## 3 Experiments
### [PERSON] and [PERSON]'s Modulus Measurement of a Double-Layer Phantom
The two coefficient matrixes \({}^{n}\mathbf{A}\) were  
\[{}^{n}A_{11}=\sum\
olimits_{k=1}^{M_{n}}\varepsilon_{3,k}v_{k} \tag{27}\] \[{}^{n}A_{12}=\sum\
olimits_{k=1}^{M_{n}}(\varepsilon_{1,k}+ \varepsilon_{2,k})v_{k}\] \[{}^{n}A_{21}=\sum\
olimits_{k=1}^{M_{n}}[\varepsilon_{1,k}(z_{k} -z_{b})(z_{k}-z_{t})+\varepsilon_{3,k}x_{k}(2z_{k}-z_{b}-z_{t})]v_{k}\] \[{}^{n}A_{22}=\sum\
olimits_{k=1}^{M_{n}}\Big{[}(\varepsilon_{2,k} +\varepsilon_{3,k})(z_{k}-z_{b})(z_{k}-z_{t})-\tfrac{1}{2}\varepsilon_{5,k}x_{ k}(2z_{k}-z_{b}-z_{t})\Big{]}v_{k}\]  
where the superscript \(n\) on the upper left corner represents the \(n\)th layer and \(n=1,2\), where \(1\) means the bottom layer and \(2\) means the top layer; \(M_{n}\) is the number of meshes in the \(n\)th layer; \(v_{k}\) is the volume of the \(k\)th mesh and \(v_{k}=LxLyLz/M\). One key step for the calculation is to segment the two layers according to the strain-gradient as described earlier, so as to determine the value of \(M_{n}\). After the two layers were divided as shown in Figure 6c, the \(M_{n}\) were calculated to be \(M_{1}=143\) and \(M_{2}=172\) respectively. According to Equation (19) and the static equilibrium, the virtual work vectors \({}^{n}\mathbf{B}\) are  
\[{}^{1}B_{1}=\tfrac{TLxLyLz}{M}\sum\limits_{d=1}^{N_{x}N_{y}}N_{d}, \tag{28}\] \[{}^{1}B_{2}=0;\]  
\[{}^{2}B_{1}=TLxLyLz\Bigg{(}1-\tfrac{1}{M}\sum\limits_{d=1}^{N_{x}N_{y}}N_{d} \Bigg{)}, \tag{29}\] Figure 7.0  
where \(N_{x}\) and \(N_{y}\) are the number of points in the \(x\) and \(y\) directions respectively; \(N_{d}\) is the \(d\)th maximum number in the \(z\)-direction of the bottom layer, where \(d=1,2,\dots,N_{x}N_{y}\). Then the coefficient matrixes \({}^{n}\mathbf{A}\) and the virtual work vectors \({}^{n}\mathbf{B}\) were substituted into the linear equations, Equation (16). The elastic constants \({}^{n}Q_{11}\) and \({}^{n}Q_{12}\) of the two layers were then solved as: \({}^{1}\mathbf{Q}=(1.1920\times 10^{6}\ \text{Pa},\,8.8362\times 10^{5}\ \text{Pa})^{\text{T}}\); \({}^{2}\mathbf{Q}=(3.2683\times 10^{5}\ \text{Pa},\,2.4166\times 10^{5}\ \text{Pa})^{\text{T}}\). The [PERSON]'s moduli and [PERSON]'s ratios were respectively \(E_{1}=439.7\ \text{kPa},\,\
u_{1}=0.426;\,E_{2}=121.4\ \text{kPa},\,\
u_{2}=0.425,\) where the subscript \(1\) means the bottom layer and the subscript \(2\) means the top layer. The [PERSON]'s moduli were also measured by tensile tests. The comparison of the results between the VFM and the tensile tests is listed in Table 1, which shows that the relative discrepancy between the two measurements is less than  
Figure 7.1 3D displacement measurement of a double-layer phantom. (**a**) OCT images: (**a-i**) a 3D OCT image with the ROI marked by a blue cube; (**a-ii**) a 2D OCT image with the ROI marked by a blue rectangle; (**b–d**) 3D displacements in the \(x,y\) and \(z\) directions respectively. Two displacement maps on the right side of each 3D image are the cross-sections along the dashed lines on (**b–d**). 5%, which is 5% better than a simple plane boundary as we used before. The [PERSON]'s ratio obtained by DVC was 0.422. The relative difference between the method of DVC and VFM is 1%. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Young’s Moduli** & **Tensile Tests** & **VMF** & **Relative Errors** \\ \hline \(E_{1}\) (kPa) & 462.1 & 439.7 & 4.8\% \\ \(E_{2}\) (kPa) & 120.8 & 121.4 & 0.5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The comparison between the tensile test and calculated constitutive parameters of a double-layer phantom by VFM. Figure 6: 3D strain fields of a double-layer phantom under compression. (**a–c**) Normal strains in the \(x\), \(y\) and \(z\) directions respectively. The purple dashed lines in (**c-i**) and (**c-ii**) indicate the interface of the two layers. (**d–f**) Shear strains. Two strain maps shown underneath each of the 3D displays are the images of the cross-sections along the dashed lines. Figure 7: Strain-gradient of \(\epsilon_{2}\) in the \(z\)-direction.