Introduction to Continuum Mechanics

Introduction to Continuum Mechanics

©Romesh C. Batra, 1998, 2000

CONTENTS

Chapter 1 Introduction

1.: What is Mechanics?1-1
2.: Continuum Mechanics1-2
3.: An example of an ad-hoc approach1-3

Chapter 2 Mathematical Preliminaries

1.: Summation convention, Dummy Indices2-1
2.: Free Indices2-3
3.: Kronecker Delta2-4
4.: Index Notation2-6
5.: Permutation Symbol2-7
6.: Manipulation with the Indicial Notation2-9
7.: Translation and Rotation of Coordinate Axes2-12
8.: Tensors2-19

Chapter 3 Kinematics

1.: Description of Motion of a Continuum 3-1
2.: Referential and Spatial Descriptions3-3
3.: Displacement Vector3-5
4.: Restrictions on Continuous Deformation of a
Deformable Body3-6
5.: Material Derivative3-9
6.: Finding Acceleration of a Particle from a given
Velocity Field3-11
7.: Deformation Gradient3-14
8.: Strain Tensors 3-21
9.: Principal Strains3-24
10.: Deformation of Areas and Volumes3-33
11.: Mass Density. Equation of Continuity3-35
12.: Rate of Deformation3-38
13.: Polar Decomposition3-45
14.: Infinitesimal Deformations 3-50

Chapter 4 The Stress Tensor

1.: Kinetics of a Continuous Media4-1
2.: Boundary Conditions for the Stress Tensor4-10
3.: Nominal Stress Tensor 4-13
4.: Transformation of Stress Tensor under Rotation of Axes4-15
5.: Principal Stresses. Maximum Shear Stress4-20

Chaper 5 The Linear Elastic Material

1.

Introduction5-1

2.

Linear Elastic Solid. Hookean Material5-1

3.

Equations of the Infinitesimal Theory of Elasticity5-7

4.

Principle of Superposition5-10

5.

A Uniqueness Theorem5-1ll

6.

Compatibility Equations Expressed in terms of the Stress
Components for an Isotropic, Homogeneous, Linear, Elastic Solid51-4

7.

Some Examples. 5-18

a): Vibration of an Infinite Plate
b): Torsion of a Circular Shaft5-22
c): Torsion of Non-Circular Cylinders5-26

Chaper 6 The Linear Elastic Material

1.: Constitutive Relation6-1
2.: Formulation of an Initial-Boundary-Varlue Problem6-3
3.: Examples6-4