The Discrete and Continuous Retardation and Relaxation Spectrum Method for Viscoelastic Characterization of Warm Mix Crumb Rubber-Modiﬁed Asphalt Mixtures

materials

Article The Discrete and Continuous Retardation and Relaxation Spectrum Method for Viscoelastic Characterization of Warm Mix Crumb Rubber-Modiﬁed Asphalt Mixtures

Fei Zhang 1, Lan Wang 2, Chao Li 2 and Yongming Xing 1,*

1 School of Science, Inner Mongolia University of Technology, Hohhot 010040, China; [email protected] 2 School of Civil Engineering, Inner Mongolia University of Technology, Hohhot 010040, China; [email protected] (L.W.); [email protected] (C.L.) * Correspondence: [email protected]; Tel.: +86-187-4816-6534

Received: 21 July 2020; Accepted: 20 August 2020; Published: 23 August 2020

Abstract: To study the linear viscoelastic (LVE) of crumb rubber-modified asphalt mixtures before and after the warm mix additive was added methods of obtaining the discrete and continuous spectrum are presented. Besides, the relaxation modulus and creep compliance are constructed from the discrete and continuous spectrum, respectively. The discrete spectrum of asphalt mixtures can be obtained from dynamic modulus test results according to the generalized Maxwell model (GMM) and the generalized Kelvin model (GKM). Similarly, the continuous spectrum of asphalt mixtures can be obtained from the dynamic modulus test data via the inverse integral transformation. In this paper, the test procedure for all specimens was ensured to be completed in the LVE range. The results show that the discrete spectrum and the continuous spectrum have similar shapes, but the magnitude and position of the spectrum peaks is different. The continuous spectrum can be considered as the limiting case of the discrete spectrum. The relaxation modulus and creep compliance constructed by the discrete and continuous spectrum are almost indistinguishable in the reduced time range of 5 3 10− s–10 s. However, there are more significant errors outside the time range, and the maximum error is up to 55%.

Keywords: master curve; discrete spectrum; continuous spectrum; relaxation modulus; creep compliance; warm mixing crumb rubber-modiﬁed asphalt mixture

1. Introduction It is well known that asphalt mixtures demonstrate linear viscoelastic (LVE) solid properties at small strain levels (typically no more than 120 uε) and over a wide range of frequencies and temperatures [1]. These viscoelastic response functions, including dynamic modulus |E*(f)|, creep compliance D(t), and relaxation modulus E(t), are used to characterize the LVE properties of hot mix asphalt (HMA). The dynamic modulus is the ratio of the amplitude of the stress and strain response of viscoelastic material to endure dynamic loading, which can be used to determine the strength properties of viscoelastic materials. Creep compliance represents the time-dependent strain response of viscoelastic material in unit stress, which can be used to determine the viscoelastic strain. Conversely, the relaxation modulus represents the time-dependent stress response of viscoelastic material in unit strain, which can be used to determine the viscoelastic stress. Different viscoelastic response functions characterize the viscoelasticity of asphalt mixtures in different forms. Different experimental tests can characterize the corresponding viscoelastic response functions, respectively. However, it is theoretically

Materials 2020, 13, 3723; doi:10.3390/ma13173723 www.mdpi.com/journal/materials Materials 2020, 13, 3723 2 of 29 feasible to convert these LVE functions to each other, since they are mathematically equivalent for each loading mode [2]. The concept of spectrum distribution function [3] plays a significant role in LVE behavior theory. It is independent of a specific time system and occupies a vital position in viscoelasticity theory. The relaxation spectrum and the retardation spectrum are the distributions of the spectrum intensity with respect to relaxation and retardation time. They are an essential characteristic of LVE materials, from which all other viscoelastic functions and their responses can be obtained [4]. These functions describe the primary response of viscoelastic materials, so it has great significance to determine and characterize them from the given experimental results. Generally, they can be divided into the continuous spectrum and discrete spectrum according to the time spectrum interval [5]. The discrete relaxation and retardation spectrum of asphalt mixtures can be characterized by mechanical models with different configurations of linear springs and dashpots [6]. The discrete relaxation spectrum can be determined by Prony coefficients from the generalized Maxwell model (GMM) model. In contrast, the Prony coefficients of the generalized Kelvin model (GKM) can be used to determine the discrete retardation spectrum [7]. A series of methods for determining the discrete spectrum from experimental data have been proposed. However, oscillations and negative spectrum intensities are unavoidable problems with the discrete spectrum due to the scattered nature of the experimental data and the time interval choice. Schapery [8] uses the collocation method to solve the discrete spectrum. In a word, using the data of the pre-selected collocation points to solve the corresponding spectrum intensity, but this is an ill-posed problem, and the solution result is not unique. The accuracy of the spectrum intensity depends on the spacing of the collocation points [9]. However, the collocation method is susceptible to the value of the discrete-time interval. To address the problems with the collocation method, Cost and Becker [10] proposed the multiple data method by applying the least-squares fitting technique to the Laplace transform domain. The recursive method proposed by Emri and Tschoegl [11–13] further improves the calculation method of the discrete spectrum. Park and Kim [14] proposed a method of pre-smoothing experimental data by power-law sequence representation and then fitting the Prony series to the pre-smoothed data. This method avoids the occurrence of negative spectrum intensity and local oscillation of the spectrum. Besides, the regularization method of quadratic programming [15], Marquardt-Levenberg program [16], Bayesian method [17] and piecewise cubic Hermite spline polynomials [18] are new methods that have been gradually developed in recent years. The discrete relaxation spectrum is an essential parameter for predicting the nonlinear viscoelastic behavior, while the continuous spectrum is not convenient for numerical calculations. Therefore, Bae [19] developed a new method to detect the traditional discrete relaxation spectrum by continuous relaxation spectrum, then allows precise determination of the discrete relaxation spectrum. Considering the problems with discrete spectrum, the continuous spectrum concept has been gradually developed in recent years. The continuous spectrum can be considered as a discrete spectrum with an infinitely dense time interval [20]. Using the continuous spectrum instead of the discrete spectrum can avoid the above problems. Although we can, obtain the continuous spectrum from a viscoelastic function requires integral inversion, which usually presents a considerable challenge. Bažant and Xi [21] obtained the continuous retardation spectrum using the inverse Laplace transform and the higher-order derivative of the creep compliance function. Mun and Zi [22] adopted a similar method, converting the frequency-domain test results into time-domain test results using an approximate method, and constructing the master curve in the time domain, and then using the master curve to obtain higher-order derivatives and finally obtain the continuous retardation spectrum. Based on the LVE theory, Sun [23] constructed continuous relaxation and retardation spectrum based on the H-N complex modulus model and compared it with the results of the discrete spectrum. Liu [24]. proposed a new method for determining the continuous relaxation spectrum. This method solves the continuous relaxation spectrum simultaneously from the master curve expressions of the storage modulus E0 and the loss modulus E”. Materials 2020, 13, 3723 3 of 29

Although the computational methods of discrete and continuous spectrum have achieved an unprecedented development, more and more people are paying great attention to the interconversion between each other [25]. There are some results comparing the viscoelastic functions constructed from the discrete and continuous spectrum. Gu [26] used the discrete spectrum of the 2S2P1D model to obtain the viscoelastic information of a fine aggregate matrix to construct discrete element models. Yu [27] used the generalized sigmoidal model (GSM) and Havriliak-Negami model (HNM) to construct master curve models and then compared the differences between the discrete and continuous spectra constructed by the different models. Relaxation modulus and creep compliance can be derived from relaxation and retardation spectra, respectively. Teltayev [28] presented the results of bending beam rheometer (BBR) tests of blown bitumen at low temperatures and analyzed the results using the developed relaxation modulus and relaxation time spectra. The framework developed by Liu [29] based on spectral methods ensures that the master curves of all viscoelastic variables complied with the LVE theory. Moreover, the relaxation and retardation spectrum are the most important parameters for the construction of viscoelastic continuous damage models (VECD) [30–32] and their probabilistic models [33,34]. This study focused on comparing the relaxation modulus and creep compliance master curves of crumb rubber-modified asphalt mixtures constructed by discrete and continuous spectrum methods and to evaluate the accuracy of the master curve model. In this paper, four types of crumb rubber-modified asphalt mixtures were produced in laboratory, under the premise of approximately satisfying the Kramers-Kronig (K-K) relations [35], the coefficients of the Prony series can be treated as a discrete spectrum according to the collection method. The inverse integral transformation [36] can be used to derive the continuous spectrum according to the master curve model of generalized Sigmoidal. Finally, the viscoelastic response functions constructed based on the discrete and continuous spectrum are compared, and the error magnitudes were analyzed.

2. Materials and Methods

2.1. Materials and Specimen Fabrication Four types of laboratory-produced mixtures were used in this study. AC-16, which is also used widely in highway construction in China, was utilized in the following text. The virgin asphalt PG 64–22 was purchased from Inner Mongolia Luda Asphalt Co., Ltd. (Jining, China). The mineral aggregate was purchased from a quarry located in Zhuozishan (Inner Mongolia Autonomous Region, China). The warm mix additive (SDYK) was purchased from MeadWestvaco Co., Ltd. (Wuxi, China). Crumb rubber was purchased from the Hubei Anjie Road & Bridge Technology Co., Ltd. (Ezhou, China). Table1 and Figure1 show the aggregate gradation of mixtures. The crumb rubber-modified asphalt binder mixing method used in this study was a wet process, in which the crumb rubber is added to the virgin asphalt binder (penetration grade 80/100) before introducing it into the asphalt mixture. The crumb rubber-modified asphalt binder was produced in the laboratory at 180 ◦C for 30 min using an open blade mixer at a blending speed of 700 rpm [37]. The percentage of crumb rubber added for the crumb rubber-modified asphalt binder was 20% by weight of virgin asphalt. For the warm mix crumb rubber- modified asphalt binder, the warm mix additive was added to crumb rubber-modified asphalt binder by mixing at 180 ◦C for 30 min with a conventional mechanical mixer. The storage stability of binder can be characterized by the difference in softening points, the results are shown in Table2. Then the crumb rubber-modified asphalt binder was prepared for manufacturing specimens.

Table 1. Aggregate stockpile blend percentages.

Aggregate 10–20 mm 5–10 mm 3–5 mm 0–3 mm Filler Blend Percentage by Weight/% 21 38 10 28 3 Materials 2020, 13, x 4 of 28

Table 1. Aggregate stockpile blend percentages.

Materials 2020, 13, 3723 Aggregate 10–20 mm 5–10 mm 3–5 mm 0–3 mm Filler 4 of 29 Blend Percentage by Weight/% 21 38 10 28 3

100 Maximum Gradations 90 Minimum Gradations Average Gradations 80 Restricted Zone Control Points 70 Maximum Density Line Gradations 60

30 Percent Passing (%) 20

0 0.075 0.15 0.3 0.6 1.18 2.36 4.75 9.5 13.2 16 19 Sieve Size (mm) Figure 1. CrumbCrumb rubber-modified rubber-modiﬁed asph asphaltalt mixtures gradation.

Table 2. The softening points of the top and bottombottom of the samples after storing atat 163163 ◦℃C for 48 h.

SofteningSoftening Point Point/°C /◦C Sample Sample BottomBottom Top TopDifference Diﬀerence H-CR-60 63.5 57.0 6.5 H-CR-60 63.5 57.0 6.5 W-CR-60W-CR-60 60.860.8 56.2 56.2 4.6 4.6 H-CR-CH-CR-C 68.168.1 59.8 59.8 8.3 8.3 W-CR-CW-CR-C 62.962.9 57.5 57.5 5.4 5.4

The crumb rubber-modified asphalt content is 5.4% and 5.6% by the weight of the total mixture Hot mix 60-mesh crumb rubber-modified asphalt mixture (HMA-60) is the mixture prepared mass for HMA-60 and HMA-C, respectively. The content of crumb rubber-modified asphalt of warm by mixing aggregates and 60-mesh crumb rubber-modified asphalt binder following the hot mixing mix asphalt mixture is the same with the corresponding hot mix asphalt mixture. HMA was mixed process, while warm mix 60-mesh crumb rubber-modified asphalt mixture (WMA-60) is the mixture and compacted at 180 ℃ and 170 ℃, respectively, while WMA was mixed and compacted at 162 ℃ prepared by mixing aggregate and warm mix 60-mesh crumb rubber-modified asphalt binder following and 152 ℃, respectively. The volume parameters of warm-mixed asphalt mixture were consistent the warm mixing process. Hot mix compound-mesh crumb rubber-modified asphalt mixture (HMA-C) with the corresponding hot-mixed mixture. Table 3 lists the volume parameters of the mixture. The is the mixture prepared by mixing aggregates and compound-mesh crumb rubber- modified asphalt raw specimens (150 mm in diameter and 178 mm in height) were fabricated using a Superpave binder following the hot mixing process. Warm mix compound-mesh crumb rubber-modified asphalt gyratory compactor (produced by IPC Company, Melbourne, Australia) and then cut and cored to mixture (WMA-C) is the mixture prepared by mixing aggregates and warm mix compound-mesh standard dimensions (100 mm in diameter and 150 mm in height). The air void content of all type crumb rubber-modified asphalt binder following the warm mixing process. mixture was 4 ± 0.5%. Three replicate specimens were fabricated and tested for every mixture type. The crumb rubber-modified asphalt content is 5.4% and 5.6% by the weight of the total mixture Prior to testing all specimens were stored in an unlit cabinet to reduce ageing, but not allowed to mass for HMA-60 and HMA-C, respectively. The content of crumb rubber-modified asphalt of warm more than two weeks. mix asphalt mixture is the same with the corresponding hot mix asphalt mixture. HMA was mixed and compacted at 180 ◦CTable and 3. 170 Volume◦C, respectively, parameters of while optimum WMA asphalt was mixedcontent. and compacted at 162 ◦C and 152 ◦C, respectively. The volume parameters of warm-mixed asphalt mixture were consistent with Gross theAsphalt corresponding hot-mixed mixture.Theoretical Table3 lists the volume parameters of the mixture. TheFlow raw OAC/% Density Void/% VMA/% VFA/% Stability/KN Mixture Density g/cm³ Value/mm specimens (150 mm in diameterg/cm³ and 178 mm in height) were fabricated using a Superpave gyratory compactorHMA-60 (produced5.4 by 2.439 IPC Company, 2.537 Melbourne, 3.86 Australia) 14.04 and72.5 then cut and 10.45 cored to standard 2.74 dimensionsWMA-60 (1005.4 mm in diameter 2.442 and 150 2.538 mm in height). 3.78 The13.93 air void72.8 content of 10.95 all type mixture 2.87 was 4 HMA-C0.5%. Three5.6 replicate 2.441 specimens were 2.546 fabricated 4.12 and tested14.15 for every70.9 mixture 11.23 type. Prior to 2.45 testing ±WMA-C 5.6 2.444 2.546 4.01 14.04 71.5 11.61 2.62 allStandard specimens were- stored in - an unlit cabinet - to reduce 3~5 ageing, ≥ but13 not65~75 allowed to more≥8 than two2~4 weeks.

Materials 2020, 13, 3723 5 of 29

Table 3. Volume parameters of optimum asphalt content.

Gross Theoretical Asphalt Flow OAC/% Density Density Void/% VMA/% VFA/% Stability/KN Mixture Value/mm g/cm3 g/cm3 HMA-60 5.4 2.439 2.537 3.86 14.04 72.5 10.45 2.74 WMA-60 5.4 2.442 2.538 3.78 13.93 72.8 10.95 2.87 HMA-C 5.6 2.441 2.546 4.12 14.15 70.9 11.23 2.45 WMA-C 5.6 2.444 2.546 4.01 14.04 71.5 11.61 2.62 Standard - - - 3~5 13 65~75 8 2~4 ≥ ≥

2.2. Experimental Plan The complex modulus test was conducted in a compression load according to AASHTO TP 79–15 [38]. A servo-hydraulic universal testing machine (UTM-100) produced by the IPC Company (Melbourne, Australia) was used to measure dynamic modulus and phase angle for all test specimens. The test was conducted in four diﬀerent temperature (5 ◦C, 20 ◦C, 35 ◦C, 50 ◦C) and seven diﬀerent frequency (25 Hz, 20 Hz, 10 Hz, 5 Hz, 1 Hz, 0.5 Hz, 0.1 Hz). The test started from the lowest temperature (5 ◦C) to the highest (50 ◦C) for each specimen. Moreover, each specimen was tested at every temperature started from the highest frequency (25 Hz) to the lowest (0.1 Hz). The maximum axial strain was controlled within 70 µε to assure that the specimens stayed within the LVE domain [38]. The raw data on dynamic modulus and phase angle for types of mixture is shown in Table S1. The information on the statistical analysis of the test results of the types of asphalt mixture is shown in Tables A1 and A2.

3. Theoretical Background

3.1. LVE Theory Asphalt Mixtures

3.1.1. The Time-Temperature Superposition Principle (TTSP) The viscoelastic of asphalt mixtures is signiﬁcantly dependent on time and temperature. In polymer physics, the TTSP [39] can be used to analyze material properties in unknown conditions based on known conditions. Then, the master curve of viscoelastic function to the reduced frequency is obtained, which would greatly reduce the test workload. the reduced frequency can be calculated following Equations (1) and (2). Williams-Landel-Ferry [40] applied the TTSP to polymers and found an empirical expression for the shift factor. It is the WLF equation. It has been found that the WLF equation can ﬁt the observed shift factor data over a wide range of temperatures from Tg~Tg + 100 for asphalt binders and mixtures, the individual expression is shown in Equation (1).

C1(T Tr) lg αT = − − (1) C + (T Tr) 2 − where: lgαT is the shift factor; C1, C2 are positive test constants of WLF for the material; Tr and T are the reference temperature and the experimental temperature respectively, ◦C. in this paper, the reference temperature Tr was taken as 20 ◦C. Reduced frequency [22] fr is the equivalent frequency of experimental temperature to the reference temperature, and it can be calculated as Equation (2) in the logarithmic axis:

lg fr = lg f + lg αT (2) where: lg fr is the reduced frequency in reference temperature; lg f is the frequency in experiment temperature. Materials 2020, 13, 3723 6 of 29

3.1.2. The Method of Construction the Master Curve of Linear Viscoelastic Response Function The generalized sigmoidal model (GSM) [41,42] and Huet-Sayegh model (HSM) [43,44] have been used successfully to characterize the mechanical behavior of asphalt mixtures, and this study uses the model to construct the dynamic modulus, storage modulus, and storage compliance master curves of asphalt mixtures. Based on the K-K relations, the phase angle, loss modulus, and loss compliance master curves can be derived from the GSM. The master curves constructed using this method, including all the viscoelastic information. whose expression is presented as follows.

The Master Curve of Dynamic Modulus and Phase Angle The master curve of dynamic modulus can be presented by GSM. The master curve of phase angle can be derived from GSM following the approximate K-K relations. The error function e f1 was applied to the test data of dynamic modulus and phase. Its equation is shown in Equation (3). The master curve model parameters in Equations (4) and (5) for each mixture type were solved simultaneously by minimizing the error function e f1 (Equation (3)). v uv u  2 t N !2 t N φ φ 1 X E∗ m,i E∗ p,i 1 X m,i p,i  e f = e f + e f = | | − | | +  −  (3) 1 E∗ φ   N E∗ m,i N  φ  i=1 | | i=1 m,i where: e f1 is the error function of dynamic modulus and phase angle; e fE∗ is the error function of dynamic modulus; e fΦ is the error function of phase angle; N is the number of the measured data points that is equal to 28; E∗ m,i is ith data point of the measured dynamic modulus, MPa; E∗ p,i is ith | | | | data point of predicted by dynamic modulus master curve, MPa; φ m,i is ith data point of the measured phase angle, ◦; φ p,i is ith data point of predicted by phase angle master curve, ◦. α lg E∗ = δ + (4) | | 1 + eβ+γ(log fr)

π d(lg E ) π eβ+γ lg fr ( ) = ∗ = φ1 fr k | | kαγ 1 (5) 2 d(lg fr) − 2 1+ 1 + λeβ+γ lg fr λ where: lg E is the logarithm of the dynamic modulus; δ is the value of the lower asymptote of the E | ∗| | ∗| master curve; α is the diﬀerence between the upper and lower asymptotes of the E master curve; β and | ∗| γ is shape coeﬃcients of the E master curve;. λ determined the model’s asymmetric characteristics; | ∗| Φ1( fr) the master curve of phase angle predicted by the GSM based on the approximate K-K relations; k is a positive correction, it is to obtain potentially more accurate prediction.

The Master Curve of Storage Modulus and Loss Modulus The master curve of storage modulus can be presented by GSM. The master curve of the loss modulus can be derived from GSM following the approximate K-K relations. The error function e f2 was applied to the test data of storage modulus and loss modulus. The equation is shown in Equation (6). The test data of storage modulus and loss modulus can be determined from the results of the complex modulus test following Equations (7) and (8). The master curve model parameters in Equations (9) and (10) for each mixture type were solved simultaneously by minimizing the error function e f2 (Equation(6)). uv uv t N !2 t N !2 1 X E0 m,i E0 p,i 1 X E00 m,i E00 p,i e f = e f + e f = | | − | | + | | − | | (6) 2 E0 E00 N E N E i=1 | 0|m,i i=1 | 00 |m,i Materials 2020, 13, 3723 7 of 29

where: e f2 is the error function of storage modulus and loss modulus; e fE0 is the error function of storage modulus; e fE00 is the error function of loss modulus; N is the number of the measured data points that is equal to 28; E is ith data point of the measured storage modulus, MPa; E is ith | 0|m,i | 0|p,i data point predicted by storage modulus master curve, MPa; E00 is ith data point of the measured | |m,i loss modulus, MPa; E00 is ith data point predicted by loss modulus master curve, MPa. | |p,i

E0 = E∗ cos(φ) (7) | |

E00 = E∗ sin(φ) (8) | | where: E0 is the storage modulus obtained from complex modulus test; E00 is the loss modulus obtained from complex modulus test; Φ is the phase angle obtained from complex modulus test:

α lgE = δ + 0 (9) 0 0 1 β +γ (log fr) λ 1 + λ0e 0 0 0

β +γ lg f π d(logE0) π e 0 0 r E00 = kE0 = k0E0α0γ0 (10) 2 d(log f ) 2 1+ 1 r β +γ lg fr λ 1 + λ0e 0 0 0 where: lgE0 is the logarithm of the storage modulus; δ0 is the value of the lower asymptote of the E0 master curve; α0 is the diﬀerence between the upper and lower asymptotes of the E0 master curve; β0 and γ0 is shape coeﬃcients of the E0 master curve;. λ0 determined the model’s asymmetric characteristics; E00 is the master curve of loss modulus predicted by the GSM based on the approximate K-K relations; k0 is a positive correction, used to obtain potentially more accurate predictions.

The Master Curve of Storage Compliance and Loss Compliance The master curve of storage compliance can be presented by GSM. The master curve of the loss compliance can be derived from GSM following the approximate K-K relations. The error function e f3 was applied to the test data of storage compliance and loss compliance. The equation is shown in Equation (11). The test data of storage compliance and loss compliance can be determined from the complex modulus test results following Equation (12). The master curve model parameters in Equations (13) and (14) for each mixture type was solved simultaneously by minimizing the error function e f3 (Equation(11)). uv uv t N !2 t N !2 1 X D0 m,i D0 p,i 1 X D00 m,i D00 p,i e f = e f + e f = | | − | | + | | − | | (11) 3 D0 D00 N D N D i=1 | 0|m,i i=1 | 00 |m,i where: e f3 is the error function of storage compliance and loss compliance; e fD0 is the error function of storage compliance; e fD00 is the error function of loss compliance; N is the number of the measured 1 data points that is equal to 28; D0 m,i is the i-th data point of the measured storage compliance, MPa− ; | | 1 D0 p,i is i-th data point predicted by the storage compliance master curve, MPa− ; D00 m,i is i-th data | | 1 | | point of the measured loss compliance, MPa− ; D00 p,i is ith data point predicted by the loss compliance 1 | | master curve, MPa− . 1 E0 E00 D = = D iD00 = i (12) ∗ 0 2 2 2 2 E∗ − E0 + E00 − E0 + E00 α00 lg D0 = δ00 (13) − 1 β γ (log fr) λ00 1 + λ00 e 00 − 00

β γ lg fr π d(logE0) π e 00 − 00 00 = 00 = 00 00 00 D k D0 k D0α γ 1 (14) − 2 d(log fr) 2 1+ β γ lg fr λ00 1 + λ00 e 00 − 00 Materials 2020, 13, 3723 8 of 29

where: D∗ is the complex compliance; E∗ is the complex modulus; D0 is the storage compliance obtained from complex modulus test; D00 is the loss compliance obtained from complex modulus test; lg D is the logarithm of the storage compliance; δ00 is the value of the higher asymptote of the lg D | 0| | 0| master curve; α00 is the diﬀerence between the upper and lower asymptotes of the lg D master curve; | 0| β00 and γ00 is shape coeﬃcients of the lg D master curve;. λ00 determined the model’s asymmetric | 0| characteristics (when λ00 = 1 the shape of master curve is symmetric); D00 the master curve of loss compliance predicted by the GSM based on the approximate K-K relations; k00 is a positive correction, used to obtain a potentially more accurate prediction.

3.2. Determine the Discrete Relaxation Spectrum and Retardation Spectrum The relaxation (retardation) time spectrum is the most general function describing the dependence of a material’s viscoelasticity on time or frequency [45]. It is the most vital part, and all viscoelastic functions can combine each other. Through studying the relaxation (retardation) time spectrum, we can obtain the distribution of the relaxation (retardation) time and the contribution of various motion modes to the macroscopic viscoelasticity [46], and providing an eﬀective way to study the microstructure of viscoelastic materials.

3.2.1. Determine the Discrete Relaxation Spectrum The GMM [47] is widely used to characterize the relaxation behavior of asphalt mixtures in the LVE range. The relaxation modulus of the GMM expressed by the Prony series is shown in Equation (15).

N N X t X t ρ ρ E(t) = Ee + E e− i = Eg E 1 e− i (15) i − i − i=1 i=1 where: E(t) is relaxation modulus, MPa.;Ee is the equilibrium modulus, MPa; Eg is the glass modulus, MPa; Ei is the relaxation strength, MPa; ρi is relaxation time, s; w is the angular frequency and is equal to 2π f . The set of parameters consisting of all discrete relaxation times and corresponding relaxation strength is the discrete relaxation spectrum [33]. It is listed in Table A3 of AppendixA. To obtain the discrete relaxation spectrum, the storage modulus and the loss modulus rewritten in the form of the Prony series, respectively. The error function e f4 was applied to the test data of storage modulus and loss modulus. The equation was shown in Equation (16). The test data of storage modulus and loss modulus can be determined from the complex modulus test results following Equations (7) and (8). The Prony series parameters in Equations (17) and (18) for each mixture type was solved simultaneously by minimizing the error function.

uv uv t N !2 t N !2 1 X E0 ps,i E0 p,i 1 X E00 ps,i E00 p,i e f = e f + e f = | | − | | + | | − | | (16) 4 E0 E00 N E N E i=1 | 0|p,i i=1 | 00 |p,i where: e f4 is the sum of error function of storage modulus and loss modulus; N is the number of the measured data points that is equal to 28; E is the i-th data point predicted by storage modulus | 0|p,i master curve of the GSM, MPa; E is the i-th data point of storage modulus predicted by Prony | 0|ps,i series, MPa; E00 is the i-th data point predicted by loss modulus master curve of the GSM, MPa; | |p,i E00 is ith data point of loss modulus predicted by Prony series. | |ps,i m 2 2 2 X w ρi Ei E0(w) = Ee + (17) 2 2 + i=1 w ρi 1 Materials 2020, 13, 3723 9 of 29

m X wρiEi E00 (w) = (18) 2 2 + i=1 w ρi 1 where: E0(w) and E00 (w) the storage modulus and the loss modulus rewritten in the form of the Prony series. Before ﬁtting the storage modulus and loss modulus with the Prony series, the GSM was used to smooth the storage modulus data, according to the generalized K-K relations, then get the master curve form of loss modulus. The expression has also been used to smooth the loss modulus data. Then the smoothed storage modulus and loss modulus data are respectively ﬁtted for the Prony series. Finally, the discrete relaxation spectrum is obtained.

3.2.2. Determine the Discrete Retardation Spectrum The GKM [7] is widely used to characterize the creep behavior of asphalt mixtures in the LVE range. The creep compliance of the GKM expressed by the Prony series is shown in Equation (19).

Xn t ! Xn t τj τj D(t) = Dg + D 1 e− = De D e− (19) j − − j j=1 j=1

1 1 where: D(t) is the creep compliance, MPa− ; Dg is the equilibrium compliance, MPa− ; Dj is retardation 1 strength, MPa− ; τj is the retardation time, s is obtained by the Park and Schapery method [2]. The detailed selection process for discrete retardation time in the Prony series of mixtures is shown in Figures A1–A4 of AppendixA. The set of parameters consisting of all discrete retardation times and retardation strengths is the discrete retardation spectrum. It is listed in Table A4 of AppendixA. To obtain the discrete retardation spectrum, the storage compliance and the loss compliance rewritten in the form of the Prony series, respectively. The error function e f5 was applied to the test data of storage compliance and loss compliance. The corresponding equation is shown in Equation (20). The test data of storage compliance and loss compliance can be determined from the complex modulus test results following Equation (12). The Prony series parameters in Equations (21) and (22) for each mixture type was solved simultaneously by minimizing the error function.

uv uv t N !2 t N !2 1 X D0 ps,i D0 p,i 1 X D00 ps,i D00 p,i e f = e f + e f = | | − | | + | | − | | (20) 5 D0 D00 N D N D i=1 | 0|p,i i=1 | 00 |p,i where: e f5 is the sum of error function of storage compliance and loss compliance; N is the number of the measured data points that is equal to 28; D0 p,i is the i-th data point predicted by storage compliance 1 | | master curve of GSM, MPa− ; D0 ps,i is the i-th data point of storage compliance predicted by Prony 1 | | 1 series, MPa− ; D00 p,i is the i-th data point predicted by loss compliance master curve of GSM, MPa− ; | | 1 D00 is the i-th data point of loss compliance predicted by Prony series, MPa . | |ps,i − n X Dj D0(w) = Dg + (21) 2 2 + j=1 w τj 1

n 1 X wτjDj D00 (w) = + (22) 2 η0w 2 + j=1 w τj 1 where: D0(w) and D00 (w) the storage compliance and the loss compliance rewritten in the form of the Prony series. Similarly, before ﬁtting the master curves of storage compliance and loss compliance with the Prony series, the GSM was used to smooth the storage compliance data; according to the generalized Materials 2020, 13, 3723 10 of 29

K-K relations, then get the expression form of the master curves of loss compliance based on GSM. The expression has also been used to smooth the loss compliance data. Finally, the smoothed storage compliance and loss compliance data are ﬁtted in the Prony series, and then the discrete retardation spectrum is also obtained.

3.3. Determine the Continuous Relaxation Spectrum and Retardation Spectrum Although the discrete-time spectrum can characterize the viscoelastic properties of asphalt mixtures, the discrete spectrum is dependent on the spacing of discrete-time. The viscoelastic response function obtained from the discrete-time spectrum is prone to oscillations in the local region, which could be solved by the continuous-time spectrum.

3.3.1. Determine the Continuous Relaxation Spectrum Integral transformation theory can be used to establish the relations between various LVE functions and the continuous-time spectrum. The continuous relaxation spectrum and the retardation spectrum can be derived from the corresponding storage modulus and storage compliance, respectively. jπ jπ The angular frequency w in storage modulus was replaced by ρ 1exp( ), and replaced by τ 1exp( ) − ± 2 − ± 2 for storage compliance. Finally, only the imaginary part is retained. Equation (23) is the expression of the continuous relaxation time spectrum. We can get it following the method proposed by Tschoegl [4].

! 2 1 jπ H(ρ) = ImE0[ρ− exp ] (23) ± π ± 2

Then we rewrite Equation (9) to form Equation (24).

B ( ) = ( ) ( ) E0 ω exp A exp 1 (24) λ [1 + λ0 exp(C + DlnαT + Dlnω)] 0

γ where A = δ ln10; A = α ln10; C = β γ log(2π); D = 0 ; 0 0 0 − 0 ln10 Substituting Equation (24) into Equation (23), and according to Euler’s Equation, we obtain the function expression Equation (25) for the continuous relaxation spectrum.

2 n o H(ρ) = exp(A)Im exp [P(ρ)]λ0 (25) ± π h i where the expressions of P(ρ), X(ρ), and Y(ρ) are shown is Equations (26)–(28).

P(ρ) = X(ρ) iY(ρ) (26) ∓ h i λ0 πD B 1 + λ0 exp(E)cos 2 X(ρ) = (27) 2 πD 1 + (λ0) exp(2E) + 2λ0 exp(E)cos 2 λ πD B 0 λ0 exp(E)sin Y(ρ) = 2 (28) 2 πD 1 + (λ0) exp(2E) + 2λ0 exp(E)cos 2

αT where E = C + Dln ρ . ( ) = [ ( )]2 + [ ( )]2 ( ) = Y(ρ) To further simplify Equation (25), when F ρ X ρ Y ρ ; G ρ arctan X(ρ) ; then Equation (25) is simpliﬁed to Equation (29).

2 1 G 1 G H(ρ) = exp (A + F 2λ0 cos ) sin (F 2λ0 sin ) (29) − π λ0 λ0 Materials 2020, 13, 3723 11 of 29

Finally, substituting the model parameters of the master curve of the storage modulus and the loss modulus into Equation (29), the function expression of the continuous relaxation time spectrum can be accurately obtained. When λ0 = 1 Equation (29) can be further simpliﬁed to the form of Equation (30).

2 H(ρ) = exp(A + X(ρ)) sin(Y(ρ)) (30) π

3.3.2. Determine the Continuous Retardation Spectrum The continuous retardation spectrum can be derived from the master curve of storage compliance. jπ The angular frequency w in the master curve of storage compliance was replaced by τ 1 exp ( ), − ± 2 and only the imaginary part was retained. Finally, the continuous retardation time spectrum was expressed as Equation (31). ! 2 1 jπ L(τ) = ImD0[τ− exp ] (31) ∓ π ± 2 Similarly, Equation (13) can be rewritten into the expression of Equation (32).

B0 D0(ω) = exp(A0) exp ( ) (32) − 1 [1 + λ00 exp(C0 + D0lnαT + D0lnω)] λ00

γ00 where A = δ00 ln10; B = α00 ln10; C = β00 + γ00 log(2π); D = . 0 0 0 0 − ln10 Then substituting Equation (32) into Equation (31), and according to Euler’s Equation, we obtain the function expression Equation (33) for the continuous retardation spectrum.

n o 2 λ00 L(τ) = exp(A0)Im exp [P(τ)] (33) ∓ π h i where The expressions of P(τ), X(τ), and Y(τ) are shown in Equations (34)–(36).

P(τ) = X(τ) iY(τ) (34) − ± h i λ00 πD0 B0 1 + λ00 exp(E0)sin 2 X(τ) = (35) 2 πD0 1 + λ00 exp(2E0) + 2λ00 exp(E0)cos 2

λ00 πD0 B0 λ00 exp(E0)sin Y(τ) = 2 (36) 2 πD0 1 + λ00 exp(2E0) + 2λ00 exp(E0)cos 2

αT where E0 = C0 + D0ln τ . ( ) = [ ( )]2 + [ ( )]2 ( ) = Y(τ) To simplify Equation (33), when F τ X τ Y τ ; G τ arctan X(τ) ; then Equation (33) was simpliﬁed to Equation (37).

2 1 G 1 G L(τ) = exp A0 + F 2λ00 cos sin (F 2λ00 sin ) (37) π λ00 λ00 Finally, substituting the model parameters of the master curve of the storage compliance and the loss compliance into Equation (37), the function expression of the continuous retardation time spectrum can be accurately obtained. When λ” = 1 Equation (37) can be further simpliﬁed to the form of Equation (38).

2 H(τ) = exp(A0 X(τ)) sin(Y(τ)) (38) π − Materials 2020, 13, 3723 12 of 29

4. Results and Discussion

4.1. Characterization of LVE Behavior of Crumb Rubber-modiﬁed Asphalt Mixtures Solving Equation (3) can get the GSM ﬁtting parameters of dynamic modulus and phase angle master curve, and also the parameter results were shown in Table4. Figure2 shows the master curves of the dynamic modulus and phase angle for the types of mixtures.

Table 4. GSM ﬁtting parameters for dynamic modulus and phase angle.

Parameters Mixture Type Error/% δ α β γ λ C1 C2 k HMA-60 1.62 2.84 0.60 0.56 0.80 8.93 87.75 0.90 1.23 − − WMA-60 2.29 2.13 0.14 0.62 0.81 19.14 202.14 0.97 0.83 − − HMA-C 2.12 2.40 0.54 0.51 0.51 19.13 187.95 0.98 0.81 − − WMA-C 2.33 2.14 0.31 0.56 0.52 14.02 138.11 1.01 0.75 Materials 2020, 13, x − − 12 of 28

35 10000

1000 20

HMA-60 Dynamic modulus

WMA-60 Dynamic modulus (°) angel Phase HMA-C Dynamic modulus 15

Dynamic modulus (MPa) modulus Dynamic WMA-C Dynamic modulus HMA-60 Phase angel WMA-60 Phase angel HMA-C Phase angel 10 WMA-C Phase angel 100 10-4 10-3 10-2 10-1 100 101 102 103 104 Reduced frequency (Hz) Figure 2. Master curve of dynamic modulus and phase angle.

Solving Equationequation (6) (6) can can get get the the GSM GSM fitting ﬁtting pa parametersrameters for storage storage modulus modulus and and loss loss modulus, modulus, and also the parameter results are showed in Table5 5.. FigureFigure3 3shows shows the the master master curves curves of of the the storage storage modulus and loss modulus for the types of mixtures.

Table 5. GSM ﬁttingfitting parameters for storage modulus and lossloss modulus.modulus.

ParametersParameters Mixture Type Error/% Mixture Type 𝜹 𝜶 𝜷 𝜸 𝝀 C1 C2 𝒌 Error/% δ0 α0 β0 γ0 λ0 C1 C2 k0 HMA-60 1.98 2.57 −0.36 −0.51 0.22 8.93 87.75 1.00 1.12 HMA-60 1.98 2.57 0.36 0.51 0.22 8.93 87.75 1.00 1.12 WMA-60 2.13 2.33 −0.12− −0.58− 0.92 19.14 202.14 1.00 0.83 WMA-60 2.13 2.33 0.12 0.58 0.92 19.14 202.14 1.00 0.83 HMA-C 1.86 2.66 −0.54− −0.53− 0.79 19.13 187.95 1.00 0.81 HMA-C 1.86 2.66 0.54 0.53 0.79 19.13 187.95 1.00 0.81 WMA-C 2.16 2.33 −0.28− −0.55− 0.67 14.02 138.11 1.00 0.82 WMA-C 2.16 2.33 0.28 0.55 0.67 14.02 138.11 1.00 0.82 − −

10000

1000

1000 HMA-60 Storage modulus WMA-60 Storage modulus HMA-C Storage modulus WMA-C Storage modulus Loss modulus (MPa)

Storage modulus (MPa) modulus Storage HMA-60 Loss modulus WMA-60 Loss modulus 100 HMA-C Loss modulus WMA-C Loss modulus

10-4 10-3 10-2 10-1 100 101 102 103 Reduced frequency (Hz) Figure 3. Master curve of storage modulus and loss modulus.

Solving equation (11) we can get the generalized model parameters of storage compliance and loss compliance master curve, and the corresponding parameter results are shown in Table 6. Figure 4 shows the master curves of the storage compliance and loss compliance for the types of mixtures.

Materials 2020, 13, x 12 of 28

35 10000

1000 20

HMA-60 Dynamic modulus

WMA-60 Dynamic modulus (°) angel Phase HMA-C Dynamic modulus 15

Solving equation (6) can get the GSM fitting parameters for storage modulus and loss modulus, and also the parameter results are showed in Table 5. Figure 3 shows the master curves of the storage modulus and loss modulus for the types of mixtures.

Table 5. GSM fitting parameters for storage modulus and loss modulus.

Parameters Mixture Type Error/% 𝜹 𝜶 𝜷 𝜸 𝝀 C1 C2 𝒌 HMA-60 1.98 2.57 −0.36 −0.51 0.22 8.93 87.75 1.00 1.12 WMA-60 2.13 2.33 −0.12 −0.58 0.92 19.14 202.14 1.00 0.83 Materials 2020, 13, 3723HMA-C 1.86 2.66 −0.54 −0.53 0.79 19.13 187.95 1.00 0.81 13 of 29 WMA-C 2.16 2.33 −0.28 −0.55 0.67 14.02 138.11 1.00 0.82

10000

1000

1000 HMA-60 Storage modulus WMA-60 Storage modulus HMA-C Storage modulus WMA-C Storage modulus Loss modulus (MPa)

Storage modulus (MPa) modulus Storage HMA-60 Loss modulus WMA-60 Loss modulus 100 HMA-C Loss modulus WMA-C Loss modulus

10-4 10-3 10-2 10-1 100 101 102 103 Reduced frequency (Hz) FigureFigure 3.3. Master curve of storage modulus and loss modulus.modulus.

SolvingSolving Equationequation (11) we can get the generalizedgeneralized modelmodel parametersparameters ofof storagestorage compliancecompliance andand lossloss compliancecompliance mastermaster curve,curve, and and the the corresponding corresponding parameter parameter results results are are shown shown in in Table Table6. Figure6. Figure4 shows4 shows the the master master curves curves of of the the storage storage compliance compliance and and loss loss compliance compliance for for the the types types of of mixtures. mixtures.

Table 6. GSM ﬁtting parameters for storage compliance and loss compliance.

Parameters Mixture Type Error/% δ” α” β” γ” λ” C1 C2 k” HMA-60 1.70 2.81 0.55 0.53 1.00 8.93 87.75 1.15 1.21 − − WMA-60 2.16 2.26 0.27 0.60 1.00 19.14 202.14 1.06 0.72 − − HMA-C 2.01 2.46 0.61 0.57 1.00 19.13 187.95 1.08 0.90 − − WMA-C 2.20 2.23 0.42 0.61 1.00 14.02 138.11 1.16 0.72 Materials 2020, 13, x − − 13 of 28

HMA-60 Storage compliance WMA-60 Storage compliance HMA-C Storage compliance WMA-C Storage compliance 1E-3 HMA-60 Loss compliance ) -1 WMA-60 Loss compliance ) HMA-C Loss compliance -1 1E-3 WMA-C Loss compliance

1E-4 Loss compliance (MPa compliance Loss Storage compliance (MPa compliance Storage 1E-4 1E-5

10-4 10-3 10-2 10-1 100 101 102 103 Reduced frequency (Hz) Figure 4. MasterMaster curve curve of of storage storage comp complianceliance and loss compliance.

Table 6. GSM fitting parameters for storage compliance and loss compliance.

Parameters Mixture Type Error/% 𝜹 𝜶 𝜷 𝜸 𝝀 C1 C2 𝒌 HMA-60 −1.70 2.81 −0.55 0.53 1.00 8.93 87.75 1.15 1.21 WMA-60 −2.16 2.26 −0.27 0.60 1.00 19.14 202.14 1.06 0.72 HMA-C −2.01 2.46 −0.61 0.57 1.00 19.13 187.95 1.08 0.90 WMA-C −2.20 2.23 −0.42 0.61 1.00 14.02 138.11 1.16 0.72

4.2. Relaxation Spectrum and Retardation Spectrum of Asphalt Mixture The relaxation (retardation) time spectrum is a useful tool for describing the viscoelasticity of asphalt and mixtures [48]. It is also the most general functional relations of viscoelasticity to time or frequency dependence. From the Boltzmann superposition principle, it is known that the full properties of the material are represented by all modes of motion that vary with the time spectrum. Moreover, the material functions measured in various experiments are based on the same relaxation (retardation) time spectrum, which is the vital framework of all viscoelastic materials. Many research results [49,50] show that the relaxation (retardation) spectrum is one of the essential characteristics of LVE materials, from which all LVE functions in the time and frequency domains can be derived.

Materials 2020, 13, 3723 14 of 29

4.2.1. The Discrete Relaxation Spectrum and Retardation Spectrum of Asphalt Mixture Figure5 shows the discrete relaxation and retardation spectrum of asphalt mixture obtained using Equations (16)–(22). The bell-shaped relaxation and retardation spectrum peak intensities correspond 3 2 3 to relaxation and retardation times located at around 10− s and 10 –10 s, respectively. The relaxation 3 spectrum peak point is consistent Gundla’s work [47]. When relaxation times more than 10− s, the discrete relaxation spectrum of HMA-60 is always below HMA-C. That is to say, the discrete relaxation spectrum strength of the former is always smaller than the latter when relaxation times 3 3 more than 10− s. When the relaxation time is less than 10− s, the data for former and latter are too few to accurately compare, it is due to the limited of discrete time points. The discrete retardation spectrum of HMA-60 is always above the HMA-C, and the former has a broader retardation spectrum width than the latter. That is to say, the discrete retardation spectrum strength of the former is always higher than the latter. Considering the discrete relaxation spectra and discrete retardation spectra of the two mixtures (HMA-60 and HMA-C), it can be learned that the elastic component of HMA-60 is smaller than HMA-C and the viscous component is larger than HMA-C, which also indicates that HMA-C only needs a short time to achieve the retardation process under loading. This is because the viscosity of the compound-mesh rubber-modified asphalt binder is greater than the 60-mesh, then the cohesion of HMA-C is also greater than HMA-60, which gives HMA-C greater strength and deformation recovery. The discrete relaxation spectrum of the WMA-60 is always below the WMA-C. That is to say, the discrete relaxation spectrum strength of the former is always smaller than the latter. While the discrete retardation spectrum of WMA-60 is always above the WMA-C, and the former has a broader retardation spectrum width than the latter.. That is to say, the discrete retardation spectrum strength of the former is always higher than the latter. Considering the discrete relaxation spectra and discrete retardation spectra of the two mixtures (WMA-60 and WMA-C), it can be learned that the elastic component of WMA-60 is smaller than WMA-C and the viscous component is larger than WMA-C. Which also indicates that WMA-C only needs a short time to achieve the retardation process under loading. This is also because the viscosity of the compound-mesh rubber-modified asphalt binder is greater than the 60-mesh, then the cohesion of WMA-C is also greater than WMA-60, which gives WMA-C greater strength and deformation recovery. Compared the HMCRMA and WMCRMA, once warm mix additive was added, the intensity of the peak relaxation spectrum decreased, the width of the relaxation spectrum decreased, and the discrete relaxation spectrum was difficult to distinguish on the left side of the peak intensity. This means that the elastic component of the mixture is reduced and the viscous component increased, so the mixture needs a shorter time to achieve the relaxation process under load, which is due to the reduced aging effect after the warm mix additive was added. Materials 2020, 13, x 14 of 28 viscosity of the compound-mesh rubber-modified asphalt binder is greater than the 60-mesh, then the cohesion of HMA-C is also greater than HMA-60, which gives HMA-C greater strength and deformation recovery. The discrete relaxation spectrum of the WMA-60 is always below the WMA- C. That is to say, the discrete relaxation spectrum strength of the former is always smaller than the latter. While the discrete retardation spectrum of WMA-60 is always above the WMA-C, and the former has a broader retardation spectrum width than the latter.. That is to say, the discrete retardation spectrum strength of the former is always higher than the latter. Considering the discrete relaxation spectra and discrete retardation spectra of the two mixtures (WMA-60 and WMA-C), it can be learned that the elastic component of WMA-60 is smaller than WMA-C and the viscous component is larger than WMA-C. Which also indicates that WMA-C only needs a short time to achieve the retardation process under loading. This is also because the viscosity of the compound-mesh rubber- modified asphalt binder is greater than the 60-mesh, then the cohesion of WMA-C is also greater than WMA-60, which gives WMA-C greater strength and deformation recovery. Compared the HMCRMA and WMCRMA, once warm mix additive was added, the intensity of the peak relaxation spectrum decreased, the width of the relaxation spectrum decreased, and the discrete relaxation spectrum was difficult to distinguish on the left side of the peak intensity. This means that the elastic component of the mixture is reduced and the viscous component increased, so the mixture needs a shorterMaterials 2020time, 13 to, 3723 achieve the relaxation process under load, which is due to the reduced aging 15effect of 29 after the warm mix additive was added.

7000 0.0030 6000

0.0025 ) 5000 -1

0.0020 4000

0.0015 3000

HMA-60 Discrete Relaxation Spectrum WMA-60 Discrete Relaxation Spectrum 0.0010 2000 HMA-C Discrete Relaxation Spectrum WMA-C Discrete Relaxation Spectrum HMA-60 Discrete Retardation Spectrum WMA-60 Discrete Retardation Spectrum 0.0005 1000 HMA-C Discrete Retardation Spectrum Relaxation Spectrum (MPa) Spectrum Relaxation WMA-C Discrete Retardation Spectrum Retardation Spectrum (MPa Spectrum Retardation

0 0.0000

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 Time (s) Figure 5. TheThe discrete discrete relaxation relaxation spectrum and re retardationtardation spectrum of asphalt mixture.

4.2.2. The The Continuous Continuous Relaxation Spectrum and Retardation Spectrum of Asphalt Mixture The continuous relaxation and retardation spectrum function of asphalt mixture are shown in Figure 66.. TheThe continuouscontinuous relaxationrelaxation andand retardationretardation spectrumspectrum alsoalso presentpresent aa typicaltypical bell-shapebell-shape inin aa the wide time domain.domain. The The continuous relaxation spectrum is asymmetric, further illustrating the advantages of using the GSM and approximate K-K rela relationstions to fit fit the master curve of the viscoelastic response function. The magnitude ofof continuouscontinuous relaxationrelaxation spectrum spectrum of of HMA-60 HMA-60 is is less less than than HMA-C HMA- 3 whenC when the the reduced reduced time time is is greatergreater than 10 −3 s sand and greater greater than than HMA-C HMA-C once once the thereduced reduced time time is less is 3 thanless than 10−3 s, 10 in− others, in other words, words, the continuous the continuous relaxation relaxation spectrum spectrum magnitude magnitude of the of former the former is less is than less thatthan of that the oflatter the in latter longer in reduced longer reduced time and time grea andter than greater the latter than thein shor latterter inreduced shorter time. reduced However, time. theHowever, magnitude the magnitude of the former’s of the continuous former’s continuous retardation retardation spectrum is spectrum always greater is always than greater the latter than in theall timelatter domains. in all time Furthermore, domains. Furthermore, the former thehas formera wide hasr retardation a wider retardation spectrum spectrumwidth than width the latter. than the In otherlatter. words, In other the words, strength the of strength the former’s of the former’scontinuous continuous retardation retardation spectrum spectrum is always is greater always than greater the latterthan thein all latter time in domains. all time domains.Considering Considering the continuo theus continuous relaxation spectra relaxation and spectra continuous and continuousretardation spectraretardation of the spectra two mixtures of the two (HMA-60 mixtures and (HMA-60 HMA-C), and it can HMA-C), be learned it can that be the learned elastic that component the elastic of HMA-60component is ofsmaller HMA-60 than is HMA-C smaller thanand the HMA-C viscous and component the viscous is component larger than is HMA-C, larger than which HMA-C, also which also indicates that HMA-C only needs a short time to achieve the retardation process under loading. This is because the viscosity of the compound-mesh rubber-modified asphalt binder is greater than the 60-mesh, then the cohesion of HMA-C is also greater than HMA-60, which gives HMA-C greater strength and deformation recovery. The magnitude of the continuous relaxation spectrum of the WMA-60 is always less than the WMA-C. In other words, the continuous relaxation spectrum strength of the former is always smaller than that of the latter, while the magnitude of the continuous retardation spectrum of the former is always greater than the latter. That is to say, the strength of the continuous Retardation spectrum of the former is always greater than the latter. Considering the continuous relaxation spectra and continuous retardation spectra of the two mixtures (WMA-60 and WMA-C), it can be learned that the elastic component of WMA-60 is smaller than WMA-C and the viscous component is larger than WMA-C, which also indicates that WMA-C only needs a short time to achieve the retardation process under loading. This is because the viscosity of the compound-mesh rubber-modified asphalt binder is greater than the 60-mesh, then the cohesion of WMA-C is also greater than WMA-60, which gives WMA-C greater strength and deformation recovery. Comparing the hot-mixed and warm-mixed crumb rubber-modified asphalt mixtures, once warm mix additive was added, the width of the relaxation (retardation) spectrum narrowed, the peak intensity decreased, and The continuous retardation spectrum peak point shifted horizontally to the left. It indicates that the mixture’s elastic component is reduced and increased viscous component, so the mixture only Materials 2020, 13, x 15 of 28 indicates that HMA-C only needs a short time to achieve the retardation process under loading. This is because the viscosity of the compound-mesh rubber-modified asphalt binder is greater than the 60-mesh, then the cohesion of HMA-C is also greater than HMA-60, which gives HMA-C greater strength and deformation recovery. The magnitude of the continuous relaxation spectrum of the WMA-60 is always less than the WMA-C. In other words, the continuous relaxation spectrum strength of the former is always smaller than that of the latter, while the magnitude of the continuous retardation spectrum of the former is always greater than the latter. That is to say, the strength of the continuous Retardation spectrum of the former is always greater than the latter. Considering the continuous relaxation spectra and continuous retardation spectra of the two mixtures (WMA-60 and WMA-C), it can be learned that the elastic component of WMA-60 is smaller than WMA-C and the viscous component is larger than WMA-C, which also indicates that WMA-C only needs a short time to achieve the retardation process under loading. This is because the viscosity of the compound-mesh rubber-modified asphalt binder is greater than the 60-mesh, then the cohesion of WMA-C is also greater than WMA-60, which gives WMA-C greater strength and deformation recovery. Comparing the hot-mixed and warm-mixed crumb rubber-modified asphalt mixtures, once warm mix additive was added, the width of the relaxation (retardation) spectrum narrowed, the peak intensity Materials 2020, 13, 3723 16 of 29 decreased, and The continuous retardation spectrum peak point shifted horizontally to the left. It indicates that the mixture’s elastic component is reduced and increased viscous component, so the mixtureneeds a only shorter needs time a toshorter achieve time the to relaxation achieve the process relaxation under process load. The under faster load. relaxation The faster mechanism relaxation is mechanismdue to the reducedis due to aging the reduced effect after aging the effect warm afte mixr additivethe warm was mix added additive [51]. was From added the molecular [51]. From point, the molecularthe primary point, reason the for primary the above reason results for can the be above attributed results to the can decrease be attributed in molecular to the weight decrease and thein molecularconcentration weight of polarand the functional concentration groups of in polar the asphalt functional binders groups [52 ].in the asphalt binders [52].

FigureFigure 6 6. TheThe Continuous Continuous relaxation relaxation spectrum spectrum and and retardation retardation spectrum spectrum of of asphalt asphalt mixture mixture.

4.2.3.4.2.3. Compared Compared the the Discrete Discrete and and Continuous Continuous Relaxation Relaxation Spectrum Spectrum and and Retardation Retardation Spectrum Spectrum of of AsphaltAsphalt Mixture Mixture TheThe inset inset in in Figure Figure 77 showsshows the the discrete discrete relaxation relaxation time time where where the the Prony Prony series series have have been been plotted plotted againstagainst the the relaxation relaxation times. times. The The continuous continuous relaxati relaxationon spectrum spectrum obtained obtained from from equation Equation (29) (29) is is also also shownshown on on the the same same plot plot as as the the dashed dashed line. line. A A simila similarr plot plot is is shown shown in in the the inset inset of of Figure Figure 88 forfor the the retardationretardation time.From time.From thethe resultsresults in in Figures Figures7 and7 and8, we 8, canwe seecan that see thethat discrete the discrete spectrum spectrum of the crumbof the crumbrubber-modified rubber-modified asphalt asphalt mixtures mixtures coincides coincides with the with continuous the continuous spectrum, spectrum, Comparing Comparing the discrete, the discrete,continuous continuous relaxation relaxation and retardation and retardation spectrum spectrum of asphalt of mixtures, asphalt themixtures, shape of the the shape discrete-time of the discrete-timespectrum is similarspectrum to thatis similar of the to continuous-time that of the continuous-time spectrum. However, spectrum. the However, peak point the of peak the discrete point ofrelaxation the discrete spectrum relaxation is closer spectrum to the longeris closer relaxation to the timelonger than relaxation the continuous time than relaxation the continuous spectrum. relaxationconversely, spectrum. the peak conversely, point of the the discrete peak point retardation of the discrete spectrum retardation is closer tospec thetrum shorter is closer retardation to the shortertime than retardation the continuous time than retardation the continuous spectrum. retard Theation most spectrum. significant The most difference significant is that difference the discrete is thatspectrum’s the discrete strength spectrum’s is more thanstrength thecontinuous is more than spectrum the continuous at the same spectrum time. These at the results sameare time. consistent These resultsMaterialswith the are 2020 results consistent, 13, x of many with other the resu scholarslts of [many22,53 ].other scholars [22,53]. 16 of 28

HMA-60 Discrete Relaxation Spectrum 6000 WMA-60 Discrete Relaxation Spectrum HMA-C Discrete Relaxation Spectrum WMA-C Discrete Relaxation Spectrum HMA-60 Continuous Relaxation Spectrum 5000 WMA-60 Continuous Relaxation Spectrum HMA-C Continuous Relaxation Spectrum WMA-C Continuous Relaxation Spectrum

4000

3000

2000

Relaxation Spectrum (MPa) 1000

10-15 10-10 10-5 100 105 1010 1015 Time (s) FigureFigure 7. TheThe discrete discrete and and continuous continuous relaxati relaxationon spectrum of asphalt mixture.

HMA-60 Discrete Retardation Spectrum WMA-60 Discrete Retardation Spectrum 0.0030 HMA-C Discrete Retardation Spectrum WMA-C Discrete Retardation Spectrum

) HMA-60 Continuous Retardation Spectrum

-1 WMA-60 Continuous Retardation Spectrum 0.0025 HMA-C Continuous Retardation Spectrum WMA-C Continuous Retardation Spectrum

0.0020

0.0015

0.0010

Retardation Spectrum (MPa 0.0005

0.0000

10-15 10-10 10-5 100 105 1010 1015 Time (s) Figure 8. The discrete and continuous retardation spectrum of asphalt mixture.

In order to compare the discrete and continuous spectra more significantly, the continuous spectrum was converted to its corresponding discrete spectrum by conversion method, and then illustrated it with the discrete spectrum obtained from Prony series in Figure 9. The discrete relaxation spectra can be efficiently calculated from continuous relaxation spectra following equation (39).

𝐸 =𝐻(𝜌)Δ𝑙𝑛𝜌 (39) where 𝐸 is discrete relaxation spectrum strength transformed from continuous spectrum; 𝐻(𝜌) is continuous spectrum; Δ𝑙𝑛𝜌 is logarithmic interval of discrete relaxation time. Similarly, the discrete retardation spectra can be efficiently calculated from continuous retardation spectra following equation (40). 𝐷 =𝐿𝜏Δ𝑙𝑛𝜏 (40) where 𝐷 is discrete retardation spectrum strength transformed from continuous spectrum; 𝐿𝜏 is continuous retardation spectrum; Δ𝑙𝑛𝜌 is logarithmic interval of discrete retardation time.

Materials 2020, 13, x 16 of 28

4000

3000

2000

Relaxation Spectrum (MPa) 1000

10-15 10-10 10-5 100 105 1010 1015 Time (s) Materials 2020, 13, 3723 17 of 29 Figure 7. The discrete and continuous relaxation spectrum of asphalt mixture.

HMA-60 Discrete Retardation Spectrum WMA-60 Discrete Retardation Spectrum 0.0030 HMA-C Discrete Retardation Spectrum WMA-C Discrete Retardation Spectrum

) HMA-60 Continuous Retardation Spectrum

-1 WMA-60 Continuous Retardation Spectrum 0.0025 HMA-C Continuous Retardation Spectrum WMA-C Continuous Retardation Spectrum

0.0020

0.0015

0.0010

Retardation Spectrum (MPa 0.0005

0.0000

10-15 10-10 10-5 100 105 1010 1015 Time (s) Figure 8.8. The discrete and continuous retardation spectrumspectrum ofof asphaltasphalt mixture.mixture.

InIn orderorder toto comparecompare thethe discretediscrete andand continuouscontinuous spectraspectra moremore signiﬁcantly,significantly, thethe continuouscontinuous spectrumspectrum waswas convertedconverted toto itsits correspondingcorresponding discretediscrete spectrumspectrum byby conversionconversion method,method, andand thenthen illustratedMaterialsillustrated 2020 itit, 13 withwith, x thethe discretediscrete spectrumspectrum obtainedobtained fromfrom PronyProny seriesseries inin FigureFigure9 .9. 17 of 28 The discrete relaxation spectra can be efficiently calculated from continuous relaxation spectra 7000 0.0035 following equation (39). HMA-60 Relaxation Spectrum E WMA-60 Relaxation Spectrum E HMA-C Relaxation Spectrum E 6000 WMA-C Relaxation Spectrum E 0.0030 𝐸 =𝐻(𝜌)Δ𝑙𝑛𝜌 HMA-60 Relaxation Spectrum E' (39) WMA-60 Relaxation Spectrum E'

HMA-C Relaxation Spectrum E' )

-1 where 𝐸 is discrete relaxation spectrum strength transformed WMA-C fromRelaxation Spectrum continuous E' 0.0025 spectrum; 𝐻(𝜌 ) is 5000 HMA-60 Retardation Spectrum D WMA-60 Retardation Spectrum D continuous spectrum; Δ𝑙𝑛𝜌 is logarithmic interval of discrete HMA-C relaxation Retardation Spectrum Dtime. WMA-C Retardation Spectrum D 0.0020 Similarly, the4000 discrete retardation spectra can be HMA-60efficiently Retardation Spectrum calculated D' from continuous WMA-60 Retardation Spectrum D' HMA-C Retardation Spectrum D' retardation spectra following equation (40). WMA-C Retardation Spectrum D' 0.0015 3000 𝐷 =𝐿𝜏Δ𝑙𝑛𝜏 (40) 0.0010 2000 where 𝐷 is discrete retardation spectrum strength transformed from continuous spectrum; 𝐿𝜏 is continuous retardation spectrum; Δ𝑙𝑛𝜌 is logarithmic interval of discrete retardation0.0005 time. Relaxation Spectrum (MPa) Spectrum Relaxation 1000

0.0000 (MPa Spectrum Retardation 0 -0.0005 10-5 100 105 1010 Time (s)

FigureFigure 9. ComparedCompared the the strength strength of discrete of discrete spectrum spectrum obtained obtained from fromcontinuous continuous spectrum spectrum and Prony and Pronyseries. series.

TheAlthough discrete the relaxation shape of the spectra discrete can spectrum be eﬃciently converted calculated from from the continuous continuous spectrum relaxation is spectrasimilar followingthe discrete Equation spectrum (39). calculated from the Prony series, there are some differences in the local region, the reasons for this are as follows. Ei0 = H(ρi)∆lnρi (39) ( ) where(1)E i0Theis discrete complex relaxation modulus test spectrum carried strength out in temperature transformed of from5 ℃–50 continuous ℃, and there spectrum; are someH errorsρi is continuousin the spectrum; test. The∆ higherlnρi is logarithmicthe temperature, interval the of greater discrete the relaxation error. time. (2)Similarly, We used the the discrete approximate retardation K-K spectra relations can rather be eﬃ cientlythan the calculated exact K-K from relations. continuous In addition, retardation the spectra following Equation (40). GSM constructed from the test results tend to show larger errors at extremely low and high frequencies. D0j = L τj ∆lnτj (40) (3) In the process of using the Prony series to calculate the discrete spectrum, the discrete time where D is discrete retardation spectrum strength transformed from continuous spectrum; L is is0j pre-selected, and the discrete spectrum intensity is obtained by the optimization algorithm,τj continuousbut retardationit is a multi-solution spectrum; ∆ problem,lnρi is logarithmic although interval we set ofconstraints discrete retardation during the time. optimization process.

4.3. Construction of Master Curves Relaxation Modulus and Creep Compliance Creep and stress relaxation are the essential phenomena of viscoelastic materials such as asphalt mixtures [54]. Creep is the phenomenon in which when a constant stress is applied, and the strain gradually increases with time. Stress relaxation is the phenomenon in which constant strain is applied, and the stress gradually decreases with time. Creep compliance and relaxation modulus are essential indicators for evaluating creep and relaxation. In the process of asphalt pavement design, creep compliance is an essential parameter for predicting strain, and relaxation modulus is an essential parameter for predicting stress [55]. However, the relaxation test has higher requirements on the instruments, and the strain is challenging to remain constant; besides, the creep tests typically require longer time to accurately obtain creep compliance. To solve this problem, the storage modulus and loss modulus master curves are obtained from the test results of the dynamic modulus test, then the discrete spectrum and continuous spectrum are obtained respectively according to the Prony series and Laplace inversion. Furthermore, the creep compliance and relaxation modulus obtained from the known discrete and continuous spectrum.

4.3.1. The Relaxation Modulus and Creep Compliance Obtained from the Discrete Spectrum According to equation (15), the relaxation modulus can be obtained from the result of the discrete relaxation spectrum. similarly, according to equation (19), the creep compliance can be obtained from the result of the discrete relaxation spectrum. All of the results are plotted in Figure 10.

Materials 2020, 13, 3723 18 of 29

Although the shape of the discrete spectrum converted from the continuous spectrum is similar the discrete spectrum calculated from the Prony series, there are some diﬀerences in the local region, the reasons for this are as follows.

(1) The complex modulus test carried out in temperature of 5 ◦C–50 ◦C, and there are some errors in the test. The higher the temperature, the greater the error. (2) We used the approximate K-K relations rather than the exact K-K relations. In addition, the GSM constructed from the test results tend to show larger errors at extremely low and high frequencies. (3) In the process of using the Prony series to calculate the discrete spectrum, the discrete time is pre-selected, and the discrete spectrum intensity is obtained by the optimization algorithm, but it is a multi-solution problem, although we set constraints during the optimization process.

4.3.1. The Relaxation Modulus and Creep Compliance Obtained from the Discrete Spectrum According to Equation (15), the relaxation modulus can be obtained from the result of the discrete relaxation spectrum. similarly, according to Equation (19), the creep compliance can be obtained from the result of the discrete relaxation spectrum. All of the results are plotted in Figure 10. Materials 2020, 13, x 18 of 28

0.01

2.4x104 10000 ) 2.2x104 -1

2x104

HMA-60 Discrete Relaxation Spectrum Modulus

Relaxation modulus (MPa) WMA-60 Discrete Relaxation Spectrum Modulus 1E-3 1.8x104 HMA-C Discrete Relaxation Spectrum Modulus -30 -25 -20 -15 -10 10 10 10 10 10 WMA-C Discrete Relaxation Spectrum Modulus Time (s) HMA-60 Discrete Retardation Spectrum Compliance 6x10-5 WMA-60 Discrete Retardation Spectrum Compliance HMA-C Discrete Retardation Spectrum Compliance ) 1000 -1 5.5x10-5 WMA-C Discrete Retardation Spectrum Compliance

5x10-5

4.5x10-5 Creep Compliance (MPaComplianceCreep

Relaxationmodulus (MPa) 1E-4 -5 4x10 (MPa Compliance Creep 10-30 10-25 10-20 10-15 10-10 Time (s)

100

10-30 10-25 10-20 10-15 10-10 10-5 100 105 1010 1015 1020 1025 1030 Time (s) Figure 10. TheThe relaxation modulus and creep complianc compliancee obtained from the discrete spectrum.

Figure 10 shows that the relaxation modulus and creep compliance obtained from the discrete spectrum. It was Z-shaped in a broad time domain, which shows that the relaxation modulus tends to the maximum magnitude in the shorter time domain and the minimum magnitude in the longer time domain; while the creep compliance tends to the minimum magnitude in the shorter time domain and the maximum magnitude in the longer time domain. The results are consistent with many previous works [24,29]. Figure 10 also shown that the relaxation modulus and creep compliance obtained from the discrete-time spectrum change significantly just in the time domain of 10−5 s to 105 s, but no significant change outside the region. Compared the relaxation modulus and creep compliance of hot mix crumb rubber-modified asphalt mixture, it found that the relaxation modulus of HMA-60 is always less than HMA-C in the entire time domain, which shows the stress relaxation ability of HMA-60 is better than HMA-C. The creep compliance of HMA-60 is higher than HMA-C in the entire time domain, which shows that the deformability of HMA-60 is better than HMA-C. This is because the viscosity of the 60-mesh rubber-modified asphalt binder is less than compound-mesh, then the cohesion of HMA-60 is also less than that of HMA-C, which gives HMA-60 a better relaxation and deformation capabilities. Compared the relaxation modulus and creep compliance of warm-mixed crumb rubber-modified asphalt mixture, it found that the relaxation modulus of WMA-60 is always less than that of WMA- C in the entire time domain, which shows the stress relaxation ability of WMA-60 is better than WMA-C. The creep compliance of WMA-60 is always higher than the WMA-C in the entire time domain, which shows that the deformability of WMA-60 is better than WMA-C. This is also because the viscosity of the 60-mesh rubber-modified asphalt binder is less than compound-mesh, then the cohesion of WMA-60 is also less than that of WMA-C, which gives WMA-60 a better relaxation and deformation capabilities. Comparing the hot-mixed and warm-mixed crumb rubber-modified asphalt mixtures, it found that once the warm mix additive was added, the relaxation modulus of crumb rubber-modified asphalt mixtures becomes smaller in the shorter time domain and larger in the longer time domain, creep compliance varies in the opposite way to the relaxation modulus. It shows that the addition of warm mix improves the asphalt mixture’s low-temperature stress relaxation ability and high-temperature deformation resistance. This is because the reduced aging effect after the warm mix additive was added. Finally, WMA-60 is recommended in the cold region, while WMA-C recommended in the hot region.

Materials 2020, 13, 3723 19 of 29

Figure 10 shows that the relaxation modulus and creep compliance obtained from the discrete spectrum. It was Z-shaped in a broad time domain, which shows that the relaxation modulus tends to the maximum magnitude in the shorter time domain and the minimum magnitude in the longer time domain; while the creep compliance tends to the minimum magnitude in the shorter time domain and the maximum magnitude in the longer time domain. The results are consistent with many previous works [24,29]. Figure 10 also shown that the relaxation modulus and creep compliance obtained from the 5 5 discrete-time spectrum change significantly just in the time domain of 10− s to 10 s, but no significant change outside the region. Compared the relaxation modulus and creep compliance of hot mix crumb rubber-modified asphalt mixture, it found that the relaxation modulus of HMA-60 is always less than HMA-C in the entire time domain, which shows the stress relaxation ability of HMA-60 is better than HMA-C. The creep compliance of HMA-60 is higher than HMA-C in the entire time domain, which shows that the deformability of HMA-60 is better than HMA-C. This is because the viscosity of the 60-mesh rubber-modified asphalt binder is less than compound-mesh, then the cohesion of HMA-60 is also less than that of HMA-C, which gives HMA-60 a better relaxation and deformation capabilities. Compared the relaxation modulus and creep compliance of warm-mixed crumb rubber-modified asphalt mixture, it found that the relaxation modulus of WMA-60 is always less than that of WMA-C in the entire time domain, which shows the stress relaxation ability of WMA-60 is better than WMA-C. The creep compliance of WMA-60 is always higher than the WMA-C in the entire time domain, which shows that the deformability of WMA-60 is better than WMA-C. This is also because the viscosity of the 60-mesh rubber-modified asphalt binder is less than compound-mesh, then the cohesion of WMA-60 is also less than that of WMA-C, which gives WMA-60 a better relaxation and deformation capabilities. Comparing the hot-mixed and warm-mixed crumb rubber-modified asphalt mixtures, it found that once the warm mix additive was added, the relaxation modulus of crumb rubber-modified asphalt mixtures becomes smaller in the shorter time domain and larger in the longer time domain, creep compliance varies in the opposite way to the relaxation modulus. It shows that the addition of warm mix improves the asphalt mixture’s low-temperature stress relaxation ability and high-temperature deformation resistance. This is because the reduced aging effect after the warm mix additive was added. Finally, WMA-60 is recommended in the cold region, while WMA-C recommended in the hot region.

4.3.2. The Relaxation Modulus and Creep Compliance Obtained from the Continuous Spectrum The continuous relaxation spectrum can obtain the relaxation modulus, and the expression is as follows (Equation (41)). Z + ! ∞ t E(t) = E + H(ρ) exp d(lnρ) (41) ∞ −ρ −∞ where E is the long terms equilibrium modulus, it can be determined by the method proposed by ∞ Liu [23], MPa; ρ is the relaxation time, s; t is the loading time, s. The trapezoidal rule was applied to calculate the relaxation modulus. The integration interval is selected as ( 30, 30), when lgρn > 30 and lgρn < 30, H(ρ) is so small that it can be regarded as 0. − − The expression is as follows (Equation (42)). n=M t t X H(ρn 1) exp + H(ρn) exp − − ρn 1 − ρn E(t) E + (ln10) − ∆lgρn (42) ≈ ∞ 2 n=1 where M is the number of subintervals in the integration interval, the value of M was 6000; ∆lgρn is the length of the subinterval interval. In this paper, the value of ∆lgρn was selected to be 0.01. Materials 2020, 13, x 19 of 28

𝑡 𝐸(𝑡) =𝐸 + 𝐻(𝜌)exp (− )𝑑(𝑙𝑛𝜌) (41) 𝜌 where 𝐸 is the long terms equilibrium modulus, it can be determined by the method proposed by Liu [23], MPa; 𝜌 is the relaxation time, s; t is the loading time, s. The trapezoidal rule was applied to calculate the relaxation modulus. The integration interval is selected as (−30, 30), when 𝑙𝑔𝜌 > 30 and 𝑙𝑔𝜌 ＜ −30, 𝐻(𝜌) is so small that it can be regarded as 0. The expression is as follows (equation (42)). 𝑡 𝑡 𝐻(𝜌 ) exp − +𝐻(𝜌 ) exp − 𝜌 𝜌 𝐸(𝑡) ≈𝐸 + (𝑙𝑛10) ∆𝑙𝑔𝜌 (42) 2 where M is the number of subintervals in the integration interval, the value of M was 6000; ∆𝑙𝑔𝜌 is the length of the subinterval interval. In this paper, the value of ∆𝑙𝑔𝜌 was selected to be 0.01. MaterialsThen2020 equation, 13, 3723 (42) can be replaced by equation (43), which is expressed as follows. 20 of 29

0.01 𝑡 𝑡 𝐸(𝑡) ≈𝐸 + (𝑙𝑛10) 𝐻(10) exp − +𝐻(10.) exp − Then Equation (42) can be2 replaced by Equation10 (43), which is expressed as10 follows.. nh 𝑡 𝑡 i E(t) E +(ln10) 0.01.H 10 30 exp t + H .10 29.99 exp t +𝐻(10 2 ) exp− − 10+𝐻30 (10 − ) exp − 10 29.99 + ⋯ (43) ≈ ∞ h 10.− − −10i.− + H 10 29.99 exp t + H 10 29.98 exp t + (43) − 10 29.99 − 10 29.98 h − − 𝑡 −io𝑡− ··· ++𝐻H (101029.99.)expexp − t ++𝐻H 10(3010exp) exp −t − 101029.99. − 1030 10

Finally, the relaxation relaxation modulus modulus can can be be obtained obtained by bythe the continuous continuous relaxation relaxation time time spectrum. spectrum. The Theresult result was waspresented presented in Figure in Figure 11. 11.

0.01 HMA-60 Continuous Relaxation Spectrum Modulus

4 WMA-60 Continuous Relaxation Spectrum Modulus 4x10 HMA-H Continuous Relaxation Spectrum Modulus WMA-H Continuous Relaxation Spectrum Modulus

3.5x104 10000 ) -1

3x104

2.5x104 Relaxation modulus (MPa) modulus Relaxation

2x104 HMA-60 Continuous Relaxation Spectrum Modulus 10-25 10-23 10-21 10-19 10-17 10-15 10-13 10-11 10-9 Time (s) WMA-60 Continuous Relaxation Spectrum Modulus 1E-3 HMA-H Continuous Relaxation Spectrum Modulus

-5 WMA-H Continuous Relaxation Spectrum Modulus 4.2x10 HMA-60 Continuous Retardation Spectrum Compliance WMA-60 Continuous Retardation Spectrum Compliance HMA-60 Continuous Retardation Spectrum Compliance 4x10-5 HMA-H Continuous Retardation Spectrum Compliance WMA-60 Continuous Retardation Spectrum Compliance WMA-H Continuous Retardation Spectrum Compliance HMA-H Continuous Retardation Spectrum Compliance

1000 ) -5 -1 3.8x10 WMA-H Continuous Retardation Spectrum Compliance

3.6x10-5

3.4x10-5

3.2x10-5 Creep Compliance (MPa

-5 1E-4 3x10 Relaxation modulus (MPa)modulus Relaxation

10-25 10-23 10-21 10-19 10-17 10-15 10-13 10-11 10-9 (MPa Compliance Creep Time (s)

100

10-20 10-15 10-10 10-5 100 105 1010 1015 1020 Time (s) Figure 11. TheThe relaxation relaxation modulus and creep compliance obtained from the continuous spectrum.

Similarly, according to Equationequation (44), the cr creepeep compliance can be obtained by continuous retardation spectrum. Z + ∞ t D(t) = Dg + L(τ) 1 exp 𝑡 d(lnτ) (44) 𝐷(𝑡) =𝐷 + 𝐿(𝜏)[1 −− exp(− −τ)]𝑑(𝑙𝑛𝜏) (44) −∞ 𝜏 where Dg is the transient equilibrium compliance, it can be determined by Park and Schapery method, where 𝐷 is the transient equilibrium compliance, it can be determined by Park and Schapery MPa; τ is the retardation time, s; t is the loading time, s. 𝜏 method,The MPa; trapezoidal is the rule retardation was applied time, to s; calculatet is the loading creep compliance.time, s. The expression is as follows. The integration interval is selected as ( 30, 30), when lgτn > 30 and lgτn < 30, L(τ) is so small that it − − can be regarded as 0 (Equation (45)).

h i n=M t t X L(τn 1) 1 exp + L(τn)[1 exp ] − τn 1 τn D(t) Dg + (ln10) − − − − − ∆lgτn (45) ≈ 2 n=1 where M is the number of subintervals in the integration interval, the value of M was 6000; ∆lgτn is the length of the subinterval interval. in this paper, the value of ∆lgτn was selected to be 0.01. Then Equation (45) can be replaced by Equation (46), which is expressed as follows.

0.01 n 30 t D(t) Dg +(ln10) 2 L 10− [1 exp 30 ] ≈ − − 10− 29.99 t 29.99 +L 10− [1 exp 29.99 ] + L 10− [1 − − 10− (46) t 29.98 t exp 29.99 ] + L 10− [1 exp 29.98 ] + ... − − 10− − − 10− o +L 1029.99 [1 exp t ] + L 1030 [1 exp t ] − − 1029.99 − − 1030 Materials 2020, 13, 3723 21 of 29

Finally, Then the creep compliance can be obtained by the continuous retardation time spectrum. The result was also presented in Figure 11. It can be learned from Figure 11 that the relaxation modulus and creep compliance obtained 8 7 from the continuous time spectrum change significantly in the time domain of 10− s~10 s, but do no 5 change significantly outside that region. When the time domain is greater than 10− s, the relaxation modulus and creep compliance calculated by the continuous time spectrum are consistent with the 5 discrete time spectrum. However, when the time domain is less than 10− s, the relaxation modulus and creep compliance calculated by the continuous time spectrum are different from the discrete time spectrum. The specific performance is that the relaxation modulus of HMA-60 in this region is greater than HMA-C, which shows that the stress relaxation ability of HMA-60 is less than HMA-C in the extremely shorter time domain (extremely low temperature). The creep compliance of HMA-60 is less than that of HMA-C in this region, which shows that the deformation ability of HMA-60 is not as good as HMA-C in low temperature. This is because the magnitude of the continuous relaxation spectrum of HMA-60 is significantly greater than HMA-C in shorter reduced times (lower temperatures), and the higher magnitude of the relaxation spectrum, the higher relaxation modulus.

4.3.3. Compared Relaxation Modulus and Creep Compliance Obtained the Discrete and Continuous Spectrum It can be learned from Figure 12 that the curve shape of the discrete spectrum modulus is similar to the continuous spectrum modulus. They have a Z-shape in the broad time domain. However, 5 5 the discrete spectrum modulus changes significantly in the time domain of 10− s~10 s, while the 8 7 continuous spectrum modulus changes significantly in the time domain of 10− s~10 s. The discrete 5 spectrum modulus less than the continuous spectrum modulus in the shorter time domain (<10− s), 5 but greater than the continuous spectrum modulus in the longer time domain (>10− s), and there is almost no difference between the discrete spectrum modulus and the continuous spectrum modulus in 5 5 the middle time region (10− s~10 s). The reason can be learned from Figure9, where the magnitude of the relaxation spectrum obtained from the continuous relaxation spectrum is greater than the magnitude of the discrete relaxation spectrum in the lower time domain and less than the discrete relaxation spectrum in the higher time domain. Materials 2020, 13, x 21 of 28

10-2

4 10 ) -1

HMA-60 Discrete Relaxation Spectrum Modulus WMA-60 Discrete Relaxation Spectrum Modulus HMA-H Discrete Relaxation Spectrum Modulus WMA-H Discrete Relaxation Spectrum Modulus HMA-60 Continuous Relaxation Spectrum Modulus -3 WMA-60 Continuous Relaxation Spectrum Modulus 10 HMA-H Continuous Relaxation Spectrum Modulus WMA-H Continuous Relaxation Spectrum Modulus HMA-60 Discrete Retardation Spectrum Compliance WMA-60 Discrete Retardation Spectrum Compliance 3 HMA-H Discrete Retardation Spectrum Compliance 10 WMA-H Discrete Retardation Spectrum Compliance HMA-60 Continuous Retardation Spectrum Compliance WMA-60 Continuous Retardation Spectrum Compliance HMA-H Continuous Retardation Spectrum Compliance WMA-H Continuous Retardation Spectrum Compliance 10-4 Relaxation modulus (MPa) modulus Relaxation Creep Compliance (MPa Compliance Creep

102

10-20 10-15 10-10 10-5 100 105 1010 1015 1020 Time (s) FigureFigure 12. Compared 12. Compared the the relaxation relaxation modulus modulus andand creep compliance compliance obtained obtained from from the thediscrete discrete and and continuouscontinuous spectrum. spectrum.

It canIt alsocan also be obtainedbe obtained in in Figure Figure 1212 that the the cu curverve shape shape of the of thediscrete discrete spectrum spectrum compliance compliance is also similar to the continuous spectrum compliance. They have a Z-shape in the broad time domain. is also similar to the continuous spectrum compliance. They have a Z-shape in the broad time However, the discrete spectrum compliance changes significantly in the time domain of 10−5 s~105 s, domain. However, the discrete spectrum compliance changes signiﬁcantly in the time domain of while the continuous spectrum compliance changed significantly in the time domain of 10−8 s~107 s. The discrete spectrum compliance higher than the continuous spectrum compliance in the shorter time domain (<10−5 s), but less than the continuous spectrum compliance in the longer time domain (>10−5 s), and there is almost no difference between the discrete spectrum compliance and the continuous spectrum compliance in the middle time region (10−5 s~105 s). The strength of the retardation spectrum obtained by the continuous retardation spectrum is smaller than the discrete retardation spectrum strength in the lower time domain and greater than the discrete retardation spectrum strength in the higher time domain. The discrete spectrum modulus (compliance) is particularly close to the continuous spectrum modulus (compliance) in the time domain of 10−5 s~103 s. To compare the relative errors of the relaxation modulus and creep compliance constructed by discrete and continuous spectrum methods, respectively. The relative errors of the relaxation modulus and creep compliance can be defined follow the equations (47) and (48), The result is presented in Figure 12. The relative error of relaxation modulus.

|𝐸(𝑡) −𝐸(𝑡)| 𝑅𝐸 = (47) 𝐸(𝑡)

where: 𝑅𝐸 is the relative errors of the relaxation modulus; 𝐸(𝑡) is the relaxation modulus calculated from discrete relaxation spectrum; 𝐸(𝑡) is the relaxation modulus calculated from continuous relaxation spectrum. The relative error of creep compliance.

|𝐷(𝑡) −𝐷(𝑡)| 𝑅𝐸 = (48) 𝐷(𝑡)

where: 𝑅𝐸 is the relative errors of the creep compliance; 𝐷(𝑡) is the creep compliance calculated from discrete retardation spectrum; 𝐷(𝑡) is the creep compliance calculated from continuous retardation spectrum. It can be learned from Figure 13 that within the range of 10−5 s~103 s, both the relaxation modulus and creep compliance calculated by the discrete spectrum are almost indistinguishable from those calculated by the continuous spectrum (the maximum error is only 4%), and once exceeded the range,

Materials 2020, 13, 3723 22 of 29

5 5 10− s~10 s, while the continuous spectrum compliance changed significantly in the time domain of 8 7 10− s~10 s. The discrete spectrum compliance higher than the continuous spectrum compliance in 5 the shorter time domain (<10− s), but less than the continuous spectrum compliance in the longer 5 time domain (>10− s), and there is almost no difference between the discrete spectrum compliance 5 5 and the continuous spectrum compliance in the middle time region (10− s~10 s). The strength of the retardation spectrum obtained by the continuous retardation spectrum is smaller than the discrete retardation spectrum strength in the lower time domain and greater than the discrete retardation spectrum strength in the higher time domain. The discrete spectrum modulus (compliance) is 5 3 particularly close to the continuous spectrum modulus (compliance) in the time domain of 10− s~10 s. To compare the relative errors of the relaxation modulus and creep compliance constructed by discrete and continuous spectrum methods, respectively. The relative errors of the relaxation modulus and creep compliance can be defined follow the Equations (47) and (48), The result is presented in Figure 12. The relative error of relaxation modulus.

Ed(t) Ec(t) REE = − (47) Ec(t) where: REE is the relative errors of the relaxation modulus; Ed(t) is the relaxation modulus calculated from discrete relaxation spectrum; Ec(t) is the relaxation modulus calculated from continuous relaxation spectrum. The relative error of creep compliance.

Dd(t) Dc(t) RED = − (48) Dc(t) where: RED is the relative errors of the creep compliance; Dd(t) is the creep compliance calculated from discrete retardation spectrum; Dc(t) is the creep compliance calculated from continuous retardation spectrum. 5 3 It can be learned from Figure 13 that within the range of 10− s~10 s, both the relaxation modulus and creep compliance calculated by the discrete spectrum are almost indistinguishable from those calculatedMaterials by 2020 the, 13 continuous, x spectrum (the maximum error is only 4%), and once exceeded22 the of 28 range, there will be a significant difference (the maximum error for relaxation modulus and creep flexibility is there will be a significant difference (the maximum error for relaxation modulus and creep flexibility 39% andis 39% 55%, andrespectively). 55%, respectively). The The reason reason can can be be learned learned from from Figure Figure9 9..

Relaxation modulus of HMA-60 60 Creep Compliance of HMA-60 Relaxation modulus of WMA-60 Creep Compliance of WMA-60 Relaxation modulus of HMA-C Creep Compliance of HMA-C 50 Relaxation modulus of WMA-C Creep Compliance of WMA-C

20 Relative Error (%) Error Relative

10-20 10-15 10-10 10-5 100 105 1010 1015 1020 Time (s) FigureFigure 13. Relative 13. Relative errors errors of theof the relaxation relaxation modulus modulus and and creep creepcompliance compliance produced by by methods methods of of the discretethe discrete and continuous and continuous spectrum. spectrum.

5. Conclusions In this paper, based on the complex modulus test, we use the discrete and continuous spectrum to construct relaxation modulus and creep compliance master curves. The two methods are consistent with the LVE theory and allow for possible asymmetries in the spectrum curve. The following conclusions can be drawn from this study: (1) According to the complex modulus test data of four types of mixtures, the master curve models of storage modulus and loss modulus are developed according to the approximate K-K relations. These master curve models share the same model parameters and allow for possible asymmetry. Similarly, the master curve model of storage compliance and loss compliance is also developed according to the approximate K-K relations. (2) The continuous and discrete relaxation spectrum can be obtained from the storage modulus master curves utilizing integral transformations and the collocation method, respectively. Similarly, the continuous and discrete retardation spectrum can be obtained from the storage compliance master curve. (3) The shape of the discrete spectrum is similar to that of the continuous spectrum. The most significant difference is the intensity of the discrete spectrum is higher than that of the continuous spectrum. (4) The relaxation modulus and creep compliance were calculated by the discrete and continuous spectrum, respectively. The difference between the discrete and continuous spectrum is negligible in the time domain of 10−5 s-103 s. However, there are significant differences outside the region. (5) HMA-60 has a better deformation capacity than HMA-C at a shorter reduced time and a worse deformation resistance than HMA-C at a longer reduced time. Moreover, the deformation capacity of WMA-60 is better than WMA-C at a shorter reduced time, and a worse deformation resistance than WMA-C is seen at a longer reduced time. Once the warm mix additive was added, the mixture’s deformation capacity was improved at a shorter reduced time. Moreover, improved the deformation resistance in longer reduced time.

Supplementary Materials: The following are available online at www.mdpi.com/xxx/s1, Table S1: Raw data on dynamic modulus and phase angle for types of mixture.

Materials 2020, 13, 3723 23 of 29

(1) According to the complex modulus test data of four types of mixtures, the master curve models of storage modulus and loss modulus are developed according to the approximate K-K relations. These master curve models share the same model parameters and allow for possible asymmetry. Similarly, the master curve model of storage compliance and loss compliance is also developed according to the approximate K-K relations. (2) The continuous and discrete relaxation spectrum can be obtained from the storage modulus master curves utilizing integral transformations and the collocation method, respectively. Similarly, the continuous and discrete retardation spectrum can be obtained from the storage compliance master curve. (3) The shape of the discrete spectrum is similar to that of the continuous spectrum. The most significant difference is the intensity of the discrete spectrum is higher than that of the continuous spectrum. (4) The relaxation modulus and creep compliance were calculated by the discrete and continuous spectrum, respectively. The difference between the discrete and continuous spectrum is negligible in the time domain of 10 5 s-103 s. However, there are significant differences outside the region. − (5) HMA-60 has a better deformation capacity than HMA-C at a shorter reduced time and a worse deformation resistance than HMA-C at a longer reduced time. Moreover, the deformation capacity of WMA-60 is better than WMA-C at a shorter reduced time, and a worse deformation resistance than WMA-C is seen at a longer reduced time. Once the warm mix additive was added, the mixture’s deformation capacity was improved at a shorter reduced time. Moreover, improved the deformation resistance in longer reduced time.

Supplementary Materials: The following are available online at http://www.mdpi.com/1996-1944/13/17/3723/s1, Table S1: Raw data on dynamic modulus and phase angle for types of mixture. Author Contributions: F.Z.: Conceptualization, investigation, methodology, writing—original draft; L.W.: Data curation, writing—review and editing; C.L.: Formal analysis; Y.X.: Conceptualization, validation, supervision. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Natural Science Foundation of China, grant number 11462018 and The APC was funded by National Natural Science Foundation of China. The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (11462018) and the Natural Science Foundation of Inner Mongolia Autonomous Region of China (NJ-2015-1). Acknowledgments: The authors would like to acknowledge Inner Mongolia Luda Asphalt Co., Ltd. and MeadWestvaco Co., Ltd. for providing the raw materials for this study. Conflicts of Interest: The authors declare no conflict of interest.

Appendix A Coeﬃcients of variation (CV) was used to assess the variability of the test results for the types of asphalt mixture. Materials 2020, 13, 3723 24 of 29

Table A1. Coeﬃcients Variation of Dynamic Modulus.

Coeﬃcients Variation of Dynamic Modulus/% Temperature/◦C 0.1 Hz 0.5 Hz 1 Hz 5 Hz 10 Hz 20 Hz 25 Hz Average 5 3.5 3.9 4.6 4.9 5.7 5.7 6.5 4.97 20 5.1 5.6 6.4 8.3 8.5 9.5 8.7 7.44 HMA-60 35 10.4 11.3 11.5 11.6 12.4 14.2 15.9 12.47 50 12.8 12.8 13.9 14.6 15.4 16.9 16.8 14.74 5 4.1 4.6 5 5.2 4.9 5.6 6.7 5.16 20 4.8 5.3 6.7 7.9 7.8 9.5 10.6 7.51 WMA-60 35 9.9 10.6 13.4 11.1 14.8 14.5 17.2 13.07 50 11.4 12.9 14.6 14.8 15.3 16.2 17.3 14.64 5 3.2 4.9 6.2 5.5 7.1 6.9 10.4 6.31 20 5.6 5.1 6.7 7.9 9.3 10.9 12.4 8.27 HMA-C 35 9.9 11.3 10.7 10.4 11.6 10.9 13.7 11.21 50 11.9 13.5 12.4 11.7 12.8 12.7 14.5 12.79 5 4.5 5.5 4.4 5.1 5.7 6.3 8.7 5.74 20 6.3 5.2 7.1 7.5 9.2 8.5 10.8 7.80 WMA-C 35 9.9 11.3 10.9 10.4 11.7 13.8 15.1 11.87 50 11.5 11.8 12.9 13.7 12.4 11.8 15.9 12.86

Table A2. Coeﬃcients Variation of Phase Angle.

Coeﬃcients Variation of Phase Angle/% Temperature/◦C 0.1 Hz 0.5 Hz 1 Hz 5 Hz 10 Hz 20 Hz 25 Hz Average 5 5.6 6.2 7.1 7.8 8.2 9.2 10.3 7.77 20 11.5 12 13.4 14.2 15.3 16.7 17.5 14.37 HMA-60 35 16.1 17.6 17.2 20.3 21.7 20.9 22.3 19.44 50 18.5 21.6 20.4 19.8 22.3 21.5 20.8 20.70 5 6 6.8 7.3 7.5 8.3 8.9 9.9 7.81 20 10.4 11.6 12.8 13.9 15.7 15.9 18 14.04 WMA-60 35 15.3 16.7 17.3 19.9 19.4 20.3 21.1 18.57 50 17.6 20.5 20.6 18.9 19.8 22 21.5 20.13 5 6.3 6.8 7.0 8.3 8.8 9.1 10.6 8.13 20 10.7 11.7 12.9 13.5 14.9 15.7 16.5 13.70 HMA-C 35 15.4 16.8 17.2 20.5 21.2 20.6 21.8 19.07 50 17.9 21.4 20.3 18.9 21.7 20.5 22.1 20.40 5 4.5 5.5 6.1 6.9 8.4 8.6 9.7 7.10 20 10.3 11.2 12.1 13.5 14.2 15.5 16.8 13.37 WMA-C 35 15.4 16.9 18.2 17.6 20.4 20.3 21.6 18.63 50 16.9 20.7 22.7 20.8 21.5 20.9 21.5 20.71

The discrete relaxation time ρi can be determined by Equation (A1):

i 6 ρ = 2 10 − (A1) i ×

The discrete retardation time τj can be determined by taking the negative reciprocal of the solution of Equation (A2): lim E(S) = 0 (j = 1, ... , n) (A2) s (1/τ ) →− j The method for determining the discrete retardation time of the crumb rubber-modiﬁed asphalt mixture is listed in Figures A1–A4. In addition, the discrete relaxation and retardation spectrum of the mixture is shown in Tables A3 and A4, respectively. Materials 2020,, 13,, xx 24 of 28

50 16.9 20.7 22.7 20.8 21.5 20.9 21.5 20.71

The discrete relaxation time 𝜌 can can bebe determineddetermined byby equationequation (A1):(A1):

𝜌 =2×10 (A1)

The discrete retardation time 𝜏 cancan bebe determineddetermined byby takingtaking thethe negativenegative reciprocalreciprocal ofof thethe solution of equation (A2): lim 𝐸(𝑆) =0 (𝑗 =1,…,𝑛) lim 𝐸(𝑆) =0 (𝑗 =1,…,𝑛) (A2) →/ The method for determining the discrete retardationion timetime ofof thethe crumbcrumb rubber-modified asphalt mixtureMaterials 2020 is listed, 13, 3723 in Figures A1–A4. In addition, the discrete relaxation and retardation spectrum25 of of 29 the mixture is shown in Tables A3 and A4, respectively.

7 10107 OperationalOperational modulusmodulus ofof HMA-60HMA-60 ) )

) 6 10106 MPa MPa MPa ( ( ( 55 | | | 1010 ) ) ) S S S ( ( ( 4 10104

3 10103

2 10102

1 10101 Operational modulus |E modulus Operational Operational modulus |E modulus Operational Operational modulus |E modulus Operational 0 10100

-1-1 1010-1 -5-5 -3-3 -1-1 1 3 5 7 1010-5 1010-3 1010-1 10101 10103 10105 10107 Compound variable -1/S FigureFigure A1. A1. ||𝐸(𝑆)|E(S)|| vs.vs.vs.−1/𝑆1/S for forfor HMA-60. HMA-60.HMA-60. −

7 10107 OperationalOperational modulusmodulus ofof WMA-60WMA-60 ) ) ) 6 10106 MPa MPa MPa ( ( ( | | | ) ) ) 55 S S S

( 10

4 10104

3 10103

2 10102 Operational modulus |E modulus Operational Operational modulus |E modulus Operational Operational modulus |E modulus Operational 1 10101

0 10100 -5-5 -3-3 -1-1 1 3 5 7 1010-5 1010-3 1010-1 10101 10103 10105 10107 Compound variable -1/S FigureFigure A2. A2. ||𝑬(𝑺)|E(S|)| | vs. −1/𝑆1/S for forfor WMA-60. WMA-60.WMA-60. −

7 10107 OperationalOperational modulusmodulus ofof HMA-CHMA-C ) ) ) 6 10106 MPa MPa MPa ( ( ( 55 | | | ) ) 10

) 10 S S S ( ( ( 4 10104

3 10103

2 10102

1 10101 Operational modulus |E modulus Operational Operational modulus |E modulus Operational Operational modulus |E modulus Operational 0 10100

-1-1 1010-1 -5-5 -3-3 -1-1 1 3 5 7 1010-5 1010-3 1010-1 10101 10103 10105 10107 Compound variable -1/S FigureFigure A3. A3. ||𝐸(𝑆)|E(S)||| vs. −1/𝑆1/S for forfor HMA-C. HMA-C.HMA-C. −

Materials 2020, 13, 3723 26 of 29 Materials 2020, 13, x 25 of 28

107 Operational modulus of WMA-C ) 106 MPa ( |

) 5

S 10 (

104

103

102

Operational modulus |E modulus Operational 101

100 10-5 10-3 10-1 101 103 105 107 Compound variable -1/S FigureFigure A4. A4. |𝐸(𝑆)|E(S)|| vs. −1/𝑆1/S for for WMA-C. WMA-C. − Table A3. Discrete relaxation spectrum of crumbcrumb rubber-modiﬁedrubber-modified asphaltasphalt mixture.mixture.

HMA-60HMA-60 WMA-60WMA-60 HMA-C HMA-C WMA-C WMA-C i i ρi ρi Ei Ei ρi ρi EEii ρρi i Ei Ei ρi ρi Ei Ei 1 0.000021 0.00002 3852.1 3852.1 0.000020.00002 2650.0 0.00002 0.00002 3087.9 3087.9 0.00002 0.00002 4910.2 4910.2 2 0.00022 0.0002 5434.55434.5 0.00020.0002 4580.1 0.0002 0.0002 5998.5 5998.5 0.0002 0.0002 5693.3 5693.3 33 0.002 0.002 6061.56061.5 0.0020.002 5233.2 0.002 0.002 5665.0 5665.0 0.002 0.002 5464.6 5464.6 4 0.02 3680.0 0.02 3208.0 0.02 4254.5 0.02 3717.2 54 0.2 0.02 1751.03680.0 0.20.02 1476.53208.0 0.02 0.2 4254.5 2239.5 0.02 0.23717.2 1920.1 65 2 0.2 566.51751.0 20.2 1476.5 610.9 0.2 2 2239.5 1054.3 0.2 21920.1 758.0 76 202 223.2566.5 202 271.7610.9 2 20 1054.3 394.9 2 20758.0 318.4 87 20020 35.8223.2 20020 105.4271.7 20 200 394.9 165.9 20 200318.4 120.9 98 2000 200 11.8 35.8 2000 200 105.4 40.4 200 2000 165.9 30.0 200 2000120.9 32.8 10 20,000 5.3 20,000 10.1 20,000 4.1 20,000 7.91 11 200,0009 2000 1.911.8 200,0002000 40.4 1.2 2000 200,000 30.0 1.5 2000 200,000 32.8 1.65 10 Ee20,000= 121.58 5.3 20,000Ee = 193.10 10.1 20,000Ee = 183.344.1 20,000 Ee =7.91209.75 11 200,000 1.9 200,000 1.2 200,000 1.5 200,000 1.65 Ee = 121.58 Ee = 193.10 Ee = 183.34 Ee = 209.75 Table A4. Discrete retardation spectrum of crumb rubber-modiﬁed asphalt mixture.

TableHMA-60 A4. Discrete retardation WMA-60 spectrum of crumb rubber-modified HMA-C asphalt mixture. WMA-C i τ D τ D τ D τ D j HMA-60j j WMA-60 j j HMA-C j WMA-Cj j 1i 0.000024 5.70 10 5 0.000023 5.69 10 7 0.000023 4.97 10 6 0.000025 8.82 10 6 τj × Dj− τj ×Dj − τj D×j − τj Dj × − 2 0.000282 5 0.00028 5 0.00028 5 0.00029 5 3.12 10− −5 2.79 10−−7 2.01 10−6− 2.32 −106 − 1 0.000024 5.70× × 105 0.000023 5.69× × 105 0.000023 4.97 ×× 10 5 0.000025 8.82 × 10× 5 3 0.00372 5.46 10− 0.00363 6.83 10− 0.00331 4.00 10− 0.00347 5.04 10− 2 0.000282 3.12× × 104 −5 0.00028 2.79× × 10−45 0.00028 2.01 ×× 10−5 4 0.00029 2.32 × 10×−5 4 4 0.0447 1.66 10− 0.0427 1.55 10− 0.0398 1.04 10− 0.0407 1.24 10− × 4 −5 × −45 × −5 4 ×−5 4 53 0.55 0.00372 3.94 5.4610 × −10 0.003630.437 3.456.83 ×10 10− 0.003310.437 4.002.41 × 1010− 0.003470.457 5.042.91 × 10 10 − × 4 × 4 × 4 × 4 64 5.1 0.0447 1.05 1.6610 × −10−4 0.04274.07 7.66 1.55 ×10 10−−4 0.03984.68 1.046.04 × 1010−4− 0.04074.27 1.246.74 × 10−104 − × 3 × 3 × 3 × 3 7 49.0 2.01 10 −4 36.3 1.23 10 −4 41.7 1.32 10−4 38 1.23 −104 5 0.55 3.94× × −10 0.437 3.45× × 10− 0.437 2.41 ×× 10 − 0.457 2.91 × 10× − 8 263.0 3.00 10 3 295 1.60 10 3 372 1.43 10 3 309 1.21 10 3 6 5.1 1.05× × −10−4 4.07 7.66× × 10−−4 4.68 6.04 ×× 10−4− 4.27 6.74 × 10×−4 − 9 2150.0 1.66 10 3 2400 1.01 10 3 2340 1.01 10 3 2340 3.28 10 4 − −3 −−3 −3− −3 − 7 49.0 2.01× × 104 36.3 1.23× × 103 41.7 1.32 ×× 10 4 38 1.23 × 10× 4 10 20,900 2.68 10− 20,900 3.34 10− 20,400 2.94 10− 20,900 1.00 10− × 5 −3 × −53 × −3 6 ×−3 6 118 204,000 263.0 1.70 3.0010 × −10 204,000295 3.941.60 ×10 10− 204,000 372 1.435.94 × 1010− 309204,000 1.212.47 × 10 10 − × 5 −3 ×5 −3 × 5 −3 ×−54 9 2150.0Dg = 4.60 1.6610 × 10 2400Dg = 5.4 1.0110 × 10 2340Dg = 4.331.01 10× 10 2340D g = 4.41 3.28 × 1010 × − × − × − × − 10 20,900 2.68 × 10−4 20,900 3.34 × 10−3 20,400 2.94 × 10−4 20,900 1.00 × 10−4 11 204,000 1.70 × 10−5 204,000 3.94 × 10−5 204,000 5.94 × 10−6 204,000 2.47 × 10−6 References Dg = 4.60 × 10−5 Dg = 5.4 × 10−5 Dg = 4.33 × 10−5 Dg = 4.41 × 10−5 1. Kim, Y.R.; Little, D.N.; Lytton, R.L. Fatigue and healing characterization of asphalt mixtures. J. Mater. Civ. Eng. References2003, 15 , 75–83. [CrossRef] 2. Park, S.W.; Schapery, R.A. Methods of interconversion between linear viscoelastic material functions. 1. Kim, Y.R.; Little, D.N.; Lytton, R.L. Fatigue and healing characterization of asphalt mixtures. J. Mater. Civ. Part I—A numerical method based on Prony series. Int. J. Solids Struct. 1999, 36, 1653–1675. [CrossRef] Eng. 2003, 15,75–83, doi:10.1061/(ASCE)0899-1561(2003)15:1(75). 3. Alavi, S.M.Z. Comprehensive Methodologies for Analysis of Thermal Cracking in Asphalt Concrete Pavements; 2. Park, S.W.; Schapery, R.A. Methods of interconversion between linear viscoelastic material functions. Part University of Nevada: Reno, NV, USA, 2014; ISBN 9781321216097. I—A numerical method based on Prony series. Int. J. Solids Struct. 1999, 36, 1653–1675, doi:10.1016/S0020- 7683(98)00055-9. 3. Alavi, S.M.Z. Comprehensive Methodologies for Analysis of Thermal Cracking in Asphalt Concrete Pavements; University of Nevada: Reno, NV, USA, 2014; ISBN: 9781321216097.

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