# Comparison of Lab vs. Backcalculated Moduli of Virgin Aggregate and Recycled Aggregate Base Layers
## 2 Methodology
### Nonlinearity of Unbound Materials  
Cyclic triaxial testing is usually used to evaluate the stiffness of unbound materials [22]. During this test, the unbound materials generally exhibited a plastic deformation in the initial cycles, which later stabilized at a certain level. This phenomenon is usually referred to as shakedown, after which a steady state and elastic response is expected from the unbound materials. The \(M_{R}\) at a certain applied load level is then defined as the ratio of applied deviatoric stress (\(\sigma_{d}\)) to the recoverable elastic strain (\(\xi_{r}\)) after the shakedown, as shown in Equation (1):  
\[M_{R}=\frac{\sigma_{d}}{\varepsilon_{r}} \tag{1}\]  
where \(M_{R}\) is the resilient modulus (psi), \(\sigma_{d}\) is the deviatoric stress (psi), and \(\varepsilon_{r}\) is the recoverable peak-to-peak strain after the shakedown. Several models have been proposed in the literature to capture the nonlinear stress-dependency of the MR to the stress state within the unbound materials caused by different applied load levels. These include models such as the K-\(\theta\) model developed by [PERSON] and [PERSON] [24] and the universal model developed by [PERSON] and [PERSON] [25]. Later, the universal model was modified during the NCHRP 1-37A project [2], as shown in Equation (2). This model has become the most commonly used model to describe the nonlinearity of unbound materials. \[M_{R}=k_{1}.p_{a}\left(\frac{\theta}{p_{a}}\right)^{k_{2}}\left(\frac{\tau_{ oct}}{p_{a}}+1\right)^{k_{3}} \tag{2}\]  
where \(\theta\) is the bulk stress (psi), \(\tau_{oct}\) is the octahedral shear stress (psi), \(p_{a}\) is the atmospheric air pressure (-14.706 psi), and \(k_{1}\), \(k_{2}\), and \(k_{3}\) are the model coefficients. The \(\theta\) and \(\tau_{oct}\) parameters in the two-dimensional (2D) axisymmetric system can be defined as in Equations (3)-(5):  
\[\theta=\sigma_{r}+\sigma_{z}+\sigma_{t} \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Material** & **Gravel 1** & **Sand 2** & **Fine 3 (\%)** & **C\({}_{\textbf{a}}\)4** & **C\({}_{\textbf{c}}\)5** & **LL 6** & **USCs 7** & **MDU 8** & **OMC 9** & **App.10** \\ \hline Sand (subgrade) & 27.6 & 59.8 & 12.6 & 33.12 & 1.24 & 19.9 & SM & 134.84 & 5.6 & 1.84 \\ \hline Clay loam & 3.1 & 37.2 & 59.7 & N/A & N/A & 36.3 & CL & 122.36 & 10 & N/A \\ \hline Sand (subbase) & 31.1 & 56.5 & 12.4 & 30.30 & 1.10 & 18.9 & SM & 137.34 & 5.3 & 1.53 \\ \hline \hline \end{tabular}  
* \({}^{1}\) Percent of gravel (\%), measured based on ASTM D6913 (2004) [14]. 2 Percent of fan sc (\%), measured based on ASTM D6913 (2004) [14]. 4 Coefficient of uniformity, measured based on ASTM D2487 (2017) [15]. 5 Coefficient of curvature, measured based on ASTM D2487 (2017) [15]. 6 Liquid limit (\%), measured based on PSI 1377-2 (1990) [16]. 7 Unified soil classification system, classified based on ASTM D2487 (2017) [15]. 8 Maximum dry unit weight (pfc), measured based on ASTM D1557 (2020) [17] and ASTM DAT718 (2015) [19]. 9 Optimum moisture content (\%), measured based on ASTM D1557 (2020) [17] and ASTM DAT718 (2015). 1810 Absorption, calculated by taking the weighted average of the absorption of the coarse and fine fractions (based on ASTM C127 (2015) [19] and ASTM C128 (2015) [20]) of the materials. For the clay loam, the ASTM D854 (2014) [21] was used. 11 Not applicable (N/A). Unit conversion: 1 pcf = 0.157 kN/m3  
\end{table} 
Table 1: \(Cont.\)\[\tau_{oct}=\sqrt{\frac{2}{3}I_{2}} \tag{4}\]  
\[J_{2}=\frac{1}{6}\Big{[}(\sigma_{r}-\sigma_{z})^{2}+(\sigma_{r}-\sigma_{t})^{2}+( \sigma_{t}-\sigma_{z})^{2}\Big{]} \tag{5}\]  
where \(\sigma_{r},\sigma_{z}\), and \(\sigma_{t}\) are the radial, vertical, and tangential shear stress components (psi), and \(J_{2}\) is the second invariant of the stress (psi\({}^{2}\)). The NCHRP 1-37A model for unbound materials was used to model the unbound layers' nonlinearity under the multi-load FWD testing.