Viscoelastic properties of warm recycled asphalt mixture based on S-φ model

Viscoelastic properties of warm recycled asphalt mixture based on S-φ model | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Viscoelastic properties of warm recycled asphalt mixture based on S-φ model Lei Feng, WU Jianing, Lan Wang, Chuanyu Shao, Tao Li This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6912958/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 13 You are reading this latest preprint version Abstract In order to study the influence of temperature, frequency and RAP content on the viscoelastic properties of warm mix recycled asphalt mixture, the viscoelastic properties of asphalt mixture with different RAP content (0%, 50%, 70%) at four temperatures and six frequencies were studied by dynamic modulus test. The results show that under the condition of high temperature and low frequency, the dynamic modulus of the two kinds of asphalt mixture after warm mix regeneration is higher than that of hot mix asphalt mixture ; under the condition of low temperature and high frequency, the dynamic modulus of the two kinds of asphalt mixture after warm mix regeneration is not significantly different from that of hot mix asphalt mixture, which indicates that the elasticity of asphalt mixture after warm mix regeneration is still higher than that of hot mix asphalt mixture at high temperature, and there is no significant difference between flexible and hot mix asphalt mixture at low temperature. Compared with 70% RAP warm mix recycled asphalt mixture, when the RAP content is 50%, the dynamic modulus increases the most at high temperature (50°C), and the phase angle decreases the least at low temperature (5°C). Therefore, 50% RAP warm mix recycled asphalt mixture has better high and low temperature performance. According to the principle of time-temperature equivalence, S (GMS model) and φ ( improved phase angle model ) with auxiliary parameter ρ are fitted synchronously to realize parameter sharing, and S and φ models can well fit the variation of dynamic modulus and phase angle with frequency. warm recycled asphalt mixture The RAP Dynamic modulus Phase Angle The master curve Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 0 Introduction There are many factors influencing the dynamic modulus and phase angle. Temperature, frequency as well as RAP content have a great influence on the dynamic modulus and phase angle of warm - mixed recycled asphalt mixture. Compared with static modulus, dynamic modulus can more accurately reflect the mechanical behavior of asphalt pavement under the changes of vehicle load [ 1 , 2 ] . Li, Harnaeni, Xi, Zheng along with Barraj et al. [ 2 – 6 ] explored the influence of temperature and frequency on asphalt mixture via dynamic modulus tests. The results show that the dynamic modulus of asphalt mixture decreases as temperature increases and as loading frequency decreases. The phase angle exhibits an opposite trend. Zhang, Gong along with Li et al. [ 7 – 9 ] established the master curves of dynamic modulus and phase angle of crumb rubber modified asphalt mixture using the generalized Sigmund model (GSM) and the phase angle model (DEM). The results show that GSM and DEM are capable of accurately predicting the variation of dynamic modulus and phase angle with loading frequency. Ma, Acharjee along with Jukte et al. [ 10 – 12 ] developed a prediction model based on dynamic shear modulus (|G*|) and phase angle (δ) utilizing the dynamic modulus master curve, Kramers-Kronig relationship, artificial neural network (ANN) and other methods. Lan [ 13 ] studied the variation of dynamic modulus and phase angle of recycled asphalt mixture with loading frequency using the CAM model (an improved model for dynamic modulus and phase angle). The results show that the dynamic modulus and phase angle master curve fitted by the improved CAM model exhibit high precision and strong regularity. Currently, many scholars have studied the high temperature viscoelastic properties of asphalt mixture [ 14 , 15 ] . However, few studies have been conducted on the viscoelastic properties of warm mix recycled asphalt mixture using a more accurate viscoelastic model. Although some scholars have employed relevant models for research, many dynamic modulus and phase angle viscoelastic models fail to satisfy the same shift factor and the accurate Kramers-Kronig (K-K) relationship among viscoelastic functions, and cannot accurately ensure the model's accuracy. In this paper, the S and φ models effectively overcome this defect. Therefore, based on the S and φ models, this paper will study the dynamic viscoelastic properties of warm mix recycled asphalt mixture through dynamic modulus tests. Compared with hot mix recycled asphalt mixture, warm mix recycled asphalt mixture can reduce the secondary aging of asphalt during mixing and compaction, facilitate the integration of the new and old asphalt interfaces, strengthen the adhesion between asphalt and aggregates, and improve the road performance of warm mix recycled asphalt mixture. Therefore, it is of great guiding importance to study the viscoelastic properties of warm mix recycled asphalt mixture using an accurate viscoelastic model for the development of warm mix recycled pavement. 1 Experiment 1.1 Raw material The base asphalt used in the experiment is Panjin 90# asphalt. The modifier is the SMC ambient temperature regenerator produced by Beijing Sierma Company. Based on the manufacturer's recommendations and the stability, flow value and other indicators obtained from the Marshall test in the early stage of the project, the dosage of the regenerator is determined to be 14%. The asphalt indicators are shown in Table 1 . Table 1 Basic technical specifications of 90# base asphalt Project Test result Technical requirements 0% 50% 70% Penetration(25℃,100g,5s)×10 − 1/mm 97.0 81.0 80.3 80–100 Softening point/℃ 47.6 50.9 52.0 ≥ 45 Elongation(5cm/min,15℃)/cm Unbroken 120.1 61.7 ≥ 50 The aggregate used in the test is basalt, and the mineral powder is limestone (the fineness ≤ 0.075). The old asphalt content of RAP with different particle sizes was obtained by combustion furnace method, as shown in Table 2 . According to the " Technical Specification for Construction of Asphalt Pavement of Highway Engineering " (JTG F40-2004), the AC − 20 gradation is adopted in the test. The specific design information is shown in Table 3 . (0%, 50%, and 70% in the table represent warm recycled asphalt mixtures with different RAP content.) Table 2 Old asphalt content of aggregates with different particle sizes Aggregate size/mm Aggregate content/% Old asphalt content/% Nominal diameter 10–20 97.8 2.2 Nominal diameter 5–10 97.3 2.7 Nominal diameter 0–5 94.2 5.8 Table 3 Aggregate gradation of AC-20 Sieve size/mm 19 16 13.2 9.5 4.75 2.36 1.18 0.6 0.3 0.15 0.075 Passing rate (%) 0% 100 90.9 71.2 54.8 35.7 29.4 21.9 14.9 10 7.9 6.2 50% 100 92 74.5 56.5 35.9 27.6 22.4 17.2 12 8.7 5.6 70% 100 92.7 76.7 53.3 35.7 27.4 22.2 17 11.9 8.7 5.6 1.2 Test scheme Firstly, three kinds of warm-recycled asphalt mixture specimens with size of ∅150 mm×H170 mm were molded by the Rotary compaction method for preparing asphalt mixture specimens (referred to as SGC) .The method of use is shown in the Fig. 1 . hen, standard specimens with dimensions of 100 mm × H150 mm were cut using a water drill. Standard specimens were subjected to uniaxial compression dynamic modulus tests at four temperatures (5℃, 20℃, 35℃, 50℃) and six frequencies (0.1Hz, 0.5Hz, 1Hz, 5Hz, 10Hz, 25Hz). 2 Result and discussion 2.1 Analysis of dynamic modulus test results 2.1.1 The effect of frequency on dynamic modulus It can be seen from Fig. 2 that as the frequency increases, the dynamic modulus of warm mix recycled asphalt mixture with varying RAP contents increases continuously. This is mainly because under the same temperature conditions, the loading frequency significantly affects the dynamic modulus of warm mix recycled asphalt mixture. As the frequency rises, the load - stress response duration of pavement decreases continuously. As the frequency increases, the strain generated by the specimen decreases and the dynamic modulus increases. In addition, it can be seen from Fig. 2 that the difference in dynamic modulus between high and low frequencies decreases with the increase of dosage under low temperature and moderate and low temperature conditions. Under moderate and high temperature conditions, the difference in dynamic modulus between high and low frequencies increases with the increase of dosage. It shows that under low temperature and moderate and low temperature, the higher the RAP content, the more obvious the influence of frequency on the bearing capacity of pavement. It is mainly because the elasticity of asphalt mixture is prominent at low temperature, and the higher the RAP content, the less significant the influence of frequency on dynamic modulus. Under the condition of medium and high temperature, the higher the RAP content, the lower the influence of frequency on the bearing capacity of the pavement. This is because the viscosity of asphalt mixture is more prominent at medium - high and high temperatures. At such temperatures, the influence of frequency on dynamic modulus is weakened, and the mineral aggregate skeleton plays a leading role. The higher the RAP content, the more prominent the skeleton effect, and the weaker the influence of frequency. 2.1.2 The effect of temperature on dynamic modulus As can be seen from Fig. 3, when the loading frequency is constant, the dynamic modulus of RAP recycled asphalt mixture with different contents decreases with the increase of temperature. This is because the warm mix recycled asphalt mixture is a viscoelastic material. With the increase of temperature, the viscosity of the binder in the asphalt mixture becomes more significant [ 10 , 16 ] , and the viscous flow is easy to occur at high temperature, resulting in the decrease of the bonding force between asphalt and aggregate. Thus, the ability to resist high temperature rutting deformation is weakened, and the dynamic modulus is reduced. Furthermore, as shown in the Fig, at the same temperature, the greater the loading frequency, the higher the dynamic modulus of RAP recycled asphalt mixture with different contents. As the temperature increases, the influence of loading frequency on dynamic modulus diminishes, which is consistent with the previous analysis results. 2.2 Analysis of phase angle test results 2.2.1 The influence of frequency on phase angle From Fig. 4, it can be seen that at lower temperatures (5°C, 20°C), the phase angle decreases with the increase in frequency. This is mainly because at lower temperatures, the asphalt binder determines the viscoelastic properties of the asphalt mixture. As the frequency increases, the deformation response time of the recycled asphalt mixture is shortened, which is manifested by the weakening of its viscosity and the enhancement of elasticity. The higher the frequency is, the less active the polymer chain is. The final performance is that the phase angle gradually decreases with the increase of frequency at lower temperature [ 17 ] . At higher temperatures (35°C, 50°C), the phase angle increases as the frequency increases. This is primarily due to the fact that at higher temperatures, the interlocking effect among the mineral aggregate skeletons determines the viscoelastic properties of the asphalt mixture. As a prominent viscous component of the asphalt mixture, at high temperatures, the asphalt mixture softens, leading to a weakening of the bonding effect among the mineral aggregate skeletons and an increase in the phase angle. The mineral aggregate generally exhibits an elastic state. In the low frequency state, the phase angle of the aggregate with large internal friction resistance is close to zero, and as the frequency increases, the phenomenon of strain lagging behind stress is prominent, resulting in the phase angle increasing as the frequency increases. 2.2.2 The effect of temperature on phase angle It can be seen from Fig. 5 that under low frequency (0.1 Hz, 0.5 Hz, 1 Hz), the phase angle of warm mix recycled asphalt mixture with varying RAP contents first increases and then decreases with the increase of temperature, peaking at 20°C. In the medium and high frequency (5 Hz, 10 Hz, 25 Hz) state, the phase angle exhibits the same variation pattern, peaking at 35°C. This is mainly because in the low frequency state, the asphalt mixture resembles a viscous material. As the temperature gradually increases, the viscosity of the binder in the asphalt mixture gradually decreases, and the viscous component becomes more prominent, causing the phase angle to increase. When the temperature increases to a specific value, the influence of the binder on the mixture can be ignored. In the medium and high frequency state, the viscosity of the asphalt mixture is difficult to reflect, but as the temperature increases, the viscosity of the mixture gradually highlights, and finally shows the same change rule as the low frequency, but the peak value of the phase angle is delayed. 2.3 The effect of RAP content on dynamic modulus and phase angle Table 4 Dynamic modulus increase ratio of 0-50% RAP and 50-70% RAP at different frequencies at low and high temperatures Table 5 0~50%RAP、50~70%RAP recycled asphalt mixture under different frequencies of low and high temperature phase Angle increase, decrease proportion It can be seen from Fig. 6(a) and Table 4 and Table 5 that the dynamic modulus increase ratio of recycled asphalt mixture with 0% ~ 50% RAP is higher than that of asphalt mixture with 50% ~ 70% RAP at low temperature and 6 frequencies. The phase angle reduction ratio of recycled asphalt mixture with 0% ~ 50% RAP is less than that of asphalt mixture with 50% ~ 70% RAP. Under the condition of low temperature, the smaller the dynamic modulus and the larger the phase angle, the more obvious the improvement effect of the low temperature crack resistance of the recycled asphalt mixture. Therefore, the order of low temperature crack resistance of warm mix recycled asphalt mixture is 0% RAP > 50% RAP > 70% RAP for the recycled asphalt mixture. From Fig. 6(b), Table 4 and Table 5 , it can be seen that at high temperature and 6 frequencies, the increase proportion of dynamic modulus of recycled asphalt mixture with 0% ~ 50% RAP is higher than that of 50% ~ 70% RAP asphalt mixture. The phase angle increase ratio of 0% ~ 50% RAP warm-recycled asphalt mixture is less than that of 50% ~ 70% RAP asphalt mixture. At high temperature, the larger the dynamic modulus, the smaller the phase angle, the better the high temperature deformation resistance of recycled asphalt mixture. Therefore, the order of high temperature resistance to permanent deformation performance of warm-recycled asphalt mixture is 70% RAP > 50% RAP > 0% RAP for the recycled asphalt mixture. Considering the high temperature stability and low temperature crack resistance of recycled asphalt mixture, 50% RAP asphalt mixture has better high and low temperature performance. 2.4 principal curve analysis 2.4.1 Basic concept of principal curve According to the principle of time-temperature equivalence, a smooth curve at the reference temperature is established by translating the dynamic modulus and phase angle at different temperatures and frequencies obtained by the dynamic modulus test. This curve is called the main curve of dynamic modulus and phase angle. Using the master curve can not only solve the defects of the instrument and equipment, but also predict the mechanical properties of asphalt mixture in a wider temperature and frequency range [ 18 ] . Based on the nonlinear least squares method, the S [ 9 ] model and the φ model are used for fitting. The S model and the φ model are expressed as : $$\lg \left| {E*\left( {{f_r}} \right)} \right|=\delta +\frac{\alpha }{{{{\left( {1+\lambda {{\exp }^{\beta +\gamma \lg \left( {{f_r}} \right)}}} \right)}^{\frac{1}{\lambda }}}}}$$ 1 $$\Phi \left( {{f_r}} \right)=\rho \frac{\pi }{2}\frac{{d\left( {\lg \left( {E*} \right)} \right)}}{{d\left( {\lg {f_r}} \right)}}= - \frac{\pi }{2}\rho \alpha \gamma \frac{{{{\exp }^{\beta +\gamma \lg {f_r}}}}}{{{{\left( {1+\lambda {{\exp }^{\beta +\gamma \lg {f_r}}}} \right)}^{1+\frac{1}{\lambda }}}}}$$ 2 In the formula : lg|E * (f r )| is the logarithm of dynamic modulus ; δ is the minimum value of dynamic modulus ; α is the difference between the maximum and minimum values of dynamic modulus ; β and γ are the fitting parameters in the S model, which are related to the material properties of asphalt mixture. λ determines the asymmetric characteristics of sigmoidal model ; f r is reduced frequency ( H Z ) ; Φ(f r ) is the main curve form of phase angle ; ρ is an improved fitting parameter. The relationship between the reduced frequency f r , the loading frequency f , and the shift factor α t is expressed as : $${f_r}=f * {\alpha _t}$$ 3 In the formula : α t is the shift factor. α t can be solved by the WLF equation, and the specific equation is expressed as : $$\lg {\alpha _t}=\frac{{ - {C_1}\left( {t - {t_0}} \right)}}{{{C_2}+\left( {t - {t_0}} \right)}}$$ 4 In the formula : C 1 and C 2 are the fitting parameters, t 0 is the reference temperature of the main curve fitting, and the reference temperature is selected as 20°C. Substituting Formulas ( 3 ) and ( 4 ) into Formulas ( 1 ) and ( 2 ), the expressions of dynamic modulus and phase angle are obtained, which are expressed as : $$\lg \left| {{E^ * }_{{\left( f \right)}}} \right|=\delta +\frac{\alpha }{{{{\left[ {1+\lambda {{\exp }^{\beta +\gamma \lg f+\gamma \frac{{ - {C_1}\left( {t - {t_0}} \right)}}{{{C_2}+\left( {t - {t_0}} \right)}}}}} \right]}^{\frac{1}{\lambda }}}}}$$ 5 $${\Phi _{\left( f \right)}}= - \frac{\pi }{2}\rho \alpha \gamma \frac{{{{\exp }^{\beta +\gamma \lg f+\gamma \frac{{ - {C_1}\left( {t - {t_0}} \right)}}{{{C_2}+\left( {t - {t_0}} \right)}}}}}}{{{{\left[ {1+\lambda {{\exp }^{^{{\beta +\gamma \lg f+\gamma \frac{{ - {C_1}\left( {t - {t_0}} \right)}}{{{C_2}+\left( {t - {t_0}} \right)}}}}}}} \right]}^{1+\frac{1}{\lambda }}}}}$$ 6 Formulas ( 5 ) and ( 6 ) need to fit eight test parameters of δ, ρ, α, β, γ, λ, C 1 and C 2 . In order to improve the accuracy of parameter fitting, Python is used to synchronously fit Formulas ( 5 ) and ( 6 ) to realize parameter sharing. The fitting results are shown in Table 6 . Table 6 Main curve fitting results of dynamic modulus and phase angle parameter 0%RAP 50%RAP 70%RAP δ 2.459 0 2.561 3 2.366 6 α 1.880 3 1.838 8 2.113 9 λ 0.941 6 0.498 4 0.372 7 β -0.321 6 -0.586 5 -0.830 6 γ -0.902 0 -0.653 5 -0.491 2 C 1 20.416 9 20.486 2 29.500 9 C 2 192.696 2 163.916 2 230.024 5 ρ 4.162 0 56.498 7 57.761 5 R 2 0.999 0.999 0.999 2.4.2 Analysis of dynamic modulus master curve It can be seen from Fig. 7(a) that the dynamic modulus of warm mix recycled asphalt mixtures with different RAP content increases in an S - shaped pattern as the loading frequency increases. This is because the warm mix recycled asphalt mixture is mainly elastic at high frequency (Low Temperature) and mainly viscous at low frequency (High Temperature). Therefore, when the reduction frequency changes from low frequency to high frequency, the dynamic modulus of recycled asphalt mixture increases continuously. Under high temperature and low frequency, the dynamic modulus of the two types of asphalt mixtures after warm mix regeneration is higher than that of hot mix asphalt mixture; under low temperature and high frequency, the dynamic modulus of the two types of asphalt mixtures after warm mix regeneration is not significantly different from that of hot mix asphalt mixture, which indicates that the elasticity of warm mix recycled asphalt mixture after warm mix regeneration is still higher than that of hot mix asphalt mixture at high temperature, and there is no significant difference between warm mix recycled asphalt mixture and hot mix asphalt mixture at low temperature. This is due to the high temperature state, the main factors affecting the dynamic modulus are the skeleton embedding and the adhesion of the aggregate and the asphalt. After the warm mix regeneration, the addition of the modifier reduces the viscosity of the asphalt, so that the asphalt can be better spread on the aggregate. At the same time, it also promotes the asphalt into the tiny voids on the surface of the stone, enhances the adhesion performance of the asphalt and the aggregate, and improves the dynamic modulus of the asphalt mixture. Under the condition of low temperature, the main factor affecting the dynamic modulus is the improvement effect of the regenerant on the old asphalt. After warm mix regeneration, on the one hand, it can reduce the secondary aging of asphalt in the process of mixing and compaction, promote the integration of new and old asphalt interfaces, and enhance the adhesion effect between asphalt and aggregate. On the other hand, the regenerant plays a solubilizing role in the old asphalt, reduces the interfacial tension between the light component and the heavy component, and improves the mutual solubility of the two, so that the proportion of light and heavy components in the warm-recycled asphalt mixture is smaller than that of the hot mix asphalt mixture, so the dynamic modulus of the warm-recycled asphalt mixture at low temperature is not significantly different from that of the hot mix asphalt mixture. 2.4.3 Analysis of main curve of phase angle It can be seen from Fig. 7(b) that the main curve of the phase angle of the warm mix recycled asphalt mixture with different RAP content is bell-shaped. With the increase of the reduction frequency, the phase angle first increases to the peak and then decreases to zero. When the reduction frequency tends to zero or infinity, the phase angle tends to zero, indicating that the warm mix recycled asphalt mixture is elastic at higher and lower frequencies. This is because the elastic component of the warm-mix recycled asphalt mixture is prominent at low temperature. When the temperature continues to rise, the binder in the recycled asphalt mixture softens. At this point, the viscous component is prominent and the phase angle increases. In addition, it can be seen from Fig. 7(b) that the peak value of the main curve of the phase angle of the different content of RAP asphalt mixture gradually moves to the left with the increase of the content. The lower the RAP content, the higher the peak value of the main curve of the phase angle. It shows that the temperature and viscoelastic ratio of asphalt mixture with different RAP content are not the same when they are close to their ultimate viscous state. The main reason is that the higher the RAP content, the more the old asphalt under the aging effect, and the ability to resist high temperature deformation is enhanced. The recycled asphalt mixture with different RAP content has different resistance to high temperature deformation. In addition, from the perspective of the influence of the change of phase angle on the high and low temperature performance of asphalt pavement, better elasticity is needed to resist pavement deformation at high temperature, and better viscosity is needed to resist pavement cracking at low temperature. When the RAP content is 50%, the phase angle in low frequency and high frequency is in a balanced state. At this time, the recycled asphalt mixture not only has good high temperature deformation resistance, but also has good low temperature crack resistance. Therefore, considering comprehensively, the high and low temperature performance of asphalt mixture with 50% RAP is the best, which is consistent with the previous analysis results. 2.4 principal curve analysis 2.4.1 Basic concept of principal curve According to the principle of time-temperature equivalence, a smooth curve at the reference temperature is established by translating the dynamic modulus and phase angle at different temperatures and frequencies obtained by the dynamic modulus test. This curve is called the main curve of dynamic modulus and phase angle. Using the master curve can not only solve the defects of the instrument and equipment, but also predict the mechanical properties of asphalt mixture in a wider temperature and frequency range [ 18 ] . Based on the nonlinear least squares method, the S [ 9 ] model and the φ model are used for fitting. The S model and the φ model are expressed as : $$\lg \left| {E*\left( {{f_r}} \right)} \right|=\delta +\frac{\alpha }{{{{\left( {1+\lambda {{\exp }^{\beta +\gamma \lg \left( {{f_r}} \right)}}} \right)}^{\frac{1}{\lambda }}}}}$$ 1 $$\Phi \left( {{f_r}} \right)=\rho \frac{\pi }{2}\frac{{d\left( {\lg \left( {E*} \right)} \right)}}{{d\left( {\lg {f_r}} \right)}}= - \frac{\pi }{2}\rho \alpha \gamma \frac{{{{\exp }^{\beta +\gamma \lg {f_r}}}}}{{{{\left( {1+\lambda {{\exp }^{\beta +\gamma \lg {f_r}}}} \right)}^{1+\frac{1}{\lambda }}}}}$$ 2 In the formula : lg|E * (f r )| is the logarithm of dynamic modulus ; δ is the minimum value of dynamic modulus ; α is the difference between the maximum and minimum values of dynamic modulus ; β and γ are the fitting parameters in the S model, which are related to the material properties of asphalt mixture. λ determines the asymmetric characteristics of sigmoidal model ; f r is reduced frequency ( H Z ) ; Φ(f r ) is the main curve form of phase angle ; ρ is an improved fitting parameter. The relationship between the reduced frequency f r , the loading frequency f , and the shift factor α t is expressed as : $${f_r}=f * {\alpha _t}$$ 3 In the formula : α t is the shift factor. α t can be solved by the WLF equation, and the specific equation is expressed as : $$\lg {\alpha _t}=\frac{{ - {C_1}\left( {t - {t_0}} \right)}}{{{C_2}+\left( {t - {t_0}} \right)}}$$ 4 In the formula : C 1 and C 2 are the fitting parameters, t 0 is the reference temperature of the main curve fitting, and the reference temperature is selected as 20°C. Substituting Formulas ( 3 ) and ( 4 ) into Formulas ( 1 ) and ( 2 ), the expressions of dynamic modulus and phase angle are obtained, which are expressed as : $$\lg \left| {{E^ * }_{{\left( f \right)}}} \right|=\delta +\frac{\alpha }{{{{\left[ {1+\lambda {{\exp }^{\beta +\gamma \lg f+\gamma \frac{{ - {C_1}\left( {t - {t_0}} \right)}}{{{C_2}+\left( {t - {t_0}} \right)}}}}} \right]}^{\frac{1}{\lambda }}}}}$$ 5 $${\Phi _{\left( f \right)}}= - \frac{\pi }{2}\rho \alpha \gamma \frac{{{{\exp }^{\beta +\gamma \lg f+\gamma \frac{{ - {C_1}\left( {t - {t_0}} \right)}}{{{C_2}+\left( {t - {t_0}} \right)}}}}}}{{{{\left[ {1+\lambda {{\exp }^{^{{\beta +\gamma \lg f+\gamma \frac{{ - {C_1}\left( {t - {t_0}} \right)}}{{{C_2}+\left( {t - {t_0}} \right)}}}}}}} \right]}^{1+\frac{1}{\lambda }}}}}$$ 6 Formulas ( 5 ) and ( 6 ) need to fit eight test parameters of δ, ρ, α, β, γ, λ, C 1 and C 2 . In order to improve the accuracy of parameter fitting, Python is used to synchronously fit Formulas ( 5 ) and ( 6 ) to realize parameter sharing. The fitting results are shown in Table 6 . Table 6 Main curve fitting results of dynamic modulus and phase angle parameter 0%RAP 50%RAP 70%RAP δ 2.459 0 2.561 3 2.366 6 α 1.880 3 1.838 8 2.113 9 λ 0.941 6 0.498 4 0.372 7 β -0.321 6 -0.586 5 -0.830 6 γ -0.902 0 -0.653 5 -0.491 2 C 1 20.416 9 20.486 2 29.500 9 C 2 192.696 2 163.916 2 230.024 5 ρ 4.162 0 56.498 7 57.761 5 R 2 0.999 0.999 0.999 2.4.2 Analysis of dynamic modulus master curve It can be seen from Fig. 7(a) that the dynamic modulus of warm mix recycled asphalt mixtures with different RAP content increases in an S - shaped pattern as the loading frequency increases. This is because the warm mix recycled asphalt mixture is mainly elastic at high frequency (Low Temperature) and mainly viscous at low frequency (High Temperature). Therefore, when the reduction frequency changes from low frequency to high frequency, the dynamic modulus of recycled asphalt mixture increases continuously. Under high temperature and low frequency, the dynamic modulus of the two types of asphalt mixtures after warm mix regeneration is higher than that of hot mix asphalt mixture; under low temperature and high frequency, the dynamic modulus of the two types of asphalt mixtures after warm mix regeneration is not significantly different from that of hot mix asphalt mixture, which indicates that the elasticity of warm mix recycled asphalt mixture after warm mix regeneration is still higher than that of hot mix asphalt mixture at high temperature, and there is no significant difference between warm mix recycled asphalt mixture and hot mix asphalt mixture at low temperature. This is due to the high temperature state, the main factors affecting the dynamic modulus are the skeleton embedding and the adhesion of the aggregate and the asphalt. After the warm mix regeneration, the addition of the modifier reduces the viscosity of the asphalt, so that the asphalt can be better spread on the aggregate. At the same time, it also promotes the asphalt into the tiny voids on the surface of the stone, enhances the adhesion performance of the asphalt and the aggregate, and improves the dynamic modulus of the asphalt mixture. Under the condition of low temperature, the main factor affecting the dynamic modulus is the improvement effect of the regenerant on the old asphalt. After warm mix regeneration, on the one hand, it can reduce the secondary aging of asphalt in the process of mixing and compaction, promote the integration of new and old asphalt interfaces, and enhance the adhesion effect between asphalt and aggregate. On the other hand, the regenerant plays a solubilizing role in the old asphalt, reduces the interfacial tension between the light component and the heavy component, and improves the mutual solubility of the two, so that the proportion of light and heavy components in the warm-recycled asphalt mixture is smaller than that of the hot mix asphalt mixture, so the dynamic modulus of the warm-recycled asphalt mixture at low temperature is not significantly different from that of the hot mix asphalt mixture. 2.4.3 Analysis of main curve of phase angle It can be seen from Fig. 7(b) that the main curve of the phase angle of the warm mix recycled asphalt mixture with different RAP content is bell-shaped. With the increase of the reduction frequency, the phase angle first increases to the peak and then decreases to zero. When the reduction frequency tends to zero or infinity, the phase angle tends to zero, indicating that the warm mix recycled asphalt mixture is elastic at higher and lower frequencies. This is because the elastic component of the warm-mix recycled asphalt mixture is prominent at low temperature. When the temperature continues to rise, the binder in the recycled asphalt mixture softens. At this point, the viscous component is prominent and the phase angle increases. In addition, it can be seen from Fig. 7(b) that the peak value of the main curve of the phase angle of the different content of RAP asphalt mixture gradually moves to the left with the increase of the content. The lower the RAP content, the higher the peak value of the main curve of the phase angle. It shows that the temperature and viscoelastic ratio of asphalt mixture with different RAP content are not the same when they are close to their ultimate viscous state. The main reason is that the higher the RAP content, the more the old asphalt under the aging effect, and the ability to resist high temperature deformation is enhanced. The recycled asphalt mixture with different RAP content has different resistance to high temperature deformation. In addition, from the perspective of the influence of the change of phase angle on the high and low temperature performance of asphalt pavement, better elasticity is needed to resist pavement deformation at high temperature, and better viscosity is needed to resist pavement cracking at low temperature. When the RAP content is 50%, the phase angle in low frequency and high frequency is in a balanced state. At this time, the recycled asphalt mixture not only has good high temperature deformation resistance, but also has good low temperature crack resistance. Therefore, considering comprehensively, the high and low temperature performance of asphalt mixture with 50% RAP is the best, which is consistent with the previous analysis results. 3 Conclusion (1) Under the condition of high temperature and low frequency, the dynamic modulus of the two kinds of asphalt mixture after warm mix regeneration is higher than that of hot mix asphalt mixture. Under the condition of low temperature and high frequency, the dynamic modulus of the two kinds of asphalt mixture after warm mix regeneration is not significantly different from that of hot mix asphalt mixture, indicating that the elasticity of asphalt mixture after warm mix regeneration is still higher than that of hot mix asphalt mixture at high temperature, and there is no significant difference between flexible and hot mix asphalt mixture at low temperature. (2) Compared with the warm-recycled asphalt mixture with 70% RAP content, when the RAP content is 50%, the dynamic modulus increases the most at high temperature (50°C), and the phase angle decreases the least at low temperature (5°C). Therefore, the warm-recycled asphalt mixture with 50% RAP has better high and low temperature performance. (3) The fitting effect of S and φ models on the main curve of dynamic modulus and phase angle is good. The fitting results show that the dynamic modulus of warm-recycled asphalt mixture with different RAP content increases in an S -shaped manner with the increase of reduction frequency, and the phase angle shows a bell-shaped change rule with the increase of reduction frequency. Declarations This study was funded by the Inner Mongolia Autonomous Region Natural Science Foundation Project(2025LHMS05037); Basic Scientific Research Business Fee Items of Universities in Inner Mongolia Autonomous Region(ZTY2025066). This manuscript has not been published or presented elsewhere in part or in entirety. All the authors have approved the manuscript and agree with submission to your esteemed journal. Consent to Participate declaration not applicable. Author Contribution F.L. and W.J. conceived the research and designed experiments. W.L. and S.C. performed laboratory tests and data curation. L.T. developed analytical models and validation. F.L. and W.J. wrote the main manuscript text. W.L. prepared figures and visualizations. S.C. and L.T. conducted formal analysis. All authors reviewed and approved the final manuscript. Data Availability Statement Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request. References ZHOU Zihao. Study on influencing factors of Dynamic modulus of Asphalt Mixture under Uniaxial Compression mode. Master’s thesis,Changsha University of Science and Technology, china,2017.DOI:10.26985/d.cnki.gcsjc.2017.000004.（in Chinese）周梓豪.单轴压缩模式下沥青混合料动态模量影响因素研究. 硕士学位论文,长沙理工大学,2017.DOI:10.26985/d.cnki.gcsjc.2017.000004. HARNAENI S R, PRAMESTI F P, BUDIARTO A, et al. Study on structural performance of asphalt concrete and hot rolled sheet through viscoelastic characterization[J]. Materials, 2020, 13(5). DOI:10.3390/ma13051133. LI C, NING F, LI Y. Effect of carbon black on the dynamic moduli of asphalt mixtures and its master curves[J]. Frontiers of Structural and Civil Engineering, 2019, 13(4): 918-925. DOI:10.1007/s11709-019-0526-6. XI L, LUO R, LIU H. Effect of relative humidity on the linear viscoelastic properties of asphalt mixtures[J]. Construction and Building Materials, 2021, 307. DOI:10.1016/j.conbuildmat.2021.124956. ZHENG K, XU J, WANG J. Viscoelasticity of Recycled Asphalt Mixtures with High Content Reclaimed SBS Modified Asphalt Pavement[J]. SUSTAINABILITY, 2023, 15(3): 2515. DOI:10.3390/su15032515. BARRAJ F, ELKORDI A. Assessment of dynamic modulus and prediction of phase angle of conventional and reclaimed asphalt mixtures incorporating different rap contents[J]. 2022. ZHANG F, WANG L, LI C, et al. Predict the phase angle master curve and study the viscoelastic properties of warm mix crumb rubber-modified asphalt mixture[J]. Materials, 2020, 13(21): 1-26. DOI:10.3390/ma13215051. Gong Guan yu. Study on Healing Characteristic of Asphalt Binder and Asphalt Mastic Based on Different Loading Modes. [D]. Beijing University Of Technology, china, 2024. DOI:10.26935/d.cnki.gbjgu.2022.000173. （in Chinese）宫官雨. 基于不同加载方式的沥青与沥青胶浆损伤自愈特性研究[D]. 北京工业大学, 2024. DOI:10.26935/d.cnki.gbjgu.2022.000173. LI Bo. Variable-order Fractional order viscoelastic Constitutive Model of Asphalt Mixture and its Application. Master’s thesis , Xiangtan University, china,2020.DOI:10.27426/d.cnki.gxtdu.2020.000509.（in Chinese）黎博.沥青混合料的变阶分数阶黏弹性本构模型与应用.硕士学位论文,湘潭大学,2020.DOI:10.27426/d.cnki.gxtdu.2020.000509. BELHAJ M, VALENTIN J, BALDO N. Accuracy of dynamic modulus models of asphalt mixtures containing reclaimed asphalt (RA)[J]. Applied Sciences, 2024, 14(22): 10505. DOI:10.3390/app142210505. ACHARJEE P K, SOULIMAN M I, KHALIFAH R, et al. Frequency- and temperature-dependent dynamic shear modulus and phase angle prediction models based on existing asphalt binder viscosity data using Artificial Neural Network (ANN)[J]. Construction and Building Materials, 2024, 414. DOI:10.1016/j.conbuildmat.2023.134772. JUKTE N R, SWAMY A K. Composite sigmoidal model for asphalt concrete phase angle mastercurve construction[J]. International Journal of Pavement Engineering, 2024, 25(1). DOI:10.1080/10298436.2024.2420249. LAN Jianli, GAO Xuekai, KONG Fansheng. Research on dynamic Viscoelastic Properties of Thermally-recycled Asphalt Mixture Based on CAM Model [J]. Silicate Bulletin, 2021,40(07):2454-2460.DOI:10.16552/j.cnki.issn1001-1625.2021.07.012.（in Chinese）兰建丽,高学凯,孔繁盛.基于CAM模型的热再生沥青混合料动态粘弹特性研究[J].硅酸盐通报,2021,40(07):2454-2460.DOI:10.16552/j.cnki.issn1001-1625.2021.07.012. LI Xiujun, ZHAO Linhao, OU Yanghuan, et al. Effect of water on high temperature stability of Asphalt Mixture measured by vacuum saturation [J]. Journal of Building Material, 2022, 25(6): 607-612（in Chinese）李秀君,赵麟昊,欧阳欢,等.真空饱水衡量水对沥青混合料高温稳定性影响[J].建筑材料学报, 2022, 25(6): 607-612. ZHANG Huaizhi, WANG Di, YANG Yanhai. Research on Discriminant Degree of Asphalt Mixture High Temperature Performance Evaluation Index [J]. Journal of Building Materials,2021,24(06):1248-1254. （in Chinese）张怀志,王迪,杨彦海.沥青混合料高温性能评价指标区分度研究[J].建筑材料学报,2021,24(06):1248-1254. ZHANG P, OUYANG L, YANG L, et al. Laboratory investigation of carbon black/bio-oil composite modified asphalt[J]. Materials, 2021, 14(17). DOI:10.3390/ma14174910. LIU Chao. Rheological properties of nano-montmorillonite (OMMT)/SBS modified asphalt. Master's thesis, Dalian Maritime University, China, 2016.（in Chinese）刘超.纳米蒙脱土(OMMT)/SBS改性沥青流变特性研究.硕士学位论文,大连海事大学,2016. MA L, WANG H, MA Y. Viscoelasticity of Asphalt Mixture Based on the Dynamic Modulus Test[J]. Journal of Materials in Civil Engineering, 2024, 36(3). DOI:10.1061/JMCEE7.MTENG-17124. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Revision Version 1 posted Editorial decision: Revision requested 27 Oct, 2025 Reviews received at journal 02 Aug, 2025 Reviews received at journal 29 Jul, 2025 Reviews received at journal 27 Jul, 2025 Reviewers agreed at journal 27 Jul, 2025 Reviewers agreed at journal 22 Jul, 2025 Reviewers agreed at journal 22 Jul, 2025 Reviews received at journal 22 Jul, 2025 Reviewers agreed at journal 22 Jul, 2025 Reviewers invited by journal 22 Jul, 2025 Editor assigned by journal 25 Jun, 2025 Submission checks completed at journal 25 Jun, 2025 First submitted to journal 17 Jun, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6912958","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":489437500,"identity":"10dba025-3bc5-4af2-9fe9-55d4f3051787","order_by":0,"name":"Lei Feng","email":"","orcid":"","institution":"Inner Mongolia University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Lei","middleName":"","lastName":"Feng","suffix":""},{"id":489437501,"identity":"a5d0b4b2-dbbd-4a08-aa4e-195aeba65f51","order_by":1,"name":"WU Jianing","email":"","orcid":"","institution":"Inner Mongolia University of Technology","correspondingAuthor":false,"prefix":"","firstName":"WU","middleName":"","lastName":"Jianing","suffix":""},{"id":489437504,"identity":"672497d7-ca7a-4edf-8419-e225c3974428","order_by":2,"name":"Lan Wang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCklEQVRIie3QMUvEMBTA8ZRCXKK3vkOxX+FJIDgc6EdJl97SE8cOgrqki3irLn4GJ+eEwLlEXQ90KBTE4Ya6yE2H2aWpo2B+Qwjh/QMJIVH0VzWE7NOteWO6Cvym1sOJJISPmEtacBO+w5z8VZLf3pQpbqsiv4NjDE7j45P9kGdvyb2eLQConSrwV6yrh/7EnRSHcvGeon4u4JTZmdq91MmVe+1NhC4FSmopmgsBAD7Z0zJNVCB5WflkYxlaIsCvUwoSw8my5E2uLIwV48hkIQeTo+VKkPza4ojRvAU9OVD+k03oLeN5ybvuy56rrNXmcwNZVtemWVf9iUfhx5EOzXtpNzAQRVH0330DoUxgpoiCeB4AAAAASUVORK5CYII=","orcid":"","institution":"Inner Mongolia University of Technology","correspondingAuthor":true,"prefix":"","firstName":"Lan","middleName":"","lastName":"Wang","suffix":""},{"id":489437506,"identity":"07b0fd00-3ee8-4e6b-9ffd-f26a333bcace","order_by":3,"name":"Chuanyu Shao","email":"","orcid":"","institution":"Inner Mongolia University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Chuanyu","middleName":"","lastName":"Shao","suffix":""},{"id":489437508,"identity":"a10c730b-2c5b-4ada-8511-359f346547e5","order_by":4,"name":"Tao Li","email":"","orcid":"","institution":"Inner Mongolia University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Tao","middleName":"","lastName":"Li","suffix":""}],"badges":[],"createdAt":"2025-06-17 09:53:14","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6912958/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6912958/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":87504402,"identity":"29284f25-ad38-4b58-a596-a64d875ef231","added_by":"auto","created_at":"2025-07-24 14:26:11","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":57447,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of the use of SGC\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-6912958/v1/bb0fd8182a59e6d3d0d3deaf.png"},{"id":87504404,"identity":"03732235-2a6a-437f-ab04-c1936a8d2685","added_by":"auto","created_at":"2025-07-24 14:26:12","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":29793,"visible":true,"origin":"","legend":"\u003cp\u003eDynamic modulus curves of recycled asphalt mixture at different frequencies\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-6912958/v1/e27df3cd816103970a1b5dbd.png"},{"id":87504806,"identity":"2c36856a-eab9-4485-a9ce-00121689fa5b","added_by":"auto","created_at":"2025-07-24 14:34:12","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":32529,"visible":true,"origin":"","legend":"\u003cp\u003eDynamic modulus curves of recycled asphalt mixture at different temperatures\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-6912958/v1/f02098637f48da2d18beb90e.png"},{"id":87504405,"identity":"fbaa3649-f0a3-4c12-a04f-d33aaa2a5d0c","added_by":"auto","created_at":"2025-07-24 14:26:12","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":34468,"visible":true,"origin":"","legend":"\u003cp\u003ePhase angle curves of recycled asphalt mixture at different frequencies\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-6912958/v1/a13f245a6ab2f9b9dfa8b405.png"},{"id":87504406,"identity":"8b23b5e9-89ac-4556-91df-63cd0093456c","added_by":"auto","created_at":"2025-07-24 14:26:12","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":41516,"visible":true,"origin":"","legend":"\u003cp\u003ePhase angle curves of recycled asphalt mixture at different temperatures\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-6912958/v1/9694fb51b361e6e5fb93a052.png"},{"id":87504409,"identity":"77fb3ab4-6da3-4bd9-b11d-8aa27a3675e8","added_by":"auto","created_at":"2025-07-24 14:26:12","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":36221,"visible":true,"origin":"","legend":"\u003cp\u003eDynamic modulus,phase angle and frequency curves of warm mixed recycled asphalt mixture at low temperature and high temperature\u003c/p\u003e\n\u003cp\u003eNote: ' DM ' is the dynamic modulus; ' PA ' is the phase Angle (same as later).\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-6912958/v1/b403d2caf59a9a5f19a787a3.png"},{"id":87504810,"identity":"3cd75454-1fd4-4166-911f-883d69bc9edf","added_by":"auto","created_at":"2025-07-24 14:34:12","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":37711,"visible":true,"origin":"","legend":"\u003cp\u003eMain curve of dynamic modulus and phase Angle of warm-mix recycled asphalt mixture\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-6912958/v1/1f6e8d7b21f8195c05fd5a40.png"},{"id":88505166,"identity":"952a786d-3d1c-4fd7-869c-1122d1962bff","added_by":"auto","created_at":"2025-08-07 07:19:34","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1088797,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6912958/v1/9bf8b07b-2968-4ab2-bcd6-03e040876643.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Viscoelastic properties of warm recycled asphalt mixture based on S-φ model","fulltext":[{"header":"0 Introduction","content":"\u003cp\u003eThere are many factors influencing the dynamic modulus and phase angle. Temperature, frequency as well as RAP content have a great influence on the dynamic modulus and phase angle of warm - mixed recycled asphalt mixture. Compared with static modulus, dynamic modulus can more accurately reflect the mechanical behavior of asphalt pavement under the changes of vehicle load\u003csup\u003e[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eLi, Harnaeni, Xi, Zheng along with Barraj et al.\u003csup\u003e[\u003cspan additionalcitationids=\"CR3 CR4 CR5\" citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]\u003c/sup\u003e explored the influence of temperature and frequency on asphalt mixture via dynamic modulus tests. The results show that the dynamic modulus of asphalt mixture decreases as temperature increases and as loading frequency decreases. The phase angle exhibits an opposite trend. Zhang, Gong along with Li et al.\u003csup\u003e[\u003cspan additionalcitationids=\"CR8\" citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]\u003c/sup\u003e established the master curves of dynamic modulus and phase angle of crumb rubber modified asphalt mixture using the generalized Sigmund model (GSM) and the phase angle model (DEM). The results show that GSM and DEM are capable of accurately predicting the variation of dynamic modulus and phase angle with loading frequency. Ma, Acharjee along with Jukte et al.\u003csup\u003e[\u003cspan additionalcitationids=\"CR11\" citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]\u003c/sup\u003e developed a prediction model based on dynamic shear modulus (|G*|) and phase angle (δ) utilizing the dynamic modulus master curve, Kramers-Kronig relationship, artificial neural network (ANN) and other methods. Lan\u003csup\u003e[\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]\u003c/sup\u003e studied the variation of dynamic modulus and phase angle of recycled asphalt mixture with loading frequency using the CAM model (an improved model for dynamic modulus and phase angle). The results show that the dynamic modulus and phase angle master curve fitted by the improved CAM model exhibit high precision and strong regularity.\u003c/p\u003e\u003cp\u003eCurrently, many scholars have studied the high temperature viscoelastic properties of asphalt mixture\u003csup\u003e[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]\u003c/sup\u003e. However, few studies have been conducted on the viscoelastic properties of warm mix recycled asphalt mixture using a more accurate viscoelastic model. Although some scholars have employed relevant models for research, many dynamic modulus and phase angle viscoelastic models fail to satisfy the same shift factor and the accurate Kramers-Kronig (K-K) relationship among viscoelastic functions, and cannot accurately ensure the model's accuracy. In this paper, the S and φ models effectively overcome this defect. Therefore, based on the S and φ models, this paper will study the dynamic viscoelastic properties of warm mix recycled asphalt mixture through dynamic modulus tests. Compared with hot mix recycled asphalt mixture, warm mix recycled asphalt mixture can reduce the secondary aging of asphalt during mixing and compaction, facilitate the integration of the new and old asphalt interfaces, strengthen the adhesion between asphalt and aggregates, and improve the road performance of warm mix recycled asphalt mixture. Therefore, it is of great guiding importance to study the viscoelastic properties of warm mix recycled asphalt mixture using an accurate viscoelastic model for the development of warm mix recycled pavement.\u003c/p\u003e"},{"header":"1 Experiment","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\u003ch2\u003e1.1 Raw material\u003c/h2\u003e\u003cp\u003eThe base asphalt used in the experiment is Panjin \u003cb\u003e90#\u003c/b\u003e asphalt. The modifier is the SMC ambient temperature regenerator produced by Beijing Sierma Company. Based on the manufacturer's recommendations and the stability, flow value and other indicators obtained from the Marshall test in the early stage of the project, the dosage of the regenerator is determined to be 14%. The asphalt indicators are shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eBasic technical specifications of 90# base asphalt\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"5\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u003cp\u003eProject\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colspan=\"3\" nameend=\"c4\" namest=\"c2\"\u003e\u003cp\u003eTest result\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\" morerows=\"1\" rowspan=\"2\"\u003e\u003cp\u003eTechnical requirements\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0%\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003e50%\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003e70%\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ePenetration(25℃,100g,5s)×10 − 1/mm\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e97.0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e81.0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e80.3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e80–100\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSoftening point/℃\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e47.6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e50.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e52.0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e≥ 45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eElongation(5cm/min,15℃)/cm\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eUnbroken\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e120.1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e61.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e≥ 50\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe aggregate used in the test is basalt, and the mineral powder is limestone (the fineness ≤ 0.075). The old asphalt content of RAP with different particle sizes was obtained by combustion furnace method, as shown in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. According to the \" Technical Specification for Construction of Asphalt Pavement of Highway Engineering \" (JTG F40-2004), the AC − 20 gradation is adopted in the test. The specific design information is shown in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. (0%, 50%, and 70% in the table represent warm recycled asphalt mixtures with different RAP content.)\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eOld asphalt content of aggregates with different particle sizes\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"3\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eAggregate size/mm\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eAggregate content/%\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eOld asphalt content/%\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eNominal diameter 10–20\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e97.8\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e2.2\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eNominal diameter 5–10\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e97.3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e2.7\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eNominal diameter 0–5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e94.2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e5.8\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c11\" colnum=\"11\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c12\" colnum=\"12\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c13\" colnum=\"13\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eAggregate gradation of AC-20\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"13\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e\u003cp\u003eSieve size/mm\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003e19\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003e16\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003e13.2\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003e9.5\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e\u003cp\u003e4.75\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c8\"\u003e\u003cp\u003e2.36\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c9\"\u003e\u003cp\u003e1.18\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c10\"\u003e\u003cp\u003e0.6\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c11\"\u003e\u003cp\u003e0.3\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c12\"\u003e\u003cp\u003e0.15\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c13\"\u003e\u003cp\u003e0.075\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\" morerows=\"2\" rowspan=\"3\"\u003e\u003cp\u003ePassing\u003c/p\u003e\u003cp\u003erate\u003c/p\u003e\u003cp\u003e(%)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e100\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e90.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e71.2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e54.8\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e35.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e29.4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e21.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e14.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c11\"\u003e\u003cp\u003e10\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e\u003cp\u003e7.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e\u003cp\u003e6.2\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e50%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e100\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e92\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e74.5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e56.5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e35.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e27.6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e22.4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e17.2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c11\"\u003e\u003cp\u003e12\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e\u003cp\u003e8.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e\u003cp\u003e5.6\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e70%\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e100\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e92.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e76.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e53.3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e35.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e27.4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e22.2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c10\"\u003e\u003cp\u003e17\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c11\"\u003e\u003cp\u003e11.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e\u003cp\u003e8.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e\u003cp\u003e5.6\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\n\u003ch3\u003e1.2 Test scheme\u003c/h3\u003e\n\u003cp\u003eFirstly, three kinds of warm-recycled asphalt mixture specimens with size of ∅150 mm×H170 mm were molded by the Rotary compaction method for preparing asphalt mixture specimens (referred to as SGC) .The method of use is shown in the Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. hen, standard specimens with dimensions of 100 mm × H150 mm were cut using a water drill. Standard specimens were subjected to uniaxial compression dynamic modulus tests at four temperatures (5℃, 20℃, 35℃, 50℃) and six frequencies (0.1Hz, 0.5Hz, 1Hz, 5Hz, 10Hz, 25Hz).\u003c/p\u003e"},{"header":"2 Result and discussion","content":"\u003ch2\u003e2.1 Analysis of dynamic modulus test results\u003c/h2\u003e\u003cp\u003e2.1.1 \u003cb\u003eThe effect of frequency on dynamic modulus\u003c/b\u003e\u003c/p\u003e\u003cp\u003eIt can be seen from Fig.\u0026nbsp;2 that as the frequency increases, the dynamic modulus of warm mix recycled asphalt mixture with varying RAP contents increases continuously. This is mainly because under the same temperature conditions, the loading frequency significantly affects the dynamic modulus of warm mix recycled asphalt mixture. As the frequency rises, the load - stress response duration of pavement decreases continuously. As the frequency increases, the strain generated by the specimen decreases and the dynamic modulus increases.\u003c/p\u003e\u003cp\u003eIn addition, it can be seen from Fig.\u0026nbsp;2 that the difference in dynamic modulus between high and low frequencies decreases with the increase of dosage under low temperature and moderate and low temperature conditions. Under moderate and high temperature conditions, the difference in dynamic modulus between high and low frequencies increases with the increase of dosage. It shows that under low temperature and moderate and low temperature, the higher the RAP content, the more obvious the influence of frequency on the bearing capacity of pavement. It is mainly because the elasticity of asphalt mixture is prominent at low temperature, and the higher the RAP content, the less significant the influence of frequency on dynamic modulus.\u003c/p\u003e\u003cp\u003eUnder the condition of medium and high temperature, the higher the RAP content, the lower the influence of frequency on the bearing capacity of the pavement. This is because the viscosity of asphalt mixture is more prominent at medium - high and high temperatures. At such temperatures, the influence of frequency on dynamic modulus is weakened, and the mineral aggregate skeleton plays a leading role. The higher the RAP content, the more prominent the skeleton effect, and the weaker the influence of frequency.\u003c/p\u003e\u003cp\u003e2.1.2 The effect of temperature on dynamic modulus\u003c/p\u003e\u003cp\u003eAs can be seen from Fig.\u0026nbsp;3, when the loading frequency is constant, the dynamic modulus of RAP recycled asphalt mixture with different contents decreases with the increase of temperature. This is because the warm mix recycled asphalt mixture is a viscoelastic material. With the increase of temperature, the viscosity of the binder in the asphalt mixture becomes more significant\u003csup\u003e[\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]\u003c/sup\u003e, and the viscous flow is easy to occur at high temperature, resulting in the decrease of the bonding force between asphalt and aggregate. Thus, the ability to resist high temperature rutting deformation is weakened, and the dynamic modulus is reduced. Furthermore, as shown in the Fig, at the same temperature, the greater the loading frequency, the higher the dynamic modulus of RAP recycled asphalt mixture with different contents. As the temperature increases, the influence of loading frequency on dynamic modulus diminishes, which is consistent with the previous analysis results.\u003c/p\u003e\u003ch3\u003e2.2 Analysis of phase angle test results\u003c/h3\u003e\u003cp\u003e2.2.1 The influence of frequency on phase angle\u003c/p\u003e\u003cp\u003eFrom Fig.\u0026nbsp;4, it can be seen that at lower temperatures (5°C, 20°C), the phase angle decreases with the increase in frequency. This is mainly because at lower temperatures, the asphalt binder determines the viscoelastic properties of the asphalt mixture. As the frequency increases, the deformation response time of the recycled asphalt mixture is shortened, which is manifested by the weakening of its viscosity and the enhancement of elasticity. The higher the frequency is, the less active the polymer chain is. The final performance is that the phase angle gradually decreases with the increase of frequency at lower temperature\u003csup\u003e[\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eAt higher temperatures (35°C, 50°C), the phase angle increases as the frequency increases. This is primarily due to the fact that at higher temperatures, the interlocking effect among the mineral aggregate skeletons determines the viscoelastic properties of the asphalt mixture. As a prominent viscous component of the asphalt mixture, at high temperatures, the asphalt mixture softens, leading to a weakening of the bonding effect among the mineral aggregate skeletons and an increase in the phase angle. The mineral aggregate generally exhibits an elastic state. In the low frequency state, the phase angle of the aggregate with large internal friction resistance is close to zero, and as the frequency increases, the phenomenon of strain lagging behind stress is prominent, resulting in the phase angle increasing as the frequency increases.\u003c/p\u003e\u003cp\u003e2.2.2 The effect of temperature on phase angle\u003c/p\u003e\u003cp\u003eIt can be seen from Fig.\u0026nbsp;5 that under low frequency (0.1 Hz, 0.5 Hz, 1 Hz), the phase angle of warm mix recycled asphalt mixture with varying RAP contents first increases and then decreases with the increase of temperature, peaking at 20°C. In the medium and high frequency (5 Hz, 10 Hz, 25 Hz) state, the phase angle exhibits the same variation pattern, peaking at 35°C. This is mainly because in the low frequency state, the asphalt mixture resembles a viscous material. As the temperature gradually increases, the viscosity of the binder in the asphalt mixture gradually decreases, and the viscous component becomes more prominent, causing the phase angle to increase. When the temperature increases to a specific value, the influence of the binder on the mixture can be ignored. In the medium and high frequency state, the viscosity of the asphalt mixture is difficult to reflect, but as the temperature increases, the viscosity of the mixture gradually highlights, and finally shows the same change rule as the low frequency, but the peak value of the phase angle is delayed.\u003c/p\u003e\u003ch2\u003e2.3 The effect of RAP content on dynamic modulus and phase angle\u003c/h2\u003e\u003cp\u003eTable 4 Dynamic modulus increase ratio of 0-50% RAP and 50-70% RAP at different frequencies at low and high temperatures\u003c/p\u003e\u003cp\u003e\u003cimg 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QxdQp5DXqxx1qnei6A9djH1x8ODB/tDstXr16kE9WbFiRXplXKB+14+R6RybbYkfTWUwHV2pA02eaJnk2hw/pnMd6WIdmD9/fnXu3Ll0XD+RfX8p19aZUvR3gHoJYEpSetlqKnx2HglK7vbbb08XtS1btqSDm4pJpeYCx0m8juUxjY7l58kESQgJDsu85557+mMn4+DpZcn9oSdn3bp1gxiJYfPmzemgW7ZsWTq4iBsxrOkjcUSe4LJ/KV8u9L07mWrnzp39KReQZMZHl4ETBXWFpJj3sy8iOW1T8/cTPWl1JX5Mpwy6XAeaTKdMuhZ/03Uk19U6wPWFG3RiqcfX5FKvrZdDEQlQfCYdFSn/fJIEZd++fSnxoNJSeXNMJ4mg433s2L1791bHjh1LOzouhIGkgnkjucjRWsB4OpKS/A6YVh+2kQqxdevW/thmTXfOdfk8xMA6GceBxd0H2832N7VEMd901lGybdu2pQM39iUnK3AXRHlzIOco79jHu3fvTvuZcdQn6gsnD/YLCTJYZr11aTbimNi0aVN/6ALiGnUByLU9fjSVQZOu1oEm0y0TdC3+putI6HIdiMSNc2OcD0e51Gvr5dC5BCgKjgv4qVOnqkceeSTtEAqZikTFq4sKR3fkyJE0jh3BzgyxM9hp00ErQL0SRIsMHS0EBw4c6E+pUqtPTBt110QliWRmGNbLdnLwxHbTzzhi5MDD2rVrB/EGlsvyWY+Go8WMfUW5Ul4MU7bRsnf+/PmUDAX2aexfWolIuvP9zD5rW5nHDQX1ihNYIKnOPw4L9WMzP5baGD+GlUGTLtaBJpdSJrmuxD/qOtLlOsD1de7cuenY5/ierku9ts6o3o7oHMKi612c+mMuxpeuYj76A1/CYlyvcvbHTP5SG/iSFvP0Lnz9MRfE8mLeWNY73vGOi+ann3G9g2PwvnydU6mvI+9i2rAYWTfvA9PyL53VY9Jo1I28zGJ/xLioKzn2c74/GM73Afuvvs9mm3q9qx9DoR5/zJ8fm22MH8PKgHhiHLpaB1CPtalM8vjZ7zEtrydtjR/1MkCUA9PQ5ToQIka62LfEEP1N5TTda+vlMsaf3golSZKKUfSXoCVJUplMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnFMgCRJUnGKSYDOnDlTjY2NpW7//v1p3I4dO9LwqlWr0vAwGzduHLyXDvny6AfLYTiWD/qPHz+e+mP+6FhuHduUb0++nuhifWD+u+++ezCN4Sdj4cKF/T5NJfZ3vj8oP8YNq1Oxn6JORN3K9xvT8jo02+V1tKn+xbSo7/n8uabjoW1i/+d1ArGfo150sQwixqZzSP246HIdIIam4zfKJ2LuWhlw7Ec89fofGB/1YzbEX0wCdPLkyWpiYiJ1q1evToV/5MiRNLx8+fKRF5xly5YN3rt9+/Y0jp10+vTp1NEfO5zhPXv2pH7GHT16tFqyZEkaZtr4+PhgWYcPHx5cCHOHDh0aLG/BggXVsWPHqpUrV6b37Nu3b1BB4mJz1113pfEbNmyoNm/enMY9UWxT0wlMFzt48GDaL4E6RPmxn06dOnXRvmW/sS+Zvm7dujQu5r/nnnvSMLZt25bqaFvs3Lkz1b96HKAMGE8XdfrAgQNpfo6lOO6oy2vXrk39bcVxw74jVo7bEPUgxhNrF8sgzpPU/VzTcdHVOkAsu3fv7g/9D3U/yiBi7loZLF68OMVHTFxvm6xYsaLfNzvOA8UkQGvWrJmUmbKDSHzAjiNRGSYuRrx33rx5qZ8DmZMZXf2AD1wYdu3a1R+6GAnRnDlz+kMXls+2kMgMq0CLFi1KCRK42Ewn4YksmwswJx/6OVlT2ejnlQpIP/FwEo8KqemjnsSFLz/QAyfASIYp/6iLOfYRiVWbzJ8/vzp37lyKpx53xMu0SPrqohxi3jbiGALnmbhBCSdOnEjJATi5nz17NvXn2l4GxN8UO6Y6LtCFOgCuI5y/64if8z3Y/5zH69peBrHdnAuabuCoI9E40OSqxN/L2IrRq4AThMxrL+uc6GWfaXzvrnyiV2lT/yi8B7y/d+ef+jE+Pp5eGcfyWS7zslymMY7+WH90dbF85s2Xnw+zbPrr28z4phgYx7xgW+jP5+WV9bJtMV89Pg1HOVFedU37IuoJ4n3MR12I+hj7l3H0twVx5PHliIN4Ynp+HCDqcz5P28SxBeLI60TsUzAPZdXFMgCxNdX9ENO6GH/sY2JsOnYj5pjW1eMgjzEQV1xnIrbZEH9RX4ImC+/tmNS6QksOmWouWkHyzyjzlpC4c2M5Ta0+3Ln3ynRSds8dT2+nVnv37k3DvR2b5uG1jhYd1r906dJBK09gmGncZW3durU6f/58uvOeCq0O0crEtvA+to/xgY8CKZM88x7WqqWpcaezadOm/tBotBBSH2jJ4+4x6k7v5DDybmk24RihhYOWw6bvPnE3SIzR6sXxwzAdZUUZcHxwnFBH6x8dtkUcP70TeXoNxM/xFMc2ZdXVMiCOuJOvy4+LrsXP9sY5mo/AOE/XY6B1jHg5fzOti3WAes721z9R4dywZcuWdN3j3MZ5YjbEX9yvwOIiQ8eFHxQ6JyWmsTO48FCR2VnRlMdBHc3YiJM5OykSpkBFn+qjKRKQ/GLBslhvVIjeXcSk5IuTakzjRDt37tzGpvRAhYpt47NWsA7eR8VDVLp6soV6TJqe2GeUMeWbi4Oa/cCJIvYDqAvsizai3lKv4kI/DPHmMccJLh/XVuzb2Pf1fQvGcexyHOcfD3SpDEDdbopl2HHRlfg5J+fnbm608xvKiBMkA3wsGrpWB7jhrt+cR+MAyQ/X1fxj/qsaf2+jOo/mOEKly5vmaJJj3Kgm2xAfT4V6812gGY9pobez0zy9rHYwP/Mg1l8fjxh3xx13DPrr28mykS8775BvZ/5++mN9vObxUUZ5OakZZZiXdX1/Uoaxb0JMy+sIy2G+wP6I97dBXv+iHkWdysskj4f48/oey4g63Ub1OKMMGI5puS6VQZzn8m1vqgN0UT5drAPgeK7XgeiPMghdKoOIrakOBOLNp1/t+Mf401uhWog7KVqBZrLlgDs47mZH3clLktR2JkAtRxLE95nyZvUng4++TH4kSV1nAiRJkopT3JegJUmSTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIAkSVJxTIBa4vjx49WOHTv6Q1cG62O9kiR1TScSIC7SY2NjqZsKF/X6vAsXLkzD+/fvT8Mxz8aNG9Mwzpw5c1ECkq83ulhGbtWqVZPeyzz5e1j/KKxn27Zt1ebNm9Mw8+fbNgzrGTYf46daL+tjvU0x6X/1hv0b8v1aL/u87lGfwDwM5/WD/d2GMh8V/6h6x/Soe22OH8TO9sf+RMSUl0uIc0ZePm0ug6Y60FQmoWvHAJrKIOJsqgNdK4NLrQOI+IkRVy3+iQ4YHx9Pr8eOHZvYsGFD6h9m3759/b4Ltm/fPhgXy6m/YuXKlf2+yXhvrPP06dMT9SJlHNPzZYH10oHp0d+kHtN04pyO+jahKc5hsZeM/c6+BeXIPsnrFsN0uRhmvmF1Dm0o76b483hHHS+5tsafY3ujLCiDODaHHdf5POhSHQh5meRini4cA2gqA4Zj+/PrS+hSGTTFH4bVAep+zFeP+0rH3/oWIDLMuJtcsmRJdfjw4dTfhGxyzZo1kzLVI0eOVIsWLUr9K1asGGSkObLSXbt29YeGe+SRR6reDuwPXXDy5MnBe4dlw8uWLUvb0YTtYXqTyLJDZOJsL+vKM+qYN7rAdIZZD/McOnRoUJ5h3bp1jeVSstWrV1cLFixI/dQbMC6cOHEi1cdcDJ87d27SvDnuhA4ePNgfmr2a4o/4qHvUmTrqGscfMQ7TlviHYb/H8bp27drq7Nmzqf9StLkOTKVLxwCayoBz/vLly1P/4sWLq6NHj6b+0PXzwFS4RkcZcK1pui5eqfhbnwCRdMQOmAo7q5f0pcoZSdCpU6cuev/69etTUsAOjQv/zp0707hIKHK7d+9O0/i4iOXlqORgmQcOHEj9dRwgccBMF+vcunVr1cum0zayXayf+O65554UU+/uI80bFYxpjOtl7Wm4l52nV4Y5cVPhSODqMcydO7c6f/58f0hN4oCeCgf8li1bBk271DHqDvWDcVw8I1ltQ/N3iPjZZuoQdbOOj1Spg+AEh67EH+bNm1ft2bOnPzQ9XasDU+nqMQDKgHM+9WCUrp8HLtVVi793Qmq1vLkRNKExjtDoogmu3hQd78mb6fKmuRBN1dE0F6+BJkDmqW8H8u2gy6ezPfm0UerLZRtju2Kbec2Xx7pZR8Sdry9ELMQQ89XjQ15Gmoxyq5cN+4MyHSbff7m8HsXrbNcUP6aqM02xtTF+1GPlGIpjrakeDNv/6EodqJdJXZeOAeRlkJ9PR+3rrp8HhtWB/BrTNM+VjL/1LUC0dESLBZkimSQtLb2CS3ebvcJOWSRNkU1oeaHJErSq5Bksd6mbNm3qD43GdtDsn7cQsVy2ITq2M2/u61WawbRRWC7Z8Cjz589PLTmxvHqrFk3x01lXHeuljOrL04X6Bsom3++0pg1r2sacOXPS/spRztP5mHU2GRY/GDesznAM1Ke1Mf5hOM45znon7pH1oK5rdWCUrhwDqJcBX6mIrzTs3bs3tWY0KeE80IRrNJ9acB7gWMnPBVc8/t6B2npk0oTS1HqRI5NkvnrYccfGcgIZbX73RlbKPJHZI9abj8/XkY/P573zzjsH/aNaCnK8P5YV763Hk9953nfffYN+sujop2O+eG++DNYRy0C9DPQ/lE2UG11eTrGfEPsdMW+9nlK3mC/EPpnNZd8Ufz4utj2PP+pWF+IPcV6IGNnmfBh5GeTHYl5PulIHUC+TPP4Y35U6MKwMYjxxoatl8ETqAGIax0O4GvGP8ae3EnUYmTbflYrWLbL0+Em9JEkl8kGIBeCjuKVLl6aPAukkSSqdLUCSJKk4tgBJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimABJkqTimAC1zKpVq/p9V8bx48erHTt29IckSeqG1idAZ86cqcbGxlK3f//+NI4LNsNTJQsbN24cvJcO+fLoB8thOJYP+kkOEPNHx3Lr2KZ8e/L1RBfrG4Z5Dh48mPpZx8KFC1P/KCxz2HzEMNV6lyxZkl6bYtL/xD6MOlEXZU055vs+19YyJuaIrUkcP1HPuhY/pioDdLEO1PctON8wbtj5N+KOY4WYGc5vtJiWn29ns6YyQL6/c3F9io73tbkMhsXPcNO1J8qF7qrXgYmW27dvX7/vgtOnT0+sXLky9W/fvv2i6bl8GvOC97KMWE79FfRv2LAh9YPh8fHx/tBE6j927Fh/6AKWT3Ezb2CeWCbbEv1N8vWFfJ1PVMSWY131cWx/PSZdQHlF2TTtE/ZtPj7qZV4/216+bHtTHQ15Peti/BhVBl2uA/m+JZbobzoPNh0r9VeMOhfORvXzaH1/52J/g32OtpdBPX4QS1MZRMyUQ8R4teJvfQvQmjVrJmWfJ0+erJYvX576Fy9eXB09ejT1N1m9enV65b3z5s1L/adOnaoWLFiQOvqb7Ny5s9q1a1d/6GK9ilDNmTOnP3Rh+WxL7+BP29dk0aJF1aFDh/pD0xd3E/VMOu6+yMCjv37nEe9hmxhmOt3u3burXkWclNHffvvt1YkTJ/pDyh0+fHjQUkZ55+UG6igo4/o0xLhYRmlKiL+UOsA5lXMnVqxYkV5zUx0r4BwWLd1tNWp/x3VnlLaXAdeRPXv29Icm27x5c3rlmjfMlYq/9QlQL4lLCUdcsM+dOzdIZqbrwIEDqVLy/nqTXRzMLH/dunVpx65duzbNlycRbAPDdGxTvA8sn4Oe9w2rFCQhvYy3PzQ9rBO9TDolJ7EtsX6G8/Xdc889aVrvDiwlY3EiYh6WdeTIkVQ52Q6G8xjoP3v2bH9I00UTLmXNib9355OSZ5JJTpBbtmxJ9Y6DnQSZulOvf11UWvwl14HpJHQkSsS9fv36VFbLli0bfKxy2T8CuQya9ncT5qMeoEtlENehvBGgCde8rVu3pv6rFX8nvgTNxZkkgAIl+SEJylGAFGScWEh08kKNCzvLaWr1IRMlccgzVnYYicTevXvTMAkS8/BaR+LB+pcuXXpRKw/DTONkGJWhSdOdEuuKbBrnz59PrTcsj1eGcyQ6sR0kY2FUa1Yg8aNS6tLNnz8/1S0ucKCfukJHuVL+1CPqE/UqTiBdVWL8pdUBYtq0aVN/aDRipxw4l9FiH+dZbsKG3TDOdvX93YRYmQddKoNt27alxJ7rE9sfn0DUEWckyFcr/s78CiwKjY6WDHBC4aLNNAo3EgB2TDRDkljkF3aSJMZxAqrfiXGXliccTcj68x3Oslgv66fjziBPvmhtiWmj7paoIGz7KHPnzk3Lj+VFjIHymM666jiZUab15emCuGCxr+Mj1EB9JAEGCSknxhAXuXz+kpQSf2l1IM5vxMS5IzfqWOG8yXmu7Ubt71zT+C6UQTQYkLxwrW36KItrK3FSV6gL4YrH39vQ1uJLVIRAV/9iGeN6yUB/zHDxhazQ22mDZeb4QhbTQm/Hpnl6d2yD+eNLW7H++njEuDvuuGPQP53tDLEsXmPZsRy2hWXFMGUS20l/Pi9dL0tPr8wT87HtsQyWF8vUaFGmUUco66hbUR8o48B8sS8R9Sifpy3yYyZizuPP62ToUvyYqgy6Wgfq+zbijI7zTsQVYhrxB5bDfIFyiffPdk31u76/62VAXHn8aGsZNMUP4muKP+aly+v61Yh/jD+9lagAZNvRihN3npfSEiRJUleYABWCZsZett0fuvDRW5t/ZSBJ0pNhAiRJkorTmS9BS5IkTZcJkCRJKo4JkCRJKo4JkCRJKo4JkCRJKo4JkCRJKo4JkCRJKo4JkCRJKo4JkCRJKo4JkCRJKo4JkCRJKo4JkCRJKo4JkCRJKo4JkCRJKs7YRE+/f5KxsbF+nyRJUjdE2mMLkCRJKkxV/T/qF/eLlUoQMgAAAABJRU5ErkJggg==\" width=\"576\" height=\"221\"\u003e\u003c/p\u003e\u003cp\u003eTable 5 0~50%RAP、50~70%RAP recycled asphalt mixture under different frequencies of low and high temperature phase Angle increase, decrease proportion\u003c/p\u003e\u003cp\u003e\u003cimg 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\" width=\"577\" height=\"222\"\u003e\u003c/p\u003e\u003cp\u003eIt can be seen from Fig.\u0026nbsp;6(a) and Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e and Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e that the dynamic modulus increase ratio of recycled asphalt mixture with 0% ~ 50% RAP is higher than that of asphalt mixture with 50% ~ 70% RAP at low temperature and 6 frequencies. The phase angle reduction ratio of recycled asphalt mixture with 0% ~ 50% RAP is less than that of asphalt mixture with 50% ~ 70% RAP. Under the condition of low temperature, the smaller the dynamic modulus and the larger the phase angle, the more obvious the improvement effect of the low temperature crack resistance of the recycled asphalt mixture. Therefore, the order of low temperature crack resistance of warm mix recycled asphalt mixture is 0% RAP \u0026gt; 50% RAP \u0026gt; 70% RAP for the recycled asphalt mixture.\u003c/p\u003e\u003cp\u003eFrom Fig.\u0026nbsp;6(b), Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e and Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, it can be seen that at high temperature and 6 frequencies, the increase proportion of dynamic modulus of recycled asphalt mixture with 0% ~ 50% RAP is higher than that of 50% ~ 70% RAP asphalt mixture. The phase angle increase ratio of 0% ~ 50% RAP warm-recycled asphalt mixture is less than that of 50% ~ 70% RAP asphalt mixture. At high temperature, the larger the dynamic modulus, the smaller the phase angle, the better the high temperature deformation resistance of recycled asphalt mixture. Therefore, the order of high temperature resistance to permanent deformation performance of warm-recycled asphalt mixture is 70% RAP \u0026gt; 50% RAP \u0026gt; 0% RAP for the recycled asphalt mixture. Considering the high temperature stability and low temperature crack resistance of recycled asphalt mixture, 50% RAP asphalt mixture has better high and low temperature performance.\u003c/p\u003e\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e\u003ch2\u003e2.4 principal curve analysis\u003c/h2\u003e\u003cp\u003e2.4.1 Basic concept of principal curve\u003c/p\u003e\u003cp\u003eAccording to the principle of time-temperature equivalence, a smooth curve at the reference temperature is established by translating the dynamic modulus and phase angle at different temperatures and frequencies obtained by the dynamic modulus test. This curve is called the main curve of dynamic modulus and phase angle. Using the master curve can not only solve the defects of the instrument and equipment, but also predict the mechanical properties of asphalt mixture in a wider temperature and frequency range\u003csup\u003e[\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]\u003c/sup\u003e. Based on the nonlinear least squares method, the \u003cb\u003eS\u003c/b\u003e\u003csup\u003e[\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]\u003c/sup\u003e model and the \u003cb\u003eφ\u003c/b\u003e model are used for fitting. The \u003cb\u003eS\u003c/b\u003e model and the \u003cb\u003eφ\u003c/b\u003e model are expressed as :\u003c/p\u003e\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\lg \\left| {E*\\left( {{f_r}} \\right)} \\right|=\\delta +\\frac{\\alpha }{{{{\\left( {1+\\lambda {{\\exp }^{\\beta +\\gamma \\lg \\left( {{f_r}} \\right)}}} \\right)}^{\\frac{1}{\\lambda }}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\Phi \\left( {{f_r}} \\right)=\\rho \\frac{\\pi }{2}\\frac{{d\\left( {\\lg \\left( {E*} \\right)} \\right)}}{{d\\left( {\\lg {f_r}} \\right)}}= - \\frac{\\pi }{2}\\rho \\alpha \\gamma \\frac{{{{\\exp }^{\\beta +\\gamma \\lg {f_r}}}}}{{{{\\left( {1+\\lambda {{\\exp }^{\\beta +\\gamma \\lg {f_r}}}} \\right)}^{1+\\frac{1}{\\lambda }}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eIn the formula : \u003cem\u003elg|E\u003c/em\u003e\u003csup\u003e\u003cem\u003e*\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e(f\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e)|\u003c/em\u003e is the logarithm of dynamic modulus ; \u003cem\u003eδ\u003c/em\u003e is the minimum value of dynamic modulus ; \u003cem\u003eα\u003c/em\u003e is the difference between the maximum and minimum values of dynamic modulus ; \u003cem\u003eβ\u003c/em\u003e and \u003cem\u003eγ\u003c/em\u003e are the fitting parameters in the \u003cb\u003eS\u003c/b\u003e model, which are related to the material properties of asphalt mixture. \u003cem\u003eλ\u003c/em\u003e determines the asymmetric characteristics of sigmoidal model ; \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e is reduced frequency ( H\u003csub\u003eZ\u003c/sub\u003e ) ; \u003cem\u003eΦ(f\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e)\u003c/em\u003e is the main curve form of phase angle ; \u003cem\u003eρ\u003c/em\u003e is an improved fitting parameter.\u003c/p\u003e\u003cp\u003eThe relationship between the reduced frequency \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e, the loading frequency \u003cem\u003ef\u003c/em\u003e, and the shift factor \u003cem\u003eα\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e is expressed as :\u003c/p\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$${f_r}=f * {\\alpha _t}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eIn the formula : \u003cem\u003eα\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e is the shift factor. \u003cem\u003eα\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e can be solved by the WLF equation, and the specific equation is expressed as :\u003c/p\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\lg {\\alpha _t}=\\frac{{ - {C_1}\\left( {t - {t_0}} \\right)}}{{{C_2}+\\left( {t - {t_0}} \\right)}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eIn the formula : \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e are the fitting parameters, \u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e is the reference temperature of the main curve fitting, and the reference temperature is selected as 20°C.\u003c/p\u003e\u003cp\u003eSubstituting Formulas ( 3 ) and ( 4 ) into Formulas ( 1 ) and ( 2 ), the expressions of dynamic modulus and phase angle are obtained, which are expressed as :\u003c/p\u003e\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\lg \\left| {{E^ * }_{{\\left( f \\right)}}} \\right|=\\delta +\\frac{\\alpha }{{{{\\left[ {1+\\lambda {{\\exp }^{\\beta +\\gamma \\lg f+\\gamma \\frac{{ - {C_1}\\left( {t - {t_0}} \\right)}}{{{C_2}+\\left( {t - {t_0}} \\right)}}}}} \\right]}^{\\frac{1}{\\lambda }}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$${\\Phi _{\\left( f \\right)}}= - \\frac{\\pi }{2}\\rho \\alpha \\gamma \\frac{{{{\\exp }^{\\beta +\\gamma \\lg f+\\gamma \\frac{{ - {C_1}\\left( {t - {t_0}} \\right)}}{{{C_2}+\\left( {t - {t_0}} \\right)}}}}}}{{{{\\left[ {1+\\lambda {{\\exp }^{^{{\\beta +\\gamma \\lg f+\\gamma \\frac{{ - {C_1}\\left( {t - {t_0}} \\right)}}{{{C_2}+\\left( {t - {t_0}} \\right)}}}}}}} \\right]}^{1+\\frac{1}{\\lambda }}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFormulas ( 5 ) and ( 6 ) need to fit eight test parameters of δ, ρ, α, β, γ, λ, C\u003csub\u003e1\u003c/sub\u003e and C\u003csub\u003e2\u003c/sub\u003e. In order to improve the accuracy of parameter fitting, Python is used to synchronously fit Formulas ( 5 ) and ( 6 ) to realize parameter sharing. The fitting results are shown in Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eMain curve fitting results of dynamic modulus and phase angle\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eparameter\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0%RAP\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003e50%RAP\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003e70%RAP\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cem\u003eδ\u003c/em\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e2.459 0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e2.561 3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e2.366 6\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cem\u003eα\u003c/em\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e1.880 3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e1.838 8\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e2.113 9\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eλ\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.941 6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.498 4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.372 7\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eβ\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e-0.321 6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e-0.586 5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e-0.830 6\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eγ\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e-0.902 0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e-0.653 5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e-0.491 2\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eC\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e20.416 9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e20.486 2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e29.500 9\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eC\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e192.696 2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e163.916 2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e230.024 5\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eρ\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e4.162 0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e56.498 7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e57.761 5\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eR\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.999\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.999\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.999\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003ch2\u003e2.4.2 Analysis of dynamic modulus master curve\u003c/h2\u003e\u003cp\u003eIt can be seen from Fig.\u0026nbsp;7(a) that the dynamic modulus of warm mix recycled asphalt mixtures with different RAP content increases in an S - shaped pattern as the loading frequency increases. This is because the warm mix recycled asphalt mixture is mainly elastic at high frequency (Low Temperature) and mainly viscous at low frequency (High Temperature). Therefore, when the reduction frequency changes from low frequency to high frequency, the dynamic modulus of recycled asphalt mixture increases continuously. Under high temperature and low frequency, the dynamic modulus of the two types of asphalt mixtures after warm mix regeneration is higher than that of hot mix asphalt mixture; under low temperature and high frequency, the dynamic modulus of the two types of asphalt mixtures after warm mix regeneration is not significantly different from that of hot mix asphalt mixture, which indicates that the elasticity of warm mix recycled asphalt mixture after warm mix regeneration is still higher than that of hot mix asphalt mixture at high temperature, and there is no significant difference between warm mix recycled asphalt mixture and hot mix asphalt mixture at low temperature.\u003c/p\u003e\u003cp\u003eThis is due to the high temperature state, the main factors affecting the dynamic modulus are the skeleton embedding and the adhesion of the aggregate and the asphalt. After the warm mix regeneration, the addition of the modifier reduces the viscosity of the asphalt, so that the asphalt can be better spread on the aggregate. At the same time, it also promotes the asphalt into the tiny voids on the surface of the stone, enhances the adhesion performance of the asphalt and the aggregate, and improves the dynamic modulus of the asphalt mixture. Under the condition of low temperature, the main factor affecting the dynamic modulus is the improvement effect of the regenerant on the old asphalt. After warm mix regeneration, on the one hand, it can reduce the secondary aging of asphalt in the process of mixing and compaction, promote the integration of new and old asphalt interfaces, and enhance the adhesion effect between asphalt and aggregate. On the other hand, the regenerant plays a solubilizing role in the old asphalt, reduces the interfacial tension between the light component and the heavy component, and improves the mutual solubility of the two, so that the proportion of light and heavy components in the warm-recycled asphalt mixture is smaller than that of the hot mix asphalt mixture, so the dynamic modulus of the warm-recycled asphalt mixture at low temperature is not significantly different from that of the hot mix asphalt mixture.\u003c/p\u003e\u003cp\u003e2.4.3 Analysis of main curve of phase angle\u003c/p\u003e\u003cp\u003eIt can be seen from Fig.\u0026nbsp;7(b) that the main curve of the phase angle of the warm mix recycled asphalt mixture with different RAP content is bell-shaped. With the increase of the reduction frequency, the phase angle first increases to the peak and then decreases to zero. When the reduction frequency tends to zero or infinity, the phase angle tends to zero, indicating that the warm mix recycled asphalt mixture is elastic at higher and lower frequencies. This is because the elastic component of the warm-mix recycled asphalt mixture is prominent at low temperature. When the temperature continues to rise, the binder in the recycled asphalt mixture softens. At this point, the viscous component is prominent and the phase angle increases.\u003c/p\u003e\u003cp\u003eIn addition, it can be seen from Fig.\u0026nbsp;7(b) that the peak value of the main curve of the phase angle of the different content of RAP asphalt mixture gradually moves to the left with the increase of the content. The lower the RAP content, the higher the peak value of the main curve of the phase angle. It shows that the temperature and viscoelastic ratio of asphalt mixture with different RAP content are not the same when they are close to their ultimate viscous state. The main reason is that the higher the RAP content, the more the old asphalt under the aging effect, and the ability to resist high temperature deformation is enhanced. The recycled asphalt mixture with different RAP content has different resistance to high temperature deformation. In addition, from the perspective of the influence of the change of phase angle on the high and low temperature performance of asphalt pavement, better elasticity is needed to resist pavement deformation at high temperature, and better viscosity is needed to resist pavement cracking at low temperature. When the RAP content is 50%, the phase angle in low frequency and high frequency is in a balanced state. At this time, the recycled asphalt mixture not only has good high temperature deformation resistance, but also has good low temperature crack resistance. Therefore, considering comprehensively, the high and low temperature performance of asphalt mixture with 50% RAP is the best, which is consistent with the previous analysis results.\u003c/p\u003e\u003c/div\u003e\u003ch2\u003e2.4 principal curve analysis\u003c/h2\u003e\u003cp\u003e2.4.1 Basic concept of principal curve\u003c/p\u003e\u003cp\u003eAccording to the principle of time-temperature equivalence, a smooth curve at the reference temperature is established by translating the dynamic modulus and phase angle at different temperatures and frequencies obtained by the dynamic modulus test. This curve is called the main curve of dynamic modulus and phase angle. Using the master curve can not only solve the defects of the instrument and equipment, but also predict the mechanical properties of asphalt mixture in a wider temperature and frequency range\u003csup\u003e[\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]\u003c/sup\u003e. Based on the nonlinear least squares method, the \u003cb\u003eS\u003c/b\u003e\u003csup\u003e[\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]\u003c/sup\u003e model and the \u003cb\u003eφ\u003c/b\u003e model are used for fitting. The \u003cb\u003eS\u003c/b\u003e model and the \u003cb\u003eφ\u003c/b\u003e model are expressed as :\u003c/p\u003e\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\lg \\left| {E*\\left( {{f_r}} \\right)} \\right|=\\delta +\\frac{\\alpha }{{{{\\left( {1+\\lambda {{\\exp }^{\\beta +\\gamma \\lg \\left( {{f_r}} \\right)}}} \\right)}^{\\frac{1}{\\lambda }}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\Phi \\left( {{f_r}} \\right)=\\rho \\frac{\\pi }{2}\\frac{{d\\left( {\\lg \\left( {E*} \\right)} \\right)}}{{d\\left( {\\lg {f_r}} \\right)}}= - \\frac{\\pi }{2}\\rho \\alpha \\gamma \\frac{{{{\\exp }^{\\beta +\\gamma \\lg {f_r}}}}}{{{{\\left( {1+\\lambda {{\\exp }^{\\beta +\\gamma \\lg {f_r}}}} \\right)}^{1+\\frac{1}{\\lambda }}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003cp\u003eIn the formula : \u003cem\u003elg|E\u003c/em\u003e\u003csup\u003e\u003cem\u003e*\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e(f\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e)|\u003c/em\u003e is the logarithm of dynamic modulus ; \u003cem\u003eδ\u003c/em\u003e is the minimum value of dynamic modulus ; \u003cem\u003eα\u003c/em\u003e is the difference between the maximum and minimum values of dynamic modulus ; \u003cem\u003eβ\u003c/em\u003e and \u003cem\u003eγ\u003c/em\u003e are the fitting parameters in the \u003cb\u003eS\u003c/b\u003e model, which are related to the material properties of asphalt mixture. \u003cem\u003eλ\u003c/em\u003e determines the asymmetric characteristics of sigmoidal model ; \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e is reduced frequency ( H\u003csub\u003eZ\u003c/sub\u003e ) ; \u003cem\u003eΦ(f\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e)\u003c/em\u003e is the main curve form of phase angle ; \u003cem\u003eρ\u003c/em\u003e is an improved fitting parameter.\u003c/p\u003e\u003cp\u003eThe relationship between the reduced frequency \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e, the loading frequency \u003cem\u003ef\u003c/em\u003e, and the shift factor \u003cem\u003eα\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e is expressed as :\u003c/p\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$${f_r}=f * {\\alpha _t}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003cp\u003eIn the formula : \u003cem\u003eα\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e is the shift factor. \u003cem\u003eα\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e can be solved by the WLF equation, and the specific equation is expressed as :\u003c/p\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\lg {\\alpha _t}=\\frac{{ - {C_1}\\left( {t - {t_0}} \\right)}}{{{C_2}+\\left( {t - {t_0}} \\right)}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003cp\u003eIn the formula : \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e are the fitting parameters, \u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e is the reference temperature of the main curve fitting, and the reference temperature is selected as 20°C.\u003c/p\u003e\u003cp\u003eSubstituting Formulas ( 3 ) and ( 4 ) into Formulas ( 1 ) and ( 2 ), the expressions of dynamic modulus and phase angle are obtained, which are expressed as :\u003c/p\u003e\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\lg \\left| {{E^ * }_{{\\left( f \\right)}}} \\right|=\\delta +\\frac{\\alpha }{{{{\\left[ {1+\\lambda {{\\exp }^{\\beta +\\gamma \\lg f+\\gamma \\frac{{ - {C_1}\\left( {t - {t_0}} \\right)}}{{{C_2}+\\left( {t - {t_0}} \\right)}}}}} \\right]}^{\\frac{1}{\\lambda }}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$${\\Phi _{\\left( f \\right)}}= - \\frac{\\pi }{2}\\rho \\alpha \\gamma \\frac{{{{\\exp }^{\\beta +\\gamma \\lg f+\\gamma \\frac{{ - {C_1}\\left( {t - {t_0}} \\right)}}{{{C_2}+\\left( {t - {t_0}} \\right)}}}}}}{{{{\\left[ {1+\\lambda {{\\exp }^{^{{\\beta +\\gamma \\lg f+\\gamma \\frac{{ - {C_1}\\left( {t - {t_0}} \\right)}}{{{C_2}+\\left( {t - {t_0}} \\right)}}}}}}} \\right]}^{1+\\frac{1}{\\lambda }}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003cp\u003eFormulas ( 5 ) and ( 6 ) need to fit eight test parameters of δ, ρ, α, β, γ, λ, C\u003csub\u003e1\u003c/sub\u003e and C\u003csub\u003e2\u003c/sub\u003e. In order to improve the accuracy of parameter fitting, Python is used to synchronously fit Formulas ( 5 ) and ( 6 ) to realize parameter sharing. The fitting results are shown in Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eMain curve fitting results of dynamic modulus and phase angle\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"4\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eparameter\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0%RAP\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003e50%RAP\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003e70%RAP\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cem\u003eδ\u003c/em\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e2.459 0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e2.561 3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e2.366 6\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cem\u003eα\u003c/em\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e1.880 3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e1.838 8\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e2.113 9\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eλ\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.941 6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.498 4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.372 7\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eβ\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e-0.321 6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e-0.586 5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e-0.830 6\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eγ\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e-0.902 0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e-0.653 5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e-0.491 2\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eC\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e20.416 9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e20.486 2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e29.500 9\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eC\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e192.696 2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e163.916 2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e230.024 5\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eρ\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e4.162 0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e56.498 7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e57.761 5\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eR\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.999\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.999\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e0.999\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003ch2\u003e2.4.2 Analysis of dynamic modulus master curve\u003c/h2\u003e\u003cp\u003eIt can be seen from Fig.\u0026nbsp;7(a) that the dynamic modulus of warm mix recycled asphalt mixtures with different RAP content increases in an S - shaped pattern as the loading frequency increases. This is because the warm mix recycled asphalt mixture is mainly elastic at high frequency (Low Temperature) and mainly viscous at low frequency (High Temperature). Therefore, when the reduction frequency changes from low frequency to high frequency, the dynamic modulus of recycled asphalt mixture increases continuously. Under high temperature and low frequency, the dynamic modulus of the two types of asphalt mixtures after warm mix regeneration is higher than that of hot mix asphalt mixture; under low temperature and high frequency, the dynamic modulus of the two types of asphalt mixtures after warm mix regeneration is not significantly different from that of hot mix asphalt mixture, which indicates that the elasticity of warm mix recycled asphalt mixture after warm mix regeneration is still higher than that of hot mix asphalt mixture at high temperature, and there is no significant difference between warm mix recycled asphalt mixture and hot mix asphalt mixture at low temperature.\u003c/p\u003e\u003cp\u003eThis is due to the high temperature state, the main factors affecting the dynamic modulus are the skeleton embedding and the adhesion of the aggregate and the asphalt. After the warm mix regeneration, the addition of the modifier reduces the viscosity of the asphalt, so that the asphalt can be better spread on the aggregate. At the same time, it also promotes the asphalt into the tiny voids on the surface of the stone, enhances the adhesion performance of the asphalt and the aggregate, and improves the dynamic modulus of the asphalt mixture. Under the condition of low temperature, the main factor affecting the dynamic modulus is the improvement effect of the regenerant on the old asphalt. After warm mix regeneration, on the one hand, it can reduce the secondary aging of asphalt in the process of mixing and compaction, promote the integration of new and old asphalt interfaces, and enhance the adhesion effect between asphalt and aggregate. On the other hand, the regenerant plays a solubilizing role in the old asphalt, reduces the interfacial tension between the light component and the heavy component, and improves the mutual solubility of the two, so that the proportion of light and heavy components in the warm-recycled asphalt mixture is smaller than that of the hot mix asphalt mixture, so the dynamic modulus of the warm-recycled asphalt mixture at low temperature is not significantly different from that of the hot mix asphalt mixture.\u003c/p\u003e\u003cp\u003e2.4.3 Analysis of main curve of phase angle\u003c/p\u003e\u003cp\u003eIt can be seen from Fig.\u0026nbsp;7(b) that the main curve of the phase angle of the warm mix recycled asphalt mixture with different RAP content is bell-shaped. With the increase of the reduction frequency, the phase angle first increases to the peak and then decreases to zero. When the reduction frequency tends to zero or infinity, the phase angle tends to zero, indicating that the warm mix recycled asphalt mixture is elastic at higher and lower frequencies. This is because the elastic component of the warm-mix recycled asphalt mixture is prominent at low temperature. When the temperature continues to rise, the binder in the recycled asphalt mixture softens. At this point, the viscous component is prominent and the phase angle increases.\u003c/p\u003e\u003cp\u003eIn addition, it can be seen from Fig.\u0026nbsp;7(b) that the peak value of the main curve of the phase angle of the different content of RAP asphalt mixture gradually moves to the left with the increase of the content. The lower the RAP content, the higher the peak value of the main curve of the phase angle. It shows that the temperature and viscoelastic ratio of asphalt mixture with different RAP content are not the same when they are close to their ultimate viscous state. The main reason is that the higher the RAP content, the more the old asphalt under the aging effect, and the ability to resist high temperature deformation is enhanced. The recycled asphalt mixture with different RAP content has different resistance to high temperature deformation. In addition, from the perspective of the influence of the change of phase angle on the high and low temperature performance of asphalt pavement, better elasticity is needed to resist pavement deformation at high temperature, and better viscosity is needed to resist pavement cracking at low temperature. When the RAP content is 50%, the phase angle in low frequency and high frequency is in a balanced state. At this time, the recycled asphalt mixture not only has good high temperature deformation resistance, but also has good low temperature crack resistance. Therefore, considering comprehensively, the high and low temperature performance of asphalt mixture with 50% RAP is the best, which is consistent with the previous analysis results.\u003c/p\u003e"},{"header":"3 Conclusion","content":"\u003cp\u003e(1) Under the condition of high temperature and low frequency, the dynamic modulus of the two kinds of asphalt mixture after warm mix regeneration is higher than that of hot mix asphalt mixture. Under the condition of low temperature and high frequency, the dynamic modulus of the two kinds of asphalt mixture after warm mix regeneration is not significantly different from that of hot mix asphalt mixture, indicating that the elasticity of asphalt mixture after warm mix regeneration is still higher than that of hot mix asphalt mixture at high temperature, and there is no significant difference between flexible and hot mix asphalt mixture at low temperature.\u003c/p\u003e\u003cp\u003e(2) Compared with the warm-recycled asphalt mixture with 70% RAP content, when the RAP content is 50%, the dynamic modulus increases the most at high temperature (50°C), and the phase angle decreases the least at low temperature (5°C). Therefore, the warm-recycled asphalt mixture with 50% RAP has better high and low temperature performance.\u003c/p\u003e\u003cp\u003e(3) The fitting effect of \u003cb\u003eS\u003c/b\u003e and \u003cb\u003eφ\u003c/b\u003e models on the main curve of dynamic modulus and phase angle is good. The fitting results show that the dynamic modulus of warm-recycled asphalt mixture with different RAP content increases in an \u003cb\u003eS\u003c/b\u003e-shaped manner with the increase of reduction frequency, and the phase angle shows a bell-shaped change rule with the increase of reduction frequency.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eThis study was funded by the Inner Mongolia Autonomous Region Natural Science Foundation Project(2025LHMS05037); Basic Scientific Research Business Fee Items of Universities in Inner Mongolia Autonomous Region(ZTY2025066).\u003c/p\u003e\u003cp\u003eThis manuscript has not been published or presented elsewhere in part or in entirety. All the authors have approved the manuscript and agree with submission to your esteemed journal.\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eConsent to Participate declaration\u003c/strong\u003e\u003cp\u003enot applicable.\u003c/p\u003e\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eF.L. and W.J. conceived the research and designed experiments. W.L. and S.C. performed laboratory tests and data curation. L.T. developed analytical models and validation. F.L. and W.J. wrote the main manuscript text. W.L. prepared figures and visualizations. S.C. and L.T. conducted formal analysis. All authors reviewed and approved the final manuscript.\u003c/p\u003e\u003ch2\u003eData Availability Statement\u003c/h2\u003e\u003cp\u003eSome or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eZHOU Zihao. Study on influencing factors of Dynamic modulus of Asphalt Mixture under Uniaxial Compression mode. Master\u0026rsquo;s thesis,Changsha University of Science and Technology, china,2017.DOI:10.26985/d.cnki.gcsjc.2017.000004.（in Chinese）周梓豪.单轴压缩模式下沥青混合料动态模量影响因素研究. 硕士学位论文,长沙理工大学,2017.DOI:10.26985/d.cnki.gcsjc.2017.000004.\u003c/li\u003e\n\u003cli\u003eHARNAENI S R, PRAMESTI F P, BUDIARTO A, et al. Study on structural performance of asphalt concrete and hot rolled sheet through viscoelastic characterization[J]. Materials, 2020, 13(5). DOI:10.3390/ma13051133.\u003c/li\u003e\n\u003cli\u003eLI C, NING F, LI Y. Effect of carbon black on the dynamic moduli of asphalt mixtures and its master curves[J]. Frontiers of Structural and Civil Engineering, 2019, 13(4): 918-925. DOI:10.1007/s11709-019-0526-6.\u003c/li\u003e\n\u003cli\u003eXI L, LUO R, LIU H. Effect of relative humidity on the linear viscoelastic properties of asphalt mixtures[J]. Construction and Building Materials, 2021, 307. 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Journal of Materials in Civil Engineering, 2024, 36(3). DOI:10.1061/JMCEE7.MTENG-17124.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"discover-civil-engineering","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Discover Civil Engineering](https://www.springer.com/journal/44290)","snPcode":"44290","submissionUrl":"https://submission.nature.com/new-submission/44290","title":"Discover Civil Engineering","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Discover Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"warm recycled asphalt mixture, The RAP, Dynamic modulus, Phase Angle, The master curve","lastPublishedDoi":"10.21203/rs.3.rs-6912958/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6912958/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn order to study the influence of temperature, frequency and RAP content on the viscoelastic properties of warm mix recycled asphalt mixture, the viscoelastic properties of asphalt mixture with different RAP content (0%, 50%, 70%) at four temperatures and six frequencies were studied by dynamic modulus test. The results show that under the condition of high temperature and low frequency, the dynamic modulus of the two kinds of asphalt mixture after warm mix regeneration is higher than that of hot mix asphalt mixture ; under the condition of low temperature and high frequency, the dynamic modulus of the two kinds of asphalt mixture after warm mix regeneration is not significantly different from that of hot mix asphalt mixture, which indicates that the elasticity of asphalt mixture after warm mix regeneration is still higher than that of hot mix asphalt mixture at high temperature, and there is no significant difference between flexible and hot mix asphalt mixture at low temperature. Compared with 70% RAP warm mix recycled asphalt mixture, when the RAP content is 50%, the dynamic modulus increases the most at high temperature (50\u0026deg;C), and the phase angle decreases the least at low temperature (5\u0026deg;C). Therefore, 50% RAP warm mix recycled asphalt mixture has better high and low temperature performance. According to the principle of time-temperature equivalence, \u003cb\u003eS\u003c/b\u003e (GMS model) and\u003cb\u003eφ\u003c/b\u003e ( improved phase angle model ) with auxiliary parameter \u003cb\u003eρ\u003c/b\u003e are fitted synchronously to realize parameter sharing, and \u003cb\u003eS\u003c/b\u003e and\u003cb\u003eφ\u003c/b\u003e models can well fit the variation of dynamic modulus and phase angle with frequency.\u003c/p\u003e","manuscriptTitle":"Viscoelastic properties of warm recycled asphalt mixture based on S-φ model","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-07-24 14:26:07","doi":"10.21203/rs.3.rs-6912958/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-10-27T10:57:24+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-08-02T09:25:51+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-07-29T11:37:49+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-07-27T18:43:08+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"332843995844048457930926098001624226488","date":"2025-07-27T14:32:13+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"271635224192167445205981006964359585279","date":"2025-07-23T01:15:35+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"287112864858756335380932536969311610607","date":"2025-07-22T13:42:51+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-07-22T13:40:16+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"25890039146492769149167394450244640883","date":"2025-07-22T13:26:44+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-07-22T13:17:37+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-06-25T13:10:52+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-06-25T13:09:59+00:00","index":"","fulltext":""},{"type":"submitted","content":"Discover Civil Engineering","date":"2025-06-17T09:41:36+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"discover-civil-engineering","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Discover Civil Engineering](https://www.springer.com/journal/44290)","snPcode":"44290","submissionUrl":"https://submission.nature.com/new-submission/44290","title":"Discover Civil Engineering","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Discover Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"a8396c4e-5768-4cb9-9cbb-ba42dcf419bb","owner":[],"postedDate":"July 24th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[],"tags":[],"updatedAt":"2026-05-14T10:11:05+00:00","versionOfRecord":[],"versionCreatedAt":"2025-07-24 14:26:07","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6912958","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6912958","identity":"rs-6912958","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}