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On the Föppl-von Kármán theory for elastic prestrained films with varying thickness

Hui Li

Abstract

We derive the variational limiting theory of thin films, parallel to the Föppl-von Kármán theory in the nonlinear elasticity, for films that have been prestrained and whose thickness is a general non-constant function. Using $Γ$-convergence, we extend the existing results to the variable thickness setting, calculate the associated Euler-Lagrange equations of the limiting energy, and analyze the convergence of equilibria. The resulting formulas display the interrelation between deformations of the geometric mid-surface and components of the growth tensor.

On the Föppl-von Kármán theory for elastic prestrained films with varying thickness

Abstract

We derive the variational limiting theory of thin films, parallel to the Föppl-von Kármán theory in the nonlinear elasticity, for films that have been prestrained and whose thickness is a general non-constant function. Using -convergence, we extend the existing results to the variable thickness setting, calculate the associated Euler-Lagrange equations of the limiting energy, and analyze the convergence of equilibria. The resulting formulas display the interrelation between deformations of the geometric mid-surface and components of the growth tensor.

Paper Structure

This paper contains 5 sections, 4 theorems, 164 equations, 1 figure.

Table of Contents

  1. Introduction
  2. An overview of the main results
  3. The Gamma-Convergence
  4. The Föppl-von Kármán Equations
  5. Convergence of Equilibria

Key Result

Theorem 3.1

Assume the energies of a sequence of deformations $u^h \in W^{1, 2}(\Omega^h, \mathbb R^3)$ satisfy: with some constant $C>0$. Then there exist rotations $\bar{R}^h \in SO(3)$ and translations $c^h \in \mathbb R^3$ such that for the normalized deformations: the following assertions hold:

Figures (1)

  • Figure 1.1: Imposing an incompatible target metrics a sheets of NIPA gels. The experiment (on the left) and the obtained film shapes (on the right) in Klein-Efrati-Sharon_2007

Theorems & Definitions (6)

  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3
  • proof : Theorem \ref{['CompactnessFvK']}
  • Theorem 5.1
  • proof