Validating the Fatigue Endurance Limit for Hot Mix Asphalt (2010)

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122 Determination of the Relaxation Modulus Construction of the damage characteristic curve requires the calculation of pseudo strains, which in turn require the relaxation modulus, E(t), of the mixture. The relaxation modulus is difficult to measure directly in the laboratory and is therefore determined from the dynamic modulus master curve using linear viscoelastic theory. The relaxation modulus is expressed as the Prony series: Where E∞ is the value of E(t) as t→∞, En are Prony series co- efficients and ρn are relaxation times. To obtain the Prony series expression of relaxation modulus from dynamic mod- ulus, the first step is to determine the storage modulus master curve using: Where |E′(ωr)| is the storage modulus, ϕ is the phase angle and ωr is the reduced frequency in rad/sec. The relaxa- tion modulus curve of each specimen is obtained from the storage modulus master curve by applying the following relation: Where tr is the reduced time, Γ is the gamma function and n is the slope of log(E′(ω)) versus log(ω) curve which is obtained at each point of reduced frequency. E t E t n n r r r r ( ) = ′ ′( ) = ′ = −( ) ⎛⎝⎜ 1 0 08 1 2 λ ω ω λ π , . cosΓ ⎞⎠⎟ = ′( ) ( ) log log 3 n d E d ω ω ′( ) = ∗( ) ( )( )E Er r rω ω ϕ ωcos ( )2 E t E E t n nn ( ) = + −⎛⎝⎜ ⎞ ⎠⎟∞ ∑ exp ( )ρ 1 Once the relaxation modulus is obtained from the storage modulus, the Prony series coefficients in Equation 1 are obtained through the following steps. First, Equation 1 is rewritten in matrix form as: Where the relaxation times tj are chosen at decade intervals along time axis and N is the number of data points used. The MATLAB optimization toolbox is used to obtain the solution for Ej with the following constraint: which provides positive values of Ej. The prony series repre- sentations of the relaxation modulus for the four mixtures are shown in Figure E1. Monotonic Characteristic Curves Pseudo Strain Calculation Monotonic tests (constant crosshead) at 20°C are per- formed on specimens at various crosshead strain rates. The pseudo strains are calculated using the strains measured from the on-specimen LVDTs. Figure E2 shows the typical stress, crosshead strain, and LVDT strain as a function of time for a monotonic test. The pseudo strain is defined as: ε τ ε τ τR R t t E E t d d d( ) = −( )∫1 6 0 ( ) min , ( ) C B C A C 1 2 0 5 2 2[ ]{ }−{ } { } ≥such that, A B C A E t E B t C j j jk j k { } = [ ]{ } = ( )− = − ⎛ ⎝⎜ ⎞ ⎠⎟ ∞ exp ( ) ρ 4 j jE j N = =1 2, , . . . , A P P E N D I X E Construction of Characteristic Curve

123 where ER is the reference modulus (which is chosen as unity), E(t) is the relaxation modulus obtained from storage modulus and expressed as Prony series and ε is the on-specimen LVDT strain observed under monotonic tests. The above integration 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 Δt and Equation 1 is substituted in Equation 6, resulting in the following form of numerical integration scheme: (7) Calculation of Pseudo Stiffness (C) and Damage Parameter (S) The pseudo stiffness, C, is defined as: C t t tR 1 8( ) = ( )( ) σ ε ( ) εR i t t j t t j N j N t c E u t t t E u ( ) = ( ) = = − − − + = + ∑ 1 0 1 1 1 0, , ( ) = − −⎛⎝⎜ ⎞ ⎠⎟∞ ∑E u E u j j jj ρ ρ exp where σ(t) is the stress history. The damage parameter, S, is obtained from the following equation: where I is the initial pseudo stiffness and where m is the slope of the linear portion of the relaxation curve. The damage characteristic curve is obtained by plotting the dam- age parameter versus the pseudo stiffness. Fatigue Characteristic Curves The following steps are used to construct the characteristic curve for the fatigue tests. Step 1: Calculate Average Strain The strain history is decomposed into the mean strain and cyclic strain components for analysis. The average strain is calculated for each cycle from each LVDT. The individual α = +1 1 m , S t C C t tR j j j j j j1 1 2 2 1 1 1( ) = ( ) −( )⎛⎝⎜ ⎞⎠⎟ −− + −∑ ε α α , ( ) +11 9α ( ) 1.0E-01 1.0E+01 1.0E+03 1.0E+05 1.0E-09 1.0E-011.0E-05 1.0E+071.0E+03 R el ax at io n M od ul us , M Pa Time, sec 67-22, Opt 76-22, Opt 67-22, Opt+ 76-22, Opt+ Figure E1. Relaxation modulus curves for all four mixtures. 0 0.1 0.2 0.3 0.4 0.5 0 50 150100 200 Time, Sec St re ss , M Pa 0 0.002 0.004 0.006 0.008 0.01 0.012 St ra in , m /m Stress Crosshead Strain OSP Strain Figure E2. Stress and strain histories for a typical monotonic test.

strains recorded by different LVDTs are then averaged to determine the mean strain history for the specimen. Figure E3 shows the mean strain during a constant amplitude fatigue test for a PG 67-22 optimum specimen. Step 2: Calculate Initial Pseudo Stiffness Initial pseudo strain is needed to determine the initial pseudo stiffness, which is used to calculate the normalized 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 pseudo stiffness is calculated as the slope of the initial linear portion of the stress-vs-pseudo strain plot. Figure E4 demon- strates the calculation of the initial pseudo stiffness for a par- ticular specimen. Step 3: Calculate Mean Pseudo Strain The mean strains determined in Step 1 are used to calcu- late the corresponding pseudo strains using the methodology described for the monotonic tests. Step 4: Calculate Cyclic Pseudo Strain The cyclic strain is determined by subtracting the mean strain from total strain. The cyclic strain is then fit using the following equation: where, p, q, ω and ϕ are regression constants. The cyclic pseudo strains are calculated using: where, |E| is the dynamic modulus of the mix at test temper- ature and frequency. Step 5: Calculate Maximum Pseudo Strain in Each Cycle The maximum pseudo strains for each cycle after the first cycle are calculated by adding the maximum cyclic pseudo strain in each cycle to the corresponding mean pseudo ε ω ϕcyR t q E t( ) = ∗ +( )cos ( )11 ε ω ϕcy t p q t( ) = + +( )cos ( )10 124 0 1000 2000 3000 0 500 1000 1500 2000 Time, sec A ve ra ge S tra in (M S) Figure E3. Mean strain for a PG 67-22 at optimum specimen. y = 1.1302x + 0.0302 R2 = 0.9966 0.00 0.50 1.00 1.50 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Pseudostrain St re ss , M Pa Figure E4. Determination of initial pseudo stiffness.

strain value. Figure E5 shows the variation of the maximum pseudo strain over the course of a typical constant ampli- tude fatigue test. Step 6: Calculate Pseudo Stiffness The pseudo stiffness is defined as: For the first cycle, pseudo stiffness is calculated at each point along the loading path. The secant pseudo stiffness is calcu- lated for subsequent cycles to represent the change in slope of the stress- pseudo strain loops: where εRmax is the maximum pseudo strain in a cycle and σmax is the stress corresponding to εRmax. Step 7: Calculate Normalized Pseudo Stiffness The normalized pseudo stiffness, C, is calculated as: 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: S t I C C t tR i i i N i i( ) = ( ) −( )⎡⎣⎢ ⎤ ⎦⎥ −−= + −∑ 1 2 2 1 1 1 ε α α 1 1 1 15( ) +α ( )a C S I R = ( )14 SR R = σ ε max max ( )13 SR R = σ ε ( )12 Equation 15a is used to calculate S during first loading cycle and Equation 15b is used during the rest of the loading cycles until failure. The parameter x is the fraction of the total stress-vs-strain cycle during which damage can grow. This is that portion of the loading curve where ten- sile stress occurs. To determine x, plots of the stress-vs- strain curves are examined. From Figures E6 and E7, it can be seen that an appropriate value for x is 4.0 (tensile stress on the loading portion of the curve is approximately 1⁄4 of the whole loop). Step 9: C-vs-S Characteristic Curve The characteristic curve is constructed by cross-plotting C and S. An example is shown in Figure E8. S t I C C tR i i i N i( ) = ( ) −( )⎡⎣⎢ ⎤⎦⎥ − − = +∑ 1 2 2 1 1 1 ε α α max t x bi− +⎛⎝⎜ ⎞⎠⎟1 1 1 15 α ( ) 125 0.0 2.0 4.0 6.0 0 500 1000 1500 2000 Time, sec M ax P se ud os tra in Figure E5. Typical plot of maximum pseudo strain versus time. Figure E6. Stress-vs-strain plots for specimen No. 14.

126 0 0.25 0.5 0.75 1 0 0.5 1 1.5 2 2.5 3 S1 C1 Figure E8. Example characteristic curve. Figure E7. Stress-vs-strain plots for specimen no. 15.