Remarks on a possible elasticity of membranes and lamellar media: disordered layers

Abstract
In this article the implications of an elasticity of layered media are considered from the following assumptions. For each state of deformation of the lamella, however large the variations in curvature radii might be, the state of lowest energy is defined by the fact that the constitutive molecules are separated in the mid-surface Ʃ of the ‘shell’ by distances equal to those of the planar situation. We call such a state, in which one moreover assumes that the molecules are perpendicular to Ʃ , a state of inextension . The energy of the states of inextension, as well as the (higher) energies of states of stretching of Ʃ and tilting of the molecules are calculated assuming a Hookean isotropic elasticity (amorphous layer). The hypothesis of inextension is reminiscent of a two dimensional liquid-like behaviour, but different. Like this latter behaviour, inextension leads to a term of splay in the free energy. But a striking difference appears when one considers the modes of deformation of the shell which preserve the distances between molecules constant in Ʃ . In the elasticity considered, these modes obey a condition of constant Gaussian curvature, which is stronger than (and implies) the usual condition of incompressibility. Stretched and tilted states are only qualitatively discussed. The equations of equilibrium are obtained in the framework of the classical theory of strongly deformed shells. The results apply to isolated layers (membranes) and to smectic and lyotropic phases, if one introduces suitable interaction forces between layers. Among the shapes of equilibrium of such layers in the inextended state, one finds minimal surfaces, surfaces of revolution and Dupin cyclides. In a confocal domain, only one Dupin cyclide satisfies the equations of equilibrium in the inextended state, whereas the other ones suffer slight distortions of different nature in the concave and the convex parts of the cyclide.

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