Oblique wrinkles

Soft Matter, 2012

Morphological instabilities and surface wrinkling of soft materials such as gels and biological tissues are of growing interest to a number of academic disciplines including soft lithography, metrology, flexible electronics, and biomedical engineering. In this paper, we review some of the recent progresses in experimental and theoretical investigations of instabilities that lead to the emergence and evolution of surface wrinkling, folding and creasing under various geometrical constraints (e.g.

A laterally confined thin elastic sheet lying on a liquid substrate displays regular undulations, called wrinkles, characterized by a spatially extended energy distribution and a well-defined wavelength λ. As the confinement increases, the deformation energy is progressively localized into a single narrow fold. An exact solution for the deformation of an infinite sheet was previously found, indicating that wrinkles in an infinite sheet are unstable against localization for arbitrarily small confinement. We present an extension of the theory to sheets of finite length L, accounting for the experimentally observed wrinkle-to-fold transition. We derive an exact solution for the periodic deformation in the wrinkled state, and an approximate solution for the localized, folded state. We find that a second-order transition between these two states occurs at a critical confinement F = λ 2 /L.

We present mechanics of surface creasing caused by lateral compression of a nonlinear neo-Hookean solid surface, with its elastic stiffness decaying exponentially with depth. Nonlinear bifurcation stability analysis reveals that neo-Hookean solid surfaces can develop instantaneous surface creasing under compressive strains greater than 0.272 but less than 0.456. It is found that instantaneous creasing is set off when the compressive strain is large enough, and the longest-admissible perturbation wavelength relative to the decay length of the elastic modulus is shorter than a critical value. A compressive strain smaller than 0.272 can only trigger bifurcation of a stable wrinkle that can prompt a setback crease upon further compression. The minimum compressive strain required to develop setback creasing is found to be 0.174. If the relative longest-admissible perturbation wavelength is long enough, then the wrinkle can fold before it creases, and the specimen can be compressed further beyond the Biot critical strain limit of 0.456. Various bifurcation branches on a plane of normalized longest-admissible wavelength versus compressive strain delineate different phases of corrugated surface configurations to form a ruga phase diagram. The phase diagram will be useful for understating surface crease, as well as for controlling ruga structures for various applications, such as designing stretchable electronics.

Physical Review Fluids, 2021

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 2018

We study what is clearly one of the most common modes of deformation found in nature, science and engineering, namely the large elastic bending of curved structures, as well as its inverse, unbending, which can be brought beyond complete straightening to turn into eversion. We find that the suggested mathematical solution to these problems always exists and is unique when the solid is modelled as a homogeneous, isotropic, incompressible hyperelastic material with a strain-energy satisfying the strong ellipticity condition. We also provide explicit asymptotic solutions for thin sectors. When the deformations are severe enough, the compressed side of the elastic material may buckle and wrinkles could then develop. We analyse, in detail, the onset of this instability for the Mooney–Rivlin strain energy, which covers the cases of the neo-Hookean model in exact nonlinear elasticity and of third-order elastic materials in weakly nonlinear elasticity. In particular, the associated theoreti...

2004

A compressively strained film on a substrate can wrinkle into intricate patterns. This Rapid Communication studies the evolution of the wrinkle patterns. The film is modeled as an elastic nonlinear plate and the substrate a viscoelastic foundation. A spectral method is developed to evolve the nonlinear system. When the initial film strains are isotropic, the wrinkles evolve into a pattern with a motif of zigzag segments, in random orientations.

Journal of Elasticity, 2011

When a thin rectangular sheet is clamped along two opposing edges and stretched, its inability to accommodate the Poisson contraction near the clamps may lead to the formation of wrinkles with crests and troughs parallel to the axis of stretch. A variational model for this phenomenon is proposed. The relevant energy functional includes bending and membranal contributions, the latter depending explicitly on the applied stretch. Motivated by work of Cerda, Ravi-Chandar, and Mahadevan, the functional is minimized subject to a global kinematical constraint on the area of the mid-surface of the sheet. Analysis of a boundary-value problem for the ensuing Euler-Lagrange equation shows that wrinkled solutions exist only above a threshold of the applied stretch. A sequence of critical values of the applied stretch, each element of which corresponds to a discrete number of wrinkles, is determined. Whenever the applied stretch is sufficiently large to induce more than three wrinkles, previously proposed scaling relations for the wrinkle wavelength and, modulo a multiplicative factor that depends on the Poisson ratio of the sheet and the applied stretch and arises from the more general and weaker nature of geometric constraint under consideration, root-mean-square amplitude are confirmed. In contrast to the scaling relations for the wrinkle wavelength and amplitude, the applied stretch required to induce any number of wrinkles depends on the in-plane aspect ratio of the sheet. When the sheet is significantly longer than it is wide, the critical stretch scales with the fourth power of the length-to-width Dedicated to the memory of Donald E. Carlson, whose insight and clarity of thought were exceeded only by his modesty and generosity.

Nature Materials, 2005

S tiff thin films on soft substrates are both ancient and commonplace in nature; for instance, animal skin comprises a stiff epidermis attached to a soft dermis. Although more recent and rare, artificial skins are increasingly used in a broad range of applications, including flexible electronics 1 , tunable diffraction gratings 2,3 , force spectroscopy in cells 4 , modern metrology methods 5 , and other devices 6-8 . Here we show that model elastomeric artificial skins wrinkle in a hierarchical pattern consisting of self-similar buckles extending over five orders of magnitude in length scale, ranging from a few nanometres to a few millimetres. We provide a mechanism for the formation of this hierarchical wrinkling pattern, and quantify our experimental findings with both computations and a simple scaling theory. This allows us to harness the substrates for applications. In particular, we show how to use the multigeneration-wrinkled substrate for separating particles based on their size, while simultaneously forming linear chains of monodisperse particles.

The buckling and wrinkling of thin films has recently seen a surge of interest among physicists, biologists, mathematicians, and engineers. This activity has been triggered by the growing interest in developing technologies at ever-decreasing scales and the resulting necessity to control the mechanics of tiny structures, as well as by the realization that morphogenetic processes, such as the tissue-shaping instabilities occurring in animal epithelia or plant leaves, often emerge from mechanical instabilities of cell sheets. Although the most basic buckling instability of uniaxially compressed plates was understood by Euler more than two centuries ago, recent experiments on nanometrically thin (ultrathin) films have shown significant deviations from predictions of standard buckling theory. Motivated by this puzzle, we introduce here a theoretical model that allows for a systematic analysis of wrinkling in sheets far from their instability threshold. We focus on the simplest extension of Euler buckling that exhibits wrinkles of finite length-a sheet under axisymmetric tensile loads. The first study of this geometry, which is attributed to Lamé, allows us to construct a phase diagram that demonstrates the dramatic variation of wrinkling patterns from near-threshold to far-from-threshold conditions. Theoretical arguments and comparison to experiments show that the thinner the sheet is, the smaller is the compressive load above which the far-from-threshold regime emerges. This observation emphasizes the relevance of our analysis for nanomechanics applications.

Nature Communications

A thin elastic sheet lying on a soft substrate develops wrinkled patterns when subject to an external forcing or as a result of geometric incompatibility. Thin sheet elasticity and substrate response equip such wrinkles with a global preferred wrinkle spacing length and with resistance to wrinkle curvature. These features are responsible for the liquid crystalline smectic-like behaviour of such systems at intermediate length scales. This insight allows better understanding of the wrinkling patterns seen in such systems, with which we explain pattern breaking into domains, the properties of domain walls and wrinkle undulation. We compare our predictions with numerical simulations and with experimental observations.