INTERNATIONAL UNION OF PURE
AND APPLIED CHEMISTRY

MACROMOLECULAR DIVISION
COMMISSION ON MACROMOLECULAR NOMENCLATURE*

DEFINITION OF TERMS RELATING TO THE NON-ULTIMATE MECHANICAL PROPERTIES OF POLYMERS

(IUPAC Recommendations 1997)

Prepared by a Working Group consisting of 
A. Kaye (UK), R. F. T. Stepto (UK), W. J. Work (USA), 
J. V. Alemán (Spain), A. Ya. Malkin (Russia)

*Membership of the Commission during the preparation of this report (1993-97) was as follows:

Titular Members:  R. E. Bareiss (Germany, to 1993); M. Barón (Argentina, Associate Member to 1995, Titular Member from 1996); K. Hatada (Japan); J. Kahovec (Czech Republic); E. Maréchal (France); W. V. Metanomski (USA); C. Noël (France, to 1993); V. P. Shibaev (Russia, to 1995, Associate Member from 1996); R. F. T. Stepto (UK, Chairman); W. J. Work (USA, Secretary).

Associate Members  contributing to this report:  J. V. Alemán (Spain, to 1995); M. Hess (Germany, from 1996); K. Horie (Japan, from 1996); J.-I.Jin (Korea, National Representative to 1993, Associate Member from 1994); O. Kramer (Denmark, from 1996); P. Kubisa (Poland, from 1996); S. Penczek (Poland, from 1994); U. W. Suter (Switzerland, to 1993).

Others contributing to this report:  H. A. Barnes (UK); R. B. Bird (USA); C. B. Bucknall (UK); J. M. Dealy (USA); J. D. Ferry (USA); R. B. Fox (USA); W. W. Graessley (USA); A. D. Jenkins (UK); A. Kaye (UK); P. Kratochvíl (Czech Republic); T. Masuda (Japan); I. Mita (Japan); D. R. Moore (UK) and Commission IV.2; K. Osaki (Japan); N. A. Platé (Russia, National Representative); J. C. Rigg (The Netherlands); R. Simha (USA); A. Sirigu (Italy, National Representative); R. I. Tanner (Australia); K. Walters (UK).

Republication or reproduction of this report or its storage and/or dissemination by electronic means is permitted without the need for formal IUPAC permission on condition that an acknowledgement, with full reference to the source along with use of the copyright symbol ©, the name IUPAC and the year of publication are prominently visible.  Publication of a translation into another language is subject to the additional condition of prior approval from the relevant IUPAC National Adhering Organization.

DEFINITION OF TERMS RELATING TO THE NON-ULTIMATE MECHANICAL PROPERTIES OF POLYMERS

SUMMARY

	The document gives definitions of terms related to the non-ultimate mechanical behaviour of polymeric materials, in particular bulk polymers and concentrated solutions and their elastic and viscoelastic properties.

	The terms which have been selected are those met in the conventional mechanical characterization of isotropic polymeric materials.  They have additionally been limited to those which can be defined precisely and with mathematical rigour.  They are arranged in sections dealing with basic definitions of stress and strain, deformations used experimentally, stresses observed experimentally, quantities relating stress and deformation, linear viscoelastic behaviour, and oscillatory deformations and stresses used experimentally for solids.

	An index, an alphabetical list of terms and a glossary of symbols are included for ease of reference.

INTRODUCTION

	This document gives definitions of terms related to the non-ultimate mechanical behaviour or mechanical behaviour prior to failure of polymeric materials, in particular bulk polymers and concentrated solutions and their elastic and viscoelastic properties.

	The terms are arranged into sections dealing with basic definitions of stress and strain, deformations used experimentally, stresses observed experimentally, quantities relating stress and deformation, linear viscoelastic behaviour, and oscillatory deformations and stresses used experimentally for solids.  The terms which have been selected are those met in the conventional mechanical characterization of polymeric materials.

	To compile the definitions, a number of sources have been used.  A number of the definitions were adapted from an International Standards Organization (ISO) manuscript on Plastics Vocabulary1.  Where possible, the names for properties, their definitions and the symbols for linear viscoelastic properties were checked against past compilations of terminology2-6.  Other documents consulted include ASTM publications 7-13.

	The document does not deal with the properties of anisotropic materials.  This is an extensive subject in its own right and the reader is referred to specialized texts 14,15 for information.

	In the list of contents, main terms separated by / are alternative names, and terms in parentheses give those which are defined in the context of main terms, usually as notes to the definitions of main terms, with their names printed in bold type in the main text.  Multicomponent quantities (vectors, tensors, matrices) are printed in bold type.  Names printed in italics are defined elsewhere in the document and their definitions can be found by reference to the alphabetical list of terms.

CONTENTS

1. BASIC DEFINITIONS

											Page

1.1	traction/stress vector								8

1.2	stress/stress tensor (true stress, plane stress)					8

1.3	deformation of an elastic solid (homogeneous deformation,
	inhomogeneous deformation, deformation gradients)			9

1.4	deformation gradient tensor for an elastic solid				9

1.5	deformation of a viscoelastic liquid or solid
	(homogeneous deformation)							10

1.6	deformation gradients in a viscoelastic liquid or solid			11

1.7	deformation gradient tensor for a viscoelastic liquid or solid			11

1.8	strain tensor (C, B, C-1, B-1, plane strain)					11

1.9	Cauchy tensor (C) for a viscoelastic liquid or solid				12

1.10	Green tensor (B) for an elastic solid (small-strain tensor)			12

1.11	Finger tensor (C-1) for a viscoelastic liquid or solid				13

1.12	rate-of-strain tensor for a viscoelastic liquid or solid in
	homogeneous deformation							13

1.13	vorticity tensor for a viscoelastic liquid or solid in
	homogeneous deformation							14

1.14	Rivlin-Ericksen tensors for a viscoelastic liquid or solid
	in homogeneous deformation.							14

2. DEFORMATIONS USED EXPERIMENTALLY

2.1	general orthogonal homogeneous deformation of an
	elastic solid (deformation gradient, deformation ratio)			16

											Page

2.2	uniaxial deformation of an elastic solid					16

2.3	uniaxial deformation ratio/deformation ratio (extension ratio)		17

2.4	uniaxial strain/engineering strain (tensile stain,
	compressive strain)								17

2.5	Hencky strain									18

2.6	Poisson’s ratio (lateral strain/lateral contraction ratio)			18

2.7	pure shear of an elastic solid							19

2.8	simple shear of an elastic solid (shear/shear strain)				19

2.9	bulk compression/volume compression/isotropic
	compression/bulk compressive strain						20

2.10	general homogeneous deformation or flow of a viscoelastic
	liquid or solid									20

2.11	homogeneous orthogonal deformation or flow of an
	incompressible viscoelastic liquid or solid (uniaxial deformation
	or flow)									20

2.12	steady uniaxial homogeneous elongational deformation or
	flow of an incompressible viscoelastic liquid or solid 
	(elongational/extensional strain rate)						21

2.13	homogeneous simple shear deformation or flow of an
	incompressible viscoelastic liquid or solid (shear, shear rate,
	steady (simple) shear flow, oscillatory (simple) shear flow)			22

3. STRESSES OBSERVED EXPERIMENTALLY

3.1	stress tensor resulting from an orthogonal deformation or flow
	(uniaxial orthogonal deformation or flow, pure shear deformation or
	flow, pure shear stress)							23

3.2	tensile stress									23

3.3	compressive stress								24
											Page

3.4	engineering stress/nominal stress						24

3.5	stress tensor resulting from a simple shear deformation or flow
	(normal stresses, shear stress)							24

3.6	first normal-stress difference/first normal-stress function			25

3.7	second normal-stress difference/second normal-stress function		25

4. QUANTITIES RELATING STRESS AND DEFORMATION

4.1	constitutive equation for an elastic solid (stored energy function)		26

4.2	constitutive equation for an incompressible viscoelastic liquid or 
	solid (Newtonian liquid, non-Newtonian liquid)				27

4.3	modulus (elastic modulus/modulus of elasticity)				28

4.4	compliance									28

4.5	bulk modulus/bulk compressive modulus					28

4.6	bulk compliance/bulk compressive compliance				29

4.7	Young’s modulus (tensile modulus, secant modulus, tangent
	modulus)									29

4.8	uniaxial compliance (tensile compliance)					30

4.9	extensional viscosity/elongational viscosity					31

4.10	shear modulus									31

4.11	shear compliance								32

4.12	shear viscosity/coefficient of viscosity/viscosity (apparent
	viscosity, zero-shear viscosity, infinite-shear viscosity)			32

4.13	first normal-stress coefficient							33

4.14	second normal-stress coefficient						33

5. LINEAR VISCOELASTIC BEHAVIOUR
											
											Page

5.1	viscoelasticity (dynamic stress or strain)					34

5.2	linear viscoelastic behaviour (liquid, solid)					34

5.3	Maxwell model/Maxwell element (spring constant, 
	dashpot constant)								35

5.4	Voigt-Kelvin model/Voigt-Kelvin element (Voigt model/Voigt
	element, spring constant, dashpot constant)					36

5.5	standard linear viscoelastic solid						36

5.6	relaxation time (relaxation spectrum)						37

5.7	stress relaxation (relaxation function)						38

5.8	retardation time (retardation spectrum)					38

5.9	creep (creep function, creep compliance)					39

5.10	forced oscillation (strain amplitude, angular velocity,
	stress amplitude, phase angle, loss angle)					40

5.11	loss factor/loss tangent (loss curve)						41

5.12	storage modulus								41

5.13	loss modulus									41

5.14	absolute modulus								42

5.15	complex modulus (complex strain, complex stress)				42

5.16	storage compliance								43

5.17	loss compliance								44

5.18	absolute compliance								44

5.19	complex compliance (complex strain, complex stress)			45

											Page

5.20	dynamic viscosity								45

5.21	out-of-phase viscosity								46

5.22	complex viscosity (complex rate of strain, complex stress)			46

6. OSCILLATORY DEFORMATIONS AND STRESSES
USED EXPERIMENTALLY FOR SOLIDS

6.1	free oscillation									48

6.2	damping curve (torsion pendulum)						48

6.3	decay constant									49

6.4	decay frequency								49

6.5	logarithmic decrement							50

6.6	forced uniaxial extensional oscillation (resonance frequency)		51

6.7	forced flexural oscillation (three-point bending/flexure, second
	moment of area, neutral axis/plane, resonance frequency,
	angular velocity of resonance frequency)					52

6.8	flexural force (flexural stress)							55

6.9	flexural deflection								55

6.10	flexural modulus								55

6.11	resonance curve								56

6.12	resonance frequency								56

6.13	half-width of resonance curve							57

7.  REFERENCES									59

8.  ALPHABETICAL INDEX OF TERMS					60
											Page

9.  GLOSSARY OF SYMBOLS							65

1.  BASIC DEFINITIONS

	In this section, quantities are expressed with respect to rectangular Cartesian 
co-ordinate axes, Ox1, Ox2, Ox3, except where otherwise stated.  The components of a vector V are denoted V1, V2, and V3 with respect to these axes.

1.1	traction
	stress vector

Recommended symbol:  t 

A vector force per unit area on an infinitesimal element of area that has a given normal and is at a given point in a body.

Unit:  Pa

Note

1.  The components of t are written as t1, t2, t3.

	2.  t is sometimes called true stress.  The term traction (or stress vector) is preferred to avoid confusion with stress tensor (see 1.2 note 5)

1.2	stress tensor
	stress

Recommended symbol: 

The tensor with components ij which are the components of the traction in the Oxi direction on an element of area whose normal is in the Oxj direction.

Unit:  Pa

Notes

1.  A unit vector area with normal n can be resolved into three smaller areas equal to n1, n2, and n3 with normals in the directions of the respective co-ordinate axes.  Accordingly, each component of the traction on the original area can be considered as the sum of components in the same direction on the smaller areas to give
.

	2.  In usual circumstances, in the absence of body couples, ij  =  ji .

	3.  For a homogeneous stress  is the same at all points in a body.

	4.  For an inhomogeneous stress ij  =  ij  (x1, x2, x3).

	5.   is a true stress because its components are forces per unit current area (cf. 3,4).

	6.  If 13 (= 31)  =  23 (= 32)  =  33  =  0  then the stress is called a plane stress.  Plane stresses are associated with the deformation of a sheet of material in the plane of the sheet.

1.3  	deformation of an elastic solid

A deformation of an elastic solid through which a mass point of the solid with co-ordinates X1, X2, X3 in the undeformed state moves to a point with co-ordinates x1, x2, x3 in the deformed state and the deformation is defined by

xi = xi (X1, X2, X3)  ,  i = 1, 2, 3

Notes

1.  A homogeneous deformation is one in which the relationships between the co-ordinates in the undeformed and deformed states reduce to

	where the fij are constants.

2.  An inhomogeneous deformation is one in which the incremental changes in the undeformed and deformed co-ordinates are related by

where ,  and where the fij are the functions of the coordinates xj.

3.  The fij in notes 1 and 2 are deformation gradients.

1.4	deformation gradient tensor for an elastic solid

Recommended symbol:  F

The tensor whose components are deformation gradients in an elastic solid.

Notes

1.  The components of F are denoted fij.

2.  See 1.3 for the definitions of fij.

1.5	deformation of a viscoelastic liquid or solid

A deformation of a viscoelastic liquid or solid through which a mass point of the viscoelastic liquid or solid with co-ordinates  at time t moves to a point with co-ordinates x1, x2, x3 at time t such that there are functions gi,  i = 1,2,3, where

gi (, t)  =  gi (x1, x2, x3, t).

Notes

1.  t often refers to some past time and t to the present time.

2.  The relationships between the total differentials of the functions gi define how particles of the material move relative to each other.  Thus, if two particles are at small distances  apart at time t and dx1, dx2, dx3 at time t then

where 
and 
i, j  =  1,2,3.

3.  The matrix with elements gij is denoted G and the matrix with elements gij is denoted G.

4.  A homogeneous deformation is one in which the functions gi are linear functions of the xj, i, j  =  1,2,3.  As a result, the gij and G are functions of t only and the equations which define the deformation become

5. Homogeneous deformations are commonly used or assumed in the methods employed for characterising the mechanical properties of viscoelastic polymeric liquids and solids.

1.6	deformation gradients in a viscoelastic liquid or solid

Recommended symbol:  fij

If two mass points of a liquid are at a small distance apart at time t then the deformation gradients are the rates of change of with respect to dxj, i, j  =  1,2,3.

Note

1.7	deformation gradient tensor for a viscoelastic liquid or solid

Recommended symbol:  F

The tensor whose components are deformation gradients in a viscoelastic liquid or solid.

Notes

1.  The components of F are denoted fij

2.  See 1.6 for the definition of fij.

3.  By matrix multiplication, F  = (G)-1G, where the matrices G and G are those defined in 1.5.

1.8	strain tensor

A symmetric tensor that results when a deformation gradient tensor is factorised into a rotation tensor followed or preceded by a symmetric tensor

Notes

1.  A strain tensor is a measure of the relative displacement of the mass points of a body.

2.  The deformation gradient tensor F may be factorised as
F = R U  =  V R,
	where R is an orthogonal matrix representing a rotation and U and V are strain tensors which are symmetric.

3.  Alternative strain tensors are often more useful.
	For example:
	the Cauchy tensor,  C  =  U2  =  FTF
	the Green tensor,  B  =  V2  =  FFT
	the Finger tensor, C-1
	the Piola tensor, B-1
	'T' denotes transpose and '-1' denotes inverse.  B is most useful for solids and C and C-1 for viscoelastic liquids and solids.

4.  If the 1,3;  3,1;  2,3;  3,2;  3,3 elements of a strain tensor are equal to zero then the strain is termed plane strain

1.9	Cauchy tensor

Recommended symbol:  C

The strain tensor for a viscoelastic liquid or solid, whose elements are 
,

where  and xi are co-ordinates of a particle at times t and t, respectively.

Notes

	1.  See 1.5 for the definition of and xi

	2.  See 1.8 for the definition of a strain tensor.

1.10  Green tensor

Recommended symbol:  B

The strain tensor for an elastic solid, whose elements are 
,
where Xi and xi are co-ordinates in the undeformed and deformed states, respectively.

Notes

1.  See 1.3 for the definition of Xi and xi.

2.  See 1.8 for the definition of a strain tensor

3.  For small strains, B may be expressed by the equation

B = I  +  2  ,

	where I is the unit matrix of order three and  is the small-strain tensor.  The components of  are 
 ,

	with uk  =  xk  -  Xk,  k  =  1,2,3, the displacements due to the deformation.

1.11  Finger tensor

Recommended symbol:  C-1

The strain tensor, for a viscoelastic liquid or solid, whose elements are
  ,
where  and xi are co-ordinates of a particle at times t and t, respectively.

Notes

1.  See 1.5 for the definition of and xi.

2.  See 1.8 for the definition of a strain tensor.

1.12  rate-of-strain tensor

Recommended symbol:  D

The time derivative of a strain tensor for a viscoelastic liquid or solid in homogeneous deformation at reference time, t.

Unit:  s-1

Notes

1.  For an inhomogeneous deformation, the material derivative has to be used to find time derivatives of strain.

2.  , where U  and V  are defined in 1.8, note 2.

3.  The elements of D are 
  ,
	where the vk are the components of the velocity v at x and time, t.

1.13  vorticity tensor

Recommended symbol:  W

The derivative, for a viscoelastic liquid or solid in homogeneous deformation, of the rotational part of the deformation-gradient tensor at reference time, t.

Unit:  s-1

Notes

1.  For an inhomogeneous deformation the material derivative has to be used.
2.  , where R is defined in 1.8, note 2.
3.  The elements of W are
  ,
	where the k are the components of the velocity v at x and time t.

1.14  Rivlin-Ericksen tensors

Recommended symbol:  An

The Rivlin-Ericksen tensor of order n, for a viscoelastic liquid or solid in homogeneous deformation, is the nth time derivative of the Cauchy strain tensor at reference time, t.

Unit:	s-n

Notes

1.  For an inhomogeneous deformation the material derivatives have to be used.

2.  , where C is defined in 1.9.
3.  A0 = I, where I is the unit matrix of order three.

4.  A1  =  =  2D, where F is the deformation-gradient tensor (see 1.7), , 'T' denotes transpose and D is the rate-of-strain tensor (see 1.12).

5.  In general, , n = 0,1,2, . . . .

2.  DEFORMATIONS USED EXPERIMENTALLY

	All deformations used in conventional measurements of mechanical properties are interpreted in terms of homogeneous deformations.

2.1  general orthogonal homogeneous deformation of an elastic solid 

A deformation, such that a mass point of the solid with co-ordinates X1, X2, X3 in the undeformed state moves to a point with co-ordinates x1,  x2,  x3 in the deformed state, with

xi  = iXi  , i  =  1, 2, 3,

where the i are constants.

Notes

1.  The relationships between the xi and Xi for orthogonal homogeneous deformations are a particular case of the general relationships given in 1.3, provided the deformation does not include a rotation and the co-ordinate axes are chosen as the principal directions of the deformation.

2.  The i  are effectively deformation gradients, or, for finite deformations, the deformation ratios characterising the deformation.

3.  For an incompressible material

					123  =  1  .
	
4.  The i  are elements of the deformation gradient tensor F (see 1.4) and the resulting Cauchy and Green tensors C and B (see 1.9 and 1.10) are

2.2  uniaxial deformation of an elastic solid 

An orthogonal, homogeneous deformation in which, say,

						1 = 
				and		2 = 3

Notes

1.  See 2.1 for the definition of i, i = 1,2,3 .

2.  For an incompressible material

2 =  3 =  1/1/2

2.3  uniaxial deformation ratio
       deformation ratio

Recommended symbol:  
 
The quotient of the length (l) of a sample under uniaxial tension or compression and its original length (l0)

				 = l / l0

Notes

1.  In tension  (>1) may be termed the extension ratio.

2.  In compression  (<1) may be termed the compression ratio.

3.   is equivalent to 1 in 2.1 and 2.2.

2.4  uniaxial strain
       engineering strain 

Recommended symbol: 

The change in length of a sample in uniaxial tensile or compressive deformation divided by its initial length
				  =  (l1 - l0) / l0

where l0 and l1 are, respectively, the initial and final lengths.

Notes

1.   =  -1, where  is the uniaxial deformation ratio (see 2.3).

2.   > 0 is referred to as (uniaxial) tensile strain.

3.   < 0 is referred to as (uniaxial) compressive strain.

2.5  Hencky strain

Recommended symbol: H

The integral over the total change in length of a sample of the incremental strain in uniaxial tensile deformation

l0 , l1 and l are, respectively, the initial, final and instantaneous lengths.

Notes

1.  See uniaxial strain (2.4).

2.  The same equation can be used to define a quantity H (< 0) in compression.

2.6  Poisson’s ratio 

Recommended symbol: 

In a sample under small uniaxial deformation, the negative quotient of the lateral strain (lat) and the longitudinal strain (long) in the direction of the uniaxial force

Notes

1.  Lateral strain lat is the strain normal to the uniaxial deformation.
	lat  = 2  - 1 = 3 - 1 (see 2.2 and 2.4).

2.  For an isotropic, incompressible material,  = 0.5.  It should be noted that, in materials referred to as incompressible, volume changes do in fact occur in deformation, but they may be neglected.

3.  For an anisotropic material,  varies with the direction of the uniaxial deformation.

4.  Poisson’s ratio is also sometimes called the lateral contraction ratio and is sometimes used in cases of non-linear deformation.  The present definition will not apply in such cases.

2.7  pure shear of an elastic solid 

An orthogonal, homogeneous deformation in which

				1 = 
				2 = 1/
				3 = 1

Note

See 2.1 for the definition of i , i = 1,2,3 .

2.8  simple shear of an elastic solid 

A homogeneous deformation, such that a mass point of the solid with co-ordinates X1, X2, X3 in the undeformed state moves to a point with co-ordinate x1, x2, x3 in the deformed state, with

			x1 = X1 + X2

			x2 = X2

			x3 = X3

where  is constant.

Notes

1.  The relationships between the xi and Xi, i = 1,2,3, in simple shear are a particular case of the general relationships given in 1.3.

2.   is known as the shear or shear strain.

3.  The deformation gradient tensor for the simple shear of an elastic solid (see 
	1.4) is

	and the Cauchy (C) and Green (B) strain tensors (see 1.9 and 1.10) are
  and   .

2.9  bulk compression 

Recommended symbol: 

The fractional decrease in volume (V) caused by a hydrostatic pressure

			 = -  V/V .

Note

	Also referred to as volume compression, isotropic compression and bulk 	compressive strain.

2.10  general homogeneous deformation or flow of a viscoelastic liquid or solid 

A flow or deformation such that a particle of the viscoelastic liquid or solid with co-ordinate vector X at time t moves to a point with co-ordinate vector X at time t with

				GX  =  GX

where G and G are tensors defining the type of deformation or flow and are functions of time only.

Notes

1.  The definition is equivalent to that given in 1.5 note 4.  Accordingly, the elements of G and G are denoted and gij(t) and those of X and X, () and (x1, x2, x3).

2.  For an incompressible material
					     det G = 1
	where det G is the determinant of G.

3. Deformations and flows used in conventional measurements of properties of viscoelastic liquids and solids are usually interpreted assuming incompressibility.

2.11  homogeneous orthogonal deformation or flow of an incompressible
        viscoelastic liquid or solid 

A deformation or flow, as defined in 2.10, such that 

  .

Notes

1.  The gii are defined in 1.5, notes 2 to 4.

2.  If  g22 = g33 =    the elongational deformation or flow is uniaxial.

3.  The Finger strain tensor for an homogeneous orthogonal deformation or flow of incompressible, viscoelastic liquid or solid (see 1.11) is

2.12  steady uniaxial homogeneous elongational deformation or flow of an incompressible viscoelastic liquid or solid 

Uniaxial homogeneous elongational flow in which

where   is a constant, and g22  =  g33  =.

Notes

1.  g11(t), g22(t) and g33(t) are elements of the tensor G defined in 1.5.

2.  From the definition of general homogeneous flow (1.5) (G´X´ = GX = constant) in the particular case of steady uniaxial elongation flow

	and differentiation with respect to time gives

	Hence,  is the elongational or extensional strain rate.

3.  The Finger strain tensor for a steady uniaxial homogeneous elongation deformation or flow of an incompressible viscoelastic liquid or solid (see 1.11) is
  .

2.13   homogeneous simple shear deformation or flow of an incompressible viscoelastic liquid or solid 

A flow or deformation such that 

where (t) is the shear.

Notes

1.  The general tensor G is defined in 1.5.

2.   is the shear rate.  The unit of  is s-1.

3.  If , where  is a constant, then the flow has a constant
	shear rate and is known as steady (simple) shear flow.

4.  If then the flow is oscillatory (simple) shear flow of frequency  and amplitude .   The unit of  is Hz.

5.  The Finger strain tensor for simple shear flow (see 1.11) is

	where  is the amount of shear given to the liquid between the times t and t.
	For steady simple shear flow  becomes 

3.  STRESSES OBSERVED EXPERIMENTALLY

	For a given deformation or flow, the resulting stress depends on the material.  However, the stress tensor (see 1.2) does take particular general forms for experimentally used deformations (see section 2).  The definitions apply to elastic solids, and viscoelastic liquids and solids.

3.1  stress tensor resulting from an orthogonal deformation or flow 

Recommended symbol: 

For an orthogonal deformation or flow the stress tensor is diagonal with

Unit:  Pa

Notes

1.  See 1.2 for the general definition of .

2.  If the strain tensor is diagonal for all time then the stress tensor is diagonal for all time for isotropic materials.

3.  For a uniaxial (orthogonal) deformation or flow 22 = 33.

4.  For a pure shear deformation or flow the stresses (11, 22, 33) are usually all different from each other.

5.  The stress tensor resulting from a pure shear deformation or flow is called a pure shear stress.

3.2  tensile stress 

Recommended symbol: 

The component 11 of the stress tensor resulting from a tensile uniaxial deformation.

Unit:  Pa

Notes

1.  The stress tensor for a uniaxial deformation is given in 3.1.

2.  The Ox1 direction is chosen as the direction of the uniaxial deformation.

3.3  compressive stress 

Recommended symbol: 

The component 11 of the stress tensor resulting from a compressive uniaxial deformation.

Unit:  Pa

Note

See notes 1 and 2 of 3.2.

3.4  	nominal stress
	engineering stress

Recommended symbol: 

The force resulting from an applied tensile or compressive uniaxial deformation divided by the initial cross-sectional area of the sample normal to the applied deformation.

Unit:  Pa

Note

The term engineering or nominal stress is often used in circumstances when the deformation of the body is not infinitesimal and its cross-sectional area changes.

3.5  stress tensor resulting from a simple shear deformation or flow 

Recommended symbol:  

For a simple shear deformation or flow the stress tensor takes the form

where 21 is numerically equal to 12.

Unit:  Pa

Notes

1.  See 1.2 for the general definition of .

2.  ii, i = 1,2,3 are denoted normal stresses.

3.  12 is called the shear stress.

3.6  first normal-stress difference
       first normal-stress function

Recommended symbol:  N1

The difference between the first two normal stresses 11 and 22 in simple shear flow

N1   =   11  -  22 .

Unit:  Pa

Notes

1.  See 3.5 for the definition of 11 and 22.

2.  For Newtonian liquids (see 4.2 note 3)  N1  =  0.

3.7  second normal-stress difference
       second normal-stress function

Recommended symbol:  N2

The difference between the second and third normal-stresses (22 - 33) in simple shear flow

N2  =  22 - 33  .

Unit:  Pa

Notes

1.  See 3.5 for the definition of 22 and 33

2.  For Newtonian liquids (see 4.2 note 3), N2  =  0.

4.  QUANTITIES RELATING STRESS AND DEFORMATION

4.1  constitutive equation for an elastic solid 

An equation relating stress and strain in an elastic solid.

Notes

1.  For an elastic solid, the constitutive equation may be written

,

	where B is the Green strain tensor  (see 1.10).
	I1, I2, I3 are invariants of B,
	with I1 = Tr(B)
	        I2 = 1/2 ((Tr(B))2 - Tr(B2))
	        I3 = det(B),
	where ‘Tr’ denotes trace and ‘det’ denotes determinant.  (Invariants are independent of the co-ordinate axes used and for symmetric tensors there are three independent invariants.)
	W is a function of I1, I2, and I3 and is known as the stored energy function and is the increase in energy (stored energy) per unit initial volume due to the deformation.

2.  For small deformations, the constitutive equation may be written
	
 = 2G + lITr () , 

	where G is the shear modulus (see 4.10),  is the small-strain tensor (see 1.10, note 3) and l is a Lamé constant

3.  The Lamé constant, (l), is related to the shear modulus (G) and Young’s modulus (E) (see 4.7) by the equation

l  =  G(2G - E)/(E - 3G),

4.  For an incompressible elastic solid, the constitutive equation may be written	

  ,

	where P is the hydrostatic (or isotropic) pressure, I3 = 1 and W is a function of I1 and I2, only.

5.  For small deformations of an incompressible, inelastic solid, the constitutive 
	equation may be written

  +  PI  =  2G

4.2  constitutive equation for an incompressible viscoelastic liquid or solid

An equation relating stress and deformation in an incompressible viscoelastic liquid or solid.

Notes

1.  A possible general form of constitutive equation when there is no dependence of stress on amount of strain is

  +  PI  =  f(A1, A2, . . . . , An)  ,

	where A1, A2, . . . . are the Rivlin-Ericksen tensors (see 1.14)

2.  For a non-Newtonian liquid (see note 3), a form of the general constitutive equation which may be used is 

  +  PI =  A12  +  A1  +  ßA2 ,

	where  is the viscosity (see 4.12) and   and ß are constants.

3.  A Newtonian liquid is a liquid for which the constitutive equation may be written

  +  PI =  A1 =  2D, 

	where D is the rate-of-strain tensor (see 1.12).  Liquids which do not obey this constitutive equation are termed non-Newtonian liquids.

4.  For cases where there is a dependence of stress on strain history the following constitutive equation may be used, namely

  ,

	where C is the Cauchy strain tensor (see 1.9) and  is a function of the invariants I1, I2 and I3 of C-1 and the time interval t-t´.   is formally equivalent to the stored-energy function, W, of a solid (see 4.1, note 4).

4.3 modulus

Recommended symbols:	general M
				in bulk compressive deformation K
				in uniaxial deformation E
				in shear deformation G

The quotient of stress and strain where the type of stress and strain is defined by the type of deformation employed.

Unit:  Pa

Notes

1.  The detailed definitions of K, E and G are given in 4.5, 4.7 and 4.10.

2.  An elastic modulus or modulus of elasticity is a modulus of a body which obeys Hooke’s law (stress  strain).

4.4  compliance 

Recommended symbols:	general C
				in bulk compressive deformation B
				in uniaxial deformation D
				in shear deformation J

The quotient of strain and stress where the type of strain and stress is defined by the type of deformation employed.

Unit:  Pa-1

Notes

1.  C = 1/M, where M is modulus (see 4.3).

2.  The detailed definitions of B, D and J are given in 4.6, 4.8 and 4.11.

4.5  bulk modulus 

Recommended symbol:  K

The quotient of hydrostatic pressure (P) and bulk compression ()

K  =  P/ .

Unit:  Pa

Notes

1.  Also known as bulk compressive modulus.

2.  For the definition of  , see 2.9.

3.  At small deformations, the bulk modulus is related to Young’s modulus (E) (see 4.7) by

	K = E/(3(1 - 2))

	where  is Poisson’s ratio  (see 2.6).

4.6  bulk compliance 

Recommended symbol:  B

The quotient of bulk compression () and hydrostatic pressure (P)

B  =  /P .

Unit:  Pa-1 

Notes

1.  Also known as bulk compressive compliance.

2.  For the definition of , see 2.9.

3.  B = 1/K, where K is the bulk modulus (see 4.5)

4.7  Young’s modulus

Recommended symbol:  E

The quotient of uniaxial stress () and strain () in the limit of zero strain

  .
Unit:  Pa

Notes

1.  The stress is a true stress, as in 3.2 and 3.3, and not a nominal stress, as in 3.4.

2.   is defined in 2.4.

3.  Young’s modulus may be evaluated using tensile or compressive uniaxial deformation (see 2.4).  If determined using tensile deformation it may be termed tensile modulus.

4.  For non-Hookean materials (see 4.3), the Young’s modulus is sometimes evaluated as:

	(i)  the secant modulus - the quotient of stress () and strain at some 
	nominal strain () in which case

E = /

	(ii) the tangent modulus  -  the slope of the stress-strain curve at some 
	nominal strain (´), in which case

E  =  (d/d)=´

4.8  uniaxial compliance

Recommended symbol:  D

The quotient of uniaxial strain () and uniaxial stress () in the limit of zero strain

 .
Unit:  Pa-1

Notes

1.  The stress is a true stress as in 3.2 and 3.3, and not a nominal stress, as in 3.4.

2.   is defined in 2.4.

3. Uniaxial compliance may be evaluated using tensile or compressive uniaxial deformation (see 2.4).  If determined using tensile deformation it may be termed tensile compliance.

4.  D  =  1/E, where E is Young’s modulus  (see 4.7).

4.9  extensional viscosity
       elongational viscosity

Recommended symbol:  E

The quotient of the difference between the longitudinal stress (11) and the lateral stress (22) and the elongational strain rate  in steady uniaxial flow

E  =  (11 - 22) / 

Unit:  Pa s

Note

See 3.1 and 2.12 for the definitions of 11, 22 and 

4.10  shear modulus 

Recommended symbol:  G

The quotient of shear stress (12) and shear strain () 

G = 12/.

Unit:  Pa

Notes

1.  See 2.8 for the definitions of    for an elastic solid and 3.5 for the definition of 12.

2.  The shear modulus is related to Young’s modulus (E) (see 4.7) by the equation

G = E/(2(1+µ))

	where µ is Poisson’s ratio (see 2.6).

3.  For elastomers, which are assumed incompressible, the modulus is often evaluated in uniaxial tensile or compressive deformation using  - -2  as the strain function (where  is the uniaxial deformation ratio (see 2.3)).  In the limit of zero deformation the shear modulus is evaluated as

  ,

	where  is the tensile or compressive stress (see 3.2 and 3.3).

4.11  shear compliance

Recommended symbol:  J

The quotient of shear strain () and shear stress (12)

J  = /12
 
Unit:  Pa-1 

Notes

1.  See 2.8 for the definition of  for an elastic solid and 3.5 for the definition of 12..

2.  J = 1/G, where G is the shear modulus (see 4.10).

4.12	shear viscosity
	coefficient of viscosity
	viscosity 

Recommended symbol: 

The quotient of shear stress (12) and shear rate  in steady, simple shear flow

  =  12/ .

Unit:  Pa s

Notes

1.  See 3.5 and 2.13 for the definitions of 12  and .

2.  For Newtonian liquids (see 4.2 note 3), 12  is directly proportional to  and  is constant.

3.  For non-Newtonian liquids (see 4.2 note 3), when 12 is not directly proportional to ,  varies with.  The value of  evaluated at a given value of  is termed the non-Newtonian viscosity.

4.  Some experimental methods, such as capillary flow and flow between parallel plates, employ a range of shear rates.  The value of  evaluated at some nominal average value of  is termed the apparent viscosity and given the symbol app.  It should be noted that apparent viscosity is an imprecisely defined quantity.

5.  Extrapolation of  or app for non-Newtonian liquids to zero  gives the zero-shear viscosity, which is given the symbol 0.

6.  Extrapolation of  and app  for non-Newtonian liquids to infinite  gives the infinite-shear viscosity, which is given the symbol .

4.13  first normal-stress coefficient 

Recommended symbol:  

The quotient of the first normal stress difference (N1) and the square of the shear rate  in the limit of zero shear rate

 .

Unit:  Pa s2

Note

See 3.6 and 2.13 for the definitions of N1 and .

4.14  second normal-stress coefficient 

Recommended symbol:  

The quotient of the second normal stress difference (N2) and the square of the shear rate  in the limit of zero shear rate

Unit:  Pa s2

Note

See 3.7 and 2.13 for the definitions of N2 and .

5.  LINEAR VISCOELASTIC BEHAVIOUR

5.1  viscoelasticity

The time-dependent response of a liquid or solid subjected to stress or strain.

Notes

1.  Both viscous and elastic responses to stress or strain are required for the description of viscoelastic behaviour.

2. Viscoelastic properties are usually measured as responses to an instantaneously applied or removed constant stress or strain or a dynamic stress or strain.  The latter is defined as a sinusoidal stress or strain of small amplitude, which may or may not decrease with time.

5.2  linear viscoelastic behaviour 

The interpretation of the viscoelastic behaviour of a liquid or solid in simple shear or uniaxial deformation such that 

P(D)  =  Q(D),

where  is the shear stress or uniaxial stress,  is the shear strain or uniaxial strain, and P(D) and Q(D) are polynomials in D, where D is the differential coefficient operator d/dt.

Notes

1.  In linear viscoelastic behaviour, stress and strain are assumed to be small so that the squares and higher powers of  and  may be neglected.

2.  See 3.5 and 2.13 for the definitions of  and  in simple shear.

3.  See 3.2 and 2.12 for definitions of  and  (E ) in uniaxial deformations.

4.  The polynomials Q(D) and P(D) have the forms: 

Q(D)  =  a(D + q0) .  .  .  .  . (D + qn) 
(a polynomial of degree n + 1)

P(D)  =  (D + p0) (D + p1)  .  . (D + pn)
(a polynomial of degree n + 1)

and
P(D)  =  (D + p0) (D + p1)  .  . (D + pn-1)
(a polynomial of degree n)

where
(i) a is a constant
(ii) q0   0, p0 > 0 and ps, qs > 0,  s = 1,  .  .  ., n.
(iii) qi<pi<qi+1 and qn<pn  (if pn exists)
pi and qi are related to relaxation and retardation times, respectively (see 5.6 - 5.9).

5.  If q0= 0, the material is a liquid, otherwise it is a solid.

6.  Given that Q(D) is a polynomial of degree n + 1; if P(D) is also of degree n + 1 the material shows instantaneous elasticity; if P(D) is of degree n, the material does not show instantaneous elasticity (i.e. elasticity immediately the deformation is applied.)

7.  There are definitions of linear viscoelasticity which use integral equations instead of the differential equation in definition 5.2.  (See, for example, ref. 11 .)  Such definitions have certain advantages regarding their mathematical generality.  However, the approach in the present document, in terms of differential equations, has the advantage that the definitions and descriptions of various viscoelastic properties can be made in terms of commonly used mechano-mathematical models (e.g. the Maxwell and Voigt-Kelvin models).

5.3  Maxwell model
       Maxwell element 

A model of the linear viscoelastic behaviour of a liquid in which

(D  +  ) = D 

where  and  are positive constants, D is the differential coefficient operator d/dt, and  and  are the stress and strain in simple shear or uniaxial deformation..

Notes

1.  See 5.2 for a discussion of  and .

2. 	The relationship defining the Maxwell model may be written

d/dt  +  (/)  =  (1/)d/dt

3.  Comparison with the general definition of linear viscoelastic behaviour (5.2) shows that the polynomials P(D) and Q(D) are of order one, q0=0, p0=/ and a=1/.  Hence, a material described by a Maxwell model is a liquid (q0 = 0) having instantaneous elasticity (P(D) and Q(D) are of the same order).

4.  The Maxwell model may be represented by a spring and a dashpot filled with a Newtonian liquid in series, in which case 1/ is the spring constant (force = 1/)·extension) and 1/ is the dashpot constant (force = (1/)·rate of extension).

5.4  Voigt-Kelvin model
       Voigt-Kelvin element 

A model of the linear viscoelastic behaviour of a solid in which
  =  (  +  D)
where  and  are positive constants, D is the differential coefficient operator d/dt, and  and  are the stresses and strain in simple shear or uniaxial deformation..

Notes

1.  The Voigt-Kelvin model is also known as the Voigt model or Voigt element.

2.  See 5.2 for a discussion of  and .

3.  The relationship defining the Voigt-Kelvin model may be written

 =  +  (d/dt).

4.  Comparison with the general definition of linear viscoelastic behaviour (5.2) shows that the polynomial P(D) is of order zero, Q(D) is of order one, aq0 = , and a  =  .  Hence, a material described by the Voigt-Kelvin model is a solid (q0>0) without instantaneous elasticity (P(D) is a polynomial of order one less than Q(D)).

5.  The Voigt-Kelvin model may be represented by a spring and a dashpot filled with a Newtonian liquid in parallel, in which case  is the spring constant (force = ·extension) and  is the dashpot constant (force = ·rate of extension).

5.5  standard linear viscoelastic solid 

A model of the linear viscoelastic behaviour of a solid in which

 (1  +  1D)  =  (2  + 2D)

where 1, 1, 2 and 2 are positive constants, D is the differential coefficient operator d/dt, and  and  are the stress and strain in simple shear or uniaxial deformation.

Notes

1.  See 5.2 for a discussion of  and .

2.  The relationship defining the standard linear viscoelastic solid may be written
1 + 1(d/dt)  =  2  + 2(d/dt)

3.  Comparison with the general definition of a linear viscoelastic behaviour (5.2) shows that the polynomial P(D) and Q(D) are of order one, q0  =  2/2, a = 2/1  and p0 = 1/2.  Hence, the standard linear viscoelastic solid is a solid (aq0>0) having instantaneous elasticity (P(D) and Q(D) are of the same order).

4.  The standard linear viscoelastic solid may be represented by:

(i)  a Maxwell model (of spring constant h2 and dashpot constant k2) in parallel with a spring (of spring constant h1) in which case 1 = h2, 1 = k2, 2 	= h1h2 and 	2 = h1k2+ h2k2.

(ii) a Voigt-Kelvin model (of spring constant h2 and dashpot constant k2) in series with a spring (of spring constant h1) in which case 1 = h1 +h2, 1 = k2, 2 = h1h2 and 2 = h1k2.

5.  The standard linear viscoelastic solid can be used to represent both creep (see 5.9) and stress relaxation (see 5.7) in materials in terms of single retardation and relaxation times, respectively.

5.6  relaxation time 

Recommended symbol:  

A time characterising the response of a viscoelastic liquid or solid to the instantaneous application of a constant strain.

Unit:  s

Notes

1.  The response of a material to the instantaneous application of a constant strain is termed stress relaxation (see 5.7).

2.  The relaxation time of a Maxwell element (5.3) is  = 1/p0  =  /.

3.  The relaxation time of a standard linear viscoelastic solid (5.5) is  = 1/p0 = 1/1.

4.  Generally, a linear viscoelastic material has a spectrum of relaxation times, which are the reciprocals of pi, i = 0, 1, …, n in the polynomial P(D) (see 5.2).

5.  The relaxation spectrum (spectrum of relaxation times) describing stress relaxation in polymers may be considered as arising from a group of Maxwell elements in parallel (see 5.7).

5.7  stress relaxation 

The change in stress with time after the instantaneous application of a constant strain.

Notes

1.  The applied strain is of the form   = 0 for t < 0 and    =  0 for t  > 0 and is usually a uniaxial extension or a simple shear (see 5.2).

2.  For linear viscoelastic behaviour, the stress takes the form

	c is a constant that is non-zero if the material has instantaneous elasticity and is the relaxation function.

3.  has the form

	where the i are functions of the pi and qi of the polynomials P(D) and Q(D) defining the linear viscoelastic material (see 5.1).

4.  The relaxation times of the material are 1/pi  (see 5.6).

5.8  retardation time 

Recommended symbol:  

A time characterising the response of a viscoelastic material to the instantaneous application of a constant stress.

Unit: s

Notes:

1.  The response of a material to the instantaneous application of a constant stress is termed creep (see 5.9).

2. The retardation time of a Voigt-Kelvin element is  = 1/q0 = / = (dashpot constant)/(spring constant).

3.  The retardation time of a standard linear viscoelastic solid (5.5) is  = 1/q0 = 2/2.

4.  Generally, a linear viscoelastic material has a spectrum of retardation times, which are reciprocals of qi, i = 0, 1, … , n in the polynomial Q(D)  (see 5.2).

5. The retardation spectrum (spectrum of retardation times) describing creep in polymers may be considered as arising from a group of Voigt-Kelvin elements in series (see 5.9).

5.9  creep 

The change in strain with time after the instantaneous application of a constant stress.

Notes

1.  The applied stress is of the form  = 0 for t<0 and  = 0 for t>0 and is usually a uniaxial stress or a simple shear stress (see 5.2).

2.  For linear viscoelastic behaviour, the strain usually takes the form

	a is a constant that is non-zero if the material has instantaneous elasticity and b is a constant that is non-zero if the material is a liquid.  is the creep function.  In addition,

J(t) = (t)/0

	is sometimes called the creep compliance.

3.  The creep function has the form

	where the summation runs from i = 0 to n for a solid and 1 to n for a liquid.  The Ai are functions of the pi and qi of the polynomials P(D) and Q(D) defining the linear viscoelastic material and the qi are the qi of the polynomial Q(D)  (see 5.1).

4.  The retardation times of the material are 1/qi (see 5.8).

5.  Creep is sometimes described in terms of non-linear viscoelastic behaviour, leading, for example, to evaluation of recoverable shear and steady-state recoverable shear compliance.  The definitions of such terms are outside the scope of this document.

5.10  forced oscillation 

The deformation of a material by the application of a small sinusoidal strain () such that

where 0 and  are positive constants.

Notes

1.   may be in simple shear (see 2.8 and 2.13) or uniaxial deformation (often denoted , see 2.4).

2.  0 is the strain amplitude.

3.   is the angular velocity of the circular motion equivalent to a sinusoidal frequency , with  = 2.  The unit of  is rad s-1.

4.  For linear viscoelastic behaviour, a sinusoidal stress () results from the 
	sinusoidal strain with

.

	0 is the stress amplitude.   is the phase angle or loss angle between stress and strain.

5.  Alternative descriptions of the sinusoidal stress and strain in a viscoelastic material under forced oscillations are:

	

5.11  loss factor
        loss tangent

Recommended symbol:  tan 

The tangent of the phase angle difference () between stress and strain during forced oscillations.

Notes

1.  tan  is calculated using
 and .  (see 5.10).

2.  tan  is also equal to the ratio of loss to storage modulus (see 5.12 and 5.13).

3.  A plot of tan  versus temperature or frequency is known as a loss curve.

5.12  storage modulus 

Recommended symbol:	general M´
				in simple shear deformation G´
				in uniaxial deformation E´

The ratio of the amplitude of the stress in phase with the strain (0 cos ) to the amplitude of the strain (0) in the forced oscillation of a material

M´  =  (0 cos )/0.

Unit:  Pa

Note

	See 5.10 for the definition of a forced oscillation in which  = 0 cos t and 
	 = 0 cos (t + ).

5.13  loss modulus 

Recommended symbols:	General M´´
				in simple shear deformation G´´
				in uniaxial deformation in E´´

The ratio of the amplitude of the stress 90° out of phase with the strain (0 sin ) 
to the amplitude of the strain (0) in the forced oscillation of a material

M´´  =  (0 sin )/0

Unit:  Pa

Note

See 5.10 for the definition of a forced oscillation in which  = 0cos t and  = 0 cos (t + ).

5.14  absolute modulus 

Recommended symbols:		general |M*|
					in simple shear deformation |G*|
					in uniaxial deformation |E*|

The ratio of the amplitude of the stress (0) to the amplitude of the strain (0) in the forced oscillation of a material

|M*|  =  0 /0  .

Unit:  Pa

Notes

1.  See 5.10 for the definition of a forced oscillation in which  = 0cos t 
	and  = 0 cos (t + ).

2.  The absolute modulus is related to the storage modulus (5.12) and the loss modulus (5.13) by the relationship

.

5.15  complex modulus

Recommended symbols:		general M*
					in simple shear deformation G*
					in uniaxial deformation E*

The ratio of complex stress (*) to complex strain (*) in the forced oscillation of material

M*  =  */*    .

Unit:  Pa

Notes

1.  See 5.10 for the definition of a forced oscillation in which  = 0cos t and 
	 = 0cos (t + ).

2.  The complex strain , where i = -1, so that the real part of the complex strain is that actually applied to the material.

3.  The complex stress , so that the real part of the complex strain is that actually experienced by the material.

4. The complex modulus is related to the storage and loss moduli through the relationships

.

5.  For linear viscoelastic behaviour interpreted in terms of complex stress and strain (see notes 2, 3)

P(D)*  = Q(D)*

	(see 5.2).  Further as D*  =  d*/dt  =  i* and D*  =  i*,

M*  =  */*  = Q(i)/P(i)

5.16 storage compliance 

Recommended symbols:		general C´
					in simple shear deformation J´
					in uniaxial deformation D´

The ratio of the amplitude of the strain in phase with the stress (0 cos ) to the amplitude of the stress (0) in the forced oscillation of a material

C´  =  (0 cos )/0 .

Unit:  Pa-1

Note

See 5.10, note 5 for the definition of a forced oscillation in which 
 = 0 cos (t - ) and  = 0 cos t.

5.17  loss compliance 

Recommended symbols:		general C´´
					in simple shear deformation J´´
					in uniaxial deformation D´´

The ratio of the amplitude of the strain 90° out of phase with the stress (0 sin ) to the amplitude of the stress (0) in the forced oscillation of a material

.

Unit:  Pa-1

Note

See 5.10 for the definition of a forced oscillation in which 
	 = 0 cos(t - ) and  = 0 cos t.

5.18  absolute compliance 

Recommended symbols:		general |C*|
					in simple shear deformation |J*|
					in uniaxial deformation |D*|

The ratio of the amplitude of the strain (0) to the amplitude of the stress (0) in the forced oscillation of a material

|C*|  = 0/0 .

Unit:  Pa-1

Notes

1.  See 5.10 note 5 for the definition of a forced oscillation in which 
	 = 0 cos(t - ) and  = 0 cos t.

2.  The absolute compliance is related to the storage compliance (5.16) and the loss 	compliance (5.17) by the relationship

.

3.  The absolute compliance is the reciprocal of the absolute modulus (5.14).

|C*|  =  1/|M*|.

5.19  complex compliance 

Recommended symbols:		general C*
					in simple shear deformation J*
					in shear deformation D*

The ratio of complex strain (*) to complex stress (*) in the forced oscillation of a material

C*  =  */*  .

Unit:  Pa-1

Notes

1.  See 5.10 for the definition of a forced oscillation in which 
	  =  0 cos (t - ) and  = 0 cos t.

2.  The complex strain * = 0ei (t-) =  0 (cos(t-) + i sin (t-)), where i = -1, so that the real part of the complex strain is that actually experienced by the material.

3.  The complex stress * = 0eit = 0 (cos t  +  i sin t), so that the real part of the complex stress is that actually applied to the material.

4.  The complex compliance is related to the storage and loss compliances (5.16 and 5.17) through the relationships

5.  The complex compliance is the reciprocal of the complex modulus

C*  =  1/M*  .

5.20  dynamic viscosity 

Recommended symbol:  ´

The ratio of the stress in phase with the rate of strain (0 sin ) to the amplitude of the rate of strain (0) in the forced oscillation of a material

´ = (0 sin )/(0) .

Unit:  Pa s

Notes

1.  See 5.10, note 5 for the definition of a forced oscillation in which  and, so that  and .

2.  See 5.2, note 6:  ´ = M´´/ may be used for evaluating the dynamic viscosity. The same expression is often used to evaluate the shear viscosity.  The latter use of this expression is not recommended.

5.21  out-of-phase viscosity 

Recommended symbol:  ´´

The ratio of the stress 90° out of phase with the rate of strain (0 cos ) to the amplitude of the rate of strain (0) in the forced oscillation of a material

´´ = (0 cos )/(0)  .

Unit:  Pa s

Notes

1.  See 5.10, note 5 for the definition of a forced oscillation in which  and, so that  and .
2.  See 5.22, note 6:  ´´ = M´/ may be used to evaluate the out-of-phase viscosity.

5.22 complex viscosity 

Recommended symbol:  *

The ratio of complex stress (*) to complex rate of strain (*) in the forced oscillation of a material

* = */*   .

Unit:  Pa s

Notes

1.  See 5.10, note 5 for the definition of a forced oscillation  in which  = 0 sin t and  = 0  cos (t + ) and the rate of strain  =  0 cos t.

2.  The complex rate of strain * = i 0eit = i 0 (cos t  +  i sin t), where 
	i = -1.

3.  The complex stress * = 0ei(t+) = 0 (cos (t+) + i sin (t+))

4.  The complex viscosity may alternatively be expressed as

* = */*  =  (0ei)/(i 0)  =  M*/i 
	where M* is the complex modulus (see 5.15).

5.  The complex viscosity is related to the dynamic and out-of-phase viscosities through the relationships

* = */* = 0 (cos  + i sin )/(i 0)  =  ´ - i ´´.

6.  The dynamic and out-of-phase viscosities are related to the storage and loss moduli (5.11 and 5.13) by the relationships * =   - i   =  M*/i   =  (M + i M)/i , so that   = M/ and    =  M/.

6.  OSCILLATORY DEFORMATIONS AND STRESSES USED EXPERIMENTALLY

There are three modes of free and forced oscillatory deformations which are commonly used experimentally, torsional oscillations, uniaxial extensional oscillations and flexural oscillations.

The oscillatory deformations and stresses can be used for solids and liquids.  However, the apparatuses employed to measure them are usually designed for solid materials.  In principle, they can be modified for use with liquids.

Analyses of the results obtained depend on the shape of the specimen, whether or not the distribution of mass in the specimen is accounted for and the assumed model used to represent the linear viscoelastic properties of the material.  The following terms relate to analyses which generally assume small deformations, specimens of uniform cross-section, non-distributed mass and a Voigt-Kelvin solid (see 5.4).   These are the conventional assumptions.

6.1  free oscillation 

The oscillatory deformation of a material specimen with the motion generated without the continuous application of an external force.

Note

For any real sample of material the resulting oscillatory deformation is one of decaying amplitude.

6.2  damping curve

The decreased deformation of a material specimen versus time when the specimen is subjected to a free oscillation.

Notes

1.  See 6.1 for the definition of a free oscillation.

2.  The term ‘damping curve’ is sometimes used to describe a loss curve (see 5.11).

3.  A damping curve is usually obtained using a torsion pendulum, involving the measurement of decrease of the axial, torsional displacement of a specimen of uniform cross-section of known shape, with the torsional displacement initiated using a torsion bar of known moment of inertia.

4.  Damping curves are conventionally analysed in terms of the Voigt-Kelvin solid (see 5.4) giving a decaying amplitude and a single frequency.

5.  Given the properties of a Voigt-Kelvin solid, a damping curve is described by the equation

X  =  Aexp(-t)·sin(t-),

	where X is the displacement from equilibrium (for torsion X  =  , the angular displacement), t is time, A is the amplitude,  is the decay constant (see 6.3),  is the angular velocity corresponding to the decay frequency (see 5.10 and 6.4) and  is the phase angle.

6.3  decay constant 

Recommended symbol:    

The exponential coefficient of the time-dependent decay of a damping curve, assuming Voigt-Kelvin behaviour

Unit:  s-1

Notes

1.  See damping curve (6.2) and the equation therefor (6.2, note 5).

2.  See Voigt-Kelvin solid (5.4).

3.  For small damping,  is related to the loss modulus (M), see 5.13, through the equation

M = 2/H .

	 is the angular velocity corresponding to the decay frequency (see 5.10 and 6.4).  H depends on the cross-sectional shape of the specimen and the type of deformation.  (For example, for the axial torsion of a circular rod of radius a and length l using a torsion pendulum (see 6.2, note 3) with a torsion bar of moment of inertia I

H = a4/(2Il)

	and M  G, the loss modulus in simple shear).

6.4  decay frequency 

Recommended symbol:  

The frequency of a damping curve assuming Voigt-Kelvin behaviour.

Unit:  Hz

Notes

1.  See damping curve (6.2) and the equation therefor (6.2, note 5).

2.  See Voigt-Kelvin solid (5.4).

3.   = /2, where  is the angular velocity corresponding to  (see 5.10).

4.  For small damping, the storage modulus (M), see 5.12, may be evaluated from  through the equation

M  =  2/H,

	where H is discussed in 6.3, note 3.  Again, for torsion, M  G, the storage modulus in simple shear.

6.5  logarithmic decrement 

Recommended symbol:  

Natural logarithm of the ratio of the displacement of a damping curve separated by one period of the displacement.

Notes

1.  Voigt-Kelvin behaviour (see 5.4) is assumed so that the displacement decays with a single period T, where

	with  the frequency and  is the angular velocity corresponding to  (see 6.4).

2.  The logarithmic decrement can be used to evaluate the decay constant  (see 6.3).  From the equation for the damping curve of a Voigt-Kelvin solid (see 6.2, note 5).

 =  ln (Xn/Xn+1)  =  (tn+1 - tn)  = ·T,

	where Xn and tn are the displacement and time at a chosen point (usually near a maximum) in the n-th period of the decay, and Xn+1 and tn+1 are the corresponding displacement and time one period later.

3.   can also be defined using displacements k periods apart, with

 =  (1/k)ln (Xn/Xn+k).

4.  For small damping,  is related to the loss tangent, tan  (see 5.11) by

tan  = M/M  =  2/  =  2/T  = /

	(See 6.3 and 6.4 for expressions for M and M).

6.6  forced uniaxial extensional oscillations

The uniaxial extensional deformation of a material specimen of uniform cross-sectional area along its long axis by the continuous application of a sinusoidal force of constant amplitude.

Notes

1.  For a specimen of negligible mass, the linear-viscoelastic interpretation of the resulting deformation gives

(A/L)Q(D)l = P(D)f0 cos t

	where P(D) and Q(D) are the polynomials in D(=d/dt) characterising the linear-viscoelastic behaviour (see 5.2), A is the cross-sectional area of the specimen, L its original length, l is here the change in length, f0 the amplitude of the applied force of angular velocity  (see 5.10, note 3) and t the time.

2.  For a Voigt-Kelvin solid (see 5.4), with P(D)=1 and Q(D)=+D, where  is the spring constant and  the dashpot constant, the equation describing the deformation becomes

(A/L)(dl/dt)  +  (A/L)l  =  f0 cos t

	or, in terms of stress and strain,

	where  = l/L is the uniaxial strain (see 2.4 and 5.10) and 0 = f0/A is the amplitude of the stress (see 5.10).  The solution of the equation is

	where  is the phase angle (see 5.10) with tan  = /.

3.  From 5.14, the absolute modulus in uniaxial deformation

|E*|  =  0/0  =  (2+22)1/2

	where   =  E,   =  E and tan   =  E/E equal to the loss tangent (see 5.11).

4.  If one end of the specimen is fixed in position and a mass m is attached to the moving end, the linear-viscoelastic interpretation of the resulting deformation gives

m·P(D)(d2l/dt2)+(A/L)Q(D)l  =  P(D)f0 cos t

	where the symbols have the same meaning as in note 1.

5.  For a Voigt-Kelvin solid (cf. note 2), the equation in note 4 describing the deformation becomes

m(d2l/dt2)+(A/L)(dl/dt)+(A/L)·l  =  f0 cos t

	with the solution

	where   and symbols have the same meaning as in notes 1 and 2.

6.  The amplitude of the strain 0 is maximal when 

2  =  A/(Lm)  =

	giving the value of the angular velocity (R) of the resonance frequency of the specimen (see 6.12) in forced uniaxial extensional oscillation.

7.  Notes 2 and 5 show that application of a sinusoidal uniaxial force to a Voigt-Kelvin solid of negligible mass, with or without added mass, results in an out-of-phase sinusoidal uniaxial extensional oscillation of the same frequency.

6.7  forced flexural oscillation

The flexural deformation (bending) of a material specimen of uniform cross-sectional area perpendicular to its long axis by the continuous application of a sinusoidal force of constant amplitude.

Notes

1.  There are three modes of flexure in common use.

	(i)	Application of the flexural force at one end of the specimen with the other 			end clamped.

	(ii)	Application of the flexural force at the centre of the specimen with the two 		ends clamped (three-point bending or flexure).

	(iii)	Application of the flexural force at the centre of the specimen with the two 		ends resting freely on supports (also known as three-point bending or 			flexure).

2.  For specimens without mass, the linear-viscoelastic interpretation of the resulting deformations follows a differential equation of the same form as that for a uniaxial extensional forced oscillation (see 6.6, note 1), namely

(HJ/L3)Q(D)y  =  P(D)f0 cos t

	where P(D), Q(D), f0,  and t have the same meaning as for a forced uniaxial extensional oscillation (see 6.6, note 1) and H is a constant.  The length of the specimen is 2L.  For mode of flexure (i) H=3, for (ii) H=24 and 	for (iii) H=6 (see note 1).  J is the second moment of area of the specimen, defined by

	where dA is an element of the cross-sectional area (A) of the specimen and q is the distance of that element from the neutral axis or plane of the specimen, lying centrally in the specimen and defined by points which experience neither compression nor extension during the flexure.  For a specimen of circular cross-section J=r2/4, where r is the radius, and for one of rectangular cross-section J=4ab3/3, where 2a and 2b are the lateral dimensions with flexure along the b dimension.  Finally, y is the flexural deflection (see 6.9) of the specimen at the point of application of the force, of either the end (mode of flexure (i)) or the middle (modes of flexure (ii) and (iii)).

3.  For a Voigt-Kelvin solid, the equation describing the deformation becomes

(HJ/L3)·y  +  (HJ/L3)(dy/dt)  =  f0 cos t

	with solution

	where  is the phase angle with

tan   =  /

	equal to the loss tangent (see 5.11).

4.  Unlike the strain in forced uniaxial extensional oscillations, those in forced flexural deformations are not homogeneous.  In the latter modes of deformation, the strains vary from point-to-point in the specimen. Hence, the equation defining the displacement y in terms of the amplitude of applied force (f0) cannot be converted into one defining strain in terms of amplitude of stress.

5.  If a mass m is attached to the specimen at the point of application of the force, the linear-viscoelastic interpretation of the resulting deformation gives

m·P(D) (d2y/dt2) + (HJ/L3)Q(D)y  =  P(D)f0 cos t

	(cf. 6.6, note 4).

6.  For a Voigt-Kelvin solid (cf. note 3 and 6.6, note 5), the equation describing the deformation becomes

m(d2y/dt2)+(HJ/L3)(dy/dt)+(HJ/L3)·y  = f0 cos t

	with the solution

	where 

7.  The flexural deflection y (see 6.9) is maximal when

2  =  HJ/(L3m)  = 

	giving the value of the angular velocity (R) of the resonance frequency of the specimen (see 6.6, note 6) in forced flexural oscillations.

8.  Notes 3 and 6 show that the application of the defined sinusoidal flexural forces (i), (ii) and (iii) (note 1) to a Voigt-Kelvin solid of negligible mass, with or without added mass at the points of application of the forces, results in out-of-plane sinusoidal flexural oscillations of the same frequency.

6.8  flexural force

Recommended symbol:  f0

The amplitude of the force applied to a material specimen to cause a forced flexural oscillation.

Unit:  N

Notes

1.  See 6.7 for the definition and interpretation of forced flexural oscillation.

2.  A related quantity is the flexural stress which is somewhat arbitrarily defined as the amplitude of the stress in the convex, outer surface of a material specimen in forced flexural oscillation.

6.9  flexural deflection

Recommended symbol:  y

The deflection of a specimen subject to a forced flexural oscillation at the point of application of the flexural force.

Unit:  m

Notes

1.  See 6.7 for the definition and interpretation of forced flexural oscillation.

2.  See 6.8 for the definition of flexural force.

6.10  flexural modulus

Recommended symbol:  |E*|

The modulus measured using forced flexural oscillations.

Unit:  Pa

Notes

1.  See 6.7 for the definition and interpretation of forced flexural oscillation.

2.  For a Voigt-Kelvin solid (see 5.4) of negligible mass, the absolute modulus can be evaluated from the ratio of the flexural force (f0) and the amplitude of the flexural deflection (y) with

f0/Y0  =  (HJ/L3)(2 + 22)1/2

	where Y0 is the amplitude of the flexural deflection (see 6.7, note 3, 6.8 and 6.9),

|E*|  =  (2 + 22)1/2

	(see 5.14 and 6.6, note 3) and the remaining symbols are as defined in 6.7, note 2.

3.  The ratio of the loss to the storage flexural modulus (E/E) is derived from the loss tangent (tan ) of the forced flexural oscillation with

tan   =  /  =  E/E

	(see 5.11 and 6.7, note 3).

4.  The flexural modulus has been given the same symbol as the absolute modulus in uniaxial deformation (see 5.14) as it becomes equal to that quantity in the limit of zero amplitudes of applied force and deformation.  Under real experimental conditions it is often used as an approximation to |E*|.

6.11  resonance curve 

Recommended symbol:  A()

The curve of the frequency dependence of the amplitude of the displacement of a material specimen subject to forced oscillations in the region of a resonance frequency.

Unit:  that of the amplitude A

Notes

1.  See 6.6 and 6.7 for the description of modes of forced oscillation commonly used.

2.  See 6.12 for the definition of resonance frequency.

6.12  resonance frequency 

Recommended symbol:  R

The frequency at a maximum of a resonance curve

Unit:  Hz

Notes

1.  See 6.11 for the definition of a resonance curve.

2.  Material specimens subject to a forced oscillation (see 6.6 and 6.7) in general have a spectrum of resonance frequencies.

3.  In cases of a single resonance frequency, the resonance frequency is proportional to the square root of the storage modulus (M) of the material (see 5.12).

4.  A material specimen which behaves as a Voigt-Kelvin solid under forced oscillations with a mass added at the point of application of the applied oscillatory force has a single resonance frequency.

5.  Under a forced uniaxial extensional oscillation the resonance frequency

	(see 6.6 for the origin of the equation and definitions of symbols).  E is the storage modulus in uniaxial extension (see 5.12).

6.  Under a forced flexural oscillation the resonance frequency

	(see 6.7 for the origin of the equation and the definition of symbols).

6.13  width of the resonance curve

Recommended symbol:  

The magnitude of the difference in frequency between two points on a resonance curve on either side of R which have amplitudes equal to (1/2)A(R).

Unit:  Hz

Notes

1.  For a material specimen which behaves as a Voigt-Kelvin solid under forced uniaxial extensional oscillation with mass added at the point of application of the applied oscillatory force,  is proportional to the loss modulus (E)  (see 5.13).

	In addition (6.6, note 6), the storage modulus (E)  (see 5.12) may be
	evaluated from

	(see 6.6 for the definition of symbols).

2.  For a material specimen which behaves as Voigt-Kelvin solid under forced flexural oscillations with added mass at the point of application of the applied oscillatory force,  is proportional to the loss modulus (E)  (see 5.13)

 .

	In addition, the storage modulus (E) (see 5.12) may be evaluated form

	(see 6.7 for the definition of symbols).

3.  For the Voigt-Kelvin behaviours specified in notes 1 and 2, the ratio of  and the resonance frequency (R) is equal to the loss tangent (tan ).

	Under forced uniaxial extensional oscillation

.

	Under forced flexural oscillation

	
	(see 5.11 for the definition of tan ).

7.  REFERENCES

1.	ISO 472, ISO TC-61, Dec. 1991 Draft, Plastics Vocabulary

2.	C.L. Sieglaff, Trans. Soc. Rheol. 1976 20 311

3.	D.W. Hadley, J.D. Weber, Rheol. Acta 1975 14 1098

4.	D.W. Hadley, J.D. Weber, Bull. Br. Soc. Rheol. 1979 22 4

5.	M. Reiner, G.W. Scott Blair, Rheology, Vol. 4, ed. F.R. Eirich,
	Academic Press, New York, 1967

6.	J.M. Dealy, J. Rheol. 1984 28 181; 1995 39 253

7.	ASTM E 6-88, ‘Standard Definitions of Terms Relating to
	Methods of Mechanical Testing, Annual Book of ASTM Standards,
	Vol. 8.03, 1989, p. 672

8.	ASTM D4092-83a, ‘Standard Definitions and Descriptions of Terms Relating
	to Dynamical Mechanical Measurements on Plastics’, Annual Book ASTM
	Standards, Vol. 8.03, 1989, p. 334

9.	ASTM D 883-88, ‘Standard Definitions of Terms Relating to Plastics,
	Annual Book of ASTM Standards, Vol. 8.01, 1989, p. 333.

10.	G. Astarita and G. Marucci, ‘Principles of Non-Newtonian 
	Fluid Mechanics’, McGraw-Hill Book Company (UK) Ltd.,
	Maidenhead, 1974.

11.	J.D. Ferry, ‘Viscoelastic Properties of Polymers’, John Wiley
	and Sons, New York, third edition, 1980

12.	R.I. Tanner, ‘Engineering Rheology’, Clarendon Press, Oxford, 1985.

13.	H.A. Barnes, J.F. Hutton and K. Walters, ‘An Introduction to
	Rheology’, Elsevier, Amsterdam, 1989

14.	J.F. Nye, ‘Physical Properties of Crystals’, Oxford University
	Press, 1969

15.	I.M. Ward and D.W. Hadley, ‘An Introduction to the Mechanical
	Properties of Solid Polymers’, John Wiley and Sons,
	Chichester, 1993

8.  ALPHABETICAL INDEX OF TERMS

absolute compliance (5.18)
absolute modulus (5.14)
angular velocity (of a forced oscillation)  (5.10)
angular velocity of resonance frequency (6.7)
apparent viscosity (4.12)

bulk compliance (4.6)
bulk compression (2.9)
bulk compressive compliance (4.6)
bulk compressive modulus (4.5)
bulk compressive strain (2.9)
bulk modulus (4.5)

Cauchy tensor (1.8, 1.9)
coefficient of viscosity (4.12)
complex compliance (5.19)
complex modulus (5.15)
complex rate of strain (5.22)
complex strain (5.15, 5.19)
complex stress (5.15, 5.19, 5.22)
complex viscosity (5.22)
compliance (4.4)
compressive strain (2.4)
compressive stress (3.3)
constitutive equation for an elastic solid (4.1)
constitutive equation for an incompressible viscoelastic liquid or solid (4.2)
creep (5.9)
creep compliance (5.9)
creep function (5.9)

damping curve (6.2)
dashpot constant (5.3, 5.4)
decay constant (6.3)
decay frequency (6.4)
deformation gradients in an elastic solid (1.3)
deformation gradients in a viscoelastic liquid or solid (1.6)
deformation gradient in the orthogonal deformation of an elastic solid (2.1)
deformation gradient tensor for an elastic solid (1.4)
deformation gradient tensor for a viscoelastic liquid or solid (1.7)
deformation of an elastic solid (1.3)
deformation of a viscoelastic liquid or a solid (1.5)
deformation ratio (2.3)
deformation ratio in the orthogonal deformation of an elastic solid (2.1)
dynamic strain (5.1)
dynamic stress (5.1)
dynamic viscosity (5.20)

elastic modulus (4.3)
elongational strain rate (2.12)
elongational viscosity (4.9)
engineering strain (2.4)
engineering stress (3.4)
extensional strain rate (2.12)
extensional viscosity (4.9)
extension ratio (2.3)

Finger tensor (1.8, 1.11)
first normal-stress coefficient (4.13)
first normal-stress difference (3.6)
first normal-stress function (3.6)
flexural deflection (6.9)
flexural force (6.8)
flexural modulus (6.10)
flexural stress (6.8)
forced flexural oscillation (6.7)
forced oscillation (5.10)
forced uniaxial extensional oscillation (6.6)
free oscillation (6.1)

general homogenous deformation or flow of a viscoelastic liquid or solid (2.10)
general orthogonal homogeneous deformation of an elastic solid (2.1)
Green tensor (1.8, 1.10)

Hencky strain (2.5)
homogeneous deformation of elastic solids (1.3)
homogeneous deformation of viscoelastic liquids and solids (1.5)
homogeneous orthogonal deformation or flow of an incompressible viscoelastic liquid
   or solid (2.11)
homogeneous simple shear deformation or flow of an incompressible viscoelastic liquid
   or solid (2.13)

infinite-shear viscosity (4.12)
inhomogeneous deformation of elastic solids (1.3)
isotropic compression (2.9)

lateral contraction ratio (2.6)
lateral strain (2.6)
linear viscoelastic behaviour (5.2)
linear viscoelastic behaviour of a liquid (5.2)
linear viscoelastic behaviour of a solid (5.2)
logarithmic decrement (6.5)
loss angle of a forced oscillation (5.10)
loss compliance (5.17)
loss curve (5.11)
loss factor (5.11)
loss modulus (5.13)
loss tangent (5.11)

Maxwell element (5.3)
Maxwell model (5.3)
modulus (4.3)
modulus of elasticity (4.3)

neutral axis (in forced flexural oscillation) (6.7)
neutral plane (in forced flexural oscillation) (6.7)
Newtonian liquid (4.2)
nominal stress (3.4)
non-Newtonian liquid (4.2)
normal stresses (3.5)

oscillatory (simple) shear flow (2.13)
out-of-phase viscosity (5.21)

phase angle (of a forced oscillation)  (5.10)
Piola tensor (1.8)
plane strain (1.8)
plane stress (1.2)
Poisson’s ratio (2.6)
pure shear deformation or flow (3.1)
pure shear of an elastic solid (2.7)
pure shear stress (3.1)

rate-of-strain tensor (1.12)
relaxation function (5.7)
relaxation spectrum (5.6)
relaxation time (5.6)
resonance curve (6.11)
resonance frequency (6.12)
resonance frequency (in forced flexural oscillation) (6.7)
resonance frequency (in forced uniaxial extensional oscillation) (6.7)
retardation spectrum (5.8)
retardation time (5.8)
Rivlin-Ericksen tensors (1.14)

secant modulus (4.7)
second moment of area (in forced flexural oscillation) (6.7)
second normal-stress coefficient (4.14)
second normal-stress difference (3.7)
second normal-stress function (3.7)
shear (2.8, 2.13)
shear compliance (4.11)
shear modulus (4.10)
shear rate (2.13)
shear strain (2.8)
shear stress (3.5)
shear viscosity (4.12)
simple shear of an elastic solid (2.8)
small-strain tensor (1.10)
spring constant (5.3, 5.4)
standard linear viscoelastic solid (5.5)
steady (simple) shear flow (2.13)
steady uniaxial homogeneous elongational deformation or flow of an incompressible
   viscoelastic liquid or solid (2.12)
storage compliance (5.16)
storage modulus (5.12)
stored energy function (4.1)
strain amplitude (of a forced oscillation) (5.10)
strain tensor (1.8)
stress (1.2)
stress amplitude (of a forced oscillation) (5.10)
stress relaxation (5.7)
stress tensor (1.2)
stress tensor resulting from an orthogonal deformation or flow (3.1)
stress tensor resulting form a simple shear deformation or flow (3.5)
stress vector (1.1)

tangent modulus (4.7)
tensile compliance (4.8)
tensile modulus (4.7)
tensile strain (2.4)
tensile stress (3.2)
three-point bending (6.7)
three-point flexure (6.7)
torsion pendulum (6.2)
traction (1.1)
true stress (1.2)

uniaxial compliance (4.8)
uniaxial deformation of an elastic solid (2.2)
uniaxial deformation or flow of an incompressible viscoelastic liquid or solid (2.11)
uniaxial deformation ratio (2.3)
uniaxial orthogonal deformation or flow (3.1)
uniaxial strain (2.4)

viscoelasticity (5.1)
viscosity (4.12)
Voigt-Kelvin element (5.4)
Voigt-Kelvin model (5.4)
Voigt element (5.4)
Voigt model (5.4)
volume compression (2.9)
vorticity tensor (1.13)

width of the resonance curve (6.13)

Young’s modulus (4.7)

zero-shear viscosity (4.12)

9.  GLOSSARY OF SYMBOLS

A()	resonance curve (6.11)
An	Rivlin-Ericksen tensors (1.14)

B	compliance in bulk compressive deformation (4.4)/
	bulk compliance/bulk compressive compliance (4.6)
B	Green tensor (1.8, 1.10)
B-1	Piola tensor (1.8)

C	compliance (general symbol) (4.4)
C	storage compliance (general symbol )(5.16)
C	loss compliance (general symbol) (5.17)
C*	complex compliance (general symbol) (5.19)
|C*|	absolute compliance (general symbol) (5.18)
C	Cauchy tensor (1.8, 1.9)
C-1	Finger tensor (1.8, 1.11)

D	compliance in uniaxial deformation (4.4)/
	uniaxial compliance/tensile compliance (4.8)
D	storage compliance in uniaxial deformation (5.16)
D	loss compliance in uniaxial deformation (5.17)
D*	complex compliance in uniaxial deformation (5.19)
|D*|	absolute compliance in uniaxial deformation (5.18)
D	rate-of-strain tensor (1.12)

E	modulus in uniaxial deformation (4.3)/Young’s 
	modulus/tensile modulus/secant modulus/tangent modulus (4.7)
E	storage modulus in uniaxial deformation (5.12)
E	loss modulus in uniaxial deformation (5.13)
E*	complex modulus in uniaxial deformation (5.15)
|E*|	absolute modulus in uniaxial deformation (5.14)
|E*|	flexural modulus (6.10)

F	deformation gradient tensor for an elastic solid (1.4) and for a 
	viscoelastic liquid or solid (1.7) 
f0	flexural force (6.8)

G	modulus in shear deformation (4.3)/shear modulus (4.10)
G	storage modulus in simple shear deformation (5.12)
G	loss modulus in simple shear deformation (5.13)
G*	complex modulus in simple shear deformation (5.15)
|G*|	absolute modulus in simple shear deformation (5.14)

J	compliance in shear deformation (4.7)/shear compliance (4.11)/
	creep compliance (5.9)
J	second moment of area (in a forced flexural oscillation)(6.7)
J	storage compliance in simple shear deformation (5.16)
J	loss compliance in simple shear deformation (5.17)
J*	complex compliance in simple shear deformation (5.19)
|J*|	absolute compliance in simple shear deformation (5.18)

K	modulus in bulk compressive deformation (4.3)/bulk modulus/
	bulk compressive modulus (4.5)

M	modulus (general symbol) (4.3)
M	storage modulus (general symbol) (5.12)
M	loss modulus (general symbol) (5.13)
M*	complex modulus (general symbol) (5.15)
|M*|	absolute modulus (general symbol) (5.14)

N1	first normal-stress difference/first normal-stress function (3.6)
N2	second normal-stress difference/second normal-stress function (3.7)

t	traction (1.1)
tan 	loss factor/loss tangent (5.11)

W	stored energy function (4.1)
W	vorticity tensor (1.13)

y	flexural deflection (6.9)

	decay constant (of a damping curve)  (6.2, 6.3)

	width of the resonance curve (6.13)

	shear/shear strain (2.8)
	shear rate (2.13)
0	strain amplitude (of a forced oscillation)(5.10)
	elongational strain rate/extension strain rate (2.12)
*	complex strain (of a forced oscillation) (5.15, 5.19)
	complex rate of strain (of a forced oscillation) (5.22)

	phase angle (of a forced oscillation)/loss angle of a forced oscillation (5.10)

	uniaxial strain/engineering strain/(uniaxial) 
	   tensile strain/(uniaxial) compressive strain (2.4)
	small-strain tensor (1.10)
H	Hencky strain (2.5)
lat	lateral strain (2.6)

	shear viscosity/coefficient of viscosity/viscosity (4.12)
´	dynamic viscosity (5.20)
´´	out-of-phase viscosity (5.21)
app	apparent viscosity (4.12)
E	extensional viscosity/elongational viscosity (4.9)
0	zero shear viscosity (4.12)
*	complex viscosity (5.22)
	infinite-shear viscosity (4.12)

	uniaxial deformation ratio/deformation ratio/
	extension ratio/compression ratio (2.3)
i	deformation gradients/deformation ratios;
			i = 1,2,3  (2.1)

	logarithmic decrement (of a decay curve) (6.5)

	Poisson’s ratio (2.6)

	decay frequency (of a damping curve) (6.4)
R	resonance frequency (6.12)

	tensile stress (3.2)
	compressive stress (3.3)
	engineering stress (3.4)
ii	normal stresses; i = 1,2,3  (3.5)
0	stress amplitude (of a forced oscillation) (5.10)
12	shear stress (3.5)
	stress/stress tensor (1.2, 3.1, 3.5)
*	complex stress (in a forced oscillation)
		(5.15, 5.19, 5.22)

	relaxation time (5.6)/retardation time (5.8)

	bulk compression/volume compression/isotropic compression/
	bulk compressive strain (2.9)

	first normal-stress coefficient (4.13)
	second normal-stress coefficient (4.14)
	creep function (5.9)
	relaxation function (5.7)

	angular velocity (of a forced oscillation) (5.10)
	angular velocity (of a decay frequency) (6.2)
R	angular velocity of the resonance frequency
	(of a forced flexural oscillation)  (6.7)