The presentation and analysis of round robin results are performed in two steps. First, in order to provide overall trends and reproducibility of the measured complex Young’s modulus and complex Poisson’s ratios, unprocessed results are compared. In a second step, an analysis based on simulations using the 2S2P1D model in 3 dimensions [28] is proposed. As small differences in testing conditions do exist between the laboratories, any direct comparison may bring non-realistic conclusions. To overcome this drawback, the 2S2P1D model is used to quantify the differences between the laboratories.
Measured complex Young’s modulus (E*) and complex Poisson’s ratios (ν*)
Black diagrams and Cole-Cole plots
Experimental results for complex moduli and complex Poisson’s ratios from the 5 laboratories are plotted in Fig. 5 in Black diagrams and Col-Cole plots. The advantage of these types of plots is that data comparison is not affected by eventual errors due to temperature appreciation for thermorhelogically simple materials [i.e. which respect the time–temperature superposition principle (TTSP)]. Two samples were tested by lab1 (lab1_sp1 and lab1_sp2) and one sample by other laboratories. In this Figure, the complex Young’s modulus and complex Poisson’s ratio results, obtained from measurements of all laboratories, appear rather close for all plots. The complex Young’s modulus values reveal more scattering in both types of axes.
As can be seen in Fig. 5, complex Poisson’s ratios values are a function of both temperature and frequency. In addition all data are situated along a unique curve on each Figure, which show that TTSP is respected for this parameter. Except for test results involving highest temperatures, the complex Poisson’s ratio phase angle values are negative, signifying that the opposite radial strain lags behind the axial strain.
The complex Poisson’s ratio values presented in Fig. 5 are between 0.25 and 0.48, which is a significant range for this parameter. Viscous and thermo-susceptibility effects are then of utmost importance and should not be ignored. From both the Black diagrams and the Cole-Cole plots it can be observed that the scattering of the complex Poisson’s ratio is smaller at low temperatures and/or high frequencies, i.e. the norm falls in the range 0.25–0.27, and the phase angle gets close to 0 (see also Fig. 8). On the other hand, at high temperatures and/or low frequencies higher dispersion appears (values between 0.35 and 0.48), whereas the phase angle progressively becomes positive (Fig. 8). It is emphasized that results appear less dispersed when the norm values are lower, even though in such conditions the smaller transverse strain values should bring higher measurement noise in the results.
Furthermore, the Poisson’s ratio values of lab2 and lab3 are closer for direction 2 (Fig. 5c–f), and often seem to be slightly lower than the one measured by lab1. One contributing factor could be that the strain gages glued around the sample may restrain transversal deformation. Based on the testing program, this question could not be properly answered.
The difference in Poisson’s ratios in both directions, that allows to check whether the tested material is isotropic regarding transversal deformation, can only be obtained from lab1 and lab3 data. Figure 6 shows the relationship between the norm of the complex Poisson’s ratio values in direction 2 (|ν
2|) and the norm values in direction 3 (|ν
3|) from 3 tests. Results from lab1 are mainly above the equality line, while those from lab3 are somewhat below. The maximum absolute difference in the norms of the complex Poisson’s ratios to the equality line is less than 0.05. As the magnitude of the applied axial strain amplitude was close to 50 μm/m, 0.05 difference in Poisson’s ratio values corresponds to a difference of about 0.19 μm in diameter amplitude variation for the tested sample. Consequently, it can be considered that the difference is quite small and situated within the accuracy range of the experimental procedure. Then, the authors believe that this small difference is related to measurement accuracy. From our results, it is not possible to conclude about the anisotropic behaviour of the material. Meanwhile, if the behaviour is anisotropic, the anisotropy doesn’t create a difference between Poisson’s ratio values in direction 2 and 3.
Master curves of complex Young’s modulus and complex Poisson’s ratios
Master curves of complex Young’s modulus and complex Poisson’s ratios can be used when TTSP is respected. Then only one variable, the equivalent frequency (freq) takes into account the effect of both temperature and frequency. Equivalent frequency is the product of shift factor a
Tref(T), which depends only on the temperature (T) and the chosen reference temperature (T
ref), by frequency (Eq. 9)
$${\text{fr}}_{\text{eq}} = a_{T} \times {\text{fr}}$$
(9)
Master curves obtained from the results of the different laboratories were considered at a reference temperature of 0 °C.
Figure 7 shows the values of obtained experimental shift factors (a
TE) as a function of temperatures established for each laboratory. The results from each laboratory follow the same trend and are relatively close.
It should be underlined that shift factor values for complex Poisson’s ratios (directions 2 and 3) and complex Young’s modulus are the same. This confirms the results presented by [3, 28–32], who already showed the validity of TTSP for Poisson’s ratio measurements. They also showed that shift factors used to build the Poisson’s ratio master curve are very close to those (aTE) obtained for the complex modulus, that can be considered as identical values.
Figure 8 shows the master curves of the complex Young’s modulus and complex Poisson’s ratios obtained for each laboratory involved in the round robin program. The norm of the complex modulus increases as a function of frequency and, inversely, it decreases as a function of temperature, which was expected. It’s phase angle φ
E increases as a function of frequency up to a given maximum, and then decreases.
Differences were observed in the complex modulus master curves between the laboratories. Contrary to what was observed in Fig. 5, the complex modulus master curves of lab3 show a clear difference with those of lab1. As Fig. 5 shows close results for lab1 and lab3, the observed differences in the master curves can be attributed to temperature measurement error between the two laboratories. This point is confirmed because a shift along the equivalent frequency axis makes the curves from the 2 laboratories identical. The shift value is 0.31, which, from Fig. 7, gives a temperature error of 3 °C between the 2 laboratories.
The techniques used to measure the testing temperature at the sample surface could mainly explain the gap. Lab1 used a PT100 rubber coated temperature probe put on the sample surface and held in place with a rubber band. In contrast, lab3 used a PT100 uncoated temperature probe, also placed on the sample surface and held in place with a rubber band. The utmost importance of a correct temperature conditioning and measurement is then again to be stressed.
Simulation and comparison using the 2S2P1D model
Presentation of 2S2P1D model and calibration from results of test Lab1_sp1
As measurements from each laboratory are not performed at exactly the same temperatures and same frequencies, it is not possible to compare the data directly. It was decided by the group to compare all data to a common reference given by the 2S2P1D (2 Springs, 2 Parabolic creep elements and 1 Dashpot in one dimension) linear viscoelastic model. The calculated 2S2P1D values can be obtained for any experimental frequency and temperature condition and compared with experimental data.
The 2S2P1D model, developed at the University of Lyon/ENTPE, is a generalization of the Huet-Sayegh model. The 2S2P1D model is based on a simple combination of physical elements (spring, dashpot and parabolic elements). The graphical representation of the 2S2P1D model is given in Fig. 9. It is widely used to model the linear viscoelastic unidimensional or tridimensional behavior of bituminous materials (including binders, mastics and mixes) [33–38].
The 2S2P1D analytical expression of complex Young’s modulus and the Poisson’s ratio, at a specific temperature, is given by Eqs. 10 and 11.
$${E}_{{{\rm 2S2P1D}}}^{{*}} {(\omega ) = E}_{{{00}}} { + }\frac{{{E}_{{0}} { - E}_{{{00}}} }}{1 + \delta ({\rm j}\omega \tau _{{E}} )^{{ - k}}+ ({\rm j}\omega \tau _{{E}} )^{{{ - h}}} + ({\rm j}\omega \beta \tau _{{E}} )^{{ - 1}}}$$
(10)
$${\nu }_{{{i/{\rm 2S2P1D}}}}^{{*}} {(\omega ) = \nu }_{{{i00}}} { + }\frac{{{\nu }_{{{i0}}} { - \nu }_{{{i00}}} }}{{{1 + \delta (j\omega \tau }_{{\nu }} {)}^{{{ - k}}} { + (j\omega \tau }_{{\nu }} {)}^{{{ - h}}} { + (j\omega \beta \tau }_{{\nu }} {)}^{{{ - 1}}} }}$$
(11)
where: j is the complex number defined by j2 = −1, ω is the angular frequency, ω = 2πf, (f is the frequency), k, h are the constant exponents such that 0 < k < h < 1, δ is the constant, E
00 the static modulus when ω → 0, E
0 the glassy modulus when ω → ∞, ν
i00, the static Poisson’s ratio in direction “i” when ω → 0 (for i = 2 and 3), ν
i0 is the glassy Poisson’s ratio in direction “i” when ω → ∞ (for i = 2 and 3), β is the parameter linked with η, the Newtonian viscosity of the dashpot, η = (E
0 − E
00) βτ
E
, τ
E
and τ
ν
are the characteristic time values, which are only parameters depending on temperature and have a similar evolution:
$${\tau }_{{E}} \left( {T} \right) = a_{{T}} \left( {T} \right).\tau_{{0E}} \quad {\rm and}\quad \tau _{\nu } \left( T \right) = a_{T} \left( T \right).\tau _{{0\nu }}$$
(12)
where a
Tref(T) is the shift factor at temperature T, τ
E
= τ
0E
and τ
ν
= τ
0ν
at reference temperature T
ref. Ten constants (E
00, E
0, δ, k, h, β, ν
i00, ν
i0, τ
0E, τ
0ν) are required to completely characterize the 3D LVE properties (with isotropy hypothesis) of the tested material at a given temperature. The evolutions of τ
E and τ
ν were approximated by the WLF equation [39] (Eq. 13). τ
0E and τ
0ν were determined at the chosen reference temperature T
ref = 0 °C. When the temperature effect is considered, the number of constants becomes twelve, including the two WLF constants (C
1 and C
2 calculated at the reference temperature).
$${\text{log}}\left( a_{T} \right) = - \frac{C_{1} \left( T - T_{{\text{ref}}}\right)}{C_{2} + T - T_{{\text{ref}}}}$$
(13)
2S2P1D constants were fitted using results from lab1_sp1 sample. The 2S2P1D and WLF constants are reported in Table 3. Simulation curves obtained from 2S2P1D model are also plotted in Figs. 5, 7 and 8 together with experimental data points.
Table 3 2S2P1D parameters and WLF constants set at 0 °C in accordance with data of lab1_sp1 tested sample
Difference between experimental results and 2S2P1D simulated values
The relative differences between the calibrated WLF values, using constants of Table 3, and the corresponding experimental data for shift factor (aT) are presented in Fig. 10. What should be observed to characterize reproducibility of the test is the difference between the different data points and not the obtained relative difference values. These last values give information on quality of the simulation for each test condition. If results from lab2 are not considered, the differences between values for other specimens are within a range of ±25 % on the whole frequency and temperature range. This value is quite small when comparing to the range of variation of the shift factor parameter, which covers more than 10 decades [from 3 × 10−6 to 4 × 10+4 (Fig. 7)]. Larger differences observed for the specimen from Lab2 up to 150 % are obtained at higher temperatures.
The complex moduli and complex Poisson’s ratios are calculated with the 2S2P1D model considering exact temperature and frequency values for each data condition. Obtained values are noted with subscript “2S2P1D lab1_sp1” indicating that the calibration was performed using specimen lab1_sp1; |E
*|2S2P1D lab1_sp1, |ν
*2
|2S2P1D lab1_sp1, |ν
*3
|2S2P1D lab1_sp1, φ
E 2S2P1D lab1_sp1, φ
ν2 2S2P1D lab1_sp1 and φ
ν3 2S2P1D lab1_sp1. Figure 11 shows the relative differences between simulated and experimental values for complex modulus absolute (norm) values (Fig. 11a) and differences between simulated and experimental values for the 5 other parameters (Fig. 11b–f).
A first glance on Fig. 11 shows that, for all equivalent frequencies, points having the lowest difference values are from test lab1_sp1. This result was expected as the calibration of the model is made from the data of this test. The rather low difference values for this specimen indicate that 2S2P1D is able to simulate correctly the observed behavior on the whole range of temperatures and frequencies. For all 6 parameters, simulation results are better for low temperature and/or high frequencies.
Comparison between results from the different tests should consider the thickness of the clouds of points (i.e. the range of variation) and not its position on the y axis. Figure 11a shows that the scattering of the relative difference in the values of modulus increases for low values of reduced frequencies (a
T
··fr) and reach an overall difference of 250 % (between +200 and −50 % at a
T
·fr = 10−6). Complex Young’s modulus values of lab3 and lab5 have the maximum deviation. This large difference can be explained for lab3 by an error in sample temperature measurements, as explained further (see Fig. 12). Figure 11c, e show that differences in the norms of the complex Poisson’s ratios are smaller than about ±0.05. As already noted in Sect. 4.1.1, this value is in the range of the accuracy limit of measurement systems and should be considered as good reproducibility.
It was noted previously that lab3 may have recorded incorrect temperature measurements. In Sect. 4.1.2, it is estimated that the temperature error is about 3 °C. 2S2P1D values for lab3 were then recalculated considering a shift in temperature of −3 °C and −2 °C. Differences between experimental values of lab3 and 2S2P1D values, calculated at −3 °C and −2 °C, are plotted in Fig. 12. Previous difference values for lab1 (determined at 0 °C) are also plotted in Fig. 12. As compared with results of Fig. 11 scattering of results are considerably reduced, which confirms the probable error of 2 to 3 °C in temperature measurement between the two laboratories, confirming the importance of accurate temperature measurements.