This section contains an outline of the theory and results of experimental studies of the elastic properties of nematics. First, a short introduction of the standard theories is given and the characteristic quantities, used to describe nematic phase elasticity are introduced. After an overview of the standard methods of measuring elastic constants, a summary of the experimental results is given. In particular, we list a collection of papers dealing with the extensively explored cyanobiphenyls and the standard substance 4-methyloxy-4'-butylbenzylideneaniline (MBBA). The next part is devoted to the less-common surface-like elastic constants, and this is followed by a sketch of the theoretical approaches to the microscopic interpretation of elastic constants and the Landau-de Gennes expansion. The section . is concluded by a brief discussion of elastic theory for biaxial nematic phases. Spatial elastic distortions of the nematic director field change the free energy of the mesophase. The relations between the spatial derivatives of the director field and the free energy density of the nematic phase are described by the elastic moduli. In de-veloping an elastic theory of the nematic mesophase, two different approaches were chosen.The phenomenological theory starts from phase symmetry considerations. The free energy is expanded in terms of combinations of director derivatives, which leave the free energy invariant under the symmetry operations of the phase. For each independent term in this expansion, an elastic modulus is introduced.The molecular theory is similar to Cauchy's description of the elastic theory of solids [ 11 and utilizes additive local molecular pair interactions to describe elasticity. The latter approach was taken by Oseen [2], who was the first to establish an elastic theory of anisotropic fluids. Oseen assumed short-range intermolecular forces to be the reason for the elastic properties, and he derived eight elastic constants in the expression for the elastic free energy density of uniaxial nematic phases. Finally, he retained only five of them, which enter the Euler-Lagrange equations describing equilibrium deformation states of the nematic mesophase, and omitted the other three.Oseen's elastic theory was developed further by Zocher [3] and re-examined by Frank [4] in a phenomenological approach.
