Deformation of an asymmetric thin film

Abstract

Experiments have investigated shape changes of polymer films induced by asymmetric swelling by a chemical vapor. Inspired by recent work on the shaping of elastic sheets by non-Euclidean metrics [Y. Klein, E. Efrati, and E. Sharon, Science 315, 1116 (2007)], we represent the effect of chemical vapors by a change in the target metric tensor. In this problem, unlike that earlier work, the target metric is asymmetric between the two sides of the film. Changing this metric induces a curvature of the film, which may curve into a partial cylinder or a partial sphere. We calculate the elastic energy for each of these shapes, and show that the sphere is favored for films smaller than a critical size, which depends on the film thickness, while the cylinder is favored for larger films.

Contrasting bending energies from bulk elastic theories

“…When used in concert with our adoption above of the traditional transpose notation for the deformation gradient (the latter is unfortunately entrenched in continuum mechanics, and serves to indicate which tensor legs the inner product refers to), one may get the mistaken impression that it might make sense to write transposes of these symmetric tensors. To be consistent, one should either replace the transpose notation, or symmetrize the bases in expressions such as (1)(2)(3)(4)(5)(6)(7)(8). We emphasize that these expressions are not general, but work for our restricted situation.…”

“…Much recent work in soft matter mechanics has explored the interplay of stretching and bending elasticity in incompatible plates and shells composed of isotropic gels or elastomers, or nematic solids [1][2][3][4][5]. This subject has inspired, among other things, fundamental questions about embeddings [6][7][8], design principles for shape-programmable materials [9][10][11][12], and insight into the complex shapes of leaves and torn plastic trash bags [13,14].…”

Some observations on variational elasticity and its application to plates and membranes

“…It will be shown in Section III B and a separate publication [15] that constitutive choices have even more marked qualitative effects on derived bending elasticity. In particular, the Green form of the bulk energy favored in some recent works [12,14,50] appears to generate bending energies in qualitative contrast with intuitively sensible definitions even at moderate strains. This paper is restricted to the relatively simple case of isotropic compatible or incompatible elasticity as considered by [12] and related works, where it is assumed that certain quantities, such as metrics, can still be defined even if no reference configuration exists in E 3 .…”

“…It will be shown in Section III B and a separate publication [15] that constitutive choices have even more marked qualitative effects on derived bending elasticity. In particular, the Green form of the bulk energy favored in some recent works [12,14,50] appears to generate bending energies in qualitative contrast with intuitively sensible definitions even at moderate strains.…”

“…In keeping with a primary audience in physics, I will assume basic knowledge of differential geometric symbols and index gymnastics in the early 20th century style of "absolute differential calculus", but no background in continuum mechanics. I will follow similar styles of notation as found in the older elasticity literature [4,7], some later mechanics literature derived therefrom [8][9][10][11], and recent physics literature [12][13][14], but will introduce a bit of new notation in an attempt to avoid ambiguity in certain common expressions. Even within this stylistic slice of the literature, it is impossible to be consistent with everybody, as things like capitalization or barring of quantities are used in opposing ways by different authors.…”