Elastic response and Ward identities in stressed nematic elastomers

Nematic elastomers exhibit a rich elastic response to external stresses. Of particular interest is the semisoft response of elastomers with an anisotropy direction $\left(z\right)$ frozen in by a double cross-linking process. This response is characterized by a stress-strain curve for stresses along $x$ perpendicular to $z$ that rises initially, exhibits a nearly flat plateau between two critical values of strain, and then rises again. This paper explores elastic response in semisoft elastomers as a function of externally applied strain. It derives general Ward identities for elastic moduli and shows that the elastic modulus measuring response to $xz$ shears vanishes at the boundaries of the semisoft plateau whereas moduli measuring response to shears perpendicular to the $xz$ plane do not. It then calculates all relevant moduli in a simple model of elastomers and verifies the general Ward-identity predictions.

Figure 1

Schematic of the engineering or first PK stress as a function of stretch ${\overline{\Lambda }}_{xx}$ showing the nearly flat semisoft plateau between critical values ${\overline{\Lambda }}_{-}$ and ${\overline{\Lambda }}_{+}$. In the plateau region, ${\Lambda }_{xz}$ is nonzero and samples adopt a polydomain structure [9], determined by boundary conditions, with coexisting regions with opposite signs for ${\Lambda }_{xz}$. The double arrows in this figure indicate the direction of the principal strain anisotropy axis.

Figure 2

Geometry for rheology measurements of elastic moduli in films of nematic elastomers. $z$ is the direction of initial nematic anisotropy and $x$ the direction of applied stress. In the parallel geometry of figure (a), the nematic director lies in the plane of the film. This configuration allows measurement of the moduli $K_{xyxy}^{\prime }$ and $K_{zyzy}^{\prime }$. In the perpendicular geometry of figure (b), the director lies normal to the plane of the film. This configuration allows measurement of $K_{xzxz}^{y\sigma }$ and $K_{yzyz}^{\prime }$. Clamps used to prevent rotations about axes in the plane of the films force deformations ${\Lambda }_{yx}^{\prime }$ and ${\Lambda }_{yz}^{\prime }$ to be zero in (a) and ${\Lambda }_{zy}^{\prime }$ and ${\Lambda }_{zx}^{\prime }$ to be zero in (b). In addition, boundary conditions at the upper and lower surfaces prevent deformations in the plane of the films, i.e., ${\Lambda }_{xz}^{\prime }={\Lambda }_{zx}^{\prime }=0$ in (a) and ${\Lambda }_{xy}^{\prime }={\Lambda }_{yx}^{\prime }=0$ in (b).

Figure 3

Elastic moduli $K_{xyxy}^{\prime }$, $K_{zyzy}^{\prime }=K_{yzyz}^{\prime }$, and $K_{xzxz}^{y\sigma }$ as a function of ${\overline{\Lambda }}_{xx}$. $K_{xyxy}^{\prime }$, and $K_{zyzy}^{\prime }$ are measured at fixed ${\underset{͇}{\Lambda }}^{\prime }$. $K_{xzxz}^{y\sigma }$ is measured at ${\sigma }_{yy}^I=0$ and all deformations except ${\Lambda }_{yy}^{\prime }$ at their equilibrium value. These curves were calculated using the minimal model discussed in Sec. V with $\tilde{r}=0.08$, $\tilde{h}=0.05$ and $w/v=1$. The unit of these elastic constants is $w$.

Figure 4

Elastic moduli $K_{xzxz}^{\sigma }$, $K_{yzyz}^{\sigma }$, $K_{xyxy}^{\sigma }$, and $K_{zyzy}^{\sigma }$ as a function of ${\overline{\Lambda }}_{xx}$. These moduli are measured at ${\sigma }_{xx}^I$ fixed and all other stress except those fixing rotational and in-plane deformations [See Fig. 2] equal to zero. These curves were calculated using the minimal model discussed in Sec. V with $\tilde{r}=0.08$, $\tilde{h}=0.05$ and $w/v=1$. The unit of these elastic constants is $w$. Note that on the stress plateau $K_{zyzy}^{\sigma }$ is different from $K_{yzyz}^{\sigma }\left(=K_{yzyz}^{\prime }\right)$.

Figure 5

(a) ${\mathring{K}}_{zxzx}^{\prime }$ and (b) ${\mathring{K}}_{yxyx}^{\prime }$, in units of $w$ as a function of ${\overline{\Lambda }}_{xx}$, for $\tilde{r}=0.08$, $\tilde{h}=0.05$ and $w/v=1$. These moduli measure rotations away from the $x$ direction defined by ${\sigma }_{xx}^I$ and have a ${\sigma }_{xx}$ contribution. Thus, at the plateau boundaries, $K_{zxzx}^{\prime }$ is equal to ${\sigma }_{xx}^{\prime }$ rather than zero.