Continuum modeling of a free-standing graphene monolayer, viewed as a two dimensional 2-lattice, requires specifications of the components of the shift vector that act as an auxiliary variable. The field equations are then the equations ruling the shift vector, together with momentum and moment of momentum equations. We present an analysis of simple loading histories such as axial, biaxial tension/compression and simple shear for a range of problems of increasing difficulty. We start by laying down the equations of a simplified model which can still capture bending effects. Initially, we ignore out of plane deformations. For this case, we solve analytically the equations ruling the auxiliary variables in terms of the shift vector; these equations are algebraic when the loading is specified. As a next step, still working on the simplified model, out-of-plane deformations are taken into account and the equations complicate dramatically. We describe how wrinkling/buckling can be introduced into the model and apply the Cauchy-Kowalevski theorem to get existence and uniqueness in terms of the shift vector for some characteristic cases. Finally, for the treatment of the most general problem, we classify the equations of momentum and give conditions for the Cauchy-Kowalevski theorem to apply.

Continuum modeling of a free-standing graphene monolayer, viewed as a two dimensional 2-lattice, requires specifications of the components of the shift vector that act as an auxiliary variable. The field equations are then the equations ruling the shift vector, together with momentum and moment of momentum equations. To introduce material linearity energy is assumed to have a quadratic dependence on the strain tensor, the curvature tensor, the shift vector, as well as to combinations of them. Hexagonal symmetry then reduces the overall number of independent material constants to nine. We present an analysis of simple loading histories such as axial, biaxial tension/compression and simple shear for a range of problems of increasing difficulty for the geometrically and materially linear case. We start with the problem of in-plane motions only. By prescribing the displacement, the components of the shift vector are evaluated. This way the field equations are satisfied trivially. Outof-plane motions are treated as well; we assume in-plane tension/compression that leads to buckling/wrinkling and solve for the components of the shift vector as well as the function present in buckling's modeling. The assumptions of linearity adopted here simplifies the analysis and facilitates analytical results.

Of interest here is the fully three-dimensional analysis of the Freedericksz transition for the twisted nematic device (TND), which is widely used in liquid-crystal display monitors. Using a coupled electromechanical variational formulation, the problem is treated as a bifurcation instability triggered by an externally applied electric field. More specifically, we study a finite thickness liquid-crystal layer, anchored between two infinite parallel plates relatively rotated with respect to each other by a given twist angle and subjected to a uniform electric field perpendicular to these bounding plates. The novelty of the proposed analysis lies in the fully three-dimensional formulation of the TND problem that considers all possible bounded perturbations about the principal solution. By scanning a wide range of the liquid crystal's material parameter space, we establish whether the Freedericksz transition is global, i.e., has an eigenmode depending solely on the layer thickness coordinate, or local (also termed the periodic Freedericksz transition), i.e., has an eigenmode with finite wavelengths in one or both directions parallel to the plate. It is found that global modes are typical for low values, while local modes appear at large values of the twist angle. Moreover, for certain TND's, the increase in twist angle can lower the critical electric field, findings that could be useful in guiding liquid-crystal selection for applications.