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122
APPENDIX E
Construction of Characteristic Curve
Determination of Once the relaxation modulus is obtained from the storage
the Relaxation Modulus modulus, the Prony series coefficients in Equation 1 are
obtained through the following steps.
Construction of the damage characteristic curve requires First, Equation 1 is rewritten in matrix form as:
the calculation of pseudo strains, which in turn require the
relaxation modulus, E(t), of the mixture. The relaxation { A} = [ B ]{C}
modulus is difficult to measure directly in the laboratory and
is therefore determined from the dynamic modulus master A j = E ( t j ) - E
curve using linear viscoelastic theory. The relaxation modulus tj
is expressed as the Prony series: B jk = exp - (4)
k
t Cj = Ej
E ( t ) = E + En exp - (1)
n
n
j = 1, 2, . . . , N
Where E is the value of E(t) as t, En are Prony series co- Where the relaxation times tj are chosen at decade intervals
efficients and n are relaxation times. To obtain the Prony along time axis and N is the number of data points used. The
series expression of relaxation modulus from dynamic mod- MATLAB optimization toolbox is used to obtain the solution
ulus, the first step is to determine the storage modulus master for Ej with the following constraint:
curve using:
1
E ( r ) = E ( r ) cos ( ( r )) (2)
min
C 2
[ B ]{C} - { A} 2
2
, such that, {C } 0 (5)
Where |E(r)| is the storage modulus, is the phase angle which provides positive values of Ej. The prony series repre-
and r is the reduced frequency in rad/sec. The relaxa- sentations of the relaxation modulus for the four mixtures are
tion modulus curve of each specimen is obtained from the shown in Figure E1.
storage modulus master curve by applying the following
relation: Monotonic Characteristic Curves
1 0.08 Pseudo Strain Calculation
E (t r ) = E ( r ), r =
tr Monotonic tests (constant crosshead) at 20°C are per-
n formed on specimens at various crosshead strain rates. The
= (1 - n ) cos (3) pseudo strains are calculated using the strains measured from
2
the on-specimen LVDTs. Figure E2 shows the typical stress,
d log E ( ) crosshead strain, and LVDT strain as a function of time for a
n=
d log monotonic test.
The pseudo strain is defined as:
Where tr is the reduced time, is the gamma function and
n is the slope of log(E()) versus log() curve which is 1 t d
R (t ) = E ( t - ) d (6)
obtained at each point of reduced frequency. ER 0 d
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1.0E+05
67-22, Opt
Relaxation Modulus, MPa
76-22, Opt
67-22, Opt+
1.0E+03
76-22, Opt+
1.0E+01
1.0E-01
1.0E-09 1.0E-05 1.0E-01 1.0E+03 1.0E+07
Time, sec
Figure E1. Relaxation modulus curves for all four mixtures.
where ER is the reference modulus (which is chosen as unity), where (t) is the stress history. The damage parameter, S, is
E(t) is the relaxation modulus obtained from storage modulus obtained from the following equation:
and expressed as Prony series and is the on-specimen LVDT
strain observed under monotonic tests. The above integration 1 1+ 1
S1( t ) = ( R , j ) (C j -1 - C j ) ( t j - t j -1 )1+
2
(9)
j 2
is evaluated numerically over the strain range up to the time
of failure. The strain history is discritized into N number of
small segments with time increment t and Equation 1 is 1
where I is the initial pseudo stiffness and = 1 + , where m
substituted in Equation 6, resulting in the following form of m
numerical integration scheme: is the slope of the linear portion of the relaxation curve. The
damage characteristic curve is obtained by plotting the dam-
N +1
age parameter versus the pseudo stiffness.
R ( t ) = ci E (u ) t -t jj -1 , t 0 = 0, t N +1 = t
t -t
j =1
(7)
u Fatigue Characteristic Curves
E (u ) = Eu - E j j exp -
j j The following steps are used to construct the characteristic
curve for the fatigue tests.
Calculation of Pseudo Stiffness (C)
and Damage Parameter (S)
Step 1: Calculate Average Strain
The pseudo stiffness, C, is defined as:
The strain history is decomposed into the mean strain and
(t ) cyclic strain components for analysis. The average strain is
C1( t ) = (8)
R (t ) calculated for each cycle from each LVDT. The individual
0.5 0.012
0.4 0.01
Stress, MPa
Strain, m/m
0.008
0.3
0.006
0.2
0.004
0.1 0.002
0 0
0 50 100 150 200
Time, Sec
Stress Crosshead Strain OSP Strain
Figure E2. Stress and strain histories for a typical
monotonic test.
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3000
Average Strain (MS)
2000
1000
0
0 500 1000 1500 2000
Time, sec
Figure E3. Mean strain for a PG 67-22 at optimum specimen.
strains recorded by different LVDTs are then averaged to Step 4: Calculate Cyclic Pseudo Strain
determine the mean strain history for the specimen. Figure E3
The cyclic strain is determined by subtracting the mean
shows the mean strain during a constant amplitude fatigue
strain from total strain. The cyclic strain is then fit using the
test for a PG 67-22 optimum specimen.
following equation:
Step 2: Calculate Initial Pseudo Stiffness cy ( t ) = p + q cos ( t + ) (10)
Initial pseudo strain is needed to determine the initial
where, p, q, and are regression constants. The cyclic
pseudo stiffness, which is used to calculate the normalized
pseudo strains are calculated using:
pseudo stiffness for the entire fatigue test. Strains captured
during the first loading cycle are used to calculate the initial
pseudo strains in that cycle using Equation 6. The initial () (
R t = q E cos t +
cy ) (11)
pseudo stiffness is calculated as the slope of the initial linear
portion of the stress-vs-pseudo strain plot. Figure E4 demon- where, |E | is the dynamic modulus of the mix at test temper-
strates the calculation of the initial pseudo stiffness for a par- ature and frequency.
ticular specimen.
Step 5: Calculate Maximum Pseudo Strain
Step 3: Calculate Mean Pseudo Strain in Each Cycle
The mean strains determined in Step 1 are used to calcu- The maximum pseudo strains for each cycle after the first
late the corresponding pseudo strains using the methodology cycle are calculated by adding the maximum cyclic pseudo
described for the monotonic tests. strain in each cycle to the corresponding mean pseudo
1.50
y = 1.1302x + 0.0302
2
R = 0.9966
1.00
Stress, MPa
0.50
0.00
0.00 0.20 0.40 0.60 0.80 1.00 1.20
Pseudostrain
Figure E4. Determination of initial pseudo stiffness.
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6.0
Max Pseudostrain
4.0
2.0
0.0
0 500 1000 1500 2000
Time, sec
Figure E5. Typical plot of maximum pseudo strain versus time.
1
strain value. Figure E5 shows the variation of the maximum 1+ t i - t i -1 1+
1
( ) (Ci-1 - Ci )
N
S (t ) = I R
2
pseudo strain over the course of a typical constant ampli- (15b)
x
max
tude fatigue test. i =1 2
Equation 15a is used to calculate S during first loading cycle
Step 6: Calculate Pseudo Stiffness and Equation 15b is used during the rest of the loading
cycles until failure. The parameter x is the fraction of
The pseudo stiffness is defined as:
the total stress-vs-strain cycle during which damage can
grow. This is that portion of the loading curve where ten-
SR = (12)
R sile stress occurs. To determine x, plots of the stress-vs-
strain curves are examined. From Figures E6 and E7, it can
For the first cycle, pseudo stiffness is calculated at each point be seen that an appropriate value for x is 4.0 (tensile stress
along the loading path. The secant pseudo stiffness is calcu- on the loading portion of the curve is approximately 1/4 of
lated for subsequent cycles to represent the change in slope of the whole loop).
the stress- pseudo strain loops:
max Step 9: C-vs-S Characteristic Curve
SR = (13)
max
R
The characteristic curve is constructed by cross-plotting C
where is the maximum pseudo strain in a cycle and max
R and S. An example is shown in Figure E8.
max
is the stress corresponding to R
max.
Step 7: Calculate Normalized Pseudo Stiffness
The normalized pseudo stiffness, C, is calculated as:
SR
C= (14)
I
where, I is the initial pseudo stiffness, calculated in Step 2.
Step 8: Calculate Damage Parameter
The damage parameter, S, is calculated by using the follow-
ing equations:
Figure E6. Stress-vs-strain plots for specimen
N
1 1+ 1
S ( t ) = I ( R ) (Ci -1 - Ci ) ( t i - t i -1 )1+
2
(15a) No. 14.
i =1 2
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Figure E7. Stress-vs-strain plots for specimen
no. 15.
1
0.75
C1
0.5
0.25
0
0 0.5 1 1.5 2 2.5 3
S1
Figure E8. Example characteristic curve.
