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

132 CHAPTER 5. SOFT ELASTICITY OF SMECTIC ELASTOMERSUsingthisexpressionforξ all √thecomponentsofthedeformation tensorcanbeobtained. Defining a(φ) = cos 2 φ+ ρ r sin2 φ produces the following matrixfor λ⎛(r−1)sin2θ⎞a(φ) 02ρ(−a(φ)+cosφ)⎜( ⎝ 1−ρ) (sin2φ 1 (r−1)sin2θr 2a(φ) a(φ) 2ρsinφ− ( 1− ρ ) )sin2φ ⎟⎠r 2a(φ)0 0 1This tensor is explicitly constructed to be a soft mode and evidently hasdet[λ] = 1. To illustrate this mode Fig. 5.5 shows how this sample deformsfor various different azimuthal angles, φ. The figure gives a view of a blockof SmC rubber down the layer normal and should be compared with Fig. 5.2.Note that even after a rotation of the director of φ = π the rubber does notFigure 5.5: An illustration of the soft mode of a SmC elastomer.In this case the layer normal remains out of the pageand the c direction together with φ is shown in the centre ofthediagram. A tilt angle of θ = 30 ◦ and an anisotropy of r = 8were chosen.return to its original configuration. Because of the tilt of the director w.r.t.the layer normal a strain λ xz < 0 is generated after φ → π and this componenthas a cosφ term. By contrast λ yx = λ yz = 0 and λ yy = 1 at φ = π; indeedλ yz depends on 2φ. At the intermediate value of φ = π/2 the elastomer hascontracted along the direction of the original anisotropy tensor and so hasdeveloped both a λ xz and λ yz components of shear, that is with displacementsin both the x and y directions. The maximum extension in the y directionoccurs at φ = π/2, when the λ yy component takes the value √ r/ρ. For thecase with θ = 30 ◦ and r = 2 this gives a maximum extension of roughly 7%.