  Smectic liquid crystals can be viewed as model systems for lamellar
structures for which there has been extensive theoretical development. We
demonstrate that a nonlinear energy description is required with respect to the
usual Landau-de Gennes elasticity in order to explain the observed layer
spacing of highly curved smectic layers. Using X-ray diffraction we have
quantitatively determined the dilation of bent layers distorted by antagonistic
anchoring (as high as 1.8% of dilation for the most bent smectic layers) and
accurately described it by the minimal nonlinear expression for energy. We
observe a 1{\deg} tilt of planar layers that are connected to the curved
layers. This value is consistent with simple energetic calculations,
demonstrating how the bending energy impacts the overall structure of a thin
distorted smectic film. Finally, we show that combined X-ray measurements and
theoretical modeling allow for the quantitative determination of the number of
curved smectic layers and of the resulting thickness of the dilated region with
unprecedented precision.
