Assessment of Thermal Stresses in Asphalt Mixtures at Low Temperatures Using the Tensile Creep Test and the Bending Beam Creep Test  
[PERSON]  
1 Faculty of Civil and Environmental Engineering, Gdansk University of Technology, 80-233 Gdansk, Poland; [EMAIL_ADDRESS] (M.J.); [EMAIL_ADDRESS] (C.S.) 1 Faculty of Civil and Environmental Engineering, Gdansk University of Technology, 80-233 Gdansk, Poland; [EMAIL_ADDRESS] (M.J.  
adding mineral fiber increased the creep rate of asphalt mixtures, and low-temperature properties were improved. The problem of thermal stresses is important not only in asphalt layers but also in concrete pavements subjected to seasonal and daily changes of temperature [12; 13]. Tensile thermal stresses that occur in the asphalt layers during cooling are very difficult to measure directly in the pavement structure. Therefore, correct analytical estimation of thermal stresses is of crucial importance for evaluation of asphalt binders and design of asphalt mixtures. Calculation of tensile thermal stresses in asphalt pavements has been a subject of research for over 50 years. Viscoelastic solutions for thermal stresses were developed in the early 1960s by [PERSON] [14]. [PERSON] [15] and [PERSON] [16]. This approach was dedicated to a specific class of polymers and used the temperature-dependent viscoelastic material parameters. The viscoelastic solution was also adopted in 1965 by [PERSON] et al. [17]. The basic viscoelastic equation for thermal stress calculation is as follows:  
\[\sigma(t)=\int_{0}^{t}E(t-\bar{\xi})\frac{\delta\varepsilon(\bar{\xi})}{\delta \bar{\xi}}d\bar{\xi} \tag{1}\]  
where: \(\sigma(t)\)-thermal stress in a viscoelastic slab at time t, \(E(t)\)-relaxation modulus of the viscoelastic slab as a function of time t, \(\varepsilon\) - thermal strain induced in the viscoelastic slab by change of temperature and calculated as:  
\[\varepsilon(\bar{\xi})=\alpha_{T}[T(\bar{\xi})-T_{0}] \tag{2}\]  
where: \(\alpha_{T}\)-linear coefficient of thermal expansion, \(\xi\)-reduced time associated with time and temperature: \(\xi=\frac{t}{a_{T}}\) and \(a_{T}\)-temperature shift factor, \(T(\bar{\xi})\)-pavement temperature at reduced time \(\bar{\xi}\), \(T_{0}\)-pavement temperature when \(\sigma(t)=0\). The viscoelastic solution presented in Equations (1) and (2) has been used for calculation of thermal stresses in asphalt layers by several researchers [18; 19; 20; 21; 22; 23; 24; 25]. This method has also been incorporated in the new AASHTO mechanistic-empirical method of pavement design [26] as well as the AASHTO PP 42-02 [27] and ASTM D6816-11 [28] standards. In 1966, [PERSON] and Brien introduced a simple quasi-elastic solution for calculation of thermal stresses [29]. Due to the fact that asphalt mixtures are viscoelastic materials with time-temperature dependent properties, the basic disadvantage of this method is that the viscoelastic nature of bituminous material and stress relaxation behavior is not fully considered. Nevertheless, mainly for its simplicity, the [PERSON] and Brien method of thermal stress calculation has been used in several research papers [30; 31; 32; 33; 34]. In this method the thermal stresses are calculated from the following equation:  
\[\sigma(T)=\frac{1}{1-\
u}\sum_{i=1}^{n}S(t,\ T_{i})\cdot\alpha_{T}\cdot\Delta T \tag{3}\]  
where: \(\sigma(T)\)-thermal stress induced in an asphalt layer at temperature T, \(\
u\)-[PERSON]'s ratio of the asphalt layer, \(S(t,\ T)\)-stiffness modulus of the asphalt layer as a function of time of loading \(t\) and temperature \(T_{i}\), \(\alpha_{T}\)-coefficient of thermal expansion of the asphalt layer, \(\Delta T\)-increment of temperature assumed in calculations, \(i=1,\dots,n\)-steps in calculations, \(T_{i}\) temperature at step \(i\) and \(T_{i}=T_{i-1}+\Delta T\). Thermal stresses can be also assessed in laboratory and the results of thermal stress calculations can be then compared to the laboratory test results. In the research of [PERSON] et al.