A Homogenized Bending Theory for Prestrained Plates

Abstract

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.

Finite Element Approximation of Large-Scale Isometric Deformations of Parametrized Surfaces

“…A quadratic bending energy. As it was proven in [7], by Poincare's and Korn's inequality using the L 2orthogonality of ιpy 3 Aq and ∇ γ ϑ for ϑ P V, for every A P R 2ˆ2 sym , and x P ω, there exists a unique ϑpx, A, ¨q P V such that for all v P V, where C 3 px, y 1 q P R 3ˆ3ˆ3ˆ3 denotes the linearized elasticity tensor associated with the quadratic form Q 3 px, y 1 , ¨q for px, y 1 q P ω ˆY, i.e.…”

“…The homogenization and dimension reduction limit by Hornung et al in [19]. The above model appears as the limit problem with a simultaneous homogenization of a microstructured thin plate and a dimension reduction from 3D volume elasticity to a 2D elastic bending model as it is studied by Hornung et al in the case where Q 3 is independent of the macroscopic variable x. Böhnlein et al [7] considered the generalized case of piecewise constant macroscopic dependence on grain domains. Skipping the dependence on the macroscopic variable one takes into account deformations ϕ δ : Ω δ Ñ R 3 of the reference configuration Ω δ ω ˆδI of a thin plate with thickness δ ą 0 and a microscopic scale parameter ε ą 0 with an elastic energy…”

“…The case where ε 2 ď δ ď ε was discussed by Velčić in [22]. Recently, Böhnlein et al extended this theory to homogenized prestrained plates in [7]. In [20] Neukamm and Olbermann considered the spatial periodic homogenization in the context of nonlinear bending functionals for plates.…”

Numerical approximations of thin structure deformations

“…where the inhomogeneity and anisotropy of the bilayer material is encoded in the tensor and m : Ω → R 2 . Note that the expression (26) does not reduce to the previous setting with target metric (18).…”

“…For the derivation in the general case, namely when G is not the identity matrix, we refer to [5,20,[24][25][26]. The first Γ-convergence results for prestrained plates were obtained in [20] for target metrics as in (3), and later extended to more general metrics in [24].…”