Period-doubling and period-tripling in growing bilayered systems

Abstract

Growing layers on elastic substrates are capable of creating a wide variety of surface morphologies. Moderate growth generates a regular pattern of sinusoidal wrinkles with a homogeneous energy distribution. While the critical conditions for periodic wrinkling have been extensively studied, the rich pattern formation beyond this first instability point remains poorly understood. Here we show that upon continuing growth, the energy progressively localizes and new complex morphologies emerge. Previous studies have often overlooked these secondary bifurcations; they have focused on large stiffness ratios between layer and substrate, where primary instabilities occur early, long before secondary instabilities emerge. We demonstrate that secondary bifurcations are particularly critical in the low stiffness ratio regime, where the critical conditions for primary and secondary instabilities move closer together. Amongst all possible secondary bifurcations, the mode of period-doubling plays a central role - it is energetically favorable over all other modes. Yet, we can numerically suppress period-doubling, by choosing boundary conditions, which favor alternative higher order modes. Our results suggest that in the low stiffness regime, pattern formation is highly sensitive to small imperfections: surface morphologies emerge rapidly, change spontaneously, and quickly become immensely complex. This is a common paradigm in developmental biology. Our results have significantly applications in the morphogenesis of living systems where growth is progressive and stiffness ratios are low.

Ultradonut topology of the nuclear envelope

“…Leonhard Euler derived the celebrated Euler buckling formula for slender columns in the mid-18th century and the buckling of plates and shells was analyzed in the early 20th century (20). In the last few decades, buckling instabilities have been used to model and understand the force-deformation response of a wide variety of biological structures including DNA (32), cytoskeletal filaments (33)(34)(35)(36), membranes (37)(38)(39)(40)(41), white blood cells (42), viruses (43), and tissues (44)(45)(46)(47). Here we present an example in which the buckling of curved membranes can explain the ultradonut topology of the nuclear envelope.…”

An Asymptotic Analysis of the Period-Doubling Secondary Bifurcation in a Film/Substrate Bilayer

“…As a result, microstructures forming on a hard thin film that is bonded to a soft substrate have received a lot of attention in recent years. Of particular interest is the period-doubling secondary bifurcation that has been observed experimentally by Pocivavsek et al [15], Brau et al [4], and Sun et al [16], and numerically simulated by Sun et al [16], Cao and Hutchinson [8], and Budday, Kuhl, and Hutchinson [5]. This phenomenon has also been analyzed theoretically based on various approximate theories by Brau et al [4], Zhao et al [17], and Zhuo and Zhang [18].…”

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php algebraic equation for the amplitude of U (n−2,1) . Thus, to determine u (3) (x) completely, we need to solve the problem for u (4) (x) completely and the problem for u (5) (x) partially in the sense that we only need to obtain the solvability condition on the problem for U (5,1) in order to fix the amplitude of U (3,1) . Finally, we remark that although, according to the above procedure, the solution for u (2) (x) should also contain the terms U (2,1) (x 2 )e ix1 + c.c., it is found from the solvability condition on U (4,1) (x 2 ) that these terms are in fact zero and are hence absent from (2.6).…”

From period-doubling to folding in stiff film/soft substrate system: The role of substrate nonlinearity

“…The compression is commonly generated by the release of substrate pre-stretch imposed prior to film attachment [7,8,14]. Further compression of the system can induce perioddoubling [22], period-tripling [23], folds [27] and localized ridges [25]. Some previous studies revealed that the substrate pre-stretch and the film/substrate modulus ratio determine the occurrence and evolution of the post-buckling modes [23][24][25]28].…”

“…The primary instability modes include wrinkling [17], creasing [18,19], and buckle delamination [20,21], while the secondary bifurcation modes consist of period-doubling [22], period-tripling [23], localized ridges [24,25] and folds [26,27]. If a film is well-bonded to a substrate, and the film modulus is much higher than that of the substrate, the buckle delamination and creasing instability can be avoided [25,28].…”