Figure 1

(a) Smectic-$A$ phase with parallel director $\hat{\mathbf{n}}$ and layer normal $\hat{\mathbf{k}}$. The polymer backbones are for the polymeric case with the rods shown as pendant. A cross-link (dots at either end of the rod) links polymers into the network. (b) Smectic-$C$ phase with $\hat{\mathbf{n}}$ tilted by $\theta$ in a direction defined by the unit vector $\hat{\mathbf{c}}$ in the plane. The third (unit) direction, $\hat{\mathbf{p}}=\hat{\mathbf{k}}\wedge \hat{\mathbf{c}}$, is into the page. It defines the direction of polarization, $\mathbf{P}=P\hat{\mathbf{p}}$. For Sm-$C$ elastomers, there is a spontaneous shear $\Lambda$ with respect to its Sm-$A$ parent. Photographs (c) and (d) [6] are of the $A$ to $C$ transition in elastomers. The cut out in (d) is the Sm-$C$ sample in Fig. 2a that is to be deformed in this paper.Reuse & Permissions

Figure 2

A block of Sm-$C$ rubber (a) initially undistorted with ${\hat{\mathbf{n}}}_0$ and ${\hat{\mathbf{c}}}_0\equiv \hat{\mathbf{x}}$. The layer normal will be taken to remain along $\hat{\mathbf{z}}$. (b) sheared by ${\lambda }_{xz}=-2\Lambda$ with respect to its relaxed state. The in-plane director has reversed, $\hat{\mathbf{c}}\to -{\hat{\mathbf{c}}}_0$ and thus so has the polarization direction: ${\hat{\mathbf{p}}}_0\to \hat{\mathbf{p}}=-{\hat{\mathbf{p}}}_0$.Reuse & Permissions

Figure 3

(a) The elastic free energy, in units of $\frac{1}{2}\mu$, against imposed shear ${\lambda }_{xz}\le 0$. The dotted line is energy cost when allowing $yx$ relaxation as well. (b) Rotation $\varphi$ of the in-plane director $\hat{\mathbf{c}}$ about the layer normal and the concomitant $yz$ shear relaxation, both starting and concluding at thresholds ${\lambda }_1$ and ${\lambda }_2$, respectively. The anisotropy is $r=2$ and the director tilt is $\theta =\pi /6$.Reuse & Permissions

Figure 4

The $yx$ and $yz$ shears and in-plane director rotation $\varphi$ against imposed shear ${\lambda }_{xz}$ for the softer case where $yx$ relaxation is permitted ($r$ and $\theta$ as before). The light vertical dotted line is at ${\lambda }_{xz}=-\Lambda$ to emphasize the asymmetry about the point where the director is transverse; see Appendix .Reuse & Permissions

Figure 5

The $yx$ and $yz$ shears and the $xx$ contraction against imposed shear ${\lambda }_{xz}$ for soft deformations ($r$ and $\theta$ as before).Reuse & Permissions

Figure 6

Soft deformations of a cube of Sm-$C$ rubber with $r=8$ and $\theta =\pi /6$ (as before) viewed along the smectic layer normal. The $\mathbf{c}$ vectors and the corresponding $\varphi$ are shown.Reuse & Permissions

Figure 7

(a) Sheet and (b) slab geometries for Sm-$C$ elastomers. Rigid clamps (shaded) are shown on the $xy$ sample surfaces. ${\lambda }_{xz}$ shear displacements are indicated by arrows. A few smectic planes are indicated in each case, with normals ${\hat{\mathbf{k}}}_0$. polarization $\mathbf{P}$ in the $y$ direction (of extent $w$) emerges from the $xz$ sample surfaces. Lines of electric flux in the slab case are captured by enveloping the small $xz$ surfaces by overhanging, split $xy$ electrodes, their charge being identified by $\pm$.Reuse & Permissions

Figure 8

A slice through a solid deforming with a textured version of Eq. (4) for $\underset{͇}{\lambda }$. Alternating $\pm {\lambda }_{yz}$ shears lead to no overall macroscopic ${\lambda }_{yz}$ but to the desired imposed ${\lambda }_{xz}$.Reuse & Permissions

Figure 9

(Color online) An initially cubic sample deforming softly with laminates of $yx$ and $yz$ shears of alternating sign and with $xz$ shear advancing from 0 to $-2\Lambda$ in steps of $\pm \pi /3$ in director rotation $\varphi$ about the $z$ axis (marked on figures; see also the $\mathbf{c}$ relevant to each laminate as it emerges at the top face of the sample). The smectic layers (a few dotted on the $\varphi =0$ snapshot) are unrotating as $\varphi$ changes and retain their normal ${\hat{\mathbf{k}}}_0=\mathbf{z}$. The alternating shear deformations lead to no corresponding macroscopic shears whereas the imposed ${\lambda }_{xz}$ does. The laminate normal ${\mathbf{s}}_{\varphi }^{\prime }$ in the deformed (target) state is shown for the example of $\varphi =\pi /3$. It starts and finishes parallel to $\hat{\mathbf{n}}$ but in general makes an angle $\chi$ with the layer normal ${\hat{\mathbf{k}}}_0$ given in the text. The deformations away from the initial shape (light reference frame given in $\varphi \ne 0$ pictures) reveal the slight contraction ${\lambda }_{xx}$ along $x$ and a compensatory lengthening along $y$, see, e.g., $\varphi =\pi /3$, for $\varphi \ne 0,\pm \pi$. The $\pm \pi /2$ case is like Fig. 8, that is, with $\chi =0$, but with ${\lambda }_{xx},{\lambda }_{yy}\ne 1$.Reuse & Permissions

Figure 10

A sample deforming by disproportionation. The macroscopic ${\lambda }_{xz}$ shear displacements are indicated by heavy arrows and consists of a fraction (lower) of weakly sheared ${\lambda }^{\left(h\right)}$ sample and the complementary (upper) fraction distorted by $-2\Lambda$. The macroscopic shear is ${\lambda }_{xz}$.Reuse & Permissions

Figure 11

(a) A sample with the layer normal at an angle $\beta$ to the long direction is cut out of a monodomain Sm-$C$. (b) The shaded region is that of elongations $\lambda$ for tension angles $\beta$ for which the sample remains soft (for $\theta =22.5°$ and $r=2$).Reuse & Permissions

Figure 12

Energy of deformation against both positive (hard) and negative (softer) shear ${\lambda }_{xz}$, and against extensions ${\lambda }_{xx}\ge 1$ (dotted curve; ${\lambda }_{xx}$ upper scale) up to strains of $\pm \Lambda =\pm 0.246$, for the values of $r$ and $\theta$ adopted in the text. Note how much lower the energy of distortion is if sympathetic relaxations and director rotation are permitted.Reuse & Permissions

Figure 13

The principal director, $\hat{\mathbf{n}}$, of the shape tensor is tilted with respect to the layer normal, $\hat{\mathbf{k}}$, by an angle $\theta$. Thereby the other two principal directions, ${\mathbf{p}}_1$ and ${\mathbf{p}}_2$, are distinguished to give a biaxial distribution and thus also a biaxial shape tensor $\underset{͇}{\ell }$, here seen in section.Reuse & Permissions