Bilayer Plates: Model Reduction, $Γ$-Convergent Finite Element Approximation and Discrete Gradient Flow
This work provides a rigorous numerical framework for simulating bilayer plates, which is important for engineers and materials scientists studying shape-morphing structures.
The authors develop a finite element discretization for bilayer plate bending with an isometry constraint, prove its Γ-convergence, and propose an energy-decreasing iterative method. Numerical experiments demonstrate large geometrically nonlinear deformations.
The bending of bilayer plates is a mechanism which allows for large deformations via small externally induced lattice mismatches of the underlying materials. Its mathematical modeling, discussed herein, consists of a nonlinear fourth order problem with a pointwise isometry constraint. A discretization based on Kirchhoff quadrilaterals is devised and its $Γ$-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.