Bilayer Plates: Model Reduction, Γ‐Convergent Finite Element Approximation, and Discrete Gradient Flow

Abstract

The bending of bilayer plates is a mechanism that allows for large deformations via small externally induced lattice mismatches of the underlying materials. Its mathematical modeling, discussed herein, consists of a nonlinear fourthorder problem with a pointwise isometry constraint. A discretization based on Kirchhoff quadrilaterals is devised and its -convergence is proved. An iterative method that decreases the energy is proposed, and its convergence to stationary configurations is investigated. Its performance, as well as reduced model capabilities, are explored via several insightful numerical experiments involving large (geometrically nonlinear) deformations.

Modeling and simulation of thermally actuated bilayer plates

“…which in turn allows us to follow the arguments of [5] for the formal dimension reduction of the plate model. A rigorous derivation for more general material models including the above St. Venant-Kirchhoff material can be found in [21] which leads to the same dimensionally reduced model up to a different prefactor in the energy density.…”

“…The heat equation decouples from the mechanical equation due to the inextensibility of the plate and we approximate it with standard finite element methods together with the backward Euler time stepping algorithm. The proposed iterative numerical method for the coupled system is roughly ten times faster than the one used in [5]. The outline of this article is as follows.…”

Dimension reduction via $$\Gamma $$ Γ -convergence for soft active materials

“…Finally, we consider the case of a bilayer governed, again, by the prototypical energy density (2.9). A similar problem has been considered in [24] and also in [8]. More precisely, we consider a model for bilayers where in the physical reference configuration Ω ε h the energy density W h = W h (z 3 , F ) is given by (2.8)-(2.9) and the spontaneous (right Cauchy-Green) strain is defined as…”

Spontaneous morphing of equibiaxially pre-stretched elastic bilayers: The role of sample geometry

“…Some of the interesting spontaneous morphing behaviour of Liquid Crystal Elastomers can be reproduced in pre-stretched bilayers, made of simple (i.e., not-active) elastic materials. It is shown in (Agostiniani and DeSimone, 2017a;Bartels et al, 2017;DeSimone, 2017) that, by "engineering" the prestretch in each of the two layers of a bilayer one can reproduce different structural models, such as bistable shells and shells with very low stiffness with the respect to twisting, as discussed in (Guest and Pellegrino, 2006) and (Guest et al, 2011), respectively. Moreover, by shaping the mid-plane of the bilayer in the form of a long, narrow rectangle, and varying the angle between the long axis of the rectangle and the uniaxial prestretch direction in one of the layers, one can produce tunable helical ribbons as in (Chen et al, 2011).…”