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3 Although classical viscoelastic fundamentals suggest a log E * = -1.249937 + 0.02923 200 - 0.001767( 200 ) 2 linear relationship between log a(T) and T (in degrees Vbeff -0.002841 4 - 0.058097Va - 0.802208 Fahrenheit), experience has shown that for precision, a Vbeff + Va (Eq. 8) second-order polynomial relationship between the logarithm 3.871977 - 0.0021 4 + 0.003958 38 - 0.000017( 38 ) + 0.00547 34 2 of the shift factor, that is log a(Ti), and the temperature in + [ -0.603313 - 0.313351 log( f ) - 0.393532 log( )] 1+ e degrees Fahrenheit (Ti) should be used. The relationship can thus be expressed as follows: where |E*| = dynamic modulus, 105 psi Log a(Ti) = aTi + bTi + c 2 (Eq. 5) = binder viscosity at the age and temperature of inter- est, 106 Poise where f = loading frequency, Hz a(Ti) = shift factor as a function of temperature Ti Va = air void content, % Ti = temperature of interest, °F Vbeff = effective binder content, % by volume a, b and c = coefficients of the second-order polynomial 34 = cumulative % retained on 19-mm sieve 38 = cumulative % retained on 9.5-mm sieve If the value of the coefficient a approaches zero, the shift 4 = cumulative % retained on 4.76-mm sieve factor equation collapses to the classic linear form. 200 = % passing 0.075-mm sieve The Witczak predictive equation (Equation 8) can be pre- 2.1.2. Dynamic Modulus: Levels sented in the same form as Equation 3 for a mixture-specific of Analysis master curve as follows: The Mechanistic-Empirical Pavement Design Guide uses the laboratory-measured E* data for the Level 1 design Log E* = + (Eq. 9) 1 + e+ (log t ) r analysis; it uses E* values predicted from the Witczak E* predictive equation in Levels 2 and 3. The master curve for where the Level 1 analysis is developed using numerical optimiza- |E*| = dynamic modulus, 105 psi tion to shift the laboratory mixture test data into a smooth master curve. Before shifting the test data, the relationship = -1.249937 + 0.02923200 - 0.001767(200 ) 2 between binder viscosity and temperature must be estab- Vbeff (Eq. 9a) -0.0028414 - 0.058097Va - 0.802208 lished. This is done by first converting the binder stiffness Vbeff + Va data at each temperature to viscosity using Equation 6. The parameters of the ASTM Ai-VTSi equation are then found = 3.871977 - 0.00214 + 0.00395838 by linear regression of Equation 7 after log-log transforma- - 0.000017(38)2 + 0.0054734 (Eq. 9b) tion of the viscosity data and log transformation of the tem- = - 0.603313 - 0.393532 log (Tr) (Eq. 9c) perature data. = 0.313351 (Eq. 9d) G * 1 4.8628 = (Eq. 6) tr = reduced time of loading at reference temperature 10 sin Va = air void content, % Vbeff = effective binder content, % by volume log log = A + VTS log TR (Eq. 7) 34 = cumulative % retained on 19-mm sieve 38 = cumulative % retained on 9.5-mm sieve where 4 = cumulative % retained on 4.76-mm sieve = binder viscosity, cP 200 = % passing 0.075-mm sieve G* = binder complex shear modulus, Pa Tr = binder RTFOT viscosity at the reference tempera- = binder phase angle, degree ture, 106 Poise A, VTS = regression parameters TR = temperature, ēRankine 2.2 FLOW NUMBER (Fn) The master curve for the Level 2 analysis is developed using the Witczak Dynamic Modulus Predictive Equation An approach to determine the permanent deformation char- (Equation 8) from specific laboratory test data. The Level 3 acteristics of paving materials is to employ a repeated dynamic analysis requires no laboratory test data for the AC binder, load test for several thousand repetitions and record the cumu- but requires those mixture properties included in the Witczak lative permanent deformation as a function of the number of predictive equation. cycles (repetitions) over the testing period. This approach was
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4 employed by Monismith and coworkers in the mid-1970s using uniaxial compression tests. Several research studies conducted by Witczak and coworkers used a temperature of 100oF or 130oF at 10, 20, or 30 psi deviator stress level. A haversine pulse load of 0.1 sec and 0.9 sec dwell (rest time) is applied for the test duration of approximately 3 hours. This approach results in approximately 10,000 cycles applied to the specimen. Several parameters describing the accumulated permanent deformation response can be obtained from the Fn test. Fig- ure 2 illustrates the typical relationship between the total cumulative permanent strain and number of load cycles. Like the creep test, the cumulative permanent strain curve is gener- ally defined by three zones: primary, secondary, and tertiary. In the primary zone, permanent deformations accumulate rapidly. The incremental permanent deformations decrease Figure 3. Regression constants "a" and "b" when reaching a constant value in the secondary zone. Finally, plotted on a log-log scale. the incremental permanent deformations again increase and permanent deformations accumulate rapidly in the tertiary zone. The starting point, or cycle number, at which tertiary flow occurs is referred to as the flow number. The intercept a represents the permanent strain at N = 1, Typical permanent deformation parameters, which are whereas the slope b represents the rate of change in perma- obtained and analyzed from the repeated load permanent nent strain as a function of the change in loading cycles [log deformation test, include the intercept (a, ) and slope (b, ) (N)]. An alternative form of the mathematical model used to parameters. The permanent deformation properties (, ) characterize the permanent strain per load repetition (pn) have been used as input for predictive design procedures. All relationship can be expressed by of the parameters derived from the linear (secondary) portion of the cumulative permanent strainrepetitions curve ignore p ( aN b ) the tertiary zone of material deformability. Thus, all four of = pn = or, pn = abN ( b-1) (Eq. 10) the parameters noted (, , b, a) are regression constants of N N a statistical model that is only based on the "linear" sec- ondary phase of the permanent strainrepetition curve. The resilient strain (r) is generally assumed to be indepen- The log-log relationship between the permanent strain and dent of the load repetition value (N). As a consequence, the the number of load cycles can be expressed by the classical ratio of permanent-to-elastic strain components of the material power model: p = aNb, where a and b are regression con- in question can be defined by stants depending on the material-test combination condi- tions. Figure 3 illustrates the relationship when plotted on a pn ab b-1 = N (Eq. 11) log-log scale. r r ab By letting = and = 1 - b, one obtains r pn = N - (Eq. 12) r In the above equation, pn is the permanent strain resulting from a single load application; that is, at the Nth application. is a permanent deformation parameter representing the constant of proportionality between permanent strain and elastic strain (i.e., permanent strain at N = 1). is a perma- nent deformation parameter indicating the rate of decrease in incremental permanent deformation as the number of load applications increases. Figure 4 illustrates the above relationship and the occur- Figure 2. Typical relationship between total cumulative rence of the flow point when the rate of decrease in perma- permanent strain and number of load cycles. nent strain is constant.