Abstract
The stability of the wrinkling mode experienced by a compressed half-space of neo-Hookean material is investigated using analytical and numerical methods to study the post-bifurcation behaviour of periodic solutions. It is shown that wrinkling is highly unstable owing to the nonlinear interaction among the multiple modes associated with the critical compressive state. Concomitantly, wrinkling is sensitive to exceedingly small initial imperfections that significantly reduce the compressive strain at which the instability occurs. The study provides insight into the connection between wrinkling and an alternative surface mode, the finite amplitude crease or sulcus. The shape of the critical combination of wrinkling modes has the form of an incipient crease, and a tiny initial imperfection can trigger a wrinkling instability that collapses into a crease.
Highlights
Surface instabilities are frequently observed when highly elastic soft materials are compressed (Tanaka et al 1987; Gent & Cho 1999; Trujillo et al 2008; Cai et al submitted) and their importance has grown along with the steady increase in applications of soft materials (Crosby 2010). Biot (1963, 1965) appears to be the first to have demonstrated the existence of wrinkling instability modes at the surface of an incompressible neo-Hookean elastic half-space
A slight local depression on the surface of the half-space reduces the overall strain at the wrinkling instability to levels similar to that seen for the sinusoidal imperfection, based on comparable values of the normalized imperfection amplitudes that have been defined
The crease can be regarded as the collapse state of a wrinkle
Summary
Surface instabilities are frequently observed when highly elastic soft materials are compressed (Tanaka et al 1987; Gent & Cho 1999; Trujillo et al 2008; Cai et al submitted) and their importance has grown along with the steady increase in applications of soft materials (Crosby 2010). Biot (1963, 1965) appears to be the first to have demonstrated the existence of wrinkling instability modes at the surface of an incompressible neo-Hookean elastic half-space. Biot (1963, 1965) appears to be the first to have demonstrated the existence of wrinkling instability modes at the surface of an incompressible neo-Hookean elastic half-space These modes occur as a bifurcation from a state of uniform compression with the unusual feature that their wavelength is undetermined—the scale of wrinkle undulations is arbitrary as long as it is short compared with any other geometric dimension of the solid. Hohlfeld & Mahadevan (2011) explored the closing and opening pathways of a finite amplitude crease under a cycle of applied compression by attaching a very thin film with bending stiffness to the surface whose purpose is to regularize the numerical model by fixing the wavelength As these authors emphasize, the free surface of any soft elastic solid is susceptible to wrinkling and creasing under compression because the mode wavelengths can be arbitrarily small and locally a surface will be effectively flat.
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