| Abstract: |
| We compute the $\Gamma$-limit of energy functionals describing mechanical systems composed of a thin nematic liquid crystal elastomer sustaining a homogeneous and isotropic elastic membrane. We work in the regime of infinitesimal displacements and model the orientation of the liquid crystal according to the order tensor theories of both Frank and De Gennes. We describe the asymptotic regime by analysing a family of functionals parametrised by the thickness of the membranes and the relative ratio of the elastic constants,
establishing that, in the limit, the system is represented by a two-dimensional integral functional interpreted as a linear membrane on top of a nematic `active foundation` involving an effective De Gennes optic tensor which allows for low order states. The latter can suppress shear energy by formation of microstructure as well as act as a pre-strain transmitted by the foundation to the overlying film.
We then compute a phase diagram of the effective macroscopic behaviour of the homogeneous volume element and finally provide numeric computations solving a boundary value problem for a structure of interest for applications. |
|