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

Abstract

Abstract. It has previously been observed experimentally and simulated numerically that when a thin film bonded to a much softer substrate is subjected to a uni-axial compression parallel to the interface, the initial buckled pattern will suffer a secondary bifurcation that doubles the period of the original pattern when the compressive strain reaches a critical value. This perioddoubling phenomenon is analyzed in this paper using an asymptotically self-consistent approach based on the exact theory of nonlinear elasticity. The predicted critical strain based on a fourterm expansion shows good agreement with that obtained using fully numerical simulations, and it is demonstrated that four is the minimum number of terms that should be included in order to give realistic predications. Although our illustrative calculations are conducted for neo-Hookean materials, the proposed approach can deal with any material models and can be extended to higher orders.

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

“…10 and sufficient growth, a period-doubling instability occurs due to nonlinearities in the substrate response [151,152], as shown in figure 7. Whereas period doubling is well understood in dynamical systems, understanding the development of a spatial period-doubling pattern is more challenging even in the absence of growth [153]. The theory of growth and Figure 7.…”

Mathematical Modeling of Morphogenesis in Living Materials

“…Nonetheless, it is known that they do not allow to capture the occurrence of secondary bifurcations, which have been observed in bi-layered materials, possibly leading to the emergence of more complex patterns characterized by sub-harmonic resonances [14]. Although out of the scopes of this work, we remind that secondary bifurcations can be studied using analytic perturbation techniques [44] or more advances numerical tools [81].…”

Elastic contact of a stiff thin layer and a half-space

“…At the same time, incorporating dynamic and nonlinear phenomena in the proposed two-parametric scheme seems to be less straightforward, e.g. see aforementioned papers [6,14] and also [21,22,23] dealing with high-frequency thickness vibration.…”

“…In addition, we refer to [4], appreciating the importance of high contrast limit. Among modern considerations on the subject we also mention [5] inspired by modelling of advance resonant devices, and [6,9,12,14,15,27] tackling a variety of vibration and stability phenomena. In the recent paper [24] a 3D problem in linear elasticity for a soft layer attached to a substrate was treated for a broad range of ratios between relative stiffnesses and wavelengths, resulting, in particular, in the justification and refinement of Winkler-Fuss hypothesis.…”