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

100 CHAPTER 4. THE ELASTICITY OF SMECTIC-A ELASTOMERSOn elimination the free energy density is given by{f = 1 2 µ 1+rλ 2 xz +λ2 +λ 2 zx + 1λ 2 zx +λ 2 + 2rλ xzλ zx√λλ 2 2 zx +λ 2+ λ2 zxr ( 1+λ 2 )}xzλ 2 . (4.79)This can be minimized w.r.t λ xz and λ zx . The first of these yields the expressionλ zxλ xz = −√ (4.80)λ2 zx +λ2. On substituting this into the expression obtained after minimisation w.r.t λ zxwe obtain the following equationλ 2 λ zx((−1+λ2zx +λ 2) √λ 2 zx +λ 2( 1+λ 2 zx +λ2) = 0. (4.81)From this equation it is clear that the following real solutions for λ zx areobtainedλ zx = ± √ 1−λ 2 ,0 (4.82)The first of these solutions is only valid for λ < 1 and gives λ xz = −λ zx ,λ yy = 1, λ zz = λ. In this solution λ corresponds to a rotation about the yaxis. This solution is disregarded here and the second, λ zx = 0, is examined.From this result it follows that the director does not rotate when a stretch inthe plane of the layers is applied. The remaining components in this case, andthe free energy density are given byλ yy = 1 (4.83)λλ zz = 1 (4.84){f = 1 2 µ λ 2 + 1 }λ 2 +1 . (4.85)The Poisson ratios of the material in this configuration are of interest experimentally.In this case the Poisson ratios are (1,0) in the (y,z) directionsrespectively. Note that since the material is incompressible the sum of thePoisson ratios must be 1.Imposed λ xzFor an imposed λ xz , as shown in Fig. 4.9 c), it is not immediately clear howgeneral a deformation matrix is required. Initially an upper triangular deformationmatrix is considered here, and followed by a more general type ofdeformation matrix ⎛ ⎞λ xx 0 λλ = ⎝ 0 λ yy 0 ⎠. (4.86)0 0 λ zz