Mechanics Lecture Notes Part III: Foundations of Continuum Mechanics

This is primarily aimed at the Graduate Student and Postgraduate Researcher (with a sub-set appropriate for Part III-IV Engineering undergraduate students).

Last updated: 24-Feb-2015

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Table of Contents

 

 

 

 

 

Last updated

 

 

 

 

 

1.

Vectors and Tensors

1

 

27-Feb-13

 

1.1

Vector Algebra

3

 

26-Mar-15

 

1.2

Vector Spaces

10

 

27-Feb-13

 

1.3

Cartesian Vectors

15

 

27-Feb-13

 

1.4

Matrices and Element Form

22

 

27-Feb-13

 

1.5

Coordinate Transformation of Vector Components

24

 

27-Feb-13

 

1.6

Vector Calculus 1 - Differentiation

30

 

27-Feb-13

 

1.7

Vector Calculus 2 Integration

50

 

27-Feb-13

 

1.8

Tensors

67

 

27-Feb-13

 

1.9

Cartesian Tensors

75

 

23-Feb-15

 

1.10

Special Second Order Tensors & Properties of Tensors

84

 

04-May-16

 

1.11

The Eigenvalue Problem and Polar Decomposition

96

 

27-Feb-13

 

1.12

Higher Order Tensors

107

 

27-Feb-13

 

1.13

Coordinate Transformation of Tensor Components

110

 

27-Feb-13

 

1.14

Tensor Calculus 1: Tensor Fields

115

 

27-Feb-13

 

1.15

Tensor Calculus 2: Tensor Functions

124

 

27-Feb-13

 

1.16

Curvilinear Coordinates

135

 

16-Aug-12

 

1.17

Curvilinear Coordinates: Transformation Laws

152

 

24-Feb-09

 

1.18

Curvilinear Coordinates: Tensor calculus

156

 

24-Feb-09

 

1.19

Curvilinear Coordinates: Curved Geometries

175

 

24-Feb-09

 

1.A

Appendix A to Chapter 1

189

 

24-Feb-09

 

1.B

Appendix B to Chapter 1

192

 

24-Feb-09

 

 

 

 

 

 

2.

Kinematics

199

 

27-Feb-13

 

2.1

Motion

201

 

21-Mar-13

 

2.2

Deformation and Strain

207

 

30-Mar-16

 

2.3

Deformation and Strain: Further Topics

233

 

16-Aug-12

 

2.4

Material Time Derivatives

239

 

27-Mar-13

 

2.5

Deformation Rates

243

 

04-Apr-13

 

2.6

Deformation Rates: Further Topics

253

 

16-Aug-12

 

2.7

Small Strain Theory

258

 

16-Aug-12

 

2.8

Objectivity

263

 

16-Aug-12

 

2.9

Rigid Body Rotations

276

 

16-Aug-12

 

2.10

Convected Coordinates

279

 

12-Jan-15

 

2.11

Convected Coordinates: Time Rates of Change

296

 

16-Aug-12

 

2.12

Pull Back, Push Forward and Lie Time Derivatives

298

 

16-Aug-12

 

2.13

Variation and Linearisation of Kinematic Tensors

308

 

16-Aug-12

 

 

 

 

 

 

3.

Stress and the Balance Principles

315

 

16-Aug-12

 

3.1

Conservation of Mass

317

 

16-Aug-12

 

3.2

The Momentum Principles

327

 

16-Aug-12

 

3.3

The Cauchy Stress Tensor

330

 

16-Aug-12

 

3.4

Properties of the Stress Tensor

335

 

16-Aug-12

 

3.5

Stress Measures for Large Deformations

341

 

16-Aug-12

 

3.6

The Equations of Motion and Symmetry of Stress

351

 

16-Aug-12

 

3.7

Boundary Conditions and the Boundary Value Problem

356

 

16-Aug-12

 

3.8

Balance of Mechanical Energy

361

 

16-Aug-12

 

3.9

The Principles of Virtual Work and Power

369

 

16-Aug-12

 

3.10

Convected Coordinates

382

 

16-Aug-12

 

 

 

 

 

 

4.

Thermomechanics

387

 

24-Mar-13

 

4.1

Classical Thermodynamics: the First Law

389

 

24-Mar-13

 

4.2

Classical Thermodynamics: the Second Law

405

 

24-Mar-13

 

4.3

Thermodynamic Functions

414

 

24-Mar-13

 

4.4

Continuum Thermomechanics

421

 

15-Aug-14

 

 

 

 

 

 

5.

Constitutive Theory

 

 

 

 

 

 

 

 

 

Answers to Selected Problems: Chapter 1

 

 

24-Mar-13

 

Answers to Selected Problems: Chapter 2

 

 

09-Apr-13

 

Answers to Selected Problems: Chapter 3

 

 

 

 

Answers to Selected Problems: Chapter 4

 

 

15-Aug-14

 

 

References

 

 

09-Dec-16