1.1 Linear viscoelasticity

Chapter 1 Rheology

1.1 Linear viscoelasticity

Rheology as the discipline is a part of continuum mechanics. It studies the flow/deformation of materials. Important part of rheology is the development of constitutive equations. These are mathematical relations between the stress tensor and various measures of deformation.

One of the most important flows, studied in rheology, is the simple shear flow. Quite often the general (tensorial) constitutive equations are reduced to relations between the shear components of the stress tensor and an appropriate deformation measure i.e. between the shear stress and the shear strain or the shear rate. Such relations are for simplicity considered in this work. When these components are infinitesimal and the relations between them are linear (either in the form of a simple integral or ordinary linear differential equations) one is working within the realm of linear viscoelasticity. A linear viscoelastic material is a material with both elastic and viscous properties. Such a material may exhibit behavior that combines liquid-like and solid-like characteristics.

1.1.1 Elasticity

If a material shows an ideal elastic behavior, it will accumulate energy during the loading time, and it will release it recovering all the deformation, as the loading force is removed. This is an ideal behavior, and no energy dissipation occurs in the described mechanism. The magnitude of the stress depends on the strain magnitude and, in the simplest case, this dependence will be linear. Such a material is called “ideal elastic solid” and it is described by the following relationship, valid for the simple shear:

τ = ⋅ (1.1) G γ

where τ is the shear stress and γ is the shear strain. The constant of proportionality G is called shear modulus of rigidity. This relationship is known as Hooke’s Law.

Theoretically, it is possible to model each type of linear viscoelastic behavior by using simple mechanical models. The mechanical model describing an ideal elastic solid is a spring coil of stiffness equal to G. Ideal response of the spring to a sudden shear is shown in Figure (1.1).

Figure (1.1) – Response of an ideal spring to a sudden shear.

1.1.2 Viscosity

In a viscous material, no recovery will occur as the load is removed because all the energy is dissipated into heat. Fluids rather than solids exhibit this type of behavior, where the magnitude of the stress is related to the strain rate, and, in the simplest case, shear stress is directly proportional to the strain rate: such a material is called “Newtonian liquid”. Its behavior, in simple shear, is described by the following relationship:

τ η γ = ⋅ (1.2)

t

γ

τ

where η is the viscosity and γ = d γ / dt is the (shear) strain rate. The mechanical analogue of the Newtonian liquid is a dashpot of viscosity η . Ideal response of the dashpot to a sudden shear is shown in Figure (1.2).

Figure (1.2) – Response of an ideal dashpot to a sudden shear.

As already said, the above mentioned behaviors are idealizations, but we can generally assume that they are valid when the shear stress and the shear rate are infinitesimal.

1.1.3 The Boltzmann superposition principle: linear viscoelasticity

As pointed above the linearity is important characteristic of the theory of linear viscoelasticity.

Another cornerstone of the theory is the assumption that the present state of a material is determined by the history of deformation or stressing. According to the Boltzmann superposition principle the effects of mechanical history are linearly additive, where the stress is described as a function of rate of strain history or alternatively the strain is described as a function of the history of rate of change of stress [1]. The following linear constitutive equations are mathematical forms of this principle:

( ) ( ' ) ( ' ) '

t

t G t t t dt

τ γ

−∞

= − ⋅ (1.3)

t

γ

τ

where ( ) γ t = d γ / ' dt is the shear rate, G(t) is the relaxation modulus and integration is carried out over all the past times (t’) up to the current time (t). Alternatively it is possible to write

( ) ( ' ) ( ' ) '

t

t J t t t dt

γ τ

−∞

= − ⋅ (1.4)

where ( ) τ t = d dt τ / ' and J(t) is called the creep compliance.

1.2 Time dependent experiments in shear

In this work, either stress or strain controlled rotational rheometers have been used. The Cartesian infinitesimal shear rate tensor for simple shear is:

0 0

0 0 0 0 0

γ

γ = γ (1.5)

from which we can obtain the two nonzero components of the stress tensor:

12 21 ^t

G t t dt ( ' ) '

τ = τ = = ⋅ τ γ

− (1.6)

All the following equations are valid for simple shear, so we will always refer to the nonzero

scalar components of the stress, strain, or shear rate. In this paragraph we’ll analyze some of

the most common experiments by which it is possible to characterize viscoelastic materials,

referring to the case of shear only. Before doing that, it is useful to define two mechanical

models: the Maxwell model and the Voigt (or Kelvin) model. Those models are too simple to

represent a real viscoelastic material, but they’re very useful to understand the transient nature

of viscoelastic behavior. Equations in the next sections are valid for the linear viscoelastic behavior only.

1.2.1 Two basic mechanical models

1.2.1.1 Maxwell Body

The Maxwell body, shown in Figure (1.3), is made of a spring and dashpot element in series.

Figure (1.3) – Maxwell body.

The behavior of this body is described by the following equation:

1 1 ( ) 1 1

( ) t ( ) t t ( ) t ( ) t

G G G

γ τ τ τ τ

λ η

= ⋅ + ⋅ = ⋅ + ⋅ (1.7)

where G is stiffness of the spring coil, η is the viscosity of the dashpot and λ = η / G has the dimensions of time and is called the relaxation time.

1.2.1.2 Voigt (Kelvin) Body

The Voigt body, shown in Figure (1.4), is made of a spring and dashpot element in parallel.

Figure (1.4) – Voigt body.

The Voigt model describes its behavior:

( ) ( ) t G ( )

t τ t

γ γ

η η

= − ⋅ (1.8)

where G and η have the same meaning as in the Maxwell model.

1.2.2 Stress Relaxation

If we imagine to apply a sudden, constant shear strain γ

to a linear viscoelastic material and we model it by the Maxwell body, we can obtain the stress by integrating equation (1.7):

( ) t

G exp( / ) t

exp( / ) t

τ = ⋅ ⋅ γ − λ τ = ⋅ − λ (1.9)

or

( ) ( ) /

exp( / )

G t = τ t γ = ⋅ G − t λ (1.10)

This type of experiment where a constant strain is applied to the sample and the resulting

stress is measured is called the “Stress Relaxation” experiment. Strain and relaxation modulus

against time, for a viscoelastic material, are shown in Figure (1.5).

Figure (1.5) – Stress relaxation experiment.

When the strain is not small the relaxation modulus will depend not only on time but also on the strain. Then it is customary to report the results in the following form [2]:

( , )

( )

o

G t γ τ t

= γ (1.11)

where ( , ) G t γ

is the nonlinear relaxation modulus. Relaxation modulus is very useful and

function in rheology. For γ

→ the nonlinear relaxation modulus ( , ) 0 G t γ

approaches the linear relaxation modulus G(t). Of course the simple relation (1.10) is not able to describe the relaxation of real viscoelastic materials. The superposition of terms (1.10) has a better chance to describe the relaxation in such materials.

Note that for a sequence of N small, sudden strains δγ , as shown in Figure (1.6), assuming that each contributes to the stress independently, the shear stress is given by [2]:

1

( )

(

) ( )

i

t G t t t

τ δγ

=

= − ⋅ (1.12)

γ

( , ) G t γ

t

For a smooth strain history and imagining that N tends to infinity, we can rewrite equation (1.12) as:

( ) ( ' ) ( ' ) ( ' ) ( ' ) '

t t

t G t t d t G t t t dt

τ γ γ

−∞ −∞

= − ⋅ = − ⋅ (1.13)

which is the mathematical expression of the Boltzmann superposition principle and previously introduced equation (1.3).

Figure (1.6) – Series of step strains.

1.2.3 Creep and recovery

If we apply a constant stress τ

to a linear viscoelastic material and use the Voigt model the strain is described by the following relationship, obtained by integrating equation (1.8):

[ ]

( ) t

1 exp( / ) t G

γ = τ ⋅ − − λ (1.14)

The strain is increasing with time, until the applied stress is kept constant: this is called the

“Creep experiment”. Again, real materials generally behave in a more complex way than described by (1.14). It is customary to report the results of this experiment in the following form:

( , )

( )

o

J t τ γ t

= τ (1.15)

where ( , ) J t τ

is the creep compliance function, it has the dimension [Pa

] and generally it is a function of time and stress. When the applied stress is infinitesimal the compliance, J, is a function of time only. The linear creep compliance given by Voigt model is:

[ ]

( ) 1

( ) 1 exp( / )

o

J t t t

G

γ λ

= τ = ⋅ − − (1.16)

Let us recall that the Boltzmann superposition principle yields the strain by equation (1.4). It is also useful to obtain additional information about the behavior of the sample by removing the stress at a certain time t and to measure the recovery of the material. This constitutes the

“Creep and Recovery” experiment. The amount of recovered strain in the linear case is given by:

( ) [ ( ) ( )]

r

t t

J t J t t

γ − = ⋅ τ − − (1.17)

The (linear) recovery compliance is defined as:

( ) ( ) /

r r o

J t t − = γ t t − τ (1.18)

Note that the amount of recovery depends on t and on the time elapsed since the stress has

been removed i.e. ( t t − . )

Let’s now examine the two following graphs, both showing a linear behavior: Figure (1.7) reports a typical compliance curve for a viscoelastic solid, while Figure (1.8) reports the same function for a viscoelastic liquid. As we can see, a viscoelastic solid under a constant stress will reach a maximum value of the compliance (strain) and will recover all the deformation as the load is removed. The viscoelastic fluid instead, under a constant load, after a certain period will start to flow at a constant shear rate, and it won’t recover all the deformation after the cessation of the load, reaching an asymptotic compliance value after sufficiently long recovery times.

Figure (1.7) – Creep behaviour of a viscoelastic solid.

Figure (1.8) – Creep behaviour of a viscoelastic liquid.

t t

( , ) J t τ

t

( , ) J t τ

t

Referring to the case of the viscoelastic fluid, which better describes the behavior of asphalt binders at medium and high temperatures, and assuming to be in the linear domain, the creep compliance function can be written as the sum of three different terms [3]:

( )

( )

o

J t J J t t

= + Ψ + η (1.19) where:

• J

is called “glassy compliance”, it represents the instantaneous elastic deformation, and it is usually very small ( ≈ 10

Pa

for polymeric systems).

• J

is called “delayed compliance” and represents the delayed elastic deformation.

• Ψ is the “normalized memory function”, describing the transient behavior of the ( ) t material: ( Ψ (0) 0, ( ) 1 = Ψ ∞ = ) [3].

• τ η

⋅ t /

is the dissipated, viscous deformation, where η

is the Zero Shear Viscosity.

This value of the viscosity is the one predicted by equation (1.2) for a Newtonian fluid, and can be experimentally obtained for asphalt binders and other materials by

measuring the viscosity at low applied shear rates, see paragraph (1.2.6).

In Figure (1.9), is shown the physical meaning of each term of equation (1.9).

0

, . J

= J glassy contribution

( )

, .

J

ψ t = J elastic contribution

0 2

/ , .

t η = J viscous contribution

Figure (1.9) – Contributions to creep compliance.

When a constant stress is applied to a viscoelastic liquid, the rate of strain approaches a limiting value and a steady flow is eventually attained. This “flow” part of the creep is governed by Newtonian viscosity, η

. Thus, the last term of equation (1.19) has to be added to the compliance function each time we are describing the behavior of a viscoelastic liquid, irrespectively of the used model, so that, for example, the creep compliance given by the Voigt model will be:

[ ]

( ) 1

( ) 1 exp( / )

o o

t t

J t t

G

γ λ

τ η

= = ⋅ − − + (1.20)

The intercept of the steady flow asymptote with the J axis is called the “Steady State compliance”, J

, and it is a measure of the elastic energy stored during the steady shear

t

J(t)

t

flow [1]. Referring to the creep curve only, after sufficiently long time, the compliance is given as:

( )

(

)

o o

t t

J t J J J

η η

= + = + + (1.21)

Equation (1.21) shows that Zero Shear Viscosity is the only parameter related to the slope of the compliance/time curve once the steady shear flow is achieved. This also implies that compliance curves can be useful to estimate the Zero Shear Viscosity. We are focusing on the possibility of estimating the ZSV from creep because it seems to be strongly correlated with rutting of asphalt paving mix [4,5]. If the steady flow conditions are achieved during the creep time, ZSV can be also estimated from the recovery zone. The Boltzmann superposition principle gives us the compliance equation for the recovery zone [6]:

( ) ( ) ( )

J t t

− = J t − J t t − (1.22) From where it follows that:

lim[ ( )] /

t

J t t η

→∞

= (1.23)

This is the asymptotic value the recovery compliance function reaches after sufficiently long time, i.e. the zero shear viscosity can be also determined from the recovery.

1.2.4 Dynamic creep

This experiment consists of applying a series of creep and recovery cycles: creep and recovery

time, applied stress and the number of cycles are the free parameters to choose, in this

experiment. What is measured is the accumulate strain and the percentage of recovered strain

per cycle and after the whole series. The accumulate strain after 100 cycles of 1 second creep

and 9 seconds recovery has been recently purposed as an alternative way to estimate the rutting performance of PMAs (see Chapter 2). Figures (1.10) and (1.11) show the compliance curves for a neat asphalt modified by a thermoplastic elastomer, respectively.

time

Strain

Figure (1.10) – Dynamic creep behaviour of a Base asphalt.

time

Strain

Figure (1.11) – Dynamic creep behaviour of a PMA.

1.2.5 Small amplitude oscillatory sweep

In this kind of experiment a small sinusoidal shear strain is applied to the sample, and the stress is measured as a function of time. The applied strain can be varied both in magnitude and frequency. This test provides important information on the linear viscoelastic properties of the tested material. Assume that the applied strain is given as:

( ) t

sin( ) t

γ = ⋅ γ ω (1.24)

where γ

is the strain amplitude and ω is the frequency. Thus the shear rate will be:

( ) t

cos( ) t

γ = ωγ ω (1.25)

The stress for sufficiently small γ

is given by the constitutive equation (1.3):

( ) t

sin( t )

τ = ⋅ τ ω δ + (1.26)

where δ is a phase shift called the “mechanical loss angle”. For a purely elastic solid, it is known that the stress is in phase with the strain ( δ =0), while for a Newtonian liquid the stress is directly proportional to the strain rate, so δ = π/2. The loss angle in a linear viscoelastic material is in the interval 0 < δ < π/2. Introducing the quantities G’( ω ) and G”( ω ), respectively called the “storage modulus” and the “loss modulus”, the stress can be written as:

[ ]

( ) t

G ' ( ) sin( ) t G ' ' ( ) cos( ) t

τ = ⋅ γ ω ⋅ ω + ω ⋅ ω (1.27)

where:

' ( )

cos( )

o

G ω τ δ

= γ ⋅ (1.28)

and:

' ' ( )

sin( )

o

G ω τ δ

= γ ⋅ (1.29)

Comparing equations (1.1) and (1.2) with equation (1.27), it can be easily shown that for a purely elastic solid G’=G and G”=0, while for a Newtonian liquid G’=0 and G”= ηω

this suggests to associate G’ with the stored energy, while G” can be associated with energy loss.

At a given frequency, both these quantities are obtained by a single measurement. The ratio of the lost energy to the accumulated one, in each cycle, is related to the phase angle:

' ' ( ) tan( ) ' ( )

G G

ω δ

ω ⁼ (1.30)

It is sometimes convenient to express the sinusoidally varying strain and stress as complex variables:

( )

*

e

τ = ⋅ τ

(1.31)

*

e

γ = ⋅ γ (1.32)

so that the modulus and the compliance can be considered complex numbers too:

( )* * ( ) ' ( ) ' '

G ω τ * G ω iG ω

= γ = + (1.33)

* 1

( )* ( ) ' ( ) ' '

* ( )*

J J iJ

G

ω γ ω ω

τ ω

= = = − (1.34)

1.2.6 Viscometry

Viscometry deals with the determination of the viscosity function. In steady viscometric

experiments, a train of increasing shear rates is applied to the sample, and the viscosity is

measured. For asphalts and other non-Newtonian materials, the viscosity/shear rate plot, in logarithmic scale, appears as the one reported in Figure (1.12).

Figure (1.12) – Viscosity curve showing shear thinning.

Until the material is in the linear viscoelastic domain (low applied shear rates), viscosity will not vary with shear rate, following equation (1.2) for Newtonian fluids. With increasing shear rate a critical shear rate is reached, where viscosity starts to decrease [7]. This kind of behavior is called “shear thinning”. There are also materials showing shear thickening, which means that they become stronger as the shear rate is increased. Thus, for a non-Newtonian material we generally have:

τ η γ γ = ( ) ⋅ (1.35)

The dependence of the viscosity on the shear rate is a manifestation of a non-linear phenomenon occurring in the material. This typical experiment is reported in this chapter because it provides a direct measurement of the zero shear viscosity, which is a fundamental linear viscoelastic parameter of each material. Experimental data can be usually fitted with different equations, here we report only the power law model [2]:

( ) m ( )

η γ = ⋅ γ

(1.36)

and the Carreau [8 equation:

1

( ) (

) [1 ( ) ]

n

a a

η γ = η

+ η η −

⋅ + ⋅ λ γ

(1.37)

that can describe viscosity behavior in neat and some modified asphalts. Figure (1.13) shows the physical meaning of Carrau’s equation parameters.

Figure (1.13) – Meaning of the Carrau equation parameters.

The three parameters Carrau equation is simpler and satisfactory for many materials:

2

( )

[1 ( ) ]

η γ = ⋅ + ⋅ η λ γ

(1.38)

The zero shear rate (limiting) viscosity can be related to the relaxation modulus. For steady simple shear, combining equations (1.2) and (1.3):

( ' ) '

t

o

G t t dt

η =

− (1.39)

The complex viscosity is defined as [1]:

( )* ( ) ' ( ) ' i '

η ω = η ω − η ω (1.40)

0 0

"( ) τ cos( )

η ω δ

= γ ⋅ (1.41)

0 0

' ( ) τ sin( )

η ω δ

= γ ⋅ (1.42) ( )* G ( )* ω

η ω ⁼ ω (1.43)

For ω → 0, the real component ' η is also converging to the zero shear viscosity η

.

1.3 Mechanical models for viscoelasticity

We have seen that both the Maxwell and the Voigt mechanical models, paragraph (1.2.1.1) and (1.2.1.2), are described by a single relaxation time and a weight. The following models combine Maxwell and Voigt elements, to increase the number of time constants with the aim of a better description of the viscoelastic behavior, i.e. better fitting of experimental data.

1.3.1 The Burger model

Burger’s body is obtained by combining Maxwell and Voigt elements in series, so it is fully described by two relaxation times and two rigidities. The Burger model equation is reported here for a constant applied stress only (creep experiment):

( )

0 1

1

1 1

( ) 1 exp( / )

o o o

t t t

G G G

γ τ λ

= + − − + λ (1.44)

where λ η

=

/ G

and λ η

=

/ G

. The Burger body is shown in Figure (1.14).

Figure (1.14) – Burger body.

1.3.2 The generalized Maxwell model

In the generalized Maxwell model, the mechanical analogue is made by N Maxwell elements in parallel, i.e. there are N pairs of material constants. Thus:

1

( )

[ exp( / )]

i

G t G t λ

=

= ⋅ − (1.45)

Figure (1.15) – Generalized Maxwell body with 5 elements.

Generalized Maxwell body is shown in Figure (1.15). The set of N pairs {G

, λ

} is called the discrete relaxation spectrum of the material. Taking N sufficiently large, it is theoretically possible to fit any relaxation data. If N tends to infinity and the distance between λ

to zero, we obtain a continuous function, H( λ ), which is called the continuous relaxation spectrum, then G(t) is represented by the following relationship:

( ) ( ) [exp( / )] (ln )

G t =

H λ ⋅ − t λ ⋅ d λ (1.46)

where Hd λ represents the infinitesimal contribution to the spectrum given by relaxation times between λ and λ+ d λ .

N pairs of {G

, λ

} are usually calculated from dynamic experiments. Unfortunately, even if the obtained constants fit well the experimental data, they probably would not have a physical significance, because the final set would not be a unique one. For polymeric systems, from ten to twenty couples of {G

, λ

} are usually required to obtain a good fitting of the G(t) experimental data [2]. To obtain such a set of data, it is necessary that dynamic experiments cover a wide interval of frequencies (or temperatures, paragraph (1.4)). It has been noticed that for some polymer melts, a long time behavior of G(t) is dominated by the {G

, λ

} where λ

is the longest relaxation time. Then in the terminal zone G(t) will decrease exponentially with time, as predicted by the Maxwell model.

Continuous spectrum is even harder to obtain because it cannot be directly measured. For the discussion of the procedure, we remand to Ferry [1].

1.3.3 The generalized Voigt model

The mechanical analogue for this model is made by N Voigt elements in series. Thus, for a viscoelastic liquid:

1

( )

[1 exp( / )]

i o

J t J t λ t

η

=

= ⋅ − − + (1.47)

Figure (1.16) – Generalized Voigt body with 3 elements.

The generalized Voigt body is shown in Figure (1.16). The set of N pairs {J

, λ

} is called the discrete retardation spectrum. The analogue of the continuous relaxation spectrum is the continuous retardation spectrum, L( λ) , it yields :

( )

( ) [1 exp( / )] (ln )

o

J t J L λ t λ d λ t

η

∞

= +

⋅ − − ⋅ + (1.48)

As it was for H( λ ), a direct measurement of L(λ) is not possible. From a qualitative point of view, the plot of L( λ ) against λ is similar to the one of J( ω )’’ , after the viscous contribution 1/ η ω

is removed. J(ω)’’ is easily obtained from dynamic experiments. This function is used for the prediction of high temperature performance of asphalt binders (see Chapter 2), according to the North American specification AASHTO M-230 [9]. The graph of this function, for neat as well as polymer modified asphalts, resembles the one of the gamma distribution function [10]:

( )

exp( )

L λ α λ = ⋅ ⋅ − αλ (1.49)

By assuming that (1.49) holds for the studied material one can calculate the shear compliance J(t). The normalized memory function is then given as [11]:

( ) ( ) [1 exp( / )]

t

L λ t λ λ d

Ψ = ⋅ − − (1.50)

The integral in (1.50) can be evaluated and the normalized memory function is given as:

( ) 1 2 t α tK

(2 α t )

Ψ = − (1.51)

where K

is the Bessel function of imaginary argument [12]. By substituting equation (1.51) into equation (1.19), we obtain:

( )

[1 2

(2 )]

o

J t J J α tK α t t

= + ⋅ − + η (1.52)

This equation represents a linear viscoelastic model governed by the given continuous retardation spectrum.

1.4 Time temperature superposition principle

We have seen that all the main material functions are time (or frequency) dependent.

Rheological behavior is strongly temperature dependent as well. Figure (1.17) shows that the effect of changing the test temperature for asphalts is equivalent to the change in time.

Materials for which this so called time temperature superposition principle holds are called thermorheologically simple materials [2].

Figure (1.17) – Effect of the temperature on asphalt.

For such kind of materials, it is possible to report experimental data obtained at different times/frequencies and temperatures on a single master curve. Referring to the generalized Maxwell model, for a thermorheologically simple material, it is possible to quantify the effect of a change in temperature by means of the Rouse theory [2], stating that a change in temperature from T

to T will change all the relaxation times by the same factor a

as follows:

( ) ( )

i

T a

T

λ = λ (1.53)

where a

is a function of both T

and T . The changes in the magnitude of material functions are also provided by Rouse theory. For example for Maxwellian rigidities the theory predicts:

( ) ( )

i i o

o o

G T G T T T

ρ ρ

= ⋅ ⋅

⋅ (1.54)

where ρ is the density of the material at temperature T . Thus:

1

( , )

( ) exp{ /[ ( ) ]}

o o i

G t T T G T t T a

T

ρ λ

ρ

= ⋅ ⋅ −

⋅ (1.55)

Introducing:

( ) ( , )

r

G t G t T T T

ρ ρ

= ⋅ ⋅

⋅ (1.56)

and:

r T

t t

= a (1.57)

one can obtain the master curve of the relaxation modulus at temperature T

from the values of G at several experimental temperatures as:

1

( )

( ) exp[ /[ ( )]

r r i o r i o

i

G t G T t λ T

=

= − (1.58)

If the curves of the relaxation modulus from experimental data at different temperatures are plotted against time, the function a

is obtained as the magnitude of the horizontal shift necessary to superpose the data obtained at temperature T

to the reference curve (at T

). The WLF (Williams, Landel, Ferry) equation describes reasonably well the behavior of the shift factor against T for T

< < + T T

100 °C:

( )

log( )

( )

T o

o

A T T

a B T T

− ⋅ −

= + − (1.59)

where A and B are constants. For temperatures below T

the Arrhenius equation, related to the

“activation energy of flow” [2], has been found useful:

1 1

log( )

o

a E

R T T

= ⋅ − (1.60)

Note that in both equations, if T T = ,

a

= . Conventional asphalt is usually considered as a 1

“simple” material, i.e. the time-temperature superposition principle (TTS) applies [13,14], while for polymer modified asphalts, the TTS doesn’t apply for all bends.

All the linear viscoelastic properties are related by means of continuous spectra, which means

that relaxation modulus is not the only viscoelastic property obeying the time-temperature

superposition principle. For example for viscosity, substituting s t t = − in equation (1.39), we ' can write:

0

( )

o T r r r

o o

a T G t dt

T η ρ

ρ

⋅

= ⋅ ⋅

⋅ (1.61) and:

( )

( )

o

T

G t dt

η =

(1.62)

Thus:

( ) ( )

( ) ( )

o o o o

T

o o o o

T T T

a T T T

ρ η η

ρ η η

= ⋅ ⋅ =

⋅ ⋅ (1.63)

considering that the ratio T

T

ρ ρ

⋅

⋅ is close to unity.

1.5 Non-linear viscoelasticity

The theory of linear viscoelasticity is basically a closed mathematical theory of viscoelastic materials, in the limit of infinitesimal strains and/or the rates of strain. One of the basic characteristics of the theory is the unicity of the infinitesimal strain. When the deformations are large this unicity is lost and one can use various tensorial measures of deformation.

Another problem of nonlinear viscoelasticity is the inherent non-uniqueness of some solutions obtained with various nonlinear phenomenological models. Basically it means that for the description of various nonlinear phenomena one might need also various nonlinear models.

One of the most successful (nonlinear) models is the Bernstein-Kearsley-Zapas model [15].

The another one is the generalized rubberlike model of A.S. Lodge [16]. This model seems to

be especially suitable for the description of polymer modified asphalt, and we will use it for

the analysis of some of our experiments, in these materials.

1.5.1 Cauchy and Finger tensors

The Cauchy and the Finger tensors are the most widely used measures of finite deformation in nonlinear viscoelastic theories. For simple shear, they’re respectively defined as:

[ ]

[ ] [ ]

1 ( ' ) ( ) 0

( , ' ) ( ' ) ( ) 1 ( ' ) ( ) 0

0 0 1

t t

C t t t t t t

γ γ

γ γ γ γ

−

= − + − (1.64)

[ ] [ ]

[ ]

1 ( ) ( ' )

( ) ( ' ) 0

( , ' ) ( ) ( ' ) 1 0

0 0 1

t t t t

B t t t t

γ γ γ γ

γ γ

+ − −

= − (1.65)

where ' t < is the running time, t is the current time and γ is the strain. Note that the Finger t tensor is obtained by inverting the Cauchy tensor:

( , ' )

( , ' )

B t t = C t t

(1.66)

1.5.2 Rubberlike liquid model

This model is presented for the analysis of the stress relaxation experiment, so that the differences from linear viscoelasticity are stressed. Assuming that, for nonlinear viscoelasticity, the effect of strain history on stress is taken into account by a simple integral in time as for the generalized viscoelastic material, we can write:

( )

( ' )

( , ' ) '

ij

t m t t C t t dt

τ =

− ⋅

(1.67)

This equation defines the so called “rubberlike liquid” model. The function m t t ( − ' ) is called rubberlike memory function and can be related to the relaxation modulus as follows:

( ) dG s ( )

m s = − ds (1.68)

The relaxation modulus of the rubberlike liquid model is independent of the strain, as the one for a linear viscoelastic material. A particular form of the memory function, for the rubberlike liquid model, is given by Lodge [17]:

1

( ' ) ( ' )

exp

i i i

G t t

m t t

λ λ

=

− = ⋅ − − (1.69)

The model leads to the expression of relaxation modulus which is identical to the one predicted by the generalized Maxwell model. Again modulus is time dependent only, which implies that the theory cannot describe properly the material behavior when the strains are large.

1.5.3 Wagner’s equation: the damping function

In the most general case, memory function will depend on time as well as on the principal invariants of Finger tensor. Thus we assume that memory function has the following form:

1 2

[( ' ), , ]

M M t t I I = − (1.70)

where, I

and I

are the first and the second principal invariant of the Finger tensor. Then one can generalize the rubberlike model (1.67) and write:

1

1 2

( )

[( ' ), , ] ( , ' ) '

ij

t M t t I I C t t dt

τ =

− ⋅

(1.71)

Wagner proposed that M can be factorized as follows:

1 2 1 2

[( ' ), , ] ( ' ) ( , )

M t t I I − = m t t − ⋅ h I I (1.72)

where h I I ( , )

is the so called “damping function” and m is the memory function depending on time only. Wagner’s constitutive equation is then:

1

1 2

( )

( ' ) ( , ) ( , ' ) '

ij

t m t t h I I C t t dt

τ =

− ⋅ ⋅

(1.73)

Note that ( m t t − ' ) is the linear viscoelastic, memory function. The damping function h I I ( , )

must be determined by experimental data, and it generally varies from one material to another.

The existence of damping function can substantially simplify the calculation of various material functions. However, it cannot be found for many materials and it is sometimes argued that its existence can lead to material instability [2]. The separability of the memory function introduces an additional step in materials’ behavior characterization. We can consider that materials are described by:

• Linear viscoelasticity

• Separable nonlinear viscoelasticity

• Non-separable nonlinear viscoelasticity

1.5.4 Separable nonlinear viscoelasticity

For a stress relaxation experiment, in the case of simple shear, the two invariants of the Finger tensor are functions of γ only, so that h I I ( , )

≡ h ( ) γ . For this experiment, in case of separable nonlinear viscoelasticity, we can write the shear stress as:

( ) t h ( ) G t ( )

τ = ⋅ γ γ ⋅ (1.74)

where ( ) G t is the linear viscoelastic relaxation modulus. The nonlinear modulus is obtained from experimental data and is given as:

( , ) ( ) ( )

G t γ = h γ ⋅ G t (1.75)

Then damping function is given as:

( , )

( ) ( )

h G t

G t

γ = γ (1.76)

It represents the vertical shift necessary to overlap the stress relaxation curves (obtained at different strain values) to that of the linear viscoelastic behavior, as shown in figure (1.19).

Figure (1.19) – Meaning of the damping function (γγγγ

< γ < γ < γ < γ

< γ < γ < γ < γ

)))).

( , ) G t γ

t

0 linear γ =

γ

γ

h ( ) γ

( )

h γ

The following equations are examples of damping functions used in the literature:

Wagner ( ) exp( h γ = − n γ ) (1.77) Osaki h ( ) γ = ⋅ a exp( − n

γ ) (1 + − ⋅ a ) exp( − n

γ ) (1.78)

Soskey ( ) 1

1

h γ a

= γ

+ ⋅ (1.79)

We can notice that ( ) h γ is decreasing as the strain is increasing, which agrees with the relaxation data shown in Figure (1.19), and that ( ) h γ → for 1 γ → . 0

The most studied model for the prediction of the damping function is the “tube model” by Doi and Edwards; its success is in part due to the fact that it gives a parameter-free prediction for

( )

h γ , and is in good quantitative agreement with a large number of experimental data [18].

What has been found is that for many polymeric liquids, there is a characteristic time that separates a fast relaxation process, mainly due to chain retraction, from a slower relaxation due to reptation, where G(t,γ) appears to be separable [19].

1.5.5 Non-separable nonlinear viscoelasticity

In non-separable nonlinear viscoelasticity, very common for polymer modified asphalts, ( , )

G t γ is not factorable into two different functions, one dependent on strain and the other one on time only, so that curves obtained at different strain values are shifted and also decrease in time with different shapes. In this case, memory function is the function to look at.

To obtain M(t, γ ), one can first construct a discrete grid on (t, γ ) domain and plot G(t, γ ) as a

three dimensional graph (function of the two independent variables). Then, G(t, γ ) surface can

be built in a nonparametric or parametric way. In nonparametric study, the grid is interpolated

by an interpolation algorithm and a smooth surface of M(t, γ ) can be constructed. In the other

case, an analytical description of G(t, γ ) surface must be obtained, and M(τ,γ) is then

calculated.