A. Ya.

Malkin 245

7
VISCOELASTICITY

7.1 INTRODUCTION
Rheological behavior related to viscoelasticity is the most relevant for the de-
scription of a majority of real materials. Viscoelastic effects exist in Newtonian
liquids (e.g., water) under special conditions of measurement, such as very high
frequencies. The other extreme of viscoelastic effects is offered by the example of
vibration damping in bells, due to “viscous” losses in metal.
In general, viscoelasticity is a combination (or superposition) of properties
characteristic for liquids (viscous dissipative losses) and solids (storage of elas-
tic energy). Therefore, a general definition of viscoelastic materials includes two
components — elastic potential and intensity of dissipative losses. However,
these two values are factors of a different dimension. The main characteristic
material constants (i.e., viscosity and modulus of elasticity) are also values of a
different dimension. The approach taken to combine elastic and viscous charac-
teristics of a material for description of its viscoelastic properties is worth a spe-
cial discussion because it leads to various models of a viscoelastic body.
Viscoelastic behavior can be considered as a slow (or delayed) development of
stresses and deformations in time, and this delay must not be confused with in-
ertial effects also characterized by a specific lag time. A very important, al-
though not explicit, word in the last sentence is “slow”. In order to discover
viscoelastic effects in regular liquids, we need to use ultra-high frequencies
(characteristic time of an experiment in this case is about 10-7 s), whereas time
delay effects, in deformations of concrete rods and plastic tubes under pressure,
246 Viscoelasticity

require years of observation (characteristic time is about 108 s). Moreover, one
can treat deformations of stones as a very slow process, realizing that it requires
geological periods of time (characteristic time is of the order of 1017 s; as we know
only the Lord has enough time to observe it).
The dimensionless criterion called the Deborah Number, De, was introduced
to be a measure of a ratio between characteristic time of observation, tobs, and the
time scale of inherent processes in a material, tinh. Then

De = tobs/tinh

The physical meaning of the value tinh is not identified here, and in fact, it can dif-
fer. In a general sense, tinh characterizes the rate of inherent rearrangement of
the material structure. Since the level of structure organization, its rupture and
restoration can vary, we can find very different values of characteristic times for
the same material. Therefore, in principle, different values of the Deborah Num-
ber are expected to exist.
The Deborah Number is especially important for viscoelastic phenomena be-
cause they always proceed in time. Since the time interval is very wide, we must
encounter a situation when the Deborah Number is of the order of 1, i.e., an ob-
server can “feel” that something happens with (or inside) a material. All this
means that complete description of viscoelastic phenomena includes experi-
ments which must occupy no less than 15 decimal orders along the time scale.
Certainly, in the majority of cases, this is not very realistic. Therefore, two gen-
eral approaches are usually taken to investigate viscoelastic phenomena:
• The measurement is done in a limited window along the time scale (for ex-
ample, from 10-1 to 103s) and the experimental data extrapolated beyond the
borders of the window
• Special methods are used to accelerate (or decelerate) viscoelastic processes
and then empirical (or semi-theoretical) approaches applied to modify the
time-scale. For example, one may increase temperature to accelerate the
process. Then, it is assumed that the change in temperature is equivalent to
some change in the time-scale (for example, what occurs at 100oC in 10 s re-
quires 104 s at 20oC).
Both general approaches are empirical, and even in the best case, if they are
based on some theoretical ideas, they may or may not be correct. Besides, any ex-
perimental data includes a scatter of points due to unavoidable errors of mea-
 A. Ya. Malkin 247

surement. The scatter of predictions (extrapolations) from this not-ideally-clear

window, with dispersed field points, is increasingly wider the farther we depart
from the borders of the window. It is thus rather dangerous to make far extrapo-
lations, even though it is very attractive to observe deformations during 10 h
and predict further development of deformations for 10 years.
Viscoelastic phenomena belong to the fundamental rheological effects describ-
ing the relationship between stresses and deformations. A general theory of
viscoelasticity is thus based on three-dimensional analysis and consideration of
concepts as tensor values. Quite adequate understanding of basic ideas of
viscoelasticity is available for uni-dimensional deformations. Especially, it is
true, for small deformations when non-linear effects of any kind do not appear.
In this case, it is not the mode of deformations which is important. That is why,
in the following Sections of this Chapter, discussion of simple deformations and
stresses does not identify the geometrical mode of deformation.
In specialized books, one can find a very rigorous and complete account of the
theory of (linear) viscoelasticity. The first comprehensive account of the linear
theory of viscoelasticity was published by B. Gross,1 followed by several books
containing mathematical foundations and main results.2-4 There are also some
textbooks containing not only main results of phenomenological theory but also
numerous ideas concerning its physical meaning and illustrating applica-
tions.5-7 It is now a homogeneous theory, based on solid mathematical back-
ground, containing all necessary theorems and answers to pertinent questions.
However, following the line of this book, we do not attempt to prove statements
of the theory and conclusions discussed below. The major points − physical ideas
used in the theory, definitions, main results and relationships, not requiring
high mathematical analysis − are included for the practical purpose of their ap-
plication in observation of material behavior.
In order to complete this introduction, it is worth mentioning that those profes-
sionals who know the theory of electrical networks may notice that ideas, con-
clusions, and relationships of the theory of viscoelasticity can be restructured
into terms of electrical networks by simple substitution of symbols. The same is
true for the theory of dielectric properties of materials. The same mathematical
structure of the theory is used, even though the physical objects differ.
248 Viscoelasticity

7.2 DEFINITIONS
This Section is devoted to the quantitative theory of effects briefly described in
Chapter 4.
Three fundamental experiments form the basis for the discussion:
• creep
• relaxation
• periodic deformations.
It is possible to discuss any other time-dependent stress or/and deformation
modes, but these three are the simplest, and they allow us to define the main
concepts used for description of viscoelastic effects.
7.2.1 CREEP

At constant stress, σ o , applied at initial time (t = 0), slow (or delayed) develop-
ment of deformations, ε(t), is observed, and this phenomenon is called creep.
The function ε(t) can be considered as consisting of three components:

t
ε (t) = ε o (t,σ o ) + Ψ(t,σ o ) + σo [7.1]
η(σ o )

where ε o is an instantaneous deformation, Ψ(t,σ o ) a function describing delayed

development of deformations, η(σ o ) viscosity, which (in a general case) can de-
pend on stress, t current time.
Another form of Eq 7.1 is

η(t) t
= I(t) = I o (σ ) + ψ(t,σ ) + [7.2]
σ η(σ )

where the value I(t) is called compliance, Io an instantaneous compliance, and

ψ(t) a creep function. Subscript zero at σ is omitted in this equation.
As pointed out above, in formulating Eqs 7.1 and 7.2, the type of deformation
(extension, shear, and so on) is not specific but it must be accepted that deforma-
tion is unidimensional.
A material is called linear viscoelastic if material parameters Io, ψ(t), and η
do not depend on stress; in the opposite case, material has non-linear
viscoelastic behavior.
 A. Ya. Malkin 249

It is necessary to separate two main cases shown in Figure 4.20:

• if ηis unlimitedly high, the last member in Eq 4.2 is absent and we deal with
a solid viscoelastic body. For linear viscoelastic solids, the value
ψ ∞ = ψ(t → ∞ ) is limited and the sum

I∞ = Io + ψ ∞ [7.3]

is called equilibrium compliance.

• For viscoelastic liquids (curve 2 in Figure 4.20), deformation increases un-
limitedly due to the linear (in time) increase of input of the last item in Eq
7.2. For viscoelastic liquids, equilibrium compliance equals zero.
Accelerated growth of deformation in creep (as shown by the curve 3 in Figure
4.20) is characteristic for non-linear viscoelastic behavior and acceleration
starts at lower deformations.
Measurements of creep in long-term loading of materials are widely used for
many fabricated goods and parts of machinery, continuously exploited under
stress; for example, pipes for transporting gases under pressure. In all these sit-
uations, creep has a detrimental effect on applied properties of materials. How-
ever, in technological practice, creep is used as a method to produce orientation
in drawing of fibers, films, and so on, with posterior fixation of the oriented
state.
7.2.2 RELAXATION

At constant deformation, ε o , set at some initial moment of time, t = 0, we can

observe slow decay of stresses in time σ(t). This phenomenon is called relax-
ation. The function σ(t) can be presented as consisting of two components

σ(t) = Φ (t,ε o ) + E ∞ (ε o )ε o [7.4]

where Φ(t,ε o ) is a function describing decay of stresses and E ∞ represents a re-

sidual (non-relaxing) or equilibrium component of stress.
Another form of Eq 7.4 is

σ(t)
= φ (t,ε ) + E ∞ (ε ) [7.5]
εo
250 Viscoelasticity

where φ(t,ε ) is a function of stresses, called relaxation function (sometimes

this function is called relaxation modulus) and E ∞ is called an equilibrium
modulus.
A material is called linear viscoelastic if φ(t) and E ∞ do not depend on initial de-
formation, ε; in the opposite case, material exhibits non-linear relaxation. As a
rule, even with non-linear behavior, E ∞ does not depend on ε, but the rate of re-
laxation represented by the function φ(t) does.
Similar to creep, it is reasonable to separate two main cases:
• if E ∞ = 0 material is a viscoelastic liquid
• if E ∞ > 0 , it means that a material can support stresses and therefore it is
considered a viscoelastic solid. In the latter case, the value

E o = φ (0) + E ∞ [7.6]

is called initial (or instantaneous) modulus.

Stress relaxation is a very important phenomenon in technological practice. If
it occurs with too slow a rate, the material is capable of storing residual (frozen)
stress, and this effect strongly influences its quality. A rapid relaxation can also
have adverse effect in use of some materials. For example, seals should exclude
gas or liquid leakages in equipments working under pressure. This can only be
achieved if a seal is continuously stressed during exploitation; relaxation leads
to loss of close contact between a seal and a solid wall of an apparatus.
7.2.3 PERIODIC OSCILLATIONS

It is the third well defined regime of deformations. The form of time-dependent

periodic deformation can be arbitrary but the theory deals with harmonic oscil-
lations. It is quite natural because a signal of any arbitrary form can be ex-
panded into Fourier series to form a harmonic function.
For mathematical convenience, periodic signals can be written in complex ex-
ponential functions exp(iωt) using Euler’s rule:

e iωt = cos ωt + i sin ωt

It can be assumed that stress, σ(t), changes as

σ(t) = σ o e iωt [7.7]

 A. Ya. Malkin 251

where σ o is an amplitude of stress, and ω its frequency of oscillations.

One may expect that deformations will change periodically, and in the first ap-
proximation, ε(t) is described by a harmonic function with some delay with re-
spect to σ(t). Then

ε (t) = ε o e i( ωt-δ ) [7.8]

where ε o is an amplitude value of deformations and δ the phase angle (i.e., a

value characterizing the phase difference in oscillations of stresses and defor-
mations); this value is also called loss angle and the physical meaning of this
term is discussed below.
If deformation, not stress, is a preset function, i.e.

ε (t) = ε o e iωt [7.9]

then stress is changing according to the equation

σ(t) = σ o e i( ωt+ δ ) [7.10]

Certainly, Eqs 7.9 - 7.10 are equivalent to Eqs 7.7 - 7.8.

Now, let us introduce the main parameter used to characterize viscoelastic
properties of a material measured by periodic oscillations. It is a dynamic
modulus of elasticity, E*, determined as

σo σo
E *=
εo
=
εo
(cos δ + isinδ) [7.11]

It is evident that E* is characterized by two parameters: its absolute value

σo
Eo = [7.12]
εo

and phase angle, δ.

Both factors can depend on frequency and (in principle) on amplitude of defor-
mation. For a linear viscoelastic body, the amplitude dependencies of Eo and
252 Viscoelasticity

δ are absent, which is an additional definition of linear viscoelastic behavior.

Dynamic modulus can be represented by two elements

E* = E′ + iE′′ [7.13]

where

σo σ
E′ = cosδ ; E′′ = o sinδ [7.14]
εo εo

These parameters are called real and imaginary components of dynamic

modulus, respectively, or E′ is called storage modulus and E′′ loss modulus.
Absolute value of dynamic modulus is expressed through E′ and E′′ as
1 2
|E *|= [(E′ ) 2 + (E′′ ) 2 ] [7.15]

Instead of E* and its components E′ and E′′, their reciprocal values can also be
used. They are called dynamic compliance

I * = I′ − iI′′ [7.16]

determined as

εo ε
I′ = cosδ ; I′′ = o sin δ [7.17]
σo σo

The relationship between complex modulus and complex compliance is sim-

ple and evident

E * I* = 1 [7.18]

and relationships between components of E* and I* are established from

Eq 7.18 according to the rules of operation with complex numbers. They can be
written as
δ
 A. Ya. Malkin 253

I′ I′′
E′ = ; E′′ = [7.19]
2 1
[(I′ ) 2 + (I′′ ) ]
2
[(I′ ) + (I′′ ) 2 ]1/ 2
2

and vice versa:

E′ E′′
I′ = ; I′′ = [7.20]
[(E′ ) + (E′′ ) ]
2 2 1/ 2
[(E′ ) + (E′′ ) 2 ]1/ 2
2

The last useful final expression in the theory of periodic oscillations is derived
from Eqs 7.14 and 7.17. It connects loss angle with components of dynamic
modulus and compliance:

E′′ I′′
tanδ = = [7.21]
E′ I′

When liquids are studied by a method of periodic oscillations, sometimes it is

more convenient to use rates of deformation rather than deformations them-
selves. If ε(t) is changing in accordance with Eq 7.8 then

dε
ε& = = ε o iωe i( ωt-δ ) = iωε
dt

A new parameter, characterizing viscoelastic properties and defined as regu-

lar viscosity by the ratio of stress to rate of deformation, can be introduced. This
parameter is called dynamic viscosity and is expressed as

σ σo
η* = = (sinδ − icosδ ) = η′ − η′′ [7.22]
ε& ε o ω

where

σo E′′ σ E′
η′ = sinδ = ; η′′ = o cosδ = [7.23]
ε oω ω ε oω ω
254 Viscoelasticity

Any pair of the above-introduced parame-

ters is a complete measure of viscoelastic
properties of a material. It can be E′ and E′′
(or E′ and tanδ), or I′ and I′′ or η′ and η′′. All
other parameters can be calculated from any
pair by means of simple algebraic operations
written above.
Concepts and definitions discussed above
can be visually interpreted by the graph
(Figure 7.1). Let deformations and stresses
be depicted by vectors with their length
equal to their amplitude values,|ε o|and|σ o|,
respectively. Vector of deformation rate is
also given. Coordinate axes are formed by
real and imaginary numbers. The angle be-
Figure 7.1. Graphic interpretation tween vectors of deformation and stress
of osci llat o r y m e asur e m e nt s: equals the loss angle δ. Let all vectors in Fig-
scheme illustrating interrelations
of a ll m ain par am e t e r s o f
ure 7.1 rotate counter-clockwise with angu-
viscoelastic behavior and their defi- lar velocity ω. It means that the angle
nitions. between the vector σ o and abscissa equals ωt,
and between the vector ε o and abscissa is (ωt − δ).
This figure allows for the following interpretation of the main parameters con-
sidered in the theory of viscoelasticity. If we project the vector of stress onto the
vector of deformation, the value of σ o cosδ and the ratio of this projection to the
length of the vector of deformation gives the real component of dynamic modu-
lus, E′. If we take the projection of σ o onto the direction perpendicular to the vec-
tor of deformation and calculate the ratio of this projection (equal to σ o sin δ) to
the length of the vector of deformation, the imaginary part of dynamic modulus,
E′′, is received.
If we project the vector of deformation onto the vector of stress and onto the
perpendicular direction to find the ratios of lengths, the definition of I′ and I′′ is
obtained. Then the analogous procedure with the vector of rate of deformation
leads to the components of dynamic viscosity.
The graphic interpretation of oscillatory measurements allows one to treat lin-
earity of viscoelastic behavior for a linear viscoelastic material:
• by changing the length of one of the vectors in Figure 7.1, all other vectors
 A. Ya. Malkin 255

change proportionally to the first

• the angle between stress and deformation vectors does not depend on their
lengths and does not change during rotation of all vectors at any angular ve-
locity.
One more geometrical interpretation of viscoelastic behavior, very useful and
utilized in practice, can be obtained from rearrangement of Eqs 7.7 and 7.8 and
by excluding time as a parameter of these equations. The direct relationship be-
tween stress and deformation then has the following form:
2 2
 σ   ε   σ  ε 
  +   = sin 2δ + 2   cosδ [7.24]
σo  εo  σo εo 

This equation is that of an ellipsis.

The following designations can be introduced in order to simplify the interpre-
tation of experimental results:

σ ε
x= ; y=
σo εo

Then Eq 7.24 can be written as

x 2 + y 2 = sin 2δ + 2xycosδ [7.25]

The characteristics (ellipsis) described by Eq 7.25 is drawn in Figure 7.2. Calcu-

lations show that the area of the ellipsis A is

A = πε o σ o sinδ [7.26]

i.e., the area of the ellipsis is proportional to the amplitude values of stress and
deformation and depends on loss angle, δ.
There are two limiting cases important for further interpretation:
• if δ = π 2, then Eq 7.25 has the form:

x 2 + y 2 = sin 2δ

it means that an ellipsis degenerates into a circle;

256 Viscoelasticity

• if δ = 0, then Eq 7.25 degenerates into

x=y

which is an equation of a straight line.

Graphic interpretation of stress-vs-deformation dependence in the form of el-
lipsis allows one to make some comments concerning inputs of elastic and
dissipative components in deformation of viscoelastic material. For this pur-
pose, let us calculate the work, W, produced during a cycle of deformation of a
viscoelastic body. This work is found as
T
W = ∫ σ(t)dε [7.27]
0

where T = 2π ω is a duration of an oscillation cycle.

Direct calculations show that the work, W, is

W = πε o σ o sinδ [7.28]

The identity of expressions in Eqs 7.26 and 7.28 is obvious and gives the inter-
pretation of the area of ellipsis in Figure 7.2 as the work produced during an os-
cillation cycle. Certainly, there is no elastic (stored) energy when a cycle is
completed, because otherwise an unlimited increase in stored energy from cycle
to cycle would be observed, which is physically impossible. That is why the en-
ergy calculated from Eq 7.28 reflects the work dissipated during a cycle of oscil-
lation. The ellipsis, as in Figure 7.2, is called a hysteresis loop because it
represents a delayed part of deformation.
Eq 7.28 allows one to propose the interpretation of a physical meaning of com-
ponents of dynamic modulus and compliance. If we substitute expressions for E′′
by I′′, the following equation is obtained:

W = πε 2o E′′ = πσ 2o I′′ [7.29]

Both values, E′′ and I′′, are measures of energy dissipation during periodical os-
cillations.
It is also possible to show that real components of dynamic modulus and com-
pliance, E′ and I′, are measures of elasticity because energy stored (and then re-
 A. Ya. Malkin 257

turned) during the cycle of oscillation is

proportional to them.
The loss angle can be found from Figure 7.2 by the
following simple method. The area of a rectangle
circumscribed around an ellipsis is equal to 4σ o ε o .
The ratio of areas of an ellipsis to a circumscribed
rectangle equals sinδ/4. Then it is possible to find
the loss angle from this ratio without the measure-
ment of amplitudes of stress and deformation. This
approach is used in some standards, and instru-
mental measurements were introduced in studies
Figure 7.2. Ellipsis in of damping characteristics of rubbers and rubber
stress - deformation coor-
dinates as a representa-
compounds. Hysteresis loop surface area (see Fig-
t ion of visc o e last ic ure 7.2) is a measure of mechanical losses on defor-
properties of material. mation.
Now, we can come back to two limiting cases of the values ofδ mentioned above.
If δ = π 2 and sinδ = 1, the energy dissipation is at its maximum, typical for liq-
uid without any elastic properties. If δ = 0 and sinδ = 0, there is no energy dissipa-
tion, corresponding to the other limiting case of an elastic material without
viscous losses. In all other situations, intermediate cases are met in which the
value of the loss angle characterizes the ratio of viscous-to-elastic properties in
viscoelastic materials. By decreasing δ, and consequently decreasing viscous
losses, material transits from pure viscous to pure elastic.
In real practice, viscoelastic materials are in the form of springs, rings, and so
on. Engineering constructions must be as highly elastic as possible (losses must
be low). Shock-absorbers, sound isolators, and materials for many other similar
applications must possess a high dissipative function, meaning that the loss an-
gle of such materials must be as close to π/2 as possible.
The above-formulated functions are used to describe viscoelastic effects and
characterize properties of real materials. However, it is necessary to emphasize
that the definition of all these functions implies that they must be defined (mea-
sured) in the range of their arguments (time or frequency as a value reciprocal to
time) from zero and to infinity. It already has been mentioned that this is unreal,
and such a requirement is the main problem in practical applications of the
viscoelasticity theory. It is not a formal point but a serious physical limitation of
the theory. It is easy to write equations including these limits, and below, many
258 Viscoelasticity

equations of this kind are presented, including the above-introduced parame-

ters, because they are frequently used in the theory of viscoelasticity. At the
same, it is very difficult to use equations in practice because input of some
viscoelastic functions (creep, relaxation, dynamic modulus) beyond the limits of
their direct measurement (beyond an experimental “window”) can lead to uncer-
tain errors in predicting deformation behavior of a real material. We shall re-
visit this problem in the following sections of this chapter.
7.3 PRINCIPLE OF SUPERPOSITION
The general theory of viscoelasticity is designed to answer two related ques-
tions:
• either functions, introduced for formal description of deformations or
stresses in the fundamental experiments (creep, relaxation, periodic oscilla-
tion), are independent characteristics of material or they are inherently con-
nected to each other
• if one or some basic functions of viscoelastic material are known (have been
measured), can one describe its deformation-vs-stress behavior in any arbi-
trary mode of deformation (for example, find evolution of deformation for ar-
bitrary history of loading)?
The answer to these questions is the main content of the theory of
viscoelasticity. And this answer is: “yes”, all fundamental functions are inher-
ently related to each other, and “yes” to the second question means that we can
describe deformational behavior of material if at least one basic viscoelastic
function has been measured beforehand.
Both positive answers are founded based on the principle of linear super-
position of stresses and/or deformations. This principle was formulated by
Boltzmann.8-11 The concept may have different forms, but the basic idea is re-
lated to the mutual independence of all consequent events happening to the ma-
terial. In fact, it means that all materials are sufficiently weak, therefore they
cannot change the mode of reaction to an external action. The material reacts to
the next action as if no former action took place. In other words, the structure
and properties of material are not changed, regardless of its deformation, and
the last statement is a real physical meaning of the Boltzmann Principle.
Now, let us write the above-stated concept in the form of mathematical sym-
bols. Let the initial stress, acting from the moment t = 0, be equal to σ o . Then, de-
formations change according to Eq 7.2. At some point of time, t′, let stress change
 A. Ya. Malkin 259

by ∆σ. The principle of linear superposition assumes that in this case, deforma-
tion changes accordingly:

 t  t − t′ 
ε (t) = σ o  I o + ϕ (t) +  + ∆σ  I o + ϕ (T − t ′ ) +
 η  η 

Stress can change at any given time which follows. For any such moment of
time and any corresponding change of stress, one can add an independent item
in the last equation for ε(t).
Certainly, stress can change continuously, and bearing this in mind, we come
to the final integral (instead of sum) formulation of the Boltzmann principle of
linear superposition:
t
 t − t′ 
ε (t) = ∫  I o + ψ(t − t ′ ) + dσ [7.30]
0 
η 

or
t
dσ  t − t′ 
ε (t) = ∫  I 0 + ψ(t − t ′ ) + η  dt ′ [7.31]
0
dt ′  

The analogous line of arguments can be used to describe changes in deforma-

tion, and in this case, Eq 7.5 is a starting-point. The final result is quite similar
to Eq 7.31, and can be written as
t
dε
σ(t) = ∫ [ϕ (t − t ′ ) + E ∞ ]dt ′ [7.32]
0
dt ′

The pair of symmetrical Eqs 7.31 - 7.32 is called the Boltzmann-Volterra

equations.12,13 They are the mathematical formulation of the principle of lin-
ear superposition.
It is possible to illustrate the behavior of viscoelastic material according to the
principle of superposition by the following example for elastic recoil (retarda-
tion) after forced deformation of a body.
260 Viscoelasticity

Figure 7.3. Deformation history (a) and the reaction (from the point A when outside force is re-
moved) of ideal elastic (b) and viscoelastic (c) materials.

Let the history of deformations be as shown in Figure 7.3a: the external force
created deformation, ε o , and then (very rapidly) the same deformation but with
the opposite sign, -ε o . When the force was acting during two short periods of
time, one could neglect partial relaxation at deformations ε o and − ε o . The ques-
tion is: what happens if, at the moment A, the external force is removed? Ideal
elastic body immediately returns to its initial state, as shown by the vertical line
from the point A in Figure 7.3b. The behavior of viscoelastic body is quite differ-
ent, as illustrated by the line ABC in Figure 7.3c. The first part of this line, AB, is
retardation from the second deformation, -ε o , not to zero but to a state deter-
mined by the first deformation, ε o . Only after that slow (delayed) action, a re-
turn to the zero state occurs.
Another very interesting (and important for technological applications) exam-
ple of influence of deformational prehistory on the behavior of a material is re-
lated to the processing of polymers (thermoplastics and rubber compounds).
During extrusion of continuous profiles, a molten material moves between a
screw and a barrel of an extruder, then it passes transient channels. Finally, it is
shaped in an outlet section of a die. It would be desirable that the shape of a final
profile is equivalent to the shape an outlet section of a die. On the contrary, the
material continues to react to all deformations which took place before the outlet
section of a die. As a result, distortion of its shape occurs; therefore, the final sec-
tion of a part can be very different than expected. The distortions can be so se-
vere that “melt fracture” (shown in Figures 4.14 and 4.15) is observed.
These and many other examples are characteristic for technological practice.
Viscoelastic materials have fading memory of the history of previous deforma-
 A. Ya. Malkin 261

tions. In this sense, the integrals 7.31 and 7.32 are called hereditary integrals
because they summarize events which took place before the current moment of
time and are responsible for the stress (or deformation) state of a material at the
current moment.
The relaxation function φ(t) is a decreasing function. Therefore, its values are
higher when the argument is smaller. It means that the changes of deformation,
which happened earlier, influence stress in lesser degree than later changes. In
the first case, the value of the argument (t-t′) in Eq 7.32, for the fixed moment of
time, t, is smaller than for events which happened later because values of t′ are
smaller. In other words, a material continuously “forgets” what happened before
and therefore integrals in Eqs 7.31 and 7.32 form a model of material with “fad-
ing memory”.
It is interesting to outline the limiting cases. They are:
• liquid which “forgets” everything immediately (energy of deformation com-
pletely dissipates); in this case, the integral 7.32 transforms to the New-
ton-Stokes Law
• solid which “remembers” everything (energy of deformation is completely
stored), and in this case, the integrals 7.31 and 7.32 transform to the Hooke
Law.
Both Eqs 7.31 and 7.32 contain deformation and stress, and each of them can
be treated as an equation either of stress or deformation. Eq 7.31 determines the
development of deformations for known evolution of stresses. It can be consid-
ered as an integral equation for σ(t) if the function ε(t) is known. The same is true
for Eq 7.32. Therefore, it is possible to exclude these functions by substituting,
for example, the function ε(t) from Eq 7.31 to the right side of Eq 7.32. After some
formal mathematical rearrangements, the relationship between rheological pa-
rameters does not contain either σ(t) nor ε(t). The resulting equation includes
creep and relaxation functions in the following form:
t
t  1 dψ(t − t ′ 
E ∞ I o + I o ϕ (t) + E ∞  + ψ(t) +
η 
∫ ϕ (t′ ) η +
0
d(t − t ′ ) 
dt ′ = 1 [7.33]

where E ∞ is an equilibrium modulus, I0 an instantaneous compliance, η viscos-

ity, φ(t) a relaxation function, and ψ(t) creep function.
Eq 7.33 shows that the relaxation and creep functions are not independent but
related to each other by the integral equation. If one of these functions is known
262 Viscoelasticity

(measured, calculated, assumed), the other can be found from Eq 7.33. This
equation formally, and quite rigorously, confirms the above-formulated state-
ment that the behavior of material, in different modes of deformations, is gov-
erned by the same inherent properties.
Eqs 7.31 and 7.32 give mathematical ground for calculation of stress-vs-defor-
mation relationship at any arbitrary path of material loading. The only essen-
tial limitation in application of these equations is the requirement of linearity of
rheological behavior of a medium, i.e., independence of all material constants
and functions entered to these equations (instantaneous compliance, equilib-
rium modulus, viscosity, relaxation and creep functions) on stresses and defor-
mations.
Certainly, Eqs 7.31 and 7.32 are rheological equations of state for viscoelastic
materials. It is necessary to remember that everything discussed above is re-
lated to “a point” in the sense as adapted for a Newtonian liquid and a Hookean
solid and in general for any rheological equation of state. In order to find
stress-deformation distribution throughout a body, one must combine these
equations with equilibrium conditions (equations of conservation, introduced in
Chapter 2) and appropriate boundary conditions as suggested in Chapters 5 and
6 for any liquid and solid.
7.4 RELAXATION AND RETARDATION SPECTRA
Relaxation, ϕ(t), is a decreasing (at least not increasing) function of time. As a
first approximation, it is reasonable to estimate it by an exponential function:

ϕ (t) = E o e -t/ θ [7.34]

where Eo is a instantaneous modulus andθ is a value called a relaxation time.

Relaxation process is described by a single exponential function called
Maxwell (or Maxwellian) relaxation. However, it is a rather rough approxima-
tion and it can be improved by increasing the number of exponents, i.e., by ex-
pansion of ϕ(t) into a sum of N exponents:
N
ϕ (t) = ∑ E i e -t/ θ i [7.35]
i =1
 A. Ya. Malkin 263

N
Eo = ∑ Ei
i =1

where Ei are called partial moduli, and θ i

are the set (or a spectrum) of relaxation
times.
Such spectrum can be called discrete and
can be drawn by a set of lines as in Figure
7.4, where each value of Ei (length of a line)
is put in correspondence to its argument
equal to θi. In the limiting case, the lines in
Figure 7.4 can fill all the graph if they are
situated very close and their tops form a
Figure 7.4. Discrete relaxation
spectrum. continuous curve. In this limiting case, we
have the transition from the sum in Eq
7.35 to the integral
∞
ϕ (t) = ∫ F(θ )e -t/ θ dθ [7.36]
0

where the value F(θ )dθ plays a role of a partial modulus as in Eq 7.35 and the
function F(θ) is called a relaxation spectrum or a distribution of relaxation
times.
There is a mathematical theorem stating that any decreasing function (in our
case, the relaxation function, ϕ(t)), of any kind can be represented by its expo-
nential image, i.e., by the integral (Eq 7.36). This statement leads to the conclu-
sion that there is an unambiguous correspondence between any relaxation
function, ϕ(t), and the relaxation spectrum, F(θ).
In principle, a relaxation spectrum can be found as a solution of the integral in
Eq 7.36. Certainly, it is always possible to know the analytical form of the func-
tion ϕ(t) in its full interval from 0 to ∞. It appears simple in a theoretical ap-
proach, but not so easy in treatment of experimental data.
There are two reasons complicating a problem of the transition from experi-
mental points to solution of the integral Eq 7.36. Both reasons already have been
mentioned above: the first is an uncertainty in behavior of an experimental
function beyond the borders of measurements (close to zero and at high values of
264 Viscoelasticity

the argument, at t → ∞), and the second concerns the natural scatter of experi-
mental points, which makes the analytical approximation of these points, and
extrapolation beyond the experimental window, an ambiguous procedure.
It is very important to know the relaxation spectrum because in many theoreti-
cal investigations a relaxation spectrum, is directly related to molecular move-
ments and thus to molecular structure of matter. Therefore, many different
methods were proposed which give a solution of this problem. Below, comments
are included concerning their validity.
7.4.1 CALCULATING CONTINUOUS SPECTRUM

The method is based on direct solving of Eq 7.36. There are two possible ways
to do so. The first consists of analytical approximation of experimental points by
appropriate formula and direct solving of Eq 7.36 using numerous published ta-
bles of conversion of exponential images. The error of approximation must be
lower than an experimental error, and many such analytical approximations
are possible. Each of them gives a different spectrum.1-6 If the error of approxi-
mation is sufficiently low, it does not yet imply that the error of a calculated
spectrum is acceptable.
It is also possible to use the second, rather old, method6 of finding a rough solu-
tion of Eq 7.36. For this purpose, one can substitute an exponential function in
Eq 7.36 by its approximate expression:

exp(-t / θ ) ≈ 1 at t / θ < 1; t < θ

exp(-t / θ ) = 0 at t / θ > 1; t > θ

The idea of this (first order) approxima-

tion is seen from Figure 7.5. The continu-
ous exponential curve is changed by an
abrupt step. Eq 7.36 becomes:
∞
ϕ (t) ≈ ∫ F(θ )dθ [7.37]
θ

Figure 7.5. Stepwise change of expo-

nential function to calculate a spectrum In fact, it means that a term on the right
(first approximation). side of Eq 7.36 is omitted on the basis that
 A. Ya. Malkin 265

the exponential decay will suppress the input from the function F(θ), assumed to
be rather low. The error of this approximation is unknown. Due to this process, a
very simple formula for calculation of F(θ) is obtained by differentiating both
sides of Eq 7.37 by the lower limit of the integral:
dϕ (t)
F(θ ) = − at t = θ [7.38]
dt

The idea of this method can be used for higher approximations but the same
main disadvantages of uncertain errors still exist.
7.4.2 CALCULATING DISCRETE SPECTRUM

The practice of treating real experimental data of rheological investigations

shows that, in fact, any relaxation curve (decay of stresses) can be approximated
by the sum of 4-5 or (in the worst case) of 7-8 exponential items. A very limited
number of relaxation times is thus needed to describe experimental points with
permissible error within scatter of measurements.
Standard computer procedures of minimizing an error in finding constants of
some analytical formula are also available. This problem was under discussion
from the beginning of application of the linear theory of viscoelasticity to real ex-
perimental data and continues to be the focus of interest. The most important
goal is to minimize non-linear functional errors to which numerous publications
were devoted.14-18 In this case, it is a sum of N exponents with unknown weights
Ei and values of θ i . In this approach, one begins with N = 1 and increases the
number of exponents. Each step permits on to decrease the error of approxima-
tion. The last step and the final number of exponents is when an error of approxi-
mation becomes lower than permissible error of experiment.
In some theories, the set of relaxation times appears to be dependent but fol-
lows the definite rule. For example, the row of relaxation times obeys the follow-
ing rule:

θ i = θ -ni
o

where θ o and n are constants (e.g., n = 2) and i are integers (i = 0, 1, 2, 3...).19 It

limits the possibility to vary the parameters of a relaxation spectrum because
there are only two independent parameters (θ o and n); moreover, since n is a re-
sult of molecular model calculations it cannot be treated as a fully independent
266 Viscoelasticity

parameter. Now the relaxation function takes the following form:

∞
ϕ (t) = ∑ E i θ -ni [7.39]
i =1

The procedure minimizes error of approximation but the set of constants under
search (Ei) is different from the general case based on Eq 7.35.
According to this approach, it is irrelevant to search for a “true” number of ex-
ponents (or relaxation times) in a discrete spectrum because the procedure of ap-
proximation should only be continued up to the limit of experimental error, after
which further effort is inconsequential. The minimal number of exponents
which correctly (within the limits of experimental error) describe results of mea-
surements should be used.
Discussion of the relaxation function, ϕ(t), can be almost repeated word for
word as regards the creep function, ψ(t). In the latter case, the resulting equa-
tion is as follows:
∞
ψ(t) = ∑ [I i (1 − e -t/ λi )] [7.40]
i =1

or in the form of the continuous spectrum:

∞
ψ(t) = ∫ Φ (λ )(1 − e -t/ λ )dλ [7.41]
0

where Φ(λ) is called a retardation spectrum or a distribution of retardation

times.
The introduction of the integral kernel in Eq 7.40 in the form (1 − e -t/ λ ) instead
of e -t/ θ is explained by the fact that ψ(t) is increasing, not decreasing (as ϕ(t)),
though its mirror reflection, or elastic recoil function, is also a decreasing func-
tion, as is shown in Figure 4.20, curve 1.
The change of an exponent function e -t/ θ for (1- e -t/ λ ) leads to small variations
in methods of approximation of spectrum. Instead of the step shown in Figure
7.5, one may write the following (first order ) approximate equalities:
 A. Ya. Malkin 267

1 − e -t/ λ ≈ 0 at t / λ < 1; t < λ

1 − e -t/ λ ≈ 1 at t / λ > 1; t > λ

Graphic interpretation of this approximation is essentially the same as in Fig-

ure 7.5, with an exception that the mirror reflection of Figure 7.5 is considered.
Discussion in relation to a relaxation spectrum, including arguments concern-
ing methods of its calculation, holds true for a retardation spectrum.
In real practice, the measurement of relaxation and retardation (or creep) is
performed for as wide a time range as possible to cover at least some decimal or-
ders of the argument. It is analogous to determining a flow curve (Chapter 5) in a
wide range of shear rates. The linear time scale is changed to logarithmic in or-
der to make experimental results easier to visualize (the same method was used
for flow curves in Chapter 5). This conversion is expressed by the following for-
mulas:
∞
ϕ (t) = ∫
-∞
H(lnθ )e -t/ θ dlnθ [7.42]

and
∞
ψ(t) = ∫
-∞
L(lnλ )e -t/ λdlnλ [7.43]

with the following obvious relationships between linear and logarithmic spectra

H(lnθ ) = θF(θ ) [7.44]

and

L(lnλ ) = λΦ (λ ) [7.45]
268 Viscoelasticity

The last problem to be discussed in this section concerns the interrelation be-
tween distribution of relaxation and retardation times. They are not equivalent,
though certainly they both originate from the same molecular phenomena and
thus are closely related to each other.
The existence of interrelation between relaxation F(θ) and retardation Φ(λ )
spectra can be proven if we take into account that F(θ) represents a relaxation
function, Φ(λ ) represents a creep function, and both are connected by Eq 7.33. It
is illustrated in Figure 7.6. One can expect that a relationship between spectra
F(θ) and Φ(λ ) also exists.20
If one of the pair of spectra is continuous, the second one is also continuous,
though they are not equivalent. If a relaxation spectrum is discrete, then a retar-
dation spectrum is also discrete. Besides, there is an interesting point rigorously
proven in the theory regarding correlation of numbers and positions of lines in a
pair of discrete spectra. In viscoelastic liquid (η < ∞ and G ∞ = 0), the number of
members (lines) in a relaxation spectrum is larger by one than the number of
members in a retardation spectrum, and the lines of a retardation spectrum are
arranged between the lines of a relaxation spectrum. For example, in
Maxwellian relaxation there is one relaxation time (Eq 7.34) but the retardation
spectrum is empty (no lines) and deformation of a Maxwellian viscoelastic liquid
at constant stress occurs without delay. In a viscoelastic solid, the number of
lines in both spectra is the same.
7.5 DYNAMIC AND RELAXATION PROPERTIES - CORRELATIONS
The dynamic characteristics (components of dynamic modulus and compli-
ance) can be correlated with the relaxation properties of material. Certainly, it
can be done based on the fundamental principle of linear superposition. In this
section, main theoretical results of correlations between dynamic and relax-
ation properties are discussed.
The components of the dynamic modulus are expressed through a relaxation
function as
∞
G′ (ω) = ω∫ ϕ (t)sinωtdt [7.46]
0
 A. Ya. Malkin 269

The inverse relationships, i.e., solution of Eqs 7.45 and 7.46, have the following
form:
∞
G′′ = ω∫ ϕ (t)cosωtdt [7.47]
0

For a relaxation function ϕ(t), which is calculated from components of the dy-
namic modulus, it has a similar form:
∞
2 G′ (ω)
π ∫0
ϕ (t) = sin ωtdt [7.48]
ω

and
∞
2 G′′ (ω)
π ∫0
ϕ (t) = cos ωtdt [7.49]
ω

Two main conclusions can be drawn from Eqs 7.46 to 7.49:

• components of the dynamic modulus can be calculated if a relaxation func-
tion was measured (and vice versa)
• accuracy of such calculations is limited by the need to find the integrals in
these equations in limits from 0 to ∞, and uncertain input of “tails" of func-
tions on the right side of these integrals.
Analogous equations can be established for the components of dynamic compli-
ance. Rigorous calculations give the following results:
∞
∂ψ
I′ (ω) = ∫ cos ωtdt [7.50]
0
∂t

∞
∂ψ
I′′ = ∫ sin ωtdt [7.51]
0
∂t
270 Viscoelasticity

The inverse relationships for a creep

function can also be written. They are
quite analogous to Eqs 7.48 and 7.49, with
an evident change of components of modu-
lus for compliance and relaxation function
for a derivative of a creep function.
The same conclusions apply to the possi-
bility of a mutual correlation between dy-
namic compliance and creep function, as
for the correlation between dynamic modu-
lus and a relaxation function as discussed
above.
Figure 7.6. Structure of interrelations The main sense of all equations given in
between relaxation and retardation this section is, first of all, to demonstrate
spectra.
the existence of relationships for all
viscoelastic characteristics under discus-
sion (they can be used for mutual calculations), and secondly, to emphasize the
fact that all these relationships are represented by the integral equations with
infinite limit, which complicates practical applications of these relationships.
Figures 7.6 and 7.7 show that components of dynamic modulus and compliance
can also be expressed through relaxation (retardation) spectra. The equations
below give a final result for G*(ω):
∞
(ωθ ) 2
E′ (ω) = ∫ F(θ ) [7.52]
0 1+ (ωθ ) 2

∞
(ωθ )
E′′ = ∫ F(θ ) [7.53]
0 1+ (ωθ ) 2

Analogous equations can be written for I * (ω), and they are

∞
1
I′ (ω) = ∫ Φ (λ ) dλ + I o [7.54]
0 1+ (ωλ ) 2
 A. Ya. Malkin 271

∞
1 (ωλ )
I′′ =
ωη
+ ∫ Φ (λ ) 1+ (ωλ )
0
2
dλ [7.55]

All these relationships have a

structure very similar to the
equations discussed above.
They show that it is possible to
establish correspondence be-
tween dynamic properties and
relaxation spectra, but real cal-
culations are rather complex
because of integral equations
with infinite limits.
Inverse transforms of Eqs
Figure 7.7. Structure of interrelations between relax- 7.52 to 7.55 cannot be ex-
ation characteristics of a viscoelastic material includ-
ing dynamic modulus and compliance. pressed in analytical form for
arbitrary dynamic functions. It
can be illustrated by constructing the first approximation formula related to re-
laxation spectrum and dynamic modulus.22,23 As in Figure 7.5, we change the
kernel in the equation for dynamic modulus stepwise:

(ωθ ) 2
≈ 0 at θ < ω-1
1+ (ωθ ) 2

(ωθ ) 2
≈ 1 at θ > ω-1
1+ (ωθ ) 2

The meaning of this approximation is il-

lustrated in Figure 7.8. Then, Eq 7.51
changes to
Figure 7.8. Stepwise change of the ker-
nel in the integral equation for dynamic
∞
modulus to calculate a spectrum (first
approximation). E′ ≈ ∫ F(θ )dθ
ω-1
272 Viscoelasticity

After differentiating it by the lower limit of integral, the following approximate

equation for calculating a relaxation spectrum by dynamic modulus is obtained:

ω2dE′ (ω)
F(θ ) ≈ at θ = ω-1
dω

Quite analogous equations can be written for all other dynamic functions un-
der discussion. Their accuracy is limited by an uncertain input of the “tail” of a
function used for calculations which has been cut off because of change in the
kernel in the integral equation by a rather rough step approximation − a com-
mon feature of all approximations used in solving integral equations applied in
the theory of linear viscoelasticity
Equations written in the last two sections and schemes in Figures 7.6 and 7.7
give the positive answer to the main question on the theory formulated above:
all viscoelastic characteristics of a material are related to each other and can be
mutually calculated. However, this answer is partially positive because of inevi-
table complications of these calculations, due to the special form of equations
connecting all viscoelastic functions.
7.6 RELATIONSHIPS BETWEEN CONSTANTS
In the main definitions and in all equations of the theory of viscoelasticity
there are some constants representing limiting cases (at t = 0 and t → ∞) of
stresses and deformations. These constants are:
η - viscosity at steady flow (constant when liquid is linear or Newtonian)
Io - instantaneous compliance
I ∞ - equilibrium compliance
E ∞ - equilibrium modulus
Eo - instantaneous modulus.
Discussion below concentrates on establishing the relationships between
these constants and relaxation characteristics of a material, such as creep and
relaxation functions, and a relaxation spectrum. To begin, it should be repeated
that some of the constants are characteristics for either viscoelastic solid or
viscoelastic liquid, which causes relationships between the constants to be dif-
ferent for these two types of viscoelastic materials.
 A. Ya. Malkin 273

7.6.1 VISCOELASTIC SOLID

The definition of a solid is

1
=0
η

It is a rather formal definition because material with viscosity of the order 1020
Pa⋅s (inorganic glass) is a solid, though formally it is a liquid having very high
viscosity. Nevertheless, this formal definition is important for rigorous classifi-
cation of viscoelastic materials.
Then, for a viscoelastic solid, when stress is preset:

1 1
Io = = [7.56]
G o G ∞ + ϕ (0)

and

I ∞ = I o + ψ( ∞ ) [7.57]

For a viscoelastic solid when deformation is preset:

E o = I-1o [7.58]

1
E∞ = = I-1∞ [7.59]
I o + ψ( ∞ )

and

ψ( ∞ ) I − Io
E o − E ∞ = ϕ (0) = = ∞ [7.60]
I o [I o + ψ( ∞ )] I∞ Io
274 Viscoelasticity

7.6.2 VISCOELASTIC LIQUID

The definition of a liquid is

η < ∞; E∞ = 0

Then, one can derive the following relationships between the constants:
∞
E o = I-1o = ϕ (0) = ∫ F(θ )dθ [7.61]
0

All other relationships vanish because E ∞ = 0.

Two of the relationships listed above are of special interest: equation express-
ing instantaneous modulus by “zero” moment and viscosity by “ first” moment of
a relaxation spectrum:
∞ ∞
E o = ∫ F(θ )dθ ; η = ∫ θF(θ )dθ [7.62]
0 0

Then, it is possible to determine a certain “average” relaxation time, θ av , as a ra-

tio η/ E o
∞

η
∫ θF(θ )dθ
θ av = = 0
∞
Eo
∫
0
F(θ )dθ

Certainly it is possible to define other characteristics of “averaged” relaxation

times, such as ratio of two consequent moments of a relaxation spectrum.
7.7 MECHANICAL MODELS OF VISCOELASTIC BEHAVIOR
Theory of viscoelasticity is treated above as a phenomenological generalization
of ideas concerning delayed effects in deformations or superposition of elastic
storage and viscous dissipative losses of energy during deformation. The con-
cepts of relaxation or slow return to the equilibrium state and creep can also be
 A. Ya. Malkin 275

introduced from more general physical reasoning and illustrated by very visual
examples.
Let the state of a system be characterized by a certain parameter x and its
value corresponding to the equilibrium state by x ∞ . Then, let us consider the
possible reaction of a system taken out of its equilibrium state. A system always
tends to return to the equilibrium state. Let us assume, according to the
Maxwell fundamental idea, that the rate of an approach, from any arbitrary
state, to the equilibrium state is proportional to the degree of divergence from
the equilibrium.
In the language of mathematics, this concept can be written in the form of a ki-
netic equation

dx 1
= − (x − x ∞ )
dt θ

where 1/θ is a kinetic factor, characterizing the rate of changes in the state of a
system.
The integral of this differential equation is

x = x o e -t/ θ + x ∞ [7.63]

where xo is an initial value of x.

It is quite evident that if stress is substituted by parameter x, the equation de-
scribes mechanical relaxation, with θ being a characteristic time of relaxation of
this process and x ∞ the residual stress, which equals zero for liquid.
Now, let us imagine a mechanical model constructed from a combination of a
spring and a plunger in a cylinder filled with a liquid (damper), with these two
elements connected in series (Figure 7.9). The spring is a model of an ideal
Hookean solid and its deformation is described by the equation
F
XH =
E

where XH is a displacement of a lower end of a spring from the equilibrium

(non-loaded) state, E is the modulus of a spring, and F is the force applied at the
bottom of the model, causing its deformation.
276 Viscoelasticity

A plunger moving in a cylinder filled with vis-

cous liquid (damper) is a model of an ideal New-
tonian liquid and its deformation is described
by the equation

dX N F
X& N = =
dt η

where dXN/dt is the rate of displacement of a

plunger, η viscous resistance to a movement of
the plunger in the cylinder, proportional to vis-
cosity of liquid, F the same force which caused
the deformation of the spring.
It is evident that the full displacement of a
Figure 7.9. Mechanical model of
Maxwellian relaxation: spring lower point of the model in Figure 7.9 is a sum of
and a plunger in a cylinder with the two components: the displacement of the
viscous liquid (damper), con-
nected in series.
bottom of a spring and the movement of a
plunger. It can be written as follows:

X = XH + XN

and

X& = X& H + X& N

After substituting expressions for displacement of the elements of the model,

the equation for deformation of the whole model is obtained:

F& F &
+ =X [7.64]
E η

Let us analyze the behavior of the mechanical model in Figure 7.9 (called
Maxwell model)when the step displacement is set up, or in other words, let us
find a relaxation function of this model. Integrating Eq 7.64 at the appropriate
boundary conditions gives
 A. Ya. Malkin 277

F(t) = X o Ee -tE/ η

or

F(t)
= ϕ (t) = Ee -t/ θ [7.65]
Xo

where Xo is an initial displacement of the bottom of the model, and θ = η E a

constant which can be called a relaxation time.
The complete equivalence of Eq 7.64 to expressions for relaxation of a liquid
having a single relaxation time (Eq 7.34) and the Maxwell concept of relaxation
as delayed approach to the equilibrium state (Eq 7.63) are quite evident. That is
why the model shown in Figure 7.9 is called the model of the Maxwell liquid. The
rheological equation of state, Eq 7.64, is called the equation of linear Maxwell
viscoelastic liquid.
The rheological properties of a linear Maxwell viscoelastic liquid are charac-
terized by the following parameters: viscosity, η, instantaneous modulus, E, and
the components of dynamic modulus which are expressed as

(ωθ ) 2 ωθ
G′ (ω) = E ; G′′ = E
1+ (ωθ ) 2
1+ (ωθ ) 2

and

tanδ = (ωθ )-1

Certainly the mechanical model drawn in Figure 7.9 is a model of liquid be-
cause equilibrium modulus is zero.
A creep function of a Maxwell liquid is described by the equation

ε (t) 1 t
ψ(t) = = +
σo E η

This equation shows that, when a constant stress is preset, a step-like defor-
mation equals σ o / E, and there is no delay in deformation.
278 Viscoelasticity

The mechanical model, repre-

sented in Figure 7.9, reflects the
viscoelastic behavior of a sin-
gle-relaxation-time liquid. In
line with phenomenological
g e n e ra l i z a t i o n o f a si n-
gle-time-relaxation in Eq 7.34
to the sum of exponents in
Eq 7.35, we can obtain the same
result using mechanical mod-
els. This aim is reached by the
same method of summarizing
exponential items, as in Eq
Figure 7.10. Mechanical model of delayed (Kel- 7.35. Indeed, let us join in paral-
vin-Voigt) creep: spring and a plunger in a cylinder lel a number of Maxwell models
with viscous liquid (damper) connected in parallel. with different values of parame-
ters. Then, the relaxation process, in this multi-relaxation-time Maxwell model,
is described by Eq 7.35 with a set of relaxation times, having different value for
every branch of a generalized model.
The rheological behavior of the generalized model is described by an equation
of the differential type containing time derivatives of stress and deformation (Eq
7.68).
Another way of constructing mechanical models consists in joining a spring
and a plunger moving in a cylinder with viscous liquid (damper), not in series, as
in Figure 7.9, but in parallel, as shown in Figure 7.10. This model is called the
Kelvin-Voigt model. The mechanical properties of both elements in this model
are the same as in the Maxwell model, but, contrary to the latter (deformations
of the components are added), stresses acting in the branches of a model are
added. The following equation describes the rheological properties of the Kel-
vin-Voigt model:

Eε + ηε& = σ [7.66]

In the standard experiment, when σ = σ o = const, the following creep function

of the Kelvin-Voigt model takes place:
 A. Ya. Malkin 279

ε (t) 1
ψ(t) = = (1 − e -t/ λ ) [7.67]
σo E

where σ o / E is the initial deformation, and consequently E-1 = Io is instantaneous

compliance, and λ = η/ E is a retardation time.
It can be easily shown that the Kelvin-Voigt model represents the viscoelastic
behavior of a solid, because the model conserves residual stresses equal to Eε o ,
where ε o is an instantaneous deformation. This model represents the properties
of a non-relaxing body (relaxation time is equal to infinity).
Generalizing the serially joined elementary models, one obtains the model of
viscoelastic solid behavior with a set of retardation times. Creep of the
multi-constant Kelvin-Voigt model is described by Eq 7.40.
In a general case deformation properties of any mechanical model constructed
from the Maxwell and Kelvin-Voigt elements can be represented by the equa-
tion:
N
d nε M d mσ
∑A
n
n
dt n
= ∑ Bm m
m dt
[7.68]

where the order (M and N) of sums in Eq 7.68 depends on the structure of a me-
chanical model, and An and Bm are material constants of a model.
Thus, any mechanical model leads to the equation describing mechanical be-
havior of a material by a differential equation and this is equation of state of
differential type, contrary to equation of state of integral type such as Eqs
7.31 and 7.32. It means that differential equations correspond to line (discrete)
relaxation spectra, while integral equations correspond to continuous spectra.
This difference is not very important because any discrete spectrum can be ap-
proximated by a smooth curve and any continuous spectrum can be approxi-
mated by a set of lines; nevertheless, this difference exists and can be used in
applications for real calculations.
Whether the model represented by Eq 7.68 corresponds to a viscoelastic liquid
or to a viscoelastic solid is determined by the junior member of the left-hand
sum. If this member is of the zero-th order, i.e., calculation of the sum starts
from n = 0, it is a model of a solid. If the zero-th member is absent and the junior
member is of the first order (n = 1), it is a model of liquid. For the Maxwell model,
280 Viscoelasticity

N = 1 (there is only one member in the left-side sum) and M = 1. For the Kel-
vin-Voigt model also, N = 1 (but we have two members in the left-side sum be-
cause the junior member corresponds to n = 0) and M = 0 (only one member is
present in the right-side sum).
Varying the order of differential operators in Eq 7.68 and values of parameters
An and Bm, we are able to describe various special cases of viscoelastic behavior of
a material with any arbitrary relaxation properties. There is one fundamental
limitation: Eq 7.68 describes mechanical properties of a linear viscoelastic ma-
terial and cannot go beyond this limit if some modification of the initial idea is
not introduced.
After the Maxwell and Kelvin-Voigt models were proposed, many attempts
were performed to apply models to explain the behavior of real physical sub-
stances. In this approach, it was thought that the above-discussed models repre-
sent not only behavior but also molecular structure of a body. For example, in
considering relaxation properties of a very long (polymeric) molecular chain, it
was proposed that any part of such chain can be represented by the Maxwell
model.
Such an approach opens wide opportunities for constructing models of real ma-
terials expressed by combinations of elementary Maxwell and Kelvin-Voigt
models. Though some of these structures are very attractive, it is, however, nec-
essary to distinguish between models of a body structure and models of material
behavior, which are not the same (both are related to each other, not quite di-
rectly and not in an obvious way). Any mechanical model is a model of behavior,
and the stress-deformation relationship, describing movement of elements of a
mechanical model, gives equations which can closely approximate deformation
behavior of material. The model can be very deceiving because a real body does
not consist of springs and dampers.
The equations of differential type, like any other rheological equations, are
written for a “point”. If deformations are small, time derivatives entering Eq
7.68 are only partial time derivatives and nothing more. If deformations become
large, it appears necessary to substitute ε in Eq 7.68 by some measure of large
deformation to use their time derivatives as discussed in Chapter 3. This offers a
model of a viscoelastic medium capable of large deformations and leads to some
new rheological phenomena which can be treated as a weak non-linear effect.
The main goal of any mechanical model is to present a visual illustration of
the concept of creep, relaxation, and viscoelastic behavior occurring simulta-
 A. Ya. Malkin 281

neously, and to demonstrate how rheological equations of differential type can

be obtained. The same goal (especially its second part — construction of rheolog-
ical models of differential type) can be reached by other methods, for example,
using models based on analogy to electrical properties. In this case, it is possible
to create electrical networks of any complexity consisting of resistors, condens-
ers and capacitors. The time dependencies of electrical current and voltage
would also be described by the same differential equations, such as Eq 7.68, and
they can be treated as electrical models of viscoelastic behavior. Such models
can also be very convenient in representing transient behavior of a viscoelastic
medium.
7.8 SUPERPOSITION
The idea of superposition (the discussion in this section must not be confused
with the Boltzmann principle of linear superposition expressed by Eqs 7.31 and
7.32) based on the influence of different factors on viscoelastic properties is
widely used in practice of investigating behavior of real materials. The basic
idea of this approach can be formulated in the following way: the same value of
any viscoelastic function can be obtained either by changing time (frequency) or
physical state of material, the latter governed by change in temperature, con-
centration of components, or other parameters. Sometimes, it can be a very un-
expected factor; for example, it can be a duration of exposure of a material to
ultraviolet radiation of the Sun (important for predicting long-term behavior of
organic glasses in illuminators of air-liners).
The idea of superposition is illustrated in Fig-
ure 7.11. Let us have two experimental points for
a creep function measured on the same time
base, t1, but at two different temperatures, T1 and
T2. The value of a creep function, ψ 2 , can be ob-
tained in two ways; first, as shown in Figure 7.11,
by direct measurement at temperature T2 during
time t1, and second, at temperature T1, but with
time base t2. If we know a coefficient aT, which
Figure 7.11. Superposition of characterizes temperature dependence of a creep
points of the creep function by function, we can shift the point (t1, ψ 2 ) to the po-
its shift along the time scale -
an illustration.
sition (t2, ψ 2 ) as shown by the arrow in Figure
7.11.
282 Viscoelasticity

The temperature coefficient aT is

t2
a T = log [7.69]
t1

or in a general form

a T = t * a T (T) [7.70]

where t* is a constant and aT(T) is a function characterizing temperature de-

pendence of viscoelastic properties. A very important supposition is that this
function is the same for any viscoelastic function; it can be proven by the exis-
tence of mutual interrelations between different functions characterizing the
viscoelastic behavior of materials.
Now, after shifting the point on the creep function (t1, ψ 2 ) into a new position
(t2, ψ 2 ), we have obtained two points on the isothermal (at T1) creep curve and we
can try to reconstruct the creep function curve as shown in Figure 7.11. In fact,
superposing results of measurements of viscoelastic functions on the limited
time (frequency) base or “window” (in Figure 7.11) (it is only one point at t1), one
has a possibility to extend the range of experimental determination of this func-
tion. It is a very important method because in real experimental practice we
rarely have a chance to make measurements beyond the range of 1-104 s or
10-2-103 Hz, except by using the method of superposition. We can extend this
range practically without limit and cover the range of 12-15 decimal orders in
time or frequency.
In many real situations, we do not know the temperature coefficient, aT(T), be-
forehand. In order to obtain its value, it is not sufficient to measure only one
point for every temperature. It is necessary to obtain a relationship with some
points having the same values of a viscoelastic function. This approach is illus-
trated in Figure 7.12. We have two sections of the G′ (ω) dependencies measured
at two temperatures. In this case, the temperature coefficient is found as a dis-
tance between two curves at a height where the values of the modulus at two
temperatures appear to be the same, E′ o .
The reduced time-scale is calculated as t/aT and the reduced frequency scale,
as ωaT. It is possible to find hundreds examples of application of the time-temper-
ature superposition in publications devoted to measuring properties of poly-
meric materials.6,7,24 In order to illustrate the strength of this method we shall
 A. Ya. Malkin 283

Figure 7.12. Superposition of two portions of the frequency dependencies of dynamic modulus
measured at two temperatures.

Figure 7.13. Example illustrating superposition of many portions of frequency dependencies of dy-
namic modulus measured at different temperatures (increasing along the direction of the arrow).

discuss one conditional (but close to reality) example of superposition of dy-

namic modulus measured at rather limited frequency range (Figure 7.13). The
experimental “window” was no wider than 3 decimal orders and therefore only
very limited portions of the full G′ (ω) dependence are known. Superposition
opens this window and we can now know the values of this dependence in a very
wide range of frequencies.
An empirical approach to time-temperature superposition is possible if all
neighboring portions of the curve have common points. We do not need to know
the aT(T) function. However, such treatment of data is not always possible, and
284 Viscoelasticity

in many cases, not even convenient. That is why it is important to know the aT(T)
function beforehand.
It was proven experimentally that two possible expressions have general
meaning for the aT(T) function. The first of them is the Arrhenius-Eyring expo-
nential equation, proposed as an analogue to the kinetic equation for the rates
of chemical reactions

a T = Ae E/ RT [7.71]

where A is a front-factor (coefficient), and E energy of activation of relaxation

processes.
It we choose some temperature To, as a reference point (i.e., if we reduce all ex-
perimental data to this selected temperature), then the Arrhenius-Eyring equa-
tion can be written as

E  1 1 
a T = exp   −   [7.72]
 R  T To 

The second expression for the aT function, widely used for time-temperature su-
perposition of experimental data in various polymeric materials (it must be em-
phasized that polymeric materials and polymer-based compositions are the
main object for application of superpositions of different type), is the so-called
Williams-Landel-Ferry (WLF) equation, which can be written as25-27
-c 1 (T − To )
loga T = [7.73]
c 2 + (T − To )

where To is the reference temperature, and c1 and c2 are constants depending on

the choice of the reference temperature.
If the glass temperature is assumed as a reference temperature the values of
these constants appear rather stable: c1 is close to 17.4 and c2 to 51.6. But in fact
it is a rough approximation and it is preferable to use individual values of the
constants which are different for various materials.
As a general rule, it is thought that the Williams-Landel-Ferry equation is true
in the temperature range from the glass transition temperature, Tg, up to
 A. Ya. Malkin 285

Tg + 100oC. The Arrhenius-Eyring equation can successfully be used in the

range of temperatures approximately at T > Tg + 100oC.
If we know the form of temperature dependence of aT, it allows us not to have to
measure the function aT(T) in the whole range of temperatures but to restrict
measurement to some points only in order to verify the constants entering these
equations.
Method of superposition is not limited to reduction of data by means of chang-
ing temperature only. The state of a material can be changed for different rea-
sons. Variations in content (concentration) of a polymer in a solution is often
used to change its relaxation properties. Then we can realize time-concentration
superposition, and so on.
The method of superposition, used to reduce experimental data along the time
(frequency) scale, is a very powerful possibility to increase the range of observa-
tions. At the same time, it must be remembered that the main principle of the
method is based on the assumption that in changing the state of a material its
relaxation spectrum changes in the same manner for all relaxation times, i.e.,
no new relaxation process appears, no process disappears, and temperature de-
pendencies of all relaxation times in a spectrum are the same. The latter as-
sumption is confirmed by the fact that the temperature dependence of any
relaxation time is the same as viscosity; and we know that viscosity is the inte-
gral representation of all relaxation times (see Eq 7.62). At the same time, it is a
rather strong assumption which may not be fulfilled, especially if superposition
is carried out for initial experimental data obtained in a wide temperature
range. In fact, many cases are known in which this basic assumption was wrong.
One of the most evident examples is a phase transition. If it takes place in the
temperature range under discussion, it definitely leads to radical changes in re-
laxation properties. The danger is rather serious for crystallizable polymers be-
cause the process of their crystallization takes place in a wide temperature
range.
The second example is that of block-copolymers:28 the moveability of blocks of
different types (i.e., possibility to relax) appears in different temperature ranges
and reflects freezing of different relaxation modes. It means that different parts
of relaxation spectrum are characterized by different temperature dependen-
cies and direct superposition of all portions obtained at different temperatures
is incorrect in principle.
286 Viscoelasticity

The third example regards mixtures of different components. In this case, it is

quite obvious that temperature dependencies of relaxation properties of various
components in a mixture, if different, restrict the possibility of superposition of
portions of viscoelastic functions measured at different temperatures.
Examples discussed above show that time-temperature superposition is not a
universal method and has definite limitations. That is why one must be very ac-
curate in treating experimental data of a newly investigated material by this
method, especially if far extrapolation, beyond an experimental window, is at-
tempted.
7.9 APPLICATIONS OF LINEAR VISCOELASTICITY
There are three main lines of applications of the theory of linear viscoelasticity.
They are as follows:
• comparison of different materials through constants and functions intro-
duced and determined based on the theory
• calculation of the response of a material and predicting its behavior in arbi-
trary deformation using results of standard experiments and their theoreti-
cal relationships
• comparison of predictions of structure and molecular theories with experi-
mental data.
7.9.1 COMPARISON OF MATERIALS BY THEIR VISCOELASTIC PROPERTIES

We can think about two ways of comparison

of different materials. First, it is possible to es-
timate the type of a material and to determine
quantitatively such qualitative definitions as,
for example, “solid”, “rigid”, “stiff”, “mild”,
“fluid” and so on. Second, it is possible to com-
pare materials of the same type by values of
their constants; e.g., it is possible to distin-
guish two rigid materials with different resis-
Figure 7.14. Frequency depend- tance to creep. These possibilities are
encies of dynamic modulus of a illustrated below.
polymer melt (A) and a lightly
cured rubber (B), prepared from As a first example, let us compare typical lin-
this polymer. ear viscoelastic characteristics (frequency de-
pendencies of dynamic modulus) of a polymer
 A. Ya. Malkin 287

melt and a slightly cured rubber prepared from the same material (Figure 7.14).
We can see that it is reasonable to select five regions of a frequency dependence
of dynamic modulus for a polymer melt. They are called:
I - flow (or terminal) zone
II - transient viscoelastic region
III - rubbery plateau
IV - transient leathery zone
V - glassy zone.
Certainly, all five parts of the E′ (ω) curve can be observed for a single sample by
varying temperature, which results in changing physical (relaxation) states of a
material. Then, one can apply the method of frequency-temperature superposi-
tion and construct the generalized E′ (ω) dependence by joining segments of the
full curve and shift along the frequency axis, such as curve A in Figure 7.14, con-
structed for a polymer which can melt at high temperatures (or flow at very low
frequencies).
The first two regions of the complete E′ (ω) dependence for a cured rubber are
absent (curve B in Figure 7.14), though all other three parts stay practically un-
changed in comparison with the melt. It means that, based on the measurement
of the E′ (ω) dependence for an unknown sample, one may find out that it is a lin-
ear polymer which can flow and be processed by regular methods or distinguish
it from rubber able to sustain higher temperatures and deformations without ir-
reversible changes of the shape.
The second example concerns the influ-
ence of molecular weight (length of a mo-
lecular chain). Figure 7.15 shows that, for
high molecular weight samples, all five re-
gions are observed in the E′ (ω) curve. De-
creasing molecular weight leads to a shift
of low-frequency regions (flow and tran-
sient viscoelastic zones) of the complete
curve but does not affect the height of the
Figure 7.15. Frequency dependencies
of dynamic modulus for samples of the
rubbery plateau and shape and position of
same polymer but of different molecu- two other regions of the full E′ (ω) depend-
lar weights. The arrow shows the di- ence. However, with a rather low molecu-
rection of increasing molecular
weight. lar weight member of the same homologous
series, we lose a transient viscoelastic zone
288 Viscoelasticity

(marked as II in Figure 7.14) and rubbery plateau (marked as III in Figure 7.14),
though the flow zone and high-frequency (glassy) part of a curve are present.
It shows that we can distinguish between polymers having different molecular
weights by measuring their viscoelastic properties and comparing the position
of the flow zone and/or the length of a rubbery plateau. It can be used as a rela-
tive method of comparison of two polymers by measuring the length of a plateau
or frequencies for any arbitrary level of modulus in a flow zone. If molecular
weight dependence of these factors was known beforehand (for calibrated series
of samples), this approach could have been used as an absolute method of mea-
suring molecular weights of polymers.
The third example (Figure 7.16) is a compari-
son of a cured rubber and a gel. In both cases,
the first two regions (I and II in Figure 7.14) of
a complete E′ (ω) dependence are absent, such
as for sample B in Figure 7.14. Addition of a
low molecular weight solvent shifts all other
parts of the curve, and in particular, lowers
the height of a rubbery plateau during transi-
tion from a cured rubber to a gel.
Figure 7.16. Frequency depend-
Figure 7.16 also represents the influence of
encies of a cured rubber (R) and a increasing density of network crosslinks in a
gel (G) prepared from the same rubber. This factor is primarily reflected in the
rubber. The influence of higher
density of a network of crosslinks height of a rubbery plateau and also in the val-
(curve R’) is also shown. ues of modulus in a glassy state.
Examples shown in Figure 7.16 demonstrate
the idea of constructing rubbery materials of different rigidity. Suppose there is
a need for a soft rubbery material for eyesight-correcting covers. In this case, we
must prepare gels having the required value of rubbery modulus (on the pla-
teau). In another case, we may need a hard rubbery material for heavy tires, and
again, properties of material are characterized by the rubbery modulus. Regu-
lating the content of a plasticizer (low molecular weight solvent) and a harden-
ing solid additive, we can reach the necessary combinations of viscoelastic
properties in different frequency ranges needed for the application. For exam-
ple, sealants require definite elasticity in the low frequency range and aircraft
tires must be elastic at high frequencies because these products work under
quite different conditions, even though both must be elastic in application.
 A. Ya. Malkin 289

The last series of examples is, in fact, comparison of materials by their

viscoelastic properties in relations to different areas of their application. In all
these cases (as in many others), the main problem consists in the necessity to
formulate special conditions of application in terms of the theory of
viscoelasticity and to establish the required level of properties. In real life it can
be constants, such as viscosity or modulus, which are integrals (moments) of a
relaxation spectrum of a material, or they can be definite values of viscoelastic
functions at one or some frequencies or moments of time.
For example, it can be very important to know values of the creep function or
the relaxation function on the definite time basis, as already discussed in Chap-
ter 4. In this and analogous situations, one does not need to know the theory of
viscoelasticity or to make any calculations, but the application conditions must
be reproducible and a simulating experiment should be performed with all nec-
essary standard requirements regarding accuracy and statistics of experimen-
tal results. Moreover, in these cases we are not restricted by the limits of
linearity because application conditions do not obey any artificial limitations.
Measurements of viscoelastic properties of a material gives a quantitative
base for reference to different relaxational states: glassy, leathery, rubbery, and
so on. At the same time, it must be remembered that, in fact one does not con-
sider the state of matter but its behavior. If some definite frequency was chosen
for measurement of dynamic modulus (for example, standard frequency of 1 Hz),
the comparison of various materials under these test conditions is straightfor-
ward. On the other hand, the same material can look quite different in other fre-
quency ranges. A typical rubber can behave like a glass at very high frequency,
as proven by frequency-temperature superposi-
tion. For example, in tire application for air-
crafts, the high frequency behavior of rubber
must be considered, since a tire must damp vi-
brations and can suddenly break if it performs
like a glassy rather than a rubbery material.
In addition to primary classification of materi-
als in accordance with their main relaxational
Figure 7.17. Typical tempera- states, some more precise conclusions can be
ture dependence of mechanical drawn from measuring the temperature de-
losses in periodic deformations
characterized by loss tangent.
pendence of the loss tangent, tanδ. A typical ex-
ample of such dependence is shown in Figure
290 Viscoelasticity

7.17. There are maxima in tanδ and each of them is treated as a relaxational
transition. The most intensive high-temperature maximum is called the main or
α-transition and identified with glass transition temperature. Others, to the left
of the main transition (in the direction of lower temperatures), are marked by
letters of the Greek alphabet, starting from β, and are called secondary transi-
tions.
It is accepted that each relaxational transition identified with maximum of tan
δ corresponds to the condition

ωθ = 1

where ωis the frequency at which temperature dependence of mechanical losses

is measured, and θ is a relaxation time.
During the experiment, the frequency is constant, and by changing tempera-
ture, one varies relaxation time. At some temperature, the above-formulated
condition is reached and it is a criterion of “transition”. It can be argued that this
condition corresponds to a maximum of tanδ; in particular, it can be illustrated
by a simple mechanical model of viscoelastic behavior including a single relax-
ation time.
One of the popular approaches to treating experimental data of the type shown
in Figure 7.17 relates the tanδ maximum with “freezing” or “defreezing” of mo-
lecular movements of different kinds. These data can be used to recognize molec-
ular process according to their position of “transition” along the temperature
scale. The other side of this approach is the idea that passing through the tem-
perature where tanδ maximum is achieved, we change the relaxation state and
spectrum of a material, and therefore the principle of temperature superposi-
tion can be applied in the temperature range between two transitions only.
In measuring temperature dependence of tanδ, we also do not need to know the
theory of viscoelasticity. It should be treated as a standard “one-point” (i.e., car-
ried out at one frequency) method of characterizing a material. This method is a
bridge for using measurements of viscoelastic properties of a material to under-
stand its molecular structure.
 A. Ya. Malkin 291

7.9.2 CALCULATION OF THE RESPONSE

This line of application of the viscoelasticity theory is restricted by the linear-

ity requirement, because, at present no non-linear theory of viscoelasticity (see
the next section of this chapter) is treated as a solid base for a long-term predic-
tion of behavior of real materials or fits the empirical experimental data.
Within the limits of linear viscoelastic behavior, the problem of calculation of
responses to arbitrary loads is completely solved by the Boltzmann superposi-
tion principle. The relationships between viscoelastic functions were discussed
above. If we know a relaxation spectrum of a material, all other calculations are
only a technical problem. The major problem is the accuracy of determination of
a relaxation spectrum based on experimental data.
We shall illustrate real limits of calculation by an example concerning relax-
ation and creep of polycarbonate,29 a typical engineering plastic widely used in
industry. The relaxation curve is approximated by the so-called Kolhrausch
function30
α
σ(t) = Ae -γt

where A, γ and α are empirical constants.

The values of the constants in this for-
mula were found by a computational pro-
cedure and they satisfy the condition:
points calculated with the set of selected
constants according to the Kolhrausch
function lie within the limits of possible
experimental error. These limits were
rather narrow, only about 5%. Then, the
creep function was calculated for three
possible sets of constants (satisfying the
above-formulated requirement). The re-
Figure 7.18. The results of calculations of
the creep function. The relaxation curve sults are represented in Figure 7.18,
was approximated by the Kolhrausch where the exact creep function is also
function with three different sets of con- drawn (dotted line). A calculated curve
stants (curves 1, 2 and 3). Dotted line is
an exact creep function. can be rather far from the true creep
function and the error large. It was dem-
292 Viscoelasticity

onstrated,29 that a real relaxation curve can be

approximated in different ways and the
method of approximation strongly influences
the results of calculations. Particularly, the re-
laxation curve can be approximated by a sum
of exponent functions (by discrete relaxation
spectrum) and only 5 members are sufficient to
cover a wide time range of relaxation.
Another example cited from the results of the
above-mentioned investigations of creep and
relaxation of polycarbonate demonstrates the
effect of non-linearity. Figure 7.19 shows two
Figure 7.19. Creep of sets of experimental points (open symbols)
polycarbonate at two levels of measured in creep at different levels of stress
stress: initial (instantaneous) de-
formation was 1% (1) and 2.5%
(different initial deformations) and curves cal-
(2). Open marks - experimental culated from a relaxation curve, the latter
points, curves - calculated from a measured in the range of linear viscoelastic be-
relaxation curve in a linear
o
viscoelastic range. 20 C. havior of a material at very low instantaneous
deformation. At initial deformation (1%), the
behavior of a material in creep is linear and
quite well predicted by the theory of linear viscoelasticity from a relaxation
curve. But the creep function becomes strongly non-linear even at initial defor-
mation (2.5%), and the experimental points radically deviate from the calcu-
lated curve.
It is also worth noting that non-linear effects do not appear from the very be-
ginning of creep; instead, here is some initial period of creep where points lie
rather close to the calculated curve. It means that linearity in viscoelastic be-
havior is limited, not by stress (or deformation) level only but by the time factor,
i.e., a material can be linear in a short range of loading and become non-linear in
long-range loading. Certainly the duration of this range depends on stress level,
as illustrated by experimental data in Figure 7.20. The limit of linear
viscoelastic behavior (the limit can be determined with some experimental er-
ror, too) is a very strong function of stress. Experimental points in Figure 7.20
can be approximated by an expression

t * = me -aσ
 A. Ya. Malkin 293

Figure 7.20. Dependence of the time, t*, corresponding to the limit of linear viscoelastic behavior
in creep on the level of stresses at different temperatures. Straight line approximating experimen-
tal points is an exponential function.

where m and a are empirical constants (the constant a can be called a stress-sen-
sitivity factor, and it equals 0.1 MPa-1 for experimental data in Figure 7.20; this
value depends on material under investigation).
The example shows that the theory of linear viscoelasticity can be applied in
practical calculations but the accuracy of predictions depends on:
• accuracy of initial experimental data
• method used for approximation of experimental points
• confidence of being inside the limits of linearity
• deformation and time ranges for which predictions are made.
All these factors are hard to estimate and this seriously limits the possibilities
of calculation based on the linear theory of viscoelasticity. It therefore should be
applied only when one is sure that the potential problems, listed above, are not
applicable; then the theory of linear viscoelasticity becomes a powerful method
of prediction based on limited experimental data.
294 Viscoelasticity

7.9.3 MOLECULAR THEORIES AND EXPERIMENTAL DATA

Any molecular theory claims to predict the behavior of real materials. One of
the broad areas for predictions is viscoelastic behavior, because relaxation ef-
fects are inherent to many materials and primarily to polymeric and colloid sys-
tems. That is why results of observations of viscoelastic behavior of different
materials are widely used for comparison with theoretical predictions based on
molecular models of materials.
As a general rule, the theory predicts a relaxation spectrum of a material, and
the main question is, which type of experiment is used for comparison with theo-
retical predictions? Frequency dependence of dynamic modulus is the most
widely-used method. It is explained by the fact that this dependence can be mea-
sured relatively simply in a very wide range of its argument.
A very important peculiarity of this line of applications of linear
viscoelasticity, which is not important in other cases, is the necessity to have re-
liable results of measurements of viscoelastic functions and their application to
very well characterized objects. The last requirement is especially important
but in many cases, it is quite difficult to have sufficient information about a sam-
ple needed for formulation and use of a molecular theory, rather than to simply
measure viscoelastic properties of the material. This is a main additional limita-
tion in comparison of experimental data with molecular theories. Nevertheless,
the reference to viscoelastic properties of a material is widely-used in modern
theoretical speculations.
7.10 NON-LINEAR VISCOELASTICITY. INTRODUCTION
Theory of linear viscoelasticity is a rather rare example of a specialized and
closed theoretical approach to describing behavior of a material. It has a rigid
mathematical structure, set of theorems, conclusions, and it can be used for
practical calculations of stress-deformation relationship in arbitrary regimes of
deformations (loading). The theory has two main limitations which were men-
tioned above. One of them is the necessity to use initial experimental data for
calculations, which can be determined only in a limited range of an argument
and with some experimental error. It can lead to errors in calculations which can
be much larger than an error of initial measurement, and as a rule, this error
cannot be estimated at all. It is a natural inherent limitation of the theory.
The second limitation is of a more fundamental nature which severely restricts
 A. Ya. Malkin 295

capabilities of the theory, especially in applications for technical purposes. That

is a limitation of linearity. Some possible definitions of the limits of linear
viscoelastic behavior were introduced in discussing fundamental experiments
of viscoelasticity (relaxation, creep, periodic oscillations). This is independence
of viscoelastic functions and constants, appearing in the theory, on the level of
stresses and/or deformations. All these requirements are, in fact, reflections of
the concept of independence of a relaxation spectrum of a material to external
influences.
Many non-linear effects are observed in deformations of viscoelastic materials.
The main effects were listed and discussed in Chapter 4, and some main types of
non-linearity, with their inherent causes, formulated. The points below are de-
signed to illustrate fundamental reasons for non-linear phenomena:
• weak (geometrical) non-linear effects due to possibility of large deformations
of medium
• strong (structural) non-linear effects caused by changes in relaxation prop-
erties (spectrum) of a medium under external influence
• breaking (phase) non-linear effects, which are results of phase or
relaxational transitions caused by deformations. This physical cause leads
to significant change in relaxation properties of a media, not related to the
properties corresponding to the initial state of a material.
All types of non-linear effects can be found in deformations of viscoelastic ma-
terials. Consequently, in constructing rheological equations of state for
non-linear viscoelastic materials, we must take into account these levels of
non-linearity. Certainly, effects of the lower level are always present when
non-linear effects of higher level are observed. One can expect that strong
non-linear phenomena related to gradual changes in a relaxation spectrum will
be accompanied by large deformations, and so on. Moreover, the transition from
one level of non-linearity to the next, as a rule, is not abrupt, especially if we bear
in mind that the majority of real materials are multi-component and conditions
of transition are different for components of a material. Certainly the threshold
for transition is different for various materials.
For example: which deformations can be called large? The general answer is:
these which are comparable with 1 (i.e., 100%). But we saw that even at defor-
mation of an order of 2.5% (i.e., 0.0025) the behavior of polycarbonate in creep
appears to be non-linear (Figure 7.19). Moreover, non-linear effects in deforma-
tion of highly filled polymeric31,32 or colloidal plastic disperse systems33 (colloidal
296 Viscoelasticity

dispersion of naphthenate aluminum in low-molecular-weight solvent)34 become

quite obvious at deformations of about 0.1%. It is explained by the existence of a
very rigid structure of a solid compound, destroyed at this deformation and lead-
ing to change in relaxation spectrum and strong non-linear effect, such as, for
example, non-Newtonian flow and dependence of dynamic modulus on ampli-
tude of deformations.
It has already been mentioned (Chapter 4) that strong and phase non-linear
phenomena are kinetic effects because changes in a relaxation spectrum and
phase transitions occur in time. These time-effects superimpose on viscoelastic
time-effects, and the rheological approach must be combined with a kinetic one
in order to understand what happens to a material and to explain and describe
what we observe in the case of “non-linear effects”.
The last concluding remark in this Introduction to non-linear viscoelasticity is:
even though there is a complete theory of linear viscoelasticity, no such non-lin-
ear theory exists, and maybe cannot exist, but there can be many theories of
non-linear viscoelastic behavior which seem to explain various non-linear phe-
nomena. At the same time, no theory exists capable of covering all non-linear
cases because of the diversity of causes of non-linear effects. For example, it is
convenient to use a flow curve equation for a non-Newtonian flow for applied cal-
culations for transportation of a liquid in pipes, neglecting that a flow curve is
only a consequence of some more general rheological equation of state not
known.
7.10.1 LARGE DEFORMATIONS IN NON-LINEAR VISCOELASTICITY

The starting point for developing the concept of viscoelastic behavior in

non-linear range is the Boltzmann-Volterra equation (7.31). It is the most gen-
eral representation of viscoelastic response at infinitesimal deformations. The
next step regards the use of large (finite) deformations discussed in Chapter 3.
It is convenient to use two tensors for large deformations: the Cauchy - Green
tensor, Cij, and the Finger tensor, C-1ij , both are functions of two time moments:
an “initial” moment of time when a point of a body is in its reference state and a
current moment of time, i.e., the moment when deformations Cij and C-1ij are mea-
sured relative to the reference state. In many cases, the initial (reference) state
is assumed to be isotropic at the point, although it is not always true. For exam-
ple, liquid crystalline polymers and reinforced plastics are very important ex-
ceptions (or to be more exact, they form another case) because these media are
 A. Ya. Malkin 297

inherently anisotropic materials and rheological description of their properties

and behavior needs special theoretical understanding.
The idea of large deformations and their role in rheological behavior of real ma-
terials can be illustrated by an example of simple shear. For this purpose let us
write the components of the Cauchy - Green and Finger tensors for
unidimensional simple shear in the plane x1 - x2. Assuming that deformations at
the reference state are absent and that shear at the moment, t, equals γ, we can
write the components of the Cauchy - Green and Finger tensors as follows

0 γ 0 -γ 2 γ 0
C ij (t) =γ γ 2 0 and C-1ij = γ 0 0
   
0 0 0 0 0 0

Then, we need to formulate the Boltzmann superposition principle for large

deformations. One of the possible ways to do it is to use a form of rheological
equation of state of rubber-like liquid as proposed by Lodge:35
t
σ ij (t) = ∫ C-1ij m(t − t ′ )dt ′ [7.74]
0

where m(t - t′) is called the memory function.

It is easy to show that the memory function is directly related to the relaxation
function by the following equations:
∞
dϕ (t)
ϕ (t) = ∫ m(s)ds; m(t) = − [7.75]
t
dt

These equations show that the memory function is the same as for the linear
viscoelasticity limit and it does not depend on deformation. Using the measure
of large deformation, instead of infinitesimal deformation, we come to some new
results: presence of diagonal members in the C-1ij tensor leads to the prediction of
the Weissenberg effect. It appears here as a natural consequence of finite defor-
mation, and it is really a second order effect because predicted normal stresses
are proportional to γ 2 . Then, using different combinations of the tensors Cij and
C-1ij , one can reach correct sign and value of normal stresses (within the limits of
298 Viscoelasticity

weak non-linearity). The statement “using different combinations” shows that

there are various possible paths for generalization of the linear theory of
viscoelasticity and we have no formal ground for unambiguous choice of either
version.
The second power of normal stresses in elastic deformation is a weak non-lin-
ear effect resulting from large deformation of a material, which is true for a
viscoelastic solid and liquid as well. The model of rubber-like liquid predicts
some new effects besides normal stresses in shear, important as reflections of
weak non-linearity. The first one is time dependence of tensile stress observed in
uniaxial extension performed at constant rate of deformation. For linear
viscoelastic liquid at constant rate of extension (and at constant rate of shear),
one expects that stress will grow to the limit of steady flow with constant (New-
tonian in shear and Trouton at extension) viscosity, and such level is reached at
t → ∞.
To illustrate what happens to rubber-like liquid, it is convenient to operate
with a single-relaxation-time model (a relaxation spectrum degenerates to a sin-
gle line relaxation time, θ) because this model gives very spectacular results.
Theory shows that in uniaxial extension of such viscoelastic (or rubber-like) liq-
uid, stress dependence on rate of deformation at t → ∞ is36

3ηε&
σ(ε& ) = [7.76]
(1 − 2εθ
& )(1+ εθ
& )

At very low rates of deformation (and the words “very low” mean that the
Weissenberg Number, We = εθ& << 1), we reach the limit of linear viscoelasticity:
the Trouton Law for extensional viscosity. Indeed, at We ≡ εθ
& << 1

σ
λ= = 3η
ε&

Increase in the rate of deformation leads to the growth of elongational viscos-

ity, as can be derived from Eq 7.76, so the theory predicts increasing extensional
viscosity. However, the most intriguing result can be obtained at We → 0.5.
Eq 7.76 predicts, in this case, an unlimited growth of stress, which is physically
impossible. In fact, it means that at We > 0.5, steady elongational flow becomes
impossible, resulting in rupture of a stream to overcome some critical rate of de-
 A. Ya. Malkin 299

formation, determined by the criterion We = 0.5.37,38

If a (linear) relaxation spectrum is not represented by a single but rather a set
of relaxation times, the expression for stress dependence on rate of deformation
becomes more complex, resulting in preclusion to reach a steady flow regime of
extension at sufficiently high rates of deformation. In the general case, the criti-
cal value (equal to 0.5) of the Weissenberg Number must be calculated through
the maximum relaxation time from a spectrum. Moreover, if a relaxation spec-
trum of a material is continuous and stretches up to infinity, then the state of
steady uniaxial flow becomes impossible at any low rate of deformation.
The prediction of unlimited growth and thus the rupture of material in uniax-
ial extension is very important because the model of rubber-like liquid explains
impossibility of increase in extension rate; for example, in the process of fiber
spinning (very much desired in technological practice) beyond the definite
threshold: a liquid jet would break, which helps to realize even weak non-linear-
ity as a result of large elastic deformations in order to observe this effect.
Now, let us discuss the next interesting and important phenomenon predicted
by the model of viscoelastic (rubber-like) liquid. This is constrained recoil of
such a liquid after cessation of shear deformation, γ r , which is a function of time.
Recoverable deformations, γ r (t), accompany flow of viscoelastic liquid and can be
measured after sudden (jump-like) cessation of flow. Recoverable deformations
at t → ∞, so-called ultimate (or “equilibrium”) recoil, γ ∞ , are calculated as

τ ηγ&
γ∞ = = [7.77]
Eo Eo

where η is viscosity (Newtonian viscosity for linear viscoelastic liquid) and E0

modulus of elasticity.
It is possible to prove that γ el can also be calculated by means of the so-called
Lodge equation:

ψ 1 γ& σ 11 − σ 22
γ el = = [7.78]
η 2σ 12
300 Viscoelasticity

where

σ 11 − σ 22
ψ1 =
2γ& 2

is called coefficient of the first difference of normal stresses (see also p. 74).
Then, one can write a new relationship between material constants of a rub-
ber-like liquid

η2
ψ1 = [7.79]
Eo

The normal stresses can be expressed as a function of the moments of a relax-

ation spectrum because both components in Eq 7.79 are such moments (see Eq
7.62). Then normal stresses appear in the model of rubber-like liquid also as a
second-order effect, which can be connected with relaxation properties of me-
dium.
7.10.2 RELAXATION PROPERTIES DEPENDENT ON DEFORMATION

There is a great variety of non-linear models which take into account the sec-
ond level of non-linearity, i.e., changing relaxation properties as a function of de-
formations.39-41 If we go by the way of generalization of Eq 7.74, we can assume
that m(t) is a function not of time only but also of deformation. Because a mem-
ory function is a material parameter of matter, its dependence on deformation
must certainly be expressed through invariants of the deformation tensor.
Then, we can write
t
σ ij = ∫ C-1ij m[(t − t ′ );I1 ;I 2 ]dt ′ [7.80]
0

where I1 and I2 are invariants of the C-1ij tensor.

Eq 7.80 is a natural generalization of a model of a rubber-like liquid. However,
practical applications of Eq 7.80 meet with a major difficulty which is common
for non-linear models of such type. It is a problem of determining a memory func-
tion with its uncertain dependencies on invariants of the deformation tensor. To
 A. Ya. Malkin 301

overcome this difficulty, Wagner42 proposed the use of a rather old observation
obtained from investigation of deformation of crosslinked rubbers. It was known
that in many practically important cases, a relaxation function (or a relaxation
modulus) at large deformations can be treated as a product of two independent
functions: the first one is a time-dependent memory function and the second is a
function of deformation (or in more general case, a function of invariants of the
deformation tensor), i.e., it is possible to separate the dependence of a memory
function on its arguments into two different dependencies based on a limited
number of arguments:

m(t,I1 ,I 2 ) = m o (t)h(I1 ,I 2 ) [7.81]

where m0(t) is a linear-limit memory function, and h(I1, I2) is called a damping
function, and the latter reflects the influence of deformation on relaxation prop-
erties of material.
It is rather difficult to separate the influence of both invariants on damping
function, h(I1,I2), because in simple (standard) experiments the two invariants
cannot be changed independently. There are some experimental result related
to large deformations in shear where a single measure of deformations, γ, can be
used. Wagner42 and Laun43 demonstrated that a damping function can be ap-
proximated by a single exponent

h( γ ) = e -nγ

and Soskey and Winter44 and Larson45 showed that experimental data can be
satisfactory fitted, if we express a damping function by a power law

1
h( γ ) =
1+ aγ b

Parameters a, b, and n in these expressions are empirical constants.

Other versions of a “non-linear” memory function are also known and their
generalization, as a function of invariants, can be found in the literature. For
our discussion, the most interesting is, first of all, the existence of the depend-
ence of relaxation properties on deformations, and secondly, the rather strong
influence of deformations, regardless of whether they are expressed by an expo-
302 Viscoelasticity

nential or a power function.

Eq 7.80 is able to predict strong non-linear phenomena. One can obtain the ex-
pression for non-Newtonian viscosity, which is written as
∞
d
η( γ& ) = ∫ ϕ (t ′ ) [ γ&h( γ& )]
0
dγ&

The final form of the dependence of non-Newtonian viscosity on rate of shear is

determined by the structure of a damping function. Regardless of its exact ex-
pression, one may state that the influence of shear appears more strongly on the
long-term side of a spectrum than at shorter relaxation times. In this sense, we
can say that shearing suppresses slow relaxation processes (they do not have
enough time to occur) and this phenomenon increasingly envelopes the part of a
relaxation spectrum (from the long-term side), the higher the rate of deforma-
tion is.
Eq 7.81 proposed by Wagner42 is popular in rheological literature because, in
the form used in Eq 7.80, it can describe many experimental data on non-linear
relaxation in the range of large deformations. Eq 7.80 also predicts non-qua-
dratic dependence of normal stress on shear rate at high rates of deformation,
i.e., it demonstrates that at high rates of deformation, the Weissenberg effect re-
flects not only weak but also strong non-linearity, as well.
Though Eq 7.80, with separation of arguments, as in Eq 7.81, looks very attrac-
tive for describing various rheological phenomena, there are at least two princi-
ple contradictions of theoretical predictions based on this approach and
experimental facts. First is the wrong prediction concerning the second differ-
ence of normal stresses: according to Eq 7.80, it equals zero, whereas, in fact, it is
not; second is the prediction of monotonous growth of normal stresses in tran-
sient shearing deformations, whereas, in fact, an overshoot takes place. It
proves that this approach is not universally acceptable.
The next step in understanding and describing non-linear viscoelastic behav-
ior is connected with so-called the K-BKZ model of rheological behavior,46,47
which incorporates both measures of large deformations, the Cauchy - Green,
Cij, and Finger, C-1ij , tensors. Again, there is some ambiguousness in constructing
rheological models because both measures of large deformations can enter rheo-
logical model as arbitrary form, but they allow one to select combinations which
give the best fit of experimental data.
 A. Ya. Malkin 303

Different forms of the K-BKZ equation exist. If we explore the idea of separa-
tion of arguments in a memory function, as discussed above, we may come to the
following, relatively simple, expression for the K-BKZ model:
t
∂W ∂W -1
σ ij = ∫ m(t − t ′ )[2 C ij (t ′ ) − 2 C ij (t ′ )]dt ′ [7.82]
0
∂I 1 ∂I 2

where W is an elastic potential depending on the invariants of the deformation

tensor, and this elastic potential is essentially the same as used in the theory of
elastic (rubbery) solids (see Chapter 6).
An elastic energy potential function in Eq 7.82 depends on the deformation and
therefore on time. The dependence W(I1,I2) must be determined experimentally
like a damping function in Eq 7.81. There is a vast number of publications de-
voted to experimental probing of the predictive strength of this model and deter-
mining elastic potential function. Meanwhile, the main advantage of Eq 7.82
over Eq 7.80 is in prediction of the non-zero second difference of normal stresses.
Freedom in combinating measures of deformations and dependency of elastic
potential on invariants of the deformation tensor provides great possibilities to
use the K-BKZ model to fit numerous experimental data and describe various
special effects in behavior of different rheological media.
Both equations (Wagner and K-BKZ) explore an idea of the influence of the de-
formation on a memory function and thus on a relaxation spectrum of a mate-
rial, but using it in a simplified form through separation of inputs of time and
deformation. It allows one to single out the linear limit of relaxation function.
Certainly, it is a particular case, but an important one. Moreover, transition to a
more general (and more complex) model of non-linear viscoelastic behavior, as a
rule, makes these equations almost unrealistic for practical applications.
There are many other, different approaches to constructing non-linear rheo-
logical equations of state for viscoelastic materials. We do not intend to review
all published theoretical ideas; only fundamental and applicable approaches
were discussed above. It is also worth mentioning that there are numerous at-
tempts to take into consideration weak (due to finite deformations) and strong
(due to deformational changes of a relaxation properties) non-linear phenom-
ena, but non-linear effects of the third level (phase non-linearity) do not appear
in rheological equations. However, the idea of fracture appears in many cases,
even as a result of a weak non-linearity as discussed for high rates of extension
304 Viscoelasticity

of a rubber-like liquid.
In the conclusion of this section, we shall shortly discuss the applied purposes
of non-linear theories of viscoelasticity. They are essentially the same as those of
the linear theory, but their applied value is even more pronounced because real
technical applications and technological operations take place at high rates of
deformation and reach large deformations. On the other hand, at present an ap-
plied meaning of non-linear theories is very limited. This is explained by diffi-
culties in determining material functions used in theories and the low reliability
of their predictions for experimental conditions other than those used to deter-
mine material functions.
A large number of investigations were performed to verify the correctness of
prediction of qualitative or (better) quantitative results of simple fundamental
experiments (simple shear, uni- and biaxial extension, transient regimes of
shear and extension) by different theories, but seldomly to solve applied techni-
cal problems. Only empirical or semi-theoretical equations for steady non-linear
phenomena, such as non-Newtonian flow of liquids or long-term creep of solids
and long-term prediction, including extrapolation, are rather widely used for
solving applied problems.
7.11 CONCLUDING REMARKS
Viscoelasticity is a combination of viscous dissipation and storage of defor-
mation energy. The phenomenon is common for practically all materials, though
its importance and potential possibility to observe viscoelastic effects is deter-
mined by the ratio of inherent time-scale (time of relaxation) and characteristic
time of deformation. Various effects explained by viscoelastic behavior are often
observed and are important for polymeric materials in the form of melts, solu-
tions, and colloidal dispersions, solids and reinforced plastics, rubbers and
foams. The main reason for its presence is their wide relaxation spectra, causing
the same order of value as duration of loading (deformation) to always exist.
There are three fundamental experiments which are treated as reflections of
viscoelastic behavior of a matter:
• creep - delayed development of deformations under action of constant force
(or stress)
• relaxation - slow decay of stresses at preserving constant deformation
• periodic oscillations - harmonic changing of stresses or deformations with
relative shift of deformation in relation to stress.
 A. Ya. Malkin 305

These experiments can be carried out in any geometrical configuration of de-

formation, primarily at shear or uniaxial extension-compression.
The experiments allow one to find material characteristics of matter - creep
function, relaxation function, dynamic modulus (and compliance), in-
stantaneous modulus, equilibrium modulus and viscosity. If these mate-
rial functions do not depend on the level of deformation (and stress), material
has a linear viscoelastic behavior; in the opposite case, material is a
non-linear viscoelastic body.
For linear viscoelastic materials, the principle of superposition
(Boltzmann principle) is valid. According to the principle, reaction of a material
to all consequent deformations (or stresses) are independent and previous defor-
mation does not influence the reaction of the material to the next one. Mathe-
matical expression of this principle is done by a pair of the Volterra equations.
Creep and relaxation functions can be represented as sums (or at a limit as an
integral) of exponential items, and exponents in these expansions are relax-
ation and retardation spectra. Calculation of relaxation times in a spec-
trum is based on solving integral equations. In principle it can be done
unambiguously if a creep or a relaxation function are known exactly in a full
range of time, from zero to infinity. Since this is impossible, we are compelled to
use approximate methods in solving integral equations, based on analytical ap-
proximation and extrapolation of experimental data. However, accuracy of
these methods is limited by the unavoidable scatter of experimental points and
the ambiguity of their extrapolation beyond the experimental time or frequency
“window”.
Characteristics of a linear viscoelastic behavior of a material are interrelated
to each other by algebraic or integral equations and can be mutually recalcu-
lated. That is why the linear theory of viscoelasticity is a closed theory, contain-
ing all necessary equations in order to estimate mechanical behavior of a
material in arbitrary stress-deformation situation based on measurements of
any fundamental characteristic of viscoelastic properties of a material.
There is an important method of extending the experimental time (frequency)
“window” based on the idea of superposition of experimental data obtained at dif-
ferent temperatures or concentration in multi-component systems at some
other external factors, which should not be confused with the Boltzmann princi-
ple of superposition. This method is based on the idea that the same value of any
viscoelastic function can be reached either by changing time (frequency) scale or
306 Viscoelasticity

varying a value of an external factor (temperature, concentration and so on).

This experimental method allows us to separate parts of time-dependent curves
by their shift along the time-scale and thus to obtain a curve over a much wider
range of arguments than can be realized in a direct experiment.
The method of superposition is related to the idea that the dependencies of re-
laxation times in a spectrum are the same for all of them, and thus this principle
is not applicable if dependencies of different relaxation times in a spectrum on
an external factor (for example on temperature) are different.
The theory of linear viscoelasticity is used for:
• obtaining objective characteristics of a material, which can be correlated
with their molecular structure and/or content
• verification of conclusions from molecular theories which give grounds for
understanding the molecular structure of a material and intermolecular in-
teractions
• calculation of mechanical behavior of a material in arbitrary regimes of their
exploitation (but at rather low levels of stresses).
The theory of linear viscoelasticity works well only within the limit of infinites-
imal deformations. In increasing deformations, fundamental assumptions of
the theory, and primarily the principle of linear superposition, become inade-
quate. The theory requires generalization because numerous effects observed at
large deformations are definitely related to viscoelasticity of a material and are
especially important in real technological practice.
The methods of generalization of the classical theory of linear viscoelasticity
can differ, depending on the proposed and assumed mechanism of non-linear-
ity. Consequently, various theories of non-linear viscoelasticity were devel-
oped. The first step consists of the introduction of a measure of large
deformations instead of infinitesimal ones, which causes ambiguity of predic-
tions of non-linear viscoelastic effects.
The concept of rubber-like liquid (viscous liquid, capable to store large elas-
tic or reversible deformations in flow) allows us to explain phenomena related to
weak non-linear behavior, such as existence of normal stresses in shear flow,
which is an effect of the second (quadratic) order. Moreover, this model predicts
that at sufficiently high rates of uniaxial extension, a steady state flow becomes
impossible and a stream is broken, due to unlimited increase of stresses.
The concept of rubber-like liquid is not sufficient to understand and to describe
the strong non-linear effects; for example, non-Newtonian viscous flow of
 A. Ya. Malkin 307

viscoelastic liquids. We need to explore the idea of changing (or modification) of

relaxation properties (or in general sense of changing spectrum) of a matter due
to large deformations as an inherent reason of strong non-linear effects. It can
be done on the basis of different theoretical or experimental considerations and
then we may arrive at different predictions concerning rheological behavior of a
material.
Efforts of numerous researches are directed primarily to comparison of predic-
tions of various existing non-linear theories with results of some principle exper-
iments in uni- or biaxial deformations. This offers the a possibility to find
material functions characterizing non-linear viscoelastic properties of a mate-
rial. However, general theories of viscoelasticity are rarely used for solving con-
crete applied problems. Only directly measured non-linear characteristics are
used for these purposes. For example, non-Newtonian flow curves are used to
solving tube transportation problems, and a creep function measured at high
(non-linear) range of stresses is used for predicting long-term deformations of
real engineering materials.
7.12 REFERENCES
1. B. Gross in Mathematical Structure of the Theories of Viscoelasticity,
Hermann, Paris, 1953.
2. D. E. Blend in The Theory of Linear Viscoelasticity, Oxford, 1968.
3. R. M. Christensen in Theory of Viscoelasticity. An Introduction, Academic Press,
New York, 1971.
4. N. W. Tschoegl in The Phenomenological Theory of Linear Viscoelasticity.
An Introduction, Springer, Berlin, 1989.
5. A. J. Staverman and R. R. Schwarzl in Die Physik der Hochpolymeren, vol. 4,
Ed. H. A. Stuart, Springer, 1956.
6. J. D. Ferry in Viscoelastic Properties of Polymers, 3rd Ed., Wiley, New York,
1980.
7. A. Tobolsky in Structure and Properties of Polymers,
8. L. Bolzmann, Pogg. Ann. Phys., 7, 624 (1876).
9. A. P. Alexandrov and Yu. S. Lazurkin, Zh. Techn. Phys., 9, 1249 (1939).
10. H. Leaderman in Elastic and Creep Properties of Filamentous Materials and
Other High Polymers, Washington, 1943.
11. T. Alfrey in Mechanical Behavior of High Polymers, New York, 1948.
12. V. Volterra in Theory of Functionals and Integrals and Integro-differential
Equations, 1931.
13. V. Volterra and J. Pérèz in Théorie générale des functions, Gauthier-Villars,
Paris, 1936.
14. M. Baumgaertel and H. H. Winter, Rheol. Acta, 28, 511 (1989); 31, 75 (1992).
308 Viscoelasticity

15. C. Ester, J. Honerkamp, and J. Weese, Rheol. Acta, 30, 161 (1991).
16. J. Honerkamp and J. Weese, Rheol. Acta, 32, 65 (1993).
17. A. Ya. Malkin, Rheol. Acta, 29, 512 (1990).
18. V. V. Kuznetsov, T. Holz, and A. Ya. Malkin, Rheol. Acta, in press.
19. P. E. Rouse, J. Chem. Phys., 21, 1272 (1953).
20. B. Gross, Quart. Appl. Math., 10, 74 (1952).
21. F. Schawarzl and A. Starerman, J. Appl. Sci. Res., A4, 127 (1953).
22. M. L. Williams and J. D. Ferry, J. Polym. Sci., 11, 169 (1953).
23. K. Ninomiya and J. D. Ferry, J. Colloid. Sci., 14, 36 (1959).
24. S. Glasstone, K. Leidler, and H. Eiring, J. Chem. Phys., 7, 1053 (1939).
25. H. Vogel, Phys. Zs., 22, 645 (1921).
26. G. Tammonn and W. Hesse, Zs. anorg. allgem. Chem., 156, 245 (1926).
27. M. L. Williams, R. F. Landel, and J. D. Ferry, J. Am. Chem. Soc., 77, 3701 (1955).
28. K. K. Lim, P. E. Cohen, and N. W. Tschoegl in Multicomponent Polymer Systems,
Ed. R. F. Gould, ACS, Washington, 1971.
29. A. Ya. Malkin, A. E. Teishev, and M. A. Kutsenko, J. Appl. Polym. Sci., 45, 237 (1992).
30. R. Kohlrausch, Pogg. Ann., 12, 397 (1847).
31. G. V. Vinogradov and A. Ya. Malkin, Inter. J. Polym. Mater., 2, 1 (1972).
32. A. Payne in Reinforcement of Elastomers, Ed. G. Kraus, Interscience, New York,
1965.
33. W. P. Pavlov, G. V. Vinogradov, W. W. Snizyn, and Y. E. Deinega, Rheol. Acta, 1, 470
(1961).
34. A. A. Trapeznikov, Rheol. Acta, 1, 617 (1961).
35. A. S. Lodge in Elastic Liquids, Academic Press, London, 1964.
36. G. V. Vinogradov and A. Ya. Malkin in Rheology of Polymers, Springer, 1980.
37. A. Ya. Malkin and G. V. Vinogradov, Vysokomol. Soed., 27A, 227 (1985).
38. A. Ya. Malkin, J. Rheol., 39 (1995), in press.
39. F. J. Lockett in Nonlinear Viscoelastic Solids, Academic Press, London, 1972.
40. R. I. Tanner in Engineering Rheology, Oxford University Press, 1985.
41. H. A. Barnes, J. F. Hutton, and K. Walters in An Introduction to Rheology,
Elsevier, 1989.
42. M. H. Wagner, Rheol. Acta, 15, 136 (1976); 18, 33 (1979).
43. H. M. Laun, Rheol. Acta, 17, 1 (1978).
44. P. R. Soskey and H. H. Winter, J. Rheol., 28, 625 (1984).
45. R. G. Larson, J. Rheol., 29, 823 (1985).
46. A. Kaye, Note No. 134, College of Aeronautics, Cranford, England, 1962.
47. B. Bernstein, E. A. Kearsby, and L. J. Zapas, J. Res. Nat. Bur. Stand, 68B, 103 (1964).