Buckling transition and boundary layer in non-Euclidean plates

Abstract

Non-Euclidean plates are thin elastic bodies having no stress-free configuration, hence exhibiting residual stresses in the absence of external constraints. These bodies are endowed with a threedimensional reference metric, which may not necessarily be immersible in physical space. Here, based on a recently developed theory for such bodies, we characterize the transition from flat to buckled equilibrium configurations at a critical value of the plate thickness. Depending of the reference metric, the buckling transition may be either continuous or discontinuous. In the infinitely thin plate limit, under the assumption that a limiting configuration exists, we show that the limit is a configuration that minimizes the bending content, amongst all configurations with zero stretching content (isometric immersions of the mid-surface). For small but finite plate thickness we show the formation of a boundary layer, whose size scales with the square root of the plate thickness, and whose shape is determined by a balance between stretching and bending energies.

Shape selection in non-Euclidean plates

“…4 and 5) there exists smooth isometric immersions x : D R → R 3 for geodesic disks of arbitrarily large radius R. Moreover, in elucidating the connection between this existence result and the non-existence results above, we provided numerical evidence that the maximum principal curvature of such immersions is bounded below by a bound which grows exponentially in √ −K 0 R. The existence of smooth, and thus also W 2,2 immersions of arbitrarily large disks have consequences for the modeling of free non-Euclidean sheets. In particular, the minimizers of the elastic energy (2.1) must converge as t → 0 to a minimizer of (2.2) [16,17], and the energy of the minimizer x * is bounded from above by…”

“…Instead of modeling non-uniform growth as isometric immersions, a "non-Euclidean" elastic energy functional has been recently introduced that incorporates g and reduces to the FvK functional when g is the identity matrix [16]. This functional has the same energy scaling as FvK with the stretching energy measuring L 2 deviations of the immersion from g. As with the FvK functional, if the thickness is sufficiently small the stretching energy dominates and we expect the elastic sheet to bend out of the plane to match the metric.…”

Curvature-Induced Instabilities of Shells

“…Then, as the boundary layer is bending dominated [27], we obtain |q −Ψ/R| ∼ κ √ Rh [24], where √ Rh is the characteristic width of the boundary layer [20]. As the perimeter of the boundary is proportional to R sin θ, we conclude that the edge work (2b) scales as W edge ∼ h 4 κ 2 (R/h) 3/2 sin θ.…”

Curvature-Dependent Elastic Properties of Liquid-Ordered Domains Result in Inverted Domain Sorting on Uniaxially Compressed Vesicles

“…Given the generality of the equations, other relevant applications can be envisaged for solving problems of tissue growth and remodeling. Introduction.-Soft tissue growth models in ideal geometries have shown shape instabilities with a special focus on morphogenesis of living systems [1][2][3]. Anisotropic or inhomogeneous growth and constraints due to boundaries are responsible for these shape instabilities.…”