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CHAPTER 2 THEORY 2.1 DYNAMIC MODULUS (E*) For linear viscoelastic materials such as HMA, the stress- to-strain relationship under a continuous sinusoidal loading is deï¬ned by its complex dynamic modulus (E*). This is a complex number that relates stress to strain for linear viscoelastic materials subjected to continuously applied sinusoidal loading in the frequency domain. The complex modulus is deï¬ned as the ratio of the amplitude of the sinu- soidal stress (at any given time, t, and angular load frequency, Ï), Ï = Ï0 sin(Ït), and the amplitude of the sinu- soidal strain ε = ε0 sin(Ït â Ï), at the same time and fre- quency, that results in a steady-state response (Figure 1): (Eq. 1) where Ï0 = peak (maximum) stress ε0 = peak (maximum) strain Ï = phase angle, degrees Ï = angular velocity t = time, seconds Mathematically, the dynamic modulus is deï¬ned as the absolute value of the complex modulus, or (Eq. 2) For a pure elastic material, Ï = 0, and it is observed that the complex modulus (E*) is equal to the absolute value or dynamic modulus. For pure viscous materials, Ï = 90°. The E* = Ï Îµ 0 0 E e e t t i t i t * sin( ) sin(= = = ââ( ) Ï Îµ Ï Îµ Ï Ï Îµ Ï Ï Ï Ï Ï 0 0 0 0 ) 2 dynamic modulus testing of HMA is normally conducted using a uniaxially applied sinusoidal stress pattern as shown in Figure 1. 2.1.1. Dynamic Modulus: Master Curve In the Mechanistic-Empirical Pavement Design Guide developed in NCHRP Project 1-37A, the modulus of HMA at all levels of temperature and time rate of load is deter- mined from a master curve constructed at a reference temperature (generally taken as 70°F). Master curves are constructed using the principle of time-temperature super- position. The data at various temperatures are shifted with respect to time until the curves merge into a single smooth function. The master curve of the modulus, as a function of time, formed in this manner describes the time dependency of the material. The amount of shifting at each temperature required to form the master curve describes the temperature dependency of the material. In general, the modulus master curve can be mathematically modeled by a sigmoidal func- tion described as (Eq. 3) where tr = reduced time of loading at reference temperature δ = minimum value of E* δ + α = maximum value of E* β, γ = parameters describing the shape of the sigmoidal function The shift factor can be shown in the following form: (Eq. 4) where a(T) = shift factor as a function of temperature t = time of loading at desired temperature tr = time of loading at reference temperature T = temperature a T t tr ( ) = Log *E e tr = + + + δ αβ γ1 (log ) Ï0 sin(Ït) ε0 sin(Ït â Ï) Ï/Ï Îµ0Ï0 Figure 1. Dynamic (complex) modulus test.
Although classical viscoelastic fundamentals suggest a linear relationship between log a(T) and T (in degrees Fahrenheit), experience has shown that for precision, a second-order polynomial relationship between the logarithm of the shift factor, that is log a(Ti), and the temperature in degrees Fahrenheit (Ti) should be used. The relationship can thus be expressed as follows: Log a(Ti) = aTi2 + bTi + c (Eq. 5) where a(Ti) = shift factor as a function of temperature Ti Ti = temperature of interest, °F a, b and c = coefficients of the second-order polynomial If the value of the coefficient a approaches zero, the shift factor equation collapses to the classic linear form. 2.1.2. Dynamic Modulus: Levels of Analysis The Mechanistic-Empirical Pavement Design Guide uses the laboratory-measured E* data for the Level 1 design analysis; it uses E* values predicted from the Witczak E* predictive equation in Levels 2 and 3. The master curve for the Level 1 analysis is developed using numerical optimiza- tion to shift the laboratory mixture test data into a smooth master curve. Before shifting the test data, the relationship between binder viscosity and temperature must be estab- lished. This is done by ï¬rst converting the binder stiffness data at each temperature to viscosity using Equation 6. The parameters of the ASTM Ai-VTSi equation are then found by linear regression of Equation 7 after log-log transforma- tion of the viscosity data and log transformation of the tem- perature data. (Eq. 6) log log η = A + VTS log TR (Eq. 7) where η = binder viscosity, cP G* = binder complex shear modulus, Pa δ = binder phase angle, degree A, VTS = regression parameters TR = temperature, ºRankine The master curve for the Level 2 analysis is developed using the Witczak Dynamic Modulus Predictive Equation (Equation 8) from speciï¬c laboratory test data. The Level 3 analysis requires no laboratory test data for the AC binder, but requires those mixture properties included in the Witczak predictive equation. η δ= ââ ââ G * sin . 10 1 4 8628 3 where |E*| = dynamic modulus, 105 psi η = binder viscosity at the age and temperature of inter- est, 106 Poise f = loading frequency, Hz Va = air void content, % Vbeff = effective binder content, % by volume Ï34 = cumulative % retained on 19-mm sieve Ï38 = cumulative % retained on 9.5-mm sieve Ï4 = cumulative % retained on 4.76-mm sieve Ï200 = % passing 0.075-mm sieve The Witczak predictive equation (Equation 8) can be pre- sented in the same form as Equation 3 for a mixture-speciï¬c master curve as follows: (Eq. 9) where |E*| = dynamic modulus, 105 psi α = 3.871977 â 0.0021Ï4 + 0.003958Ï38 â 0.000017(Ï38)2 + 0.00547Ï34 (Eq. 9b) β = â 0.603313 â 0.393532 log (ηTr) (Eq. 9c) γ = 0.313351 (Eq. 9d) tr = reduced time of loading at reference temperature Va = air void content, % Vbeff = effective binder content, % by volume Ï34 = cumulative % retained on 19-mm sieve Ï38 = cumulative % retained on 9.5-mm sieve Ï4 = cumulative % retained on 4.76-mm sieve Ï200 = % passing 0.075-mm sieve ηTr = binder RTFOT viscosity at the reference tempera- ture, 106 Poise 2.2 FLOW NUMBER (Fn) An approach to determine the permanent deformation char- acteristics of paving materials is to employ a repeated dynamic load test for several thousand repetitions and record the cumu- lative permanent deformation as a function of the number of cycles (repetitions) over the testing period. This approach was δ Ï Ï1.249937 0.02923 0.001767( ) 0. 2 = â + â â 200 200 002841 0.058097 .8 2208 a Ï4 0 0â â + V V V V beff beff a â ââ â â â Log E* = + + + δ αβ γ1 e tr(log ) log E * . . . ( )= â + â1 249937 0 02923 0 001767200 200 2Ï Ï â â â + 0 002841 0 058097 0 8022084. . .Ï V V V a beff beff Va + â + â3 871977 0 0021 0 003958 0 0000174 38. . . . (Ï Ï Ï Ï38 2 34 0 603313 0 313351 0 00547 1 ) . [ . . log( + + â â e f ) . log( )]â0 393532 η (Eq. 8) (Eq. 9a)
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 deï¬ned by three zones: primary, secondary, and tertiary. In the primary zone, permanent deformations accumulate rapidly. The incremental permanent deformations decrease reaching a constant value in the secondary zone. Finally, the incremental permanent deformations again increase and permanent deformations accumulate rapidly in the tertiary zone. The starting point, or cycle number, at which tertiary ï¬ow occurs is referred to as the ï¬ow number. Typical permanent deformation parameters, which are obtained and analyzed from the repeated load permanent deformation test, include the intercept (a, μ) and slope (b, α) parameters. The permanent deformation properties (α, μ) have been used as input for predictive design procedures. All of the parameters derived from the linear (secondary) portion of the cumulative permanent strainârepetitions curve ignore the tertiary zone of material deformability. Thus, all four of the parameters noted (α, μ, b, a) are regression constants of a statistical model that is only based on the âlinearâ sec- ondary phase of the permanent strainârepetition curve. The log-log relationship between the permanent strain and the number of load cycles can be expressed by the classical power model: εp = aNb, where a and b are regression con- stants depending on the material-test combination condi- tions. Figure 3 illustrates the relationship when plotted on a log-log scale. 4 The intercept a represents the permanent strain at N = 1, whereas the slope b represents the rate of change in perma- nent strain as a function of the change in loading cycles [log (N)]. An alternative form of the mathematical model used to characterize the permanent strain per load repetition (εpn) relationship can be expressed by (Eq. 10) The resilient strain (εr) is generally assumed to be indepen- dent of the load repetition value (N). As a consequence, the ratio of permanent-to-elastic strain components of the material in question can be deï¬ned by (Eq. 11) By letting and α = 1 â b, one obtains (Eq. 12) In the above equation, εpn is the permanent strain resulting from a single load application; that is, at the Nth application. μ is a permanent deformation parameter representing the constant of proportionality between permanent strain and elastic strain (i.e., permanent strain at N = 1). α 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- rence of the ï¬ow point when the rate of decrease in perma- nent strain is constant. pn r - = N ε ε μ α μ ε = ab r pn r r b-1 = ab N ε ε ε âââ ââ â â â = â â = â p pn b pn bor, ε ε εN = (aN ) N abN ( )1 Figure 2. Typical relationship between total cumulative permanent strain and number of load cycles. Figure 3. Regression constants âaâ and âbâ when plotted on a log-log scale.
2.3 FLOW TIME (Ft) Figure 5 shows a typical relationship between the calcu- lated total compliance and time measured in a static creep test. This ï¬gure shows that the total compliance can be divided into three major zones: (1) primary, (2) secondary, and (3) tertiary. In the primary zone, the strain rate decreases; in the secondary zone, the creep rate is constant; and in the tertiary zone, the creep rate increases. 5 The ï¬ow time, Ft, is therefore deï¬ned as the time when shear deformation, under constant volume, starts. The ï¬ow time is also viewed as the minimum point in the relationship of rate of change of compliance versus loading time. Figures 6 and 7 show typical static creep test plots. Figure 6 shows the total axial strain versus loading time on a log-log scale. The estimation of compliance parameters a and m are obtained from the regression analysis of the linear portion of the curve. Figure 7 shows a plot of the rate of change in com- pliance versus loading time in log-log scale along with the calculated value of the ï¬ow time. Figure 4. Permanent deformation parameters α and μ and the ï¬ow number. D(t) time Secondary Tertiary Primary Flow Time Defines When Shear Deformation Begins Figure 5. Typical test results between the calculated total compliance and time. Figure 6. Total axial strain vs. time from an actual static creep / ï¬ow time test. Figure 7. Typical plot of the rate of change in compliance vs. loading time.