Introduction

Since asphalt mixtures are viscoelastic materials, the operating conditions, such as velocity of the travelling load and pavement temperature, have a great influence on the strain state (Graczyk 2010). These conditions may vary within a large range (Rafalski 2007). Reflecting the stress and strain states of asphalt pavement taking into account both the temperature effect and the load velocity requires applying an appropriate computational model. The most important issue is the choice of the proper constitutive model used in order to describe the material response against mechanical and thermal loadings. Furthermore, the identification of material parameters is essential (Radziszewski et al. 2014). It can be done based on laboratory tests or by using an identification procedure proposed by Zbiciak et al. (2017). Among many proposals of rheological models in the literature, the most popular is the Burgers model. However, analysis of a wide range of time–temperature operating conditions can require the use of more complex models such as a generalised Maxwell model or Huet–Sayegh model (Zbiciak and Michalczyk 2014). One of previous paper (Zbiciak et al. 2017) shows comparison of results obtained for the Burgers and the Huet–Sayegh model in one temperature. On this basis, it was decided to use in this study only Huet–Sayegh, which performs better.

In this study, the identification of viscoelastic constitutive models is based on both laboratory tests results and mixture composition (Sybilski et al. 2010). Thus, the process of identification of Huet–Sayegh parameters has been carried out.

Materials and methods

Viscoelastic models of asphalt–aggregate mixtures

In order to describe the rheological properties of asphalt–aggregate mixture, Huet–Sayegh (HS) model has been used. The HS model contains non-classical linear viscoelastic elements whose constitutive properties are defined by fractional derivatives (Podlubny 1999; Butera and Di Paola 2014; Di Mino et al. 2016; Grzesikiewicz et al. 2013; Zbiciak 2012). The fractional derivatives are used in solution of many physical problems (Rzodkiewicz et al. 2009), including viscoelastic behaviour of certain materials from rheological point of view.

In the literature, the HS rheological model is usually defined via its complex stiffness modulus (frequency response), that is (Kim 2009),

$$E^{ * } \left( {i{\kern 1pt} \omega } \right) = E_{\text{o}} + \frac{{E_{\infty } - E_{\text{o}} }}{{ 1+ \delta \,\left( {i{\kern 1pt} \omega \,\tau } \right)^{ - k} + \left( {i{\kern 1pt} \omega \,\tau } \right)^{ - h} }} ,$$
(1)

where ω denotes the frequency of excitation; \(E_{\text{o}}\) is the static modulus (as ω → 0); E∞ is the glass transition modulus (as ω → ∞); and \(h{\kern 1pt} ,\;k\) are parameters obeying the conditions 0 < k < h < 1; δ is a dimensionless constant; and τ is the characteristic time.

The complex stiffness modulus \(E^{ * } \left( {i{\kern 1pt} \omega } \right)\) constitutes the amplitude of the harmonic steady-state stress response \(\sigma^{ * } \left( {i{\kern 1pt} \omega } \right) = E^{ * } \left( {i{\kern 1pt} \omega } \right)\,\,e^{{i\omega {\kern 1pt} t}}\) excited by the complex strain \(\varepsilon^{ * } \left( {i{\kern 1pt} \omega } \right) = e^{{i\omega {\kern 1pt} t}}\). The frequency response of any linear system can be characterised by the magnitude \(\left| {E^{ * } \left( {i{\kern 1pt} \omega } \right){\kern 1pt} } \right|\), sometimes called the dynamic stiffness modulus Edyn(ω), and the phase ϕ(ω). Thus, the following representation of the complex quantity holds

$$E^{*} \left( {i{\kern 1pt} \omega } \right) = E_{dyn} \left( \omega \right)\,\,e^{i\varphi \left( \omega \right)} ,\quad E_{dyn} \left( \omega \right): = \left| {E^{ * } \left( {i{\kern 1pt} \omega } \right){\kern 1pt} } \right| .$$
(2)

The magnitude and phase of the system frequency response, when using a logarithmic frequency scale, are called Bode plots. Moreover, the magnitude as a function of phase is called a Nichols or Black chart. Another characteristic of linear systems results from decomposition of the frequency response into real and imaginary parts

$${E^{ * } \left( {i\omega } \right) = E^{\prime}\left( \omega \right) + i\,E^{\prime\prime}\left( \omega \right)},$$
(3)
$$E^{\prime}\left( \omega \right) = \text{Re} \, E^{ * } = E_{dyn} \left( \omega \right)\,\,{ \cos }\,\varphi \,,\quad E^{\prime\prime}\left( \omega \right) = \text{Im} \, E^{ * } = E_{dyn} \left( \omega \right)\,\,{ \sin }\,\varphi ,$$
(4)

where E′ denotes the storage modulus representing elastic properties of the material and E′′ is the loss modulus used to represent viscous properties. A parametric graph visualising the real part of the frequency response versus its imaginary part is called a Nyquist or Cole–Cole plot.

It can be proved that the fractional-order differential equation (FDE) describing the HS stress–strain (\(\sigma { - }\varepsilon\)) relationship is as follows:

$$\dot{\sigma }\left( t \right) + \delta {\kern 1pt} \tau^{ - k} \,\text{D}^{\,1 - k} \sigma \left( t \right) + \tau^{ - h} \,\text{D}^{\,1 - h} \sigma \left( t \right) = E_{\infty } {\kern 1pt} \dot{\varepsilon }\left( t \right) + E_{\text{o}} \left[ {{\kern 1pt} \delta {\kern 1pt} \tau^{ - k} \,\text{D}^{\,1 - k} \varepsilon \left( t \right) + \tau^{ - h} \,\text{D}^{\,1 - h} \varepsilon \left( t \right)} \right] ,$$
(5)

where the initial condition \(\sigma \left( 0 \right) = \sigma_{\text{o}}\) is applied. The following definition of the fractional derivative of the order α ∊ (0, 1) for a function z(t) can be applied to Eq. (5) (Grzesikiewicz et al. 2013):

$$\text{D}^{\alpha } {\kern 1pt} z\,\left( t \right): = \frac{z\left( 0 \right)}{{\varGamma \left( {1 - \alpha } \right)\,t^{\alpha } }}\, + \frac{1}{{\varGamma \left( {1 - \alpha } \right)}}\,\,\int\limits_{0}^{t} {\,\frac{{\dot{z}\left( \xi \right)}}{{\left( {t - \xi } \right)^{\alpha } }}} \,{\text{d}}\xi ,$$
(6)

where Γ( · ) denotes the gamma function.

Identification of the asphalt mixture models based only on the stiffness moduli results by using the HS structure is difficult. In order to make this possible, an artificial creep curve was constructed which takes into account the correlation between the secant modulus values obtained in creep tests and the stiffness modulus determined in the cyclic test, where the appropriate relationship between the time and frequency of the load occurs. Thus, the laboratory results of the stiffness modulus tests were converted into points lying on a curve creep, as described in detail in a previous study of the authors (Zbiciak et al. 2017). A similar approach may also be used in analysing the rheological properties of asphalt (Brzezinski and Krainski 2016).

The fundamental problem is that there are no known analytical solutions of the HS creep function. On the other hand, all the characteristics of any linear fractional model (step response, impulse response, etc.) can be obtained by applying numerical algorithms for determining the inverse Laplace transforms.

As it was stated in Zbiciak and Grzesikiewicz (2011), numerical determination of the creep characteristics of the HS model requires the understanding of its transfer function E*(s), which results directly from Eq. (1) when substituting \(i{\kern 1pt} \omega\) by s (see Eq. 7a). The transfer function E*(s) characterises the response of the system and is defined as the ratio of the Laplace transforms of output and input signals (Eq. 7b)

$$E^{ * } \left( s \right) = \left. {E^{ * } \left( {i{\kern 1pt} \omega } \right)\,} \right|_{{i{\kern 1pt} \omega = s}} ,$$
(7a)
$$E^{ * } \left( s \right) = \frac{{\sigma^{ * } \left( s \right)}}{{\varepsilon^{ * } \left( s \right)}} ,$$
(7b)

where \(\sigma^{ * } \left( s \right) = \text{L}\,\left\{ {{\kern 1pt} \sigma \left( t \right)} \right\}\) and \(\varepsilon^{ * } \left( s \right) = \text{L}\,\left\{ {{\kern 1pt} \varepsilon \left( t \right)} \right\}\) are the Laplace transforms of the stress (output) and the strain (input) states, respectively.

Moving back to differential formulation of the HS model, it should be emphasised that the transfer function E*(s) results directly from Eq. (5) by applying the Laplace transform and using Eq. (7b).

The creep response of the Huet–Sayegh model can be obtained by applying an inverse Laplace transform \(\text{L}^{{\,{\kern 1pt} - 1}}\) as it is schematically visualised in Fig. 1, where H(t) denotes a Heaviside step function modelling the excitation and J(t) is the creep function defined as the strain response to a stress step excitation. In the case of the Huet–Sayegh model, obtaining an analytical form of the inverse Laplace transform is problematic. Thus, the process should be carried out numerically by using the algorithms described in the literature (Valsa and Brancik 1998; Liu 2001).

Fig. 1
figure 1

Input–output relationship to obtain a creep response

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Creep master curve development

The temperature influences the viscosity of the material (Barzinjy and Zankana 2016). The frequency range of the measured components of the complex modulus (dynamic modulus and phase angle) covers only a small part of the total frequency range at different temperatures. The reduced variables method can be applied in order to reduce the experimental data to single curves (master curves). Master curves cover a wide range of frequencies at a chosen reference temperature. The same idea can be applied to the creep tests where the reduced variables method allows for the visualisation of all the experiments in a single creep master curve covering all times of loadings (at a particular reference temperature).

The method of reduced variables is also known as the temperature–frequency or the temperature–time superposition principle (TTSP) (Kim 2009; Wang 2010). Application of the TTSP requires for the data, collected at different temperatures, to be shifted relative to the time of loading or frequency. Experimental curves representing components of the complex modulus or creep curves obtained at various temperatures can be aligned to form a single master curve. The required shift at a given temperature is defined by the shift factor.

Results and discussion

In order to model the time–temperature relationships for pavement asphaltic layers, we used various equations known in the literature (Zbiciak and Michalczyk 2014). The most popular is the William–Landel–Ferry (WLF) equation. Assuming the WLF equation, we can derive relationship between the shift factor a(T) and temperature T presented in Eq. (8a). It should be substituted into Eq. (1) via parameter τ, which is a function of temperature (Eq. 8b). A summary description of the procedure in one temperature is presented in Zbiciak et al. (2017). The data for asphaltic mixture were obtained from tests conducted in TN-248 study (Sybilski et al. 2010).

$${ \log }\left[ {a\,\left( T \right)} \right] = - \frac{{C_{1} \,\left( {T - T_{\text{o}} } \right)}}{{C_{2} + T - T_{\text{o}} }},$$
(8a)
$$\tau = \tau_{\text{o}} \,a\,\left( T \right),$$
(8b)

where C1 and C2 are fitting parameters, \(T_{\text{o}}\) denotes the reference temperature and \(\tau_{\text{o}} = \tau \left( {T_{\text{o}} } \right)\) is a constant to be determined for an arbitrarily chosen reference temperature.

The identification results of the HS rheological model for asphalt mixtures used in analysis of the pavement structure are summarised in Tables 1 and 2. Moreover, Figs. 2, 3 and 4 visualise selected curve fitting results presenting experimental creep points and fitted curves for binder course mixture.

Table 1 Coefficients of HS model and WLF shift function—binder course mixture (AC16W 50/70)
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Table 2 Coefficients of HS model and WLF shift function—base course mixture (AC16P 50/70)
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Fig. 2
figure 2

Curve fitting results (creep curves at different temperatures) based on a mixture composition for AC16W 50/70 (semilogarithmic and linear scales)

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Fig. 3
figure 3

Creep master curve based on mixture composition for AC16W 50/70 at a reference temperature of 10 °C (semilogarithmic and linear scales)

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Fig. 4
figure 4

Dynamic characteristics of AC16W 50/70 rheological model fitted based on mixture composition

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Conclusion

In this paper, the problem of characterisation of asphalt mixtures constitutive models, with application to pavement structures, was investigated. In order to describe the rheological properties of asphalt mixture, Huet–Sayegh model was used, because it suitably describes mixture properties in a wide range of time–temperature loading conditions. Identification was done based on laboratory tests data and mixture composition. Based on the results, the following conclusions were drawn:

  • The HS model can satisfactorily describe the rheological properties of asphalt mixes. What distinguishes it is the fact that its parameters are relatable to the construction of complex modulus and phase angle master curves, the Black diagram and the Cole–Cole diagram.

  • In the time–temperature superposition and the shift factor equations study, the WLF equation generally produced very good results compared to the other equations and showed the highest correlation with measured shift factor data.

  • The accuracy of the new method based on mixture composition is satisfactory. The main purpose is to avoid labour-intensive and cost-consuming laboratory tests at the early design stages.

  • In particular, by applying numerical “extraction” of mixture creep curve from experimental mixture data, the procedure performed much better than classical empirical equations.

Since the analysis was performed only on narrow range of mix designs (Sybilski et al. 2010), it is recommended that the validation be extended to mix designs that include different types of binder and aggregate types. The analysis should also be extended to mixtures containing polymer-modified bitumens.