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Kinematically incompatible Föppl-von Kármán plates: analysis and numerics

Edoardo Fabbrini, Andrés Alessandro León Baldelli, Pierluigi Cesana

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.

Kinematically incompatible Föppl-von Kármán plates: analysis and numerics

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.

Paper Structure

This paper contains 24 sections, 6 theorems, 54 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature review and comparison with existing work
  3. Notation
  4. Mathematical analysis of FvK equations
  5. Variational formulation
  6. Proof of Theorem \ref{['2412291203']}.
  7. Non-dimensional formulation
  8. Discrete Numerical Formulation
  9. Numerical results
  10. Solver verification
  11. Test 1: kinematically compatible plate under transverse pressure
  12. Test 2: kinematically incompatible plate with two disclinations
  13. Parametric study varying slenderness and load ratio
  14. The effect of the aspect ratio $\beta$
  15. The effect of the dimensionless external load $\gamma$
  16. ...and 9 more sections

Key Result

Theorem 2.1

(Existence) Let $\Omega \subset \mathbb{R}^2$ be a domain with Lipschitz boundary. Let $\theta \in H^{-2}(\Omega)$ be defined as in 202412111457 and $p \in H^{-2}(\Omega)$. Then, Eq. 2406171748 admits a solution in $H^2_0(\Omega) \times H^2_0(\Omega)$.

Figures (13)

  • Figure 1: Profiles of $v(\xi_1, 0)$ (LEFT) and $w(\xi_1, 0)$ (RIGHT) for $\xi_1 \in [-1, 1]$. Variational formulation (VAR) as in Eqs. \ref{['eq:202412271313']} and \ref{['eq:202412271314']} (blue thick line), analytical solution (black thin line). In the inset, the difference between: VAR and BNRS17 as in Brenner:Von_Karman (orange line), VAR and CMN18 as in Carstensen (green line).
  • Figure 2: Profile of $v(\xi_1, 0)$ for $\xi_1 \in [-1, 1]$. Variational formulation (VAR) as in Eqs. \ref{['eq:202412271313']} and \ref{['eq:202412271314']} (blue thick line), analytical solution (black thin line). In the inset, the difference between: VAR and BNRS17 as in Brenner:Von_Karman (orange line), VAR and CMN18 as in Carstensen (green line).
  • Figure 3: Membrane and bending energies and coupling term. Membrane (blue circles). Bending (orange triangles). Coupling (green triangles). Membrane energy according to Kirchhoff-Love can be computed exactly and is equal to $\mathscr{E}_{\text{m}}^{\text{KL}} = \frac{\beta^4}{32\pi}$ (black thin line). Bending energy predicted by the Kirchhoff-Love theory, $\mathscr{E}_{\text{b}}^{\text{KL}} = \frac{\pi}{384 c_{\nu}} (\gamma\beta^4)^2$ (black thick line). Plot in log-log scale.
  • Figure 4: Dimensionless $\sigma_{rr}$ computed for $\beta = 10$.
  • Figure 5: Profiles of the normalized Airy potential (LEFT) and normalized transverse deflection (RIGHT). Functions evaluated for $\xi_1 \in [-1, 1]$ and $\xi_2 = 0$ and normalized with respect to their maximum value in the same range of $(\xi_1, \xi_2)$. $\beta = 10$ (black line), $\beta = 100$ (orange dashed line).
  • ...and 8 more figures

Theorems & Definitions (7)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Remark 3.2
  • Proposition A.1
  • Proposition A.2: properties of the operators $B$, $C$ and $\Lambda_{\theta}$
  • Lemma A.3