Nematic director fields and topographies of solid shells of revolution

Abstract

We solve the forward and inverse problems associated with the transformation of flat sheets with circularly symmetric director fields to surfaces of revolution with non-trivial topography, including Gaussian curvature, without a stretch elastic cost. We deal with systems slender enough to have a small bend energy cost. Shape change is induced by light or heat causing contraction along a non-uniform director field in the plane of an initially flat nematic sheet. The forward problem is, given a director distribution, what shape is induced? Along the way, we determine the Gaussian curvature and the evolution with induced mechanical deformation of the director field and of material curves in the surface (proto-radii) that will become radii in the final surface. The inverse problem is, given a target shape, what director field does one need to specify? Analytic examples of director fields are fully calculated that will, for specific deformations, yield catenoids and paraboloids of revolution. The general prescription is given in terms of an integral equation and yields a method that is generally applicable to surfaces of revolution.

Defective nematogenesis: Gauss curvature in programmable shape-responsive sheets with topological defects

“…This form has been used successfully to design surfaces with uniform finite Gauss curvature (such as spherical caps) and arbitary surfaces of revolution, 46,47 and underpins recent work on inverse design. 40 However, if we attempt to apply the above derivative formula at a topological defect in the pattern we encounter a problem: the derivatives are divergent.…”

Likely striping in stochastic nematic elastomers

“…Note that the upper bound in (71) is independent of µ (1) and µ (2) . This is due to the fact that this bound is obtained from the condition ∂ 2 w ∂θ 2 (λ, 0, 0) > 0, which by (67), only involves the neoclassical component of the strain-energy function. The lower bound in (71) satisfies a −1/6 ≤ a 1/12 µ (1) /µ (2)…”

“…For monodomain nematic solids, where the director field does not vary from point to point, a well-known constitutive model is provided by the phenomenological neoclassical model developed in Bladon et al [10], Warner et al [11] and Warner and Wang [12]. This model has been applied to predict large deformations in various applications [13][14][15][16][17]. In general, the neoclassical theory extends the molecular network theory of rubber elasticity [18] to liquid crystal elastomers, whereby the constitutive parameters are directly measurable experimentally, or derived from macroscopic shape changes [19,20].…”

Evolving, complex topography from combining centers of Gaussian curvature

“…Despite having the intrinsic metric induced by U n , it is still hard to determine the deformed shape for general director patterns because of the lack of bending information. With the help of symmetry, in particular circular symmetry, deformed shapes such as cones, spherical caps, and more general surfaces of revolution [13], along with their director patterns, have been described [14][15][16][17].…”