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Characterizing Asphalt Mixtures Resistance to Crack Propagation Using the SCB Test: Weibull Distribution and Entropy Approach By: Ahmed Soliman PhD student, UMass Dartmouth Asphalt Mixture & Construction Expert Task Group Fall River, MA May 8 th , 2018
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Page 1: Characterizing Asphalt Mixtures Resistance to Crack ...

Characterizing Asphalt Mixtures Resistance to Crack Propagation Using the SCB Test:

Weibull Distribution and Entropy Approach

By: Ahmed SolimanPhD student, UMass Dartmouth

Asphalt Mixture & Construction Expert Task GroupFall River, MAMay 8th, 2018

Page 2: Characterizing Asphalt Mixtures Resistance to Crack ...

Acknowledgement This work would not have been possible without the help of

Dr. Raymond N. Laoulache.

The main idea of this research came up after a conversationwith Professor Donald Christensen about WeibullDistribution.

I would like also to Thank my advisors:Professor Walaa Mogawer

Professor Donald ChristensenProfessor Ramon Bonaquist

7/20/2018 2

Page 3: Characterizing Asphalt Mixtures Resistance to Crack ...

OutlineIntroduction and Problem Statement Research ObjectivesFitting Weibull Distribution to SCB DataInitial Complex Stiffness Modulus & Shannon EntropyMaterials and Mix DesignSCB Testing and New Approach to Analyze the DataValidating the New ApproachConclusions

7/20/2018 3

Page 4: Characterizing Asphalt Mixtures Resistance to Crack ...

Introduction and Problem Statement

SCB Test at Intermediate temperature

7/20/2018 4

01234567

0 1 2 3 4

Loa

d (K

N)

Load Line Displacement (mm)

𝐽𝐽𝑐𝑐 = (−1𝑏𝑏 )(

𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑) 𝐹𝐹𝐹𝐹 = 𝐴𝐴

𝐺𝐺𝑓𝑓|𝑚𝑚|

𝐽𝐽𝑐𝑐= critical value of the fracture resistance,b= sample thickness, a= the notch depth,U= the strain energy to failure

FI= flexibility Index, A=0.01,𝐺𝐺𝑓𝑓= fracture energy,m= post-peak load slope

Was carried out for elastic and elastic-plasticmaterials with rounded smooth notch.

Viscoelastic materials can be treated as non-linearelastic

Dissipated energy for asphalt mixtures as viscoelasticmaterials is unknown during loading.

Amount of dissipated energy changes by changingtemperature and it is not constant for all mixtures.

Empirical

Page 5: Characterizing Asphalt Mixtures Resistance to Crack ...

Weibull DistributionProbability Density Function (PDF) Cumulative Density Function (CDF)

7/20/2018 5

f x = x/𝜂𝜂 β−1𝑒𝑒− x/𝜂𝜂 β𝛽𝛽/𝜂𝜂 P x = 1 − 𝑒𝑒− x/𝜂𝜂 β

X >0

F(x)

x

β>1β=1β<1

η=1

Area under the curve= 1

0

0.2

0.4

0.6

0.8

1

1.2

P(x)

x

β>1β=1β<1

η=1

X >0

Page 6: Characterizing Asphalt Mixtures Resistance to Crack ...

Research Objectives

1) Fitting Weibull distribution to the relationship betweenload and load line displacement for SCB test.

2) Deriving mathematical equation for initial complexstiffness modulus of asphalt mixtures “Zo”.

3) Deriving mathematical equation for Shannon entropy “H”(A parameter that represents the mechanical behavior ofasphalt mixtures).

4) Develop a new approach to characterize crackpropagation resistance of asphalt mixtures using the SCBtest.

7/20/2018 6

Page 7: Characterizing Asphalt Mixtures Resistance to Crack ...

Fitting Weibull Distribution to SCB Data

F = W 𝑥𝑥/η β−1𝑒𝑒− 𝑥𝑥/η β𝛽𝛽/η

F= load (kN), W= Work of fracture (Joules),x= Load line displacement (mm),β= Shape parameter, η= Scale parameter (mm)

7/20/2018 7

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5

Loa

d (k

N)

Load Line Displacement (mm)

Raw

Weibull Fit

Page 8: Characterizing Asphalt Mixtures Resistance to Crack ...

(𝐹𝐹/A)/(x/𝑟𝑟𝑝𝑝) = (𝑟𝑟𝑝𝑝/A)(W/η)𝑒𝑒− 𝑥𝑥/η β𝛽𝛽/η

Initial Complex Stiffness Modulus

F = W 𝑥𝑥/η β−1𝑒𝑒− 𝑥𝑥/η β𝛽𝛽/η

7/20/2018 8

𝐹𝐹/ 𝑥𝑥 𝛽𝛽−1 = W 1/η β−1𝑒𝑒− 𝑥𝑥/η β𝛽𝛽/ηOut of more than 200 specimens, the average 𝛽𝛽 value was 1.99 and the standard

deviation was 0.24

𝐹𝐹/𝑥𝑥 = (W/η)𝑒𝑒− 𝑥𝑥/η β𝛽𝛽/η

Complex Stiffness Modulus of Asphalt Mixtures under indirect tensile stress (Z)

At x =0, the initial modulus can be written as

𝑍𝑍0 = W𝑟𝑟𝑝𝑝𝛽𝛽/A𝜂𝜂2

https://www.sciencedirect.com/science/article/pii/S0142112316302468

Assume rp=1% Notch

Page 9: Characterizing Asphalt Mixtures Resistance to Crack ...

Introduction to Shannon Entropy

Z = (𝑟𝑟𝑝𝑝/A)(W/η)𝑒𝑒− 𝑥𝑥/η β𝛽𝛽/η𝑍𝑍0 = W𝑟𝑟𝑝𝑝𝛽𝛽/A𝜂𝜂2

7/20/2018 9

𝑍𝑍/𝑍𝑍0 = 𝑒𝑒− 𝑥𝑥/η β

The probability that the material will experience total failure “Percentage of drop in stiffness” at a load line displacement value (𝑥𝑥) can be expressed as:

𝑃𝑃 𝑥𝑥 = 1 − 𝑍𝑍/𝑍𝑍0 = 1 − 𝑒𝑒− 𝑥𝑥/η β

Cumulative Density function of Weibull

Distribution β>10

0.2

0.4

0.6

0.8

1

1.2

P(x)

Load line displacement, mm

β>1

η=1

Page 10: Characterizing Asphalt Mixtures Resistance to Crack ...

Entropy & Shannon EntropyEntropy is a physical property that indicates the

molecular state of a system (a measure of disorder).The mechanical (effective) work of a system is a

function of its entropy and internal energy.In statistical mechanics, Shannon entropy can be used

as an indication to the mechanical behavior of asystem with unknown state.

Physical entropy can be estimated from Shannonentropy by multiplying it by a constant.

7/20/2018 10

Page 11: Characterizing Asphalt Mixtures Resistance to Crack ...

Shannon Entropy

7/20/2018 11

We want to pick a random ball from each box and return it back for four timesP=1 P=0.75 P=0.25 P=0.5 P=0.5

What is the probability that the choice will match what is in the box?P=1*1*1*1= 1 P=0.75*0.75*0.75*0.25= 0.1 P=0.5*0.5*0.5*0.5= 0.06

Probability(Knowledge)

High Medium Low

Shannon Entropy

HighLow Medium

𝑆𝑆𝑆𝑑𝑑𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 Entropy H = −�𝑃𝑃𝑖𝑖𝑙𝑙𝑆𝑆𝑙𝑙𝑏𝑏𝑃𝑃𝑖𝑖𝐻𝐻 = −[1 ∗ 𝑙𝑙𝑆𝑆𝑙𝑙101]= 0 𝐻𝐻 = −[0.75 ∗ 𝑙𝑙𝑆𝑆𝑙𝑙100.75+

0.25*𝑙𝑙𝑆𝑆𝑙𝑙100.25]= 0.244𝐻𝐻 = −[0. 5 ∗ 𝑙𝑙𝑆𝑆𝑙𝑙100.5+

0.5*𝑙𝑙𝑆𝑆𝑙𝑙100.5]= 0.3

Page 12: Characterizing Asphalt Mixtures Resistance to Crack ...

Shannon Entropy

A unique number for each distribution (SCB sample)depending on β and 𝜆𝜆.

7/20/2018 12

𝐻𝐻 = −�−∞

∞𝑓𝑓 �𝑥𝑥 𝑙𝑙𝑆𝑆 𝑓𝑓 �𝑥𝑥 𝑑𝑑 �𝑥𝑥

= 𝛾𝛾 1 − 1𝛽𝛽

+ ln 𝜆𝜆𝛽𝛽

+ 1Euler-Mascheroni constant (0.577)

Properties of Shannon Entropy:

This unique number can be used as an indication to themechanical properties (viscoelasticity) of mixtures.

𝜆𝜆 = 𝜂𝜂/𝑟𝑟𝑝𝑝

�𝑥𝑥 = 𝑥𝑥/𝑟𝑟𝑝𝑝𝑓𝑓 �𝑥𝑥 = 𝑟𝑟𝑝𝑝f �𝑥𝑥

Page 13: Characterizing Asphalt Mixtures Resistance to Crack ...

Materials and Mix Design

Sieve Size (mm) 12.5 9.5 4.75 2.36 1.18 0.6 0.3 0.15 0.075 Binder Content (%)

% Passing by Weight 100 98 85 58 42 27 15 9 6 6.5

7/20/2018 13

Incorporated BinderTemperature ºC

-15 -5 5 10 15 20 25 30 35 40 45

PG64-22 (2 Sources)

PG 58-28 (2 Sources)

PG 64-28

HiMA

Formulated PG58-28

Page 14: Characterizing Asphalt Mixtures Resistance to Crack ...

SCB Testing and Analysis

7/20/2018 14

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

-20 0 20 40 60

Com

plex

Initi

al S

tiffn

ess m

odul

us Z

o (N

/mm

2)

Temperature °C

PG58-28 Source A

PG58-28 Source B

PG64-22 Source B

PG64-22 Source C

PG64-28

HiMA

Formulated PG58-28

Max R² =0.99Min R² = 0.96

It is necessary to change testing temperature based on the used binder

This plot helps to chose appropriate mixture based on the placement region

Page 15: Characterizing Asphalt Mixtures Resistance to Crack ...

SCB Testing and Analysis

7/20/2018 15

2.4

2.6

2.8

3

3.2

3.4

3.6

-20 0 20 40 60

Shan

non

Ent

ropy

-H

Temperature °C

PG58-28 Source A

PG58-28 Source B

PG64-22 Source B

PG64-22 Source C

PG64-28

HiMA

Formulated PG58-28Max R² =0.96Min R² = 0.81

Failure due to indirect tension

Failure dueto shearThis plot helps to chose appropriate

mixture based on the placement region

Truncate data points beyond the peak Shannon Entropy

Page 16: Characterizing Asphalt Mixtures Resistance to Crack ...

SCB Testing and Analysis

7/20/2018 16

0.00

0.20

0.40

0.60

0.80

1.00

1.20

2.4 2.6 2.8 3 3.2 3.4 3.6

Com

plex

Initi

al S

tiffn

ess m

odul

us Z

o (N

/mm

2)

Shannon Entropy-H

PG58-28 Source A

PG58-28 Source B

PG64-22 Source B

PG64-22 Source C

PG64-28

HiMA

Formulated PG58-28

Max R² =0.97Min R² = 0.88

Page 17: Characterizing Asphalt Mixtures Resistance to Crack ...

Validating the New Methodology

7/20/2018 17

Binder Content

R² = 0.9694

R² = 0.9667

R² = 0.9287

R² = 0.9651

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0 10 20 30 40 50 60

Zo (N

/mm

2)

Temperature °C

PG64-22 Source C

PG64-22 Source C-0.4%

PG64-22 SourceC+0.4%

PG64-22 Source C+1%

Page 18: Characterizing Asphalt Mixtures Resistance to Crack ...

Validating the New Methodology

Binder Content

7/20/2018 18

R² = 0.9196

R² = 0.9421

R² = 0.9587

R² = 0.9729

2.4

2.6

2.8

3

3.2

3.4

3.6

0 10 20 30 40 50 60

Shan

non

Ent

ropy

-H

Temperature °C

PG64-22 Source C

PG64-22 Source C-0.4%

PG64-22 Source C+0.4%

PG64-22 Source C+1%

Page 19: Characterizing Asphalt Mixtures Resistance to Crack ...

Validating the New Methodology

Binder Content

7/20/2018 19

R² = 0.8849

R² = 0.9383

R² = 0.9398

R² = 0.945

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

2.4 2.6 2.8 3 3.2 3.4 3.6

Zo (N

/mm

2)

Shannon Entropy-H

PG64-22 Source C

PG64-22 Source C-0.4%

PG64-22 SourceC+0.4%

PG64-22 Source C+1%

Page 20: Characterizing Asphalt Mixtures Resistance to Crack ...

Validating the New Methodology

RAP Content

7/20/2018 20

R² = 0.9694

R² = 0.9151

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 10 20 30 40 50 60

Zo (N

/mm

2)

Temperature °C

PG64-22 Source C

PG64-22 Source C+50%RAP

Page 21: Characterizing Asphalt Mixtures Resistance to Crack ...

Validating the New Methodology

RAP Content

7/20/2018 21

R² = 0.9196

R² = 0.9368

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

0 10 20 30 40 50 60

Shan

non

Ent

ropy

-H

Temperature °C

PG64-22 Source C

PG64-22 Source C+50%RAP

Page 22: Characterizing Asphalt Mixtures Resistance to Crack ...

Validating the New Methodology

RAP Content

7/20/2018 22

R² = 0.8849

R² = 0.897

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

Zo (N

/mm

2)

Shannon Entropy-H

PG64-22 Source C

PG64-22 Source C+50%RAP

Page 23: Characterizing Asphalt Mixtures Resistance to Crack ...

Validating the New Methodology

NCAT ALF Mix

7/20/2018 23

R² = 0.8454

R² = 0.8081

R² = 0.8815

R² = 0.8699

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 10 20 30 40 50

Zo (N

/mm

2)

Temperature °C

GTR

RAP

RAP/RAS

SBS

Page 24: Characterizing Asphalt Mixtures Resistance to Crack ...

Validating the New Methodology

NCAT ALF Mix

7/20/2018 24

R² = 0.9019 R² = 0.7234

R² = 0.7671

R² = 0.7433

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

0 10 20 30 40 50

Shan

non

Ent

ropy

-H

Temperature °C

GTR

RAP

RAP/RAS

SBS

Page 25: Characterizing Asphalt Mixtures Resistance to Crack ...

Validating the New Methodology

NCAT ALF Mix

7/20/2018 25

R² = 0.9478

R² = 0.9247

R² = 0.9267 R² = 0.8782

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.4 1.6 1.8 2 2.2 2.4

Zo (N

/mm

2)

Shannon Entropy-H

GTR

RAP

RAP/RAS

SBS

Page 26: Characterizing Asphalt Mixtures Resistance to Crack ...

Validating the New Methodology

7/20/2018 26

NCAT ALF Mix

Laboratory and Field Evaluation of Florida Mixtures at the 2012 National Center for Asphalt Technology Pavement Test TrackJ. Richard Willis, Adam J. Taylor, and Tanya M. Nash

SBS

GTR

RAP

RAP/RAS

Page 27: Characterizing Asphalt Mixtures Resistance to Crack ...

Conclusions

Weibull distribution can be used to fit crack propagation data from theSCB test.

The initial complex stiffness modulus and Shannon entropy (ameasure of the mechanical behavior) can be derived from the Weibullfitted distributions.

Correlations between testing temperature and initial complex stiffnessmodulus and Shannon entropy are useful to choose appropriatemixture based on the placement region.

Based on this study, asphalt mixtures should be compared at the samestate (Shannon entropy value), or at the same initial complex stiffnessmodulus. This might require testing at multiple temperatures.

7/20/2018 27

Page 28: Characterizing Asphalt Mixtures Resistance to Crack ...

THANK YOU!

7/20/2018 28

Page 29: Characterizing Asphalt Mixtures Resistance to Crack ...

SCB Testing and Analysis

7/20/2018 29

0

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6

f(x)

Load-line displacement(x)-mm

25C

35C

45C

55C

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6P(

x)

Load-line displacement(x)-mm

25C

35C

45C

55C

Page 30: Characterizing Asphalt Mixtures Resistance to Crack ...

What is Next?

Initial complex stiffness modulus and Shannonentropy can be predicted for other mixture and bindertests using similar approach.

Initial complex stiffness modulus from different testscan be correlated using basics mechanics of materials.

Shannon entropy from different tests might becorrelated.

A master curve can be developed from different testswith different modes of failures and used in pavementdesign.

7/20/2018 30

Page 31: Characterizing Asphalt Mixtures Resistance to Crack ...

What is Next?

Beam Fatigue

7/20/2018 31

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 100,000 200,000 300,000 400,000 500,000 600,000

Flex

ural

Stif

fnes

s (N

/mm

2)

Cycle No.

Raw

Weibull

Page 32: Characterizing Asphalt Mixtures Resistance to Crack ...

What is Next?

Texas Overlay

7/20/2018 32

0

100

200

300

400

500

600

700

800

0 50 100 150 200 250

Max

Loa

d/C

ycle

(lb)

Cycle No.

RawWeibull

Page 33: Characterizing Asphalt Mixtures Resistance to Crack ...

What is Next?

Cyclic tension (Pull-Pull)

7/20/2018 33

0

50

100

150

200

250

300

350

400

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Stre

ss (k

Pa)

Displacement (mm)

Raw

Weibull

Page 34: Characterizing Asphalt Mixtures Resistance to Crack ...

What is Next?

Flow Number

7/20/2018 34

0

10000

20000

30000

40000

50000

60000

0 200 400 600 800 1000 1200 1400

Mic

roSt

rain

Cycle No.

Raw

Weibull

Page 35: Characterizing Asphalt Mixtures Resistance to Crack ...

What is Next?

HWTD

7/20/2018 35

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5000 10000 15000 20000

Rut

dep

th (m

m)

No. Of Passes

Raw

Weibull

Page 36: Characterizing Asphalt Mixtures Resistance to Crack ...

What is Next?

TSRST

7/20/2018 36

0

100

200

300

400

500

600

700

800

0 5 10 15 20 25 30 35

Loa

d

Temperature-C

Raw

Weibull

Results shifted by 25.8°C

Page 37: Characterizing Asphalt Mixtures Resistance to Crack ...

What is Next?

G* at multiple frequencies

7/20/2018 37

0.E+00

5.E+06

1.E+07

2.E+07

2.E+07

3.E+07

3.E+07

0 20 40 60 80 100 120

G*

(Pa)

Frequency (rad/sec)

Raw

G*

10°C

Page 38: Characterizing Asphalt Mixtures Resistance to Crack ...

What is Next?

LAS

7/20/2018 38

0.E+00

1.E+07

2.E+07

3.E+07

4.E+07

5.E+07

0 5 10 15 20 25 30 35

G*

(Pa)

Strain (%)

Raw

Weibull

Page 39: Characterizing Asphalt Mixtures Resistance to Crack ...

What is Next?

BBR

7/20/2018 39

0200400600800

1000

0 50 100 150 200 250 300

Stiff

ness

-S

Time-S

Raw

Weibull

0

0.1

0.2

0.3

0.4

0 50 100 150 200 250 300

Slop

e-m

Time-S

Raw

Weibull


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