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

4.3. FINITE DEFORMATION EXAMPLES 111a)b)11.20.810.80.60.4λ xxλ yyλ zzcos φ0 1 2 3 4λ zxd/d 00.60.40.200 1 2 3 4 5λ zxFigure 4.15: An example of the minimisation of the free energyin the case of imposed λ zx . a) The deformation tensorcomponents, b) the layer spacing. The example is for the caseb = 5, r = 3.4.3.4 Fundamental deformations for finite BNote that where as in §4.3.2 the deformation could be broken up into twodeformations, an imposed λ xx and an imposed λ xz plus a rotation, this is notpossible here. In this case we require a third deformation: a stretch along thelayer normal (λ zz ). The decomposition is now illustrated by the imposed λ zxcase as considered above.λ = W ·D (4.134)where we define⎛ ⎞ ⎛cosγ 0 sinγW = ⎝ 0 1 0 ⎠;D = ⎝−sinγ 0 cosγ1 0 λ xz0 1 00 0 1⎞⎛⎠⎝λ xx 0 0⎠.⎞10λ zzλ xx010 0λ zzThe deformation matrix is thus⎛λ xx cosγ 0⎞λ zz (λ xz cosγ +sinγ)λ = ⎝ 01λ xxλ zz0 ⎠. (4.135)−λ xx sinγ 0 λ zz (cosγ −λ xz sinγ)From this deformation matrix we can calculate the director( sinγλ −T ·n 0 = ,0, cosγ )λ zz λ zz(4.136)Note that in this case care must be taken to normalise the director. Therotation required to reduce this deformation to the form of an imposed λ zx