Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

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vi Contents

x Preface

to attract attention and to be instructive and helpful to the reader. Necessary mathematics and physics are explained in the text. The approach used in each of the eight chapters will enable the reader to work through the chapters in order of appearance, each topic being presented in a logical sequence and based on the preceding topics.

A proper understanding of the subject matter requires knowledge of tensor algebra and tensor calculus. For most of the derivations throughout the text I have used symbolic notation with those clear bold-faced symbols which give the subject matter a distinguished beauty. However, for higher-order tensors and for final results in most of the derivations I have used index notation, which provides the reader with more insight. Terminology is printed in bold-face where it appears for the first time while the notation used in the text is defined at the appropriate point.

For those who have not been exposed to the necessary mathematics I have included a chapter on tensor algebra and tensor calculus. It includes the essential ideas of linearization in the form of the concept of the directional derivative. Chapter 1 summarizes elementary properties which are needed for the vector and tensor manipulations performed in all subsequent chapters and which are necessary to many problems that arise frequently in engineering and physics.

It is the prime consideration of Chapter 2 to use tensor analysis for the description of the motion and finite deformation of continua. The continuum approach is introduced along with the notion 'Lagrangian' (material) and 'Eulerian' (spatial) descriptions. In a systematic way the most important kinematic tensors are provided and their physical significance explained. The push-forward and pull-back operations for material and spatial quantities and the concept of the Lie time derivative are introduced. The concept of stress is the main topic of Chapter 3. Cauchy's stress theorem is introduced, along with the Cauchy and first Piola-Kirchhoff traction vectors, and the essential stress tensors are defined and their interrelationships discussed. In Chapter 4 attention is focused on the discussion of the balance principles. Both statics and dynamics are treated. Based upon continuum thermodynamics the entropy inequality principle is provided and the general structure of all principles is summarized as the master balance (inequality) principle. Chapter 5 deals with important aspects of objectivity, which plays a crucial part in nonlinear continuum mechanics. A discussion of change of observer and superimposed rigid-body motions is followed by a development of objective (stress) rates and invariance of elastic material response.

Chapters 6 and 7 form the central part of the book and provide insight in the construction of nonlinear constitutive equations for the description of the mechanical and thermomechanical behavior of solids. These two chapters show the essential richness of the field. They are written for those who want to gain experience in handling material models and deriving stress relations and the associated elasticity tensors that are fundamental for finite element methods. Several examples and exercises are aimed at enabling the reader to think in terms of constitutive models and to formulate more com-

Preface xi

plex material models. All of the types of constitutive equations presented are accessible for use within finite element procedures.

The bulk of Chapter 6 is concerned with finite elasticity and finite viscoelasticity. It includes a discussion of isotropic, incompressible and compressible hyperelastic materials and provides constitutive models for transversely isotropic and composite materials which are suitable for a large number of applications in practical engineering. An approach to inelastic materials with internal variables is given along with instructive examples of hyperelastic materials that involve relaxation and/or creep effects and isotropic damage mechanisms at finite strains. The main purpose of Chapter 7 is to provide an introduction to the thermodynamics of materials. This chapter is devoted not only to the foundation of continuum thermodynamics but also to selected topics of statistical thermodynamics. It starts with a statistical approach by summarizing important physical aspects of the thermoelastic behavior of molecular networks (for example, amorphous solid polymers), based almost entirely on an entropy concept, and continues with a systematic phenomenological approach including finite thermoelasticity and finite thermoviscoelasticity. The stress-strain-temperature response of so-called entropic elastic materials is discussed in more detail and based on a representative example which is concerned with the adiabatic stretching of a rubber band. Typical thermomechanical coupling effects are studied.

Chapter 8 is designed to cover the essential features of the most important variational principles that are very useful in formulating approximation techniques such as the finite element method. Although finite elements are not treated in this text, it is hoped that this chapter will be attractive to those who approach the subject from the computational side. It shows the relationship between the strong and weak forms of initial boundary-value problems, presents the classical principle of virtual work in both spatial and material descriptions and its linearized form. Twoand three-field variational principles are also discussed. The present text ends where conventionally a book on the finite element method would begin.

There are numerous worked examples adjacent to the relevant text. These have the goal of clarifying and supplementing the subject matter. In many cases they are straightforward, but provide an essential part of the text. The symbol • is used to denote the end of an exercise or a proof. The end-of chapter exercises are for homework. The (almost) 200 exercises provided are designed to supplement the text and to consolidate concepts discussed in the text. Most of them serve the purpose of stimulating the reader to further study and to reinforce and develop practical skills in nonlinear continuum and solid mechanics, towards the direction of computational mechanics. In many cases the solutions of selected exercises are given and frequently used later in further developments. Therefore, it should be instructive for the reader to work through a reasonable number of exercises.

Numerous references to supplementary material are suggested and discussed briefly

xii Preface

throughout the book. However, for a book of this kind it is not possible to give a comprehensive bibliography of the field. Some of the references listed serve as a starting point for more advanced studies.

The material in this book is based on a sequence of courses that I have taught at the University of Technology in Graz and Vienna. The mechanics and thermodynamics of solids are relevant to all branches of engineering, to applied mechanics, mathematics, physics and material science, and it is a central field in biomechanics. This book is primarily addressed to graduate students, researchers and practitioners, although it has also proven to be of interest to advanced undergraduates. Although I have tried to provide a textbook that is self-contained and appropriate for self-instruction, it is desirable that the reader has a reasonable background in elementary mechanics and thermodynamics.

I feel that Chapters 2-5 and parts of the remaining chapters are well suited for a complete course on nonlinear continuum mechanics lasting two semesters or three quarters. A one semester or one quarter course in the nonlinear mechanics of solids would focus on Chapter 6, while a one quarter course in the thermodynamics of solids could be based on Chapter 7. Chapter I is the core for a course that provides the student with the necessary background in vector and tensor analysis. Chapter 8 is certainly not designed to train specialists in variational principles, but to form a basic one quarter course at the graduate level.

I believe that the present textbook, in providing many applications to engineering science, is not too advanced mathematically. Of course, some of the results presented may be derived with the help of more advanced mathematics using theorems and proofs. I hope that this book will help pure engineers to teach nonlinear continuum mechanics and solid mechanics.

Naturally, as the author, I take full responsibility for not doing it better. Comments and criticisms will be welcome and greatly appreciated. I have learnt that the spirit of modern continuum mechanics and the underlying mathematics are as important to the design of powerful finite element models as are insights in the theoretical foundation of constitutive models and variational principles. A successful transfer of that combination to the reader would indicate that my objective has been achieved.