (B) top view of conical sheet confined between rigid, planar plates (courtesy: E. Sharon); (C) top view of simulated flat sheet confined to a "spherical Winkler" substrate (describe in the text); (D) a polygonal patch cut from a thin spherical shell floated at a (planar) air/liquid interface (from ref. [7]).…”

Universal collapse of stress and wrinkle-to-scar transition in spherically confined crystalline sheets

“…17 and 22. We find that the compression-free region where scars emerge is LðαÞ < r < W , where the stress is σ rr =γ = W =r ; σ θθ =γ = 0 for LðαÞ < r < W ; [18] and LðαÞ=W = ðα * =αÞ 1=3 . In the unscarred center of the sheet, 0 < r < LðαÞ, the stress field is purely tensile and is described by Eq.…”

“…The second group of studies, inspired by classical elasticity of plates and shells (15,16), has focused on the mechanical, longwavelength response, which is independent of the crystalline order. In this case, the small scale on which elasticity theory breaks down is determined by the sheet's thickness, and the deformations that relax the elastic cost of confinement are wrinkles and crumples (17), folds (18), and blisters (19), which do not conform to the imposed spherical shape. In comparison with the hard singularities of crystalline defects, these deformations are "soft," because their characteristic scales decrease with the sheet's thickness slowly enough so that elasticity theory is valid throughout the sheet.…”