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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 20C 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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123 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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124 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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125 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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126 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.