Foppl-vonKarman板理论中皱曲薄板的能量

Energy scaling law for a single disclination in a thin elastic sheet

We consider a single disclination in a thin elastic sheet of thickness $h$. We prove ansatz-free lower bounds for the free elastic energy in three different settings: First, for a geometrically fully non-linear plate model, second, for three-dimensional nonlinear elasticity, and third, for the Föppl-von Kármán plate theory. The lower bounds in the first and third result are optimal in the sense that we find upper bounds that are identical to the respective lower bounds in the leading order of $h$.

Variational problems in thin elastic structures

For two different scenarios regarding thin elastic structures, described by 2d-Föppl-von Kármán plate models, we obtain energy scaling laws. Firstly, assuming the reference geometry being that of a singular excess-cone, we obtain fairly optimal upper- and lower energy bounds and we highlight how those bounds scale wrt. the thickness-parameter $h.$ Secondly, we consider the half sphere, while being indented by a thin object at the top and perpendicular to its surface. In this situation we provide an energy-scaling law, for radial symmetric admissible maps, this time, depending on $h$ and the in…

通过脊线连接向错

Connecting disclinations by ridges

本文在Föppl-von Kármán近似下研究薄弹性片向错问题，证明凸位移假设下极小值在向错间形成脊线，并给出匹配至对数因子的能量标度律及Monge-Ampère测度单调性推广。

拉梅问题中的褶皱：一个Γ-收敛方法

Wrinkling in the Lamé problem: a $Γ$-convergence approach

本文基于拉梅问题分析，研究薄弹性环在径向拉伸下的褶皱模式，通过Γ-收敛方法分析渐进行为，并建立重新缩放后的能量函数的Γ-收敛结果，同时证明了极限功能的极小值的存在性和性质。

Dimension reduction for thin films prestrained by shallow curvature

We are concerned with the dimension reduction analysis for thin three-dimensional elastic films, prestrained via Riemannian metrics with weak curvatures. For the prestrain inducing the incompatible version of the Föppl-von Kármán equations, we find the $Γ$-limits of the rescaled energies, identify the optimal energy scaling laws, and display the equivalent conditions for optimality in terms of both the prestrain components and the curvatures of the related Riemannian metrics. When the stretching-inducing prestrain carries no in-plane modes, we discover similarities with the previously describe…

Models for elastic shells with incompatible strains

The three-dimensional shapes of thin lamina such as leaves, flowers, feathers, wings etc, are driven by the differential strain induced by the relative growth. The growth takes place through variations in the Riemannian metric, given on the thin sheet as a function of location in the central plane and also across its thickness. The shape is then a consequence of elastic energy minimization on the frustrated geometrical object. Here we provide a rigorous derivation of the asymptotic theories for shapes of residually strained thin lamina with nontrivial curvatures, i.e. growing elastic shells in…

Blister patterns and energy minimization in compressed thin films on compliant substrates

This paper is motivated by the complex blister patterns sometimes seen in thin elastic films on thick, compliant substrates. These patterns are often induced by an elastic misfit which compresses the film. Blistering permits the film to expand locally, reducing the elastic energy of the system. It is natural to ask: what is the minimum elastic energy achievable by blistering on a fixed area fraction of the substrate? This is a variational problem involving both the {\it elastic deformation} of the film and substrate and the {\it geometry} of the blistered region. It involves three small parame…

Plates with incompatible prestrain

We study the effective elastic behavior of incompatibly prestrained plates, where the prestrain is independent of thickness as well as uniform through the thickness. We model such plates as three-dimensional elastic bodies with a prescribed pointwise stress-free state characterized by a Riemannian metric $G$ with the above properties, and seek the limiting behavior as the thickness goes to zero. Our results extand the prior analysis in M. Lewicka, M. R. Pakzad ESAIM Control Optim. Calc. Var. 17 (2011), no. 4. We first establish that the $Γ$-limit is a Kirchhoff type bending. Further, we show t…

The coarsening of folds in hanging drapes

We consider the elastic energy of a hanging drape -- a thin elastic sheet, pulled down by the force of gravity, with fine-scale folding at the top that achieves approximately uniform confinement. This example of energy-driven pattern formation in a thin elastic sheet is of particular interest because the length scale of folding varies with height. We focus on how the minimum elastic energy depends on the physical parameters. As the sheet thickness vanishes, the limiting energy is due to the gravitational force and is relatively easy to understand. Our main accomplishment is to identify the "sc…