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).

 

Please send any errors, typos, comments, etc. to pa.kelly@auckland.ac.nz (I am always interested to hear who is reading these notes, and where you are from)

 

To reference theses notes, see here.

 

Return to Mechanics Lecture Notes

 

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

 

13-Apr-18

 

1.3

Cartesian Vectors

15

 

27-Sep-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

 

24-Nov-17

 

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-Mar-18

 

1.17

Curvilinear Coordinates: Transformation Laws

160

 

05-Mar-18

 

1.18

Curvilinear Coordinates: Tensor calculus

164

 

05-Mar-18

 

1.19

Curvilinear Coordinates: Curved Geometries

183

 

05-Mar-18

 

1.A

Appendix A to Chapter 1

197

 

05-Mar-18

 

1.B

Appendix B to Chapter 1

200

 

05-Mar-18

 

complete pdf

 

 

 

 

 

 

 

 

 

2.

Kinematics

199

 

27-Feb-13

 

2.1

Motion

201

 

28-Feb-13

 

2.2

Deformation and Strain

207

 

30-Mar-16

 

2.3

Deformation and Strain: Further Topics

233

 

23-Jul-09

 

2.4

Material Time Derivatives

239

 

27-Mar-13

 

2.5

Deformation Rates

243

 

04-Apr-13

 

2.6

Deformation Rates: Further Topics

253

 

13-Mar-09

 

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

 

20-Mar-18

 

2.11

Convected Coordinates: Time Rates of Change

304

 

20-Mar-18

 

2.12

Pull Back, Push Forward and Lie Time Derivatives

307

 

20-Mar-18

 

2.13

Variation and Linearisation of Kinematic Tensors

327

 

20-Mar-18

 

complete pdf

 

 

 

 

 

 

 

 

 

3.

Stress and the Balance Principles

315

 

16-Aug-12

 

3.1

Conservation of Mass

317

 

21-May-14

 

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

 

28-Jun-18

 

3.10

Convected Coordinates

382

 

16-Aug-12

 

complete pdf

 

 

 

 

 

 

 

 

 

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

 

complete pdf

 

 

 

 

 

 

 

 

 

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