Glass E-Book

Eric Le Bourhis

• Herausgeber: Wiley-VCH
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 534

Veröffentlichungsjahr: 2014

Beschreibung

Glass is a material with essentially unlimited application possibilities. This second edition of a comprehensive reference in glass science, points out the correlation between the performance of industrial processes and practice-relevant properties, such as strength and optical properties. Interdisciplinary in his approach, the author discusses both the science and technology, starting with an outline of history and applications, glass structure, and rheology.
The sections on properties include mechanical strength and contact resistance, ageing, mechanics of glass processes, the production and control of residual stresses, high-tech products, and current research and development. Applications include glazing, packaging, optical glass, glass fibers for reinforcement, and abrasive tools. The development of touchscreen technology showed how important were the design and resistance of thin flexible glass and these new thin aluminosilicate glasses are also discussed.

Leseprobe

CONTENTS

Cover

Related Titles

Title Page

Copyright

Foreword

Preface to the Second Edition

Preface to the First Edition

Symbols and Definitions (Units in Parentheses)

Physical Constants

List of Abbreviations

Chapter 1: Introduction

Chapter 2: Glass, A Ceramic Material1)

2.1 Four Classes of Materials

2.2 Materials Properties

2.3 Selecting Materials

2.4 Performance Indices

2.5 Shape Factors in Mechanical Design

Chapter 3: Glass Prehistory and History

3.1 Natural Glasses

3.2 Early Glasses

3.3 First Optical Glasses

3.4 Modern Glasses

Chapter 4: Applications of Glass

4.1 Glazing

4.2 Containers

4.3 Optical Glass

4.4 Glass Fibres for Insulation and Reinforcement

4.5 Abrasive Tools

4.6 Glass Manufacturers

Chapter 5: Glass Structure

5.1 Introduction

5.2 Silica Glass and Related Glasses

5.3 Borate Glass and Related Glasses

5.4 Organic and Chalcogenide Glasses

5.5 Metallic Glasses

5.6 Avoiding Crystallization

5.7 Vitroceramic Fabrication

5.8 Glass Surface

Chapter 6: Glass Rheology

6.1 Viscosity

6.2 Glass Transition and Its Observation

6.3 Viscous Response of Glass

6.4 Viscoelastic Response of Glass

6.5 Thermal Tempering of Glass

6.6 Transient Stresses

6.7 Chemical Tempering of Glass

Chapter 7: Mechanical Strength of Glass

7.1 Theoretical Strength

7.2 Tensile Resistance of Glass

7.3 Stress Concentration and Griffith Energy Balance

7.4 Linear Elasticity Crack Tip Stress Field

7.5 SIF under Non-uniform Stress

7.6 Toughness Measurement

7.7 Influence of Residual Stress on Strength and Fragmentation

7.8 Statistical Weibull Analysis

Chapter 8: Contact Resistance of Glass

8.1 Sharp and Blunt Contact

8.2 Sharp Contact Resistance

8.3 Scratch Resistance

8.4 Abrasion Resistance

8.5 Introducing a Controlled and Critical Surface Flaw

8.6 Cutting and Drilling of Glass

Chapter 9: Ageing of Glass

9.1 Fatigue in Glass

9.2 Stress Corrosion

9.3 Charles and Hillig Theory

9.4 Lifetime under Static Fatigue

9.5 Applications

9.6 NiS Phase Transformation

9.7 Crack Healing

Chapter 10: Mechanics of Glass Processes

10.1 Introduction

10.2 Float Process

10.3 Fusion Draw

10.4 Container Process

10.5 Fibre Process

Chapter 11: Production Control of Residual Stresses

11.1 Introduction

11.2 Residual Stresses in Flat Glass

11.3 Basics of Photoelasticity in Flat Glass

11.4 Stress Meters

Chapter 12: High-Tech Products and R&D

12.1 Market Trend–Driven R&D

12.2 Flat Displays

12.3 Thin-Film Technology

12.4 Residual Stresses in Thin Films

12.5 Summary

Chapter 13: Conclusion

Appendix A: Light Absorption, Dispersion and Polarization

A.1 Electromagnetic Spectrum

A.2 Light Absorption

A.3 Light Dispersion

A.4 Light Polarization

Appendix B: Atomic Structure and Bond Formation

B.1 Atomic Structure

B.2 Mendeleev Table

B.3 Bond Formation

Appendix C: Thermal Expansion and Elasticity

C.1 The α–E Trend

C.2 Qualitative Approach

C.3 Expansion Modelling

C.4 Differential Expansion Measurement

Appendix D: Falling Sphere Viscometer and Fining of Glass

D.1 Falling Sphere

D.2 Fining of Glass

Appendix E: Theoretical Strength of a Solid

Appendix F: Weibull Analysis

Appendix G: Photoelastic Set-Up for Lectures

G.1 Set-Up for Photoelastic Projection

G.2 Example of a Beam under Flexion (Transient Stresses)

G.3 Example of Tempered Specimens (Residual Stresses)

Appendix H: Instrumented Nanoindentation Applied to Thin Films

H.1 Instrumented Nanoindentation

H.2 Indentation Strain Field

H.3 Hardness, Yield Stress and Representative Flow Stress

H.4 Coating–Substrate Composite Response

H.5 Time-Dependent Response

H.6 Elastic–Plastic Ratios

Appendix I: Strain and Stress

I.1 Stress and Strain

I.2 Stress and Strain Tensors

I.3 Uniaxial Tensile Test

I.4 Simple Shear

I.5 Plane Stress

I.6 Hydrostatic Pressure and Stress Deviator

I.7 Generalized Hooke's Law

I.8 Kelvin and Maxwell Models

I.9 Generalized Maxwell Model

Appendix J: Flow and Plasticity in Glass

J.1 Introduction

J.2 From Newtonian to Non-Newtonian Flow

J.3 From Homogeneous to Heterogeneous Flow

Appendix K: Finite Element Analysis

K.1 FEM of the Pressing of a Parison

K.2 FEM of the Precision Moulding of a Glass Lens

K.3 FEM of Fracture

K.4 FEM of Contact Loading

Appendix L: X-Ray Diffraction Analysis of Thin-Film Residual Stresses

L.1 Thin-Film Stress and Strain

L.2 X-Ray Diffraction Method

L.3 The

e

–sin

2

ψ Method

Appendix M: Diffusion

M.1 Diffusion Laws

M.2 Steady-State Diffusion

M.3 Non-Steady-State Diffusion

Glossary

References

Index

End User License Agreement

List of Tables

Table 2.1

Table 2.2

Table 2.3

Table 2.4

Table 2.5

Table 3.1

Table 4.1

Table 5.1

Table 5.2

Table 5.3

Table 6.1

Table 6.2

Table 6.3

Table 6.4

Table 6.5

Table 7.1

Table 7.2

Table 7.3

Table 8.1

Table 8.2

Table 8.3

Table 10.1

Table 10.2

Table 10.3

Table 11.1

Table 11.2

Table 11.3

Table 12.1

Table 12.2

List of Illustrations

Figure 1.1

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.12

Figure 2.13

Figure 2.14

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9

Figure 3.10

Figure 3.11

Figure 3.12

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 4.14

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11

Figure 5.12

Figure 5.13

Figure 5.14

Figure 5.15

Figure 5.16

Figure 5.17

Figure 5.18

Figure 5.19

Figure 5.20

Figure 5.21

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

Figure 6.9

Figure 6.10

Figure 6.11

Figure 6.12

Figure 6.13

Figure 6.14

Figure 6.15

Figure 6.16

Figure 6.17

Figure 6.18

Figure 6.19

Figure 6.20

Figure 6.21

Figure 6.22

Figure 6.23

Figure 6.24

Figure 6.25

Figure 6.26

Figure 6.27

Figure 6.28

Figure 6.29

Figure 6.30

Figure 6.31

Figure 6.32

Figure 6.33

Figure 6.34

Figure 6.35

Figure 6.36

Figure 6.37

Figure 6.38

Figure 6.39

Figure 7.1

Figure 7.2

Figure 7.3

Figure 7.4

Figure 7.5

Figure 7.6

Figure 7.7

Figure 7.8

Figure 7.9

Figure 7.10

Figure 7.11

Figure 7.12

Figure 7.13

Figure 7.14

Figure 7.15

Figure 7.16

Figure 8.1

Figure 8.2

Figure 8.3

Figure 8.4

Figure 8.5

Figure 8.6

Figure 8.7

Figure 8.8

Figure 8.9

Figure 8.10

Figure 8.11

Figure 8.12

Figure 8.13

Figure 8.14

Figure 8.15

Figure 8.16

Figure 8.17

Figure 8.18

Figure 8.19

Figure 8.20

Figure 8.21

Figure 8.22

Figure 8.23

Figure 8.24

Figure 8.25

Figure 8.26

Figure 8.27

Figure 8.28

Figure 8.29

Figure 8.30

Figure 9.1

Figure 9.2

Figure 9.3

Figure 9.4

Figure 9.5

Figure 9.6

Figure 9.7

Figure 9.8

Figure 10.1

Figure 10.2

Figure 10.3

Figure 10.4

Figure 10.5

Figure 10.6

Figure 10.7

Figure 10.8

Figure 10.9

Figure 10.10

Figure 10.11

Figure 10.12

Figure 10.13

Figure 10.14

Figure 10.15

Figure 10.16

Figure 10.17

Figure 10.18

Figure 10.19

Figure 10.20

Figure 11.1

Figure 11.2

Figure 11.3

Figure 11.4

Figure 11.5

Figure 11.6

Figure 11.7

Figure 11.8

Figure 11.9

Figure 11.10

Figure 11.11

Figure 11.12

Figure 12.1

Figure 12.2

Figure 12.3

Figure 12.4

Figure 12.5

Figure 12.6

Figure 12.7

Figure 12.8

Figure 12.9

Figure 12.10

Figure 12.11

Figure 12.12

Figure 12.13

Figure 12.14

Figure 12.15

Figure 12.16

Figure 12.17

Figure 12.18

Figure 12.19

Figure 12.20

Figure 12.21

Figure 12.22

Figure 12.23

Figure 12.24

Figure 12.25

Guide

Cover

Table of Contents

Begin Reading

Begin Reading

Begin Reading

Chapter 1

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Glass

Mechanics and Technology

Second Edition

Eric Le Bourhis

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: applied for

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>.

© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.

Print ISBN: 978-3-527-33705-7

ePDF ISBN: 978-3-527-67942-3

ePub ISBN: 978-3-527-67944-7

Mobi ISBN: 978-3-527-67945-4

oBook ISBN: 978-3-527-67946-1

Foreword

It is a pleasure to introduce this book which approaches glass from the scientific and technological sides. This is an important contribution to the glass literature since few books address both these aspects together. Glass mechanics has appeared as a preliminary for further development of glass functionality and performance. This issue is complex and requires detailed understanding of glass science and industry. This textbook gives a comprehensive approach to glass mechanics taking constant care to keep as close as possible to the applications. The author shares his own experiences both in the academic and industrial fields, resulting in an illustrative and relevant description of glass research and development. This book will be of great interest as training material for people entering the glass industry as well as future engineers and researchers in the field.

FranceJune 2007 and September 2014

AubervilliersRené Gy

Preface to the Second Edition

It has been seven years since this book was published in October 2007. On revising the original edition, I have benefitted from the first edition as a textbook, from comments from colleagues, students and readers. In this period, a lot of new material has been published; this second edition of Glass adds these results of new research and research and development in the field. A new International Journal of Applied Glass Science has been launched to provide further “source of information dealing with the application of glass science and engineering across the entire material spectrum” (Fuller and Steen, Int. J. Appl. Glass Sci. 1, 1 [2010]). In fact, the mechanics of glass attracts much interest both from academia and industry and further emphasizes the focus of the present project (see also the Preface of first edition and Introduction). Much insight has been gained meanwhile on the relationship between mechanical response and structure in terms of Poisson ratio, fragility and network connectivity (see Chapters 5 to 7 and Appendix I) as well as on the brittle nature of glass fracture (Chapter 7). The development of smartphones, tablet PCs and televisions with touch screen technology shows how important are the design and resistance of thin flexible glass. From 2010, several companies have been developing chemically strengthened thin aluminosilicate glasses for these applications (see Chapters 6, 7, 8 and 12). There has also been interest in elevated-temperature response (contact resistance and thin-film stress evolution; see Chapters 8 and 12 and Appendix L) as it is of utmost importance in the processing of functionalized glass articles (Chapters 10 and 12). Further historical perspectives are proposed (Chapter 3) and help getting insights into the industrial steps. For the preparation of this second edition, I should like to acknowledge again support from Wiley VCH, in particular, R. Weber and L. Fenske.

This book is dedicated to my wife, sons and parents.

Futuroscope-ChasseneuilAugust 2014

Eric Le Bourhis

Preface to the First Edition

The project of writing this book started from lectures given to future engineers in materials science. The prime idea was to present both science and technology that participate in industrial R&D. I chose the glass industry as an example having working experiences in both industrial and academic sides. The science–technology approach and concepts developed and illustrated in the book can be utilized in other contexts and for other materials. Fracture and ageing, photoelasticity, rheology, contact resistance, diffusion and production tools issues are shared by many fields. For teaching purposes, appendices suggest exercises and experimental approaches. It is hoped to offer an overview of the background science and technology used in industry and R&D. The book focuses on mechanics, which has to be considered throughout the industrial process.

Preparing this book, I am indebted to many colleagues from industry and academe for their support, discussions and motivations. I would like to thank R. Gy, H. Arribart, J. Prieur, S. Valladeau, E. Bodiot, O. Gaume, N. Brown, J. Zhang, C. Bleuze, J.M. Poujade, P.H. Guering, M.H. Chopinet, E. Barthel, L. Lesage (Saint Gobain group), A. Alliche (ECP Paris), R. Vacher, C. Levelut, D. Cavaillé, J. Pelous (Montpellier University), M.M. Chaudhri (Cambridge University), C.R. Kurkjian (Rutgers University), T. Rouxel, J.C. Sangleboeuf, V. Keryvin (Rennes University), G. Patriarche, L. Largeau (LPN Marcoussis), P. Gadaud (ENSMA Poitiers), D. Mousserion, P.O. Renault, P. Goudeau, J. Bonneville, C. Jaouen, C. Templier, T. Cabioc'h, F. Tranchant, C. Coupeau (Poitiers University), L. Pranevicius and S. Tamulevicius (Kaunas University). I should like to acknowledge support from Wiley VCH, in particular K. Sora and W. Würst.

The author also wishes to thank S. Valladeau for his help in carrying out again fractography and photoelasticity experiments, and R. Gy, E. Barthel and F. Mammeri for critical reading and suggestions on parts of the book. Some students worked with me to develop research and applicative studies, and their contributions have been of utmost importance. I would like to thank M. Bustarret, C. Teles, M. Beaurain, N. Tournerie, D. Metayer, F. Mammeri, D. Faurie and N. Chemin, who shared investigation interests during their training period.

This book is dedicated to my wife, sons and parents.

Futuroscope-ChasseneuilSeptember 2007

Eric Le Bourhis

Symbols and Definitions (Units in Parentheses)

a

Contact radius

(m)

a

Crack size

(m)

a

Distance

(m)

a

Acceleration

(m s

−2

)

a

0

Atomic diffusion jump

(m)

a

c

Atomic diffusion interface path

(m)

a

Mark diagonal

(m)

A

Cation field intensity

(m

−2

)

A

Crack surface

(m

2

)

A

Elastic anisotropy

A

p

Projected surface

(m

2

)

A

c

Contact projected surface

(m

2

)

B

Brittleness index

(m

−1/2

)

B

,

B

Magnetic field

(T)

BON

Bridging oxygen number

c

(

x

,

t

)

Concentration

(m

−3

)

c

Central plastic zone radius

(m)

c

Radial crack length

(m)

c

l

Lateral crack length

(m)

c

v

Specific heat (constant volume)

(J kg

−1

K

−1

)

c

P

Specific heat (constant pressure)

(J kg

−1

K

−1

)

c

R

Cooling rate

(K s

−1

)

C

Brewster constant

(B)

C

g

Atomic packing density

De

Deborah number

D

Diffusion coefficient

(m

2

s

−1

)

E

,

E

Electric field

(V m

−1

)

E

Photon energy

(J)

E

Young's modulus (2(1 +

ν

)

G

)

(Pa)

E

f

Film Young's modulus

(Pa)

E

i

Indenter Young's modulus

(Pa)

E

p

Potential energy

(J)

E

s

Specimen Young's modulus

(Pa)

E

s

Substrate Young's modulus

(Pa)

f

Friction coefficient

(N m

−1

s)

F

Load, force

(N)

F

c

Critical load for fracture

(N)

F

m

Maximum force

(N)

g

Gravitational acceleration

(m s

−2

)

g

(

T

)

Growth rate

(m s

−1

)

Δ

g

Energy barrier for diffusion jump

(J at

−1

)

Δ

g

′

Energy barrier for diffusion at nucleus interface

(J at

−1

)

Δ

g

″

Energy barrier for diffusion at particle interface

(J at

−1

)

G

Shear modulus

(Pa)

G

′

Conservation or storage modulus

(Pa)

G

″

Loss modulus

(Pa)

G

(

t

)

Shear relaxation modulus

(Pa)

Δ

G

Energy barrier for diffusion jump

(J mol

−1

)

Δ

G

′

Energy barrier for diffusion at nucleus interface

(J mol

−1

)

Δ

G

″

Energy barrier for diffusion at particle interface

(J mol

−1

)

Δ

G

v

Molar enthalpy of the transformation from liquid to crystal

(J mol

−1

)

h

Penetration

(m)

h

Total penetration

(m)

h

c

Contact penetration

(m)

h

e

Elastic penetration

(m)

h

l

Depth of lateral crack

(m)

h

m

Maximum penetration

(m)

h

p

Plastic penetration

(m)

h

r

Residual penetration

(m)

h

v

Viscous penetration

(m)

H

Hardness or mean pressure

(Pa)

H

v

Vickers hardness (0.927p)

(Pa)

H

(

t

)

Heaviside function (=1 for

t

≥ 0; = 0 for

t

< 0)

I

Second moment

(m

4

)

I

0

Incidence angle

(rad)

J

Diffusion flux

(m

−2

s

−1

)

J

Shear compliance

(Pa

−1

)

J

(

t

)

Shear compliance

(Pa

−1

)

k

Thermal conductivity

(W m

−1

K

−1

)

k

Wave vector

(m

−1

)

K

Stress intensity factor (SIF)

(Pa m

1/2

)

K

c

Fracture toughness

(Pa m

1/2

)

L

Length

(m)

m

Fragility

m

Weibull modulus

m

,

M

Mass

(kg)

M

Torque

(N m)

n

Optical index

n

Volumic concentration

(m

−3

)

n

Surface concentration

(m

−2

)

n

(

T

)

Nucleation rate

(m

−3

s

−1

)

N

Normal force

(N)

p

Polarization vector

(C m)

p

Pressure

(Pa)

P

Property

P

0

Hydrostatic pressure

(Pa)

P

s

Probability

Q

Heat

(J)

r

Position or distance

(m)

r

Radius

(m)

r

c

Cation radius

(m)

r

0

Interatomic distance

(m)

r

0

Oxygen radius

(m)

R

Radius

(m)

RD(

r

)

Radial distribution

(m

-1

)

S

Surface, section

(m

2

)

S

Stiffness

(N m

−1

)

S

Entropy

(J K

−1

)

t

Time

(s)

t

f

Film thickness

(m)

t

H

Glass thickness

(m)

T

Tangential force

(N)

T

Temperature

(K)

T

f

Glass fictive temperature

(K)

T

g

Glass transition temperature

(K)

T

L

Liquidus temperature

(K)

T

m

Melting temperature

(K)

T

R

Relaxation time (

η

/

G

)

(s)

T

S

Solidus temperature

(K)

Δ

T

Temperature variation

(K)

tan(δ)

Loss factor

U

Fractured specimen total energy

(J)

v

Velocity

(m s

−1

)

v

f

Free volume (per atom)

(m

3

)

v

l

Limit velocity

(m s

−1

)

v

*

Minimal volume for atomic jump

(m

3

)

V

Volume

(m

3

)

V

m

Molar volume

(m

3

mol

−1

)

w

Angular velocity

(rad s

−1

)

w

c

Critical energy for a particle to form

(J)

Y

Yield stress or yield pressure

(Pa)

Y

R

Representative flow stress

(Pa)

z

Impacter (indenter) penetration

(m)

Z

Atomic number

Z

Cation valence

α

Thermal expansion

(K

−1

)

α

f

Thermal expansion coefficient of the film

(K

−1

)

α

l

Thermal expansion coefficient of the liquid

(K

−1

)

α

s

Thermal expansion coefficient of the solid

(K

−1

)

α

s

Thermal expansion coefficient of the substrate

(K

−1

)

β

Phase shift

(rad)

β

Angle between indenter flank and specimen surface

(rad)

γ

Shear strain

δ

Loss angle

(rad)

δ

Relative retardation

(m)

δ

Volumetric strain

ε

Strain

ε

1

,

ε

2

,

ε

3

Principal strains

ζ

,

ζ

cl

,

ζ

cs

,

ζ

sl

Surface energies

(J m

−2

)

η

Viscosity

(Pa s)

η

R

Viscosity at reference temperature

(Pa s)

θ

Wetting angle

(rad)

θ

Distortion

(rad)

λ

Plastic zone size

(m)

λ

Wavelength

(m)

λ

m

Wavelength at maximum light intensity

(m)

µ

Coefficient of friction (COF)

ν

Frequency

(Hz)

ν

Poisson's ratio

ν

E

Atomic transition frequency, Einstein frequency

(Hz)

ν

f

Film Poisson's ratio

ν

I

Indenter Poisson's ratio

ν

s

Specimen Poisson's ratio

ν

s

Substrate Poisson's ratio

ξ

Reduced time

(s)

ρ

(r)

Atomic density

(m

−3

)

ρ

0

Average atomic density

(m

−3

)

ρ

Density

(kg m

−3

)

ρ

w

Water density

(kg m

−3

)

ρ

/

ρ

w

Relative density

σ

Stress

(Pa)

σ

1

,

σ

2

,

σ

3

Principal stresses

(Pa)

σ

c

Core stress (

σ

(

z

= 0))

(Pa)

σ

m

Mean stress

(Pa)

σ

M

Maximum stress

(Pa)

σ

f

Film stress

(Pa)

σ

f

Ultimate stress

(Pa)

σ

r

Radial stress

(Pa)

σ

s

Substrate stress

(Pa)

σ

s

Surface residual stress

(Pa)

σ

t

Tangent (hoop) stress

(Pa)

τ

Shear stress

(Pa)

υ

Reciprocal dispersion

ϕ

Shape factor

ϕ

i

Volume fraction

ϕ

(

T

)

Shift factor

ϕ

Heat flow

(J s

−1

)

ψ

1

(

t

)

Shear relaxation function

ψ

2

(

t

)

Hydrostatic relaxation function

Ω

Atomic volume

(m

3

)

Shorthand Notation

Gradient operator

∂/∂

t

, ∂/∂

x

Partial derivatives

For an

x

only or

r

only space variation = d/d

x

or d/d

r

(m

−1

)

Overbar or overdot = d/d

t

Time derivative

(s

−1

)

Double overdot = d

2

/d

t

2

Second time derivative

(s

−2

)

′ = d/d

V

Volume derivative

(m

−3

)

[]

Concentration

(mol l

−1

or m

−3

)