Viscoelastic Mechanical Model of Asphalt Concrete Considering the Influence of Characteristic Parameter of Fiber Content

Our tests use petroleum asphalt 70 and polyester fibers with a mean diameter of 18.5μm. To study the influence of the volume ratio V_f and aspect ratio R_a of the fibers, the fiber length was set to 3mm, 6mm, and 9mm, respectively; the aspect ratio was set to 162, 324, 486, and 649, respectively. For the polyester fibers of the length 3mm, 9mm, and 12mm, the volume ratio was set to 0.351, 0.34, and 0.346 in turn. For those of the length 6mm, the volume ratio was set to 0.174, 0.348, 0.519, and 0.686 in turn.

The aggregates were screened, cleaned, and dried, and then mixed with limestone powder back to AC-13 asphalt concrete with medium grading. Through standard Marshall test [15], the optimal asphalt content (OAC) was determined for the matrix of asphalt mix, and for the asphalt mix with each volume ratio and aspect ratio of the fibers.

The designed polyester fiber-reinforced asphalt mix was compressed into specimens of 300mm×300mm×50mm, and then cut into small beams of 250mm×30mm×35mm. Then, creep tests were carried out at 15℃ on a multi-function material testing machine. The specimens were loaded with weights. The test environment is illustrated in Figure 1. The creep load is 10% of the bending failure load of the beam under the same conditions.

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Figure 1. Creep test environment

The creep tests were performed on eight groups of small beams. Each group contains three small beams. During the tests, the midspan deflection d(t) of the small beam being tested was collected by a dial gauge, which is connected to the data acquisition system, and the time-deflection curve was plotted. When the small beam entered accelerated creep phase, the load was removed, and the midspan deflection was collected 30min after the unloading. Based on the midspan deflection, the bottom flexural-tensile strain can be calculated by:

$\varepsilon (t)=\frac{6hd(t)}{{{L}^{2}}}$            (1)

where, h is the height of the specimen (m); L is the span of the specimen (m).

Referring to the Burgers model [7, 13] in Figure 4, the viscoelastic creep equation of the fiber-reinforced asphalt concrete can be expressed as:

Loading phase:

$\varepsilon=\sigma_{0}\left(\frac{1}{\mathrm{E}_{1}}+\frac{t}{\eta_{1}}+\frac{1-e^{-t \tau}}{\mathrm{E}_{2}}\right)$       (2)

Unloading phase:

$\varepsilon=\sigma_{0}\left[\frac{t_{0}}{\eta_{1}}+\frac{\left(1-e^{-t_{0} \tau}\right) e^{-\tau\left(t-t_{0}\right)\quad}}{\mathrm{E}_{2}}\right]$               (3)

where, Ε₁ and η₁ are the elastic modulus of elastic element 1 and viscosity of viscous element 4 considering fiber influence, respectively; Ε₂ and η₂ are the elastic modulus of elastic element 2 and viscosity of viscous element 3 considering fiber influence, respectively; $\tau=\frac{E_{2}}{\eta_{2}}$.

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Figure 4. Burgers model

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Figure 5. Modified Burgers model

The greater the value of Ε₁, the stronger the elastic deformation resistance of the asphalt concrete; the greater the value of η₁, the smaller the permanent deformation induced by the same load in the same period; the greater the value of η₂/Ε₂ the slower the growth of viscous deformation over time, and the shallower the rutting depth on the asphalt concrete pavement; the greater the value of η₂/Ε₂, the slower the development of asphalt concrete deformation, and the stronger the resistance to viscoelastic deformation [1].

Drawing on the modified Burgers model (Figure 5) [8], the viscoelastic creep equation of fiber-reinforced asphalt concrete can be expressed as:

Loading phase:

$\varepsilon(t)=\sigma_{0}\left(\frac{1}{\mathrm{E}_{1}}+\frac{1-e^{-B t}}{A B}+\frac{1-e^{-t \tau}}{\mathrm{E}_{2}}\right)$              (4)

Unloading phase:

$\varepsilon (t)={{\sigma }_{0}}\left[ \frac{1-{{e}^{-B{{t}_{0}}}}}{AB}+\frac{(1-{{e}^{-{{t}_{0}}\tau }}){{e}^{-(t-{{t}_{0}})\tau\quad }}}{{{E}_{2}}} \right]$               (5)

where, σ₀ is the test stress; t is the test time; t₀ is the total loading time.

Based on the creep test results, the parameters of the viscoelastic mechanical model (2)-(5) were initialized through the nonlinear fitting approach of Origin 8.5. By adjusting model parameters, the fitting approach that maximize the matching between theoretical results and the test data was adopted to determine the parameters of the viscoelastic mechanical model. On this basis, the authors analyzed the influence of fiber volume ratio and fiber aspect ratio on model parameters. Then, the FCCP was introduced to reflect the combined effect of fiber volume ratio and fiber aspect ratio, and used to build the constitutive equation of the fiber-reinforced asphalt concrete. Finally, the viscoelasticity of the fiber-reinforced asphalt concrete was examined by the creep equation and the constitutive equation.

4.1 Influence of fiber volume ratio and fiber aspect ratio on viscoelastic model parameters and viscoelastic performance

Table 1 shows the viscoelastic model parameters of the fiber-reinforced asphalt concrete with different fiber volume ratios and fiber aspect ratios, which are fitted by formula (2). It can be observed that Ε₁ first increased and then decreased, with the growing fiber volume ratio. The peak of Ε₁ appeared at the fiber volume ratio of 0.348. In this case, the fiber-reinforced asphalt concrete has a strong resistance to the elastic deformation under instantaneous load. When the fiber volume ratio reached 0.686, the Ε₁ of the fiber-reinforced asphalt concrete was smaller than that of the asphalt concrete in the matrix, indicating that the fibers can enhance the elasticity of the fiber-reinforced asphalt concrete only under a proper volume ratio.

With the increase of fiber volume ratio, the η₁ first increased and then declined, suggesting the good thickening effect of a proper dosage of fibers. When the fiber volume ratio was 0.348, the η₁ more than doubled from that of the asphalt concrete in the matrix. In this case, the fibers exhibit a good thickening effect, and the fiber-reinforced asphalt concrete is good at resisting permanent deformation. After the fiber volume ratio reached 0.686, the fiber-reinforced asphalt concrete had a smaller η₁ than the asphalt concrete in the matrix, that is, an excessive number of fibers would weaken the ability of the asphalt concrete to withstand permanent deformation.

Table 1. Relationship between parameters of formula (2) and fiber volume ratio and fiber aspect ratio

Table 2. Relationship between parameters of formula (3) and fiber volume ratio and fiber aspect ratio

The relaxation time η₁/Ε₁ and delay time η₂/Ε₂ of the fiber-reinforced asphalt concrete both increased and then decreased with the growth of fiber volume ratio. Therefore, a proper dosage of fibers could enhance the resistance of the asphalt concrete to rutting deformation. The relaxation time and delay time reached the peak at the fiber volume ratio of 0.348. In this case, the fibers significantly enhance the resistance of the asphalt concrete to rutting deformation [17].

Ε₁, η₁, η₁/Ε₁ and η₂/Ε₂ all increased first and then dropped with the growth of fiber aspect ratio, and peaked at the fiber aspect ratio of 324. In this case, the fiber-reinforced asphalt concrete boasts a strong capability to resist elastic deformation, viscous flow deformation, and rutting deformation.

Table 3 shows the viscoelastic model parameters of the fiber-reinforced asphalt concrete with different fiber volume ratios and fiber aspect ratios, which are fitted by formula (3). It can be observed that, Ε₂ and η₂ were much greater than those in the creep loading phase; the delay time η₂/Ε₂ was longer than that in the creep loading phase; η₁ was much smaller than that in that phase.

This is because the asphalt concrete suffers from creep failure in the accelerated creep phase. During the load-induced bending creep deformation, the internal aggregates of each small asphalt concrete beam slip, causing material damage and performance attenuation. The actual permanent deformation, which is characterized by the parameters of the loading phase model, is greater than the theoretical permanent deformation, which is characterized by the parameters of the unloading phase model. Besides, the actual deformation recovery rate, which is characterized by the parameters of the loading phase model, is smaller than the theoretical deformation recovery rate, which is characterized by the parameters of the unloading phase model. Therefore, the actual deformation features of the unloaded asphalt concrete cannot be characterized by the mechanical model parameters obtained in the creep loading phase.

Ε₂, η₁ and η₂ all increased first and then decreased with the growth of fiber volume ratio and fiber aspect ratio. The peak values were observed at the fiber volume ratio of 0.348 and the fiber aspect ratio of 324. The variation law was the same as that in the creep loading phase.

It can also be seen from Table 2 that the correlation between test data and formula (3) in the creep unloading phase was weaker than that in the creep loading phase. Overall, formulas (2) and (3) can characterize the viscoelastic deformation features of asphalt concrete excellently during the loading phase.

Table 3 shows the viscoelastic model parameters of the fiber-reinforced asphalt concrete with different fiber volume ratios and fiber aspect ratios, which are fitted by formula (4). It can be observed that formula (4) is more closely correlated with the test data in the creep loading phase than formula (2). The parameters of formula (4), namely, Ε₁, Ε₂, η₂, A and B all first increased and then decreased with the growth of fiber volume ratio and fiber aspect ratio. The peak values were observed at the fiber volume ratio of 0.348 and the fiber aspect ratio of 324. In this case, the fiber-reinforced asphalt concrete boasts a strong capability to resist elastic deformation, viscous flow deformation, and rutting deformation.

Table 4 shows the viscoelastic model parameters of the fiber-reinforced asphalt concrete with different fiber volume ratios and fiber aspect ratios, which are fitted by formula (5). It can be observed that the parameters of formula (5), namely, Ε₂, η₂, A and B all first increased and then decreased with the growth of fiber volume ratio and fiber aspect ratio. The peak values were observed at the fiber volume ratio of 0.348 and the fiber aspect ratio of 324. The change law was the same as that in formula (4). The only difference between formulas (5) and (4) lies in the variation of the value of the same parameter.

Compared with those in the creep loading phase, the model parameters Ε₂ and B were relatively large, and η₂ and A were relatively small in the creep unloading phase. This phenomenon indicates that the asphalt concrete recovers slower from viscoelastic deformation, and deforms more significantly than the theoretical permanent deformation, after being damaged in creep loading. Although formulas (4) and (5) describe different creep phases from formulas (2) and (3), the changes of their parameters demonstrate the same influence law of creep loading damages on material performance.

It can also be learned from Table 4 that formula (5) is much less correlated with the test data in the creep unloading phase than formula (4), and slightly less correlated with the test data in that phase than formula (3). Hence, formulas (4) and (5) are not as good as formulas (2) and (3) in characterizing the deformation features of unloaded asphalt concrete. To sum up, formulas (4) and (5) work well in characterizing the deformation features of the asphalt concrete and fiber-reinforced asphalt concrete in the loading phase, while formulas (2) and (3) work well in characterizing the deformation features of the asphalt concrete and fiber-reinforced asphalt concrete in the unloading phase.

Table 3. Relationship between parameters of formula (4) and fiber volume ratio and fiber aspect ratio

Table 4. Relationship between parameters of formula (5) and fiber volume ratio and fiber aspect ratio

4.2 Viscoelastic model and viscoelasticity analysis

The above analysis shows that fiber volume ratio V_f and fiber aspect ratio R_a are important impactors of the viscoelastic performance of the asphalt concrete. This paper chooses the FCCP $\lambda_{f}=V_{f} \times R_{a}$ to represent the overall impact of V_f and R_a [18]. The model parameters of formulas (2)-(5) all exhibit the same change law with the growth of FCCP: increasing first before declining. By nonlinear fitting of our test data, the relationship between the model parameters of formulas (2) and (3) and the FCCP can be expressed as:

Loading phase:

$\mathrm{E}_{1}\left(\lambda_{f}\right)=603.55+454.29 \lambda_{f}-201.49 \lambda_{f}^{2}$              (6)

$\mathrm{E}_{2}\left(\lambda_{f}\right)=109.81+82.27 \lambda_{f}-34.66 \lambda_{f}^{2}$                (7)

${{\eta }_{1}}({{\lambda }_{f}})=298232.9+627942.9{{\lambda }_{f}}-290013\lambda _{f}^{2}$             (8)

${{\eta }_{2}}({{\lambda }_{f}})=29864.95+33547.72{{\lambda }_{f}}-13724.8\lambda _{f}^{2}$            (9)

Unloading phase:

$\mathrm{E}_{2}\left(\lambda_{f}\right)=343.43+103.25 \lambda_{f}-46.58 \lambda_{f}^{2}$              (10)

${{\eta }_{1}}({{\lambda }_{f}})=247449.9+275225.8{{\lambda }_{f}}-136606\lambda _{f}^{2}$            (11)

${{\eta }_{2}}({{\lambda }_{f}})=109657.3+89963.02{{\lambda }_{f}}-39147.1\lambda _{f}^{2}$             (12)

Substituting formulas (6)-(9) and formulas (10)-(12) into formulas (2) and (3), respectively, the creep equation of the fiber-reinforced asphalt concrete can be established considering the influence of the FCCP:

Loading phase:

$\varepsilon (t,{{\lambda }_{f}})={{\sigma }_{0}}\left[ \frac{1}{{{E}_{1}}({{\lambda }_{f}})}+\frac{t}{{{\eta }_{1}}({{\lambda }_{f}})}+\frac{1-{{e}^{-t\tau }}}{{{E}_{2}}({{\lambda }_{f}})} \right]$              (13)

Unloading phase:

$\varepsilon\left(t, \lambda_{f}\right)=\sigma_{0}\left[\frac{t_{0}}{\eta_{1}\left(\lambda_{f}\right)}+\frac{\left(1-e^{-t_{0} \tau}\right) e^{-\left(t-t_{0}\right)\quad}}{E_{2}\left(\lambda_{f}\right)}\right]$           (14)

where, $\tau=E_{2}\left(\lambda_{f}\right) / \eta_{2}\left(\lambda_{f}\right)$.

Finding the derivative of formulas (13) and (14) relative to time t, respectively, the differential constitutive equation of the fiber-reinforced asphalt concrete can be established considering the influence of the FCCP:

Loading phase:

$d\varepsilon (t,{{\lambda }_{f}})/dt=\dot{\varepsilon }(t,{{\lambda }_{f}})={{\sigma }_{0}}\left[ \frac{1}{{{\eta }_{1}}({{\lambda }_{f}})}+\frac{{{e}^{-t\tau }}}{{{\eta }_{2}}({{\lambda }_{f}})} \right]$                   (15)

Unloading phase:

$d \varepsilon\left(t, \lambda_{f}\right) / d t=\dot{\varepsilon}\left(t, \lambda_{f}\right)=\sigma_{0}\left[\frac{\left(e^{-t_{0} \tau}-1\right) e^{-\left(t-t_{0}\right) \tau\quad}}{\eta_{2}\left(\lambda_{f}\right)}\right]$                 (16)

Figure 6 shows the relationship between creep strain and the FCCP obtained by formula (13). Under the same loading time and different stress levels, the creep strain always decreased first and then increased with the growth of the FCCP, and reached the minimum at the FCCP of 1.128. This is very close to the test results. The greater the stress, and the longer the loading, the more significant the creep strain of the fiber-reinforced asphalt concrete.

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Figure 6. Relationship between creep strain and FCCP

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Figure 7. Relationship between creep rate and FCCP

Figure 7 shows the relationship between creep rate and the FCCP obtained by formula (15). It can be observed that, the creep rate first decreased and then increased, with the growth of the FCCP, and minimized at the FCCP of 1.128. This is also very close to the test results. The greater the stress, the faster the creep. With the elapse of the loading time, the creep rate continued to slow down.

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Figure 8. Relationship between creep strain and FCCP after unloading

Figure 8 shows the relationship between creep strain of the fiber-reinforced asphalt concrete after unloading with the FCCP. It can be inferred that the residual creep deformation after unloading increased with the loading stress. The residual creep deformation first increased and then decreased, with the growth of the FCCP. After fixing the time of loading and unloading, the FCCP was 1.128 corresponding to the minimum residual creep deformation of the fiber-reinforced asphalt concrete, under different loading stresses. In this case, the fiber-reinforced asphalt concrete has a strong ability to recover from creep deformation. Formula (14) also reveals that the residual creep deformation increases with the time of creep loading.

Figure 9 shows the relationship between creep rate the fiber-reinforced asphalt concrete after unloading with the FCCP. It can be inferred that, due to the inertia of elastic deformation recover after unloading, the greater the loading stress, the faster the recovery from creep deformation. With the growth of the FCCP, the creep recovery rate first increased and then decreased. After fixing the time of loading and unloading, the FCCP was 1.128 corresponding to the maximum creep recovery rate of the fiber-reinforced asphalt concrete, under different loading stresses. In this case, the fiber-reinforced asphalt concrete has a strong ability to recover from creep deformation. Formula (16) also reveals that the longer the loading time, the greater the damages of the fiber-reinforced asphalt concrete induced by creep load, and the smaller the recovery rate from the creep deformation.

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Figure 9. Relationship between creep rate and FCCP after unloading

By nonlinear fitting of our test data, the relationship between the model parameters of formulas (4) and (5) and the FCCP can be expressed as:

Loading phase:

$\mathrm{E}_{1}\left(\lambda_{f}\right)=489.33+193.48 \lambda_{f}-97.46 \lambda_{f}^{2}$                (17)

$\mathrm{E}_{2}\left(\lambda_{f}\right)=110.91+59.49 \lambda_{f}-27.82 \lambda_{f}^{2}$               (18)

$\eta_{2}\left(\lambda_{f}\right)=327619+914046 \lambda_{f}-412653 \lambda_{f}^{2}$            (19)

$\mathrm{A}\left(\lambda_{f}\right)=36674.63+95928.89 \lambda_{f}-41400.4 \lambda_{f}^{2}$                   (20)

$\mathrm{B}\left(\lambda_{f}\right)=1.76 \times 10^{-4}+1.975 \times 10^{-3} \lambda_{f}-8.8 \times 10^{-4} \lambda_{f}^{2}$                 (21)

Unloading phase:

$\mathrm{E}_{2}\left(\lambda_{f}\right)=340.60+164.24 \lambda_{f}-72.98 \lambda_{f}^{2}$                (22)

${{\eta }_{2}}({{\lambda }_{f}})=108072.8+96344.6{{\lambda }_{f}}-41409.7\lambda _{f}^{2}$                   (23)

$\mathrm{A}\left(\lambda_{f}\right)=9729.09+18998.79 \lambda_{f}-7340.91 \lambda_{f}^{2}$                (24)

$\mathrm{B}\left(\lambda_{f}\right)=1.916 \times 10^{-3}+5.062 \times 10^{-3} \lambda_{f}-2.28 \times 10^{-3} \lambda_{f}^{2}$            (25)

Substituting formulas (17)-(21) and formulas (22)-(25) into formulas (4) and (5), respectively, the creep equation of the fiber-reinforced asphalt concrete can be established considering the influence of the FCCP:

Loading phase:

$\varepsilon\left(t, \lambda_{f}\right)=\sigma_{0}\left[\frac{1}{\mathrm{E}_{1}\left(\lambda_{f}\right)}+\frac{1-e^{-B\left(\lambda_{f}\right) t}}{A\left(\lambda_{f}\right) B\left(\lambda_{f}\right)}+\frac{1-e^{-t \tau}}{\mathrm{E}_{2}\left(\lambda_{f}\right)}\right]$          (26)

Unloading phase:

$\varepsilon\left(t, \lambda_{f}\right)=\sigma_{0}\left[\frac{1-e^{-\mathrm{B}\left(\lambda_{f}\right) t_{0}}}{A\left(\lambda_{f}\right) B\left(\lambda_{f}\right)}+\frac{\left(1-e^{-t_{0} \tau}\right) e^{-\left(t-t_{0}\right) \tau\quad}}{\mathrm{E}_{2}\left(\lambda_{f}\right)}\right]$                   (27)

where, $\tau=E_{2}\left(\lambda_{f}\right) / \eta_{2}\left(\lambda_{f}\right)$.

Finding the derivative of formulas (26) and (27) relative to time t, respectively, the differential constitutive equation of the fiber-reinforced asphalt concrete can be established based on formulas (4) and (5) considering the influence of the FCCP:

Loading phase:

$d\varepsilon (t,{{\lambda }_{f}})/dt=\dot{\varepsilon }(t,{{\lambda }_{f}})={{\sigma }_{0}}\left[ \frac{{{e}^{-B({{\lambda }_{f}})t}}}{A({{\lambda }_{f}})}+\frac{{{e}^{-t\tau }}}{{{\eta }_{2}}({{\lambda }_{f}})} \right]$                (28)

Unloading phase:

$d\varepsilon (t,{{\lambda }_{f}})/dt=\dot{\varepsilon }(t,{{\lambda }_{f}})={{\sigma }_{0}}\left[ \frac{({{e}^{-{{t}_{0}}\tau }}-1){{e}^{-(t-t{}_{0})\tau\quad }}}{{{\eta }_{2}}({{\lambda }_{f}})} \right]$           (29)

The creep equation and differential constitutive equation of the fiber-reinforced asphalt concrete derived from formula (4), and those derived from formula (2) can yield consistent results on the correlations of creep strain and creep rate with the FCCP, stress, and loading time. Similarly, The creep equation and differential constitutive equation derived from formula (5), and those derived from formula (3) can yield consistent results on the correlations of post-unloading creep strain and creep rate with the FCCP, stress, and loading time.