Four-mm Dynamic Shear Rheometry

508 Captions

Figure 1. Photo. Four-mm DSR. In this photo, 25 mg of asphalt binder is shown between 4-mm diameter parallel plates with a 1.75-mm gap for oscillatory shear testing by dynamic shear rheometry.

Figure 2. Graph. Four-mm DSR—aged asphalt binder and complex shear modulus master curves from corrected and uncorrected data. Dynamic shear rheometry performed at low temperature and high frequency requires an instrument compliance correction (blue) to reveal the true low temperature rheology. This is shown as a plot of the complex shear modulus versus reduced frequency where the corrected master curve (blue) has a higher modulus at high frequency than the uncorrected master curve. The axes ranges are 10⁷ to 10¹⁰ Pascals for the complex modulus and 10² to 10¹⁴ radians per second for the reduced frequency. The reference temperature is 25 degrees Celsius.

Figure 3. Equation. Shear modulus. The equation calculates G as equal to tau divided by gamma.

Figure 4. Equation. Measured strain. The equation calculates measured gamma as equal to sample gamma plus instrument gamma, which equals tau divided by sample G plus tau divided by instrument G.

Figure 5. Graph. Example: determination of instrument compliance from the slope of the linear fit of the anglular displacement and torque measurements. The linear best-fit equation of the plot of angular displacement (y-axis) vs. torque (x-axis) is used to determine slope, which is the instrument compliance. The plot shows a linear fit with a slope (machine compliance) of 3.0245 micro radians per gram-centimeter, an intercept of zero, and an R² of 1. The axes ranges are 0–3000 micro radians for the angular displacement and 0–1000 g-cm for the torque.

Figure 6. Equation. Storage modulus. The equation calculates G prime sub s as equal to the numerator G prime sub m times open parenthesis one minus the term consisting of the fraction J sub tool divided by k sub g, said fraction times G prime sub m close parenthesis; minus the term consisting of the fraction J sub tool divided by k sub g, said fraction times G double prime sub m squared; said numerator divided by the denominator open parenthesis 1 minus the term consisting of the fraction J sub tool divided by k sub g, said fraction times G prime sub m close parenthesis squared; plus open parenthesis fraction J sub tool divided by k sub g, said fraction times G double prime sub m, close parenthesis squared.

Figure 7. Equation. Loss modulus. The equation calculates G double prime sub s as equal to G double prime sub m divided by open parenthesis one minus the term consisting of the fraction J sub tool divided by k sub g, said fraction times G prime sub m close parenthesis squared; plus open parenthesis J sub tool divided by k sub g times G double prime sub m close parenthesis squared.

Figure 8. Equation. Tangent phase angle. The equation calculates the tangent of delta as equal to G double prime sub m divided by G prime sub m times open parenthesis one minus the term consisting of the fraction J sub tool divided by k sub g, said fraction times G prime sub m close parenthesis; minus J sub tool divided by k sub g times G double prime sub m squared.

Figure 9. Graph. Master curves of G* combining data collected on DSR with 4-, 8-, and 25‑mm parallel plates for a PAV-aged asphalt. This graph depicts the G asterisk omega master curve for a pressure aging vessel-aged asphalt using 4-mm dynamic shear rheometry at temperatures ranging from negative thirty to thirty degrees Celsius in black, and the G asterisk omega master curve of the same asphalt using 8-mm and 25-mm parallel plates at intermediate and high temperatures ranging from zero to eighty degrees Celsius in red. The machine compliance corrections for the 4-mm parallel plate data were included for the master curve shown in black.

Figure 10. Graph. Four-mm DSR—relaxation modulus G(t) and the slope at 2 h. The graph depicts the measurement of m-value and creep stiffness S(t) by 4-mm dynamic shear rheometry at the true low performance-grade temperature. The plot shows the logarithm of the relaxation modulus (Pascals) versus the logarithm of reduced time (seconds) and the slope at 7200 s, m_r, in units of log-Pascals divided by log-seconds)

Figure 11. Graph. BBR—creep stiffness and m-value at 60 s. The graph depicts the measurement of m-value and creep stiffness S(t) by the Bending Beam Rheometer at ten degrees Celsius above the true low performance-grade temperature. The plot shows the logarithm of the creep stiffness (Pascals) versus the logarithm of loading time (seconds) and the slope at 60 s, m_c, in units of log-Pascals divided by log-seconds.

Figure 12. Graph. Correlation between BBR S(60s) and 4-mm DSR G(60s). The graph depicts the correlation between 60-s Bending Beam Rheometer creep stiffness data on the y-axis versus 60-s dynamic shear rheometry relaxation modulus data on the x-axis. The correlation coefficient for the two data sets is 0.95.

Figure 13. Graph. Correlation between BBR m_c(60s) and 4-mm DSR m_r(60s). The graph depicts the correlation between 60-s Bending Beam Rheometer creep stiffness slope data on the y-axis versus 60-s dynamic shear rheometry relaxation modulus slope data on the x-axis. The correlation coefficient for the two data sets is 0.86.

Figure 14. Graph. PG+10 °C and PG+20 °C frequency sweeps. The graph depicts G prime omega master curves at a reference temperature of performance-graded plus 10 °C (red) and performance-graded plus 20 °C (blue). Plotting the two master curves is required as the first step in the calculation of the 60-s dynamic shear rheometry relaxation modulus and slope.

Figure 15. Graph. G' master curve at a reference temperature PG+10 °C. The graph depicts how Microsoft® Excel solver is subsequently used to horizontally shift the performance-grade temperature plus 20 °C frequency sweep to overlap the performance-grade temperature plus 10°C frequency sweep. The plot shows the leftward shift necessary to overlay the two master curves. The vertical axis shows the logarithm of G-prime, and the horizontal axis shows the logarithm of frequency (radians per second).

Figure 16. Equation. Shear relaxation modulus. The equation calculates G open parenthesis t close parenthesis as approximately equal to G prime open parenthesis omega close parenthesis when omega equals two divided by pi times time.

Figure 17. Graph. Relaxation modulus master curve to determine m_r(60 s) and G(60s). The graph depicts the preparation of the relaxation modulus master curve from the figure 15 equation. The first derivative of the second order polynomial is used to determine the slope of the logarithm of the relaxation modulus at 60 s. The graph shows the logarithm of the relaxation modulus (Pascals) plotted versus the logarithm of reduced time (seconds). The data are fit with a second order polynomial, and the location on the relaxation master curve at a reduced time of 60 s is marked.