Master Curve Stiffness Properties

8/6/2019 Master Curve Stiffness Properties

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A Generalized Logistic Function todescribe the Master Curve StiffnessProperties of Binder Mastics and Mixtures

Geoffrey M. Rowe, Abatech Inc.Gaylon Baumgardner, Paragon Technical Services

Mark J. Sharrock, Abatech International Ltd.

4545thth

Petersen Asphalt Research ConferencePetersen Asphalt Research ConferenceUniversity of WyomingUniversity of WyomingLaramie, Wyoming, July 14Laramie, Wyoming, July 14 --1616

Generalized logistic

/ 1log( )1(*)log( +++= e E

Richards curve


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Master curve functionsObjectives

Review how robust mastercurve forms are fordifferent material typesMaterials

Polymers Asphalt binders Asphalt mixes

Hot Mix AsphaltMastics and filled systems

Observation different functional forms offer moreflexibility with complex materials

Need for evaluation

Work with various roofing materialsand materials used for dampingindicated that application of somestandard sigmoid functions would notdescribe functional form for materials


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OverviewShifting

Free shifting Gordon and ShawFunctional form shifting

Master curve functional formsCA Sigmoid

MEPDGRichards etc

DiscussionRelevance to materials

Master curves A system of reduced variables to describethe effects of time and temperature on thecomponents of stiffness of visco-elasticmaterials

AlsoThermo-rheological simplicity

Time-temperature superpositionProduces composite plot called mastercurve


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Simple master curveUse of EXCEL spreadsheet to manuallyshift to a reference temperature

Simple master curve

Isotherms

1.0E+03

1.0E+04

1.0E+05

1.0E+06

1.0E+07

1.0E+08

1.0E+09

1.0E- 02 1.0E- 01 1.0E+00 1. 0E+01 1.0E+02 1.0E+03 1. 0E+04

Freq. (Hz)

G * , P a

10 15 20 25 30 35 40 45 50 MC, Tref = 40 C

a(T)

Example asphaltbinder 15 PEN


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Two parts curve and shiftsShift factor relationship ispart of master curvenumerical optimization

Master Curve

1.0E+03

1.0E+04

1.0E+05

1.0E+06

1.0E+07

1.0E+08

1.0E+09

1 .0 E- 02 1 .0 E- 01 1 .0 E+0 0 1. 0E+01 1 .0 E+02 1 .0E+03 1 .0E+0 4

Reduced Freq. (Hz), T ref = 40 C

G * , P a

10 15 20 25 30 35 40 45 50

Shift factors

0.1

1

10

100

1000

10000

0 10 20 30 40 50 60

Temperature, C

S h i f t f a c

t o r , a

( T )

Both curves can be fitted tofunctional forms to describeinter-relationships

Sifting schemes

Shifting schemes improve accuracyEnable assessment of model choiceCan look at error analysis


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Shifting choicesUse a shift not dependent upon a model

Free shifting Gordon and Shaws scheme good for this

Model shiftingShift data using underlying functional modelMakes shift easier when less data available

Assumption is that model form is suitable fordata

Gordon and Shaw Method

Gordon and Shaw methodrelies upon reasonablequality data with sufficientdata points in eachisotherm to make theerror reduction process inoverlapping isothermswork wellGordon and Shaw usedsince good referencesource for computer code


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Master Curve ProductionShifting Techniques (Gordon/Shaw)

Determine an initial estimate of the shift using WLF parameters andstandard constantsRefine the fit by using a pairwise shifting technique and straight linesrepresenting each data setFurther refine the fit using pairwise shifting with a polynomialrepresenting the data being shiftedThe order of the polynomial is an empirical function of the number of data points and the decades of time / frequency covered by theisotherm pairThis gives shift factors for each successive pair, which are summedfrom zero at the lowest temperature to obtain a distribution of shiftswith temperature above the lowestThe shift at T ref is interpolated and subtracted from everytemperatures shift factor, causing T ref to become the origin of theshift factors

Gordon and Shaw After 1 st estimate thepolynomial expression isoptimized using nonlineartechniques1st pairwise shift startsfrom coldest temperatureisothermProcedure is done for bothE and E

Could do on just E*, E(t),G(t), D(t), G* if these areall that is available butdefault is to do on loss andstorage parts of complexmodulus 1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

Log Frequency, rads/sec

L o g

E ' , M P a

All IsothermsShifted 1st PairPoly. fit - 5th Order to 1st Pair

Shift = -1.31

3

40

30

20

10


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Gordon and ShawEach pairwiseshift isdetermined

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

-7.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

Log Frequency, rads/sec

L o g

E ' , M P a

All IsothermsShifted 10 CShifted 20 CShifted 30 CShifted 40 C

Shift =-1.31

3

40

30

2010

Shift =-2.35-1.04=-3.39

Shift =-1.31-1.04=-2.35

Shift =-3.39-1.15=-4.54

Summed pairwise shift for E'

Gordon and Shaw E

Implementation E shift


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Gordon and Shaw E Implementation E shift

Gordon and Shaw stats

+/- 95% confidencelimits (t-statistic) based on Gordonand Shaw book Gives values forboth E, E and

averageComparison of shiftfactors also plotted


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Gordon and Shaw shifts factorsIf shift factorsare verydifferent for E and E thenshifting maynot haveworked verywellMaybe need toconsider someother type of

shifting

Model shifting

Shifting to underlying modelIf material behavior is known, it canassist the shift by assumption of underlying model

Why would I do this?Example EXCEL solver used to giveshift parameters


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Model shiftingWhy?

If data is limited to extent thatGordon and Shaw will not work orvisual technique is difficultFor example mixture data collectedas part of MEPDG does not havesufficient data on isotherms to allow

Gordon and Shaw to work well in allinstances 4 to 5 points per decadeis best

Typical mix data

Example mixdata setcollected forMEPDGanalysisNote on logscale data has

non-equalgaps with onlytwo points perdecade


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Model fitModel shift providesthe result to be usedin a specific analysis

Models

Why we needed to consider differentmodels?

Working with some complex materialswe noted that the symmetric sigmoiddoes not provide a good fit of the data

We then started a look at other fittingschemes


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Complex materials Asphalt materials can be formulatedwhich have complex master curves

Roofing compoundsThin surfacing materialsDamping materialsJointing/adhesive compounds

HMA with modified binders

Example thin surfacing on PCC

Material mixedwithaggregate andused as a thinsurfacingmaterial onconcretebridge decks


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Models on these productsOn the three previous examples itwas observed that the master curveis not a represented by a symmetricsigmoid or CA style master curveNeed to consider something else!

Christensen-Anderson

CA, CAMIdea originally developed byChristensen and published in AAPT(1992)Work describes binder master curveand works well for non-modifiedbinders


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Asphalt binder models, SHRPChristensen-Anderson -CA model (1993)

Relates G*( ) to G g, cand R Model for phase angleModel works well fornon-modified bindersModel is similar for G(t)or S(t) formatRelates to a visco-elasticliquid whereas materialsshown in previous slidesshow more solid typebehavior

Sigmoid - logistic

Standard logistic (Verhulst,1838)

Originally developed by aBelgium mathematicianUsed in MEPDGHas symmetrical properties

Applied to a wide variety of problems

Pierre Franois Verhulst


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Mix models - Witczak Basic sigmoidfunctionBasis of Witczak model for asphaltmixture E* dataParametersintroduced to move

sigmoid to typicalasphalt mixproperties

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-6 -4 -2 0 2 4 6

x

y

)1(1

xe y

+

=

Witczak modelWitczak model parametersdefine the ordinates of thetwo asymptotes and thecentral/inflection point of thesigmoid, as follows:

10 = lower asymptote10(+ ) = upper asymptote10(/)= inflection point

Empirical relationships existto estimate and Model is limited in shape to asymmetrical sigmoidSigmoid has characteristicsof a visco-elastic solid

log( *) ( log ) E e t r = +

++

1


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Other modelsStandard logistic will not work for all asphaltmaterials - what other choices do we have?

CASChristensen-Anderson modified by Sharrock

Allows variation in the glassy modulus useful forfilled systems below a critical amount of filler wherethe liquid phase is still dominant. Have used forroofing materials and mastics.

Gompertz (1825)Works well for highly filled/modified systems. Filledmodified joint materials and sealants.

Richards model (1959) Allows a non-symmetrical model format. Gives abetter fit for some jointing compounds and hot-mix-asphalt.

Weibull (1939) Allows non-symmetric behavior Added as an additional method

Note these are being used to describe the shape of master curve

+ d

+=

+

F

E

D x

eC B A E *)log(

Sigmoid - generalized logistic

Generalized logistic(Richards, 1959)

Introduces an extraparameter to allow non-symmetrical slopeParameter introduced thatallows inflection point tovary

Analysis also yields Standard logistic (as usedin MEPDG) and Gompertz(as special case) whenappropriate by data


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Example thin surfacing on PCC

7.0

7.5

8.0

8.5

9.0

9.5

-9.0 -7.0 -5.0 -3.0 -1.0 1.0 3.0

Log Frequency (Reduced at 16 o C)

L o g

G *

Logistic SigmoidGompertz SigmoidWeibull SigmoidIsothermsG* - Shifted

0.0E+00

2.0E+08

4.0E+08

6.0E+08

8.0E+08

1.0E+09

1.2E+09

1.4E+09

1.6E+09

1.8E+09

2.0E+09

1.0E-09 1.0E-07 1.0E-05 1.0E-03 1.0E-01 1.0E+01 1.0E+03

Frequency (Reduced at 16 oC)

G *

Logistic Sigmoid

Gompertz Sigmoid

Weibull Sigmoid

Isotherms

G* - Shifted

Best fit is Gompertz

G*=Pa

Example roofing product

3.00

3.50

4.00

4.50

5.00

5.50

6.00

6.50

7.00

7.50

8.00

-12.00 -10.00 -8.00 -6.00 -4.00 -2.00 0.00 2.00

Log Frequency (Reduced at 16 oC)

L o g

G *


0.00E+00

1.00E+07

2.00E+07

3.00E+07

4.00E+07

5.00E+07

6.00E+07

7.00E+07

1.00E-11 1.00E-09 1.00E-07 1.00E-05 1.00E-03 1.00E-01 1.00E+01 1.00E+03

Frequency (Reduced at 16oC)

G *


Low stiffness=Weibull, high stiffness=Gompertz, best fit=Gompertz

G*=Pa


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Example adhesive product

6.00

6.50

7.00

7.50

8.00

8.50

9.00

9.50

-12 .00 -10 .00 -8 .00 -6.00 -4.00 -2.00 0 .00 2 .00 4 .00Log Frequency (Reduced at 16 oC)

L o g

G *

Logistic SigmoidRichard's SigmoidGompertz SigmoidWeibull SigmoidIsothermsG* - Shifted

6.00E+00

2.00E+08

4.00E+08

6.00E+08

8.00E+08

1.00E+09

1.20E+09

- 12. 00 -1 0. 00 - 8. 00 - 6. 00 - 4. 00 - 2. 00 0. 00 2 .00 4 .0 0

Log Frequency (Reduced at 16 oC)

L o g

G *

Logistic SigmoidRichard's SigmoidGompertz SigmoidWeibull SigmoidIsothermsG* - Shifted

Low stiffness=Gompertz, high stiffness=logistic, best fit=GompertzHigh stiffness appears to have some errors!

G*=Pa

Prony series/D-S fits

In each of the examplesthe data is fitted toProny series relaxationand retardation spectrawith good fitsIf data is extended byuse of a sound

functional form model,this enables extensionof calculations inregions not tested bythe rheologymeasurements


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Generalized logisticGeneralized logistic curve(Richards) allows use of non-symmetrical slopesIntroduction of additionalparameter T

When T = 1 equationbecomes standardlogisticWhen T tends to 0 then equation becomesGompertzT must be positive foranalysis of mixturessince negative values will

not have asymptote andproduces unsatisfactoryinflection in curveMinimum value of inflection occurs at 1/e or 36.8% of relativeheight

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-6 -4 -2 0 2 4 6

x

y T=-0.5T=0.0 GompertzT=0.6T=1.0 L ogisticT=2.0

T M x BTe y / 1)(( )1(

1

+=

B=M=1

Minimum inflection

Standard logistic inflection

Typical range in inflection values

HMA Standard vs. Generalized

Based on standard (logistic) sigmoid thegeneralized sigmoid formats are:-

/ 1log( )1(*)log( ++

+=e

E )log(1

*)log(

+++=

e E

)(log1*)log( M Be

A D E

++= T M BT

A D E

/ 1))(logexp(1(*)log(

++=

Standard logistic Generalized logisticMEPDG FORMAT

ALTERNATE FORM AT

=D, =A, = BM = -B, =T

= lower asymptote+ = upper asymptote

(/) = inflection point/frequency


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1

1.5

2

2.5

3

3.5

4

4.5

5

-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

x

y

Generalized logistic example

Lower asymptoteEquilibrium modulus = 98 MPa

Upper asymptote =Equilibrium modulus = 22.3 GPa

Extension to HMA mixturesDo the generalized sigmoid enable a betterdefinition of HMA mixesLook at ALF mixtures


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Generalized logistic example

1

1.5

2

2.5

3

3.5

4

4.5

5

-15 - 14 -13 - 12 - 11 - 10 -9 - 8 -7 - 6 -5 - 4 -3 - 2 -1 0 1 2 3 4 5 6 7 8 9 1 0

x

y

Model fit

Compare errorsfrom differentmodel fits to assistwith determinationof correct form of

shifting

ALF1 AZCR 70-22

Equilibrium modulus

Equilibrium modulus


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Model fitDifferentmodifiers mayneed differentmodels todefine mixbehavior

ALF7PG70-22 + Fibers

Equilibrium modulus

Equilibrium modulus

Error

Need to develop betterway of consideringerrors since most of error tends to occur atlimits

rms% values tend to below because of adequatefit for large amount of central data 0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

-10. 00 -8. 00 -6.00 -4. 00 -2. 00 0.00 2. 00

log frequency

e r r o r

%


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Applied to ALF studyWhen methods appliedto ALF data only onedata set previousexample is close tosymmetrical standardlogisticMost data sets are betterrepresented by Richardsor Gompertz (specialcase of Richards - threeexamples)

In most cases inflectionpoint is lower than(Gglassy + G equilbrium )/2

Data qualityMore recent testing on mastercurves for mixes enables moredata points to be collected andwith better data quality furtherassessment of models can beconsideredNumber of testpoints/isotherm in presentMEPDG scheme is limitedresulting in numericalproblems in some shiftingschemesNeed in many cases toassume model as part of shiftdevelopment

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E-01 1.0E+00 1.0E+01 1.0E+02

Frequency, Hz

E ' o r

E " , M P a

E ' E "


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Objective of better modelsLeads to better calculations

Spectra calculations and interconversionsBetter definition of low stiffness and highstiffness properties are critical if consideringpavement performanceWork with damping calculationsWork looking at obtaining binder properties frommix dataPhase angle interrelationships

Considerable evidence that we should be using anon-symmetrical sigmoid function

Other non-symmetrical models


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Francken and Verstaeten, 1974

Non-symmetricalsigmoid modelE*=E R*(f R )

R* - sigmoidfunction variesbetween 0 and 1

Bahia et al., 2001

NCHRP-Report 459 Bahia et al.


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SummaryStandard logistic symmetric sigmoidal

Data on more complex materials clearly does not conformObvious when looking at phase angle data versus reduced frequencyFor HMA same conclusion when apply free shifting

A few cases the generalized logistic gave a result close to the standardlogistic

Standard binder uses a non-symmetric function to describebehavior CA model this aspect is missing with the standardlogistic in HMA model

Generalized logistic non-symmetric sigmoidProvides a more comprehensive analysis toolBuilds on work of Fancken et al. and Bahia et al.Parameter introduced to allow variation of inflection point

Anticipated to become more important with more complex modifiedbindersIn most cases considered for HMA the inflection point is below themean of the Glassy and Equilibrium modulus valuesGeneralized logistic reduces to Gompertz at lower acceptable value of T or

Master Curve Stiffness Properties

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