A Novel Master Curve Formulation with Explicitly Incorporated Temperature Dependence for Asphalt Mixtures: A Model Proposal with a Case Study
Abstract
1. Introduction
2. Background
Formulations | Advantages | Disadvantages | References |
---|---|---|---|
GS model (Equation (1)) | These are the most straightforward expressions; their mathematical structure is simple and only requires 4 parameters. Thus, only 4 experimental measurements need to be made for their calibration. This calibration procedure can even be quickly conducted using trial-and-error techniques. | These expressions employ frequency as the only state variable, so they cannot reproduce the stiffness dependence on temperature (or other analogous variables) without incorporating shift factors. When the experimental measurements of the stiffness modulus are carried out in a narrow range of frequencies, it is possible for the calibration of the equations to not converge properly. | [21,22,23] |
CS model (Equation (2)) | |||
SLG model (Equation (3)) | |||
Uniaxial model (Equation (4)) | |||
Generalized logistic sigmoidal model: | Due to the incorporation of an additional fitting parameter (i.e., λ), this model allows adjustment to asymmetric curves. | This formulation holds the limitations of the traditional models (Equations (1)–(4)). Compared to traditional models, this formulation requires an additional experimental measurement for its calibration. Inadequate calibration of λ can lead to the construction of unrealistic non-symmetric master curves. | [28] |
Christensen–Anderson–Marasteanu model: | This model allows the direct specification of the minimum and maximum stiffness moduli through the parameters and , respectively. It permits the high-accuracy reproduction of viscoelastic behaviour. | This formulation holds the limitations of the traditional models (Equations (1)–(4)). Also, the trend of the simulated curves depends on one location parameter () and two tuning parameters ( and ). Counterintuitively, using few fitting parameters increases the complexity of model calibration because if the experimental data are noisy, the associated error can be increasingly propagated. | [29] |
Modified Christensen–Anderson–Marasteanu model: | Regarding the original Christensen–Anderson–Marasteanu model, this enhanced version allows the incorporation of the rheological index (), which can be used as a rotation factor to couple time–temperature–aging shift functions. | This formulation holds the limitations of the original Christensen–Anderson–Marasteanu model. | [4] |
Continuous relaxation spectrum model: | This formulation excels by the incorporation of the continuous relaxation spectrum (through the relaxation time—τ—and angular frequency—ω—parameters) into the calculation of the stiffness modulus. Thus, the model considers the linear viscoelastic properties of asphalt materials. Also, thanks to the use of incremental constitutive relations, the numerical stability of this model is extremely high. | As in the traditional formulations (Equations (1)–(4)), this model does not consider the effects of the temperature without incorporating shift factors. Due to the use of multiple integral equations, this model increases its mathematical complexity excessively, which makes its implementation difficult in more extensive simulations. Therefore, the calibration of this model involves the exploration of non-trivial solutions. | [15] |
Modified Havriliak–Negami model: | This model is especially suitable for asphalt mixtures that develop complex liquid-like and solid-like phases throughout the frequency spectrum. | This formulation holds the limitations of the traditional models (Equations (1)–(4)). The model is complex both from a mathematical and physical standpoint. On the one hand, this formulation should not be calibrated with a simple iteration for error minimization. On the other hand, the physical interpretation of the fitting parameters is non-intuitive, as is the case with conventional models. | [30,31] |
3. Mathematical Modelling
4. Case Study
4.1. Materials
4.2. Results
Fitting Parameters | Accuracy | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
(-) | MAPE (%) | |||||||||
Asphalt Mixtures | HMA-0 | 2.457944 | −0.591082 | 1.103802 | 4.271044 | 0.523182 | −3.721008 | 0.000375 | 0.9920 | 7.34 |
HMA-15 | 2.452728 | −0.731821 | 1.197044 | 4.222030 | 0.599018 | −3.725021 | 0.000410 | 0.9917 | 7.37 | |
HMA-30 | 2.418613 | −0.715055 | 1.169234 | 4.159372 | 0.599273 | −3.705805 | 0.000383 | 0.9916 | 7.21 | |
HMA-45 | 2.140254 | −0.399350 | 1.033227 | 4.008499 | 0.374255 | −3.704619 | 0.000330 | 0.9905 | 7.90 | |
WMA-0 | 2.058495 | 0.127287 | 0.774247 | 4.317418 | −0.176631 | −3.721431 | 0.000266 | 0.9937 | 6.73 | |
WMA-15 | 2.332162 | −0.247539 | 0.906308 | 4.253895 | 0.260827 | −3.720097 | 0.000292 | 0.9923 | 7.09 | |
WMA-30 | 0.938964 | 0.769965 | 0.643573 | 4.225295 | −1.208403 | −3.716841 | 0.000244 | 0.9912 | 7.86 | |
WMA-45 | 2.147436 | −0.449215 | 1.100028 | 4.089946 | 0.396154 | −3.705144 | 0.000392 | 0.9895 | 8.48 | |
Statistical Description | Minimum | 0.938964 | −0.731821 | 0.643573 | 4.008499 | −1.208403 | −3.725021 | 0.000244 | ||
Maximum | 2.457944 | 0.769965 | 1.197044 | 4.317418 | 0.599273 | −3.704619 | 0.000410 | |||
Average | 2.118325 | −0.279601 | 0.990933 | 4.193437 | 0.170959 | −3.714996 | 0.000337 | |||
Median | 0.854465 | 0.706769 | 2.602772 | 2.303886 | −2.456511 | −1.852188 | 0.000353 | |||
Std | 0.468890 | 0.474661 | 0.185565 | 0.095495 | 0.571656 | 0.007880 | 0.000059 | |||
Skewness | −2.309464 | 1.474897 | −0.865064 | −0.863506 | −2.040599 | 0.388598 | −0.386632 | |||
Kurtosis | 5.801566 | 1.973254 | −0.486327 | 0.066587 | 4.247876 | −2.042569 | −1.667467 |
5. Discussion
5.1. Convergence Challenges
5.2. Benchmarking Against the Traditional Approach
5.3. Graphical Representation
5.4. Contributions to the Literature and Practical Applications
5.5. Research Limitations
5.6. Future Research Lines
6. Summary and Conclusions
- The development of the TSI model from thermodynamics-based principles (such as Eyring’s rate theory and activation free energy) and the TTSP proves that it is possible to accurately simulate the stiffness modulus of asphalt mixtures by explicitly accounting for both temperature and frequency as state variables. Specifically, the TSI model improves upon the traditional GS model by introducing three thermal-related fitting parameters (i.e., , , and ), which enable a more precise and physically grounded characterization of the temperature-dependent behaviour of asphalt mixtures.
- The fitting parameters and capture the influences of the activation entropy and activation enthalpy, respectively. Meanwhile, introduces transition rate control and asymmetry into the model, enabling a more realistic representation of thermo-rheological behaviour in asphalt mixtures with complex mechanical responses; the preceding is particularly useful to account for effects caused by polymer modification, aging, and additive incorporation.
- For the addressed case study, the TSI model delivers highly accurate predictions of the asphalt mixture’s stiffness modulus, with R2 scores above 0.98 and MAPE values below 8.5%. In particular, the TSI model demonstrated exceptional computational stability, with a non-convergence rate of just 0.04% across 8000 runs, making it 4 to 45 times more stable than traditional master curve models such as GS, uniaxial, CS, and SLG. This level of stability is remarkably high, especially considering that the calibration was performed over a narrow frequency spectrum (i.e., from 0.5 to 8 Hz), a condition under which conventional models often struggle. This superior performance is attributed to its dual-sigmoidal structure and foundation on the (also stable) GS model, allowing for localized fitting and improved robustness during calibration.
- A benchmarking analysis confirmed that the proposed TSI model offers greater reliability than the traditional approach, which builds a master curve at a reference temperature and further relies on shift factors (e.g., Arrhenius or WLF formulations) to estimate stiffness at target temperatures. While the traditional method produced larger prediction errors (e.g., exceeding 200% at 40 °C), the TSI model maintained consistent accuracy across all loading conditions, with generally lower MAPE values and higher R2 scores, revealing its superior robustness and predictive capability.
- The TSI model lays the foundation for advancing beyond the traditional master curve concept toward a new notion introduced in this study, i.e., the SPS. Basically, the SPS can be defined as a continuous hypersurface representing the full range of possible stiffness modulus values within the boundaries of the experimentally assessed loading conditions.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Formulations | Best Temperature Range | Unreliable Temperature Range | Notes | References |
---|---|---|---|---|
Arrhenius (Equation (6)) | Low temperatures (glassy, below ) | At or above (rubbery) | Assumes constant activation energy; suitable for glassy regime. | [36,45] |
WLF (Equation (7)) | Around and above | Deep in glassy region (low T) | Empirical constants; best near Tg. | [17,18] |
Kaelble: | Below and around (sub-) | Far above (highly rubbery region) | More symmetric than WLF for sub-Tg; useful for describing both sides of Tg. | [38,41] |
Log-linear: | Wide temperature range (empirical) | May lack physical basis at extremes | Simple, flexible; used for practical fitting. | [17] |
Quadratic polynomial: | Wide temperature range (empirical) | May overfit or lack physical meaning | Useful for complex blends or modified asphalts. | [17,39] |
Asphalt Mixtures a | |||||||||
---|---|---|---|---|---|---|---|---|---|
HMA-0 | HMA-15 | HMA-30 | HMA-45 | WMA-0 | WMA-15 | WMA-30 | WMA-45 | ||
Gravimetric composition | Fine NAs (%) | 47.80 | 47.70 | 47.50 | 47.25 | 47.80 | 47.75 | 47.60 | 47.40 |
Coarse NAs (%) | 47.80 | 40.55 | 33.25 | 25.99 | 47.80 | 40.59 | 33.32 | 26.07 | |
Fine RCA (%) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
Coarse RCA (%) | 0 | 7.16 | 14.25 | 21.26 | 0 | 7.16 | 14.28 | 21.33 | |
Asphalt binder (%) | 4.40 | 4.60 | 5.00 | 5.50 | 4.40 | 4.50 | 4.80 | 5.20 | |
Laboratory characterization | Mixing temperature (°C) | 160.00 | 160.00 | 160.00 | 160.00 | 120.00 | 120.00 | 120.00 | 120.00 |
Compaction temperature (°C) | 140.00 | 140.00 | 140.00 | 140.00 | 110.00 | 110.00 | 110.00 | 110.00 | |
Bulk density (kg/m3) | 2361.04 | 2351.37 | 2331.33 | 2322.25 | 2363.12 | 2333.98 | 2337.34 | 2317.63 | |
VFA (%) | 69.69 | 70.54 | 68.62 | 66.27 | 70.58 | 70.84 | 69.40 | 67.75 | |
VMAs (%) | 14.32 | 15.41 | 15.71 | 16.10 | 14.65 | 15.50 | 15.85 | 16.28 | |
VTMs (%) | 4.34 | 4.54 | 4.93 | 5.43 | 4.31 | 4.52 | 4.85 | 5.25 | |
Marshall flow value (mm) | 3.17 | 3.65 | 6.32 | 6.86 | 2.62 | 2.96 | 6.29 | 6.39 | |
Marshall stability (kN) | 15.13 | 14.57 | 13.74 | 12.88 | 15.93 | 15.22 | 14.60 | 13.89 |
Asphalt Mixtures | Number of Non-Convergent Runs | ||||
---|---|---|---|---|---|
GS Model | CS Model | SLG Model | Uniaxial Model | TSI Model | |
HMA-0 | 2 | 22 | 10 | 4 | 0 |
HMA-15 | 1 | 12 | 17 | 2 | 0 |
HMA-30 | 2 | 19 | 24 | 3 | 0 |
HMA-45 | 1 | 17 | 26 | 1 | 0 |
WMA-0 | 2 | 13 | 21 | 3 | 1 |
WMA-15 | 1 | 19 | 11 | 1 | 0 |
WMA-30 | 2 | 22 | 16 | 2 | 0 |
WMA-45 | 2 | 14 | 18 | 2 | 2 |
Total | 13 | 138 | 143 | 18 | 3 |
Non-convergence (%) | 0.16 | 1.73 | 1.79 | 0.23 | 0.04 |
Traditional model/proposed model ratio (-) | 4 | 43 | 45 | 6 |
Asphalt Mixtures | Fitting Parameters | |||
---|---|---|---|---|
HMA-0 | 3.907667 | −5.1 | 5.1 | 4.165368 |
HMA-15 | 3.853796 | −5.1 | 5.1 | 4.108047 |
HMA-30 | 3.804261 | −5.1 | 5.1 | 4.051765 |
HMA-45 | 3.673245 | −5.1 | 5.1 | 3.886199 |
WMA-0 | 3.961070 | −5.1 | 4.740254 | 4.262092 |
WMA-15 | 3.908768 | −5.1 | 5.1 | 4.160014 |
WMA-30 | 3.858396 | −5.1 | 4.983891 | 4.141897 |
WMA-45 | 3.731134 | −5.1 | 5.1 | 3.933298 |
Asphalt Mixtures | Arrhenius Model | WLF Model | |
---|---|---|---|
(J/mol) | (-) | (K) | |
HMA-0 | 285,678.66 | 11.97 | 76.61 |
HMA-15 | 288,933.55 | 12.10 | 76.54 |
HMA-30 | 314,577.16 | 13.13 | 76.35 |
HMA-45 | 513,774.30 | 38.10 | 68.76 |
WMA-0 | 263,700.42 | 14.30 | 96.19 |
WMA-15 | 293,735.47 | 12.29 | 76.51 |
WMA-30 | 261,119.83 | 14.09 | 95.77 |
WMA-45 | 513,693.15 | 31.54 | 70.89 |
Asphalt Mixtures | Arrhenius Model | WLF Model | ||||
---|---|---|---|---|---|---|
HMA-0 | 5.5553 × 102 | 6.2733 × 101 | 5.6158 × 10−4 | 8.2233 × 102 | 6.2733 × 101 | 3.3214 × 10−3 |
HMA-15 | 5.9701 × 102 | 6.5762 × 101 | 5.1568 × 10−4 | 8.8825 × 102 | 6.5762 × 101 | 3.1187 × 10−3 |
HMA-30 | 1.0528 × 103 | 9.5351 × 101 | 2.6339 × 10−4 | 1.6254 × 103 | 9.5351 × 101 | 1.8790 × 10−3 |
HMA-45 | 8.6333 × 104 | 1.7087 × 103 | 1.4257 × 10−6 | 4.2690 × 1010 | 3.0466 × 106 | 2.6027 × 10−9 |
WMA-0 | 3.4162 × 102 | 4.5625 × 101 | 9.9883 × 10−4 | 4.3861 × 102 | 4.5625 × 101 | 3.4546 × 10−3 |
WMA-15 | 6.6392 × 102 | 7.0500 × 101 | 4.5472 × 10−4 | 9.9453 × 102 | 7.0500 × 101 | 2.8353 × 10−3 |
WMA-30 | 3.2267 × 102 | 4.3951 × 101 | 1.0687 × 10−3 | 4.1402 × 102 | 4.3951 × 101 | 3.6779 × 10−3 |
WMA-45 | 8.6178 × 104 | 1.7067 × 103 | 1.4288 × 10−6 | 2.9222 × 108 | 1.5148 × 105 | 1.1457 × 10−7 |
Asphalt Mixtures | GS Model (20 °C) | Arrhenius Model | WLF Model | ||||
---|---|---|---|---|---|---|---|
5 °C | 10 °C | 40 °C | 5 °C | 10 °C | 40 °C | ||
HMA-0 | 1.66 | 16.52 | 6.35 | 205.77 | 16.52 | 6.35 | 205.71 |
HMA-15 | 1.56 | 17.43 | 6.48 | 205.76 | 17.42 | 6.47 | 205.78 |
HMA-30 | 1.47 | 17.82 | 6.86 | 206.62 | 17.82 | 6.86 | 206.62 |
HMA-45 | 1.37 | 22.42 | 11.61 | 213.28 | 22.42 | 11.60 | 213.25 |
WMA-0 | 0.79 | 9.08 | 4.40 | 204.81 | 9.10 | 4.40 | 204.77 |
WMA-15 | 1.33 | 17.22 | 6.83 | 202.47 | 17.22 | 6.81 | 202.49 |
WMA-30 | 1.78 | 11.93 | 3.69 | 209.99 | 11.95 | 3.69 | 209.93 |
WMA-45 | 1.20 | 24.70 | 12.81 | 214.49 | 24.73 | 12.82 | 214.48 |
Asphalt Mixtures | GS Model (20 °C) |
---|---|
HMA-0 | 0.9470 |
HMA-15 | 0.9522 |
HMA-30 | 0.9542 |
HMA-45 | 0.9458 |
WMA-0 | 0.9887 |
WMA-15 | 0.9621 |
WMA-30 | 0.9523 |
WMA-45 | 0.9514 |
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Martinez-Arguelles, G.; Casas, D.; Peñabaena-Niebles, R.; Guerrero-Bustamante, O.; Polo-Mendoza, R. A Novel Master Curve Formulation with Explicitly Incorporated Temperature Dependence for Asphalt Mixtures: A Model Proposal with a Case Study. Infrastructures 2025, 10, 227. https://doi.org/10.3390/infrastructures10090227
Martinez-Arguelles G, Casas D, Peñabaena-Niebles R, Guerrero-Bustamante O, Polo-Mendoza R. A Novel Master Curve Formulation with Explicitly Incorporated Temperature Dependence for Asphalt Mixtures: A Model Proposal with a Case Study. Infrastructures. 2025; 10(9):227. https://doi.org/10.3390/infrastructures10090227
Chicago/Turabian StyleMartinez-Arguelles, Gilberto, Diego Casas, Rita Peñabaena-Niebles, Oswaldo Guerrero-Bustamante, and Rodrigo Polo-Mendoza. 2025. "A Novel Master Curve Formulation with Explicitly Incorporated Temperature Dependence for Asphalt Mixtures: A Model Proposal with a Case Study" Infrastructures 10, no. 9: 227. https://doi.org/10.3390/infrastructures10090227
APA StyleMartinez-Arguelles, G., Casas, D., Peñabaena-Niebles, R., Guerrero-Bustamante, O., & Polo-Mendoza, R. (2025). A Novel Master Curve Formulation with Explicitly Incorporated Temperature Dependence for Asphalt Mixtures: A Model Proposal with a Case Study. Infrastructures, 10(9), 227. https://doi.org/10.3390/infrastructures10090227