Metric-induced Morphogenesis and Non-Euclidean Elasticity: Scaling Laws and Thin Film Models

Shapes of leaves with parallel venation. Modelling of theEpipactissp. (Orchidaceae) leaves with the help of a system of coupled elastic beams

“…What is more, those authors have described the effect of applying auxin at leaf edges to produce ripples and emphasize more explicitly growth in the leaf. It is conjectured that in the developmental processes of naturally growing tissues, the process of growth provides a mechanism for the spontaneous formation of non-Euclidean metrics and consequently leads to complicated morphogenesis of thin films (including surfaces of leaves) exhibiting waves, ruffles and non-zero residual stress ( Lewicka , 2011 ). What we have attempted here is to provide a more physical, granular model which is from the very beginning adapted to the strong anisotropy of Epipactis leaves and concerned with local forces and torques.…”

Convex integration for the Monge–Ampère equation in two dimensions

“…When k = 1 then E h (u) models the elastic energy (per unit thickness) of the deformation u of a thin three-dimensional film with midplate ω and thickness 2h and prestrained by g h . Questions on the asymptotics of minimizing configurations to E h as h → 0, in function of the scaling exponent β in: inf E h ∼ Ch β , received a lot of attention, particularly via techniques of dimension reduction and Γ-convergence, starting with the seminal paper [5], see also [8] and references therein. (i) if γ ≥ 4, then inf E h ≤ Ch β , for every β < 2 + γ 2 , (ii) if γ ∈ 4k 3k+4 , 4 , then inf E h ≤ Ch β for every β < 4k+γ(k+4) 2k+4 , (iii) if γ ∈ 0, 4k 3k+4 , then inf E h ≤ Ch 2γ .…”

Peer Review #1 of "Shapes of leaves with parallel venation. Modelling of the Epipactis sp. (Orchidaceae) leaves with the help of a system of coupled elastic beams (v0.2)"

“…What is more, those authors have described the effect of applying auxin at leaf edges to produce ripples and emphasize more explicitly growth in the leaf. It is conjectured that in the developmental processes of naturally growing tissues, the process of growth provides a mechanism for the spontaneous formation of non-Euclidean metrics and consequently leads to complicated morphogenesis of thin films (including surfaces of leaves) exhibiting waves, ruffles and non-zero residual stress (Lewicka, 2011). What we have attempted here is to provide a more physical, granular model which is from the very beginning adapted to the strong anisotropy of Epipactis leaves and concerned with local forces and torques.…”