# Local Stiffness Assessment of Geogrid-Stabilized Unbound Aggregates in a Large-Scale Testbed
## 5 Results and Discussion  
### Estimation of Poisson's Ratio, Applied Bulk Stress, and Small-Strain Elastic Modulus  
During the repeated load triaxial tests, the resilient moduli (\(M_{R}\)) were estimated by averaging the resilient strain and deviator stress values from the last five cycles out of the 100 load repetitions applied at each stress state using Equation (1):  
\[M_{R}=\frac{\sigma_{d}}{\varepsilon_{r}} \tag{1}\]  
where \(\sigma_{d}\) is the deviator stress and \(\varepsilon_{r}\) is the recoverable resilient strain. The computed resilient modulus from each stress state is displayed in Figure 6, which shows the increase in resilient modulus with bulk stress and illustrates the stress-hardening characteristic of the unbound aggregate material. The stress-dependent moduli were characterized using the Hicks and Monismith (K-\(\theta\)) model for M\({}_{R}\), as expressed in Equation (2) [47]. A high coefficient of determination (0.94) was obtained. \[M_{R}=31.029\,\theta^{0.9417} \tag{2}\]  
where \(\theta\) is bulk stress applied to the specimen. Figure 7 displays the shear and compressional wave signals obtained after each test stage of the repeated load triaxial test. The compressional wave signals were only recorded  
Figure 5: LWD and DCP tests: (**a**) LWD device on the testbed surface of the testbed; and (**b**) Locations where DCP and LWD tests were conducted. Figure 6: Resilient moduli determined at each of the 15 stress states illustrated along with the regression curve using the k-\(\theta\) model. from Stress States 4 to 15. At lower stress levels, the confining pressure was not sufficient to provide enough support to the P-wave sensor, which was necessary for the generation of compressional waves. The first arrival times of the elastic wave signals gradually decreased as the magnitude of the bulk stress applied to the specimen increased, up to Stage 15. The elastic wave velocities, for shear and P waves, can be calculated using Equation (3) with the known distance between the source and the receiver wave transducers. \[\mathrm{V_{P}}\mathrm{\ or\ V_{S}}=\frac{L}{t_{tip-tip}} \tag{3}\]  
where \(\mathrm{V_{P}}\) and \(\mathrm{V_{S}}\) are the compressional and shear wave velocities, respectively, \(t_{tip-tip}\) is the first arrival time of the elastic wave, and L is the distance between the two transducers. While the modulus of the specimen is a function of the applied bulk stress, shear wave velocity (\(\mathrm{V_{s}}\)) is a function of the small-strain shear modulus and can be expressed by Equation (4) [48]:  
\[\mathrm{V_{S}}=\left(\frac{G_{max}}{\rho}\right)^{0.5} \tag{4}\]  
where \(\rho\) is the density of the specimen and \(G_{max}\) is the small strain shear modulus. Therefore, the shear wave velocity can be expressed in terms of the applied bulk stress. Figure 8 presents the correlation between the applied bulk stress and the shear wave velocity, which is provided in Equation (5):  
\[\mathrm{V_{S}}=48.169\ \theta^{0.324} \tag{5}\]  
In reverse order, the bulk stress applied to the unbound aggregate specimen can also be estimated using the shear wave velocity as follows:  
\[\theta=\left(0.0208V_{S}\right)^{3.086} \tag{6}\]  
The Poisson's ratio of an isotropic material can be calculated from its shear wave velocity and compressional wave velocity as follows:  
\[v=\frac{0.5\Big{(}\frac{V_{P}}{V_{S}}\Big{)}^{2}-1}{\Big{(}\frac{V_{P}}{V_{S} }\Big{)}^{2}-1} \tag{7}\]  
Figure 7: Recorded elastic waves: (**a**) Shear waves from Stages 1 to 15; and (**b**) Compressional waves from Stages 4 to 15. Figure 9 illustrates the Poisson's ratio of the unbound aggregate material with regards to the applied bulk stress. The Poisson's ratio is a function of density, gradation, and applied confining pressure. The ratio decreases as the confining pressure increases [49; 50; 51; 52; 53]. Therefore, [PERSON]'s ratio can be expressed as a function of the bulk stress applied to the specimen as expressed in Equation (8). A high coefficient of determination (0.928) was achieved.