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Register a new userConcentrated Gaussian curvature in curved creases of actuated spiral nematic solids
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Liquid crystal elastomers/glasses are active materials that can have significant metric change upon stimulation. The local metric change is determined by its director pattern that describes the ordering direction and hence the direction of contraction. We study logarithmic spiral patterns on flat sheets that evolve into cones on deformation, with Gaussian curvature localized at tips. Such surfaces, Gaussian flat except at their tips, can be combined to give compound surfaces with GC concentrated in lines. We characterize all possible metric-compatible interfaces between two spiral patterns, specifically where the same metric change occurs on each side. They are classified as hyperbolic-type, elliptic-type, concentric spiral, and continuous-director interfaces. Upon the cone deformations and additional isometries, the actuated interfaces form creases bearing non-vanishing concentrated Gaussian curvature, which is formulated analytically for all cases and simulated numerically for some examples. Analytical calculations and the simulations agree qualitatively well. Furthermore, the relaxation of Gaussian-curved creases is discussed and cantilevers with Gaussian curvature-enhanced strength are proposed. Taken together, our results provide new insights in the study of curved creases, lines bearing Gaussian curvature, and their mechanics arising in actuated liquid crystal elastomers/glasses, and other related active systems.
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Within the framework of continuum theory, we draw a parallel between ferromagnetic materials and nematic liquid crystals confined on curved surfaces, which are both characterized by local interaction and anchoring potentials. We show that the extrinsic curvature of the shell combined with the out-of-plane component of the director field gives rise to chirality effects. This interplay produces an effective energy term reminiscent of the chiral term in cholesteric liquid crystals, with the curvature tensor acting as a sort of anisotropic helicity. We discuss also how the different nature of the order parameter, a vector in ferromagnets and a tensor in nematics, yields different textures on surfaces with the same topology as the sphere. In particular, we show that the extrinsic curvature governs the ground state configuration on a nematic spherical shell, favouring two antipodal disclinations of charge +1 on small particles and four $+1/2$ disclinations of charge located at the vertices of a square inscribed in a great circle on larger particles.
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We describe a general family of curved-crease folding tessellations consisting of a repeating lens motif formed by two convex curved arcs. The third author invented the first such design in 1992, when he made both a sketch of the crease pattern and a vinyl model (pictured below). Curve fitting suggests that this initial design used circular arcs. We show that in fact the curve can be chosen to be any smooth convex curve without inflection point. We identify the ruling configuration through qualitative properties that a curved folding satisfies, and prove that the folded form exists with no additional creases, through the use of differential geometry.
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