Front. Mater. Frontiers in Materials Front. Mater. 2296-8016 Frontiers Media S.A. 1751897 10.3389/fmats.2026.1751897 Original Research Research on factors influencing high temperature deformation of asphalt mixture and prediction modeling Hu et al. 10.3389/fmats.2026.1751897 Hu Zhichen 1 Writing – review and editing Methodology Writing – original draft Conceptualization Ye Kai 1 Writing – original draft Investigation Writing – review and editing Zuo Zhaonan 1 Writing – original draft Data curation Conceptualization Investigation Huang Richeng 2 Writing – review and editing Conceptualization Investigation Liao Defu 1 Conceptualization Investigation Writing – original draft He Xinyi 2 Methodology Writing – review and editing Formal Analysis Rong Hongliu 2 * Investigation Writing – original draft Supervision Engineering Department, Guangxi G-Energy Engineering Consulting Group Co. Ltd., Nanning, China School of Civil and Architectural Engineering, Guangxi University, Nanning, China Correspondence: Hongliu Rong, ronghongliu@gxu.edu.cn 09 02 2026 2026 13 1751897 24 11 2025 23 12 2025 02 01 2026 Copyright © 2026 Hu, Ye, Zuo, Huang, Liao, He and Rong. 2026 Hu, Ye, Zuo, Huang, Liao, He and Rong https://creativecommons.org/licenses/by/4.0/ This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This study addresses the root causes of rutting defects by examining the influence of environmental temperature, load, water-heat coupling, and the maximum nominal particle size of aggregates on the permanent deformation of asphalt mixtures. Through testing rutting specimens under varying experimental conditions, the patterns of permanent deformation were analyzed. The results indicate that temperature, water-heat coupling, load, and aggregate nominal particle size are significant factors affecting rutting depth and dynamic stability, with load exerting the greatest impact, followed by water-heat coupling. Using machine learning algorithms, a predictive model was developed to estimate rutting depth and dynamic stability, enabling accurate evaluation of the high-temperature deformation capacity of asphalt mixtures.

 asphalt mixture high temperature deformation influencing factors prediction model rutting time curve The author(s) declared that financial support was received for this work and/or its publication. This study was financially supported by Guangxi Key Technologies R&D Program (Grant No. GuikeAB24010355). section-at-acceptance Structural Materials Introduction

In recent years, China’s highway network has expanded significantly, surpassing 5 million kilometers in total length by the end of 2023. Notably, asphalt pavement accounts for over 90% of high-grade highways. As a viscoelastic material, asphalt mixtures exhibit temperature-dependent mechanical properties, leading to varying performance across different thermal conditions and associated pavement distresses. Insufficient high temperature stability of asphalt mixture leads to permanent deformation of asphalt pavement under the combined effect of environment and traffic load (Zhengqi et al., 2023), which is one of the most prominent and hazardous bottleneck problems of asphalt pavement in China, especially in the high temperature areas in the south of our country, the rutting of pavement seriously affects the comfort and safety of road traffic (Jie et al., 2019; Xingyang et al., 2023; Xunli et al., 2024; Yu et al., 2024). Large-scale rutting distress significantly compromises road ride comfort, driving safety, and pavement service life, leading to substantial economic losses in highway maintenance and broader national infrastructure development. Against this backdrop, investigating the factors influencing the high-temperature performance of asphalt mixtures is crucial for advancing the understanding of their thermal behavior. Such research not only provides a theoretical foundation for materials science and road engineering but also facilitates technological advancements in pavement design and maintenance.

In contemporary research on the high-temperature performance of asphalt mixtures, Sousa et al. (1991) demonstrated that the shear stress in asphalt mixtures increases markedly under combined horizontal and vertical loading. They attributed high-temperature instability in asphalt pavements primarily to the surface layer’s shear stress exceeding its critical threshold. Qiaosheng et al. (2010) observed that as stress increases, the elastic strain in asphalt concrete pavements increases, the time to shear damage is reduced, and the strain develops more rapidly, indicating that excessive loading significantly contributes to permanent deformation in asphalt pavements. Tang-Baoli et al. (2021) assessed the high-temperature performance of asphalt mixtures through rheological number testing and rutting sensitivity testing under repeated compression-shear loading conditions, demonstrating that the rheological number, cumulative permanent deformation (as a function of rheological cycles), and rutting deformation rate serve as effective evaluation indices. Zhengqi et al. (2014) through repeated loading creep tests, demonstrated that elevated temperatures accelerate cumulative deformation development, indicating that temperature rise directly exacerbates high-temperature deformation in asphalt pavements.

In recent years, machine learning techniques have been increasingly applied to predict the high-temperature performance of asphalt mixtures due to their strong capability in handling multivariable and nonlinear problems. Random Forest (RF) models have been widely adopted because of their robustness to overfitting and ability to evaluate variable importance in high-dimensional datasets (Belgiu and Drăguţ, 2016). Support Vector Machine (SVM) models have demonstrated high prediction accuracy for nonlinear mechanical behavior, particularly under small-sample conditions (Ming-guang et al., 2004). In addition, Gradient Boosting Decision Tree (GBDT) models have shown superior performance in capturing complex interactions among temperature, loading, and material-related factors through iterative error minimization (Zhang et al., 2024).Despite these advances, existing machine learning models still exhibit certain limitations. Tree-based models such as RF and GBDT rely on fixed ensemble structures and may suffer from reduced generalization performance when multicollinearity or noise is present, while SVM models require careful kernel and parameter selection, leading to increased computational cost (Du, 2022; Liao et al., 2025; Liu et al., 2024).To address these limitations, this study employs both Linear Regression (LR) and Multilayer Perceptron (MLP) models for high-temperature deformation prediction. The LR model provides a transparent and interpretable framework for quantifying the linear contribution of individual influencing factors, serving as a baseline model for comparison (Liu et al., 2023). In contrast, the MLP model enables flexible structural optimization through multilayer architecture design and backpropagation-based weight updating, allowing it to capture high-order nonlinear relationships among multiple influencing factors. Furthermore, MLP performs implicit variable selection by adjusting connection weights during training, thereby improving prediction efficiency and scalability when dealing with complex deformation mechanisms (Liu et al., 2024).In summary, while RF, SVM, and GBDT models are effective for high-temperature performance prediction, the combined use of LR and MLP in this study provides both interpretability and nonlinear modeling capability, offering a more comprehensive and efficient framework for analyzing permanent deformation behavior of asphalt mixtures.

In summary, numerous scholars have studied the effects of various factors, such as temperature, load, number of loadings, gradation, and aggregate properties, on the high-temperature performance of asphalt mixtures. However, these studies typically examine each factor in isolation rather than considering the combined influence of multiple factors simultaneously. This lack of integrated research hinders efforts to comprehensively identify which specific factor has the greatest impact on high-temperature performance, and consequently limits the development of effective strategies to address rutting defects at their root cause. To fundamentally address rutting distress and enhance the long-term durability of asphalt pavements, this study systematically investigates the effects of ambient temperature, loading conditions, hydrothermal coupling, and maximum nominal aggregate particle size (MNPS) on the permanent deformation of asphalt mixtures. By examining the interactions among these key variables, this research provides a holistic understanding of the complex mechanisms that govern rutting behavior under high-temperature conditions. Three gradations—AC-13, AC-16, and AC-20—were prepared using conventional petroleum asphalt as the binder to simulate typical field conditions. Rutting tests were conducted under varying thermal and loading conditions to evaluate permanent deformation, with rutting depth (RD), dynamic stability (DS), and the difference in rutting depth at R60-R45 serving as key performance indicators. Furthermore, to account for the complex interactions between these influencing factors, machine learning algorithms were employed to model rutting depth and dynamic stability. This paper establishes a robust predictive framework based on the application of algorithms such as the Multilayer Perceptron (MLP) and Linear Regression (LR), capable of capturing the non-linear relationships between multiple input variables and output performance metrics. This machine learning models were trained using the aforementioned influencing factors as input variables, with the aim of offering a more accurate, data-driven approach for predicting the permanent deformation behavior of asphalt mixtures under high-temperature conditions.

 Materials and methods Materials Asphalt and aggregate

70# asphalt was selected as the reference asphalt, and its technical indexes are shown in Table 1.

Test results of technical indicators for ordinary asphalt.

Items Unit Specification limits Value Test methods
Penetration (25 °C, 100 g,5s) 0.1 mm 60–80 72.1 T0604-2011
Softening point °C ≥46 49.7 T0606-2011
Rotational viscosity (135 °C) Pa·s <3 1.205 T0625-2011
Ductility (10 °C, 5 cm/min) cm ≥100 >100 T0605-2011
Density (15 °C) g/cm3 — 1.032 T0603-2011

This study employed limestone as both coarse and fine aggregates, with the results of their performance tests presented in Tables 2, 3.

Testing results of basic technical indicators of limestone coarse aggregate.

Items Specification limits Value Test methods
Crushing value (%) ≤28 22 T0314
Los Angeles Abrasion (%) ≤30 21.5 T0317
Apparent specific gravity (g/m3) ≥2.5 2.86 T0304
Water absorption (%) ≤3.0 0.56 T0307
Needle and flake content (%) ≤18 9.6 T0312
Ruggedness (%) ≤12 2.3 T0314

Testing results of basic technical indicators of limestone fine aggregate.

Items Specification limits Value Test methods
Apparent relative density ≥2.5 2.546 T0304
Robustness (>0.3 mm Part) (%) ≥12 13.7 T0314
Silt content (<0.075 mm Content) (%) ≤3 1.1 T0354
Sand equivalent (%) ≥60 72.96 T0334
Methylene blue value (g/kg) 25≤ 20.6 T0349
Angularity (Flow time) (s) ≥30 54 T0345

The mineral powder used in the test was limestone, with its fundamental performance indicators as shown in Table 4.

Testing results of basic technical indicators of mineral powder.

Items Specification limits Value Test methods
Apparent density (g/cm3) ≥2.5 2.703 T0352
Hydrophilicity coefficient (Ring and ball method 5 °C) <0.8 0.6 T0353
Plasticity index (135 °C) <4 3 T0354
Moisture content (%) ≤1 3
Heat stability No change upon heating Actual measurement records T0355
Mix design

Mineral gradation

In accordance with the requirements of JTGE20-2011, the Marshall test method was used for the mix design. The aggregate gradation types used were AC-13, AC-16, and AC-20, as shown in Figure 1.

Optimal oil-to-stone ratio

(a) AC-13 grading curve; (b) AC-16 grading curve; (c) AC-20 grading curve.

Three line graphs labeled (a), (b), and (c) each depict the relationship between passing percentage and sieve aperture size in millimeters. They compare maximum, medium, and minimum gradation limits with composite gradation. Each graph shows an increasing trend, with variations among the gradation limits. The sieve sizes range from 0.075 to 16 mm in graph (a) and (b), and up to 26.5 mm in graph (c).

Based on domestic and international research experience on asphalt mixtures, five different oil-to-stone ratios were initially proposed: 3.9%, 4.2%, 4.5%, 4.8%, and 5.1%. The test results are shown in Figure 2.

Marshall test results, (a) The gross volume relative density, (b) Flowing Value (FL), (c) Stability (MS), (d) Asphalt saturation (VFA), (e) Porosity (VV), (f) Mineral Porosity (VMA).

Graph illustration showing six different plots of various properties versus the asphalt-aggregate ratio. (a) Bulk volume relative density increases. (b) Flowing value rises. (c) Stability peaks around 4.2%. (d) Asphalt saturation climbs steadily. (e) Porosity decreases. (f) Mineral porosity shows a dip, reaching its lowest around 4.8%. Each graph displays a specific correlation trend between the property and asphalt-aggregate ratio with marked data points.

The common asphalt-aggregate ratio range corresponding to the bulk relative density, MS, FL, VV and VMA of asphalt mixtures with different asphalt-aggregate ratios was analyzed, so as to accurately calculate the optimum asphalt-aggregate ratio of asphalt mixtures. The common asphalt-aggregate ratio range for conventional petroleum asphalt mixtures is presented in Figure 3.

Common area of asphalt mixture with different asphalt stone ratios.

Chart showing variation of asphalt mix properties with Optimum Asphalt Content (OAC) ranging from 3.9% to 5.1%. Properties include Density, Stability, Flow Value, VMA, VV, and VFA. Dashed lines mark OAC min at 4.581% and OAC max at 4.972%.

The optimal oil-to-stone ratio for the remaining asphalt mixtures was determined using this procedure and summarized in Table 5.

The optimal oil stone ratio for asphalt mixtures at all levels.

Gradation type Optimal oil-to-stone ratio
AC-13 4.76%
AC-16 4.45%
AC-20 4.27%
Specimen preparation

Experimental studies were conducted using ordinary asphalt to prepare asphalt mixture rutting test specimens of three different gradations—AC-13, AC-16, and AC-20—in accordance with specified standards. The preparation of the asphalt mixture rutting test specimens is shown in Figure 4. The technical approach is illustrated in Figure 5.

(a) Asphalt mixture rutting sample forming machine; (b) Asphalt mixture rutting test piece.

(a) An orange mechanical press compressing a sheet of material with a speckled appearance. (b) Six rectangular trays filled with dark granulated material, arranged in two columns.

Technical approach.

Flowchart detailing research on high-temperature deformation of asphalt mixtures. It includes inputs such as temperature, load, hydrothermal coupling temperature, and maximum nominal aggregate size, leading to factors x1, x2, x3, and x4. These contribute to output variables y1 (rut depth) and y2 (dynamic stability). The chart focuses on modeling and correlation studies. Experimental protocol and evaluation metrics Study on the effect of temperature on the high-temperature deformation of type AC asphalt mixtures

During summer months, asphalt pavement surface temperatures exhibit significant elevation above ambient air temperatures. When ambient temperatures exceed 30 °C, pavement surfaces typically reach 40 °C–50 °C. In southern China’s extreme climate regions where summer air temperatures surpass 40 °C, pavement surfaces can attain 65 °C–70 °C. To comprehensively evaluate temperature effects, this study employs four test temperatures (40 °C, 50 °C, 60 °C, and 70 °C) that encompass both typical and extreme service conditions. Tests were conducted at temperatures of 40 °C, 50 °C, 60 °C, and 70 °C. For each temperature, three identical asphalt mixture rutting specimens were tested simultaneously under the same conditions. The specimens were labeled A1, A2, and A3.

To quantitatively assess the impact of ambient temperature variations on asphalt mixture’s deformation resistance and high-temperature stability, two parameters - rutting depth change rate and dynamic stability change rate - were employed to quantify the influence of test temperature variations on both rutting depth and dynamic stability, as formulated in Equation 1, 2. R D C − T = R D T 2 R D T 1 − 1 × 100 %

Equation: R D C − T - Rate of change of rut depth in asphalt mixture specimens during temperature change from T1 to T2 (%); R D T 1 - Rutting depth of asphalt mixture specimens at T1 temperature (mm); R D T 2 - Rutting depth of asphalt mixture specimens at T2 temperature (mm). D S C − T = D S T 2 D S T 1 − 1 × 100 %

Equation: D S C − T - Rate of change of dynamic stability of asphalt mixture specimens during temperature change from T1 to T2 (%); D S T 1 - Dynamic stability of asphalt mixture specimens at T1 temperature (Cycles/mm); D S T 2 - Dynamic stability of asphalt mixture specimens at T2 temperature (Cycles/mm).

 Study on the effect of load on the high-temperature deformation of type AC asphalt mixtures

Based on literature review (Liming et al., 2018; Qiang and Fujian, 2014; Weidong et al., 2020; Zhengqi et al., 2017), the tire pressure of heavy trucks (≥30 tons capacity) on highways typically reaches approximately 1.3 MPa. To systematically evaluate loading effects, this study examines three pressure levels: (i) 0.7 MPa representing current Chinese specification standards, (ii) 1.0 MPa reflecting typical highway operating conditions, and (iii) 1.3 MPa simulating heavy-load vehicle impacts.

The rutting depth change rate and dynamic stability change rate serve as quantitative indicators to assess the impact of varying tire loads on asphalt mixture performance. The corresponding calculation methods are provided in Equations 3, 4. R D C − L = R D L 2 R D L 1 − 1 × 100 %

Equation: R D C − L - Rate of change of rut depth in asphalt mixture specimens during the change of tire load from L1 to L2 (%); R D L 1 - Rutting depth of asphalt mixture specimens under L1 tire loading (mm); R D L 2 - Rutting depth of asphalt mixture specimens under L2 tire loading (mm). D S C − L = D S L 2 D S L 1 − 1 × 100 %

Equation: D S C − L - Rate of change of dynamic stability of asphalt mixtures during tire load change from L1 to L2 (%); D S L 1 - L1 Dynamic stability of asphalt mixture specimens under tire loading (Cycles/mm); D S L 2 - L2 Dynamic stability of asphalt mixture specimens under tire loading (Cycles/mm).

 Study on the effect of hydrothermal coupling on the high-temperature deformation of AC-Type asphalt mixtures

Southern China’s summer climate combines heavy rainfall with high temperatures, creating hydrothermal conditions that accelerate asphalt pavement rutting (Yi et al., 2022). To investigate this coupled effect, specimens were subjected to water immersion testing under elevated temperatures. The experimental procedure involved: (i) Immersing asphalt mixture specimens in a water bath under a controlled immersion pressure of 0.1 MPa at test temperature (water temperature control precision ±0.5 °C) for 6 h, with each temperature gradient held constant for 4 h, followed by (ii) Removing surface moisture from the specimens and placing them in an oven for drying before conducting rutting tests in dry conditions (post-immersion) to isolate the hydrothermal aging effects (Haipeng et al., 2016). The rutting tests were performed using specimens measuring 300 mm in length, 300 mm in width, and 50 mm in thickness (rutting plates), under a loading frequency of 21 back-and-forth cycles per minute.

Two quantitative indicators - the rutting depth increase rate and dynamic stability reduction rate - were employed to characterize the hydrothermal coupling effects on asphalt mixtures across different temperatures. The corresponding calculation methods are presented in Equations 5 and 6. R D R − T = R D H − T R D N H − T − 1 × 100 %

Equation: R D R − T - Rate of increase in rutting depth of asphalt mixture specimens at temperature T (%); R D NH − T - Rutting depth of asphalt mixture specimens without hydrothermal coupling at temperature T (mm); R D H − T - Rutting depth of asphalt mixture specimens subjected to hydrothermal coupling at temperature T (mm). D S D − T = D S N H − T D S H − T − 1 × 100 %

Equation: D S D − T - Dynamic stability reduction rate of asphalt mixture specimens at temperature T (%); D S N H − T - Dynamic stability of asphalt mixture specimens without hydrothermal coupling at temperature T (Cycles/mm); D S H − T - Dynamic stability of asphalt mixture specimens subjected to hydrothermal coupling at temperature T (Cycles/mm).

 Study on the effect of nominal aggregate size on high-temperature deformation of AC-Type asphalt mixtures

To investigate the influence of nominal maximum aggregate size (NMAS) on high-temperature deformation characteristics, three standard gradations were selected based on typical pavement layer applications: AC-13 (representing surface layers), AC-16, and AC-20 (both commonly used in intermediate/binder layers). This selection enables systematic evaluation of how aggregate size affects rutting resistance across different structural layers.

The rutting depth increase rate and dynamic stability reduction rate serve as quantitative indicators to evaluate the influence of aggregate nominal particle size on asphalt mixture performance. These parameters are calculated using Equations 7, 8, respectively. R D C − M = R D L 2 R D L 1 − 1 × 100 %

Equation: R D C − M - Rate of change in rutting depth of asphalt mixture specimens during the change in nominal aggregate grain size from M1 to M2 (%); R D L 1 - Rutting depth of asphalt mixture specimens at M1 nominal particle size (mm); R D L 2 - Rutting depth of asphalt mixture specimens at M2 nominal particle size (mm). D S C − M = D S M 2 D S M 1 − 1 × 100 %

Equation: D S C − M - Rate of change in dynamic stability of asphalt mixture specimens during the change from M1 to M2 aggregate nominal particle size (%); D S M 1 - Dynamic stability of asphalt mixture specimens at nominal particle size of M1 aggregate (Cycles/mm); D S M 2 - Dynamic stability of asphalt mixture specimens at nominal particle size of M2 aggregate (Cycles/mm).

 Evaluation indicators

Rutting depth, dynamic stability (dynamic stability is defined as the number of wheel passes per unit of rutting depth (mm) at a specified temperature. It reflects the permanent deformation resistance of the asphalt mixture under dynamic loading, with higher values indicating greater stability.) and R (60min rutting depth and 45min rutting depth difference, you can evaluate the deformation resistance of asphalt mixture) as the evaluation indexes of this test, R is calculated as shown in Equation 9. The smaller the R, indicating that the deformation resistance of asphalt mixture is stronger, and vice versa, indicating that the deformation resistance of asphalt mixture is poorer. R = R D 60 − R D 45

Equation: R -Difference in rut depth between 60 min and 45 min (mm); R D 60 - Rut depth at 60 min into the test (mm); R D 45 - Rut depth at 45 min into the test (mm).

 Model establishment

The high-temperature performance of asphalt mixtures is influenced by multiple interacting factors, often exhibiting nonlinear behavior. Consequently, linear function models may be inadequate for accurate prediction. Machine learning algorithms, known for their strong nonlinear computational capabilities, offer a promising alternative. These algorithms can process complex relationships without requiring an explicit mathematical model of the system, making them well-suited for predicting the high-temperature performance of asphalt mixtures. In this study, a machine learning approach is employed to predict high-temperature performance using four input variables: temperature, load, hydro-thermal coupling, and aggregate nominal particle size. The output variables are rut depth and dynamic stability, which serve as key indicators of performance.

Model parameter determination

To establish a quantitative relationship between the influencing factors and the high-temperature performance of asphalt mixtures, initial parameter processing was conducted. Factors such as temperature, load, hydro-thermal coupling, and aggregate nominal particle size, which are inherently difficult to quantify, were assigned discrete values (1, 2, 3, 4) to differentiate their effects on high-temperature performance prediction. The adjustable range of intervals was used as a design reference value, ensuring greater operability and practicality in real-world applications (Liu et al., 2023; Xu et al., 2021). The performance ranges compiled for each material composition are presented in Table 6.

Compilation of data on factors affecting high-temperature deformation of asphalt mixtures.

Influencing factors Index interval ranges Interval compilation
Temperature (°C) Compiled numbers 1 2 3 4
Temperature index range ( − ∞ , 40] (40, 50] (50, 60] (60, 70]
Load (MPa) Compiled numbers 1 2 3
Load index range [0.4, 0.7] (0.7, 1] (1, 1.3]
Hydro-thermal coupled temperature (°C) Compiled numbers 1 2 3 4
Thermo-hydro coupling temperature index range ( − ∞ , 40] (40, 50] (50, 60] (60, 70]
NMAS (mm) Compiled numbers 1 2 3
Particle size index range 13 16 20
Modeling of asphalt mixture high-temperature performance

To address the significant magnitude disparity among input variables (which may impair model learning), normalized all inputs to the [0,1] range using Equation 10 prior to weight initialization. To evaluate the impact of normalization on model performance, the convergence speed and prediction accuracy of multilayer perceptron models were compared with and without normalization. Results indicate that normalization significantly enhances the model’s convergence rate, reducing the number of training epochs required. Furthermore, normalized input data substantially improved overall prediction accuracy, with mean squared error (MSE) decreasing across both training and validation sets. This demonstrates that data normalization is crucial for enhancing the efficiency and accuracy of multilayer perceptron models. x * = x − x min x max − x min , x * ∈ 0 , 1

Equation: x * - Normalized value of sample data; x - Sample raw data values; x max - Maximum value in the sample data; x min - Minimum value in the sample data.

The Multilayer Perceptron (MLP) regression model, a canonical artificial neural network architecture, consists of input, hidden, and output layers. Trained via gradient descent optimization, the model iteratively minimizes prediction errors through backpropagation-based weight and bias adjustments. To ensure reliable predictions, the operational boundaries of the MLP model were defined based on the range of input data used during training and validation. The temperature range for input variables was specified between 30 °C and 70 °C, while the load range was set between 50 kN and 200 kN, reflecting the experimental and environmental conditions relevant to this study. Predictions made outside these boundaries, especially in extrapolated ranges, may yield reduced accuracy due to the lack of representative training data for these conditions. Future extensions of the model should incorporate broader datasets to expand the operational boundaries and improve generalization across wider temperature and load ranges.

A 4-10-5-2 multilayer perceptron (MLP) regression neural network model was constructed to predict the high-temperature performance of asphalt mixtures. The model consists of an input layer (4 nodes), two hidden layers (10 and 5 nodes, respectively), and an output layer (2 nodes). The architecture is illustrated in Figure 6.

MLP neural network model structure diagram.

Diagram of a neural network with an input layer, two hidden layers, and an output layer. Inputs are Temperature (x1), Loads (x2), Hydro-Thermal Coupling Temperature (x3), and MNPS (x4). Outputs are Rut Depth (y1) and DS (y2). Each layer is fully connected to the next.

Linear Regression (LR) is a supervised learning algorithm that identifies the optimal linear relationship (or hyperplane in higher dimensions) by minimizing the discrepancy between predicted and actual values, typically via the least squares method. This model is then used to predict outcomes for new data points. Equations 11–13 show the regression analyses (Siqi and Fenglan, 2023). Q = ∑ i = 0 n y i − β 0 − β i x i

Where is the sum of the squares of the difference between the theoretical and observed values, and the function is derived to find the extreme points: ∂ Q ∂ β 0 = 2 ∑ i = 0 n y i − β 0 − β i x i − 1 = 0 ∂ Q ∂ β 1 = 2 ∑ i = 0 n y i − β 0 − β i x i − x i = 0

The above is the least squares solution to find the extreme points of the squared loss function.

 Results and analysis Effect of temperature on high temperature deformation of asphalt mixture

The average of A1, A2, and A3 was taken as the final experimental measurement value. The test results are shown in Figure 7.

Rut depth and dynamic stability, (a) (b) AC-13; (c) (d) AC-16; (e) (f) AC-20.

Six bar graphs display the relationship between temperature and two variables: rut depth and dynamic stability (DS). Graphs (a), (c), and (e) show an increase in rut depth with temperature for samples A1, A2, A3, and their average. Graphs (b), (d), and (f) illustrate a decrease in DS with temperature for the same samples. Each graph uses a distinct color for each sample and the average. Temperature is measured on the x-axis and rut depth or DS on the y-axis.

Figure 6 shows that while the rutting depth and dynamic stability of asphalt mixture specimens vary significantly at different test temperatures, their overall trends with temperature changes are consistent. As the temperature increases from 40 °C to 70 °C, the rutting depth rises, while dynamic stability decreases, reflecting a decline in the asphalt mixture’s high-temperature stability. This deterioration occurs because higher temperatures soften the asphalt and lower its modulus, while also weakening the adhesion between the asphalt and aggregates. These factors allow aggregate particles to move more easily within the mixture, resulting in greater rutting depth and reduced stability and resistance to deformation.

To quantify the effect of high temperatures on the asphalt mixture’s resistance to deformation, the indicator R was introduced. The calculation results are displayed in Figure 8.

RD60-RD45 variation of matrix asphalt mixture.

Line graph showing the relationship between temperature in degrees Celsius and R in millimeters for three materials: AC-13 (blue squares), AC-16 (green circles), and AC-20 (yellow triangles). R increases with temperature for all materials, with AC-13 showing the steepest rise.

Figure 8 illustrates that as the temperature increases, the R-values of AC-13, AC-16, and AC-20 asphalt mixtures rise to varying extents, indicating a gradual reduction in high-temperature deformation resistance. Among these, AC-20 shows the greatest resistance to high-temperature deformation, followed by AC-16, while AC-13 exhibits the weakest performance. For all three asphalt mixture gradations, the most significant increases in R-values occur between 60 °C and 70 °C.

The rate of change in rutting depth and dynamic stability for three gradations of asphalt mixture under different test temperatures is shown in Figure 9.

Change rate, (a) rutting depth of asphalt mixture; (b) dynamic stability of asphalt mixtures.

Two bar graphs comparing the influence of temperature variation on asphalt types AC-13, AC-16, and AC-20. Graph (a) shows increase in rutting depth across temperature ranges. Graph (b) shows reduction rate in dynamic stability. Each bar is color-coded to its respective asphalt type, with noticeable changes observed between temperature intervals of forty to sixty and sixty to seventy degrees Celsius.

As shown in Figure 9, both the rate of increase and the rate of decrease in rutting depth of the asphalt mixture increase with rising temperature. When the temperature rises from 60 °C to 70 °C, the increase in rutting depth is most pronounced, and the decline in dynamic stability is also most significant. This indicates that the asphalt mixture specimen is most sensitive to temperature at this point, with its high-temperature stability exhibiting considerable variation with temperature changes.

 Effect of load on high temperature deformation of type asphalt mixtures

Rutting tests were conducted under 0.7Mpa, 1.0Mpa and 1.3Mpa loading conditions, and the test results are shown in Figure 10.

Rut depth and dynamic stability, (a) (b) AC-13; (c) (d) AC-16; (e) (f) AC-20.

Six bar charts labeled (a) to (f) display data for rut depth and DS cycles per millimeter across different loads in MPa. Each chart compares A1, A2, A3, and the average. Charts (a), (c), and (e) show rut depth in millimeters, while charts (b), (d), and (f) display DS cycles per millimeter, all plotted against loads of 0.7, 1, and 1.3 MPa. Different colors represent each data set.

Figure 10 shows that under a tire pressure of 1.0 MPa, the rutting depth of AC-13 asphalt mixtures is 1.53 times greater, while the dynamic stability is 0.588 times lower compared to conditions at 0.7 MPa. Notably, the dynamic stability at 0.7 MPa meets the requirements of standard specifications.

As the tire load increases, the rutting depth progressively grows, and the dynamic stability decreases, indicating a reduction in the asphalt mixture’s resistance to deformation. This occurs because higher loads generate greater internal stresses within the mixture. When these loads approach or exceed the material’s bearing capacity, they can cause structural damage and fatigue. This, in turn, leads to more severe surface deformation, higher rutting depth, reduced overall stability, and weakened structural strength, ultimately decreasing dynamic stability.

To further evaluate the effect of varying tire loads on the deformation resistance of asphalt mixtures, the R-index was used as a quantitative indicator. The results of these calculations are shown in Figure 11.

RD60-RD45 variation of asphalt mixture.

Line graph showing relationship between loads (in MPa) and R (in mm) for three datasets: AC-13, AC-16, AC-20. AC-13 increases rapidly, AC-16 moderately, and AC-20 gradually.

Figure 11 demonstrates that increasing tire loads lead to progressive elevation of R-values for AC-13, AC-16, and AC-20 asphalt mixtures, signifying a gradual decline in deformation resistance. Among these mixtures, AC-20 exhibits the strongest high-temperature deformation resistance, followed by AC-16, while AC-13 shows the poorest performance.

Figure 12 presents the rut depth change rates and dynamic stability change rates for the three asphalt mixture gradations under varying tire load conditions.

Change rate, (a) rutting depth; (b) dynamic stability.

Two bar graphs compare different asphalt concrete mixes under varying loads. Graph (a) shows rut depth growth rates for AC-13, AC-16, and AC-20; AC-13 has the highest rates. Graph (b) presents DS degradation rates for the same mixes, with AC-13 again showing higher values, particularly under higher loads. Both graphs highlight performance under loads from 0.7 to 1.3 Mpa.

Figure 12 illustrates that under identical test conditions, increasing the tire pressure from 0.7 MPa to 1.0 MPa caused rut depth to rise by 42.96%, 30.49%, and 21.59% for AC-13, AC-16, and AC-20 asphalt mixtures, respectively. Correspondingly, dynamic stability decreased by 56.14%, 41.29%, and 29.67%. When the tire pressure was further raised from 1.0 MPa to 1.3 MPa, rut depth increased by 57.53%, 44.56%, and 38.62% for the respective mixtures. These findings clearly demonstrate a positive correlation between tire pressure and the rate of change in both rut depth and dynamic stability.

As the load increases, asphalt mixture specimens are subjected to higher pressure and shear stress, leading to greater deformation, as reflected by increased rut depth and decreased dynamic stability. This mechanical behavior occurs because higher loads enhance asphalt binder flow within the mixture, accelerating pavement deformation. At the same time, the structural integrity of the asphalt mixture is weakened, directly reducing its dynamic stability.

 Effect of hydrothermal coupling on the high temperature deformation of asphalt mixtures

Hydrothermal coupling rutting tests were conducted at 40 °C, 50 °C, 60 °C and 70 °C. The hydrothermal coupling process and test results are shown in Figures 13, 14.

Asphalt mixture specimens immersed in water.

Seed trays submerged in water within a container. The trays contain soil, and there is an electrical component with red wires on a foam block to the right side.

Rut depth and dynamic stability, (a) (b) AC-13; (c) (d) AC-16; (e) (f) AC-20.

Six bar charts depict the variation of rut depth and DS with temperature for different samples A1, A2, A3, and their averages. Charts (a), (c), and (e) show rut depth increasing with temperature. Charts (b), (d), and (f) display decreasing DS values as temperature rises. Each dataset is color-coded: A1 in blue or green, A2 in orange or red, A3 in pink or yellow, and averages in corresponding colors.

As shown in Figure 14, under the same hydrothermal coupling temperature, the rutting depths of the three graded asphalt mixture specimens follow the order: AC-13 > AC-16 > AC-20, while their dynamic stability ranks as AC-20 > AC-16 > AC-13.

With increasing hydrothermal coupling temperature, the rutting depth of the asphalt mixtures increases, while dynamic stability decreases, indicating a reduction in deformation resistance. In other words, as the temperature rises, the dynamic stability of the asphalt mixtures declines, resulting in poorer high-temperature stability. This occurs because the strength of the asphalt mixture is partially dependent on the adhesion between the asphalt binder and aggregates. Asphalt, being a viscoelastic material, softens and transitions toward a plastic state as the hydrothermal coupling temperature increases, which weakens adhesion and reduces the mixture’s strength. Additionally, hydrothermal coupling penetrates the asphalt-aggregate interface, further aggravating damage to the asphalt mixture.

The rutting test results of asphalt mixture specimens subjected to hydrothermal coupling were compared to those tested under ambient conditions, as shown in Figures 15, 16.

Rut depth and dynamic stability, (a) (b) AC-13; (c) (d) AC-16; (e) (f) AC-20.

Six panel chart comparing rut depth and DS values at different temperatures. Panels (a), (c), and (e) show increased rut depth in non-thermo-hydro coupled conditions versus thermo-hydro coupled conditions, at 40, 50, 60, and 70 degrees Celsius. Panels (b), (d), and (f) display DS values for hydrothermal and non-hydrothermal coupling, with a decline in non-hydrothermal coupling as temperature rises. Each panel includes distinct color-coded bars for comparison.

Comparative variation of RD60-RD45 asphalt mixture.

Line graph showing the relationship between temperature in degrees Celsius and R in millimeters for different conditions. It includes six lines, representing AC-13, AC-16, and AC-20 in both Thermo-Hydro and Non-Thermo-Hydro states. R increases with temperature across all conditions. The legend indicates specific line styles and colors for each condition.

Figures 15, 16 demonstrate that hydrothermal coupling leads to increased rutting depth and R-value in all three graded asphalt mixtures compared to uncoupled specimens, while dynamic stability shows varying degrees of reduction. These results indicate that hydrothermal coupling significantly reduces both deformation resistance and high-temperature stability. The pronounced decline in water damage resistance of AC-20 under hydrothermal coupling, as shown in Figure 14, can be attributed to microscale mechanisms, including changes in asphalt-aggregate interfacial adhesion, void structure evolution, and asphalt aging behavior. Hydrothermal coupling accelerates the weakening of interfacial bonding due to combined thermal expansion, moisture intrusion, and adhesive force degradation. Simultaneously, moisture migration into interconnected voids under hydrothermal conditions exacerbates pore water pressure, further compromising the structural integrity of the mixture. Additionally, elevated temperatures and moisture conditions may intensify the oxidative aging of asphalt, increasing stiffness and brittleness and reducing cohesive and adhesive properties. Furthermore, Figure 16 reveals that at identical hydrothermal coupling temperatures, AC-20 asphalt mixtures exhibit particularly pronounced degradation in water damage resistance.

The rutting depth increase rates and dynamic stability reduction rates of the three graded asphalt mixture specimens under varying hydrothermal-coupled temperature conditions are presented in Figure 17.

Change rate, (a) rut depth; (b) dynamic stability change rate.

Two line graphs, labeled (a) and (b), plot data for AC-20, AC-16, and AC-13 against temperature from 40 to 70 degrees Celsius. Graph (a) depicts rut depth growth rate, while graph (b) depicts DS degradation rate. In both graphs, rates increase with temperature, with AC-20 showing the highest increase, followed by AC-16 and AC-13.

As shown in Figure 17, with increasing hydrothermal coupling temperature, both the rutting depth increase rate and dynamic stability reduction rate of AC-13, AC-16, and AC-20 asphalt mixture specimens. The rate of increase in rutting depth and the rate of decrease in dynamic stability can be used to measure the effect of hydrothermal coupling on the rutting depth and dynamic stability of asphalt mixtures. that is, the rutting depth and dynamic stability of the asphalt mixture respectively increase and decrease to a certain extent under hydrothermal coupling. Moreover, elevated temperatures accelerate the action of moisture, thereby hastening asphalt ageing and reducing the adhesion between asphalt and aggregates, inflicting greater damage upon the strength and durability of the asphalt mixture.

Furthermore, at identical hydrothermal coupling temperatures, AC-20 asphalt mixture specimens exhibited markedly higher rates of rutting depth increase and dynamic stability reduction under hydrothermal coupling compared to AC-13 and AC-16 asphalt mixtures. This effect intensified with rising hydrothermal coupling temperatures.

The AC-13 asphalt mixture specimens exhibited the lowest rates of rutting depth increase and dynamic stability reduction under hydrothermal coupling. Attributable to AC-13’s smallest nominal aggregate size and reduced surface texture and angularity. This configuration maximises the bonding area between asphalt and aggregate, minimises mixture void ratio, and consequently enhances adhesion between asphalt and aggregate, resulting in superior resistance to water damage.

 Effect of aggregate size on asphalt mixture’s high-temperature deformation

The test results of asphalt mixtures with three different gradations, AC-13, AC-16 and AC-20, under the standard rutting test are shown in Figure 18.

Rut test results, (a) rut depth; (b) rutting stability.

Two bar graphs compare data across different MNPS values. Graph (a) depicts rut depth in millimeters for categories A1, A2, A3, and average at MNPS values 13, 16, and 20. Graph (b) shows DS in cycles per millimeter for categories A1, A2, A3, and average at the same MNPS values. Both graphs use different colors to represent each category.

Figure 18 illustrates that asphalt mixtures with larger nominal aggregate sizes exhibit reduced rutting depths and improved dynamic stability. This improved performance can be attributed to two primary mechanisms: (1) larger aggregate particles provide a greater load-bearing contact area, effectively distributing vehicular loads and minimizing deformation; and (2) coarse aggregates enhance the structural stability and durability of the asphalt matrix, collectively mitigating rutting and preserving pavement integrity under repeated loading.

To quantitatively assess the impact of maximum nominal aggregate size on the resistance to permanent deformation in asphalt mixtures, the parameter R was introduced. The computational results of this analysis are presented in Figure 19.

Rut depth at different time periods.

Line graph showing rut depth in millimeters on the y-axis versus MNPS in millimeters on the x-axis. Three data sets, RD45 (squares), RD60 (circles), and RD60-RD45 (triangles), decrease in value from left to right.

Figure 19 illustrates an inverse correlation between aggregate nominal size and parameter R in asphalt mixtures, indicating that larger aggregate sizes improve resistance to permanent deformation. This enhancement can be attributed to two primary mechanisms. First, larger aggregates increase the filler effect, contributing to improved structural stability. Second, they reduce asphalt mobility by enhancing particle interlock. These combined effects lead to higher compressive strength and improved deformation resistance in the mixture.

The rate of change in rut depth and the rate of change in dynamic stability of asphalt mixtures with three different aggregate nominal sizes under standard test conditions are shown in Figure 20.

The relationship between the rutting depth and dynamic stability change rate of different nominal particle sizes of aggregates.

Bar chart showing the rate of change in rut depth growth and DS change, measured in percentage. The rut depth growth rate is shown in orange, below zero for both intervals (13-16 mm and 16-20 mm). The DS change rate in green is above sixty percent for the same intervals. The chart is labeled MNPS (mm) on the x-axis and Rate of Change (%) on the y-axis.

Figure 20 shows that under identical test conditions, increasing the aggregate nominal size from 13 mm to 16 mm reduces rutting depth by 29.33% and improves dynamic stability by 65.34%. Further increasing the aggregate size to 20 mm results in additional reductions of 34.54% in rutting depth and 64.18% in dynamic stability. These findings demonstrate that larger aggregate sizes generally enhance rutting resistance and dynamic stability. However, the marginal benefits decrease as aggregate size increases, with the performance difference between AC-16 (16 mm) and AC-20 (20 mm) mixtures being less significant. This indicates a potential threshold beyond which further increases in aggregate size yield diminishing improvements in permanent deformation resistance.

The selection of aggregate nominal size should thus be carefully considered, as excessively large sizes can negatively impact the workability of asphalt mixtures during construction and the long-term performance of pavements. In mixture design, engineers must balance aggregate size with key project-specific factors, such as traffic volume, loading conditions, and environmental influences, to optimize structural performance and service life. An improperly selected nominal size may lead to reduced compaction uniformity and ultimately compromise pavement durability.

 Modeling of high temperature performance prediction of asphalt mixtures

In this study, 80% of the sample data is allocated for neural network training, while the remaining 20% is used for model testing. To enhance the convergence speed and accuracy of the neural network, the stochastic gradient descent (SGD) method is employed for training. The learning rate is set to 0.001, and the model is trained over 4,000 iterations. The training and prediction results of the developed model are presented in Figures 21, 22.

LR model training and testing results (a) rut depth; (b) dynamic stability.

Two scatter plots compare training and test sets with sample values. The left plot shows a range from zero to fourteen millimeters, while the right plot extends to seven thousand millimeters. Both use blue circles for training data and yellow triangles for test data, with red dashed lines indicating trends.

MLP Regression model training and testing results (a) rut depth; (b) dynamic stability.

Two scatter plots showing training and test values versus sample values. Plot (a) ranges from zero to fourteen millimeters, and plot (b) ranges from zero to seven thousand millimeters. Training set data is marked with orange circles and test set data with green triangles. Both plots include red dashed trend lines indicating a positive correlation.

The training and testing results of the two models (Figures 21, 22) show that the data points are well-distributed on both sides of the ideal fitting line, demonstrating strong correlations between predicted and observed values for both the training and testing datasets. This indicates high fitting accuracy for both machine learning models. Both the linear regression model and the multilayer perceptron neural network model exhibit better predictive performance for the dynamic stability of asphalt mixtures compared to rutting depth. Notably, the multilayer perceptron neural network model slightly outperforms the linear regression model in predicting rutting depth.

In order to further evaluate the fitting accuracies of the two prediction models on the rutting depth and dynamic stability of asphalt mixtures, the Root Mean Error (RMSE), the Mean Absolute Error (MAE) and the coefficient of determination (R 2) were selected as the evaluation indexes. RMSE, MAE and R 2 are used to evaluate the degree of dispersion between the predicted value and the measured value, and the smaller the value is, the higher the prediction accuracy is. The three evaluation index points are calculated as shown in Equations 14–16. R M S E = 1 n ∑ i = 1 n y ˙ i − y i 2 M A E = 1 n ∑ i = 1 n y ˙ i − y i R 2 = 1 − ∑ i = 1 n − 1 y i − y ˙ i 2 ∑ i = 1 n − 1 y i − y ¯ i 2

Equation: y i · - The ith projected value; y i − - Average of predicted values; y i - The ith sample value.

The results of the calculation of the root mean square error, the mean absolute error and the coefficient of determination are shown in Figure 23.

Calculation results of evaluation indicators, (a) (b) RMSE and MAE; (c) R.2.

Three bar charts comparing performance metrics for different models. Chart (a) displays RMSE and MAE for LR and MLP on training and test sets, with RMSE generally higher. Chart (b) shows RMSE and MAE with larger values for the same models, with RMSE again generally higher. Chart (c) depicts \(R^2\) values for rut depth and DS, with MLP showing higher values on the test set. Each chart includes distinct color-coded legends.

Figure 23 demonstrates that the MLP model exhibits consistently lower root mean square error and mean absolute error values compared to the LR model, while its R 2 approaches unity more closely than the LR model. These results indicate superior performance of the MLP model over the LR model in predicting both rutting depth and dynamic stability of asphalt mixtures, with enhanced capability to capture the underlying trends in both parameters.

 Conclusion and prospects Conclusion

This study investigated multiple factors influencing permanent deformation in the asphalt mixtures, using rutting depth and dynamic stability as key performance indicators. The principal research findings can be summarized as follows:

The rutting depth and dynamic stability of asphalt mixtures are generally linearly correlated with test temperature, hydrothermal coupling temperature, load, and aggregate nominal particle size. Temperature, hydrothermal coupling, load, and aggregate nominal particle size significantly influence the rutting depth and dynamic stability of asphalt mixtures, with load having the greatest impact on both. Followed by water-heat coupling.

Two machine learning models were developed: a Linear Regression (LR) model and a Multilayer Perceptron (MLP) neural network, using four input parameters (temperature, loading magnitude, hydrothermal coupling temperature, and aggregate nominal particle size) to predict rut depth and dynamic stability. Both models showed strong correlations between predicted and actual values, with symmetric data distributions around the ideal fit line. However, the MLP model outperformed the LR model in predictive accuracy based on RMSE, MAE, and R 2 metrics.

This study examined the effects of hydrothermal coupling on the deformation resistance of asphalt mixtures, focusing on water damage and temperature effects. However, the interaction between high temperature and heavy loads, which often coexist in real-world pavements, is critical as it likely exacerbates rutting. Elevated temperatures reduce asphalt binder viscosity, increasing deformation susceptibility, while repeated heavy loads induce cumulative plastic strains. Although not explicitly quantified here, the trends in dynamic stability and rutting depth highlight their influence. Future research should adopt quantitative methods, such as factorial designs or numerical modeling, to better understand deformation mechanisms under coupled conditions.

Prospects

This study employed solely a rutting tester for research purposes, resulting in relatively limited evaluation metrics. Subsequent investigations should incorporate additional assessment criteria for comparative analysis to validate the experimental findings.

The application of strain sensing technology to actual road surfaces enables monitoring of internal strain responses within pavement structures under the coupled effects of vehicular loads and complex environmental conditions. Analysing these strain trends facilitates real-time evaluation of pavement condition.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

 Author contributions

ZH: Writing – review and editing, Methodology, Writing – original draft, Conceptualization. KY: Writing – original draft, Investigation, Writing – review and editing. ZZ: Writing – original draft, Data curation, Conceptualization, Investigation. RH: Writing – review and editing, Conceptualization, Investigation. DL: Conceptualization, Investigation, Writing – original draft. XH: Methodology, Writing – review and editing, Formal Analysis. HR: Investigation, Writing – original draft, Supervision.

 Conflict of interest

Authors ZH, KY, ZZ, and DL were employed by Engineering Department, Guangxi G-Energy Engineering Consulting Group Co. Ltd.

The remaining author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

 Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

 Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Edited by: Chen Li, Inner Mongolia University, China

Reviewed by: Ling Xu, Fuzhou University, China

Xinye Jiang, Chang’an University, China

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