Covariant elasticity for smectics A

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Abstract
A theory of static deformations of smectics A is presented, taking into account the large curvature deformations of the layers, but assuming that the variation in the thickness of the layers is small. The director n(r) and the layer phase function Φ(r) (Φ is constant on a given layer) are chosen as independent variables, but they have to be related by a relation, to which corresponds a Lagrange multiplyer. Permeation is taken into account. With the help of these variables, the bulk and boundaries equilibrium equations can be expressed simply, and the physical meaning of the Lagrange multiplyer is straightforward. The formulation is applied to the case of the stability of a cylindrical structure. It is found that the cylinders are unstable versus certain helical and sinusoidal deformations