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Calculus of Variations on Thin Prestressed Films

Asymptotic Methods in Elasticity

  • Book
  • © 2023

Overview

Authors:
  1. Marta Lewicka

    1. Mathematics Department, University of Pittsburgh, Pittsburgh, USA

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  • Studies asymptotic theories in prestrained elasticity from a rigorous analytical perspective
  • Provides the necessary background information from differential geometry and calculus of variations
  • Will be of interest to researchers in both mathematics and engineering

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications (PNLDE, volume 101)

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-ix
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  2. Introduction

    • Marta Lewicka
    Pages 1-5

  3. Tools in mathematical analysis

    1. Front Matter

      Pages 7-7
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    2. Γ-Convergence

      • Marta Lewicka
      Pages 9-19

    3. Korn’s Inequality

      • Marta Lewicka
      Pages 21-64

    4. Friesecke–James–Müller’s Inequality

      • Marta Lewicka
      Pages 65-92

  4. Dimension reduction in classical elasticity

    1. Front Matter

      Pages 93-93
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    2. Limiting Theories for Elastic Plates and Shells: Nonlinear Bending

      • Marta Lewicka
      Pages 95-138

    3. Limiting Theories for Elastic Plates and Shells: Sublinear and Linear

      • Marta Lewicka
      Pages 139-172

    4. Limiting Theories for Elastic Plates: Linearized Bending

      • Marta Lewicka
      Pages 173-221

    5. Infinite Hierarchy of Elastic Shell Models

      • Marta Lewicka
      Pages 223-231

    6. Limiting Theories on Elastic Elliptic Shells

      • Marta Lewicka
      Pages 233-250

    7. Limiting Theories on Elastic Developable Shells

      • Marta Lewicka
      Pages 251-265

  5. Dimension reduction in prestressed elasticity

    1. Front Matter

      Pages 267-267
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    2. Limiting Theories for Prestressed Films: Nonlinear Bending

      • Marta Lewicka
      Pages 269-316

    3. Limiting Theories for Prestressed Films: von Kármán-Like Theory

      • Marta Lewicka
      Pages 317-348

    4. Infinite Hierarchy of Limiting Theories for Prestressed Films

      • Marta Lewicka
      Pages 349-374

    5. Limiting Theories for Weakly Prestressed Films

      • Marta Lewicka
      Pages 375-423

  6. Back Matter

    Pages 425-448
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Keywords

About this book

This monograph considers the analytical and geometrical questions emerging from the study of thin elastic films that exhibit residual stress at free equilibria.  It provides the comprehensive account, the details and background on the most recent results in the combined research perspective on the classical themes: in Differential Geometry – that of isometrically embedding a shape with a given metric in an ambient space of possibly different dimension, and in Calculus of Variations – that of minimizing non-convex energy functionals parametrized by a quantity in whose limit the functionals become degenerate.


Prestressed thin films are present in many contexts and applications, such as: growing tissues, plastically strained sheets, engineered swelling or shrinking gels, petals and leaves of flowers, or atomically thin graphene layers.  While the related questions about the physical basis for shape formation lie at the intersection of biology, chemistry and physics, fundamentally they are of the analytical and geometrical character, and can be tackled using the techniques of the dimension reduction, laid out in this book.


The text will appeal to mathematicians and graduate students working in the fields of Analysis, Calculus of Variations, Partial Differential Equations, and Applied Math.  It will also be of interest to researchers and graduate students in Engineering (especially fields related to Solid Mechanics and Materials Science), who would like to gain the modern mathematical insight and learn the necessary tools.

Authors and Affiliations

  • Mathematics Department, University of Pittsburgh, Pittsburgh, USA

    Marta Lewicka

About the author

Marta Lewicka is a mathematician specializing in the fields of Analysis and Partial Differential Equations. She has contributed results in the theory of hyperbolic systems of conservation laws, fluid dynamics, calculus of variations, nonlinear potential theory, and differential games. She is a Fellow of the American Mathematical Society and holds Professor’s scientific title awarded by the President of the Republic of Poland. She works at the University of Pittsburgh, USA.

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