PREDICTION OF WÖHLER CURVE USING ONLY ONE 4PB FATIGUE TEST. IS IT POSSIBLE? A.C. PRONK; M.D. GAJEWSKI & W. BANKOWSKI. Introduction

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1 Introduction PREDICTION OF WÖHLER CURVE USING ONLY ONE 4PB FATIGUE TEST. IS IT POSSIBLE? A.C. PRONK; M.D. GAJEWSKI & W. BANKOWSKI This note or report is the follow up of the publication Application of a material fatigue damage model in 4PB tests which is offered to the International Journal for Pavement Engineering. In that publication the results are discussed of 4PB fatigue tests, carried out by IBDiM (Poland), on a normal asphalt concrete mixture. In total 8 tests were done at a frequency of 0 Hz, one temperature of 0 o C and at three deflection levels in pure sinusoidal mode. These deflection levels resulted in initial strain levels of around 40, 90 and 40 micro strain. The responses, expressed as a stiffness modulus Mbeam and phase lag beam for the beam, are compared with calculated responses using the Asphalt Concrete Pavement-Fatigue (ACP-F) model as a material model. The ACP-F model has six parameters. Three parameters are related to the thixotropic behaviour of asphalt, two parameters are related to the fatigue damage and the last one is an endurance limit for fatigue. When the applied strain amplitude is below the endurance limit no fatigue damage will occur. In that publication a linear relationship between the fatigue damage parameters and the applied strain amplitude was adopted which was based on findings in UPP (or DT/C) tests on another mix. After finishing the publication the suspicion raised that a cubic relationship would be more appropriate. This assumption is the main difference between the publication and this note. It should be marked explicitly that there are no homogenous tests (DT/C or UPP) available which will concord this assumption. However, it will be shown at the end that with this assumption the findings are in agreement with each other. A short overview of the involved theory is given in Annex. ACP-F model The ACP-F model is a material model describing the evolutions of the loss and storage modulus in an Uni-axial Push & Pull test (UPP) applying a pure sinusoidal constant strain signal. Another common name for the UPP test is the Dynamic Tension & Compression test (DT/C). Due to the non-homogenous strain/stress distribution in a 4PB test, the ACP-F model has to be integrated over the whole specimen (beam) leading to calculated evolutions for the weighted loss modulus Lbeam, weighted storage modulus Sbeam, the weighted modulus Mbeam and weighted phase lag beam of the beam. These four evolutions are, by varying the six model parameters and the initial loss- and storage modulus, fitted on the measured evolutions for a certain interval. This interval is taken from No = 3,000 cycles to N cycles. N is visually determined as the point where the measured dissipated energy ratio for the beam starts to deviate from a straight line through the origin. After fitting the ACP-F model on the measured evolutions, the calculated evolution for the dissipated energy ratio for the beam is also compared to the measured dissipated energy ratio. The point where the two evolutions starts to deviate from each other is visually determined and defined as the fatigue life NPH. It should be noted that the fatigue life NPH and the common used fatigue life Nf,50 (50% reduction in initial modulus Mbeam) are always bigger than N. An overall view of all the comparisons is given in a separate file on the 4PB platform. Figure is an example of the fit results.

2 Modulus [MPa] Phase lag [o] Beam modulus M from 4PB Beam modulus M from ACP-F model Beam Loss Modulus L from 4PB test Beam Loss Modulus L from ACP-F model Material modulus M at the surface in the mid section using ACP-F model Beam Phase lag from 4PB test Beam Phase lag from ACP-F model Material Surface Phase lag in Midsection using ACP-F model Beam Cycle N Figure. Comparison between measured data for modulus Mbeam (full yellow line) and phase lag beam (full blue line) and the calculated values (dotted lines) according to the ACP-F model as a function of the cycle number N. Also the evolutions for material modulus M (pink dotted line) and material phase lag (yellowe dotted line) at the top and bottom surface in the mid section according to the ACP-F model are given.

3 Fit Protocol The leading quantities in the ACP-F model are the weighted loss modulus Lbeam and the weighted storage modulus Sbeam for the beam. Instead of these two quantities the differences in the modulus Mbeam and the phase lag beam for the beam can also be used. In the fitting protocol all four quantities are fitted. The summed squared differences between the measured data and the fitted model values for these four greats/quantities on the chosen interval are minimized using the solver option in the Excel program. In order to get a balanced influence each summation is divided by the variance on the interval for that quantity. The goal function F is given in equation in which the captured parameters are the values according to the model. F µ beam N N N N beam N N0 N N0 µ beam L N L N S N S N Var L N.. N Var S N.. N beam 0 beam 0 N N N N M beam N M beam N beam N $ beam N N N0 N N0 Var M N.. N Var N.. N beam beam 0 beam 0 () The measured data in the 4PB test are quantities for the response of the beam as a whole. The ACP-F model is a material model valid for a unit volume. Therefore the evolutions of the quantities have to be integrated over the whole beam for the comparison with the measured data. This procedure is presented at the 3 rd 4PB workshop. For the measurements used in this note the following protocol is chosen starting with seed values for the involved parameters. Initial loss and storage modulus (Lo and So) are taken equal to the first measured data for these quantities. The initial value for the strain endurance is 50 micro strain Seed values for the thixotropic behaviour are for the parameter Tg, for the parameter Tg and s - for the time decay. For the fatigue damage parameter Tg and Tg the values are and respectively. Furthermore it appears to be better to first minimizing the squared differences for the loss modulus Lbeam alone by varying the parameters tg, tg, and Lo. Than these values were taken constant and the remaining parameters were varied for minimizing the summed squared differences for the storage modulus Sbeam, the modulus Mbeam and the phase lag beam. Finally all four quantities are summed and minimized by varying all eight parameters. In all cases, including the measured data for beam 54, the maximum difference between a measured data point and a fitted data point was in the order of % of the measured value (see graphs in the accompany figures file). So, in view of the comparison the results using the ACP-F model are very good. In table rest values for the goal function F are given for the two assumptions of the dependency of on the applied strain amplitude and the number of data points used for the fitting. Int. Journal for Pavement Engineering: γ, = Tgγ,. (ε ε endurance ): Rest value I This report: γ, = Tgγ,. ε (ε ε endurance ): Rest value II As can be seen in table the rest values are of the same order. 3

4 Table. Number of data points used for fitting and rest values for the goal function F. Beam Data points No-N Rest value I Rest value II Initial complex stiffness modulus In table the calculated modulus Mo and phase lag o are given after the fitting of the measured data on the interval from N = 3,000 to N cycles. The data recorded at cycle 078, representing the weighted beam modulus, are also given in table. Table. Initial stiffness modulus and stiffness modulus for the beam at 078 cycles. Beam [m/m] Mo [MPa] o [ o C] M078 [MPa] 078 [ o C] cycle 5. Mean.3 GPa 5.9 Stand.Dev. 0.9 GPa.3 cycle 5 4

5 S beam [MPa] L beam [MPa] M beam [MPa] Phase lag [ o ] It should mentioned that before cycle 078 the desired deflection amplitude was not yet reached (still increasing deflection amplitude). The process control was not perfect and should be optimized in order to be able to measure at cycle 00 the initial values as requested in the CEN norm. As can be seen in table the differences between the fitted initial modulus and phase lag and the modulus and phase lag at cycle 078 are small. In figures and 3 the initial values are given as a function of the applied strain amplitude. Modulus Phase Lag Strain amplitude [m/m] Figure. The initial fitted values for the modulus Mbeam and phase lag beam Storage Loss Strain amplitude [m/m] Figure 3. The initial fitted values for the storage modulus Sbeam and loss modulus Lbeam. The variations in the values are rather small. Looking at the modulus M and storage modulus S it seems that there is tendency for higher values at lower strain values. 5

6 Tg [-] Tg [-] Results for the thixotropic parameters Tg, Tg and The fit results for the thixotropic behaviour are given in table 3 and graphically presented in figures 4 and 5 as a function of the applied strain amplitude. The variance in the calculated values is not small. For each strain level there are one or more extreme values. Table 3. Calculated values for the parameters Tg, Tg and Beam Strain [m/m] Tg [-] Tg [-] [/s] E E E E E E E E+06.34E E E+05.80E E+05.99E E E+05.48E+06.76E E E+06.4E E+05.89E E E+05.99E E E E E E+05.88E E E E+05.67E E E+05.90E E+05.6E+06.84E E+05.80E E E E+05.05E E+05.E+06.E E E E+04.E+06 Tga Tga 8.E+06 8.E+05 6.E+05 4.E+05.E+05 6.E+06 4.E+06.E+06 0.E+00 0.E Strain amplitude [m/m] Figure 4. Calculated parameters Tg and Tg as a function of the strain amplitude. In Annex the results are plot as a function of the initial complex stiffness modulus. Another point might be the amount of bitumen for each beam which can differ a lot. 6

7 [/s].3e+05.e E E E E+04.0E Strain amplitude [m/m] Figure 5. Calculated decay parameter as a function of the applied strain amplitude. Results for the fatigue parameters Tg, Tg and endurance limit endurance The fit results for the fatigue performance are given in table 4 and graphically presented in figures 6a, 6b and 7 as a function of the applied strain amplitude. Table 4. Calculated values for the parameters Tg, Tg and endurance Beam Strain [m/m] Tg [-] Tg [-] endurance [m/m] E+.97E E+ 3.00E E+.4E E+ 3.66E E+ 4.45E E+ 4.0E E+.85E E+ 3.07E E+ 4.86E E+ 3.65E E+ 4.53E E+ 4.7E E+ 4.00E E+ 4.88E E+3.49E E+ 4.0E E E E E In figure 6a the extreme high values belong to beam 54 which is really an outlier. Therefore the results for beam 54 are excluded in figure 6b. Nevertheless the results for beam 5 differ also much from the results for the other beams. The values for the endurance limit (figure 7) are quite high with a mean value around 70 micro strain in contrast with the common accepted 50 micro strain. This higher value resulted in a faster decrease of the complex material stiffness modulus as can be seen in the figures for the material modulus Mmat in the accompanying file. 7

8 Endurance limit [m/m] Tg [-] Tg [-] Tg [-] Tg [-] Tgg Tgg.E+4 8.E+4 8.E+3 7.E+4 6.E+4 6.E+3 5.E+4 4.E+4 4.E+3 3.E+4.E+3.E+4.E+4 0.E+00 0.E Strain amplitude [m/m] Figure 6a. Calculated parameters Tg and Tg as a function of the strain amplitude Results for beam 54 included. 3.E+3 Tgg Tgg.6E+4.E+4.E+3.E+3 8.0E+3 4.0E+3 0.E E Strain amplitude [m/m] Figure 6b. Calculated parameters Tg and Tg as a function of the strain amplitude Results for beam 54 excluded Endurance Strain amplitude [m/m] Figure 7. Endurance limit endurance as a function of the applied strain amplitude. 8

9 Discussion of the ACP-F model parameters The results for the six parameters are given in table 3 (thixotropy: Tg, Tgand in table 4 (fatiguetg, Tg and endurance). For the calculation of the mean values and standard deviations the results for beam 54 are excluded. The fatigue life NPH for beam 54 was outside a statistical 99.9% prediction range based on the remaining seventeen tests. Nevertheless the calculated ACP-F parameters for beam 54 with reference to thixotropy are in the range of the values for the other beams. The overall variations in the six parameters are rather high in view of the mean value (beam 54 excluded) and the standard deviation for each parameter. However, for the six green marked beams in table the parameters Tg, Tg and are rather close to each other. The mean values and standard deviations for these six beams are Tg = ( ) 0 5, Tg = ( ) 0 6 and = ( ) 0 4 [s - ] respectively. In case of the parameters Tg and Tg there is more variation. It looks if there are more groups with more or less the same values. Note that the values for beam 54 are much higher, indicating a higher decrease during the evolutions in the loss and storage modulus leading to a short fatigue life. There are five beams (4, 4, 45, 48 and 50) with more or less the same values for Tg and Tg. The mean values and standard deviations for these four beams are Tg = ( ) 0 and Tg = ( ) 0 3 respectively. However, the variation in the endurance strain is high (see table 4). Although the fits with the ACP-F model are really good, it seems that the involved ACP-F parameters are not constant or at least the variations between the beams is rather high. 9

10 Fatigue life [c] Fatigue lives NPH The fatigue lives NPH and the traditional definition Nf,50 are given in table 5 together with the N value used for the fit interval from 3000 to N cycles. Table 5. Fatigue lives NPH and Nf,50 and the values N used for fit interval. Beam Strain [m/m] N [c] NPH [c] Nf,50 [c] These results are presented in figure 8 as a function of the applied strain amplitude (Wöhler curve). The equation differs a bit from the one presented in the publication. This is due to the fact that the strain values differ a bit by taking into account one decimal more. As shown in figure 8 beam 54 is an outlier even if a 99.9 % confidence interval for a new observation is used..e+06 Nph Beam 54 Min90 Max90 N Power (Nph) N PH =.350 * 0 7 * R² = 0.97.E+05.E Strain Amplitude [m/m] Figure 8. Wöhler curve for the fatigue life NPH and the 90% confidence interval. 0

11 Fatigue life [c] Nph Nf,50 Wöhler Nph Wöhler Nf,50.E+06.E+05.E N PH =.350 * 0 7 * R² = 0.97 N f,50 =.5784 * 0 7 * R² = 0.96 Strain Amplitude [m/m] Figure 9. Wöhler curve for the fatigue lives NPH and Nf,50 (beam 54 is excluded) Dissipated energies The integral equations for the evolutions of the loss and storage modulus can be solved analytical in the case of a strain controlled fatigue test (e.g. a UPP or DT/C test in controlled deflection mode using a pure sinusoidal load; see Annex ). In 4PB tests a controlled deflection mode is used. This will not guarantee a constant strain at the surface in the midsection of the beam during the fatigue test. The induced error is small and is ignored in this note. By taking into account the strain distribution in the 4PB beam, it is possible by a numerical integration to calculate the weighted loss and storage modulus for the beam. After fitting the evolutions for the beam data, the evolutions of the complex modulus at the surface in the midsection of the beam is also calculated (see figure ). In table 6 the accumulated dissipated energies at the surface of the beam in the midsection Wdis.mat up from N = 0 to fatigue life NPH are given. Three values are given. The Num. value (column 3) is calculated by a numerical integration of the calculated loss modulus Lmat. times times the squared actual strain at that point over the interval using the calculated data points after the fitting procedure. The Anal. Value (column 4) is calculated by integration over the interval from t = 0 to t = NPH/f using the analytical expression for the evolution of the loss modulus Lmat. times the product of and the squared adopted constant strain for that test (column ). See also Annex. In column 5 the product e is given of the fatigue life NPH and the dissipated energy in the first cycle. Finally in column 6 the ratios of the values in columns 4 and 5 are calculated.this last value is similar to the factor used by Wim van Dijk for the ratio of the dissipated energy in the beam and the initial dissipated energy (AAPT paper). As can be seen in table 6 this value might be considered as a constant value (valid for this mix under the applied conditions).

12 Table 6. The total dissipated energy Wdis.mat. up to NPH at the surface of the beam in the midsection. Ratio = = Beam Wdis.mat Num. Wdis.mat Anal. NPH*Wdis0 Wdis.mat.Anal /(NPH.Wdis0) Code m/m] [MPa] [MPa] [MPa] [-] Mean (-beam 54) 0.8 St.Dev. (- beam 54) As can be seen in table 6 the accumulated dissipated energies in column 3 and 4 do not differ much. More interesting is the fact that the ratio in column 6 is very close to a constant value of 0.8. This implicate that the accumulated dissipated energy Wdis.mat. can be estimate as 0.8 times the product of the fatigue life NPH and the initial dissipated energy Wdis.mat. in the first cycle. The last quantity is for the first cycle equal to times the squared product of the applied strain amplitude and the measured loss modulus Lbeam for the beam in the first cycle. This finding of a constant ratio of 0.8 (valid for the applied circumstances, temperature and frequency) is similar with the findings of W. van Dijk for the ratio of the dissipated energy for the beam up to cycle Nf,50 (see also SPDM 78 and Wim s paper on dissipated energy approach for the AAPT). Another finding by W. van Dijk was the following relationship, which is known as the z dissipated energy law: Wdis. beam A. N f,50. This type of relationship is also applicable to the dissipated energy Wdis.mat. at the surface of the beam in the midsection at fatigue life NPH. z Wdis. mat A. N PH () This relationship is graphically presented in figure 0.

13 Fatigue life [c] Dissipated energy [MPa] Num Anal 00.0 W dis.mat.num. = * N PH R² = E+04.E+05 Fatigue life N PH [c] W dis.mat.anal. = * N PH R² = 0.95 Figure 0. The accumulated dissipated energy Wdis.mat. up to cycle NPH at the surface in the midsection of the beam as a function of cycle NPH. These two experimental findings (Wdis.mat = 0.8 times the product of the initial Wdis.mat. and NPH & Wdis.mat. = A.NPH z ) make it possible to construct the Wöhler curve in another way (like the method used by W. van Dijk). A mean value of 3350 MPa is taken for the loss modulus L0. z W dis.mat = A N PH = 0.03 N 0.6 PH = 0.8 π ε L 0 N PH = 0.8 ε 3350 N PH (3) This is done in figure together with the Wöhler construction using directly the fatigue lives NPH and strain values (see figure 8.). As shown the difference between the two trend lines is extremely small for the applied strain amplitude range from 40 to 40 micro strain. Nevertheless one still have to use more tests at different strain levels in order to compute the coefficients A and z. Another possibility is using the analytical expression for the evolution of the material loss modulus. However, for that option the model parameters of the ACP-F model have to be constants (material parameters). In the next paragraph an alternative manner for obtaining constant values will be outlined in the next paragraph..e+06 Nph Wim van Dijk Directly: N PH =.350 * 0 7 * R² = 0.97.E+05 Wim van Dijk: N PH = 6.98 * 0 6 * -5..E Strain Amplitude [m/m] Figure. Wöhler curves NPH versus. Directly and according to W.van Dijk findings 3

14 Prediction of the Wöhler by using only one 4PB test. As shown in the previous paragraph the accumulated dissipated energy for the material at the surface of the beam in the midsection Wdis.mat can be written as: Wdis. mat L0 N PH (4) The accumulated dissipated energy can also be written as (see Annex ): L D D 0 C B C B f e e C B C B NPH / f 0 (5) Equation (5) can be rewritten as: L D * C B C B D PH 0 C B N / f C B N PH / f f e e (6) The exponential functions times the parameter * turned out to be constants (see Annex ). Therefore equation (6) can be rewritten as: N PH Wdis. mat f L0 * (7) The fatigue parameter * is given by equation (8). * 4 endurance f Tg (8) Combining equations (4), (7) and (8) leads to equation (9) for the prediction. Wdis. mat L0 N PH = f L0 * = f L ( ) 0 4 f Tg Finally equation (0) is obtained which can be used for the prediction of the Wöhler curve. 4 Tg N PH endurance endurance / (0) The quantities, and Tg are calculated using the values obtained in the fitting procedure. This has been done for all 7 beams (beam 54 excluded) leading to 7 predictions for the Wöhler curve. The easiest way is to choice 3 values for the strain (40, 90 and 40 micro strain) and calculate with equation (0) the predicted fatigue lives for the 3 strain values. Finally the power trend line is calculated through these 3 points. The 7 trend lines are given in figure together with the 90% confidence interval from figure 8. Mark that in equation (0) the strain is given in m/m and not in m/m.!!! (9) 4

15 Fatigue life [c].e+06.e+05.e Strain amplitude [m/m] Figure. Power trend lines for the individual beams and the 90% confidence interval. As expected the Wöhler curve for beam 54 (red line) falls far outside the 90% confidence interval. All other individual Wöhler curves, except the pone for beam 5, fall inside the 90% confidence interval. Moreover most of the slopes are close to the mean value of -5.. Conclusion: It is possible to predict the Wöhler curve using only one 4PB test. However the dependency of the fatigue parameters, * must be known!!!! 5

16 Objective procedure for the determination of NPH As mentioned before the determination of the fatigue life NPH is just visual and therefore not really objective as for example the definition for the fatigue life Nf,50. This fatigue life is defined as the number of repetitions at which the stiffness modulus for the beam (Mbeam) has decreased with 50% with respect to the initial value. The initial value is defined as the modulus measured in the 00 th cycle (CEN standard). In this project this last value was not available. Instead the modulus M0 (at t = 0) following out of the fit procedure is adopted as the initial value. In table 7 the ratios of the stiffness modulus for the material at the surface of the beam in the midsection Mmat and the weighted modulus for the whole beam Mbeam with respect to the initial modulus M0 are given for the (visual) determined fatigue life NPH. Table 7. Ratios Mmat/M0 and Mbeam/M0 for all beams at the fatigue life NPH. Beam Mmat/M0 Mbeam/M Mean St.dev The standard deviation is not really small but it seems that, at least for this asphalt mix and circumstances, a definition of 5% decrease in the stiffness modulus for the beam is a reasonable alternative. 6

17 Annex. ACP-F model Instead of the modulus value M and phase lag, the loss modulus L and storage modulus S are chosen as the variables. The proposed changes in the loss modulus and the storage modulus are represented by the following integral equations (A) and (A). dw dw M t sin t L t M e d d t t. dis. 0 0 d d 0 0 * dis t sin (A) t t dw dw M tcos t S t M cos e d d d * dis. t dis. (A) 0 0 d 0 0 In which M{t} is the modulus of the complex stiffness modulus [MN/m ] at time t [s], {t} is the phase lag between strain and stress at time t, L{t} is the loss modulus [MN/m ], S{t} is the storage modulus [MN/m ],,,,, * are parameters. dwdis./d is the dissipated energy per time [MN/m /s]. For a sinusoidal controlled strain test the dissipated energy per cycle Wdis. is given by equation A3. W M L (A3) dis. sin It should be noted that the dissipated energy Wdis. itself is completely transformed into heat. The energy dissipated during a test will consists out of at least two parts: System losses which can be ignored if good calibrated equipment is used. Dissipated energy (Wdis.) which is transformed into heat The creation of micro cracks and micro defects, which will lead to fatigue damage in the form of a decrease in the modulus M and an increase in phase lag will have a relation with energy. The same is assumed to be true for the thixotropic process. In the ACP-F model it is assumed that the energies related to fatigue damage and thixotropic behaviour can be written in the form of the dissipated energy Wdis. as has been done in equations () and (). The first integrals in the right hand of these equations represent the thixotropic phenomenon and the second integrals the fatigue da mage phenomenon. The rate in dissipated energy for a sinusoidal strain controlled fatigue test is approximated by equation A4 which represents the rate in dissipated energy per cycle. dw d dis. W f f M f L (A4) dis. sin( ) { } In strain controlled uni-axial push-pull tests (UPP) or dynamic tension-compression tests (DT/C) it was found that for the mixes used in those tests the coefficients and were linear depending on the strain amplitude. The coefficient * depended on the square of the applied strain amplitude and the coefficients and were nil if the strain amplitude was below a certain value endurance (endurance limit). Above this value the coefficients were linear depending on the difference between the applied strain amplitude and this endurance limit. For the mix at issue no UPP tests were available. The same linear relationships are used for the parameters and of the thixotropic behaviour. However, based on findings for the slope of the Wöhler curve a cubic relationship is adopted for the parameters and of the fatigue damage process(application of a material fatigue damage model in 4PB tests) The following notations are used. 7

18 f f Tg ; f f Tg ; ; * 3 * 3 * ; f f Tg f f Tg * 4 * 4 endurance endurance (A5) The solutions for equations A and A are given by equations A6 and A7. Bt L{ t} M{ t}sin { t} M sin e cosh C t Dsinh C t (A6) 0 0 S{ t} M{ t}cos { t} M cos 0 0 * * 0sin Bt Bt M 0 e sinh C t * e cosh C t E sinh C t C (A7) In which * * * B, * * C B, * * * D, C The final (fitted) parameters in the ACP-F model are given by: * * * E. C The initial values for the modulus (M0) and the phase lag (0). The parameters Tg, Tg and for the thixotropic phenomenon. The parameters Tg, Tg and the endurance limit endurance for the fatigue. The positive sign in front of the integral describing the thixotropic effect in equation () seems odd. However, in the first phase of a (strain controlled) fatigue test the increase in the phase lag and thus sin() can be higher than the decrease in the modulus M (due to fatigue). Often for a short time an increase in the loss modulus is noticed in fatigue tests. The accumulated dissipated energy Wdis.mat at the surface of the beam in the midsection over N cycles is given by (in case of the dissipated energy related to the whole beam the subscript beam is used): N t t N/ f dwdis. mat Wdis. mat dis. mat dis. mat d T W dw d d f L{} d (A8) The loss modulus L{} is rewritten as: L C B C B 0 L{ } D e D e (A9) The accumulated dissipated energy up to cycle NPH is represented by: N/ f L D 0 Wdis. mat. f L{} d f e e C B C B 0 D C B C B NPH / f f L0 * * e B B CD Cosh( C ) C BD Sinh( C ) 0 NPH / f 0 (A0) 8

19 It was experimentally found that, following the example of Wim van Dijk, this amount of dissipated energy could also be expressed as: z Wdis. mat. A N PH (A) And for the asphalt mix at issue, the amount is, again following the example of Wim van Dijk, also approximated by: Wdis. mat. 0.8 L0 N PH (A) Combining equations (A) and A() gives an alternative way for the construction of the Wöhler curve: ( 0.8 π L 0 A ) z ε z N PH = (A3) Note: Given the values for B, C and NPH the second exponential term in equation A0 can be neglected for t = NPH/f. Finally equation (A0) can be rewritten as: L D NPH / f D 0 C B C B f e e f L0 * C B C B 0 (A4) The quantity can be considered as a constant as shown in the next table. Table A. The quantities and as calculated for the individual 4PB test data. Beam Mean St.Dev

20 [/s] Tg [-] Tg [-] Annex Presentation of the ACP-F parameters for thixotropic behaviour Tga Tga.E E+06.0E E E E E E E E E+06.0E+06.0E+05.0E E E Modulus M 0 [MPa].6E+05 Beta.4E+05.E+05.0E E E E+04.0E E Modulus M 0 [MPa] 0