StudentStudent ID

HUNG D. NGUYEN4120762

INTRODUCTION TO SOME RHEOLOGICAL MODELS

1. The Modified Sigmoidal Model

Mathematically, the |G*| equation of Sigmoidal model is shown as the following:

Where, log() is log reduced frequency, is the difference between the value upper and lower asymptote, and define the shape of between the asymptotes. However, as the limiting modulus (Go) is equal to 1 GPa, the above equation can be transformed as (Garcia Thomson, 2007):

Where, Max is the Go at 1 GPa (Bonaquist and Christensen, 2005). The other parameters are as previous defined.

The Generalized Logistic Sigmoidal Model

Rowe et al. (2008) introduced the generalization of the Sigmoid model, called the Generalized Logistic Sigmoidal model to predict the |G*| of bitumen and can be shown as the following:;

Where, the parameters are previously defined above. The parameter allows the curve to take a non-symmetrical shape. Like the modified Sigmoidal model, the Go can bee taken as 1 GPa, therefore the following equation can be used:

2. The Christensen Anderson Model (CA)

The |G*| equation of the Christensen and Anderson(CA) model can be presented as (Christensen and Anderson, 1992):

|G*|=Gg[1+(c/)log2/R]R/log2

Where, c is the crossover frequency and R is a rheological index. The other parameters are as defined previously.

The Christensen Anderson and Marasteanu(CAM)

The Christensen Anderson and Marateanu(CAM) model was developed to improve the descriptions of unmodified and modified bitumens. The|G*| equation can be shown as follows (Marasteanu and Anderson, 1990):

|G*|=Gg[1+(c/)v]w/v

The introduction of w parameter addresses the issue of how fast or how slow the |G*| data converse into two asymptotes(the 45o asymptote and the Go asymptote ) as the frequency goes to zero or infinity.

3. The 2S2P1D Model

The |G*| of the 2S2P1D model can be shown in the following mathematical expression (Olard and Benedetto, 2003):

Figure 1 : The 2S2P1D Model

ANALYSIS OF THE RELATIONSHIP OF COMPLEX MODULUS G* WITH TEMPERATURE, FREQUENCY AND

PHASE ANGLE BY USING CONTINUOS DYNAMIC FREQUENCY SCANNING.

(Guilian Zou, Xiaoning Zhang, Jian Xu and Fengxia Chi)

Where, I is complex number(i2=-1), is frequency, k and h are exponents with 0<k<h<1, is constant, G is the static modulus, Go is the glassy modulus, is constant and defined by =(G-Go), is the Newtonian viscosity and is the characteristic time, function of temperature. The 2S2P1D model’s representation can be shown in Fig.1.

1. Experiment Procedure

Rheological measurements :- Using an advanced rheometer (AR-2000)- The size of sample : 50x10x10 mm- Frequency range : 1.2x107 ~ 100 Hz- Temperature range : -40 to 100oC- The torque can be controlled from 0.1 to 200000 Nm

Material- Two neat asphalt binders designed as A-70 and A-90 were used

(one modified with styrene-butadiene-styrene and one an epoxy asphalt)

- Detail properties of two asphalt binders are described in the following table

Table 1 : Properties of Asphalt BindersProperties Unit A-70 A-90Penetration(25oC, 100g, 5s) 0.1mm 63 84Penetration Index / -0.8 -1.08T R&B oC 51.2 47.2Ductility (15oC, 5cm/min) cm >100 >100 60

oC Pa s 254 154

135oC Pa s 0.391 0.31

Table 2 : Gradations of AggregationSieves size (mm)13.0 9.50 4.75 2.36 1.18 0.6 0.3 0.15 0.075

Types of asphalt mixture

Percent passing (%)

AC-5 100 100 97.7 74.1 56.5 42.7 22.0 14.3 8.5AC-10 100 97.5 65 48 34.5 25 17 11 6.5

2. Description of dynamic rheological master curve

In this study, Williams, Landel and Ferry (WLF) equation and Arrhenius equation is not applicable. Another equation proposed by Wang Duanyi is used :

Log a(T) = ko + k1T + k2T2

Where T = temperature and ko, k1, k2 = material parametersFig.1 show a group of complex modulus

mas master curves for AC-5(A70) mixture in the the temperature range from -20 to 90oC.

Fig. 2 show storage modulus and loss modulus of AC-5(A70) mixture.- In the middle of frequency zone : loss modulus is above storage modulus master curve, indicating that the viscous component is prominent in this zone- In the high and low zone : loss modulus is beneath storage modulus master curve, indicating that the elastic component is prominent.

In this study, Christensen-Anderson-Marasteanu(CAM) rheological model is used with the equation for complex modulus master curve followed by

r = log[2me/k/1 + (2m2/k – 1)/Ge*/G*

g] (1)and the equation for the phase angle of the model followed by = 90I – (90I - m){1+[log(ft/f’)/Rd]2}-m d/2

Reference :

Zou G., Zhang X., Xu J. and Chi F. (2010) Morphology of Asphalt Mixture Rheological Master Curve Journal of Material in Civil Engineering@ASCE/August 2010 Volume 8

Thom N. (2008) Principles of pavement engineering Thomas Telford Publishing Ltd 2008

Airey G.D., Hainin M.R. and Yusoff N.I.Md (2010) Predictability of Complex Modulus Using Rheological Models Asian Journal of Scientific Research 3 (1): 18-30

Fig.5 illustrates the complex modulus master curve for equation (1). Gg

* is the maximum asymptotic modulus in shear, representing the response at very high frequencies. Gg

* is the equilibrium modulus, representing the minimum modulus at very low frequencies.

Fig. 6 shows the morphological parameters for the master curve of G*