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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
λωγβλαδ /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 model“Free shifting”Gordon and Shaw’s 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 temperature’s 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 E”Could 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 E’and 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
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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
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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
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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
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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
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Asphalt binder models, SHRPChristensen-Anderson -CA model (1993)
Relates G*(ω) to Gg, ωcand 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 François Verhulst
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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:
10δ = lower asymptote10(δ+α) = upper asymptote10(β/γ) = inflection point
Empirical relationships exist to estimate δ α β and γModel is limited in shape to a symmetrical sigmoidSigmoid has characteristics of a visco-elastic solid
log( *) (log )Ee tr
= ++ +δ
αβ γ1
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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 materials and sealants.
Richards model (1959)Allows a non-symmetrical model format. Gives a better fit for some jointing compounds and hot-mix-asphalt.
Weibull (1939)Allows non-symmetric behaviorAdded as an additional method
Note – these are being used to describe the shape of master curve
+ d
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡−+=
⎟⎠⎞
⎜⎝⎛ +
−F
EDx
eCBAE *)log(
Sigmoid - generalized logistic
Generalized logistic (Richards, 1959)
Introduces an extra parameter to allow non-symmetrical slopeParameter introduced that allows inflection point to varyAnalysis also yields –Standard logistic (as used in MEPDG) and Gompertz (as special case) when appropriate 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 16oC)
Log
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 16oC)
G*
Logistic SigmoidGompertz SigmoidWeibull SigmoidIsothermsG* - 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 16oC)
Log
G*
Logistic SigmoidGompertz SigmoidWeibull SigmoidIsothermsG* - Shifted
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*
Logistic SigmoidGompertz SigmoidWeibull SigmoidIsothermsG* - Shifted
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 16oC)
Log
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 -10.00 -8.00 -6.00 -4.00 -2.00 0.00 2.00 4.00
Log Frequency (Reduced at 16oC)
Log
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 examples the data is fitted to Prony series – relaxation and retardation spectra with good fitsIf data is extended by use of a “sound”functional form model, this enables extension of calculations in regions not tested by the rheology measurements
21
Generalized logisticGeneralized logistic curve (Richard’s) allows use of non-symmetrical slopesIntroduction of additional parameter T
When T = 1 equation becomes standard logisticWhen T tends to 0 –then equation becomes GompertzT must be positive for analysis of mixtures since negative values will not have asymptote and produces unsatisfactory inflection in curveMinimum value of inflection occurs at 1/e –or 36.8% of relative height
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 LogisticT=2.0
TMxBTey /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 the generalized sigmoid formats are:-
λωγβλαδ /1log( )1(
*)log( +++=
eE
)log(1*)log( ωγβ
αδ +++=
eE
)(log1*)log( MBe
ADE −−++= ω TMBT
ADE /1))(logexp(1(*)log(
−ω−++=
Standard logistic Generalized logisticMEPDG FORMAT
ALTERNATE FORMAT
δ=D, α=A, β= BMγ= -B, λ=T
δ = lower asymptoteδ+α = upper asymptote
−(β/γ) = 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 better definition of HMA mixesLook at ALF mixtures
23
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 10
x
y
Model fit
Compare errors from different model fits to assist with determination of correct form of shifting
ALF1AZCR 70-22
Equilibrium modulus
Equilibrium modulus
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Model fit
Different modifiers may need different models to define mix behavior
ALF7PG70-22 + Fibers
Equilibrium modulus
Equilibrium modulus
Error
Need to develop better way of considering errors since most of error tends to occur at limitsrms% values tend to be low because of adequate fit 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
erro
r %
25
Applied to ALF study
When methods applied to ALF data only one data set – previous example – is close to symmetrical – standard logisticMost data sets are better represented by Richards or Gompertz (special case of Richards - three examples)In most cases inflection point is lower than (Gglassy + Gequilbrium)/2
Data quality
More recent testing on master curves for mixes enables more data points to be collected and with better data quality further assessment of models can be consideredNumber of test points/isotherm in present MEPDG scheme is limited resulting in numerical problems in some shifting schemesNeed in many cases to assume model as part of shift development
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",
MPa
E' E"
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Objective of better modelsLeads to better calculations
Spectra calculations and interconversionsBetter definition of low stiffness and high stiffness properties are critical if considering pavement performanceWork with damping calculationsWork looking at obtaining binder properties from mix dataPhase angle interrelationshipsConsiderable evidence that we should be using a non-symmetrical sigmoid function
Other non-symmetrical models
27
Francken and Verstaeten, 1974
Non-symmetrical sigmoid modelE*=E∞ × R*(fR)
R* - sigmoid function – varies between 0 and 1
Bahia et al., 2001
NCHRP-Report 459 – Bahia et al.
28
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 standard logistic
Standard binder – uses a non-symmetric function to describe behavior – CA model – this aspect is missing with the standard logistic 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 pointAnticipated to become more important with more complex modified bindersIn most cases considered for HMA the inflection point is below the mean of the Glassy and Equilibrium modulus valuesGeneralized logistic reduces to Gompertz at lower acceptable value of T or γ