Shell theories arising as low energy Gamma-limit of 3d nonlinear elasticity

Abstract

We discuss the limiting behavior (using the notion of -limit) of the 3d nonlinear elasticity for thin shells around an arbitrary smooth 2d surface. In particular, under the assumption that the elastic energy of deformations scales like h 4 , h being the thickness of a shell, we derive a limiting theory which is a generalization of the von Kármán theory for plates.

Models for elastic shells with incompatible strains

“…It follows (via Korn's inequality) that for a flat plate S ⊂ R 2 , the space B consists precisely of symmetrized gradients of all the in-plane displacements: B = {sym ∇w; w ∈ W 1,2 (S, R 2 )}. When S is strictly convex, rotationally symmetric or developable without flat regions, it has been proved in [13,24] that B = L 2 (S, R 2×2 sym ), i.e. it contains all symmetric matrix fields on S with square integrable entries.…”

“…The strains B h as in (3.8) The proofs follow through a combination of arguments in [10,13], which we do not repeat here but instead comment on the functional (3.3) and its relationship with the prestrained von Kármán equations for plates.…”

“…We now present the main Γ -convergence result for the shallow shell regime 0 < α < 1. The proofs which consist of tedious modifications of the arguments in [10,13] are outlined in the electronic supplementary material, appendix. 4 , where I γ ,h is given as in (2.5).…”

“…elastic plates with no initial curvature, using analogies to thermoelasticity [7,8], perturbation analysis [3,9] and rigorous asymptotic analysis [10]. This follows a programme similar to the derivation of the equations for the nonlinear elasticity of thin plates and shells [11][12][13][14][15][16] and a linearized theory [17] for residually strained Kirchhoff plates [18]. However, most laminae are naturally curved in their strain-free configurations.…”

“…Finally, in §8, we conclude with a discussion of the present results and prospects for the future. As the proofs of the theorems consist of tedious yet minor (though necessary) modifications of the arguments detailed in [10,13,14], we refer the interested reader to the electronic supplementary material, where they are given for completeness.…”