Experimental Investigation of Indirect Tensile Strength of Hot Mix Asphalt with Varying Hydrated Lime Content at Low Temperatures and Prediction with Soft-Computing Models

Abstract

If asphalt pavements are exposed to cold weather conditions and high humidity for long periods of time, cracking of the pavement is an inevitable consequence. In such cases, it would be a good decision to focus on the filler material, which plays an important role in the performance variation in the hot asphalt mixtures used in the pavement. Although the use of hydrated lime as a filler material in hot asphalt mixtures is a common method frequently recommended to eliminate the adverse effects of low temperature and to keep moisture sensitivity under control in asphalt pavements, the sensitivity of the quantities of the material cannot be ignored. Therefore, in this study, an amount of filler in the mixture was replaced with hydrated lime (HL) filler additive at different rates of 0%, 1%, 2%, 3% and 4%. These asphalt briquettes, designed according to the Marshall method, have optimum asphalt contents for samples with specified HL content. In this study, where the temperature effect was examined at five different levels of −10 °C, −5 °C, 0 °C, 5 °C and 25 °C, the samples were produced in two different groups, conditioned and unconditioned, in order to examine the effect of water. The indirect tensile strength (ITS) test was applied on the produced samples. Experimental study showed that HL additive strengthened the material at low temperatures and made it more resistant to cold weather conditions and humidity. In the second part of the study, two different prediction models with varying configurations were introduced using nonlinear regression and feed-forward neural networks (FFNNs) and the best prediction performance among these was investigated. Examination of the performance measures of the prediction models indicated that ITS can be accurately predicted using both methods. As a result of comparing the developed models with the experimental data, the model provides significant contributions to the evaluation of the relationship between the ITS values obtained with the specified conditioning, temperature changes and HL contents.

1. Introduction

The fact that applications in the pavement industry require high-cost investments causes the authorities serving this field to constantly search for innovative materials and methods in order to provide a safe and comfortable service to the users for many years without compromising quality. In particular, the performance of hot asphalt mixtures decreases over time due to environmental conditions and traffic loads acting on the pavement. In this context, in regions where low temperatures are common, crack formation begins as the durability of the pavement decreases, and water seeping through the cracked tissues increases the extent of the damage and becomes one of the most common and costly problems in asphalt pavements. As a solution to these problems, the use of additives to improve the quality of asphalt mixtures is a very common method [1]. The strength of the bond between aggregate and bitumen in the mixture is one of the significant indicators of the pavement quality. However, both environmental conditions and dense traffic loads typically have a negative impact on this bonding. As a solution to the performance degradation caused by wheel loads, climatic conditions and moisture’s effect on the pavement, the use of hydrated lime instead of filler material in asphalt mixtures has a very important place in both current road construction applications and studies in the literature. In general, the benefits of using hydrated lime as a modifier material in bituminous pavements can be divided into four different categories. These categories are mechanical performance, moisture sensitivity, road service life and synergistic benefits.

1.1. Effect of Hydrated Lime on Mechanical Performance

Hydrated lime’s ability to make asphalt mixtures more stiff, durable and resistant to rutting is a reflection of its good performance as an active mineral filler. Unlike most mineral fillers, hydrated lime is chemically active. When the hydrated lime is blended into the mixture, it reacts with the bitumen and removes undesired particulate components. Therefore, it makes the mixture more resistant to rutting and fatigue cracks [2,3]. Hydrated lime improves the strength against pavement stresses [4,5,6,7,8] through increasing the modulus of elasticity [9,10] and distributing the stresses due to traffic loads [4,5,7].

The study by Al Ashaibi et al. [11] showed that HL-modified pavements have about 1.5% less deformation under traffic loading only and 39% lower stress levels, depending on climatic conditions, than those without HL additives. This is expected to help save on crack repair, workload and maintenance costs and efforts in the long run.

The factors affecting the mixture performance are listed as follows:

• Aggregate type, source and gradation;
• Binder type and amount;
• Use of additives;
• Variation in environmental conditions,;
• Planned service life;
• Pavement loading conditions.

Considering these factors, hydrated lime is featured as a multi-functional additive material [2,12].

The effects of hydrated lime on the improvement in the mechanical properties are investigated in many studies. At high temperatures, it has a positive impact on the permanent deformation and fatigue strength of the mixtures [5,12,13,14,15]. Al-Suhaibani et al. [16], Niazi and Jalili [9], Lesueur and Little [13], Mohammad et al. [14], Kennedy and Ping [17], Si et al. [18], Huang et al. [19] and Shahrour and Saloukeh [20] remarked that hydrated lime has a significant impact on the reduction in permanent deformations in the mixtures. This effect helps HMA to improve stiffening against rutting [4,5,7,13,21].

Tension and compression stresses are produced in pavement due to traffic loads as well as variation in the air temperature. Asphalt pavement tends to shrink in cold weather conditions while the friction between the pavement and the layer below resists the shrinkage of the pavement [2,22]. When tension stresses due to cold weather conditions exceed the strength threshold, transverse cracks on the pavement surface appear [3]. These cracks in the perpendicular direction to the highway axis are referred to as “low-temperature cracks”. Water leaking in these cracks causes severe damage to the pavement through expanding cracks and leading to deterioration of the asphalt mix in the short term. Moreover, water leaking through cracks causes crumbling in the layers below, thus causing potholes on the highway. Hydrated lime decreases cracking potential through its positive effect on the strength against low-temperature cracks [7,13,23]; it reduces expansion of micro-cracks through controlling the freeze–thaw cycle [24]. It is remarked that hydrated lime under proper conditions improves the cracking strength despite increasing low-temperature stiffening while reducing the brittleness [7].

Orozco et al. [25] investigated the variation in the mechanical properties of hot asphalt mixtures using three different anti-stripping agents and performed indirect tensile strength, indirect tensile modulus of stiffness and uniaxial cyclic compression tests on the samples prepared for this purpose. The results of the study showed that the inclusion of hydrated lime as a mineral filler (1.68% by weight of dry aggregate) can improve the mechanical performance of HMAs by reducing moisture sensitivity and permanent deformation, with the advantage of a slight increase in hardness.

Hydrated lime prevents oxidation of the binder, thus improves aging performance [4,5,26,27,28,29,30,31], while reducing aging stiffening [12,13,14,17,32].

1.2. Effect of Hydrated Lime on Moisture Sensitivity

Use of hydrated lime as an additive in hot asphalt mixtures is proven to improve long-term performance [2,3,26,27]. A vast number of studies are available in the literature on the use of hydrated lime as an anti-stripping additive [12,19,28,33,34,35,36,37,38,39]. When hydrated lime is added to the asphalt mixtures, the aggregate–bitumen bond is strengthened since it reacts with the aggregate. Meanwhile, hydrated lime also reacts with the asphalt and prevents the production of water-soluble soap, which causes stripping. Thus, a reaction with highly polar molecules occurs, and newly produced molecules lose their sensitivity against water.

The study by Iwański et al. [40] showed that hydrated lime was substituted for limestone filler in warm–semi-warm asphalt mixtures at 0%, 15%, 30% and 45% by weight. In the study, the effects of hydrated lime additions on air void content, moisture and frost damage resistance were observed and it was found that asphalt concrete reached the required moisture and frost resistance levels with an optimum hydrated lime substitution of 30%.

Currently, the positive effects of the use of hydrated lime in asphalt concretes continue to be the subject of many laboratories’ applied studies. As a matter of fact, a recent 5-year field study on the A84 highway in Normandy, France, investigated the application of HL in asphalt concrete containing reclaimed asphalt pavement (RAP) [41]. The fact that HL solves the aging problem in the bituminous binder, protects the material against moisture damage and provides fatigue resistance supports many studies in the literature [42]. One of the impressive results obtained in the study by Bouron et al. [41] is that in the areas where the presence of RAP is prone to moisture damage, the presence of HL compensates for this damage. Hydrated lime mitigated the negative effects of different additives, making it a material that can be utilized in many future innovative additive applications. This suggests that HL will be a popular admixture for many years to come.

1.3. Effect of Hydrated Lime on Pavement Life

Modeling and field studies have shown that hydrated lime is effective in extending the life of asphalt pavements and providing low life cycle costs [13,22] and that the contribution of HL in cost–benefit calculations provides significantly positive economic outcomes [43].

Since the moisture damages the structural integrity of the pavement and causes a reduction in the effective service life, it is remarked that the use of hydrated lime extends the pavement service life by approximately 38% [44,45].

The addition of hydrated lime can be seen as a cheap insurance policy to maximize pavement performance and reduce life cycle costs. In combination with polymer additives, it is able to meet expectations at the highest level for many years. Cost–benefit analyses show that lime is cost-effective [45].

1.4. Synergistic Effect of Hydrated Lime

One of the major benefits resulting from the addition of hydrated lime to hot mix asphalts is the achievement of a high-performance product. Although the benefits can be defined in terms of the individual additives used individually, synergistic operation can result in multiple improvements of the final product. Synergistic benefits can be demonstrated when lime is used in combination with polymer additives. Research shows that the combination of lime and polymers can, in some cases, result in higher performance improvements than when used alone [2].

When hydrated lime was used in combination with other additives (styrene-butadiene-styrene polymers), it significantly improved the performance of the mixture (especially resistance to moisture damage) and showed high compatibility with other additives [46].

Mohammed et al. [14] emphasized that one of the benefits of hydrated lime when added to hot mix asphalts is its “working together” feature. Although the benefits of lime are evident when used alone, they increase even more when used with polymer additives, which provides a significant improvement compared to its use alone.

A survey of the literature shows that in the studies on the analysis of the experimental data related to the performance evaluation of bituminous mixture containing additives, soft-computing methods are typically used [47,48,49]. Gandhi et al. [50] investigated the use of an artificial neural network technique on the prediction of the indirect tensile strength of the mixtures containing anti-stripping agents (including HL). In the proposed model [50], five input parameters are selected, including bitumen source, aggregate source, additive material, conditioning and amount of bitumen. Analysis of the model results proved the usability of the model while pointing out that conditioning is the least effective parameter among the others on the indirect tensile strength. Firouzinia and Shafabakhsh [51] investigated the thermal sensitivity of hot asphalt mixtures containing nano-silica additive materials. They observed thermal sensitivity of mixtures with varying amounts of nano-silica additives through multi-layer perceptron (MLP) and radial basis function (RBF). Results revealed that MLP leads to better results than the RBF method. Krcmarik et al. [52] developed a numerical model on the prediction of the indirect tensile strength of low-temperature asphalt mixtures based on the experimental data provided by 201 specimens containing hot mix asphalt and warm mix asphalt. They compared the performance of four prediction methods: (1) nationally and (2) locally calibrated Pavement ME Design Software v2.6.2.1, (3) linear regression and (4) feed-forward back-propagation neural networks. The best prediction performance was achieved with a neural network approach.

The effects of the hydrated lime additive to the HMA at low temperatures considering water sensitivity are within the scope of this study. There are different amounts of the hydrated lime added to the mixtures while reducing the filler content to varying degrees. An experimental dataset is prepared with 0%, 1%, 2%, 3% and 4% hydrated lime additive at low temperatures (5 °C, 0 °C, −5 °C and −10 °C) in addition to room temperature (25 °C) in conditioned and unconditioned states. Three specimens correspond to each hydrated lime content, temperature and conditioning state, forming (5 × 5 × 3 × 2) a total database of 150 specimens. We aim for characterization of tension stresses due to temperature variations and fatigue in the mixtures. Optimum bitumen amount, mixture parameters (practical specific weight (D_p), void filled with asphalt (VFA), Marshall Stability (MS), air voids (Va), voids in mineral aggregate (VMA) values) and indirect tensile strength of the materials that are prepared via application of Marshall testing procedures are determined. In this study, mathematical models are developed using the experimental results considering various additive content, temperature and conditioning. Based on the experimental data, a nonlinear regression model and a feed-forward back-propagation neural network (FFNN) were developed to predict the indirect tensile strength using temperature, conditioning state and additive content as input parameters. Two datasets were analyzed: the total dataset, which includes data at 25 °C, and a reduced dataset that excludes the 25 °C data to assess the potential effect of the temperature gap on model performance. This gap between 25 °C and the next lower temperatures introduced concerns about potential lever bias, prompting the evaluation of both datasets.

A nonlinear regression model was employed, exploring different degrees of nonlinearity to seek a balance between simplicity and accuracy. The model’s formulation allowed for varying levels of complexity, from linear to higher-order terms, to better capture the nonlinear relationships in the data, particularly at lower temperatures. The FFNN was trained with varying neuron configurations to optimize predictive accuracy, accounting for the complexities introduced by the gap in temperature ranges.

By means of the proposed models, it is possible to evaluate the experimental data comparatively and determine the optimum conditions. Models based on the experimental results enable practical evaluation of the work performed in the laboratory. The accuracy of the prediction models was compared and evaluated through statistical measures.

2. Material Properties

In this study, gradation of the aggregates and optimum asphalt cement (AC) values belonging to the mixtures with varying HL contents were determined in the experimental study carried out by Yardım et al. [53]. Yardım et al. [54] implemented prediction models to estimate Marshall Stability based on the test results obtained in experiments, while Dundar et al. [55] introduced a prediction model for ITS. The aggregates used in this study were provided from Isfalt A. Ş. Ümraniye Asphalt Plant. The source of this material is the limestone quarries of Ömerli, İstanbul, which are located in the northwestern part of Türkiye. CaO at 52.9% comprises the main element oxide in the limestone of this region, and calcite contributes 75% of its mineralogical composition. Limestone used for production has a grey–dark grey color [56]. No. 2, No. 1 and stone powder aggregates were washed, dried and sifted on 3/4″, 1/2″, 3/8″, No. 4, No. 10, No. 40, No. 80 and No. 200 sieves of the shaking machine in accordance with the standard [57]. The weight of the material passing from each sieve was calculated and their distribution was determined using the Wearing Course Type-1 mixture ratios of Highway Technical Specifications of General Directorate of Turkish Highways [57]. Aggregate No 2 was taken as 15%, No 1 as 40% and stone powder as 45%. The coarse and fine aggregates and filler material had ratios of 51.3%, 41.9% and 6.8%, respectively, in the mixture. The properties of the aggregate are given on

Table 1. Aggregate specific gravities.

Aggregate Type	Apparent Spec. Gravity (g/cm3)	Volume Specific Gravity (g/cm3)	Water Absorption (%)
Coarse Aggregate	2.729	2.698	0.42
Fine Aggregate	2.754	2.693	0.82
Filler	2.735
Aggregate Mixture	2.740	2.699

. Figure 1 shows that gradation of the mixture aggregate is within specification limits.

The 50/70 penetration bitumen provided from Izmit Oil Refinery (Tüpraş) was used as the AC. The properties and criteria of this binder that were found using the tests applied are shown in

Table 2. Characteristics of the bitumen.

Properties	Standards	Test Results	Specification
Specific gravity (g/cm3), at 25 °C	ASTM D 70 [58]	1.009
Flash point (Cleveland) (°C)	ASTM D 92 [59]	345	>230
Penetration (0.1 mm), at 25 °C, 100 g, 5 s	ASTM D 5 [60]	59.8	50–70
Ductility (cm), at 25 °C, 5 cm/min	ASTM D 113 [61]	>100	>100
Thin film heating loss (%), at 163 °C, 5 h	ASTM D 1754 [62]	11	<80
Penetration percentage after heating loss (%)	ASTM D 5 [60]	65.9	>54
Ductility after heating loss (cm)	ASTM D 113 [61]	65.4	>50
Softening point (°C)	ASTM D 36 [63]	48.5	45–55

. Optimum ACs were determined for different HL contents using the Marshall method.

In this study, the filler material percentage of the aggregate mixture with the determined gradation was decreased gradually, replacing it with the same amount of HL. The additive used in the experiment is the powdered calcium lime, produced with the code S-KK 80-T at the Bartın Lime Factory, which is located in northern Türkiye. The properties of this material, which has one of the lowest densities on the Turkish market, are given in

Table 3. Properties of hydrated lime.

Properties	Standards	Test Results	Standard Limit Value [12]
Chemical Properties
Total CaO (%)	TS EN 459-1 [64]
	TS 32 EN 459-2 [65]	85.78	>80
MgO (%)	TS EN 459-1 [64]
	TS 32 EN 459-2 [65]	3.52	<5
Total CaO + MgO (%)	TS EN 459-1 [64]	89.3	>80
Loss on Ignition (%)	TS 32 EN 459-2 [65]	22.51
SO3 (%)	TS EN 459-1 [64]
	TS 32 EN 459-2 [65]	1.47	<2
CO2 (%)	TS EN 459-1 [64]
	TS 32 EN 459-2 [65]	3.89	<7
Physical Properties
Fineness over 90 microns (%)	TS EN 459-1 [64]
	TS 32 EN 459-2 [65]	6	<9
Density(kg/m3)	TS 32 EN 459-2 [65]	472	<600

.

3. Indirect Tensile Strength Test Procedures

In the scope of this study, the effects of hydrated lime on the low-temperature performance of hot mix asphalt were investigated. For this purpose, hydrated lime was added to the mixtures in various percentages (0%, 1%, 2%, 3% and 4%) by replacing the same amount of the filler material. Optimum AC content for the specimens in the experimental program was determined by Yardım et al. [53]. In total, 75 unconditioned specimens (3 sets for 25 unique properties) with varying amounts of hydrated lime were tested at 25 °C, 5 °C, 0 °C, −5 °C and −10 °C. Likewise, another 75 specimens were tested in the conditioned status. With the indirect tensile strength test, the potential for deterioration and formation of thermal/fatigue cracks of bitumen mixtures were determined (see Figure 2 and Figure 3).

The preparation steps of the specimens used in the experiments are listed below:

• The aggregate was weighed using suitable sieves according to the selected aggregate gradation. The weighted aggregate was kept in the oven overnight until the moisture disappeared. Dry hydrated lime additive was added to dry aggregate at this stage.
• The next day, dried aggregate and bitumen in the determined ratio were mixed. The prepared mixture was placed in a standard Marshall briquette mold.
• Each mixture briquette in the mold was compressed with 50 blows on both surfaces. Without removing the material from the mold, it was left at room temperature overnight to cool.
• Practical specific gravity values were calculated through measuring the height and weight of the specimens removed from the molds.
• If the specimen was in the conditioned status, it was kept in a 60 °C water bath for overnight and the next day; it was then removed from the water bath and kept at room temperature overnight. If the specimen did not have a conditioning status, it was directly passed to the next step.
• The specimen was kept overnight in the oven at the test temperature (25 °C, 5 °C, 0 °C, −5 °C, −10 °C) planned in the study.
• The specimen was loaded until the stability value was reached and recorded using the indirect tensile strength apparatus in the Marshall test device which is manufactured by Yuksel Kaya Makina Co. Ltd. in Ankara, Türkiye.
• Using the stability value obtained from the device and the dimensions of the specimens, the indirect tensile strength value was calculated in kg/cm².

4. ITS Experimental Results

The specimens produced for the ITS experiment were prepared in accordance with the test manufacturing processes of Marshall’s design [66] and the process is illustrated in Figure 4. In the test apparatus, pressure was applied to the specimen in the vertical direction until deterioration occurred at a deformation threshold. Deterioration is defined as deformation in which there is no further increase in the applied load (i.e., the greatest recorded load value). In this study, the maximum load values determined by test results were evaluated considering specimens’ areas in stress units. The potential for crack formation in asphalt mixtures due to stresses produced by the effects of conditioning status, hydrated lime content (Figure 5) and temperature were investigated. All of the experimental results are plotted in Figure 6.

Results revealed that ITS values under low temperature are greater than ITS values observed under room temperature. This can be explained by the fact that mixtures under low temperature have a greater stiffness modulus than the ones under high temperature.

Conditioning status, which reflects the water sensitivity of the specimens, was applied in the scope of this study. Considering the experimental results, the ITSs of unconditioned specimens are greater than conditioned specimens in all temperature conditions. It is inferred that conditioning increases the sensitivity of the specimens to water.

In the scope of the study, four different amounts of hydrated lime were added to the conventional (additive-free) mixture. Investigation of the effects of the additive lime on indirect tensile strength revealed that it improves strength at low temperatures. Moreover, results remarked the significance of the sensitivity of the strength with respect to the amount of additive. Mixtures with 1% and 2%, specifically, have better strength results in comparison with the specimens with larger amounts of hydrated lime.

Hydrated lime has higher voids of dry compacted filler (Rigden air voids) than mineral fillers, with typical values ranging from 60 to 70% when mineral fillers have values closer to 30–34%. The difference comes from the higher porosity of the hydrated lime particles (Figure 5): For mineral filler, the porosity essentially comes from the voids between the particles. For hydrated lime, the porosity inside the particles sums up to the porosity between the particles, hence leading to a much higher value [3]. This shows that when hydrated lime is used in hot asphalt mixtures at certain ratios (1% and 2%), high ITS values are obtained and cracking risk is prevented, but when the HL ratio in the mixture exceeds these levels, the total void ratio in the mixture increases and performance decreases.

5. Data Analysis and Results

Experimental studies provide significant knowledge on the relationship between the indirect tensile strength of the mixtures and their physical features. However, since it is not feasible to perform experiments that represent all possible configurations, they are often supported with prediction models based on the experimental findings. The complexity of the prediction model is dependent on the dimension of the defined problem. Conventional statistical methods are useful when the number of involved parameters is few, and mathematical representation of the relationship between input and output is relatively simple. Artificial intelligence techniques are commonly used in establishing such a relationship for the problems where mathematical representation is not straightforward [67,68].

In this study, nonlinear regression and a feed-forward neural network (FFNN) were employed to develop prediction models for the ITS of hot mix asphalt as a function of temperature (T), hydrated lime content ($HL$%) and conditioning status (C). Nonlinear regression allows for capturing complex relationships between input variables and the output, particularly when interactions or quadratic effects are present. The FFNN is well suited for modeling highly nonlinear relationships in complex datasets.

The coefficient of determination ($R^2$) has been accepted as a standard value for determining the quality of the prediction model. However, the appropriateness of its use depends on the content of the dataset from which the prediction model is tested. If the observation value is y, the prediction is $\widehat{y}$ and the mean of the observations is $\overline{y}$, the coefficient of determination can be expressed with Equation (1).

$$R^2=1-{\displaystyle \frac{\sum {(y-\widehat{y})}^2}{\sum {(y-\overline{y})}^2}}$$

(1)

The numerator in this definition is the sum of the squares of the residuals ($SSR$). Therefore, a model that produces a lower $SSR$ can be considered to have good predictive strength. Minimizing the $SSR$ value means maximizing the $R^2$ value. However, evaluating the predictive strength of the model on $R^2$ alone is not appropriate when the relationship between observation and prediction variables is nonlinear [69].

The mean square of residuals ($MSE$) is calculated as the $SSR$ divided by the number of observations n and is a meaningful measure of the predictive strength of the model. The root of this value ($RMSE$) gives the standard deviation of the residuals. By definition, $R^2$ evaluates the model fit for a particular dataset. Considering RMSE, a measure of dispersion, in conjunction with $R^2$ would be more appropriate to assess the usefulness of the prediction model [70].

5.1. Nonlinear Regression Analysis

A nonlinear regression analysis was conducted to explore the relationship between temperature, hydrated lime content and conditioning state in predicting the indirect tensile strength of hot mix asphalt. Regression offers a clear and interpretable approach to understanding how these input variables interact and influence the ITS. The nonlinear regression models were fitted using an iterative optimization algorithm that minimizes the difference between the observed and predicted values of ITS. This approach is based on the Levenberg–Marquardt algorithm, which is commonly used in nonlinear least squares problems. The algorithm combines the features of the Gauss–Newton method and gradient descent, making it particularly effective in situations where the model involves nonlinearity. It works by adjusting the parameters iteratively to minimize the sum of the squared residuals between the observed data and the model’s predictions [71].

The nonlinear regression analysis for predicting ITS used a model defined as:

$$ITS(b,T,C,HL)=b_1+b_2·T+b_3·C+b_4·HL+b_5·T^m+b_6·{HL}^m+b_7·{(T+HL)}^m$$

(2)

where T represents temperature, C is the conditioning state and HL is the hydrated lime content. The exponent m was varied from 0 to 4, and the dataset was split, with 80% used for training and 20% held out for testing. The goal was to capture complex relationships between ITS and the input variables, with both quadratic and interaction terms.

The performance metrics for models with different m values are presented in

Table 4. $R^2$ and $RMSE$ values for nonlinear regression models with varying exponent m, based on test set results.

m	R2	RMSE
0	0.94	1.9
1	0.95	1.6
2	0.98	0.6
3	0.99	0.5
4	0.98	0.6

: $m=0$ (linear) achieved an $R^2$ of 0.94 with a high $RMSE$ of 1.9, indicating that it fails to represent nonlinear behaviour in the dataset, as also shown by the significant deviations from the measured ITS in the figure (blue line). Introducing first-order nonlinearity (m = 1) improves the fit slightly, with an $R^2$ of 0.95 and an $RMSE$ of 1.6, yet it still cannot fully capture the curvature in the data. When $m=2$ is used, the model achieves an $R^2$ of 0.98 and a reduced $RMSE$ of 0.6, aligning closely with the measured ITS, as depicted by the black line in the Figure 7. This indicates that quadratic terms effectively represent the diminishing effects of temperature and lime content, as well as their interaction.

For $m=3$, the model achieves the best performance, with an $R^2$ of 0.99 and an $RMSE$ of 0.5. Figure 7 shows that the cubic model (cyan line) fits the measured ITS data slightly better, particularly at peaks and troughs. However, the improvement over the quadratic model is marginal, suggesting that the additional cubic terms introduce complexity without substantial gains in accuracy. With $m=4$, the performance metrics revert slightly, with an $R^2$ of 0.98 and an $RMSE$ of 0.6, similar to the quadratic model. The magenta line in the figure indicates that quartic terms do not provide significant advantages and only add unnecessary complexity.

The coefficients of the $m=2$ model further highlight the relationships between ITS and input variables. The intercept, $b_1$, is estimated at $16.43$, indicating a significant baseline effect on ITS. The coefficient for temperature, $b_2$, is −0.26, suggesting a significant decrease in ITS with increasing temperature (p-value = $1.47\times {10}^{-9}$). The coefficient for conditioning, $b_3$, is −0.60, but it is not statistically significant (p-value = 0.14), suggesting a weaker effect in this context. Hydrated lime content, $b_4$, has an estimated coefficient of 1.22, indicating a significant positive influence on ITS (p-value = 0.017). The quadratic term for temperature, $b_5$, is −0.01, reflecting a significant accelerating decrease in ITS at higher temperatures (p-value = 5.68 × 10⁻⁶). The quadratic term for hydrated lime content, $b_6$, is −0.60, and is also significant (p-value = $2.52\times {10}^{-5}$), showing diminishing returns at higher lime contents. The interaction term, $b_7$, is estimated at 0.0005, with a significant positive effect (p-value = 0.0005), indicating that temperature and lime content interact to influence ITS in a more complex manner than their individual effects.

$$\begin{array}{cc}\hfill ITS(b,T,C,HL)& =16.43-0.26·T-0.60·C+1.22·HL\hfill \\ \hfill & -0.01·T^2-0.60·{HL}^2\hfill \\ \hfill & +0.00005·{(T+HL)}^2\hfill \end{array}$$

(3)

The quadratic model ($m=2$, Equation (3)) thus emerges as the most balanced choice, achieving high accuracy ($R^2=0.98$, $RMSE=0.6$) while maintaining interpretability. Figure 7 and Table 4 demonstrate that the quadratic model effectively captures the nonlinear behavior in ITS, making it suitable for predicting ITS across varying conditions of temperature, conditioning and lime content.

5.2. Feed-Forward Neural Network (FFNN)

An artificial neural network (ANN) is a numerical framework that imitates the learning process of the human brain, which receives, stores and processes information to infer between reasons and consequences. Similar to the human brain’s neurons, artificial neural networks associate the connection between input and output data through layers of neurons.

The first layer of neurons constitutes the input layer, which consists of the same number of neurons as the input parameters. The neurons in the input layer receive and pass the data to the hidden layer(s). The number of hidden layers and neurons in a single hidden layer are typically selected through a trial and error process to establish a model with reliable results and the minimum size. Hidden layers process the received information and pass it to the output layer. The output of the k-th node of the u-th layer $y_k$ is expressed as

$$\begin{array}{c}\hfill \begin{array}{c}{y}_k^u={\varphi }^u\left(\sum _{j=1}^J(w_{kj}^u·y_j^{(u-1)})+b_k^u\right)\hfill \end{array}\end{array}$$

(4)

where $w_{kj}$ is the synaptic weight between k-th node of the u-th layer and j-th node of the $(u-1)$-th layer and $b_k$ is the bias added to the k-th neuron of the u-th layer. The resulting product is filtered through an activation function $\varphi$. The sigmoid function is often used as an activation function, among others. Proper selection of activation filter function may significantly impact the performance of the neural network [72]. Within a process called ’training’, weights and biases at each neuron are sought to lead to the most accurate prediction.

The output layer consists the same number of neurons as the number of output parameters. The output neurons represent the prediction of the neural network for the relevant output parameter. In this phase, the error of the prediction E is evaluated by

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle E=\frac{1}{2}\sum _{i=1}^N\sum _{k=1}^K{(y_{ik}-y_{ik}^{exp})}^2}\hfill \end{array}\end{array}$$

(5)

where $y_{ik}$ is the output of the k-th node of the output layer for the i -th sample in the database of N samples in total, while K is the number of nodes in the output layer and $y_{ik}^{exp}$ is the observed output.

The rest of the training process can be described as an optimization problem in which the aim is to minimize the error between predicted and observed output measure. To search for an optimum solution error is propagated to neurons in the preceding layers, and weight and biases are adjusted to the values so that minimum error is reached [73]. The Levenberg–Marquardt algorithm has been evaluated as the most efficient algorithm to train small- and medium-sized neural networks [74].

The experimental program consisted of 150 observations, obtained from triplicate tests, which correspond to 50 unique data points. In the FFNN prediction model, the average value of the triplicates, representing each unique combination of hydrated lime content ($HL$), temperature (T) and conditioning state (C), was used as the response variable. The model was developed with a single hidden layer where the number of neurons was varied to optimize predictive accuracy. In Figure 8, the proposed FFNN model is illustrated with three input parameters and one output as described in the previous section.

A neuron sensitivity analysis was conducted by varying the number of neurons (n) in the hidden layer from 2 to 10. The ’trainbr’ (Bayesian regularization) algorithm was used for training the network, which is particularly effective at minimizing overfitting in smaller datasets. The model’s performance was assessed using Root Mean Squared Error ($RMSE$) and coefficient of determination ($R^2$), and the results were compared for both the total dataset (which includes the 25 °C data points) and a reduced dataset (excluding the 25 °C data points) of 40 datapoints to evaluate the impact of the potential lever bias due to the temperature gap.

The cross-validation process employed in this study is illustrated in Figure 9. In this approach, the complete dataset is split into five subsets (folds). During each iteration, a different fold is used as the test set, while the remaining four folds are used for training. While cross-validation provides a useful estimate of model performance, it may yield an optimistic view of predictive power, as each data point is included in the training set across iterations.

The results in the

Table 5. Average $R^2$ and $RMSE$ values from 5-fold cross-validation test sets for FFNN models with varying neurons, comparing the complete and reduced datasets.

n	Complete Dataset		Reduced Dataset
	R2	RMSE	R2	RMSE
2	0.92	1.2	0.49	1.4
3	0.92	1.3	0.53	1.4
4	0.89	1.4	0.02	1.6
5	0.92	1.2	0.37	1.7
6	0.92	1.2	0.09	1.6
7	0.86	1.4	0.12	1.6
8	0.92	1.2	0.56	1.4
9	0.91	1.2	0.18	1.5
10	0.92	1.3	0.16	1.6

show the $R^2$ and $RMSE$ values for FFNN models with varying numbers of neurons, as evaluated on both the complete dataset and the reduced dataset (excluding the 25 °C data points). The complete dataset consistently achieves higher $R^2$ values (around 0.92) and lower $RMSE$ values (around 1.2 to 1.3), indicating stronger model performance compared to the reduced dataset, which generally exhibits lower $R^2$ values and higher $RMSE$ values, particularly for models with more than three neurons.

Interestingly, the 2–3 neuron models perform similarly across both datasets (Figure 10), with $R^2$ values of 0.92 and 0.49–0.53 and $RMSE$ values of 1.2–1.3 and 1.4, respectively. This consistency suggests that these simpler models can maintain similar predictive performance regardless of whether the 25 °C data points are included. This indicates that the impact of the temperature gap between 25 °C and lower temperatures (a potential source of lever bias) is minimal, as the models are able to generalize well across different temperature ranges.

The findings also highlight that simpler models, such as those with 2–3 neurons, are not only effective but are also less prone to overfitting, achieving a good balance between model simplicity and accuracy. These models demonstrate robust predictive performance on both datasets, suggesting that they can capture the underlying relationships between input variables and the ITS without being significantly affected by the absence of data at 25 °C.

6. Conclusions and Discussion

ITS is a very important parameter that is used for evaluation of thermal crack formation in asphalt mixtures due to environmental effects. Particularly at low temperatures, selection of the pavement material types and amounts, design and maintenance to achieve the desired performance during its service life are raised as essential issues. In this study, the variation in ITS of the mixtures that contained hydrated lime as an additive under different temperature levels and water conditioning effects were measured. For this purpose, HMA specimens containing different HL amounts that were reduced from the filler content were prepared. Three specimens of each possible combination of five different HL amounts at five different temperature levels were produced in both conditioned and unconditioned states.

In the second part of the study, alongside the nonlinear regression analysis, a prediction model using an FFNN was developed based on the experimental results. It was determined that the model predictions showed a high correlation with the experimental results for the given material properties and application method.

The nonlinear regression and FFNN models developed for predicting the indirect tensile strength (ITS) showed strong performance and high correlation with the experimental results. The nonlinear regression model, incorporating both interaction and quadratic terms, provided a more comprehensive representation of the relationship between temperature, hydrated lime content and conditioning state. This model demonstrated its effectiveness in capturing the nonlinear behavior of the material, particularly at lower temperatures.

The FFNN model, using a single hidden layer with 5–6 neurons, exhibited high predictive accuracy. The model trained with the total dataset, including the 25 °C data points, achieved an $R^2$ of 0.94 and an $RMSE$ of 1.16. Stratification by temperature made only a slight improvement in the performance of the total dataset model, indicating that the 25 °C data provided sufficient robustness. However, stratification proved more beneficial for the reduced dataset (without the 25 °C data), where it generally helped improve the model’s performance across different configurations.

Conclusions of this study are summarized below:

• An HL addition replacing filler material has a positive impact on the ITS. However, HL addition in large amounts leads to a decrease in performance since dry porosity of the HL is greater than the mineral filler. Experimental investigation showed that HMAs with 1% and 2% HL content have relatively greater ITS.
• It was determined that water conditioning effects play a key role in performance evaluation. ITSs of conditioned specimens are typically less than the unconditioned specimens. However, when the HL additive and conditioning status were evaluated together, it was observed that the water resistance increased when HL additive was used.
• Temperature changes material properties and it affects the performance of different materials working together. At low temperatures, the resistance of HMA against cracking is considered as the performance criterion. In the present study, the ITS values obtained at low temperatures (5 °C, 0 °C, −5 °C and −10 °C) are higher than the ones obtained at room temperature (25 °C). The ITS value at −10 °C, which is the lowest temperature value used in the study, is highest among the rest. The HL additive makes the surface more resistant against crack formation.

The nonlinear regression and FFNN models developed in this study provide reliable predictions of indirect tensile strength, significantly reducing the need for extensive laboratory testing, which requires substantial time and effort. The results demonstrate that the developed models can effectively predict ITS for intermediate hydrated lime content in both conditioned and unconditioned specimens, under the same laboratory conditions and material properties. The quadratic model ($m=2$) emerged as the most suitable approach, balancing accuracy and simplicity, making it practical for estimating ITS values across varying conditions. The high predictive performance achieved with this model indicates its potential contribution to the relevant literature.

The analysis of the regression model revealed important insights into the influence of each input parameter on ITS. Temperature was found to have a significant negative impact, with decreasing ITS as temperature increases, which is consistent with expected material behavior. Conditioning state, while included as a parameter, showed a weaker contribution compared to temperature and lime content, indicating that the overall effect of moisture conditioning is less pronounced under the given conditions. Hydrated lime content had a significant positive impact, enhancing ITS, particularly at lower temperatures, although quadratic terms revealed diminishing returns at higher lime contents. The interaction term between temperature and lime content was also significant, suggesting a complex interplay between these variables that affects ITS more than their individual contributions. These insights highlight that temperature and lime content are the most influential factors for ITS prediction, while conditioning has a secondary effect under these experimental conditions.

The FFNN model, implemented with k-fold cross-validation for robustness, also demonstrated a strong predictive capability for ITS. Various configurations were tested, and it was found that FFNN models with 2 to 3 neurons in the hidden layer provided the most accurate results, balancing simplicity and accuracy. The FFNN model trained on the complete dataset achieved an R² of 0.92 and an RMSE of 1.2, while similar performance was observed in the reduced dataset, though with slightly lower accuracy due to fewer data points. Results from both the complete and reduced datasets indicate that the gap between 25 °C and lower temperatures did not introduce a significant bias lever, as similar predictive performance was achieved across datasets. This suggests that the absence of intermediate temperature data did not adversely affect model accuracy.

The materials used in this study were sourced from specific quarries as described in the methodology section, meaning there may be variations in ITS when using materials from different sources. The experiments were designed to simulate low temperatures between −10 °C and 10 °C, along with 25 °C representing room temperature. While the current design successfully captures ITS behavior at these levels, future experimental designs could benefit from including lower temperatures (below −10 °C) and finer temperature steps to enhance prediction model accuracy further. A larger dataset covering a broader temperature range would likely lead to improved performance of the models, regardless of the method applied.

Future research directions based on this study can be summarized as follows:

• Investigating the applicability of the developed models in field conditions to compare model predictions with real-world ITS measurements.
• Exploring the effects of different additives, such as polymers or organic compounds, on ITS under various climatic conditions, including freeze–thaw cycles and high temperatures.
• Developing new asphalt mix formulations that combine hydrated lime with other materials to enhance both the mechanical performance and durability of pavements.
• Conducting long-term performance evaluations, cost analyses and environmental impact assessments of asphalt pavements incorporating hydrated lime or other additives.

Moving forward, future research could focus on integrating environmentally sustainable additives, analyzing their impact on ITS and comparing the results with existing findings. A multidisciplinary approach to material development and environmental sustainability could yield innovative solutions for asphalt pavement design and maintenance. The identified significance of temperature and lime content in predicting ITS underscores the importance of these parameters in mix design, offering guidance for optimizing pavement performance in low-temperature conditions.