Curvature elasticity the Oseen-Zocher-Frank equations

2 Curvature elasticity the Oseen-Zocher-Frank equations 

We have shown that for an incompressible fluid and an isothermal 

We may therefore expand the free energy per unit volume of a deformed liquid crystal relative to that in the state of uniform orientation as 

We neglect higher order terms in the expansion as we are concerned only with infinitesimal deformations. Since these deformations relate to changes in the orientation of the director, may appropriately be called the curvature strain tensor (Frank ). In order to define the components of this tensor, let us choose a local right-handed system of cartesian coordinates with n parallel to z at the origin. The components of strain are then 

We ignore dnjdx, nj y and dnjdz. The three types of deformation are shown in fig. 2.3.12. The curvature strain tensor may therefore be written as 

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