From wrinkling to global buckling of a ring on a curved substrate

Abstract

We present a combined analytical approach and numerical study on the stability of a ring bound to an annular elastic substrate, which contains a circular cavity. The system is loaded by depressurizing the inner cavity. The ring is modeled as an Euler-Bernoulli beam and its equilibrium equations are derived from the mechanical energy which takes into account both stretching and bending contributions. The curvature of the substrate is considered explicitly to model the work done by its reaction force on the ring. We distinguish two different instabilities: periodic wrinkling of the ring or global buckling of the structure. Our model provides an expression for the critical pressure, as well as a phase diagram that rationalizes the transition between instability modes. Towards assessing the role of curvature, we compare our results for the critical stress and the wrinkling wavelength to their planar counterparts. We show that the critical stress is insensitive to the curvature of the substrate, while the wavelength is only affected due to the permissible discrete values of the azimuthal wavenumber imposed by the geometry of the problem. Throughout, we contrast our analytical predictions against finite element simulations.Comment: 34 pages, 9 figure

Bioinspired Multiscale Wrinkling Patterns on Curved Substrates: An Overview

“…A soft shell bonded on a rigid core subjected to volumetric growth tends to form creasing morphology while a thin stiff shell on a soft core tends to form wrinkles first that can further transform into period doubles, folds, and ridges depending on the core-shell materials properties and the substrate curvature [140][141][142]. By simplifying the cylindrical core-shell system to a 2D problem of a ring on an annular substrate, Lagrange et al [143] provided a solution for the hoop stress in the ring with accounting for both the curvature and the finite size of the substrate. Depending on the dimensionless thickness and stiffness ratio, two types of instability modes including local wrinkling of the ring and global buckling of the structure were identified.…”

Shape Memory Effect in Micro-Sized Shape Memory Polymer Composite Chains

“…The SME has occurred in macroscale and microscale in recent years, and also demonstrated at nanoscale in SMPs [14,15]. Hence, the prospect of SME applying in microscale or even nanoscale patterning receives great attention, including adaptive programmable materials [15], switchable molecule-based materials [16], photoactive electroactive applications [7,8], soft electronics [17,18], biomedical applications (e.g., cell mechanobiology [19], biomimetic 4D printing [3], switchable dual pH-Responsiveness [10], robust microcarriers [20], soft microgrippers [21], and micropatterned containers [22]). At present, a number of SMPs are available in the market and many more are under developing [1].…”

Selection of hexagonal buckling patterns by the elastic Rayleigh-Taylor instability

“…Soft elastic solids are attractive for applications as they can be actuated by forces that are otherwise too weak to induce significant elastic strains such as their own weight (Mora et al, 2014), electric forces (Arun et al, 2006;Wang et al, 2011;Huang et al, 2012;Bense et al, 2017), magnetic forces (Danas and Triantafyllidis, 2014), adhesive forces (Ghatak et al, 2000;Mönch and Herminghaus, 2001), or even the capillary forces present at a curved solid-fluid interface (Mora et al, 2010(Mora et al, , 2013. As they undergo large strains, soft elastic solids display a non-linear response and are prone to a variety of buckling instabilities (Biot, 1963;Tanaka et al, 1987;Mora et al, 2011;Ciarletta et al, 2013;Lagrange et al, 2016); some of these instabilities are discontinuous and are therefore difficult to approach analytically: this is the case of the creasing instability for example (Hong et al, 2009;Cao and Hutchinson, 2011;Hohlfeld and Mahadevan, 2012;Ciarletta and Fu, 2015). This is also the case of the elastic Rayleigh-Taylor instability, which we investigate in this paper.…”