Curvature delays growth-induced wrinkling

Abstract

Wrinkling patterns can be induced by the growth of a thin elastic film over a soft elastic substrate. While there is a good understanding of how this pattern is initiated on a flat geometry, wrinkling patterns over a curved surface are more complicated. Here, we consider this phenomenon within the framework of large deformation morphoelasticity by investigating surface wrinkling of a growing thin elastic film bonded to a large elastic cylinder. The system has two important dimensionless parameters: the ratio η of the film thickness by the cylinder radius and the relative stiffness of the two layers ξ. Depending on the values of ξ and η we identify four different regimes for which we find the critical growth and wrinkling mode number. By combining asymptotic methods with numerical computations we determine the effect of the curvature on the bifurcation and establish that it always induces a delay at the bifurcation: larger growth is needed on a curved surface to induce the same wrinkling instability. These results are crucial to understand pattern formation on surface with varying curvatures.

Growth and remodelling of living tissues: perspectives, challenges and opportunities

“…In this simple two-layer system, it is well appreciated that the pattern adopted by the system depends on a number of important factors such as the relative stiffnesses of the two layers [140], the thickness of the thin layer, the growth of the top layer [141], the curvature of the foundation [142], the adhesion energy between the layers [143,144], the imperfection of the substrate [145], the anisotropic response [146,147], the surface tension and pressure [148] and the nonlinear elastic response of the materials [149]. For small ratios of layer μ l to foundation μ s stiffnesses, m l =m s , 10, as the wrinkling patterns develop, the system localizes this initial deformation and a fold or crease appears as observed in many biological systems.…”

Bioinspired Multiscale Wrinkling Patterns on Curved Substrates: An Overview

“…Compared with planar substrate with zero curvature, the wrinkling of thin films on curved substrates has some special features: (i) The critical strain for the wrinkling of thin films on curved substrates increases with the curvature and wrinkling patterns tend to form on substrates with smaller curvatures when other parameters are constant [ 49 , 140 , 156 , 166 ]. Alain et al [ 166 ] theoretically demonstrated that curvature delays growth-induced wrinkling compared with flat substrate. Crosby et al [ 156 ] found that wrinkled patterns can be induced on oxide PDMS spheres having big radii under swelling-induced compressive stress, while it is hard to form wrinkles on the spheres with small radii.…”

Influence of the growth gradient on surface wrinkling and pattern transition in growing tubular tissues

“…It should be pointed out that surface wrinkling of sinusoidal profiles may be incurred in both planar and curved film-substrate structures under appropriate stimuli such as axial compression [38,39], growth [40][41][42][43] or swelling [44,45], if the film is stiffer than the substrate, or equivalently, the β should exceed a critical value. Furthermore, for planar structures, this critical value is proven to be around 1.74 [38,46], while for curved systems it is dependent on the geometrical parameters ranging from 1.18 to 1.6 [21].…”