Variational modelling of nematic elastomer foundations

Abstract

We compute the Γ-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 vanishing thickness of the membranes and the 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.

Gamma-convergence results for nematic elastomer bilayers: relaxation and actuation

“…solid elastic crystals [4] besides liquid crystal elastomers. This paper complements and completes the analysis developed in [10] on planar-constrained nematic bilayers by exploring the full three-dimensional elastic model allowing for out-of plane elastic deformations.…”

“…There, we approximate our relaxed target biaxial optic tensor by a weakly converging oscillating sequence. The key elements for the construction of the optic sequence draw heavily from [10] and are recalled in Lemma 3. The careful estimates of error terms and boundary layers require lengthy calculations which we omit, referring the interested reader to [10] for explicit details.…”

“…The key elements for the construction of the optic sequence draw heavily from [10] and are recalled in Lemma 3. The careful estimates of error terms and boundary layers require lengthy calculations which we omit, referring the interested reader to [10] for explicit details. We treat the regime p ∈ (−1, 0) as a particular characterised by the decoupling of membrane deformations from bending modes.…”

“…The program is to explicitly compute the effective stress relaxation induced by a local accommodation of the optical texture under mechanical deformation, a mechanism which is responsible of the emergence of fine scale, possibly periodic, optical patterns. In energetic terms, this amounts to first computing locally-optimal nematic textures at microscale and then performing the dimension reduction to derive a macroscopic two-dimensional model, as described by the following one-variable model [10,11,7,15]…”

“…2) Section 3 is devoted to the analysis of relaxation results for thin and large NLCE bilayers. This part of the paper builds upon material produced in [10] which we systematically refer to. Finally, in Section 4 we analyse limit functionals for thin and small NLCE bi-layers.…”