Modulus = Stress(t) / StrainCompliance = Strain(t) / Stress

Spring

Dashpot

Figure 1. The two basic building blocks of viscoelasticity. The spring is elastic and the dashpot is viscous.

The next step is to consider how each of these elements behaves when subjected to anoscillating strain. First we'll consider the case of the spring which is portrayed in Figure 2.

0°

90°+~.~.

•').70

Figure 2. Simplified diagram of an oscillating spring.

2

The strain can be written as:

(3) 'Y = 'Yosin wt

The term 'Yo is the amplitude or maximum strain, and w represents the angular velocity.Substituting equation (3) into equation (1) leads us to the expression:

(4) TE '" G'Yo sin wt

It is clear that the strain and the resulting shear stress acting on the spring are in-phase with eachother (i.e., they are both sine waves).

The next consideration is that of an oscillating dashpot which can be visualized in Figure 3.

Figure 3. Simplified diagram of an oscillating dashpot.

+ w 900

An inspection of equation (2) leads us to first taking the derivative of the assigned strain definedin equation (3):

(5) ,y = d)' / dt = w'Yocos wt

If this expression is substituted into equation (2), the resulting shear stress acting on the dashpotis:

(6) TV '" 1jw'Yocos wt

Contrary to the spring response, it is apparent that the applied strain and the resulting shear stresson the dashpot are 90° out-of-phase (i.e., the strain is a sine wave and the stress is a cosinewave). We can map these results in the following manner:

3

time

1:El'------~------~time

time

Figure 4. The assigued strain and stress responses as a function of time for a spring and a dashpot.

At this point the elastic and viscous limits are apparent and a measure of the phase-shiftindicates how viscoelastic the substance is under given test conditions. If the phase-shift is 0·,then the substance can be described as purely elastic. If the phase-shift is 90·, then the substancecan be classified as purely viscous. A phase-shift of 45· would indicate for example that thematerial is 50% elastic and 50% viscous.

4

Ley de Hooke de la Elasticidad

Ley de Newton de la Viscosidad

2.2 Kelvin-Voigt Model

In order to further picture exactly what viscoelasticity is, mechanical models have beenintroduced consisting of springs and dashpots. The Kelvin-Voigt model corresponds to a springand dashpot in parallel and is portrayed in Figure 5.

Figure 5. Diagram of the Kelvin-Voigt mechanical model.

The spring has a constant modulus of G and the dashpot has a constant viscosity of 1/. Anystrain 'Y applied to this system will result in the exact same strain being applied to the dashpotYv and spring 'YE' The total stress necessary to deform the system is equal to the spring stressplus the dashpot stress.

(7) 'Y = 'Yv = 'YE

(8) T = TE + TV

If we substitute equations (I) and (2) into this expression, then the constitutive equationdescribing the Kelvin-Voigt mechanical model can be written as:

(9) T = G'Y + 1/d'Y/dt Kelvin-Voigt Equation of State

If a sinusoidal strain 'Y = 'Yosin wt is applied, then the resulting solution is:

(10) T = G'Yosin wt + 1/w'Yocos wt

It is clear that the shear stress consists of two parts: a) The in-phase or sine-wave contributionand b) The out-of-phase or cosine-wave contribution. As reasoned earlier, the in-phase part isthe elastic component and the out-of-phase part is the viscous component.

5

2.3 Maxwell Model

The Maxwell model corresponds to a mechanical system consisting of a spring and dashpot inseries. This is diagramed in Figure 6.

Figure 6. Diagram of the Maxwell mechanical model.

The spring has a constant modulus of G and the dashpot has a constant viscosity of "I. Becausethese elements are in series, the stress on each element will be equal to the total stress. The totalstrain exerted on the system is equal to the dashpot strain plus the spring strain.

(11) 'T = 'Tv = 'TE

(12) "I - v» + "IE

If the derivative of equation (12) is taken with respect to time, then the strain rate can beexpressed as:

(13) d'Y/dt = d'Yv/dt + d'YE/dt

Taking the derivative of equation (1) and substituting it and equation (2) into equation (13), thefollowing constitutive equation is found:

(14) lIG(d'T/dt) + 'T/"I = d'Y/dt Maxwell Equation of State

If a sinusoidal strain "I = 'Yosin wt is applied, then the following differential equation is theresult:

(15) lIG(d'T/dt) + 'T/"I = w'Yocos wt

6

The steady-state solution of this first-order differential equation is:

(16) T = {G(Aw)2'Yo/[1 + (AW)2]) sin wt + {GAw'Yo/[1 + (AW)2]}COS wt

The term A is the relaxation time and defined as A = r¡/G. Although the final solution is more complicated than that of the Kelvin-Voigt model, it is nevertheless apparent that the solution can be broken into two parts: a) The in-phase or sine-wave contribution and b) The out-of-phase or cosine-wave contribution. The in-phase part is the elastic component and the out-of-phase part is the viscous component.

2.4 General Approach to Forced-Oscillation

When applying a sinusoidal strain to a substance, the angular velocity w and strain amplitude 'Yo are assigned parameters. The resulting shear stress is measured as a function of time and varies sinusoidally with a measurable phase-shift and amplitude. Experimentally, we can thus write the expressions:

(17) 'Y = 'Yosin wt (Assigned)

(18) T = Tosin(wt + o) (Measured)

where o is the phase-shift and TO is the shear stress amplitude. Equation (18) can be expanded trigonometrically as:

(19) T = TO[COSO sin wt + sino cos wt]

As in the cases of the Kelvin-Voigt and Maxwell models, there are two contributions corresponding to the elastic and viscous compQnents.

The magnitude of the complex modulus G is defined as:

and can be readily calculated. The complex modulus is a measure of the substance's total resistance to strain. Upon consideration of the in-phase sin wt term, the elastic component can be defined as the storage modulus G':

(21) G' = G*coso = (TO/'YO)coso

The viscous component can be likewise defined from the out-of-phase cos wt term as the loss modulus G":

(22) G" = G*sino = (To/'Yo)sino

If the test substance is purely elastic (i.e., a spring), then the phase angle o = 0 0 and G* = G', G" = O. If the test substance is purely viscous (i.e., a dashpot), then the phase-angle is 90 0 and G" = G*, G' = O.

As an alternative to the complex modulus, the complex viscosity r¡ * is defined as:

(23) r¡*= G*/w = TO/'YOW

and is a measure of the magnitude of the total resistance to a dynamic shear (i.e., the maximum shear rate is 'Yow and the maximum shear stress is TO). It can likewise be broken into two components, the storage viscosity r¡" (the elastic component) and the dynamic viscosity r¡' (the viscous component):

(24) r¡' = G"/w = (To/'Yow)sino

(25) r¡" = G' /w = (TO/'YOW)coso

7

The stress response of equation (19) can be thus written in terms of the moduli or viscosities:

(26) T = G''Yosin wt + G"'Yocos wt

(27) T = 1J"'Yowsin wt + 1J' 'Yowcos wt

The magnitudes of these parameters can be readily calculated by applying a known strain amplitude and frequency to the test substance and measuring the subsequent stress amplitude and phase-shift. The next step in this anal ysis is to match the predicted model terms with the measured terms.

2.5 Matching Tenns and Making Sense Out oC the Models

The Kelvin-Voigt model of Figure 5 will be considered first. The stress response of this mechanica1 model is expressed in equation (10). If the terms of equation (10) are matched with those of the general equation (26), the magnitudes of the loss modulus and the storage modulus for the Kelvin-Voigt mode1 can be written as:

(28) G' = G G" = 1JW

The elastic response is thus entirely due to the spring and the viscous response is entirely due to the dashpot. If one measures G' and G" experimentally as a function of frequency and G' remains constant while G" increases proportionally with the frequency, then the substance behaves in accordance with the Kelvin-Voigt model. Such a material can be described as being a viscoelastic salid since it always retums to its initial equilibrium position (Le., if one imposes a stress on the model and then re1eases the stress, the model will eventually retum to its pre-stressed position.) A viscoelastic solid is said to possess memory. The time-scale necessary for the response is the retardation time A = 1J/G. If the dashpot viscosity is great, then the dampening of the response to an imposed stress is great. Most rubbers and gels are viscoelastic solids.

The Maxwell model of Figure 6 will now be considered. The stress response of this mechanical model is expressed in equation (16). If the terms of equation (16) are matched with those of the general equation (26), the magnitudes of the 10ss modulus and storage modulus for the Maxwell model can be written as:

(29) G' = G(Aw)2/[1 + (AW)2] G" = GAw/[1 + (AW)2]

In this case the characteristic time-scale is referred to as the relaxation time and is again defined as A = 1J/G. The elastic and viscous responses are dependent on the magnitude of the dimensionless term AW. There are two distinct limits when G' and G" are measured as a function of frequency:

Case (1): (AW)2 < < < 1

Case (2): (AW)2 > > > 1

G' = G(AW)2 G" = 1JW

G' = G G" = G2/1JW

In case (1) where the frequency is low, the storage modulus scales as the frequency squared whereas the 10ss modulus is proportional to the frequency. In this regime, the viscous component is greater than the elastic component since the model has enough time to respond to a given strain. In case (2) the frequency is high and the dashpot does not have enough time to respond to the given strain. In this regime, the elastic component dominates and the substance behaves like a spring. If a material behaves according to these two limits, then it can be said to be a Maxwellian viscoelastic liquido Such a material has no singular equilibrium position and thus no memory since it does not retum to its original pre-stressed position. Any applied deformation is permanent. It is readily apparent that if the dashpot viscosity is great, the model will act as a spring and if the spring modulus is great, the mode1 will act as a dashpot. Polymer solutions are viscoelastic liquids.

8

T = (Mod)Yosin(wt) + (Visc)wYocos(wt)

The results for both of these models are diagramed in Figure 7 in order to graphicallydemonstrate how the moduli behave in each case.

G"= TjCJ)

log G

log G

-----7"'-----G'= G

log ACJ)

-------G'

G"

log ACl)

Figure 7. The behavior ofG' and G" as a function offrequency for the Kelvin-Voigt model (above) and for theMaxwell model (below).

It is not practical to expect that a given substance should behave as either of these models. Infact most materials behave like various models in conjunction with a spectrum of relaxationtimes. The models portrayed in this report only provide a basic physical foundation forexplaining viscoelastic effects. Nevertheless, it has been demonstrated that viscoelastic materialscan behave as either solids or liquids.

9

5. EVALUATION

With the above mentioned type of instrument, it is possible to meke e

defined strain and measure a defined stress. Wa know that these two

signals can be in-phase or out-of-phase depending on the Viscoelastic

behavior of the measured material.

d = ((;, sin [w t lr.- to-' (L = 1.0 sin (w t + cJ )

0'0= max. amplitude of strainrr-l-o = max. amp l i tuda of stress

Now we have to find a way to describe these Viscoelestic properties.

Tha relationship between stress and strain is defined as:

T = G*' «using the complex modulus G*.

The complex modulus G* includes the complete informetion of the

Viscoelsstic properties; the elastic component, the viscous component

as well as the phase shift between stress end strein.

G* = G' + iG"

where G' is the storaga mmmm(elastic)modulus and Gil is the loss modulus.

The storege modulus G' is e meesurement for the energy stored end

recovered in the materiel, while the loss modulus is e measurement for

the energy lost es heet in the material.

This relationship cen be visualized in the following greph:

Gil

G'

elastic viscous

The absolute magnitude of the complex modulus G* can now be calculated

as the peak stress dividad by the peak strain:

IG* I =

Out of this value, the storaga and the loss modulus can be calculated

using trigonometric identities:

G' = IG*/cos eS

G" = IG*/ sin eS

A useful parameter which is a measurement of the ratio of energy lost

to energy sto red is the loss tangent:

ten b = G" / G'

As an alternative to G*, the phasa relationship can also be dascribed

by e complex viscosity:

~ .. = ~' - i~"

wi th ~' = G" / w and ~" = G' / w

When both tha amplituda of strain and the frBquancy bacomes very small,

a ViscoBlastic fluid will bahave more lika e newtonian fluid. In this

case, tha dynamic viscosity approaches the steady-shBar viscosity

lim~'=~ w-o

3. Experimental

The dynamic properties of viseoelastie substanees ean be readily measured with a rheometer that ean be made to oseillate at an assigned frequeney and strain amplitude. The major experimental requirement is the ability to measure the shear stress as a function of time sinee the strain varies with time. This information can then be used to calculate the various viscoelastic parameters as outlined in Section 2.4 of this reporto It should be noted that a controlled-stress rheometer oseillates with an assigned frequency and stress amplitude, and the strain is measured as a funetion of time.

Various f¡xture geometries are possible with a rheometer, but the most common are concentric-cylinder, cone-and-plate, and parallel-plate. In order to characterize a given substance, various tests are possible. The material can be subjected to a:

(1) Strain sweep in order to locate the sample's linear viscoelastic region.

(2) Frequeney sweep in order to charaeterize the sample's degree of viseoelastieity at various time scales.

(3) Time sweep in order to traek sueh kinetie phenomena as euring and gel-formation.

(4) Temperature sweep to study how the viseoelastieity alters as a funetion of temperature. This is also useful for monitoring temperature-initiated reaetions.

The following sections offer brief deseriptions of relating strain and shear stress to the measuring geometry and of interpreting strain and frequeney sweeps.

3.1 Relating Strain and Stress to the Fixture Geometry

Depending upon the sample' s physical charaeteristies, different geometries are available in order to optimize dynamie measurements. Since the assigned variable is in actuality the angle of deformation '" and the measured quantity is the torque M, equations must be utilized in order to calculate the shear stress T and the strain 'Y. The three most common geometries are pictured in Figure 8 along with derivations of the strain as a function of the assigned deformation angle for each conf¡guration.

10

Cone-and-plate

-v- r lfJ~ z: r-t(lco e

-=:.-'~ ~e)R

h

-~-~------

Figure 8. Three common geometries and how to calculate their respective strains as a function of the assigneddeformation angle.

The shear stress as a function of the measured torque M for each geometry can be calculatedfrom the following expressions.

(30) T = 3MI2'll"R3 (Pa) Cone-and-plate

(31) T = 2MhrR3 (pa) Parallel-plates

(32) T = M/2'll"Ri2L (Pa) Concentric-cylinders

The expression for the parallel-plate fixture is valid when the applied strain lies within thesample's linear-viscoelastic range (i.e., the moduli are independent of strain). Note that thestress changes as a function of the radius, and that the maximum stress and strain values exist atthe edge of the plate.

Concentric-cylinders are beneficial when the test sample has a low viscosity (i.e., dilute andsemi-dilute polymer solutions).

The major advantages of a cone-and-plate are that the strain and stress are constant throughoutthe gap (when 0 :s; 4°) and that it is easy to load and clean. A cone-and-plate fixture iscommonly used for medium and high viscosity materials with extended linear-viscoelastic regions(i.e., the moduli are independent of strain).

A parallel-plate geometry is recommended when a test sample contains particles or has alimited linear-viscoelastic region. The strain can be readily adjusted by altering the gap h or theplate radius R. This is the main advantage parallel-plates have relative to a cone-and-plate (seethe strain equations of Figure 8). The strain can be minimized by increasing the plate gap anddecreasing the plate radius. Of course there is a price to pay and one should be aware that theseactions consequently decrease the torque signal - thus increasing measurement uncertainty.Optimization of the parallel-plate system is an operator intensive trial-and-error procedure. Asalready pointed out, parallel-plates are employed when testing particulate samples. The generalrule when setting the gap is that it should be at least three times greater than the particle diameter.

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3.2 Strain Sweep

A strain sweep is a dynamic test where the moduli are measured as a function of strain at aconstant frequency. The usual aim of such a test is to measure the point where the "stiffness" or"strength" of the material is initially effected by the amount of strain. If the moduli and phase-shift remain constant as a function of strain, then the material is classified as linear-viscoelasticand the test can be considered to be non-destructive (i.e., the internal structure of the sample hasnot been disrupted). This is similar to Hooke's Law in which the force required to deform anobject is directly proportional to the strain. When the strain is too great, the proportionalityfactor decreases with increasing strain and the object behaves non-linearly.

A simple example to illustrate the above is to consider the stretching of a rubber band. If theapplied strain is small, then the force necessary to stretch it will remain proportional to thedeformation. If the rubber band is stretched too far, it will lose its strength due to the destructionof crosslinks until it finally breaks.

The strain region where the moduli are dependent on the strain is defined as the non-linearviscoelastic region and the test is destructive. By performing a strain sweep, one can thusmeasure the critical strain at which the substance becomes non-linear viscoelastic. Thisinformation can be used towards predicting how stable or strain-resistant a particular product is.As the intermolecular structure is increased (i.e., permanent instead of attractive crosslinks), thelinear viscoelastic region will be extended. A practical example is presented in Figure 9.

Delta (degrees)Complex ModulU81000 r---"-'---'-=-=--=---------=--=-==.:..:..::..;

80

100

80

40

20+Do'l& ~

10.110 L..---..J'---'-'-'-LLlJ..L_---'--'--'-l-l-LLLL---..J'----'--'-'-J...UuJ o

0.01 10Strain

Figure 9. Strain sweep of pudding in order to determine its linear viscoelastic region.

In this case, a strain sweep was performed on pudding with a parallel-plate fixture in order todetermine its linear viscoelastic region and thus the strain range where the pudding retained itselasticity. Common sense leads us to the assumption that pudding is elastic under low strain.This can be demonstrated by placing a spoon in pudding and then nudging it a bit. The result -the spoon will spring back after the strain is released! In fact, children are delighted to show thatpudding can be made to swing. If the spoon is pushed too far, the strain will be too great and thespoon will not spring back. We can conclude in this case that the pudding changes from anelastic to a viscous substance after a critical strain is surpassed. This result makes sense when

12

one recalls that pudding can be made to flow in the mouth. Figure 9 shows that the linearviscoelastic region extends to almost 0.2, at which point the storage modulus G or pudding"rigidity" falls off sharply with increasing strain. This is also reflected in the change of the phaseshift 0 from the elastic region (0 < 45°) to the viscous region (0 > 45°) with increasing strain.These results are desirable since: I) The pudding should remain relatively stiff at zero or lowstrain; and 2) The pudding should flow readily when subjected to high strain.After all, the pudding should feel like a thick liquid as opposed to bubble gum while beingswallowed.

3.3 Frequency Sweep

A frequency sweep is a dynamic test in which the response of a material is measured as afunction of frequency at a constant strain amplitude. This test is normally performed such thatthe strain amplitude lies within the linear viscoelastic region so that the measured properties arestrain independent. A frequency sweep provides a fingerprint of a viscoelastic substance undernon-destructive conditions. As discussed earlier in this report, one obtains direct informationregarding the viscous and elastic behavior of the test material.

Typical results of a frequency sweep performed on a viscoelastic solid are shown in Figure10. The test substance was a hair gel [2].

.M:.::o..::d..::ul.:..i-'.(.:..Pa=')e.- ......::C..::o:.::m:sp..::18..::X_V..::I..::SC:.;O:..:S.:..lt..<.Y-'(.:..P..::as.:..;)loo0~ 1000

100

100

10

j-e-G' -b-e" -+-Eta-I

10110 L....--'--'-'-..J...L.LI..l.L....---'--'-'-..J...L.LI..l.'---'---'.....J....w..LUJ 1

0.1 100w (1/s)

Figure 10. Frequency sweep of a hair gel.

As the word 'gel' implies, hair gel should be highly elastic so that after styling the hair it stays inplace. In order to judge how elastic the hair gel is, a frequency sweep was performed at lowstrain. It is clear from the results in Figure 10 that the elastic component G' (storage modulus) isfar greater than the viscous component G" (loss modulus). Based on this result, one can describethe hair gel as exhibiting extreme elasticity at low strain. This result agrees with common sensesince the gel is intended to hold the hair in place.

Typical results of a frequency sweep performed on a viscoelastic liquid appear in Figure II.The test substance was a bath soap containing collagen which can be considered as a polymersolution (i.e., collagen is the polymer and bath soap is the solvent) [2].

13

ccM..:O:...:d__U_II ..:C..:o:...:m2P:...:le:::x-'-V_18_C_O_8_I--',ty1000F 10

100

10

I-a-- Q' -A-G" -+-Ete:'

10W (118)

Figure 11. Frequency sweep of a bath soap containing collagen.

Collagen is a natural polymer that exhibits elastic effects due to its long-chain nature. It servesno purpose as a cleanser but acts instead as a temporary therapeutic means for making the skinfeel "more elastic" and thus the user "younger and refreshed". Any measurable elasticity in thebath soap is thus due to collagen. The frequency sweep portrayed in Figure 11 reveals that theloss modulus G" is greater at low frequency but that the elastic component G' increases rapidlywith increasing frequency. This behavior is typical of a viscoelastic liquid. In a separate study,the storage modulus G' of various concentrated collagen solutions was directly correlated tomolecular-weight [3].

4. Summary

The intention of this paper was to introduce the theory and practice of forced-oscillation.Background information was provided about the terms viscous, elastic and viscoelastic. Simplemechanical models were then constructed in order to illustrate how viscoelasticity manifests itself.The Maxwell model was presented as a viscoelastic liquid and the Kelvin-Voigt as a viscoelasticsolid. A general approach to forced-oscillation was outlined and terms were matched with themechanical models. This exercise was intended to give the various moduli a more immediate andpractical meaning. This exercise also illustrated the tremendous difference between the Maxwelland Kelvin-Voigt models in the behavior of the moduli G' and G".

An experimental section described various fixture geometries and their advantages. Equationsnecessary for calculating the strain and shear stress as a function of the assigned deformationangle and the measured torque were also derived for three fixture geometries. In order to roundout the paper with a practical ring, real strain and frequency sweeps were presented along withinterpretations of the results.

5. References

[1] H.A. Barnes, J.F. Hutton and K. Walters, "An Introduction to Rheology". ElsevierScience, Amsterdam, 1989.

[2] D.A. Holland, Rheology 91,2, June 1991, 108-1l2.[3] Haake Information Report V91-40E, "Elasticity and Its Effect upon Mold-Filling" 1991.

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