# Mathematical Foundations of Elasticity

### The Authors of Mathematical Foundations of Elasticity eBook

J. Marsden and T. Hughes are the editors of Mathematical Foundations of Elasticity PDF Book

• GEOMETRY AND KINEMATICS OF BODIES
• BALANCE PRINCIPLES
• CONSTITUTIVE THEORY
• LINEARIZATION
• HAMILTONIAN AND VARIATIONAL PRINCIPLES
• METHODS OF FUNCTIONAL ANALYSIS IN ELASTICITY
• SELECTED TOPICS IN BIFURCATION THEORY

## Preface to Mathematical Foundations of Elasticity PDF

This book treats parts of the mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis.

It is intended for mathematicians, engineers, and physicists who wish to s~e this classical subject in a modern setting and to see some examples of what newer mathematical tools have to contribute.

#### Disclaimer

There are three things that every beginner in elasticity theory should know. The first is that "Kirchhoff" has two h's in it. The second is that Hooke's law will not be found as a basic axiom (it "really" means you are working with the linearized theory).

The third is that rese? ranchers in elasticity theory are very opinionated, even when they are wrong. During our own work in this field, we have refused to fight.

And in keeping with this pacifist approach, we now issue these general disclaimers: This book is neither complete nor unbiased.

Furthermore, we have not mentioned many deep and highly erudite works, nor have we elucidated alternative approaches to the subject.

Any historical comments we make on subjects prior to 1960 are probably wrong, and credits to some theorems may be incorrectly assigned.

Excellent historical sketches are available in the works of Truesdell , Sokolnikoff , and Sneddon  cited in the bibliography.

#### The Two-Track and Box Approach

To a mathematician, a tensor t is a section of a certain bundle over a manifold. To an engineer or physicist, a tensor tick is an object dressed in indices.

This is one of many unfortunate paper barriers that have retarded the growth of, and interest in, mathematical elasticity.

The beginner should learn to speak both languages and to ignore notational disputes.

For example, beginners who are already trained in some geometry and who realize that V f is a vector, while df is a one-form, will recognize at once that the deformation gradient F is not a gradient at all, but is simply the derivative of the deformation.

They may also recognize that the rate of deformation tensor is just the Lie derivative of the Riemannian metric on space and that the Cauchy-Green tensor is the pull-back of the Riemannian metric on space by the deformation.

To aid the reader in this linguistic endeavor we have tried to present as many formulas as possible in both languages.

This is done through numerous boxes that summarize the important formulas both ways. These boxes are also used to isolate more advanced or optional material.

#### Subjects Covered

The first two chapters cover the background geometry which is developed as it is needed and use this discussion to obtain the basic results on kinematics and dynamics of continuous media.

• Chapter 3 narrows the discussion to elastic materials.
• Chapter 4 on linearization gives a systematic way to linearize a nonlinear field theory along with a basic mathematical tool the inverse function theorem.
• Chapter 5 deals with variational principles.
• Chapter 6 presents a relatively self-contained account of the use of functional analysis (such as elliptic theory and semigroups) in elasticity.
• Chapter 7 introduces the bifurcation theory. We originally planned to include a chapter on numerical methods as well, but space and timeliness did not allow us to do so.

#### Level and Background

The book is written at the beginning graduate level, with occasional excursions to the research frontier. Some parts, such as the first five chapters and parts of the remainder, are accessible to good undergraduates.

To read this book one should have a solid background in advanced calculus (for example, J. Marsden [I974aJ is adequate).

One should also be prepared to invest considerable time in learning geometry and functional analysis as the book is read.

Most of what is needed are in this book, but it may be useful to consult some of the references that follow.

#### The Use of Geometry and Functional Analysis

We have found differential geometry helpful in sorting out the foundations of the subject. Deeper analytical facts about elasticity require a serious knowledge of functional analysis, including partial differential equations.

The reader should realize that many researchers understand one or the other of these subjects, but very few understand both because of the large investment of time and effort involved.

Therefore, one should adjust one's aspirations and depth of reading accordingly. For example, if one's goal is t6 get to modern research in the buckling of shells as fast as possible, it may be a mistake to start on page I. It is obvious that a large part of any book is irrelevant to such a specific endeavor.

Rather, one should jump directly into the current literature (for example, see Section 7.2) and use this book to complete the necessary background.

On the other hand, if one has the time to go through the requisite geometry, the insights gained into nonlinear elasticity will be worthwhile.

Examples of how geometry is used in elasticity are discussed in Section 6 of the introductory chapter. Likewise, abstract functional analysis is often accused of not shedding any light on "practical" problems of elasticity.

Recent progress in constitutive inequalities and numerical methods demonstrates that this view is incorrect.

#### Point of Departure and Interdependence of Chapters

Because of the large amount of geometry involved in the first three chapters, we have written an introductory chapter to enable readers to bypass parts of Chapters 1-3.

After studying Sections 1-5 of the introductory chapter, such readers should be ready to undertake Chapters 4-7. These four chapters do contain some dependence on Chapters 1-3, but this dependence is minimal and may be bypassed if one has a background in elasticity obtained from other sources.

We also recommend the introductory chapter for readers who intend to seriously study Chapters 1-3 to keep their work in perspective.

Chapters 4-7 are in a logical order, but it is not necessary to have full mastery of one before proceeding. To this end, ample cross-references are given.

#### Notation

We have adopted a reasonably simple system used by some current workers. This is summarized in a brief glossary.

Readers should understand that if they hear a lecture on elasticity, the conventions will probably differ from those here or their own.

Here boldface type is used to distinguish abstract tensors from their components. For example, (J means the abstract Cauchy stress tensor, while (Jab represents its components.

The only other nonstandard notation is the use of block boldface for the fourth-order elasticity tensors, such as C, whose components are denoted CABeD.

And A, whose components are denoted AaAbB. Occasionally the same symbol has two meanings in the book when the intended meaning is clear from the context.

We find this preferable to a multitudinous proliferation of alphabets and fonts that are impossible to reproduce in the classroom.

#### Things We Fuss Over; Things We Don't

Most mathematicians, physicists, and engineers now agree that the distinction between a linear transformation and a matrix is worth fussing over.

We believe that one should also distinguish tensors from tensor components. However, we do not fuss over whether Euclidean space should be written as 1R3 or not.

To abstract 1R3 properly, we believe that manifolds should be used. They are unquestionably the appropriate setting for tensor analysis.

Resistance to the use of abstract manifolds is frequently encountered, simply because most work in elasticity occurs in 1R3.

In the literature, 1R3 is often replaced by abstract vector spaces. This arena is not suitable for general tensor analysis.

Indeed, as Einstein has so profoundly taught us, deep insights can be gained b removing one's blinders to see the theory in the grander time-proven Conte) of covariant formulations.

This is why we encourage the use of manifolds. We do not fuss over measure-theoretic questions that are often used t introduce mass and force densities, for example.

If one understands the Radon Nikodym derivative, it takes only a few minutes to understand this even though: technical intricacies may be nontrivial.

We chose not to go into measure-theoretic formalism because it requires a lengthy exposition that would divert u from our main goal.

#### Numbering Conventions

Within Chapter I, the eleventh item of the thin section is referred to as 3.11 and the third section is referred to as Section 3 In subsequent chapters this item is referred to as 3.11, Chapter I, and the third section is referred to as Section 1.3. Similar conventions apply to discussion enclosed inboxes. Figures are given their full labels for editorial reasons. Formulas are numbered within a section when it helps the exposition.

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References

A relatively large bibliography is included at the back of this book. Specific references cited in the text are listed by author and year like this: Ball [l977b]. It is wise for beginning students to consult a few key books regularly. We recommend the following:

• an introductory modern text on continuum mechanics, such as Malvern  or Gurtin [198Ib]
• one of the classical texts on elasticity such as Love , Sokolnikoff , Landau and Lifshitz , Green and Adkins , or Green and Zerna 
• the encyclopedic treatise of Truesdell and Noll  (which has a massive bibliography)
• a modern book on manifolds and tensor analysis, such as Abraham Marsden and Ratiu , Bishop and Goldberg , Schutz , Spivak [1975J, or Warner [1971J, and a classical one such as Eisenhart [1926J, Schouten [1954), or Synge and Schild [1956);
• a book on functional analysis such as Balakrishnan [1976), Oden , or Yosida [1971).

More advanced readers should consult other contemporary works for comparisons and other points of view. For example, we find the following additional references useful:

• Kondo [1955) for an early attempt at the use of geometry in elasticity
• Truesdell and Toupin [1960), Rivlin [1966aJ, and Eringen [I975J on basic principles
• Gurtin [I972a] on linear elasticity; (d) Knops and Wilkes [I 973J on elastic stability
• Fichera [1972a and b] and Knops and Payne  on existence and uniqueness theorems
• Bloom  on the use of geometry in dislocation theory
• Naghdi  on general shell theory and Ciarlet  on the derivation of plate theory from three-dimensional elasticity; (h) Antman [1972a],  on rod theory and bifurcations in elasticity.