Richards curve - Abatech Curve.pdf · describe the Master Curve Stiffness ... Richards curve. 2...

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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. 45 45 th th Petersen Asphalt Research Conference Petersen Asphalt Research Conference University of Wyoming University of Wyoming Laramie, Wyoming, July 14 Laramie, Wyoming, July 14- 16 16 Generalized logistic λ ω γ β λ α δ / 1 log ( ) 1 ( *) log( + + + = e E Richards curve

Transcript of Richards curve - Abatech Curve.pdf · describe the Master Curve Stiffness ... Richards curve. 2...

Page 1: Richards curve - Abatech Curve.pdf · describe the Master Curve Stiffness ... Richards curve. 2 Master curve functions ... sigmoid or CA style master curve

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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 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 %

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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

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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.

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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 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