## 2 Equipment and Methods
### Vfm  
The VFM is based on the principle of virtual work, which has  
\[-\int_{V}\sigma:\varepsilon^{*}dV+\int_{S_{f}}\mathbf{T}\cdot\mathbf{u}^{*}dS+ \int_{V}\mathbf{b}\cdot\mathbf{u}^{*}dV=\int_{V}\rho\mathbf{a}\cdot\mathbf{u}^ {*}dV \tag{1}\]  
where \(\sigma\) is the real stress tensor; \(\varepsilon^{*}\) is the virtual strain tensor; \(\mathbf{u}^{*}\) is the virtual displacement vector which should meets the boundary conditions; \(\mathbf{T}\) is the external force vector on the surface of \(S_{f}\); \(\mathbf{b}\) is the body force vector; \(\mathbf{a}\) is the acceleration vector; \(\rho\) is the density; \(V\) is the volume; \(S\) is the surface; \"\(\cdot\)\" means the contraction operation of the second order tensor, i.e., \(\sigma\varepsilon^{*}=\sum_{i,j\
eq i,j}\mathbf{u}^{*}_{i,j}\), \(i,j=1,2,3\). In this paper, the body force \(\mathbf{b}\) can be neglected because it is relatively small compared with the external force \(\mathbf{T}\); the samples are compressed statically, so the acceleration \(\mathbf{a}\) can also be neglected. Hence, the Equation (1) can be simplified as  
\[-\int_{V}\sigma:\varepsilon^{*}dV+\int_{S_{f}}\mathbf{T}\cdot\mathbf{u}^{*}dS=0 \tag{2}\]  
For convenience, the stress tensor and strain tensor are written as vectors:  
\[\boldsymbol{\sigma}=\left(\sigma_{x},\sigma_{y},\sigma_{z},\sigma_{yz},\sigma _{xx},\sigma_{xy}\right)^{\text{T}}=\left(\sigma_{1},\sigma_{2},\sigma_{3}, \sigma_{4},\sigma_{5},\sigma_{6}\right)^{\text{T}} \tag{3}\]  
\[\boldsymbol{\varepsilon}=\left(\varepsilon_{x},\varepsilon_{y},\varepsilon_{ z},2\varepsilon_{yz},2\varepsilon_{zx},2\varepsilon_{xy}\right)^{\text{T}}= \left(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}, \varepsilon_{5},\varepsilon_{6}\right)^{\text{T}} \tag{4}\]  
According to elastic mechanics, the constitutive equations of 3D isotropic elastic materials are  
\[ \begin{array}{ll}\sigma_{1}=Q_{11}\varepsilon_{1}+Q_{12}(\varepsilon_{2}+ \varepsilon_{3}),&\sigma_{4}=\frac{1}{2}(Q_{11}-Q_{12})\varepsilon_{4},\\ \sigma_{2}=Q_{11}\varepsilon_{2}+Q_{12}(\varepsilon_{1}+\varepsilon_{3}),& \sigma_{5}=\frac{1}{2}(Q_{11}-Q_{12})\varepsilon_{5},\\ \sigma_{3}=Q_{11}\varepsilon_{3}+Q_{12}(\varepsilon_{1}+\varepsilon_{2}),& \sigma_{6}=\frac{1}{2}(Q_{11}-Q_{12})\varepsilon_{6},\end{array}  \tag{5}\]  
where the relationships between the two independent elastic parameters \(Q_{11}\), \(Q_{12}\) and [PERSON]'s modulus \(E\) and [PERSON]'s ratio \(\
u\) are  
\[Q_{11}=\frac{(1-\
u)E}{(1+\
u)(1-2\
u)},\ \ \ Q_{12}=\frac{\
u E}{(1+\
u)(1-2\
u)}. \tag{6}\]  
Then, Equation (2) becomes  
\[\int_{V}Q_{ij}\varepsilon_{j}\varepsilon_{i}^{*}dV=\int_{S_{f}}\mathbf{T} \cdot\mathbf{u}^{*}dS\ \ \ i,j=1,2,\cdots,6, \tag{7}\]  
where \(Q_{ij}\) is elastic constant. If the material is homogenous, the Equation (7) can be expressed as  
\[ \begin{array}{l}Q_{11}\!\int_{V}\left(\varepsilon_{1}\varepsilon_{1}^{*}+ \varepsilon_{2}\varepsilon_{2}^{*}+\varepsilon_{3}\varepsilon_{3}^{*}+\frac {1}{2}\varepsilon_{4}\varepsilon_{4}^{*}+\frac{1}{2}\varepsilon_{5}\varepsilon_ {5}^{*}+\frac{1}{2}\varepsilon_{6}\varepsilon_{6}^{*}\right)dV\\ +Q_{12}\!\int_{V}\left(\begin{array}{c}\varepsilon_{1}\varepsilon_{3}^{*}+ \varepsilon_{3}\varepsilon_{1}^{*}+\varepsilon_{2}\varepsilon_{3}^{*}+ \varepsilon_{3}\varepsilon_{2}^{*}+\varepsilon_{1}\varepsilon_{2}^{*}+ \varepsilon_{2}\varepsilon_{1}^{*}\\ -\frac{1}{2}\varepsilon_{4}\varepsilon_{4}^{*}-\frac{1}{2}\varepsilon_{5} \varepsilon_{5}^{*}-\frac{1}{2}\varepsilon_{6}\varepsilon_{6}^{*}\end{array} \right)dV=\int_{S_{f}}\mathbf{T}\cdot\mathbf{u}^{*}dS.\end{array}  \tag{8}\]  
\[{}^{n}\mathbf{Q}=\left( \begin{array}{c}{}^{n}Q_{11}\\ {}^{n}Q_{12}\end{array} \right) \tag{18}\]  
where \({}^{n}Q_{11}\) and \({}^{n}Q_{12}\) represent the two independent elastic parameters of the \(n\)th layer.