Further Theoretical Proofs for the Existence of a Relation Between the Elastic Constants K'11 (K11), (K'33)(K33) and K13 in Nematics: Three- and One-Dimensional Cases

The elaboration of the Ericksen's result for the inequalities in nematics has confirmed the proportionality between the second-order splay-bend elastic constant K13 and the difference between the elastic constants of bend K'33 and splay K'11 as well as the sign of K13 in accordance with the relation 2K13 = K'33 - K'11 obtained previously (Mol. Cryst. Liq. Cryst., 168, 7, 1989). On the basis of detailed theoretical calculations for the one-dimensional case it is demonstrated that this relation follows from the require- ment for a positive value of the elastic energy expressed with extremals and gradients of extremals when all possible symmetric and antisymmetric solutions are included.