  The nonlinear elastic properties of nematic liquid crystals have acquired new
interest with the recent experimental observation of bulk modulated nematic
phases which are composed by achiral molecules. We extend the
Oseen-Zocher-Frank's elastic theory for nematic liquid crystals by including
gradients of the nematic strain tensor in the elastic deformation energy. The
invariants of the elastic tensor fields, up to the fourth order in the nematic
director spatial derivatives, are calculated. An alternative approach that
consists in the extension of the linear elastic energy to higher powers of the
nematic strain tensor, as in classical elasticity of solids, is also developed.
The twist-bend nematic modulated phase is investigated by both approaches and
the results are critically compared. The conical angle of the twist-bend phase
is calculated as function of the elastic constants. Surface-like effects are
considered. Finally, we demonstrate that a splay-bend nematic phase with small
oscillations of the nematic director around an axis is prohibited.
