VISCOELASTICITY 1. The Nature of Viscoelastic Behaviour

Viscoelasticity Page 1 of 27

University of NottinghamDepartment of Mechanical, Materials and Manufacturing Engineering

POLYMER ENGINEERING

1. The Nature of Viscoelastic Behaviour

A viscoelastic material exhibits properties characteristic of both a solid and a fluid. Theoccurrence of viscoelastic properties in a material depends to a large extent on theenvironmental conditions, particularly temperature, and the type of loading regime applied tothe material. In general, most polymers exhibit viscoelastic behaviour at in-servicetemperatures when the load is applied over a period of time. It is therefore important toconsider such properties when designing with these materials. The time-dependence of aviscoelastic material is better understood by considering the material as a combination of bothan elastic solid and a viscous fluid as follows:

Elastic solid + Viscous fluid(recoverable) (non-recoverable)

.E dtd /.

(Hooke’s law) (Newton’s law)

=

Viscoelastic solid

],[ tF

This is the expression for a general non-linear viscoelastic solid where the stress is a generalfunction (F) of the strain and time.

At small strains (typically < 1%) the strain and time response can be separated, giving thegeneral equation for a linear viscoelastic material as follows:

VISCOELASTICITY


Linear viscoelastic solid

)(. tG

G(t) is the time-dependent modulus of the material and at any point in time stress isproportional to strain.

Fig. 1 illustrates the results of a series of constant load experiments for polypropylene at roomtemperature. The increase in strain with time is a result of the viscoelastic behaviour of thematerial. For the low loads and at short time periods, the curves are spaced fairly evenlyalong the y-axis, hence the material may be considered to be linear viscoelastic in this range.

Figure 1: Creep curves for polypropylene at 20 oC (after Crawford).Each curve represents the variation in strain with time after the application of a constant load.

In polymers time-dependent viscoelastic behaviour shows itself in a number of ways,however, there are two manifestations which are particularly important in design. These arecreep (including creep recovery) and stress relaxation.

1.1 Creep and creep recovery

In an elastic material when a load is applied instantaneously and held constant thesubsequent deformation or strain is also instantaneous. This deformation is fully recoverablewhen the load is removed. This is not the case for a viscoelastic material. The response ofsuch a material to a constant load applied instantaneously is shown in Fig. 2. There is an initialinstantaneous elastic deformation followed by a delayed time-dependent deformation i.e.creep of the material. There may also be some permanent flow of the material particularly athigh loads. On removal of the load the reverse process takes place. A degree of instantaneousrecovery is followed by a delayed recovery i.e. creep recovery. Permanent flow during loadingresults in a residual deformation even when the load has been removed.

Creep deformation, ε(t), can be represented by the following equation:

)(.)( 0 tJt


where σ0 is the applied (constant) stress and J(t) is the Creep Compliance. This is a functionof time comprising three parts, J1, + J2 + J3, corresponding to the immediate elasticdeformation (ε1), the delayed elastic deformation (ε2), and the permanent flow (ε3).

J3 can be neglected for rigid polymers at ordinary temperatures and low loads. Amorphouspolymers show a J3 at raised temperatures whilst highly crystalline and cross-linked polymersshow no J3, even at significant loads and raised temperature.

Figure 2: Creep and creep recovery

The time dependence of the creep compliance, J(t), is better shown on a log-log plot as shownin Fig. 3.

Figure 3: Variation of creep compliance with time

At short times the creep compliance is constant with a relatively low value (~10-9 m2/N). It istermed the unrelaxed compliance (JU = J1 ) and the material is regarded as being in the glassystate. The creep compliance increases over time to a constant relaxed value (JR = J1 + J2,typically 10-5 m2/N) when the material is in a rubbery state. Some materials do not reach arelaxed state but continue to flow until the material fails. When testing time scales are veryshort or very long the material appears to behave elastically with either a low or highcompliance respectively. Between the glassy and rubbery states, i.e. at intermediate times, thecreep compliance is time dependent and the material is regarded as being viscoelastic. Thedegree to which this time dependence shows itself depends on the time scale of loadingrelative to some characteristic time parameter for the material. In creep this characteristic timeis called the retardation time, τ', and varies for different materials depending on the molecularstructure.

σ

εt

t

σ0

ε1

ε2+ ε3

ε3

ε2

ε1

log t

log10 J(t)m2/N

glassy viscoelastic rubbery flow

J1+ J2

J1

-5

-6

-7

-8

-9

τ'


1.2 Stress relaxation

Whereas creep involves maintaining a constant load on the material and observing thedeformation, stress relaxation involves the instantaneous application of strain, which is heldsteady whilst the stress in the material is observed. Under these conditions the stress increasesinstantaneously and then relaxes slowly over a period of time to some steady-state value, asshown in Fig. 4.

Figure 4: Stress relaxation

The time dependent variation of stress can be represented by the following equation:

)(.)( 0 tGt

where ε0 is the applied (constant) strain, and G(t) is termed the Stress Relaxation Modulusand, as with the creep compliance, is a time dependent function. On a log-log plot thevariation with time is as shown in Fig. 5.

Figure 5: Variation of stress relaxation modulus with time

As in creep, the same regions of behaviour exist. In the glassy state, at short times, thematerial has a high modulus (~109 N/m2) and is stiff. At long times, the modulus is low (~105

N/m2) and the material is rubbery. The presence of viscous flow affects the limiting value ofthe modulus. If flow exists the modulus reduces over time to an infinitessimal value and the

σ

εt

t

ε0

log t

log10 G(t)N/m2 glassy viscoelastic rubbery flow

9

8

7

6

5

τ


limiting stress decays to zero. Where there is no flow a relaxed equilibrium modulus resultsafter a long time. At intermediate times the material behaves viscoelastically with a time-dependent modulus. The time scale of relaxation again depends on the molecular structure ofthe material and is characterised by the relaxation time, τ.

Although the molecular processes governing creep are similar to those governing relaxation,in general the retardation time and relaxation time have different values.

2. Modelling Viscoelastic Behaviour

Simple mechanical elements such as springs (elastic) and dashpots (viscous) can be combinedto form a material model with viscoelastic properties. Although the models do not tell usanything about the molecular and physical processes taking place, i.e. they are purelyphenomenological, they are particularly useful for predicting the response of a material undercreep and relaxation conditions and even under complex loading situations. In addition, theycan give a clearer insight into the general nature of viscoelastic response.

In this section, three simple material models will be described:

Maxwell modelKelvin (or Voigt) modelStandard linear solid model

The response of these models under both creep and stress relaxation conditions will beanalysed. All models are linear i.e. at any point in time, stress is proportional to strain.

2.1 Basic Elements

Linear spring - Elastic component Linear dashpot – Viscous component

11 . E Hooke’s law 22 . Newton’s law

E = spring stiffness η = dashpot viscosity

ε1 E

σ1

ε2 η

σ2

Viscoelasticity

2.2 Maxwell Model: Spring and dashpot in series.

Equilibrium: 21

Strain compatibility: 21

Governing equation:

E

Maxwell model solution for creep (constant stress σ0):

Governing equation becomes:

0

Integrating with respect to time: constt .0

When t = 0, = σ0 / E, therefore:E

t 00 .

Figure 6: Response of Maxwell model fo

Maxwell model

J(t) = ε/σ0

E

ε2ησ2

σ, ε

ε1σ1

σ

ε

ε

CREEP RECOVE

Maxwell

RealMaterial

creep compliance

Page 6 of 27

r creep, creep recovery and stress relaxation

= 1/E + t/η

t

t

t

RY RELAXATION

σ

σ

ε

t

t

t

Viscoelasticity

Under creep conditions the Maxwell model shows an instantaneous elongation of the springfollowed by a linear variation of strain with time, deriving from the delayed extension of thedashpot (Fig. 6).

Creep recovery - When the stress is removed the model shows instantaneous recovery of theelastic spring and a permanent deformation depending on the duration of the initial loadingi.e. the initial creep strain of the dashpot.

Maxwell model solution for stress relaxation (constant strain ε0):

Governing equation becomes: 0

E

Integrating (given σ = E.ε0 at t = 0): )/exp(.. 0 tE

where τ = η/E, the relaxation time, and the stress decays exponentially (Fig. 6).

Maxwell Model Summary: Relaxation behcreep and creep recovery behaviour inadequa

2.3 Kelvin (or Voigt) Model: Spring and

Eq

Str

Go

Kelvin model solution for creep (constant st


Solution:

where τ' = η/E, the retardation time.

Maxwell model r

G(t) = σ/ε0 =

where

E ε2ησ2

σ, ε

ε1σ1

elaxation modulus

E . exp (-t/τ)

Page 7 of 27

aviour acceptable as a first approximation, butte.

dashpot in parallel.

uilibrium: 21

ain compatibility: 21

verning equation: .. E

ress σ0):

..0 E

)/exp(10

tE

τ = η/E

Viscoelasticity

Figure 7: Response of Kelvin model for

Under creep conditions the Kelvin model slimiting value, εlim = σ0 / E, with a retardation t

Creep recovery

Removal of stress gives governing equation:

Solving with initial strain ε0:

where t is the elapsed time since the stress wato removal of the stress. This is an exponentthe creep, with the same retardation time, τ'.

Kelvin model solution for stress relaxation


This is the response of an elastic material, ie.7.

Kelvin Model Summary: Acceptable first orbut inadequate for stress relaxation.

Kelvin model c

J(t) = ε/σ0 =E1

where

σ

ε

ε

CREEP RECO

Kelvin

RealMaterial

reep compliance

. [ 1 - exp (-t/τ') ]

Page 8 of 27

creep, creep recovery and stress relaxation

hows an exponential increase in strain up to aime, τ' (Fig. 7).

0.. E

)/exp(.0 t

s removed, and ε0 is the strain immediately priorial recovery of the strain (Fig. 7), ie. a reversal of

(constant strain ε0):

0. E

there is no relaxation of stress, as shown in Fig.

der approximation for creep and creep recovery,

τ'= η/E

t

t

t

VERY RELAXATION

σ

σ

ε

t

t

t


For a good description of creep, creep recovery and stress relaxation, a combined Maxwelland Kelvin model is required. The simplest combination is called the Standard Linear Solidmodel.

2.4 Standard Linear Solid (or Zener) Model: Spring and Kelvin model in series.

By considering equilibrium of stresses and compatibility ofstrains, the governing equation for this model is as follows:

21211 EEEEE

This is a linear equation in stress and strain and their firstderivatives, and can be solved by integration for conditionsof creep or stress relaxation (see McCrum, section 4.3.1).

SLS model solution for creep (constant stress σ0):

Integration gives: )/exp(12

0

1

0

tEE

where τ' = η/E2, the retardation time.

The strain is seen to be made up of two components – an instantaneous deformationcorresponding to the spring, and a delayed response corresponding to the Kelvin element. Theoverall response is illustrated in Fig. 8. The creep compliance changes from an unrelaxedvalue, JU = 1/E1, at t = 0, to a relaxed value, JR = 1/E1 + 1/E2, at infinite time.

Creep recovery - When the stress is removed the spring element recovers instantaneously,whilst the Kelvin element exhibits a delayed recovery. Solving the governing equation withthe stress removed results in the following expression for the delayed recovery:

)/exp(. tc

where τ' = η/E2, the retardation time and εc is the creep strain in the Kelvin element fromprevious loading.

SLS model creep compliance

J(t) =1E

1+

2E1

. [ 1 - exp (-t/τ') ]

where τ' = η/E2

E1

η

σ, ε

E2


Figure 8: Response of Standard Linear Solid model for creep, creep recovery and stress relaxation

SLS model solution for stress relaxation (constant strain ε0):

Integration of the governing equation in this case gives:

)/exp(.1221

01

tEEEE

E

where τ = η/(E1 + E2), the relaxation time.

The stress relaxes exponentially with time from a high initial value to a lower equilibriumvalue (see Fig. 8). The relaxation time depends on both the spring stiffness, E1, and theKelvin element parameters, η and E2. This is in contrast with the retardation time in creep,which depends only on the Kelvin element parameters. In general, the relaxation time is lessthan the retardation time.

The relaxation modulus changes from an unrelaxed value, GU = E1, at t = 0, to a relaxed value,GR = E1 E2 / (E1 + E2), at infinite time.

Standard Linear Solid Model Summary: The model provides a good qualitative descriptionof both creep and stress relaxation behaviour of polymeric materials.

SLS model relaxation modulus

G(t) =21

1

EEE

[E2 + E1. exp (-t/τ) ]

where τ = η/(E1 + E2)

t

t

t

σ

ε

ε

CREEP RECOVERY RELAXATION

σ

σ

ε

StandardLinearSolid

RealMaterial

t

t

t


3. Modelling Real Materials - Multiple Element Models

The Standard Linear Solid model gives a good qualitative fit to the creep and relaxationbehaviour of polymers. The model, however, incorporates a single retardation time and thisresults in creep taking place over a fairly narrow time range i.e. the creep compliance log plotis relatively steep. By changing the retardation time the curve can only be shifted along thetime axis and not broadened (see Fig. 9).

Figure 9: Comparison of standard linear solid model with real model for creep

Real materials tend to have a relatively broad creep compliance curve spreading over severalorders of magnitude of time. This behaviour can be modelled by combining a number ofelements into a multiple model. Fig. 10(a) shows a series combination of a spring and severalKelvin elements. Such a model has a finite number of components giving a finite number ofretardation times. By choosing different values for the model parameters in each Kelvinelement a broad creep compliance curve can be obtained.

The expression for creep compliance of the multiple model is given by:

)/exp(111

)(1

i

n

i io

tEE

tJ

where τ'i= ηi /Ei , the retardation time for the i’th Kelvin element.

Four or five elements are often sufficient to obtain a good fit to real material data. However,the idea may be extended further by replacing the discrete multiple model with a continuousintegral form for the creep compliance:

0

)()/exp(1)( djtJtJ U

where j(τ') is called the retardation time spectrum and is a weighting function which definesthe concentration of Kelvin elements with retardation times between τ' and τ' + dτ'. Theretardation time spectrum can be calculated directly from measured creep data, and providesa quantitative mathematical model describing the viscoelastic behaviour of real materials.

log t

log J(t)

real material

high τ'

low τ'


(a) (b)

Figure 10: Multiple element models for real materials(a) Series model for creep; (b) Parallel model for stress relaxation

A similar description for the stress relaxation modulus is obtained by considering a multiplemodel comprising a discrete number of Maxwell elements in parallel with a spring as shownin Fig. 10(b). As with the model for creep compliance this model may be extended to anintegral form incorporating a function called the relaxation time spectrum, g(τ):

0

)()./exp()( dgtGtG R

4. Complex Loading Histories - Boltzmann Superposition Principle

Creep, creep recovery and stress relaxation are responses to simple loading histories. Atheoretical model is required which will allow calculation of the response to more complexloading histories. The Boltzmann Superposition Principle is such a theory applying to linearviscoelastic materials.

Boltzmann proposed:

(i) The response of a material is a function of the entire loading history.

(ii) Each loading step makes an independent contribution to the final deformation andthe final deformation can be obtained by the simple addition of each contribution.

Consider the staged loading programme shown in Fig. 11.

Response to: Δσ0 at t = 0 is ε0(t) = Δσ0 J(t)Δσ1 at t = t1 is ε1(t) = Δσ1 J(t – t1)Δσ2 at t = t2 is ε2(t) = Δσ2 J(t – t2)Δσ3 at t = t3 is ε3(t) = Δσ3 J(t – t3)

where J(t - ti) is the creep compliance of the material obtained from a simple single steploading creep test. The contribution of each step is the product of the incremental stress andthe creep compliance function, which depends only on the interval in time between the instantat which the incremental stress is applied and the instant at which the creep is measured.

E0

η1

σ, ε

E1

E2

En

η2

ηn σ, ε

E0

E1 E2 En

η1 η2 ηn


Figure 11: Stress response to staged loading programme

The final deformation is given by the sum of these responses:

)()( ii ttJt

For continuous changes in load this sum is generalised to an integral as follows:

t

dtJt )()()(

which is usually written as:

t

dd

dtJt

)(

)()(

where J(t - λ) is the creep compliance after the time interval t - λ. The time variable λ is integrated over the interval -∞ to t to take into account all previous loading histories.

In the same way the stress response (or relaxation) to a complex strain history may be derivedfrom:

t

dd

dtGt

)(

)()(

where G(t - λ) is the stress relaxation modulus after the time interval t - λ.

σ

εt

t

Δσ0

t1 t2 t30

Δσ1

Δσ2

Δσ3


5. Dynamic Loading

This section considers the response of polymeric materials to dynamic, cyclic loading. For alinear viscoelastic solid, under such a loading regime and once equilibrium is reached, thestrain will be out of phase with the stress response. Suppose an oscillatory strain of frequencyω is generated in a specimen:

t sin0

The stress response of the material will be of the form:

)sin(0 t

ie. the strain lags behind the stress by a phase angle δ. This is illustrated in Fig. 12.

Figure 12: Response of linear viscoelastic material to cyclic loading (after McCrum)

The expression for stress given above can be expanded to give:

sincoscossin0 tt

From this expression it is clear that the stress consists of two components: one in phase withthe strain and one out of phase by 90o. This can be re-written as:

tGtG cossin 210 where

cos0

01 G

and

sin0

02 G

G1 is referred to as the storage modulus, and defines the energy stored due to the appliedstrain. G2 is the loss modulus, and determines energy dissipation. From these expressions, thetangent of the phase angle can be written:

1

2tanGG


This is called the loss factor, and is equal to zero for materials that are purely elastic. Lossfactor, storage modulus and loss modulus vary with the frequency of loading, as illustrated inFig. 13. The curve is similar to the modulus-time curve, exhibiting glassy (high frequency),rubbery (low frequency) and viscoelastic (intermediate frequency) regions.

Figure 13: Storage modulus, loss modulus and loss factor as a function of frequency (after Ward)

6. Effect of Temperature

6.1 Time-Temperature Equivalence

This section examines the effect of temperature on mechanical properties. Typical sets ofcreep and stress relaxation data are shown in Fig. 14. These figures show the behaviour of asemi-crystalline polymer, polyethylene, and an amorphous polymer, polyisobutylene. It isclear that for both materials, the properties vary greatly with temperature. In particular, thecreep compliance increases with increasing temperature, whilst the stress relaxation modulusdecreases with increasing temperature. It is also apparent that the temperature dependenceis more pronounced for the amorphous polymer than for the semi-crystalline polymer.

Figure 14: Temperature dependence of polyethylene – left, and polyisobutylene – right (after McCrum)


Fig. 15 shows the typical variation in short termmodulus (ie. after 10 seconds) with temperaturefor an amorphous polymer. The curve is asimilar shape to the modulus-time curve atconstant temperature, ie the material starts in aglassy state at low temperature, passes through aviscoelastic or transition region to a rubberyplateau, and finally flow takes place at hightemperature. The temperature of transition fromglassy to rubbery behaviour is the glasstransition temperature, Tg.

The similar shapes of the modulus-time andmodulus-temperature curves leads to theconcept of time-temperature equivalence. Forinstance, if one tests at high speed, ie short time,one sees a glassy material with high modulus.

This is also the case when testing at low temperature. On the other hand, if one tests over along time scale one sees a rubbery material with low modulus which may flow. This is alsothe case for high temperature testing. Thus the entire spectrum of relaxation times is shiftedalong the time axis as the temperature is changed.

By changing the temperature of testing it may be possible to obtain the full relaxation curveversus time at some reference temperature providing the relationship for shifting or the shiftfactor is known. This possibility may be deduced from Fig. 14, which suggests that the fullstress relaxation curve for polyisobutylene may be obtained by shifting the curves at differenttemperatures. This shifting process is called the principle of reduced variables where thetwo independent variables (time and temperature) are reduced to a single variable (reducedtime at a given temperature, also called time-temperature superposition).

Figure 16: Creep compliance versus time for a polymer tested at two temperatures, where T > T0 (afterMcCrum)

Fig. 16 shows typical creep compliance curves for experiments carried out at two differenttemperatures. This shows that on a log-time scale, the curves are shifted by a constant factorfor a change in temperature. This is known as the logarithmic shift factor, log (aT) whilst aT

is known simply as the shift factor.

The temperature dependence of aT characterises the temperature dependence of viscoelasticbehaviour for amorphous polymers. Various models have been developed to predict thisrelationship, as discussed in the following sections.

Temp (oC)

G(t=10s)N/m2

109

108

107

106

40 80 120 160 200

AmorphousAmorphous

Tg

Figure 15: Variation in short term moduluswith temperature for an amorphous polymer


6.2 Williams, Landel and Ferry (WLF) Equation

By studying a large number of amorphous polymers, Williams et al showed that log aT can bedescribed empirically by the following equation:

)T-(T+C

)T-(TC-=a 0

2

01

T0

0log

where C10 and C2

0 are constants and T0 is a reference temperature.

If T0 is taken as the Tg, then C1g and C2

g have nearly universal values for all amorphouspolymers (17.44 and 51.6 in units of oK respectively). Although the coefficients can vary forsome polymers, the form of the WLF equation has been found to fit data on many systemsover a temperature range Tg to Tg + 100oC. Although derived purely on phenomenologicalgrounds, the WLF equation has been found to have a molecular basis from the concept offree volume and viscosity of liquids.

The WLF equation is very successful for describing the behaviour of amorphous polymersabove Tg in the rubbery state. The presence of free volume allows the molecules to relax to anew configuration. Below Tg in the glassy state, however, the motions are hindered by theclose presence of other molecules and for relaxation to take place a potential barrier must besurmounted. In this region therefore the kinetics of relaxation are better described on thebasis of barrier or transition state theories. Such theories are also applicable to semi-crystalline polymers, where even above Tg the crystalline regions interfere with the freediffusion of molecules.

6.3 Transition State Theory & Arrhenius Temperature Dependence

The temperature dependence of the shift factor for mechanisms governed by transition statetheory can be described using the Arrhenius equation (for more details, see McCrum section4.3.3):

)/exp()exp(

00RTH

H/RT==a

T

TT

ie.

T

1-

T1

.RH

=aT0

ln

where ΔH = activation energyR = gas constant

A comparison of the Arrhenius and WLF relationships is shown in Fig. 17 (with T0 = Tg). Athigh temperatures the two are approximately the same as each other, but at lowertemperatures they differ significantly.

1/T

log aT

Arrhenius

W L F

1/Tg

Figure 17: Comparison of Arrhenius andWLF relationships


6.4. Limitations of Time-Temperature Equivalence

The WLF and Arrhenius temperature dependencies do not always fully describe therelaxation behaviour of polymers. As discussed earlier, they describe a horizontal scaling orshifting of the time axis and do not take into account the effect of temperature on the relaxedand unrelaxed compliances (or moduli). In semi-crystalline polymers and amorphouspolymers at high temperature, the structure becomes less rigid and an additional vertical shiftof the compliance curve is needed as shown in Fig. 18.

Figure 18: Schematic illustrating the dependency of unrelaxed andrelaxed compliance on temperature (after Ward)

In most cases the relative change in relaxed modulus is similar to the relative change inunrelaxed modulus. This leads to McCrum’s reduction procedure for superposition of creepdata at different temperatures:

)()(

)()(

2

1

2

1

TJTJ

TJTJ

U

U

R

R

where JU(T1) and JR(T1) are the unrelaxed and relaxed compliances at temperature T1, andJU(T2) and JR(T2) are the corresponding values at temperature T2. As illustrated in Fig. 18, thecorrection for changes in temperature is therefore a vertical shift of the full compliance curve,together with the usual horizontal shift of time scale.

References/Further Reading:

R J Crawford (1998) Plastics Engineering, Butterworth-Heinemann. Ch. 1 – “Generalproperties of plastics”, Ch. 2 – “Mechanical behaviour of plastics”.

N G McCrum, C P Buckley & C B Bucknall (1997) Principles of Polymer Engineering,Oxford Science Publications. Ch. 4 – “Viscoelasticity”.

P C Powell, A Jan Ingen Housz (1998) Engineering with polymers, Stanley Thornes(Publishers) Ltd. Ch. 5 – “Stiffness of polymer products”.

I M Ward, D W Handley (1993) Mechanical Properties of Solid Polymers, John Wiley &Sons. Ch. 4 – “Principles of Linear Viscoelasticity”, Ch 6 – Time-Temperature Equivalence”.

Polymer Engineering\Viscoelasticity\Viscoelasticity.doc


Worked Example 1 - Maxwell and Kelvin Models

A Maxwell model, with spring constant E = 1.5 GPa and dashpot viscosity η = 150 x 109 Ns/m2, issubjected to an instantaneous and continuous stress of 10 MPa. Calculate the % strain after a periodof 200 seconds.

What is the % strain in a Kelvin model when subjected to a stress of 30 MPa for the same period oftime ? Assume the same values for E and η.



Worked Example 2 - Standard Linear Solid

A material can be modelled as a Standard Linear Solid with an unrelaxed modulus GU = 1 GPa and arelaxed modulus GR = 0.5 GPa.

Given that the stress in the material reduces from 20 MPa to 15 MPa over a period of 500 secondswhen a constant strain is applied, determine the dashpot viscosity.

Solution

SLS Model Relaxation Modulus: G(t) =21

1

EEE

[E2 + E1. exp (-t/τ) ]

where τ = η/(E1 + E2)

Unrelaxed state: t = 0 GU = E1 = 1 GPa (1)

Relaxed state: t = ∞ GR = E1E2 / (E1 + E2) = 0.5 GPa (2)

From (1) and (2): E1 = E2 = 1 GPa

At t = 0, stress: σ(0) = GU.ε0 = 20 MPa (3)

At t = 500, stress: σ(500) = G(500).ε0 = 15 MPa (4)

Divide (4) by (3) to remove unknown ε0:

G(500) / GU = 15/20 = 0.75

But: G(500) =21

1

EEE

[E2 + E1. exp (-500/τ) ]

Therefore: G(500)/GU =21 EE

1

[E2 + E1. exp (-500/τ) ]

Substitute for E1 and E2 and re-arrange:

exp (-500/τ) = 0.5

Taking loge of each side gives τ = 721 s

Therefore: η = τ (E1 + E2) η = 1.44 x 1012 Ns/m2



Worked Example 3 - Boltzmann Superposition Principle - Complex step loading

A stress of 20 MPa is applied to a specimen at time t = 0 for a period of 100 seconds. The stress isthen increased instantaneously to 40 MPa and held for a further period of 100 seconds after whichtime it is completely removed.

Calculate the % strain in the specimen at a time of 300 seconds from application of the original load.The material obeys a Kelvin model with spring constant E = 0.5 GPa and dashpot viscosity η = 1011

Ns/m2.



Worked Example 4 - Boltzmann Superposition Principle - Complex step and ramp loading

A stress of 20 MPa is applied to a specimen at time t = 0 for a period of 50 seconds. The stress isthen ramped from 20 MPa to 40 MPa over a further period of 50 seconds after which time it iscompletely removed.

Calculate the % strain in the specimen at a time of 150 seconds from application of the original load.The material obeys a Kelvin model with spring constant E = 0.5 GPa and dashpot viscosity η = 1011

Ns/m2.

Solution

Loading History:

At t = 0: Δσ = 20 MPa

At t = 100s: -2Δσ = -40 MPa

Ramp load between 50s & 100s

Kelvin model compliance:

J(t) =E1

. [ 1 - exp (-t/τ') ]

where τ'= η/E = 200s

Boltzmann Superposition Principle gives:

ε(t) = Δσ. J (150 – 0) - 2Δσ. J (150-100) + 100

50

)150(

ddd

J

1st Part: Step load (Δσ) & unload (2Δσ)

At t = 150s: ε1 = Δσ [ J(150) – 2 J(50)] =9

6

10x0.510x20

[1 – exp (-3/4) – 2 + 2exp (-1/4) ]

ie. ε1 = 0.04 x 0.085 = 0.341%

2nd Part: Ramp load

J(150-λ) =

150exp1

E1

&50

1020 6xdd

= 0.4 x 106

At t = 150s: ε2 =

100

50

6100

50 200150

exp1104.0

)150(

dEx

dJdd

=100

509

6

200150

exp200105.0104.0

xx

ie. ε2 = 8 x 10-4 [100 – 200 exp(-1/4) – 50 + 200 exp(-1/2)] = 1.24%

At t=150s, total strain =ε1 + ε2 = 1.58%

σ

t (s)

Δσ

0 50 100 150

2Δσ



Worked Example 5 - Time-temperature equivalence

The 10 year creep compliance is required for a new semi-crystalline polymer at room temperature(300 K). This is to be obtained from a test at 350 K over a shorter time period.

Using the Arrhenius equation, calculate the equivalent test period. Appropriate values for theactivation energy and the universal gas constant are as follows:

ΔH = 120 kJ mol-1

R = 8.31 J mol-1 K-1

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VISCOELASTICITY 1. The Nature of Viscoelastic Behaviour