Derivation of the master curve
G.1 Principle G.1 Principle
The stiffness modulus of bituminous mixes obey the frequency-temperature superposition principles. This The stiffness modulus of bituminous mixes obey the frequency-temperature superposition principles. This means that a master curve of the complex modulus can be constructed for a reference temperature from means that a master curve of the complex modulus can be constructed for a reference temperature from experimental data over a limited range of frequencies and temperatures, provided that a shift factor is used to experimental data over a limited range of frequencies and temperatures, provided that a shift factor is used to calculate the reference frequency.
calculate the reference frequency.
Key Key A
A experimental rangeexperimental range B
B range range covered covered by by the the master master curvecurve
Figure G.1 — Derivation of the master curve Figure G.1 — Derivation of the master curve
The aim of the master curve is to determine the stiffness modulus at any arbitrary combination of loading The aim of the master curve is to determine the stiffness modulus at any arbitrary combination of loading frequency and test temperature. For this purpose, the stiffness has to measured at various frequencies and frequency and test temperature. For this purpose, the stiffness has to measured at various frequencies and temperatures. These results are used to determine the parameters in an Arrhenius equation, which is used temperatures. These results are used to determine the parameters in an Arrhenius equation, which is used often in this case.
often in this case.
In this annex, the determination of the master of the stiffness is based on the approach of Arrhenius.
In this annex, the determination of the master of the stiffness is based on the approach of Arrhenius.
Alternative a
Alternative a pproaches can pproaches can also be also be used, used, e.g. the e.g. the Christensen-Anderson Christensen-Anderson model. model. In all In all cases in cases in the rthe report oeport onn the determination of the mastercurve, the theoretical background of the used model shall be incorporated.
the determination of the mastercurve, the theoretical background of the used model shall be incorporated.
G.2 Theoretical background G.2 Theoretical background
The basis form of the master curve is:
The basis form of the master curve is:
( ( ))
C is is the the experimental experimental constant, constant, in in K;K;
T
T is is the the actual actual temperature, temperature, in in K;K;
T
T ss is is the the arbitrary arbitrary chosen chosen reference reference temperature, temperature, in in K;K;
R
R is is the the Universal Universal gas gas constantconstant
==
8,314 4 J/K/mole; 8,314 4 J/K/mole;∆
∆
H H is is the the activation activation energy, energy, in in kJ/mole;kJ/mole;α
α TT is is the the shift shift factor;factor;
f
f is is the the loading loading frequency frequency at at temperaturetemperature T T , in Hz;, in Hz;
f
f ss is is the the loading loading frequency frequency at at temperaturetemperature T T ss, in Hz., in Hz.
Equation G.1 is called the Arrhenius Equation. In literature, various values for
Equation G.1 is called the Arrhenius Equation. In literature, various values for C C are mentioned. The are mentioned. The C C -value-value depends on the activation energy which depends on the characteristics oh the bitumen and filler. The
depends on the activation energy which depends on the characteristics oh the bitumen and filler. The C C -value-value for a specific asphalt mix can be determined with the following equation:
for a specific asphalt mix can be determined with the following equation:
(
S mixmix is the is the stiffness modulus stiffness modulus E as E as a function a function of loading of loading time and time and temperature, in temperature, in MPa;MPa;
η determined by shifting the stiffness modulus measured at temperature
determined by shifting the stiffness modulus measured at temperature T T yy at the same loading time (or at the same loading time (or frequency). Equalising the stiffness modulus
frequency). Equalising the stiffness modulus S S mix,Amix,A determined at a loading time determined at a loading time t t 11 and temperature and temperature T T XX to a to a stiffness modulus
stiffness modulus S S mix,Bmix,B determined at a loading time determined at a loading time t t 22 and temperature T and temperatureT YY finally results in: finally results in:
(
C -value will be different. For this reason, the-value will be different. For this reason, the C C -value shall be determined at a combination of low temperature-value shall be determined at a combination of low temperature and high loading frequency, a combination of high temperature and low loading frequency and a middle
fictive temperature T T fictfict..
( (
//))
273273T fictfict is the is the fictive asphalt concrete fictive asphalt concrete temperature where the temperature where the stiffness shall be stiffness shall be determined, in °Cdetermined, in °C;;
T
T aa is is the the actual actual temperature, temperature, in in °C;°C;
f
f chachar r is the characteristic lis the characteristic loading frequency which is oading frequency which is used to determine the used to determine the master curve, in Hz;master curve, in Hz;
f
f is is the the loading loading frequency, frequency, in in Hz.Hz.
Using linear regression the coefficients of Equation (6) can be determined for a chosen characteristic loading Using linear regression the coefficients of Equation (6) can be determined for a chosen characteristic loading frequency: several test temperatures shall be performed. This test shall consist of response measurements at a range of several test temperatures shall be performed. This test shall consist of response measurements at a range of nominal frequencies (e.g. 1 Hz, 3 Hz, 5 Hz, 8 Hz, 10 Hz, 20 Hz, 30 Hz and 60 Hz and subsequently again at nominal frequencies (e.g. 1 Hz, 3 Hz, 5 Hz, 8 Hz, 10 Hz, 20 Hz, 30 Hz and 60 Hz and subsequently again at 1 Hz). The loading mode in this pre-test shall be constant deflection representative for a maximum strain enough; at higher frequencies 200 load cycles shall be applied. In order to avoid premature fatigue damage, enough; at higher frequencies 200 load cycles shall be applied. In order to avoid premature fatigue damage, the total number of
the total number of applications for all frequencies tapplications for all frequencies t ogether shall not exceed ogether shall not exceed 3 3 000. At low tem000. At low tem peraturesperatures ((Θ Θ
≤≤
10 °C), there shall be a short rest period of about 10 min before the actual fatigue test starts. 10 °C), there shall be a short rest period of about 10 min before the actual fatigue test starts.In order to develop a representative master curve, at least 3 specimens shall be tested. From each specimen, In order to develop a representative master curve, at least 3 specimens shall be tested. From each specimen, the stiffness modulus of the mix shall be measured at 4 temperatures and at each temperature at
For an AC 16 bin/base 40/60, the following parameters are found assuming
For an AC 16 bin/base 40/60, the following parameters are found assuming T T ss
==
20 °C and 20 °C and f f char char==
8 Hz: 8 Hz:
C C==
9 175 K 9 175 K
cc11==
10,139 533 26 10,139 533 26
cc22== −−
0,023 403 440,023 403 44
cc33== −−
0,001 008 160,001 008 16
cc44==
0,000 000 55 0,000 000 55G.4 Test report G.4 Test report
The test report includes the following information:
The test report includes the following information: