Time–temperature superposition for viscoelastic materials with application to asphalt–aggregate mixes
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Abstract
This paper presents a new approach to characterisation of asphalt mixtures constitutive models, with application to pavement structures. It is wellknown fact that temperature, frequency and time of loading have a great influence on the mechanical properties of bituminous mixtures. For this reason, the properties are usually presented in the frequency domain as complex numbers having real and imaginary parts. This convenient representation in terms of complex modulus and phase angle leads to the dynamic modulus master curve. The effective use of the theory of linear viscoelasticity to characterise constitutive model of asphaltic material is shown in this paper. Viscoelastic constitutive model is represented by a combination of rheological schemes, and its identification is based on both laboratory tests results and mixture composition. The temperature–frequency or temperature–time superposition principle being applied in order to produce master curves of mechanical properties is illustrated with real experimental data. Further, the process of identification of Huet–Sayegh parameters is carried out using bestfitting methods implemented in MATLAB. Fractional rheological model is used as it needs only a small number of elements to fully characterise the response of asphalt materials.
Keywords
Asphalt–aggregate mixtures Complex moduli Curve fitting Master curve ViscoelasticityIntroduction
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 nonclassical 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.
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.
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).
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
Coefficients of HS model and WLF shift function—binder course mixture (AC16W 50/70)
Identification method  \(E_{\text{o}}\) (MPa)  E_{∞} (MPa)  k  h  δ  \(\tau { (}10^{ \circ } {\text{C)}}\)  C _{1}  C _{2} 

4 PB results  67  20,295  0.366  0.592  5.334  2.250  19.182  136.907 
Mixture composition  37  25,726  0.433  0.885  6.689  0.977  14.437  102.100 
Coefficients of HS model and WLF shift function—base course mixture (AC16P 50/70)
Identification method  \(E_{\text{o}}\) (MPa)  E_{∞} (MPa)  k  h  δ  \(\tau { (}10^{ \circ } {\text{C)}}\)  C _{1}  C _{2} 

4 PB results  82  27,503  0.339  0.646  4.658  2.320  19.998  138.710 
Mixture composition  78  27,701  0.438  0.802  5.763  0.822  18.490  146.600 
Conclusion

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 labourintensive and costconsuming 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 polymermodified bitumens.
Notes
Acknowledgements
This research was conducted by using the funds from Polish Government allocated to statutory aims of the Faculty of Civil Engineering. The data for asphaltic mixtures were obtained from TN248 study (Sybilski et al. 2010).
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