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1

A Generalized Logistic Function to describe the Master Curve Stiffness Properties 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 Wyoming

Laramie, Wyoming, July 14Laramie, Wyoming, July 14--1616

Generalized logistic

/1log( )1(

*)log( +++=

eE

Richards curve

2

Master curve functions

ObjectivesReview how robust mastercurve forms are for different material typesMaterials

PolymersAsphalt bindersAsphalt mixes

Hot Mix AsphaltMastics and filled systems

Observation different functional forms offer more flexibility with complex materials

Need for evaluation

Work with various roofing materials and materials used for damping indicated that application of some standard sigmoid functions would not describe functional form for materials

3

OverviewShifting

Free shifting Gordon and ShawFunctional form shifting

Master curve functional formsCASigmoid

MEPDGRichards etc

DiscussionRelevance to materials

Master curvesA system of reduced variables to describe the effects of time and temperature on the components of stiffness of visco-elastic materials Also

Thermo-rheological simplicityTime-temperature superposition

Produces composite plot called master curve

4

Simple master curve

Use of EXCEL spreadsheet to manually shift 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*,

Pa

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

a(T)

Example asphalt binder 15 PEN

5

Two parts curve and shifts

Shift factor relationship is part of master curve numerical optimization

Master Curve

1.0E+03

1.0E+04

1.0E+05

1.0E+06

1.0E+07

1.0E+08

1.0E+09

1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04

Reduced Freq. (Hz), Tref = 40 C

G*,

Pa

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

Shift

fact

or, a

(T)

Both curves can be fitted to functional forms to describe inter-relationships

Sifting schemes

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

6

Shifting choices

Use a shift not dependent upon a modelFree shiftingGordon and Shaws scheme good for this

Model shiftingShift data using underlying functional modelMakes shift easier when less data availableAssumption is that model form is suitable for data

Gordon and Shaw Method

Gordon and Shaw method relies upon reasonable quality data with sufficient data points in each isotherm to make the error reduction process in overlapping isotherms work wellGordon and Shaw used since good reference source for computer code

7

Master Curve ProductionShifting Techniques (Gordon/Shaw)

Determine an initial estimate of the shift using WLF parameters and standard constantsRefine the fit by using a pairwise shifting technique and straight lines representing 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 the isotherm pairThis gives shift factors for each successive pair, which are summed from zero at the lowest temperature to obtain a distribution of shifts with temperature above the lowestThe shift at Tref is interpolated and subtracted from every temperatures shift factor, causing Tref to become the origin of the shift factors

Gordon and ShawAfter 1st estimate the polynomial expression is optimized using nonlinear techniques1st pairwise shift starts from coldest temperature isothermProcedure is done for both E and ECould do on just E*, E(t), G(t), D(t), G* if these are all that is available but default is to do on loss and storage parts of complex modulus 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

Log

E', M

Pa

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

Shift = -1.31

3

40

30

2010

8

Gordon and Shaw

Each pairwise shift is determined

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 3.0Log Frequency, rads/sec

Log

E', M

Pa

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

9

Gordon and Shaw E

Implementation E shift

Gordon and Shaw stats

+/- 95% confidence limits (t-statistic) based on Gordon and Shaw bookGives values for both E, E and averageComparison of shift factors also plotted

10

Gordon and Shaw shifts factors

If shift factors are very different for Eand E then shifting may not have worked very wellMaybe need to consider some other type of shifting

Model shifting

Shifting to underlying model If material behavior is known, it can assist the shift by assumption of underlying model

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

11

Model shifting

Why?If data is limited to extent that Gordon and Shaw will not work or visual technique is difficultFor example mixture data collected as part of MEPDG does not have sufficient data on isotherms to allow Gordon and Shaw to work well in all instances 4 to 5 points per decade is best

Typical mix data

Example mix data set collected for MEPDG analysisNote on log scale data has non-equal gaps with only two points per decade

12

Model fit

Model shift provides the result to be used in a specific analysis

Models

Why we needed to consider different models?

Working with some complex materials we noted that the symmetric sigmoid does not provide a good fit of the dataWe then started a look at other fitting schemes

13

Complex materials

Asphalt materials can be formulated which have complex master curves

Roofing compoundsThin surfacing materialsDamping materialsJointing/adhesive compoundsHMA with modified binders

Example thin surfacing on PCC

Material mixed with aggregate and used as a thin surfacing material on concrete bridge decks

14

Example roofing productRoofing material

8.75 % Radial SBS Polymer61.25 % Vacuum Distilled Asphalt30 % Calcium Carbonate Filler

Master curve considered in range -24 to 75oC this range gives a good fit in linear visco-elastic regionAfter 75oC structure in material starts to change and material is not behaving in a thermo-rheologically simple manner

Example adhesive product

Master curve for a material used for fixing road markers

15

Models on these products

On the three previous examples it was observed that the master curve is not a represented by a symmetric sigmoid or CA style master curveNeed to consider something else!

Christensen-Anderson

CA, CAMIdea originally developed by Christensen and published in AAPT (1992)Work describes binder master curve and works well for non-modified binders

16

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

Relates G*() to Gg, cand RModel for phase angleModel works well for non-modified bindersModel is similar for G(t) or S(t) formatRelates to a visco-elastic liquid whereas materials shown in previous slides show more solid type behavior

Sigmoid - logistic

Standard logistic (Verhulst, 1838)

Originally developed by a Belgium mathematicianUsed in MEPDGHas symmetrical propertiesApplied to a wide variety of problems

Pierre Franois Verhulst

17

Mix models - Witczak

Basic sigmoid functionBasis of Witczak model for asphalt mixture E* dataParameters introduced to move sigmoid to typical asphalt mix properties

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

xey +

=

Witczak modelWitczak model parameters define the ordinates of the two asymptotes and the central/inflection point of the sigmoid, as follows:

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

Empirical relationships exist to estimate and Model is limited in shape to a symmetrical sigmoidSigmoid has characteristics of a visco-elastic solid

log( *) (log )E e tr= +

+ +

1

18

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

CASChristensen-Anderson modified by SharrockAllows variation in the glassy modulus useful for filled systems below a critical amount of filler where the liquid phase is still dominant. Have used for roofing materials and mastics.

Gompertz (1825)Works well for highly filled/modified systems. Filled modified joint mate