Non-linear buckling and symmetry breaking of a soft elastic sheet sliding on a cylindrical substrate

https://doi.org/10.1016/j.ijnonlinmec.2015.02.013Get rights and content

Highlights

  • We study the axial compression of an elastic sheet sliding on a cylindrical substrate.

  • The sheet always buckles into one symmetric fold, permitting no periodic solution.

  • Upon further compression, the sheet tips over into a recumbent fold.

  • The fold remains stable also in the absence of compressive forces.

  • Theory and experiments with neoprene sheets are in excellent agreement.

Abstract

We consider the axial compression of a thin sheet wrapped around a rigid cylindrical substrate. In contrast to the wrinkling-to-fold transitions exhibited in similar systems, we find that the sheet always buckles into a single symmetric fold, while periodic solutions are unstable. Upon further compression, the solution breaks symmetry and stabilizes into a recumbent fold. Using linear analysis and numerics, we theoretically predict the buckling force and energy as a function of the compressive displacement. We compare our theory to experiments employing cylindrical neoprene sheets and find remarkably good agreement.

Introduction

When you roll up your sleeves to get some work done, you will not pay attention to the intricate folding patterns which form around your arms. However, these patterns are not only interesting for graphics designers but of eminent importance for biology and technology alike [1], [2]: one can find them in the twinkling of an eye [3] as well as in the development of organs such as the brain [4], [5], the intestine [6], [7], or the kidney [8]. Technological applications include structures for optics [9] or microfluidics [10] to name just a few.

One common theme in these examples is that they consist of coupled layers which undergo morphological changes in response to external or internal constraints such as a simple compression or volumetric growth. The materials involved range from swelling hydrogels [11], [12] to supported graphene [13], [14] and many others. A well-studied setup in this context consists of a stiff membrane attached to a flat elastic or fluid bulk material [15], [16], [17]. The interplay between the bending of the sheet and the response of the bulk leads to the formation of wrinkles when the sheet is compressed uniaxially. Beyond a critical compression the wrinkles vanish and localized folds appear in the sheet. Interestingly, the shape of the sheet on the fluid can be found analytically [18], [19], [20] as long as the sheet does not touch itself [21].

When the substrate is not flat, translational invariance is broken and a whole plethora of folding patterns can be found [1], [2], [22], [23]. In this paper we study a particular type of such a system in a cylindrical geometry. An elastic cylindrical membrane is wrapped around a solid cylinder of same radius and compressed parallel to the axis of symmetry. This simple system is of potential relevance for situations as diverse as intestinal inversion [24], the folding of your sleeve, or even the neck of hidden-necked turtles (see Fig. 1). In contrast to the aforementioned flat system we allow the membrane to stretch azimuthally to accommodate to the external stress. Without the solid cylinder constraint the membrane would behave like a hollow cylindrical shell whose mechanics has been studied extensively in the literature [25], [26]: when compressed the shell develops regular patterns, such as periodic, axisymmetric undulations [27] or the trapezoidal patterns found by Yoshimura in the 1950s [28]. As is easily confirmed by compressing an empty can of soda, axisymmetric modes of deformation are typically unstable. As we will see below, the behavior becomes fundamentally different when the shell enwraps a solid cylinder of same size.

Similar to a ruck in a rug [29], [30], [31], [32] we will consider the case in which the sheet can slide on the cylinder. The only coupling between the sheet and the substrate is via the hard cylinder constraint. There is no elastic response between the two as is typically the case for cylindrical core-shell materials [33]. To simplify the theoretical treatment we assume that the sheet is unstretchable in the axial direction. This assumption will be validated by experiments with neoprene sheets and finite element simulations.

We start with the presentation of our model in Section 2. The resulting shape equations are axisymmetric and can be linearized and solved in lowest order of the axial compression as shown in Section 3. In Section 4 solutions of the full non-linear system are found numerically and compared to experiments with neoprene sheets.

Section snippets

Model

We consider a cylindrical elastic sheet of thickness h which enwraps a cylinder of radius R0 (see Fig. 1(c)). The axis of symmetry is oriented along the X-axis whereas we use R as the variable for the radial displacement. When the sheet is compressed with a fixed displacement ΔX, it buckles out of its reference configuration and forms a fold which tips over for large compressions (see Fig. 1(d)). In the following we will use the angle-arc length parametrization which describes the shape of the

Linearization

Owing to the strong non-linearities, we are unaware of any analytical, closed-form solution of Eqs. (7a), (7b), (7c), (7d), (7e), (7f). Nonetheless, certain unknowns such as the buckling force and the fold length can already be obtained in good approximation using linearized equations. To derive them, we start from Eqs. (7a), (7b), (7c), (7d), (7e), (7f) of a single fold and eliminate ρ by taking the derivative of Eq. (7f). This yieldspρ=12sinψ,ψ=fsinψpρcosψ.We defineϵ=(Δxsc)1/2and

Numerical solutions and experiments

To find the shape of the sheet for high deformations we solve the Hamilton equations (7a), (7b), (7c), (7d), (7e), (7f) numerically using a standard shooting method [37]: for a fixed scaled force f and a trial value for pρ=fn at s=0 the equations are integrated with a fourth-order Runge Kutta method. For any trial fn, the values of ψ, ρ, and pψ at s=0 are obtained from the boundary conditions (9a), (9b), (9c), (9d). The integration is stopped as soon as ρ=0 is reached again. Every fn which

Conclusion

Euler buckling is a well-studied phenomenon with numerous applications and occurs in various situations in nature and engineering [34]. In its simplest form it deals with the situation of translatory invariance in one direction, such as the buckling on planar substrates [15]. However, buckling often occurs in curved geometries [1], [2] inducing non-isometric deformations of the sheet. In this paper we have exemplified the effect of curvature on wrinkling by considering an Euler-type buckling in

Acknowledgments

This work was supported by the Swiss National Science Foundation Grant no. 148743 (N.S.). The authors thank Romain Lagrange for fruitful discussions.

References (40)

  • G. Domokos et al.

    Symmetry-breaking bifurcations of the uplifted elastic strip

    Physica D

    (2003)
  • X. Chen et al.

    Buckling patterns of thin films on curved compliant substrates with applications to morphogenesis and three-dimensional micro-fabrication

    Soft Matter

    (2010)
  • B. Li et al.

    Mechanics of morphological instabilities and surface wrinkling in soft materials: a review

    Soft Matter

    (2012)
  • D.P. Richman et al.

    Mechanical model of the brain convolutional development

    Science

    (1975)
  • D.C. van Essen

    A tension-based theory of morphogenesis and compact wiring in the central nervous system

    Nature

    (1997)
  • Z.B. Wang

    Unlocking the full potential of organic light-emitting diodes on flexible plastic

    Nat. Photon.

    (2011)
  • K. Efimenko

    Nested self-similar wrinkling patterns in skins

    Nat. Mater.

    (2005)
  • J. Dervaux et al.

    Shape transition in artificial tumors: from smooth buckles to singular creases

    Phys. Rev. Lett.

    (2011)
  • K. Zhang et al.

    Adhesion and friction control localized folding in supported graphene

    J. Appl. Phys.

    (2013)
  • L. Pocivavsek

    Stress and fold localization in thin elastic membranes

    Science

    (2008)
  • View full text