Lewicka Marta, Mahadevan L, Pakzad Mohammad Reza
Department of Mathematics , University of Pittsburgh , 301 Thackeray Hall, Pittsburgh, PA 15260, USA.
School of Engineering and Applied Sciences , Harvard University , Cambridge, MA 02138, USA ; Department of Physics , Harvard University , Cambridge, MA 02138, USA.
Proc Math Phys Eng Sci. 2014 May 8;470(2165):20130604. doi: 10.1098/rspa.2013.0604.
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 non-trivial curvatures, i.e. growing elastic shells in both the weakly and strongly curved regimes, generalizing earlier results for the growth of nominally flat plates. The different theories are distinguished by the scaling of the mid-surface curvature relative to the inverse thickness and growth strain, and also allow us to generalize the classical Föppl-von Kármán energy to theories of prestrained shallow shells.
诸如树叶、花朵、羽毛、翅膀等薄片状物体的三维形状是由相对生长引起的微分应变驱动的。生长通过薄壳上给定的黎曼度量的变化来实现，该度量是作为中心平面位置及其厚度的函数。然后，形状是受挫几何物体上弹性能最小化的结果。在这里，我们对具有非平凡曲率的残余应变薄片状物体（即弱弯曲和强弯曲状态下生长的弹性壳）的形状渐近理论进行了严格推导，推广了早期关于名义上平板生长的结果。不同的理论通过中面曲率相对于厚度倒数和生长应变的缩放来区分，并且还使我们能够将经典的弗普尔 - 冯·卡门能量推广到预应变浅壳理论。
