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Introduction to Continuum Mechanics

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T

ME185

AF

Introduction to Continuum Mechanics

Panayiotis Papadopoulos

DR

Department of Mechanical Engineering, University of California, Berkeley

Copyright c 2008 by Panayiotis Papadopoulos

i

T

Introduction

AF

This set of notes has been written as part of teaching ME185, an elective senior-year
undergraduate course on continuum mechanics in the Department of Mechanical Engineering
at the University of California, Berkeley.
Berkeley, California

DR

August 2008

ii

P. P.

1 Introduction

T

AF

Contents

1

1.1

Solids and fluids as continuous media . . . . . . . . . . . . . . . . . . . . . .

1

1.2

History of continuum mechanics . . . . . . . . . . . . . . . . . . . . . . . . .

2

2 Mathematical preliminaries

3

2.1

Elements of set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.2

Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.3

Points, vectors and tensors in the Euclidean 3-space . . . . . . . . . . . . . .

8

2.4

Vector and tensor calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

3 Kinematics of deformation

19

Bodies, configurations and motions . . . . . . . . . . . . . . . . . . . . . . .

19

3.2

The deformation gradient and other measures of deformation . . . . . . . . .

28

3.3

Velocity gradient and other measures of deformation rate . . . . . . . . . . .

48

3.4

Superposed rigid-body motions . . . . . . . . . . . . . . . . . . . . . . . . .

53

DR

3.1

4 Basic physical principles

61

4.1

The divergence and Stokes’ theorems . . . . . . . . . . . . . . . . . . . . . .

61

4.2

The Reynolds’ transport theorem . . . . . . . . . . . . . . . . . . . . . . . .

63

4.3

The localization theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

4.4

Mass and mass density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

4.5

The principle of mass conservation . . . . . . . . . . . . . . . . . . . . . . .

69

4.6

The principles of linear and angular momentum balance . . . . . . . . . . . .

70

4.7

Stress vector and stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . .

73

iii

The transformation of mechanical fields under superposed rigid-body motions

85

4.9

The Theorem of Mechanical Energy Balance . . . . . . . . . . . . . . . . . .

88

4.10 The principle of energy balance . . . . . . . . . . . . . . . . . . . . . . . . .

91

4.11 The Green-Naghdi-Rivlin theorem . . . . . . . . . . . ...
DRAFT
ME185
Introduction to Continuum Mechanics
Panayiotis Papadopoulos
Department of Mechanical Engineering, University of California, Berkeley
Copyright
c
2008 by Pa nayiotis Papadopoulos
i
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