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.
Author(s): Marta Lewicka
Series: Progress in Nonlinear Differential Equations and Their Applications, 101
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 447
City: Cham
Contents
Chapter 1 Introduction
Part I Tools in mathematical analysis
Chapter 2 Γ-convergence
2.1 Definition, examples and fundamental properties
2.2 Example of Γ-convergence in linearised elasticity
2.3 Bibliographical notes
Chapter 3 Korn’s inequality
3.1 Korn’s inequality and First Korn’s inequality
3.2 Variants of Korn’s inequality with different boundary conditions
3.3 Proof of Korn’s inequality: preliminary estimates
3.4 Proof of Korn’s inequality
3.5 Korn’s constant under tangential boundary conditions
3.6 Approximation theorem and Korn’s constant in thin shells
3.7 Killing vector fields and Korn’s inequality on surfaces
3.8 Blowup of Korn’s constant in thin shells
3.9 Uniformity of Korn’s constant under tangential boundary conditions in thin shells
3.10 Proofs of uniformity of Korn’s constant under tangential boundary conditions in thin shells
3.11 Bibliographical notes
Chapter 4 Friesecke-James-Müller’s inequality
4.1 Liouville’s theorem and quantitative rigidity estimate
4.2 First order truncation result
4.3 Local rigidity estimate
4.4 Proof of Friesecke-James-Müller’s inequality
4.5 Approximation theorem and Friesecke-James-Müller’s constant in thin shells
4.6 Friesecke-James-Müller’s inequality in the plane
4.7 Rigidity estimates in conformal setting
4.8 Bibliographical notes
Part II Dimension reduction in classical elasticity
Chapter 5 Limiting theories for elastic plates and shells: nonlinear bending
5.1 Set-up of three dimensional nonlinear elasticity
5.2 Elasticity on shells
5.3 Kirchhoff’s theory for shells: compactness and lower bound
5.4 Second order truncation result
5.5 Proof of second order truncation result
5.6 Kirchhoff’s theory for shells: recovery family
5.7 Kirchhoff’s theory for shells: Γ-limit and convergence of minimizers
5.8 Non-compactness by wrinkling
5.9 Compactness beyond Kirchhoff’s scaling
5.10 Lower bound beyond Kirchhoff’s scaling
5.11 Bibliographical notes
Chapter 6 Limiting theories for elastic plates and shells: nonlinear bending
6.1 Von Kármán’s theory for shells: recovery family
6.2 Von Kármán’s theory for shells: Γ-limit and convergence of minimizers
6.3 Linear elasticity for shells: recovery family, Γ-limit and convergence of minimizers
6.4 Shells with variable thickness
6.5 Convergence of equilibria
6.6 Von Kármán’s equations
6.7 Bibliographical notes
Chapter 7 Limiting theories for elastic plates: linearised bending
7.1 Linearised Kirchhoff’s theory for shells
7.2 Linearised Kirchhoff’s theory for plates
7.3 Matching infinitesimal to exact isometries on plates
7.4 Linearised Kirchhoff’s theory for plates: recovery family for Lipschitz displacements
7.5 Density result on plates and linearised Kirchhoff’s theory: recovery family, Γ-limit and convergence of minimizers
7.6 Elastic shallow shells
7.7 Linearised Kirchhoff’s theory for shallow shells
7.8 Matching isometries on shallow shells
7.9 Convexity of weakly regular displacements
7.10 Density result and recovery family on shallow shells
7.11 Bibliographical notes
Chapter 8 Infinite hierarchy of elastic shell models
8.1 Heuristics on isometry matching and collapse of theories
8.2 Matching infinitesimal to second order isometries on surfaces of revolution
8.3 Bibliographical notes
Chapter 9 Limiting theories on elastic elliptic shells
9.1 Linear problem symw = B on elliptic surfaces
9.2 Matching infinitesimal to exact isometries on elliptic surfaces
9.3 Density result on elliptic surfaces
9.4 Collapse of theories beyond Kirchhoff’s scaling for elliptic shells: recovery family, Γ-limit and convergence of minimizers
9.5 Bibliographical notes
Chapter 10 Limiting theories on elastic developable shells
10.1 Developable surfaces
10.2 Matching properties on developable surfaces
10.3 Density result on developable surfaces
10.4 Collapse of theories beyond Kirchhoffs scaling for developable shells: recovery family, Γ-limit and convergence of minimizers
10.5 Bibliographical notes
Part III Dimension reduction in prestressed elasticity
Chapter 11 Limiting theories for prestressed films: nonlinear bending
11.1 Three dimensional non-Euclidean elasticity
11.2 Prestressed thin films
11.3 Kirchhoff-like theory for prestressed films: compactness and lower bound
11.4 Kirchhoff-like theory for prestressed films: recovery family
11.5 Identification of Kirchhoff’s scaling regime
11.6 Coercivity of Kirchhoff-like energy for prestressed films
11.7 Effective energy density under isotropy condition
11.8 Application to liquid crystal glass
11.9 More examples
11.10 Connection to experiments
11.11 Bibliographical notes
Chapter 12 Limiting theories for prestressed films: von Kármán-like theory
12.1 Energy quantisation and approximation lemmas
12.2 Von Kármán-like theory for prestressed films: compactness and lower bound
12.3 Von Kármán-like theory for prestressed films: recovery family
12.4 Identification of von Kármán’s scaling regime and coercivity of von Kármán-like energy for prestressed films
12.5 Two examples
12.6 Beyond von Kármán’s regime: an example
12.7 Bibliographical notes
Chapter 13 Infinite hierarchy of limiting theories for prestressed films
13.1 Energy quantisation and identification of scaling regimes for prestressed films beyond von K´arm´an’s regime
13.2 Higher order theories for prestressed films: compactness and preliminary lower bound
13.3 Identification of lower bound’s curvature term
13.4 Higher order theories for prestressed films: lower bound
13.5 Higher order theories for prestressed films: recovery family
13.6 Convergence of minima and coercivity of linear elasticity-like energies
13.7 Bibliographical notes
Chapter 14 Limiting theories for weakly prestressed films
14.1 Weakly prestressed films
14.2 Von Kármán and linear elasticity-like theories for weakly prestressed films
14.3 Compactness and lower bounds
14.4 Von Kármán and linear elasticity-like theories for weakly prestressed films: recovery family
14.5 Elimination of out-of-plane displacements
14.6 Identification of scaling regimes
14.7 Linearised Kirchhoff-like theory for weakly prestressed films
14.8 Matching isometries on weakly prestressed films
14.9 Uniqueness of minimizers to linearised Kirchhoff-like energy for prestressed films
14.10 Critical points in radially symmetric case
14.11 Bibliographical notes
References
Terminology and notation
Index
