Kinematically incompatible F\"oppl-von K\'arm\'an plates: analysis and numerics

TL;DR

This paper studies thin plates with out-of-plane deformations caused by kinematic incompatibilities like disclinations, providing existence theorems and a finite element numerical approach for analyzing such complex behaviors.

Contribution

It introduces new mathematical theorems ensuring equilibrium solutions for incompatible plates and develops a Discontinuous Galerkin finite element method for numerical analysis.

Findings

Existence and regularity of solutions are proven for plates with disclinations.

A finite element code in FEniCS effectively models the equilibrium states.

Numerical experiments demonstrate the method's applicability to various scenarios.

Abstract

We investigate thin plates where out-of-plane deformations arise due to membrane kinematic incompatibility of rotational type, specifically Volterra wedge disclinations, which are commonly observed in metal plates and graphene. We present theorems that guarantee the existence and regularity of equilibrium solutions in the presence of a finite number of disclinations and a dead load, for clamped plates. To solve the equilibrium equations, we implement a numerical code in the FEniCS environment and apply it to a series of parametric test studies. Our Finite Element method follows the Discontinuous Galerkin approach with C0 elements.