Ciarlet, Philippe G. - Differential Geommetry with Applications to Elasticity
This book is based on lectures delivered over the years by the author at the
Universit´e Pierre et Marie Curie, Paris, at the University of Stuttgart, and at
City University of Hong Kong. Its two-fold aim is to give thorough introduc-tions to the basic theorems of differential geometry and to elasticity theory incurvilinear coordinates.
The treatment is essentially self-contained and proofs are complete. The
prerequisites essentially consist in a working knowledge of basic notions of anal-ysis and functional analysis, such as differential calculus, integration theory
and Sobolev spaces, and some familiarity with ordinary and partial differential
equations.
In particular, no apriori knowledge of differential geometry or of elasticity
theory is assumed.
In the first chapter, we review the basic notions, such as the metric tensor
and covariant derivatives, arising when a three-dimensional open set is equipped
with curvilinear coordinates. We then prove that the vanishing of the Riemann
curvature tensor is sufficient for the existence of isometric immersions from a
simply-connected open subset of R n equipped with a Riemannian metric into
a Euclidean space of the same dimension. We also prove the corresponding
uniqueness theorem, also called rigidity theorem.
In the second chapter, we study basic notions about surfaces, such as their
two fundamental forms, the Gaussian curvature and covariant derivatives. We
then prove the fundamental theorem of surface theory, which asserts that the
Gauß and Codazzi-Mainardi equations constitute sufficient conditions for two
matrix fields defined in a simply-connected open subset of R 2 to be the two
fundamental forms of a surface in a three-dimensional Euclidean space. We also
prove the corresponding rigidity theorem.
In addition to such “classical” theorems, which constitute special cases of the
fundamental theorem of Riemannian geometry, we also include in both chapters
recent results which have not yet appeared in book form, such as the continuity
of a surface as a function of its fundamental forms.
The third chapter, which heavily relies on Chapter 1, begins by a detailed
derivation of the equations of nonlinear and linearized three-dimensional elastic-ity in terms of arbitrary curvilinear coordinates. This derivation is then followed
by a detailed mathematical treatment of the existence, uniqueness, and regu-larity of solutions to the equations of linearized three-dimensional elasticity in
curvilinear coordinates. This treatment includes in particular a direct proof of
the three-dimensional Korn inequality in curvilinear coordinates.
The fourth and last chapter, which heavily relies on Chapter 2, begins by
a detailed description of the nonlinear and linear equations proposed by W.T.
Koiter for modeling thin elastic shells. These equations are “two-dimensional”,
in the sense that they are expressed in terms of two curvilinear coordinates
used for defining the middle surface of the shell. The existence, uniqueness, and
regularity of solutions to the linear Koiter equations is then established, thanks
this time to a fundamental “Korn inequality on a surface” and to an “infinites-imal rigid displacement lemma on a surface”. This chapter also includes a brief
introduction to other two-dimensional shell equations.
Interestingly, notions that pertain to differential geometry per se,suchas
covariant derivatives of tensor fields, are also introduced in Chapters 3 and 4,
where they appear most naturally in the derivation of the basic boundary value
problems of three-dimensional elasticity and shell theory.
Occasionally, portions of the material covered here are adapted from ex-cerpts from my book “Mathematical Elasticity, Volume III: Theory of Shells”,
published in 2000 by North-Holland, Amsterdam; in this respect, I am indebted
to Arjen Sevenster for his kind permission to rely on such excerpts. Other-wise, the bulk of this work was substantially supported by two grants from the
Research Grants Council of Hong Kong Special Administrative Region, China
[Project No. 9040869, CityU 100803 and Project No. 9040966, CityU 100604].
Last but not least, I am greatly indebted to Roger Fosdick for his kind
suggestion some years ago to write such a book, for his permanent support
since then, and for his many valuable suggestions after he carefully read the
entire manuscript.
Universit´e Pierre et Marie Curie, Paris, at the University of Stuttgart, and at
City University of Hong Kong. Its two-fold aim is to give thorough introduc-tions to the basic theorems of differential geometry and to elasticity theory incurvilinear coordinates.
The treatment is essentially self-contained and proofs are complete. The
prerequisites essentially consist in a working knowledge of basic notions of anal-ysis and functional analysis, such as differential calculus, integration theory
and Sobolev spaces, and some familiarity with ordinary and partial differential
equations.
In particular, no apriori knowledge of differential geometry or of elasticity
theory is assumed.
In the first chapter, we review the basic notions, such as the metric tensor
and covariant derivatives, arising when a three-dimensional open set is equipped
with curvilinear coordinates. We then prove that the vanishing of the Riemann
curvature tensor is sufficient for the existence of isometric immersions from a
simply-connected open subset of R n equipped with a Riemannian metric into
a Euclidean space of the same dimension. We also prove the corresponding
uniqueness theorem, also called rigidity theorem.
In the second chapter, we study basic notions about surfaces, such as their
two fundamental forms, the Gaussian curvature and covariant derivatives. We
then prove the fundamental theorem of surface theory, which asserts that the
Gauß and Codazzi-Mainardi equations constitute sufficient conditions for two
matrix fields defined in a simply-connected open subset of R 2 to be the two
fundamental forms of a surface in a three-dimensional Euclidean space. We also
prove the corresponding rigidity theorem.
In addition to such “classical” theorems, which constitute special cases of the
fundamental theorem of Riemannian geometry, we also include in both chapters
recent results which have not yet appeared in book form, such as the continuity
of a surface as a function of its fundamental forms.
The third chapter, which heavily relies on Chapter 1, begins by a detailed
derivation of the equations of nonlinear and linearized three-dimensional elastic-ity in terms of arbitrary curvilinear coordinates. This derivation is then followed
by a detailed mathematical treatment of the existence, uniqueness, and regu-larity of solutions to the equations of linearized three-dimensional elasticity in
curvilinear coordinates. This treatment includes in particular a direct proof of
the three-dimensional Korn inequality in curvilinear coordinates.
The fourth and last chapter, which heavily relies on Chapter 2, begins by
a detailed description of the nonlinear and linear equations proposed by W.T.
Koiter for modeling thin elastic shells. These equations are “two-dimensional”,
in the sense that they are expressed in terms of two curvilinear coordinates
used for defining the middle surface of the shell. The existence, uniqueness, and
regularity of solutions to the linear Koiter equations is then established, thanks
this time to a fundamental “Korn inequality on a surface” and to an “infinites-imal rigid displacement lemma on a surface”. This chapter also includes a brief
introduction to other two-dimensional shell equations.
Interestingly, notions that pertain to differential geometry per se,suchas
covariant derivatives of tensor fields, are also introduced in Chapters 3 and 4,
where they appear most naturally in the derivation of the basic boundary value
problems of three-dimensional elasticity and shell theory.
Occasionally, portions of the material covered here are adapted from ex-cerpts from my book “Mathematical Elasticity, Volume III: Theory of Shells”,
published in 2000 by North-Holland, Amsterdam; in this respect, I am indebted
to Arjen Sevenster for his kind permission to rely on such excerpts. Other-wise, the bulk of this work was substantially supported by two grants from the
Research Grants Council of Hong Kong Special Administrative Region, China
[Project No. 9040869, CityU 100803 and Project No. 9040966, CityU 100604].
Last but not least, I am greatly indebted to Roger Fosdick for his kind
suggestion some years ago to write such a book, for his permanent support
since then, and for his many valuable suggestions after he carefully read the
entire manuscript.
