A Design Principle for Actuation of Nematic Glass Sheets

Abstract

A continuum mechanical framework is developed for determining a) the class of stressfree deformed shapes and corresponding director distributions on the undeformed configuration of a nematic glass membrane that has a prescribed spontaneous stretch field and b) the class of undeformed configurations and corresponding director distributions on it resulting in a stress-free given deformed shape of a nematic glass sheet with a prescribed spontaneous stretch field. The proposed solution rests on an understanding of how the Lagrangian dyad of a deformation of a membrane maps into the Eulerian dyad in three dimensional ambient space. Interesting connections between these practical questions of design and the mathematical theory of isometric embeddings of manifolds, deformations between two prescribed Riemannian manifolds, and the slip-line theory of plasticity are pointed out.

On a general multi-layered hyperelastic plate theory of growth

“…Thus, the efficiency of the FvK-type plate model derived in this subsection is verified. The dependence of the growth bifurcation parameter λ c on the material and geometrical parameters can be seen clearly from (53). Some discussions on this issue can be found in Du et al [43].…”

“…During the past years, the problem of shape-programming of soft material samples has attracted extensive research interests (e.g. Dias et al [51], Jones and Mahadevan [52], Acharya [53], Nojoomi [54]). In our previous works [25,28,29], some explicit formulas for shape-programming of single-layered hyperelastic plates through differential growth have been derived.…”

A theoretical scheme for shape-programming of thin hyperelastic plates through differential growth

“…In this section, we focus on the case that the hyperelastic plate has the stress-free state in κ t , i.e., all the components in S ( 0 ) and S ( 1 ) are zero. It is clear that in the stress-free condition, both the plate equations (14) and the boundary conditions (16) are automatically satisfied (some recent studies on growth-induced deformations of hyperelastic bodies with stress-free states can be found in Acharya et al [28], Sadik et al [21], Acharya [29], and Chen and Dai [36]). Next, we begin to study the relations between the growth functions and the geometrical properties of the target surface S .…”

“…Acharya et al [28] studied the problem of prestrain in a 2D membrane that would result in a desired 3D stress-free surface, where the activation prestrain is interpreted as an isometric immersion of the Riemannian manifold into the 3D space. A common mathematical framework was further developed in Acharya [29] for solving the forward and inverse problems of design of thin nematic glass sheets subjected to opto-thermal stretching. Wang et al [30] focused on the plane-strain problems and derived some explicit analytical formulas for 2D shape-programming of hyperelastic plates through differential growth.…”

Shape Morphing of Planar Liquid Crystal Elastomers

“…4). Discussion-It is worth mentioning that, due to the purely geometric nature of this problem, the system that we consider has analogues in other physical and engineering contexts [26]. The networks of director and directorperpendicular fields that we obtain for PLCEs are known as Hencky-Prandtl (HP) nets [27,28].…”