Post-critical phase diagram of growing bilayers

Growth-induced morphological instabilities are ubiquitously observed in biological systems across various length scales [1]. Indeed, instabilities in the form of wrinkling, folding and creasing are critical for certain biological functions. A common material arrangement consists of a thin stiff layer mounted on top of a thicker substrate, where either the thin outer layer, the thicker inner layer, or both can grow at specific rates. Depending on the level of differential growth between film and substrate (G_f/G_s) and the relative elastic moduli (E_f/E_s), different patterns such as sinusoidal wrinkling, period doubling, period quadrupling, or creases can form. In this regard, an understanding of the morphological phase diagram of growing bilayers with regards to the moduli ratio and differential growth ratio would facilitate physical understanding and also contribute to the development of new diagnostics.

The existing work on growth in bilayers mainly focuses on the initial or intermediate post-critical regime [2], and often on the case of growth in the thin film only [3], i.e. G_s = 0. The complete bifurcation landscape of bilayers in the deep post-critical regime remains an open question, especially how the post-critical phase diagram evolves as the ratio of elastic moduli (E_f/E_s) and differential growth (G_f/G_s) changes. Here, we derive wrinkling phase diagrams of growing bilayers with E_f/E_s = 1.5 − 50, which is in the range of most biologically observed bilayers for different film/substrate growth ratios G_s/G_f = 0 − 10. Our analysis is based on a hyperelastic, plane strain finite element model which is solved using a generalised path-following algorithm (numerical continuation) to explore the bifurcation structure beyond the first wrinkling instability and to trace critical boundaries. Phase diagrams are constructed in the parameter space of E_f/E_s-G_s/G_f-g_p, where g_p is a growth parameter, and these diagrams break the design space into flat, sinusoidally wrinkled, period doubling/quadrupling, and creasing regimes. In addition, we uncover a relation between E_f/E_s and G_s/G_f that defines the boundary of supercritical to subcritical wrinkling (i.e. onset of Biot wrinkling).