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.

Simple and extensible plate and shell finite element models through automatic code generation tools

“…In both cases we use the PSRI discretisation. As expected [58], the results confirm that Marguerre-von Kármán model can safely be used to approximate the Naghdi model for sufficiently small loadings, namely for curvatures of the order of a 2 /t. More surprisingly, for the present test the discrepancies with respect the fully nonlinear model remain tolerable also for k 50 a 2 /t.…”

“…Starting from the most general model, the nonlinear Naghdi shell model [71], the linear Reissner-Mindlin plate model [80,70] is obtained as a special case of the linear Naghdi shell model [72]. In passing, we present also the so-called Marguerre-von Kármán shallow shell model [67], because (in its unshearable version) it is extensively used in the literature to study nonlinear phenomena in plates and shallow shells [65,58,12]. We review below the basic notations used in this work.…”

The Monge–Ampère constraint: Matching of isometries, density and regularity, and elastic theories of shallow shells

“…For shells of thickness h and depth h α , subject to applied forces that scale with the shell depth as h α+2 , as the thickness of the shell h → 0, depending on the choice of α, various limiting theories arise. The regimes α = 1 and α > 1 can be treated in a similar manner as discussed in another context in [20] and are not of interest to us in this paper.…”

“…which reduces the problem to studying deformations of the flat plate Ω h relative to the prestrain tensor a h , see [19,20] for a discussion of this topic.…”

“…when 0 < α < 1, so that nearly isometric deformations might be expected, but implemented via nonlinear constraints, rather than linear ones as appearing in the α ≥ 1 regimes. In the vanishing thickness limit, using the basic methods of Γ-convergence which were developed in this context in [9,10], we derive a new thin film model (see also previous results in [19,20]) corresponding to a situation where the second order infinitesimal isometries on a shallow shell reference configuration are given by the out-of-plane displacement h α v 0 . In analogy with the results in [10] for plates with flat geometry v 0 = 0 in the energy scaling regime 2 < β < 4, we recover a Monge-Ampère constraint det ∇ 2 v = det ∇ 2 v 0 as a second order isometry constraint on the limiting displacements of Sobolev regularity W 2,2 .…”