# Non-Newtonian Behaviors of Crumb Rubber-Modified Bituminous Binders
## 1 Introduction  
Bitumen is a complex mix of various hydrocarbons that is produced from the natural deposits or the residue of the crude oil distillation process [1; 2]. The complex chemical composition of bitumen determines its various physical and mechanical properties. Bitumen is usually categorized as a viscoelastic material, which means its mechanical behavior is time of loading and temperature dependent. The primary application of bituminous binders is in road construction and other uses include roofing products, sealing and insulating materials. Viscosity, defined as a measure of resistance to deformation or flow, is one of the most important properties of bituminous binders used as paving materials. Profound knowledge of the viscosity of these binders at manufacturing and construction temperatures (generally above 135 \({}^{\circ}\)C) provides vital information about their pumpability, mixability and workability [3; 4]. Workability can be defined as a property that describes the ease with which hot mix asphalt can be placed, worked by hand, and compacted. The viscosity of bitumen should be controlled within a certain range to facilitate its application in production and construction as well as to guarantee the bitumen grade [5]. Generally, bituminous binders can exhibit Newtonian or non-Newtonian behaviors under different conditions. The dominant characteristic of a Newtonian behavior in simple shear experiments conducted at constant temperature and pressure is the exerted shear stress that is constantly proportional to the applied shear rate. In contrast, any deviation from the Newtonian behavior (i.e., viscosity is dependent on applied shear rate and shear stress) is called non-Newtonian. Non-Newtonian fluids have two main categories: shear-thinning (pseudoplastic) fluids and shear-thickening (dilatant) fluids. Shear thinning is the behavior of fluids whose viscosity decreases with the rate of shear strain while the viscosity increases with the rate of shear strain for shear-thickening fluids. Apart from the applied shear rate, shearing time and temperature also have effects on the non-Newtonian behaviors [6]. Different mathematical equations were developed to describe the flow curves (viscosity measured over a range of shear rates or shear stresses under isothermal conditions) of various products, among them the Cross model, the Carreau model and the Sisko mode which are the modified versions of the Cross model [7]. The temperature dependence of viscosity can be characterized by the William-Landel-Ferry (WLF) function and the Arrhenius-type equation or some empirical equations [8]. However, these equations developed based on different theories have different applicable conditions. Misuse of these equations may result in biased predictions and explanations [9]. This issue will be addressed in the results section. Practically, two measures of viscosity in asphalt paving industry have been used: absolute viscosity and apparent viscosity [10]. Performing absolute viscosity assumes the material as a Newtonian fluid whose properties are independent of applied shear rate or stress level. Unmodified bituminous binders usually exhibit Newtonian behavior at relatively elevated temperatures (generally above the softening point) or at very low shear rates. However, considering the large service temperature range of these binders or short loading times, their viscosity is not always an absolute value at intermediate and low temperatures. Rather, it varies with the applied shear rate and shear stress. That is why it is an apparent value which needs a well-defined temperature, shear rate and shear stress. For the majority of highly modified bituminous binders, they exhibit obvious non-Newtonian behaviors due to the interaction between the bitumen and the modifier [11].