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Models for elastic shells with incompatible strains.

Marta Lewicka1, L Mahadevan2, Mohammad Reza Pakzad1

  • 1Department of Mathematics , University of Pittsburgh , 301 Thackeray Hall, Pittsburgh, PA 15260, USA.

Proceedings. Mathematical, Physical, and Engineering Sciences
|May 9, 2014
PubMed
Summary
This summary is machine-generated.

Differential growth drives the 3D shapes of thin structures like leaves and wings. This study rigorously derives theories for growing elastic shells, generalizing existing models for flat plates and curved shells.

Keywords:
Gamma convergencecalculus of variationsnon-Euclidean platesnonlinear elasticity

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Area of Science:

  • * Physics
  • * Materials Science
  • * Mathematical Modeling

Background:

  • * The three-dimensional shapes of thin lamina (e.g., leaves, wings) arise from differential growth and induced strain.
  • * Growth involves variations in the Riemannian metric across the sheet's plane and thickness.
  • * Resulting shapes are a consequence of elastic energy minimization in geometrically constrained systems.

Purpose of the Study:

  • * To rigorously derive asymptotic theories for the shapes of residually strained thin lamina with non-trivial curvatures.
  • * To generalize existing theories for growing elastic shells in both weakly and strongly curved regimes.
  • * To extend classical Föppl-von Kármán energy theories to prestrained shallow shells.

Main Methods:

  • * Derivation of asymptotic theories based on differential growth and elastic energy minimization.
  • * Analysis of thin lamina with non-trivial curvatures (growing elastic shells).
  • * Generalization of existing models for flat plates to curved shell structures.

Main Results:

  • * Developed rigorous asymptotic theories for the shapes of residually strained thin lamina.
  • * Characterized growing elastic shells in weakly and strongly curved regimes.
  • * Generalized classical Föppl-von Kármán energy to prestrained shallow shells.

Conclusions:

  • * The study provides a robust theoretical framework for understanding the morphogenesis of thin, growing structures.
  • * The derived theories accurately describe the shapes of complex elastic shells.
  • * This work advances the understanding of how growth and mechanics interplay to create biological and engineered forms.