Dielectric Response of Asphalt Mixtures and Relationship to Air Voids and Stiffness

Abstract

Asphalt mix air void content is a dominant parameter for asphalt mix design. The air void content of the mix affects the mechanical property of stiffness, while both characterize compacted asphalt mix materials. On the other hand, asphalt mix as a composite material can be characterized by its dielectric value. Considering the above, the aim of the present paper is to develop a simple methodology for the characterization of asphalt mix materials using their dielectric properties through an investigation of the interaction of dielectrics and air voids, as well as air voids and stiffness. For this purpose, an experimental laboratory study was conducted, which involved the compaction of asphalt mixes with different aggregate types and air void content. Upon this, the specimens were tested for their air void content, the dielectric constant, and the stiffness modulus. The analysis of the results showed strong correlations between the three characteristics. These findings were further verified with a new set of specimens and laboratory measurements. The final goal is to use the developed methodology for the estimation of asphalt mix stiffness considering that the effect of air content on the resulting stiffness cause indirect relationships between stiffness and dielectrics.

1. Introduction

Pavement performance is highly dependent on mix design and structural design. Asphalt mix design is a process that involves the combination and selection of various sizes and types of aggregate with asphalt binder to produce an asphalt mix with the desired strength, stability, durability, and resistance to environmental factors. The key steps and considerations in asphalt mix design are: (1) selection of the materials (aggregates and asphalt binder), (2) testing of the materials, (3) mix and compaction, (4) volumetric analysis, (5) mix tests (such as the Marshall Stability and Flow test, Hveem Stabilometer test, or Superpave tests to evaluate the mix’s properties). The final goal is to establish the optimum binder content meeting the requirements, amongst others, of the volumetric characteristics. The volumetric characteristics are critical for determining the quality and performance of flexible pavements. One of the key volume characteristics is air voids, defined as the percentage of the total volume of the compacted asphalt mix that is occupied by air. In case of an asphalt mix with a high percentage of air voids, the air enters the body of the asphalt mix, causing oxidation of the asphalt and, therefore, faster aging of the asphalt mixture [1]. Moreover, the asphalt mix is more prone to cracking and rutting. On the contrary, a low percentage of air voids may result in rutting and permanent deformation under traffic loads, as there will not be the required void ratio for the binder to permeate and cover the aggregates. The above highlights the importance of determining the percentage of air voids in the asphalt mix in the context of mix and pavement design. Furthermore, effective assessment and management of air voids is a common practice procedure in terms of pavement quality control and quality assurance and is related to the compacted density of the asphalt layers. Non-uniform zones of mix lead to segregation, raveling, fatigue cracking, and premature failure of the asphalt layers [2,3,4].

There are a variety of methods for the determination of the air void content. The most common procedure for the determination of air voids in the lab is through the determination of the bulk specific gravity (Gmb) and the theoretical maximum specific gravity (Gmm). For the determination of the bulk specific gravity of laboratory-prepared specimens or cores extracted from in-service pavements, three methods are suggested depending on the air void percentage and water absorption [5]. For asphalt mix samples with anticipated air voids less than 10% and water absorption less than 2% the saturated surface dry (SSD) method is used. For asphalt mix samples with anticipated air voids less than 10% and water absorption more than 2% the method demands the coating of the sample with sealing material. Finally, for asphalt mix samples with anticipated air voids more than 10%, the dimensional method is applied. For the determination of the theoretical maximum specific gravity, the mixture should be loose and free of any large clumps, while the test procedure requires a pycnometer and a vacuum pump. The above-mentioned procedures are moderately simple to execute, but time-consuming, and are applied regularly, especially on laboratory-prepared specimens.

For air void assessment of in-service pavements’ asphalt layers, ground-penetrating radar (GPR) offers a valuable tool [6]. It is a non-destructive method that offers continuous, rapid, and efficient assessment and has been used for a variety of pavement surveys. The basic operating principle of this system relies on the transmission of extremely high-frequency radio wave pulses through a transducer or antenna. The emitted radio waves are reflected by the separating surfaces of the asphalt mixture or pavement layers with different characteristics. The antenna receives these signals and stores them in the digital control unit. Depending on the time it takes for the reflected signal to return to the antenna, the depth of each layer and other characteristics can be determined. This time depends on the conductivity of each material, which is a function of the dielectric constant [7]. The greater the dielectric constant, the less time it will take for the signal to return to the antenna. The dielectric constant of a composite material, such as asphalt mix, is a function of the dielectric constant values of each individual material and their proportion to the mix. The dielectric constant value of air is equal to 1.0, while the dielectric constant values of aggregates and asphalt range from 4.5 to 6.5 and from 2.6 to 2.8, respectively. During compaction of the asphalt layers, the air voids decrease, and as such, the dielectric value increases. There are numerous studies that have dealt with the development of correlation relationships between the dielectric constant value derived from GPR measurement technique and the air void content of asphalt mix, offering an indirect way of estimating asphalt mix air void content and density through dielectric measurement [8,9,10,11,12,13]. An alternative method to obtain the dielectric values of asphalt mix specimens, both those prepared in the laboratory and those from core sampling, is through a reflection/transmission method applied in the laboratory [14].

Another dominant factor that primarily regulates the asphalt mix and, consequently, pavement behavior and performance, is the stiffness modulus of the asphalt layers. The stiffness modulus is required for both the determination of the pavement response and the prediction of pavement damage under the influence of traffic loads and environmental conditions. It can be defined as the relationship between applied stress and induced strain under loading. Laboratory determination of the stiffness modulus can be accomplished through several experimental setups. The fact that all these procedures require both specialized equipment and a large amount of time led to the development of estimation algorithms [15]. It is well recognized that the asphalt mix stiffness modulus is affected by the mix characteristics, with the air void content having the most significant effect. Therefore, it is included in all estimation algorithms either directly or indirectly [16]. In principle, a higher amount of air voids results in lower stiffness and vice versa, considering a constant percentage of asphalt [17].

Considering the above, the interrelation of asphalt mix air void content with the stiffness modulus and the dielectric constant can be easily observed. On these grounds, the present research aims to develop a methodology for the characterization of asphalt mix materials using their dielectric properties. For this purpose, the interaction of dielectrics and air voids, as well as that of air voids and stiffness, are investigated taking into consideration asphalt mixes that differentiate as to the air void content and the aggregate type. The final goal is to use the developed methodology for the estimation of HMA stiffness considering that the effect of the air content on the resulting stiffness cause indirect relationships between stiffness and the dielectric constant.

It is worthwhile to mention that the research findings are derived from a comprehensive laboratory experiment. Marshal specimens were prepared considering a wide range of voids, which define three categories of mixtures: dense, open, and open-graded friction course mixes and different types of aggregates. The specimens were tested in the laboratory for the determination of air voids, stiffness, and dielectrics. The created data samples were analyzed to document the relationships under investigation. Finally, the applicability of the developed methodology was verified and discussed using an additional set of data.

2. Materials and Methods

2.1. Sample Preparation

Specimens with different aggregate types and a wide range of air void contents were created. To ensure consistency and reproducibility of the specimen preparation, various aspects were taken into consideration and appropriate actions took place. It was ensured that the aggregate gradation was consistent and followed the specified mix design. Consistent mixing temperatures were maintained for both the aggregates and the asphalt binder, and it was ensured that the mixing time was standardized to achieve uniform coating of the aggregates with the binder. Consistent compaction equipment was used, namely, the Marshall hammer, and the number of blows was standardized for each asphalt mix type. Also, the compaction temperature was controlled. Finally, the laboratory equipment used was properly calibrated.

In total, 28 different mixes were produced, representing asphalt mixes intended for the construction of the wearing course. For these mixes, four different aggregate types were used: diabase, gabbro, spilite, and slag. For each aggregate type, the mixtures were classified into three categories: dense mixes, open mixes, and open-graded friction course mixes (

Table 1. Composition of asphalt mixtures.

Sieve Size	Dense-Graded	Open-Graded	Open-Graded Friction Course
	Percent Passing
19.0 mm (3/4 in.)	100	100	100
12.5 mm (1/2 in.)	90–100	85–100	90–100
9.5 mm (3/8 in.)		60–90	60–100
4.75 mm (No. 4)	44–74	20–50	15–40
2.36 mm (No. 8)	28–58	5–25	4–12
1.18 mm (No. 16)		3–19
300 μm (No. 50)	5–21	0–10
75 μm (No. 200)	2–10	100	2–5

).

All the batches of asphalt mixes were produced using a polymer modified Bitumen (PmB) 25–55/75 modified binder that ranged between approximately 3% and 5%.

Table 2. Asphalt properties.

Test	Results	Results after RTFOT
Penetration at 25 °C (pen)	51	50
Softening point (°C)	79	82
Ductility at 25 °C (mm)	>100
Brookfield viscosity 160 °C (cPs)	454
Fraass point (°C)	−10

presents the results of asphalt testing.

The pen is the measurement unit of the penetration test, which measures the depth to which a standard needle will penetrate the asphalt sample under specific conditions. One pen equals 0.1 mm. The RTFO test is designed to simulate the short-term aging that occurs in asphalt binders during the hot-mixing process and the placement of asphalt concrete. The primary goal of the RTFO test is to assess the durability of asphalt binders by simulating the oxidative aging that occurs when the asphalt is mixed at high temperatures and exposed to air during production and construction. This helps in understanding how the binder will behave in the early life of the pavement. As seen in Table 2, penetration and softening point results are nearly the same before and after RTFOT. This may be attributed to the fact that polymer additives generally improve the binder’s resistance to oxidation and hardening during the RTFO test. Thus, modified binders show lower evolutions of penetration and softening point.

Table 3. Aggregate characteristics.

Test	Diabase	Gabbro	Spilite	Slag	Standard
Loss Angeles abrasion test (LA)	13.2	15	19.1	15.2	ASTM C131
Sand equivalent (SE)	79	68	70	70	AASHTO T176
Flakiness Index (FI)%	14.4	14.7	12.0	13.3	ASTM D4791
Aggregate Abrasion Value (A.A.V.)%	3.41	3.8	3.3	2.9	ASTM D7428
Polished Stone Value (PSV)	60	57	58	64	EN 1097-8:2020

presents the main aggregates’ characteristics.

Varying mixes were used for the fabrication of specimens following the Marshall mix design method. They were compacted with the Marshall hammer [18] by applying 75 blows on each side of the specimen for the dense-graded mixes and 50 blows on each side of the specimen for the open-graded and open-graded friction course mixes. In total, 84 different specimens were created for laboratory testing.

Table 4. Specimens’ characteristics.

Aggregate Type	Dense-Graded		Open-Graded		Open-Graded Friction Course
Diabase Gabbro Spilite Slag	Air voids (%)	No of specimens for each aggregate type	Air voids (%)	No of specimens for each aggregate type	Air voids (%)	No of specimens for each aggregate type
	3–5	3	5–8	3	15–18	3
			8–11	3	18–21	3
			11–15	3	21–25	3
	Total	12		36		36

presents information about the specimens.

In total, 12 dense-graded, 36 open-graded, and 36 open-graded friction course specimens were created for testing in the laboratory and further elaboration.

2.2. Dielectrics Measurements

Dielectric constant measurement of the asphalt mix specimens was enabled through an open-ended coaxial probe embedded in the Percometer device [19]. The Percometer device is non-destructive, with an operation frequency of 50 MHz, which limits the penetration depth close to the material surface (penetration depth ranges from 3 to 5 cm). It can measure various surfaces with the adjustment of the appropriate probe so that full contact between the probe and surface is accomplished. In the present study, two probes were used, one with a flat surface and one with a curved surface (Figure 1). The flat probe was used for the measurements on the tops and bottoms of the specimens, while the curved one was used for the measurements on the sides of the specimens.

Prior to the measurements, a calibration process needed to be conducted by measuring the dielectric constant either in an open circuit (air), in a short circuit (metallic material), or, finally, in deionized water. In this study, the calibration process was performed in air, which has a dielectric constant value equal to 1.

Measurements were performed on six locations on the tops, six locations on the bottoms, and six locations along the sides of the specimens. No significant deviations occurred between the recorded dielectric constant values (ε_r) at the six locations of each surface. This was also the case for the ε_r values in all locations of the same specimen (total 18 in each specimen). This was expected, since the specimens were prepared and compacted in the laboratory in a controlled manner, ensuring the homogeneity of the material. Due to this homogeneity, the characterization of each specimen regarding its dielectric properties was achieved by considering the average value of ε_r of all measurements as representative.

2.3. Air Voids Measurements

For the determination of the air void content of the specimens, the theoretical maximum specific gravity (G_mm) was determined according to ASTM D2041. For the determination of the bulk specific gravity (G_mb), the SSD method was used for specimens with anticipated air voids less than 10% (ASTM D2726) and the dimensional method for specimens with anticipated air void contents of more than 10% (ASTM D3549).

The basic steps in the SSD methods are the following:

• Drying the specimens: The asphalt mix specimens were left to dry at room temperature until they reached a constant weight, meaning all moisture had been removed.
• Heating of the specimens: The specimens were heated in an environmental chamber at 25 °C. After heating, they were weighted, and the weights were recorded as m₁ (dry weight of the specimen).
• Soaking the specimens: The specimens were placed in water with a temperature of 25 °C for a sufficient period to ensure full saturation (Figure 2). After full saturation, the sample was weighted, and the result was recorded as m₂ (weight of the saturated specimen in water).
• Surface drying: After soaking, the specimen was removed from the water and the excess water was allowed to drain off. A towel was used to blot the aggregate and remove any free surface water. The goal was to achieve the SSD condition, where the pores are saturated, but the surface is dry. The specimen was weighted in air, and the result was recorded as m₃ (weight of the saturated specimen on air).
• Calculation of the G_mb: the bulk specific gravity was calculated according to Equation (1).

G_mb = m₁/(m₃ − m₂)

(1)

Caution was given to specimens tested with the SSD method regarding whether the water absorption exceeded 2%. None of the specimens exceeded this value, so the SSD method was considered suitable.

In the dimensional method, the bulk specific gravity is calculated based on the dimensions of the specimen, namely, diameter and thickness. For the fabrication of the specimens, the Marshall mold was used, so the diameter (d) of all specimens was 10.16 cm. For the determination of the thickness, for each specimen, four measurements were made with a caliper at locations that that each echoed 1/4 of the circumference of the circle, and the average of the measurements was recorded as the thickness of the specimen (h). The accuracy of the measurement was of the order of 0.1 cm. Finally, each sample was weighed, and its mass (m₁) was recorded to the nearest 0.1 g. The bulk specific gravity was calculated according to Equation (2).

G_mb = m₁/h × π × (d²/4)

(2)

Figure 2. Experimental setup for the SSD method.

Figure 2. Experimental setup for the SSD method.

Once the G_mm and G_mb were determined, the content of air voids (Va), expressed in percent (%), was derived from the following equation:

Va = (G_mm − G_mb)/G_mm × 100

(3)

2.4. Stiffness Measurements

The determination of the stiffness modulus (S_m) was accomplished with the Nottingham Asphalt Tester (NAT) device and the IT-CY (Indirect Tensile to Cylindrical Specimens) test procedure. The testing temperature was set at 20 °C, which is a weighted temperature representative of the local climatic conditions and is considered for pavement design purposes in Greece. Specimens were placed in a controlled temperature chamber for at least four hours to reach the desired temperature. Then, each specimen was placed on the subframe, and it was ensured that it was properly aligned (Figure 3). The loading time was set at 124 ms according to the EN 12697-26. Each specimen was tested twice (Figure 4). In the second round, the specimen was rotated so that the load was applied along a diameter which had an inclination of 90° with respect to the first measurement.

The mean value of the two measurements was used for stiffness characterization in terms of ITSM (Indirect Tensile Stiffness Modulus), unless the difference of the second measurement from the first was out of the limits of +20% to −15%. In this case, the measurement was repeated.

3. Data Analysis and Results

3.1. Data Samples

Three data samples were established from the measurements for each aggregate type, namely, dielectric constant, air voids, and stiffness modulus.

Table 5. Descriptive statistics for mixtures with aggregate type diabase.

	εr	Va (%)	Sm at 20 °C (MPa)
Mean	4.36	16.69	7286.74
Standard error	0.12	1.64	679.24
Range	1.80	19.99	9431.00
Minimum	3.55	6.38	3217.00
Maximum	5.35	26.37	12,648.00

,

Table 6. Descriptive statistics for mixtures with aggregate type gabbro.

	εr	Va (%)	Sm at 20 °C (MPa)
Mean	4.42	13.79	7469.14
Standard error	0.10	1.35	488.13
Range	1.30	17.10	6549.50
Minimum	3.60	4.95	4461.50
Maximum	4.90	22.05	11,011.00

,

Table 7. Descriptive statistics for mixtures with aggregate type spilite.

	εr	Va (%)	Sm at 20 °C (MPa)
Mean	4.76	15.53	5326.86
Standard error	0.14	1.38	496.91
Range	1.60	16.85	7323.50
Minimum	3.85	5.78	2551.50
Maximum	5.45	22.63	9875.00

and

Table 8. Descriptive statistics for mixtures with aggregate type slag.

	εr	Va (%)	Sm at 20 °C (MPa)
Mean	6.99	17.02	5434.95
Standard error	0.19	1.70	581.91
Range	2.28	22.14	9169.50
Minimum	5.83	5.34	2759.00
Maximum	8.10	27.48	11,928.50

present the descriptive statistics of the data.

The dielectric values (ε_r) of asphalt mixes with diabase, gabbro, and spilite did not seem to differ significantly, ranging from 3.55 to 5.45, and the mean values were 4.36, 4.42, and 4.76, respectively. However, this was not the case for asphalt mixes with slag. Both the range of the ε_r values and the mean were higher. The higher ε_r values of specimens with slag aggregates were due to the nature of the aggregate, which was highly conductive. The range of Va values captured the different mixes that were created: dense-graded, open-graded, and open-graded friction course. The higher values of the stiffness modulus (S_m) corresponded to dense-graded mixes, while the lower values corresponded to open-graded friction course mixes. S_m mean values of mixes with diabase and gabbro were of the same order (~7286 and ~7469 MPa, respectively). The same was valid for mixes with spilite and slag (~5326 and ~5434 MPa, respectively).

3.2. Correlation of Material Characteristics

For the development of relations between ε_r and Va and between Va and S_m, linear regression was applied. The mathematical expression of the relation between ε_r and Va is given in Equation (4), while that for the relation between Va and S_m is given in Equation (5). Constants a–d and the correlation index (R²) are given in

Table 9. Constants of ε_r–Va and Va–S_m linear regression.

	Va = a × εr + b			Sm = c × Va + d
Aggregate Type	a	b	R2	c	d	R2
Diabase	−12.68	71.96	0.93	−396.0	13,898	0.92
Gabbro	−12.50	69.08	0.85	−343.6	12,206	0.90
Spilite	−9.302	59.79	0.86	−348.7	10,742	0.94
Slag	−8.63	77.35	0.89	−307.7	10,671	0.80

. The results of the ANOVA (analysis of variance) are presented in

Table 10. ANOVA for ε_r–Va and Va–Sm regression (aggregate type gabbro).

εr–Va	df	SS	MS	F	Significance F
Regression	1	1049.354786	1049.354786	246.0236146	2.50252 × 10−12
Residual	19	81.03994804	4.265260423
Total	20	1130.394734
Va–Sm	df	SS	MS	F	Significance F
Regression	1	1.77 × 108	1.77 × 108	205.0108	1.24 × 10−11
Residual	19	16,435,579	865,030.5
Total	20	1.94 × 108

,

Table 11. ANOVA for ε_r–Va and Va–Sm regression (aggregate type diabase).

εr–Va	df	SS	MS	F	Significance F
Regression	1	650.1880215	650.1880215	107.6590565	2.89518 × 10−9
Residual	19	114.7471733	6.03932491
Total	20	764.9351948
Va–Sm	df	SS	MS	F	Significance F
Regression	1	90,331,444.77	90,331,444.77	176.1709149	4.63556 × 10−11
Residual	19	9,742,229.306	512,748.9109
Total	20	100,073,674.1

,

Table 12. ANOVA for ε_r–Va and Va–Sm regression (aggregate type spilite).

εr–Va	df	SS	MS	F	Significance F
Regression	1	684.2454526	684.2454526	113.3941124	1.89449 × 10−9
Residual	19	114.6502523	6.034223807
Total	20	798.8957049
Va–Sm	df	SS	MS	F	Significance F
Regression	1	97,184,286.93	97,184,286.93	283.0571112	7.19273 × 10−13
Residual	19	6,523,423.643	343,338.0865
Total	20	103,707,710.6

and

Table 13. ANOVA for ε_r–Va and Va–Sm regression (aggregate type slag).

εr–Va	df	SS	MS	F	Significance F
Regression	1	1071.867543	1071.867543	151.0067661	1.7328 × 10−10
Residual	19	134.8647074	7.098142495
Total	20	1206.732251
Va–Sm	df	SS	MS	F	Significance F
Regression	1	114,285,374.8	114,285,374.8	77.73262985	3.84258 × 10−8
Residual	19	27,934,499.65	1,470,236.824
Total	20	142,219,874.5

.

Va = a × ε_r + b

(4)

S_m = c × Va + d

(5)

The correlation index values were higher than 0.8, indicating that simple formulas can be used for the Va estimation based on a simple and quick testing method for measuring the ε_r. Also, the stiffness modulus can be estimated through the air void content derived from the dielectric value with great accuracy.

The ANOVA gives information about the levels of variability within the regression model. df is the number of degrees of freedom associated with the sources of variance. SS is the sum of squares. The smaller the residual SS compared with the total SS, the better the model fits the data. MS is the mean square. F is the F statistic, or F-test for the null hypothesis. It is used to test the overall significance of the model. Significance F is the p-value of F. The significance F value gives an idea of how reliable (statistically significant) the results are. If the F-statistic is significant (less than 0.05), it indicates that the model explains a significant portion of the variance in the dependent variable, meaning at least one of the predictors is significantly associated with the outcome. In every regression model, the residual SS was significantly smaller than the total SS. Moreover, the value of significance F was much smaller than 0.05, suggesting that the model fit the data well.

Furthermore, the defined constants included in Table 9 show that there was not a single relationship expressing the interaction between ε_r and Va. Constants a and b for mixes with aggregate types diabase and gabbro were close enough, but this did not apply for the other mixes with the aggregate types spilite and slag. The same was true for the interaction between Va and Sm. Therefore, the aggregate type of the HMA mixes seems to influence the above relationships and must be taken into consideration. The relationships (2) and (3) are illustrated in Figure 5, Figure 6, Figure 7 and Figure 8 for each aggregate type.

The above figures describe the interrelation of the three asphalt mix material characteristics. They illustrate that high dielectric values correspond to low air void content, while high stiffness values correspond to low air void content and inversely. Thus, higher dielectric values correspond to mixes with higher stiffness modulus values. This pattern can be explained by the fact that the increase in the stiffness modulus is achieved by increasing the volume percentage of the elements that contribute to the bearing capacity of the asphalt mixture (aggregate and asphalt) and/or by reducing the air void content. These elements are aggregates and asphalt, materials that have higher dielectric constant values than the third element of the mixture, which is air. Since air has a dielectric constant value equal to 1 and aggregates and asphalt have higher dielectric constant values, a decrease in the air void content results in higher dielectric values of the asphalt mix and higher stiffness modulus values as well.

The interaction of the three characteristics that arises from the above methodology provides the possibility of estimating stiffness through a simple measurement in the laboratory of the dielectrics. The process is indicated in the above figures with the black dashed arrows. Through the measurement of dielectrics, the air void content can be estimated, and then stiffness can be obtained within a satisfactory confidence level. Even though dielectrics and stiffness reflect two different things, i.e., the component proportions of asphalt mix and a mechanical property, respectively, there seems to be an interaction between them. Since there is, on the one hand, a well-established relationship between air void and stiffness and, on the other hand, a relationship between air void and dielectrics, it is proven that the effect of the air content on resulting stiffness cause indirect relationships between stiffness and dielectrics.

It is worthwhile to mention that, although a direct relationship between dielectrics and stiffness has not yet been established, there is some indication that materials of poorer condition produce lower dielectric constant values. Hence, low dielectric constant values may indicate material of lower stiffness [20]. The findings of this research provide evidence that supports this statement.

3.3. Verification

The applicability of the developed formulas (Equations (2) and (3)) was investigated through a verification process. For this purpose, new and different mixes were produced using the four aggregate types and classified into the same three categories (dense-, open-, and open-graded friction course). In total, 28 different specimens were created for the determination of air void content, stiffness (at 20 °C), and dielectricity.

Figure 9 shows the correlation between the measured air void content (Va-meas) and the predicted air void content (Va-pred) that resulted from measuring the dielectric constant and implementing Equation (2) for each aggregate type.

The correlations of Va-meas and Va-pred values correspond to correlation indices (R²) equal to 0.97, 0.91, 0.92, and 0.94 for mixes with diabase, gabbro, spilite, and slag, respectively, indicating very good applicability of the formula (2) to the new data. A paired-samples t-test was performed to determine whether the differences between the measured and predicted values were statistically significant (

Table 14. Paired-samples t-test for Va-meas–Va-pred.

Va-Meas–Va-Pred		95% Confidence Interval of the Difference
	Mean	Lower	Upper	Sig. (2 Tailed)
Diabase	0.556	−0.808	1.919	0.357
Gabbro	−0.477	−2.353	1.399	0.557
Spilite	−0.614	−2.196	0.967	0.379
Slag	−1.617	3.678	0.445	0.103

). In every case, the p-value (sig.) was greater than 0.05, meaning that the differences were not significant. Therefore, the developed formulas seem to be adequate for the prediction of the air void content of asphalt mixes.

Figure 10 shows the correlation between the measured stiffness (Sm-meas) at 20 °C and the predicted stiffness (Sm-pred) that resulted from measuring the dielectric constant and implementing Equations (2) and (3) for each aggregate type.

The correlations of Sm-meas and Sm-pred values corresponded to correlation indices (R²) equal to 0.91, 0.96, 0.94, and 0.78 for mixes with diabase, gabbro, spilite, and slag, respectively, indicating very good, or fair to good for the case of slag, applicability of the developed procedure to the new data. A paired-samples t-test was performed to determine whether the differences between the measured and predicted values were statistically significant (

Table 15. Paired-samples t-test for Sm-meas–Sm-pred.

Sm-Meas–Sm-Pred		95% Confidence Interval of the Difference
	Mean	Lower	Upper	Sig. (2 Tailed)
Diabase	38.286	−877.152	953.696	0.922
Gabbro	−58.571	−559.258	442.115	0.784
Spilite	45	−642.782	732.782	0.878
Slag	−26.428	−1425.841	1372.984	0.965

).

As shown in Table 11, the measured Sm values do not differ significantly from the values predicted from the model, since the p-value (sig.) is greater than 0.05. Therefore, the developed formulas seem to be adequate for the prediction of the stiffness of asphalt mixes.

4. Conclusions

In the present work, a methodology was developed for the characterization of asphalt mix materials in terms of air void content and stiffness by using the dielectric properties of the materials. Asphalt mix specimens with four different aggregate types and a wide variety of air void contents were produced. Laboratory testing on a total of 84 specimens was performed for the determination of the air void content, dielectric constant, and stiffness modulus (ITSM). These data were used for the development of relationships between the three characteristics. New mixes were produced (28 total) for verification purposes. The conclusions are presented subsequently:

• In general, the dielectric constant values are lower when the air void content of the mix is higher. This is due to the fact that the dielectric constant of asphalt mixes is a function of the values of the dielectric constants of its individual components. Therefore, by increasing the percentage of voids, with a value of dielectric constant ε_r = 1, their volume content in the mix increases, resulting in a decrease in the dielectric constant of the asphalt mix.
• Asphalt mixes with the aggregate type slag experience higher ε_r values due to the nature of the aggregate, which is highly conductive.
• The dielectric constant and air void content can generally be modeled mathematically by a linear relationship. This relationship is not unique, but is dependent on the aggregate type. For the four aggregate types considered in this study, the correlation index values were higher than 0.85, indicating a strong relationship between these two material characteristics.
• The stiffness modulus decreased with the increase in the percentage of voids, and their mathematical relationship presented high values of the coefficient of determination, ranging from 0.8 to 0.94. This reduction was due to the increase in the percentage of air by volume of the mix, which, however, could not receive a load, and therefore did not contribute to the load-bearing strength of the asphalt mix, a fact that is reflected in the reduction in the stiffness modulus.
• Taking into account, on the one hand, the relationship that was developed between the dielectric constant and the percentage of voids (where high values of the dielectric constant correspond to a smaller percentage of voids), and, on the other hand, the relationship between the stiffness measure and the percentage of voids (where small values of the percentage of voids correspond to larger values of the stiffness modulus), the increase in the dielectric constant also implies an increase in the stiffness modulus. It is noted that, regarding the samples with the slag aggregate type, the value of the coefficient of determination was lower, possibly due to the increased dielectric conductivity presented by these aggregates.
• The applicability of the developed formulas was investigated through a verification process. Analysis showed that the developed formulas seemed to be adequate for the prediction of stiffness of the asphalt mimes considered in this study.
• By measuring dielectrics in the laboratory, a rough but instant estimation of voids or even stiffness is possible, facilitating in this way the quality control process for asphalt mix materials.

This methodology could be applied widely for laboratory work, but it must be pointed out that the suggested equations correspond to specific bitumen and aggregate types. By enriching the laboratory samples with additional types of bitumen or aggregate, the developed equations that describe the relationships between dielectrics and air voids, as well between air voids and stiffness, will change, and then they could be considered as generic. Other factors that could affect the relationships between the dielectric constant, air void content, and stiffness modulus could be temperature and frequency. Further research is anticipated to attempt to extend this process using field data.