A Comprehensive Scientific Theory for Hot Mix Asphalt Design Aiming Accuracy and Simplicity for Asphalt Pavement Construction and Mix Quality Check

Abstract

This chapter deals with the first attempt to derive a scientific, revolutionary, hot mix asphalt design method. The full rutting equation was derived as a function of the damage wheel pass number (N_d) in the previous paper by Huh. Since N_d is hard to measure, the single stroke breakdown energy (W_1d), is disclosed to replace N_d. By using this W_1d, new performance equations are proposed as a function of the W_1d. The effectiveness of the new linear performance equation derived is experimentally validated. These performance equations in the form of master equations could constitute a scientific, rigorous, hot mix asphalt design theory. Since the formulation of master equations requires much effort and time, a simple design variable representing master equations is proposed as a design variable. This is an important discovery in this paper. Comparison of the suggested variable (W_1d) among various asphalt mixes can predict both rutting and fatigue cracking simultaneously. It is hoped that this noble HMA design method can be a promising alternative for currently available empirical methods like Marshall, Gyratory Compactor, Indirect Tensile, BMD, etc. to obtain a better design value.

1. Introduction

All asphalt mix design methods available so far have been empirical for the past 90 years. This paper first time deals to derive the scientific hot mix design method. The gap between empirical and scientific methods is found that the formal methods rely on experiences and observations, while the latter method relies on scientific findings reaching performance equations and experimental validation of those equations. This means that this novel hot mix design method depends on performance equations of asphalt mixes, not experiences yielding empiricism.

1. Introduction – review of the previous finding.

2. Development of New Asphalt Mix Performance Equations.

3. Validation of Mix Performance Equation Containing W_1d

4. Master Performance Equations for Rigorous Design Eqs.

5. A Novel HMA Design Theory for the Asphalt Pavement Construction and the Asphalt Mix Quality Check

6. Conclusion

The compacted asphalt concrete mix at the rutting temperature (40–80°C) is a typical multiphase material comprised of solid phase (aggregate and filler), viscous melt phase (asphalt binder), and air phase (air void). Thus, the melted asphalt mix becomes a strong multiphase material that usually forms the capillary pressure due to presence of the interphase tension between solid (aggregate and filler) and viscous melt medium (asphalt binder), and various particle curvatures. The capillary pressure force in the multiphase materials creates the network solid structure by combining every binder-coated particle together. Thus, the solid structure of compacted asphalt mixes requires repeated loadings (or wheel pass numbers) to breakdown the solid structure formed at the breakdown (0r the damage) point. The nature of the solid network structure formation of the compacted asphalt mix itself at the rutting temperature is the same as the one at the fatigue cracking temperature except only temperature difference.

The existence of the capillary pressure force in a multiphase material has been studied before. Gillespie and Settineri [1] computed an adhesion force comprising a liquid medium bridge connecting two identical spherical particles by using the capillary pressure force. Mehrotra and Sastry [2] also derived a mathematical expression of the adhesion force for a pendular liquid bridge medium connecting two unequal particles. These papers and others claim the existence of the capillary pressure force in multiphase materials.

The full rut performance equation in the term of a damage wheel pass number (N_d) has been derived in the view of breakdown (or damage) of the solid structure. A detailed description is given in the earlier paper by Huh [3] where Eq. (1) and Eq. (2) below are derived to explain the full rut depth quantitatively. Eq. (1) tells that the full rut depth at a fixed wheel pass number (h_N) is only a function of the damage wheel pass number (N_d) which is a characteristic property of a given mix. In addition, Eq. (1) is claimed to be independent of wheel pass speed, temperature, and asphalt binder viscosity (Huh [3]). According to Eq. (1), the full rut depth at a fixed N (h_N) can be determined if N_d is known.

$$h_N=\mathit{Log}\left(\frac{m}{{N_d}^B}\right)\cdot N^{\beta }+C\cdot {\left[\mathit{Log}\left(\frac{m}{{N_d}^B}\right)\right]}^D\cdot N^{\delta }$$E1

Here, h_N is the full rut depth at a fixed N, N is the number of wheel passes, N_d is the wheel pass number reaching a damage point (or a damage wheel pass number), and B, C, D, b, and d, are material constants.

All usual rutting variables affecting the full rut depth at a fixed N (h_N) are hidden in the N_d variable as shown in Eq. (2).

$${\mathrm{N}}_{\mathrm{d}}=\mathrm{F}\left(\text{measuring device},\text{wheel weight},\text{wheel speed},\text{temperature},\text{asphalt}\mathrm{mix}\text{property}\right).$$E2

In Eq. (2), the asphalt mix property in the parenthesis is characterized by physical variables such as aggregate (kind, size, and gradation), asphalt binder (viscosity, elasticity, degree of oxidation, and adhesion), filler (kind and size), and air void (percentage) of a given asphalt mix. Thus, the estimation of N_d by Eq. (2) is difficult due to the many variables involved. Kaloush and Witczak [4] who spent much time and effort to evaluate N_d experimentally reported that its evaluation was time-consuming, and repeatability was not to be assured. In predicting the full rut depth by Eq. (1), direct estimation of N_d is preferred, instead of using Eq. (2).

In Eq. (2), if the measuring device, wheel speed, wheel weight, and temperature are fixed to certain values, then, N_d measured solely represents the asphalt mix property for a given mix as shown in Eq. (3).

$${\mathrm{N}}_{\mathrm{d}}=\mathrm{F}\left(\text{asphalt}\mathrm{mix}\text{property}\right).$$E3

2. Development of new asphalt mix performance equations

The N_d in Eq. (1) under the condition of Eq. (3) is nothing but the structural damage wheel pass number characterized by the property of a given asphalt mix. It is proposed that the number of wheel passes reaching the damage point (N_d; or a damage wheel pass number) in the solid structure of rut deformation is identical to the number of loadings required in reaching the fatigue cracking failure (N_f) under the form of Eq. (1). Note that both involve breakdown of the solid structure, and the only difference is temperature.

Numerous empirical fatigue cracking models are available in the fatigue cracking study for asphalt concrete mixes up to now. Among them, two major types of fatigue cracking models have received much attention. One relates fatigue cracking to the strain and the modulus of the asphalt concrete mixes, as shown in the work of Finn et al. [4]. The other relates the fatigue cracking failure to the dissipated energy consumed up to the structural damage point. In this study, the latter model is chosen for modification because the general solid structure study tells that energy dissipation is usually involved in its failure.

Dijk [5] studied the accumulated dissipated energy per unit volume up to the structural failure point (W_fat) with respect to the repetition number of applied small load in its structural failure (N_fat), where the bending beam specimens of asphalt concrete mixes in the solid-state are tested at the room temperature. He reported that, for a given asphalt mix, both W_fat and N_fat are related to each other in the form of Eq. (4) shown below, and this equation is governed independently with respect to the test method (2- or 3-point bending beam test), temperature change (10–40°C), frequency change (10–50 Hz), mode of deformation control, or magnitude of the load applied. That is,

$$W_{\mathit{fat}}=A\cdot {N_{\mathit{fat}}}^Z$$E4

where A and z are physical constants, N_fat is the repetition number of the applied load at the fatigue failure point, and W_fat is the total dissipated energy accumulated by a small incremental energy in reaching the structural failure point. Dijk and Visser [6] studied Eq. (4) further by fitting more data obtained from the bending beam failure test (at −10 to 10°C), where specimens were made of all kinds of asphalt concrete mixes. In their tests, temperature was not fixed and possible material changes were not considered. Here, in this study, the energy-related failure model of Dijk [3], Eq. (4), is chosen as the fatigue cracking model. But it is modified for measurement to be easy, time to be saved, and effort to be less. It is assumed that the ratio of accumulated dissipated energy at failure (W_fat) among various asphalt mixes is proportional to the ratio of the single stroke breakdown energy (W_1d) among the same mixes. That is,

$${\mathbf{W}}_{\mathbf{fat}}\left(=\sum _{\mathbf{i}=\mathbf{1}}^{{\mathbf{N}}_{\mathbf{d}}}{\mathbf{W}}_{\mathbf{i}}\right)\propto {\mathrm{W}}_{\text{1d}}$$E5

Here, W_i is the incremental, dissipated energy at each repetition. By the assumption, N_fat corresponds to N_d, and W_fat matches to W_1d. Now, Eq. (4) can be changed into the form of Eq. (6).

$${\mathrm{N}}_{\mathrm{d}}\mathrm{=}{\left(\frac{\mathrm{1}}{\mathrm{A}}\right)}^{\mathrm{1}/\mathrm{z}}·{{\mathrm{W}}_{\mathrm{fat}}}^{\mathrm{1}/\mathrm{z}}\mathit{=}\mathit{P}·{{\mathrm{W}}_{\mathrm{1}\mathrm{d}}}^{\mathrm{R}}$$

$$\mathrm{Log}\left({\mathrm{N}}_{\mathrm{d}}\right)=\mathrm{Log}\left(\mathit{P}\right)+\mathrm{R}·\mathrm{Log}\left({\mathrm{W}}_{\mathrm{1}\mathrm{d}}\right)$$E6

where N_d is the wheel pass number reaching the structural damage point, W_1d is the energy required to breakdown of a pavement concrete mix specimen by a single stroke, and P (= (1/A)^1/z) and R (= 1/Z) are material constants. Eq. (6) is the new fatigue cracking performance equation based on the single stroke breakdown energy (W_1d), which is claimed, first time here. Eq. (6) could eliminate the uncertainty involved in incremental testing by Eq. (4). Eq. (6) indicates that the accumulated incremental fatigue cracking equation (Eq. (4)) is changed into the single stroke fatigue cracking equation (Eq. (6)).

Now, one wants to insert the new fatigue cracking damage expression, Eq. (6), into the rut damage wheel pass number (N_d) in the full rutting equation, Eq. (1), to change the N_d variable hard to measure into W_1d simple to estimate. Then, one obtains the full rutting performance equation, Eq. (1), in terms of the single stroke breakdown energy (W_1d). The resultant equation is demonstrated in Eq. (7). Note that N is fixed here.

$${\mathrm{H}}_{\mathrm{N}}=\left[\mathrm{S}–\mathrm{T}\mathrm{Log}\left({\mathrm{W}}_{\text{1d}}\right)\right]{\mathrm{N}}^{\mathrm{b}}+\mathrm{C}{\left[\mathrm{S}–\mathrm{T}\mathrm{Log}\left({\mathrm{W}}_{\text{1d}}\right)\right]}^{\mathrm{D}}{\mathrm{N}}^{\mathrm{d}}$$E7

where h_N is the full rut depth (h) at a fixed N, W_1d is the single stroke breakdown energy, and S, T, C, D, b, d are constants. Eq. (7) shows that the number of wheel passes at the damage point (N_d) in Eq. (1), which is difficult to measure and time-consuming in measurement, has been changed into W_1d in Eq. (7), which is relatively easy to measure. Note that a fatigue cracking variable (W_1d) is incorporated into the full rut depth equation. This is the natural consequence because both rutting and fatigue cracking involve breakdown of the solid structure that is built on the same capillary pressure at different temperatures. Thus, the full rutting equation, Eq. (7), works if and only if the fatigue cracking equation, Eq. (6), works well. This implies that the single stroke damage energy (W_1d) evaluated for the fatigue cracking performance equation at the fatigue cracking temperature range (or 10–20°C) can be used to evaluate the full rutting performance equation (Eq. (1)) for the same mix at the rutting temperature range (or 50–70°C) as well. They are interrelated based on Eq. (7).

The full rut depth expression in Eq. (7) works at a fixed N that is chosen arbitrarily among the entire N. For the rut depth comparison, the wheel pass number at the damage point (N_d) is a desirable choice for N, because it represents a unique property of a given asphalt mix in the full rutting curve. The linear ((a)) and the full ((b)) rutting deformations, and N at N_d are demonstrated in Figure 1.

Figure 1.

(a) Linear rut depth, (b) Full rut depth, and N = N_d for different asphalt mixes [7].

Thus, Eq. (7) can be rewritten in terms of the structural damage point by using Eq. (6) at N=N_d, as

$${\mathrm{H}}_{\mathrm{Nd}}=\left[\mathrm{S}–\mathrm{T}\mathrm{Log}\left({\mathrm{W}}_{\text{1d}}\right)\right]{{\mathrm{N}}_{\mathrm{d}}}^{\mathrm{b}}+\mathrm{C}{\left[\mathrm{S}–\mathrm{T}\mathrm{Log}\left({\mathrm{W}}_{\text{1d}}\right)\right]}^{\mathrm{D}}{{\mathrm{N}}_{\mathrm{d}}}^{\mathrm{d}}$$E8

Eq. (8) can be rewritten as Eq. (9) by inserting Eq. (6), N_d = P·W_1d^R, into Eq. (8),

$${\mathrm{H}}_{\mathrm{Nd}}=\left[\mathrm{E}-\mathrm{F}\mathrm{Log}\left({\mathrm{W}}_{\text{1d}}\right)\right]{{\mathrm{W}}_{\text{1d}}}^{\mathrm{j}}+\mathrm{C}{\left[\mathrm{E}-\mathrm{F}\mathrm{Log}\left({\mathrm{W}}_{\text{1d}}\right)\right]}^{\mathrm{D}}{{\mathrm{W}}_{\text{1d}}}^{\mathrm{k}}$$E9

Since the rut depth at the damage point does not deviate much from the consolidated deformation curve (h₁) mentioned in the reference [1], the approximate form of Eq. (9) can be written by modifying the consolidated deformation term [1] as

$${\mathrm{h}}_{\mathrm{Nd}}\approx {\left[\mathrm{E}-\mathrm{F}\mathrm{Log}\left({\mathrm{W}}_{\text{1d}}\right)\right]}^{\mathrm{O}}{{\mathrm{W}}_{\text{1d}}}^{\mathrm{P}}.$$E10

If a small N (=N_s) immediately after the post compaction region is chosen, the rut depth solely corresponds to the consolidated deformation (h₁) [1] with the negligible dissipated deformation (h₂) [1]. In this case, further simplification is made to yield only the linear consolidated deformation [1]. That is,

$${\mathrm{h}}_{\mathrm{Ns}}/{{\mathrm{N}}_{\mathrm{s}}}^{\mathrm{b}}\approx {\mathrm{h}}_{\mathrm{Ns}}=\mathrm{S}–\mathrm{T}\mathrm{Log}\left({\mathrm{W}}_{\text{1d}}\right).$$E11

Eq. (11) is the linear rut depth equation for different asphalt mixes at the fixed small number of wheel passes immediately after the post compaction region. Note that even in this simplest rutting equation, the equation holds the property relating to solid fatigue cracking variable (W_1d), different from the traditional linear rutting equation (h = αN^β).

At crack temperatures (0°C-(−40°C)), some molecules in straight (or modified) asphalt binders of asphalt mixes will undergo phase changes such as local crystallization or glass transition. Once any phase change in asphalt mixes occurs at low temperatures, their physical properties become different from those of the original ones with no phase change. Thus, special attention is required when using the fatigue cracking equation (Eq. (6)) at low temperatures.

Eq. (6) can be used for either cracks or low-temperature cracks with W_1d that have changed due to phase change in the asphalt binders. Thus, at each low temperature of (−15°C), (−30°C), or (−45°C), W_1d for all asphalt mixes should be measured to find the low-temperature performance because measuring temperatures are all different, regardless of phase change. Of course, rutting is out of concern at these temperatures, and only the fatigue cracking equation (Eq. (6)) must be considered.

Note that Eq. (11) is a simple linear rutting equation to prove its effectiveness. For the validation study, a laboratory wheel tracking device is used to measure the rut depth at N = 2100 (a relatively small number of wheel tracking), performed at 60°C, while the single stroke damage energy (W_1d) is measured at 20°C which belongs to the fatigue cracking temperature. Note that both rutting and fatigue cracking properties are simultaneously involved even in this simple rutting equation.

3. Validation of a rut performance equation containing W_1d

3.1 Data measured by the laboratory wheel tracking test

The aggregate size and gradation used are shown in Table 1. The 10,450 g (95.0 wt.%) of the aggregates and 550 g (5.0 wt.%) of the modified asphalt binder are used to prepare a single mix specimen (11 kg) (100 wt.%) for the laboratory wheel tracking experiment. The aggregate weight percentage includes 2.0 wt.% of fillers (the limestone powder). Aggregates were heated to 190°C in a temperature-controlled oven for 2 hours to eliminate all moisture.

19 mm modified asphalt mixes		13 mm porous asphalt mixes
Aggregate size (mm)	Weight %	Aggregate size (mm)	Weight %
19 ∼ 13.2	25.00%	—	—
13.2 ∼ 9.5	41.08%	13.2 ∼ 9.5	48.89%
9.5 ∼ 4.75	17.88%	9.5 ∼ 4.75	31.08%
4.75 ∼ 2.36	2.84%	4.75 ∼ 2.36	4.41%
2.36 ∼ 0.6	3.68%	2.36 ∼ 0.60	5.64%
0.6 ∼ 0.3	1.60%	0.60 ∼ 0.30	2.26%
0.3 ∼ 0.15	1.77%	0.30 ∼ 0.15	2.34%
0.15 ∼ 0.07	1.02%	0.15 ∼ 0.075	1.11%
Filler	5.13%	Filler	4.26%
Sum	100%	Sum	100%
Binder Content	5.8%	Binder Content	4.2%

Table 1.

Aggregate size and distribution.

The modified asphalt binder is prepared by adding 3, 4, 5, and 6% of the polymer additive to 97, 96, 95, and 94% of the straight asphalt binder (AC-20), and all are mixed in a high shear, small mixer at 170°C for 1 hour. By mixing 13 mm porous and 19 mm regular aggregates (10,450 g) with four different modified binders (550 g) in a high shear mixer at 170°C, four different 19 mm modified mixes and four different 13 mm porous asphalt mixes are prepared.

These are poured into the wheel tracking mold (30 × 30 × 5 cm) at 160°C and are compacted 55 times by the compaction roller with a diameter of 46 cm, compaction pressure of 5 kg/cm², and tracking speed of 23 cm/s. The compact height is consistent such that the air voids of the compacted mixes are in the range of 4 to 6%. Before performing wheel tracking experiments, the prepared specimens in the tracking mold are cured for more than 18 hours at room temperature and then put in the environmental chamber at 60°C for 6 hours to achieve a homogeneous temperature distribution in the mixed specimen.

After 6 hours of specimen exposure at 60°C, the wheel tracking test begins inside the environmental chamber at 60°C. In the test, the width of the steel tracking wheel is 5 cm, its diameter is 20 cm, and the wheel pressure is 70 ± 1 kg. The wheel tracking distance is 23 ± 1 cm, and the speed is 42 ± 1 passes per minute. At 21, 105, 210, 315, 630, 945, and 1260 cycles of wheel tracking, the deformation of the specimens is measured and recorded. Dynamic stability is obtained by computing the slope of the deformation curve in the tracking time range of 45–60 minutes. The test results are shown in Tables 2 and 3.

13 mm porous asphalt mixes
Measurement			Wheel tracking depth (mm)				Remarks
			1 spec	2 spec	3 spec	4 spec
Number of wheel passes		25	0.52	0.42	0.44	0.45
		105	0.90	0.70	0.72	0.72
		210	1.06	0.84	0.86	0.87
		315	1.16	0.92	0.95	0.95
		630	1.33	1.07	1.10	1.14
		945	1.43	1.15	1.19	1.25
		1260	1.50	1.21	1.25	1.34
Regression results by h = αN^β
α			0.2997	0.2300	0.2372	0.2168
β			0.2293	0.2360	0.2360	0.2563
r^2			0.9930	0.9953	0.9963	0.9992
test item		Unit	Analysis results				Specification
			1	2	3	4
Wheel tracking evaluation	Dynamic Stability	Passes/mm	9000	10,500	10,500	7000	KSF-2374: 2000
	Deformation Rate	mm/minute	0.0047	0.0040	0.0040	0.0060

Table 2.

Wheel tracking test results for 13 mm porous mix specimens.

19 mm modified asphalt mixes
Measurement			Wheel tracking depth (mm)				Remarks
			1 spec	2 spec	3 spec	4 spec
Number of wheel passes		25	0.40	0.41	0.47	0.25
		105	0.61	0.64	0.73	0.47
		210	0.69	0.73	0.85	0.59
		315	0.76	0.79	0.93	0.65
		630	0.85	0.87	1.04	0.76
		945	0.90	0.93	1.11	0.82
		1260	0.94	0.96	1.16	0.86
Regression results by h = αN^β
α			0.2424	0.2597	0.2798	0.1379
β			0.1928	0.1870	0.2024	0.2614
r^2			0.9959	0.9932	0.9952	0.9899
Test item		Unit	Analysis results				Specification
			1	2	3	4
Wheel tracking evaluation	Dynamic stability	Passes/mm	15,750	14,400	12,600	15,750	KSF-2374: 2000
	Deformation rate	mm/minute	0.0027	0.0020	0.0033	0.0027

Table 3.

Wheel tracking results for 19 mm modified mix specimens.

3.2 Data measured by Marshall and indirect tensile testing

Marshall specimens for both kinds of asphalt concrete mixes (13 mm porous asphalt concrete mixes and the 19 mm modified ones) are prepared in Section 3.1 above. Each asphalt concrete mix specimen was designed to be 1100 grams. The 13 mm porous asphalt mix specimen required 4.2% modified asphalt binders, while the 19 mm modified mix required 5.8% modified asphalt binders. The heated aggregates and a modified binder to make one specimen were poured into the Marshall mixing bowl and mixed for 3 minutes, and then the mix was back into the oven at 170°C. After 1 hour in the oven, it was poured into the Marshall mold and compacted 50 times on both sides by the Marshall compactor to make a Marshall specimen. The Marshall specimens were immersed in a temperature-controlled water bath at 60°C for 30 minutes, and then Marshall stability was tested for each specimen in the environmental chamber at 60°C.

A downward force was applied on each specimen at a speed of 50.8 mm/minute. The estimated Marshall stability (MS) and the corresponding Marshall breakdown energy were computed and are listed in Table 4. Likewise, the indirect tensile (IDT) test for each Marshall specimen at room temperature with a speed of 50.8 mm/minute was performed. The peak load and the IDT breakdown energy measured are also listed in Table 4.

Kinds of mixes		Marshall stability (kgf)		Break-down energy in Marshall Stability (kgf-mm)	Maximum IDT load (kgf)		Break-down energy in IDT (kgf-mm)
		Measured	Average	Average	Measured	Average	Average
13 mm – surface course specimen in modified porous asphalt mixes	1	1166.0	1135	3076	1050.1	982	742.4
		1081.5			910.5
		1157.5			986.0
	2	1265.5	1161	1696	1189.0	1305	1093
		1155.0			1341.5
		1063.0			1558.5
	3	785.5	894	1991	980.5	973	1112
		906.5			1063.0
		991.0			958.0
	4	927.0	979	1903	991.0	930	839.1
		989.5			901.0
		1021.0			876.5
19 mm- surface course specimen in modified asphalt mixes	1	1626.5	1447	3774	1950.0	1761	1976
		1739.5			1593.0
		1556.5			1956.5
		1396.5			1739.5
		1208.0			—
		1156.5			—
	2	1546.5	1657	3547	2151.0	2040	1786
		1835.5			1791.5
		1845.0			2143.0
		1577.5			2184.5
		1573.0			—
		1564.0			—
	3	1890.5	1787	4042	2190.0	2068	1528
		1828.5			1960.0
		1977.0			2090.0
		1741.0			2152.5
		1566.5			—
		1718.0			—
	4	1975.0	1702	7042	1841.0	1929	2174
		1777.0			1856.0
		1355.0			2089.0

Table 4.

Data Measured by MS and IDT.

3.3 Analysis of rut prediction by new mix design and Marshall stability

To validate the rut depth equations derived, Eq. (11) is adopted because it is relatively simple. Note that the rutting equation, Eq. (11), still contains the fatigue cracking variable W_1d, which belongs to the solid property. Traditionally, no rutting equation contains the solid fatigue cracking property (W_1d) in its expression. The existence of the solid property is a natural consequence because the asphalt mix after compaction, even at the rutting temperature, forms a solid structure, and breakdown appears with repeated loading [1]. Of course, the fatigue cracking has solid property, too, as shown in Eq. (6). The traditional empirical rutting equation often described by h = α N^β (where h is the rut depth, N is the number of wheel passes, and α and β are physical parameters) does not possess the solid property, in its expression.

The rut depth at 1260 cycles in Tables 2 and 3 (the small wheel tracking number after the post compaction) with respect to W_1d from Table 4 are selected to regress Eq. (11). Additionally, a relatively small N (=1260 cycle) and nearly the same β make the term h_N/N^β become almost h_N because N^β is nearly constant.

The excellent curve-fitting result shown in Figure 2 is regarded to be a great surprise under all assumptions made to come up with Eq. (11). Figure 2 clearly validates the legitimacy of both rutting and fatigue cracking assumptions to formulate Eq. (11), although all hypotheses made in driving Eq. (6), Eq. (7), Eq. (8), Eq. (9), Eq. (10), and Eq. (11) need more extensive validation.

Figure 2.

Rut depth at N = 1260 cycles predicted by using Eq. (9).

Even though only eight different asphalt mixes are chosen (four regular modified and four porous modified asphalt mix specimens), the accuracy shown in regression by Eq. (11) encourages the promising future of all performance equations derived. Even though more extensive validation may be needed in the future, the concept introduced seems to provide a reasonable scientific foundation.

The empirical Marshall stability (MS) data are correlated to the rut depth data at N = 1260 cycles to compare the prediction result of the scientifically derived Eq. (11).

The rut depth at N = 1260 cycles predicted by the MS for the four porous and the four regular modified asphalt mixes are demonstrated in Figure 3.

Figure 3.

Rut depth prediction at N = 1260 cycles by MS value.

The figure shows poor rut prediction accuracy (r² = 0.5048) in the MS method compared to the scientifically derived rut equation, Eq. (11), (r² = 0.9542) as shown in Figure 1. This comparison clearly demonstrates a wise choice of performance equations derived for rutting and fatigue cracking. The empirical MS mix design method is still widely used in most of the European and Asian countries as a practical mix design method because no accurate design method is known so far and is relatively simple to use. Thus, despite considerable errors found in the empirical MS method, it is still around in the asphalt mix design area.

The peak IDT load is usually regarded as the stiffness of a given mix and is an alternative to the MS method. Its rut prediction is manifested in Figure 4, which shows a poor result (r² = 0.6644) like the MS method (r² = 0.5048) shown in Figure 3.

Figure 4.

Prediction of rut depth by the peak IDT load.

Figure 5 exhibits below the relationship of the peak IDT at 25°C to the peak MS value at 60°C. The relatively good fitting (r² = 0.9436) between the two simply displays that the empirical MS value just represents the stiffness of the asphalt mixes such as the peak IDT representing the stiffness of the asphalt mixes.

Figure 5.

Relationship between peak IDT load and MS.

4. Master performance equations for rigorous design equations

Eqs. (6) and (9) derived above represent asphalt mix performance equations (or the rigorous asphalt mix design equations), but they cannot take roll of master equations because they fail to provide constants in their equations. The reason is that constants P and R in Eq. (6) and constants E, F, C, D, j, and k in Eq. (9) are missing. With these constants specified, Eq. (6) for fatigue cracking and Eq. (9) for full rutting truly become the master performance (or the rigorous asphalt mix design) equations.

To estimate the constants (P and R) in the fatigue cracking equation (Eq. (6)), experiments should be performed to measure N_d (number of load repetitions reaching the fatigue cracking failure) and W_1d (single stroke breakdown energy) for a given asphalt mix. Obtaining a N_d requires the fatigue cracking experiment to perform repetitive loadings at a fixed loading weight and a speed for a given mix specimen to be broken down, and W_1d should be measured for the same mix specimen. Through many experiments for diverse asphalt mix specimens, many pairs of N_d and W_1d are available that can be regressed by using Eq. (6) to get P and R. By inserting these values into Eq. (6), one yields the master equation of the fatigue cracking (Eq. (6)). Once the master equation is formulated, estimation of W_1d for a certain asphalt mix can provide its fatigue cracking life (N_d).

This procedure can be applied identically in making the master equation of the full rutting equation (Eq. (9)). Here, the rut depth (h_N) is measured at the wheel-pass number at a damage (an inflection, or a boundary) point for a given rut mix specimen by using one of the lab wheel tracking devices. Note that the same W_1d obtained in the fatigue cracking experiments can be used for the same asphalt mix. After so many pairs of h_N and W_1d are obtained for many corresponding asphalt mixes, Eq. (9) is used to regress h_N with respect to W_1d to obtain constant values of E, F, C, D, j, and k. These constants are inserted into Eq. (9) to formulate the master full rutting equation. If one measures a W_1d for a certain asphalt mix and it is put into the master full rutting equation, the rut depth can be predicted for the given mix.

These master performance equations can become rigorous hot mix asphalt design (or asphalt mix performance) equations.

Further attention is required for the W_1d. Once a W_1d is measured for a given asphalt mix will predict both the full rut depth and the fatigue cracking life simultaneously by using both corresponding master equations.

However, for the estimation of N_d by the fatigue cracking experiments and h_N by the full rutting experiments to formulate the master performance equations of Eq. (6) and Eq. (9), so many experimental works are required for measurements such as much effort of specimen preparation, set up of testing procedure, and tremendous amount of time consumption. Thus, another simple hot mix asphalt (HMA) design theory instead of formulating master performance equations will be suggested based on the performance equations of asphalt mixes in the next section.

5. A novel HMA design theory for the asphalt pavement construction and the asphalt mix quality check

The rigorous HMA design equations (or performance equations), Eq. (6) and Eq. (9), developed in the previous section are depicted schematically in Figure 6. It is noted that those design equations are the only function of the single stroke damage energy (W_1d). Each W_1d in the figure has a pair of data; one is the full rut depth (h_N), and the other is the fatigue cracking life (N_d).

Figure 6.

Eqs. (6) and (9) are schematically drawn.

W_1d can predict the fatigue cracking life (N_d) by using Eq. (6), and then Eq. (6) yields the full rut depth (h_N) in Eq. (9). This relationship is depicted mathematically in Eq. (12) below.

$${\mathrm{N}}_{\mathrm{d}}={\mathrm{F}}_1\left({\mathrm{W}}_{\text{1d}}\right);{\mathrm{h}}_{\mathrm{N}}={\mathrm{F}}_2\left({\mathrm{N}}_{\mathrm{d}}\right)={\mathrm{F}}_2\left[{\mathrm{F}}_1\left({\mathrm{W}}_{\text{1d}}\right)\right]={\mathrm{F}}_2\left({\mathrm{W}}_{\text{1d}}\right).$$E12

Figure 6 or Eq. (12) depicts that each single stroke damage energy (W_1d) determines both corresponding rutting and fatigue cracking value simultaneously. As indicated in Figure 6, the higher W_1d specifies the lower rut depth and the longer fatigue cracking life, and, reversely, the lower W_1d manifests the higher rut depth and the less fatigue life. That is, W_1d alone represents both performance values of rutting and fatigue cracking. Thus, the comparison of W_1d among different asphalt mixes is enough in finding different performance standings instead of evaluating rutting (Eq. (9)) and fatigue cracking (Eq. (6)) separatory. The direct use of W_1d in place of using Eqs. (6) and (9) reduces tremendous amount of experimental works in formulating master equations. This yields W_1d to be the new HMA mix design value. Thus, an innovative mix design method can be built upon W_1d measured for comparison among different asphalt mixes. In addition, W_1d can be used as the quality check of a new asphalt mix invented among others.

Measurement of W_1d for a given asphalt mix and then comparing it to other W_1d values to find its relative performance standing will give rise to the innovative, simple HMA mix design method. It is genuinely called the scientific, performance-based, HMA design method because W_1d itself represents the performance of rutting and fatigue cracking equations.

Figure 7 reiterates the different accuracies between the empirical Marshall Stability (MS) method and the single stroke damage energy (W_1d) method. In the asphalt mix design method for rutting as well as fatigue cracking performances, the W_1d method should be preferred over the MS method.

Figure 7.

Different accuracies are shown between MS and W_1d.

Since W_1d (or N_d) is a function of temperature, measured values will vary with temperature, especially if the material is changed. Asphalt mix materials (containing straight or modified asphalt binders) can undergo local crystallization or glass transition at low temperatures. Thus, the following temperatures should be considered for low-temperature measurement of W_1d. At these temperatures, an asphalt mix design criteria must be set differently.

1. 15°C (no phase change occurs from 10–70°C).

   Evaluation of rutting or fatigue cracking resistance simultaneously by measuring W_1d for all asphalt concrete mixes at 15°C.

2. −15°C (some molecules in asphalt binders undergo local crystallization).

   The W_1d of all asphalt concrete mixes should be measured to assess crack resistance.

3. −30°C (some molecules in the asphalt binder underwent a glass transition).

The W_1d of all asphalt concrete mixes should be measured to assess low-temperature crack resistance.

6. Conclusions

In the previous study, Huh [3], the full rutting performance equation (Eq. (1)) has been derived as a function of only a damage wheel pass number (N_d) that is hard to measure. It is explained that N_d in Eq. (1) is set to be identical to the fatigue cracking repetition number at failure (N_f) in the fatigue cracking study.

A fatigue cracking model in the literature is adopted and modified to provide a new variable easily measurable in the laboratory. It is known to be the single stroke damage energy (W_1d) that changes N_d (hard to measure) in the full rutting equation into W_1d (easily measurable).

Change of the variable constitutes two performance equations (the full rutting equation and the modified fatigue cracking equation). These two equations turned out to become rigorous mix design equations. To use them in practice, the master equations (specifying constants in two performance equations) must be formulated accordingly. The formulation requires tremendous amount of experimental works.

It is noted that two master performance equations display as a function of a single variable (W_1d) that controls both performance equations simultaneously by one single value. Thus, instead of using both master performance equations as rigorous mix design equations, direct comparison of the single variable (W_1d) can do the same role of using two equations.

Thus, measuring W_1d and comparing it to all others will constitute an innovative, simple, accurate, revolutionary, performance-based asphalt mix design method dealing with both regular and recycled asphalt mixes. This is the extraordinary discovery in the hot mix asphalt design method proposed for the first time.

Note that W_1d itself for a particular mix represents mix compositions, binder types, aggregate gradations, and environmental factors influencing the performance of mixes. Accurate mix design can save a tremendous amount of construction budgets due to avoiding early pavement damage or undesirable mix design criteria. Depending on the mix design value, a particular asphalt pavement life could be either 6 or 20 years. W_1d determines the life expectancy. That is why the proper mix design method is important.

Current mix design methods, Marshall Stability, Gyratory Compactor, Volumetric Design, Rice method, Balanced Mix Design, Vheem’s Design, Semi-Circular Bending Test (SCB), and Disc-shaped Compact Tension test (DCT) are available in the industry. All these are either empirical, contain considerable errors, or fail to relate their design values to asphalt pavement performances. The scientifically proposed asphalt mix design method proposed in this study could be the best alternative to solve problems involved in all design methods mentioned.

Development of a new asphalt mix design method like measurement of one-stroke damage energy (W_1d) for all kinds of asphalt mixes, choosing a measuring equipment (a single equipment for all performance temperatures), specimen preparations, procedures of measurement, analysis of measured data, and overall setup of mix design criteria require further research support for actual implementation in the industry.