In this paper, we study an elastic bilayer plate composed of a nematic liquid crystal elastomer in the top layer and a nonlinearly elastic material in the bottom layer. While the bottom layer is assumed to be stress-free in the flat reference configuration, the top layer features an eigenstrain that depends on the local liquid crystal orientation. As a consequence, the plate shows non-flat deformations in equilibrium with a geometry that non-trivially depends on the relative thickness and shape of the plate, material parameters, boundary conditions for the deformation, and anchorings of the liquid crystal orientation. We focus on thin plates in the bending regime and derive a two-dimensional bending model that combines a nonlinear bending energy for the deformation, with a surface Oseen-Frank energy for the director field that describes the local orientation of the liquid crystal elastomer. Both energies are nonlinearly coupled by means of a spontaneous curvature term that effectively describes the nematic-elastic coupling. We rigorously derive this model as a Γ-limit from three-dimensional, nonlinear elasticity. We also devise a new numerical algorithm to compute stationary points of the two-dimensional model. We conduct numerical experiments and present simulation results that illustrate the practical properties of the proposed scheme as well as the rich mechanical behavior of the system.

We discuss a 1D variational problem modeling an elastic sheet on water, lifted at one end. Its terms include all forces that are relevant in the experiment. By studying a suitable Gamma-limit, we identify a parameter regime in which the sheet is inextensible, and the bending energy of the sheet is negligible. In this regime, the problem simplifies to one with an explicit solution. In order to prove Γ−convergence, we introduce a retardation argument in order to deal with the possibly infinite bending energy of the ansatz. This model involves a variational problem set in an unbounded domain and non reflexive topology, and hence requires special care. KEYWORDS calculus of variations, elasto-capillarity, gamma convergence, thin elastic sheets MSC CLASSIFICATION 49 SETTING AND MODELThis article models and analyzes an experiment in which a thin sheet on water is lifted at one end. Against expectations, the profile of the thin sheet on one side of the contact point and the profile of the liquid gas interface on the other side of the contact point are exactly symmetric (see lecture notes in https://blogs.umass.edu/softmatter/lecture-notes/ specifically the course by Benny Davidovitch, lecture 4 for experimental results, or Deepak Kumar and Russell 1 ). This is counter intuitive since the forces acting on the liquid gas interface are the gravitational pull of the liquid and the effect of surface tension. On the other hand, the forces acting on the thin film are elastic forces, surface tension, and the gravitational pull of the liquid. Furthermore, the surface tension coefficients of the three different interfaces (liquid-gas, gas-solid, liquid-solid) are in principle different. This article provides a mathematical treatment of the problem: starting from well-established first principles we deduce the solution and prove that the profiles are symmetric.In order to justify our model from first principles, we derive our model as a Γ limit of functionals with positive thickness. The upper bound construction involves a highly oscillatory isometry, which in general can have infinite bending energy (proportional to the W 2,2 norm). In order to make this negligible in the limit, we introduce a retardation argument. Along the way, it is necessary to prove existence an uniqueness of problem (2) on a half-plane. Since the variational problem gives us a bound in the W 1,1 topology in which the unit ball is not weakly compact, we have to rewrite the problem in parametric coordinates, which gives a W 1,∞ bound. Since in this topology the unit ball is weakly compact, we can apply the direct method. This idea, along with a diagonal sequence-argument to deal with the unbounded domain, allows us to prove well-posedness. This is done through proposition 2.The interaction of thin sheets with surface tension have been the focus of many recent works including Deepak Kumar and Russell, 2 and works dealing with wrinkling phenomena. 3-7 More broadly, Vella and Mahadevan 8 and Vella et al 9 treat the interaction of surface tension with mechanical p...

The presence of prestrain can have a tremendous effect on the mechanical behavior of slender structures. Prestrained elastic plates show spontaneous bending in equilibrium—a property that makes such objects relevant for the fabrication of active and functional materials. In this paper we study microheterogeneous, prestrained plates that feature non-flat equilibrium shapes. Our goal is to understand the relation between the properties of the prestrained microstructure and the global shape of the plate in mechanical equilibrium. To this end, we consider a three-dimensional, nonlinear elasticity model that describes a periodic material that occupies a domain with small thickness. We consider a spatially periodic prestrain described in the form of a multiplicative decomposition of the deformation gradient. By simultaneous homogenization and dimension reduction, we rigorously derive an effective plate model as a $$\Gamma $$ Γ -limit for vanishing thickness and period. That limit has the form of a nonlinear bending energy with an emergent spontaneous curvature term. The homogenized properties of the bending model (bending stiffness and spontaneous curvature) are characterized by corrector problems. For a model composite—a prestrained laminate composed of isotropic materials—we investigate the dependence of the homogenized properties on the parameters of the model composite. Secondly, we investigate the relation between the parameters of the model composite and the set of shapes with minimal bending energy. Our study reveals a rather complex dependence of these shapes on the composite parameters. For instance, the curvature and principal directions of these shapes depend on the parameters in a nonlinear and discontinuous way; for certain parameter regions we observe uniqueness and non-uniqueness of the shapes. We also observe size effects: The geometries of the shapes depend on the aspect ratio between the plate thickness and the composite period. As a second application of our theory, we study a problem of shape programming: We prove that any target shape (parametrized by a bending deformation) can be obtained (up to a small tolerance) as an energy minimizer of a composite plate, which is simple in the sense that the plate consists of only finitely many grains that are filled with a parametrized composite with a single degree of freedom.

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