title: Models for elastic shells with incompatible strains.
abstract: 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.
authors.affiliation: Department of Mathematics , University of Pittsburgh , 301 Thackeray Hall, Pittsburgh, PA 15260, USA.
authors.forename: Marta
authors.initials: M
authors.lastname: Lewicka
authors.affiliation: School of Engineering and Applied Sciences , Harvard University , Cambridge, MA 02138, USA ; Department of Physics , Harvard University , Cambridge, MA 02138, USA.
authors.forename: L
authors.initials: L
authors.lastname: Mahadevan
authors.affiliation: Department of Mathematics , University of Pittsburgh , 301 Thackeray Hall, Pittsburgh, PA 15260, USA.
authors.forename: Mohammad Reza
authors.initials: MR
authors.lastname: Pakzad
keywords: Gamma convergence; calculus of variations; non-Euclidean plates; nonlinear elasticity