Statistical models of elasticity in main chain and smectic liquid ...

130 CHAPTER 5. SOFT ELASTICITY OF SMECTIC ELASTOMERSwhere the soft mode l −1/2n ′ ·l 1/20 is parametrised by the angle ξ and is independentof W R which can freely be set in order to obtain whatever final directoris desired, that is n = W R · n 0 . This general, final director is not confinedto the circle about w 0 , see Fig. 5.4 a) and b) for an illustration of the procedure.Having decided where the final director is to point, one then appliesa)b)w 0n 0k 0ξn’Rnn 0Figure 5.4: a) shows the first stage of the process: a rotationabout w 0 , b) shows the second stage: a general rotation aboutthe axis R.the body rotation W R · W w0 (ξ) to the softly deformed sample to completethe deformation Eq. (5.28).All of the rotations can be separated out of the soft mode leaving just asymmetric deformation by using the polar decomposition theoremλ = l 1/2n ′ ·l −1/20 = U ·S, (5.29)where U is a rotation matrix and S is a symmetric matrix. The rotation axisfor this decomposition must be in the n ′ ∧n 0 direction. This information canbe used to construct U T ·λ and demand that it is symmetric to find S. Theresulting rotation angle is given bytanα = (1−√ r) 2 n 0 ·n ′√ 1−(n 0 ·n ′ ) 2(1+r)−(n 0 ·n ′ ) 2 (1− √ r) 2 (5.30)Now a particular example of Eq. (5.28) is considered. Here rigid clampingconstraints are not included so there is no formation of microstructure.5.3.3 Example: Imposed λ yyTo illustrate the soft modes an elongation in any direction could be imposed,provided the director has scope to rotate into that direction and thereby to