# Non-Newtonian Behaviors of Crumb Rubber-Modified Bituminous Binders
## 3 Results and Discussion
### Yield Stress  
As observed in previous subsection, bituminous binders modified by wax additives do not show a practical or show an infinite viscosity approaching zero shear rate. The reason for this special non-Newtonian behavior can be attributed to the yield stress characteristic which is a property associated with numerous types of complex fluids. For example, paint can only flow with the help of brush, but remains fixed on a vertical wall despite the force of its own gravity. The yield stress is defined as a critical stress that must be applied to the material before flow or continuous deformation is observed [7]. Below the yield stress, the solid-like bituminous binder will deform elastically which results from elastic elements being stretched in the shear field. Above the yield stress, such elastic elements begin to break down causing shear-thinning behaviors and consequent flow. Although many researchers have doubt about whether the true yield stress exists [35], the concept of yield stress, if defined properly, is very useful in a whole range of applications, such as stirring, mixing and pumping.  
To determinate the yield stress of the wax-modified bituminous binders, the Herschel-Bulkley model was applied to fit the measured data (shear stress versus shear rate) and extrapolate to a zero-shear rate. The Herschel-Bulkley model [7] describes non-Newtonian behavior after yielding and is basically composed of a power low model with a yield stress term as shown in Equation (3):  
\[\sigma=\sigma_{y}+K\omega^{n} \tag{3}\]  
In Equation (3) above \(\sigma\) is the shear stress; \(\sigma_{y}\) is the yield stress; \(K\) is the consistency; and \(n\) is the flow behavior index. The parameter \(n\) describes the material as shear thinning when \(n<1\), or shear thickening when \(n>1\), or Newtonian when \(n=1\).  
The yield stress determined through the Herschel-Bulkley model was shown in Figure 6. As analyzed before, for bituminous binders having practical ZSV do not have a yield stress. This is also verified in Figure 6a as the fitted equations do not have the yield stress term. The yield stresses of wax modified binder 70/100-W and CRMB-W are clearly plotted in Figure 6b with the values of 190.60 Pa and 4.22 Pa, respectively. The significant difference between the yield stresses shows additive W has different extents of effect on the base bitumen and CRMB. This is possibly due to the different microstructures of these two binders which stem from the potentially complex interaction between bitumen, rubber and warm mix additives. Generally, larger yield stress indicates more stable internal structure of the material [35]. The yield stress brings both positive and negative effects to the material  
Figure 5: Measured and model predicted viscosity values for different binders at 50 \({}^{\circ}\)C.  
depending on the application conditions. For instance, higher yield stress can be a hindrance if the binder is being pumped or mixed while it will benefit the binder making it more stable during the service stage on the pavement. One also should note that the predicted yield stress is highly dependent on the instrument resolution, measuring conditions and analysis method.