  We compute effective energies of thin bilayer structures composed by soft
nematic elastic-liquid crystals in various geometrical regimes and functional
configurations. Our focus is on order-strain interaction in elastic foundations
composed of an isotropic layer attached to a nematic substrate. We compute
Gamma-limits as the layers thickness vanishes in two main scaling regimes
exhibiting spontaneous stress relaxation and shape-morphing, allowing in both
cases out-of-plane displacements. This extends the plane strain modelling of
[*], showing the asymptotic emergence of fully coupled macroscopic
active-nematic foundations. Subsequently, we focus on actuation and compute
asymptotic configurations of an active plate on nematic foundation interacting
with an applied electric field. From the analytical standpoint, the presence of
an electric field and its associated electrostatic work turns the total energy
into a non-convex and non-coercive functional. We show that equilibrium
solutions are min-max points of the system, that min-maximising sequences pass
to the limit and, that the limit system can exert mechanical work under applied
electric fields.
  [*]: P. Cesana and A. A. Le\'on Baldelli. "Variational modelling of nematic
elastomer foundations". In: Mathematical Models and Methods in Applied Sciences
14 (2018)
