Correlation of Test Parameters

4.4.1 Correlation between Uniaxial and Shear Stiffness

Theoretically |G*| = |E*|/2(1+μ*) and if Poisson ratio μ* is taken as 0.5 the

|E*| is 3 times larger than |G*|. Figure 14 shows that the actual relationship between the two stiffness parameters does not agree with the theoretical relationship. This discrepancy is caused by several factors, such as non-linearity, anisotropy, measurement errors, etc. Therefore, |E*| and |G*| are not interchangeable and one cannot be obtained from another by using theoretical formulations.

The analysis of the correlation between |E*| and |G*| within each frequency/temperature combination separately indicated that the largest variations were between the 40°C/10Hz and 54°C/5Hz combinations, see Table 34. All regressions were performed using power model functional form. The best correlation between |E*| and |G*| was within the plant mixtures. This is a surprising result, because raw material mixtures had the least testing variation. The variable laboratory aging of raw material mixtures may be the source of these differences; test specimens for the shear and axial stiffness testing were prepared separately, which may have introduced variation in the prepared specimens. Because asphalt plant mixtures were

taken from the same truck, all stiffness variation caused by varying asphalt plant aging is eliminated.

y = 0.0249x^1.2817 R² = 0.9547 10

100 1000 10000

10 100 1000 10000

Unconfined |E*| (MPa)

|G*| (MPa)

Equality

|G*|=|E*|/3.0

Figure 14. SST |G*| and Unconfined |E*|, (Plant Mixtures).

Table 34 Correlation between |E*| and |G*|.

R² between |E*| and| G*|

Parameter Raw Materials

Plant Loose

1^st Coring (normalized)

2^nd Coring (normalized)

|E*| vs. |G*| 0.19-0.44 0.72-0.91 0.22-0.51 0.10-0.48

As shown in Figure 14, the relationship between |E*| and |G*| can be expressed by a simple power law model that has a generalized form as given in Eq. (2):

E b

a G*| | *|

| = (2)

Since |E*| and |G*| of mixtures are correlated by stiffness of the asphalt binder, this provides a possibility to predict |G*|mix based on |E*|mix or vice versa. Then by solving |G*| of binder with the Hirsch model (Christensen et. al, 2004) for the |E*|mix, the mixture shear stiffness can be estimated by using the solved binder stiffness in the Hirsch model. The whole process can be conveniently accomplished using Solver

function provided by the Microsoft Excel software. Analysis indicated that a very good linear correlation exists between the measured and estimated |G*| of mixture (R²=0.9487 for plant mixture). Therefore, for practical purposes it is accurate enough to back-calculate |G*| of binder, and then calculate the |G*| of mixture from the axial data, or vice versa. Table 35 gives the model coefficients for different test data sets to convert |E*| to |G*|.

Table 35. Model Coefficients for Stiffness Conversion Test Data Sets a b R²

Raw Materials 0.0526 1.1896 0.79 Plant Mixtures 0.0249 1.2817 0.95 1^st Coring 0.0268 1.1675 0.67 2^nd Coring 0.1133 1.0643 0.75

4.4.2 Correlation between Triaxial and IDT Strength

The triaxial shear strength test results were analyzed using Mohr-Coulomb failure theory to obtain c and φ. The relationship between the IDT strength St and the triaxial shear strength parameters c, φ based on the elastic theory is expressed by Eq.(3). This is further explained in the paper “Relationship between Triaxial Shear Strength and Indirect Tensile Strength” by Pellinen and Xiao (2005).

St

c φ

φ cos

sin 2−

= (3)

By using the measured cohesion c and friction angle φ as input parameters for Eq. (3) the IDT strength was predicted and then compared to the measured IDT strength. Figure 15 shows that for practical purposes the predicted and measured values agree well.

Table 36 shows the correlation coefficient between cohesion and IDT strength for the raw material and plant mixtures separately. For the raw material mixtures, correlation is fair but for the plant mixtures it is good to excellent. The best correlation

is obtained at the equivalent loading conditions; 7.5 vs. 0.06 mm/min, and 50 vs. 0.39 mm/min.

y = 1.0823x + 0.8043 R² = 0.8568

0 50 100 150 200 250

0 50 100 150 200 250

Predicted IDT St, kPa

Measured IDT St, kPa

Figure 15. Correlation of cohesion and IDT Strength.

Table 36. Correlation between cohesion and IDT Strength Correlation R² Raw

Material Plant Mixtures IDT S_t

Parameter Loading time

mm/min 0.06 0.06 0.39

7.5 0.68 0.96 0.85

c 50 ⎯ 0.78 0.97

For practical purposes, the IDT tensile strength can be used independently replacing the triaxial strength test. If no information is available for the friction angle, a ratio of 1.80 of the cohesion to IDT tensile strength is a good approximation.

4.4.3 Correlation of Triaxial Strength Parameters

Table 37 shows the correlation coefficients (R²) between different strength parameters obtained from the triaxial testing. Cohesion and unconfined compressive strength have a good to excellent correlation, which is facilitated by the quite narrow

range of friction angle values, i.e., 37 to 47°, measured from the mixtures. The correlation of friction angle to cohesion is poor to fair.

Table 37. Correlation between Triaxial Strength Parameters Correlation R² Raw Material Plant Mixtures

7.5 7.5 50 Parameter Loading time

mm/min S_u φ S_u φ S_u φ

7.5 0.97 0.0 0.95 0.61 0.85 0.62 c 50 ⎯ ⎯ ⎯ 0.50 0.96 0.64

4.4.4 Correlation between Stiffness and Strength

Many engineering materials possess different strength and stiffness properties.

Literature suggests that for asphalt mixtures the correlation of stiffness and strength is mix dependent (Pellinen, 2003).

A nonlinear power model regression was conducted to analyze the correlation between stiffness and strength of Indiana mixtures. The strength data used in the analysis included 7.5 mm/min loading time triaxial testing, and 0.06 and 0.39 mm/min loading time IDT testing. Overall, the correlation between stiffness and strength ranged from very poor to fair when all 11 mixtures were compared together (see Appendix 7), and especially for the field cores the correlation was very poor to non-existent. This confirms that the stiffest mixtures are not necessarily the strongest mixtures.

However, additional analysis was preformed to see if the correlation improves when the two different mix types were separated. The |E*| of dense-graded mixtures had a good to excellent correlation (R²= 0.75 to 0.96) to cohesion and fair correlation to IDT strength (R² = 0.48 to 0.74), while the correlation for the SMA mixtures was negative, meaning that the stiffer mixtures were weaker than the softer mixtures. The confined axial stiffness |E*| did not correlate to strength properties. Again, the field cores did not show any correlation between stiffness and strength.

For |G*|, the plant-produced mixtures had better correlation to strength than the raw material mixtures. In addition, the higher test temperature gave better correlations than the lower temperature, although all correlation values are somewhat lower than that for the axial stiffness