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
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
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
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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
Weibull DistributionProbability Density Function (PDF) Cumulative Density Function (CDF)
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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
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.
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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)
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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
(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
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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
Introduction to Shannon Entropy
Z = (𝑟𝑟𝑝𝑝/A)(W/η)𝑒𝑒− 𝑥𝑥/η β𝛽𝛽/η𝑍𝑍0 = W𝑟𝑟𝑝𝑝𝛽𝛽/A𝜂𝜂2
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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
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.
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Shannon Entropy
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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
Shannon Entropy
A unique number for each distribution (SCB sample)depending on β and 𝜆𝜆.
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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
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
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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
SCB Testing and Analysis
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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
SCB Testing and Analysis
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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
SCB Testing and Analysis
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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
Validating the New Methodology
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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%
Validating the New Methodology
Binder Content
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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%
Validating the New Methodology
Binder Content
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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%
Validating the New Methodology
RAP Content
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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
Validating the New Methodology
RAP Content
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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
Validating the New Methodology
RAP Content
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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
Validating the New Methodology
NCAT ALF Mix
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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
Validating the New Methodology
NCAT ALF Mix
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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
Validating the New Methodology
NCAT ALF Mix
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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
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
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.
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THANK YOU!
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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
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.
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What is Next?
Beam Fatigue
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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
What is Next?
Texas Overlay
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0
100
200
300
400
500
600
700
800
0 50 100 150 200 250
Max
Loa
d/C
ycle
(lb)
Cycle No.
RawWeibull
What is Next?
Cyclic tension (Pull-Pull)
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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
What is Next?
Flow Number
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0
10000
20000
30000
40000
50000
60000
0 200 400 600 800 1000 1200 1400
Mic
roSt
rain
Cycle No.
Raw
Weibull
What is Next?
HWTD
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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
What is Next?
TSRST
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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
What is Next?
G* at multiple frequencies
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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
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
What is Next?
BBR
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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