## 2 Development of FBG Stress Sensor
### Measuring Directions of Sensing Units  
According to the measurement principle of the RSR method, the buried sensor needs to measure six normal stresses in different directions to determine the stress state of the measuring point. Therefore, the sensor design needs to consider the angle arrangement of the sensing unit.  
In the _oxyz_ coordinate system, the stress state of a point can be represented by three normal stress components (\(\sigma_{x}\), \(\sigma_{y}\), \(\sigma_{z}\)) and three shear stress components (\(\tau_{xy}\), \(\tau_{yz}\), \(\tau_{zx}\)). If the stress state is known, the normal stress, \(\sigma_{n}\), in any direction in the coordinate system can be expressed as follows:  
\[\sigma_{n}=l^{2}\sigma_{x}+m^{2}\sigma_{y}+n^{2}\sigma_{z}+2 lm\tau_{xy}+2 mn\tau _{yz}+2 nl\tau_{zx} \tag{5}\]  
where \(l\), \(m\), and \(n\) are the direction cosines of the angle between the normal stress direction and the \(x\), \(y\), and \(z\) axis, respectively. Based on Equation (5), conversely, the stress state of a point can be calculated from the six normal stresses, \(\sigma_{ni}\), (\(i\) = 1,2,3,... 6) measured by the sensor. The mathematical expression is:  
\[\left\{\sigma_{x},\sigma_{y},\sigma_{z},\tau_{xy},\tau_{yz},\tau_{zx}\right\} ^{T}=[K]^{-1}\{\sigma_{n1},\sigma_{n2},\sigma_{n3},\sigma_{n4},\sigma_{n5}, \sigma_{n6}\}^{T} \tag{6}\]  
Figure 5: Finite element modeling of sensing unit.  
Figure 6: Variation of FBG strain with loading pressure.  
where \([K]\) is defined as:  
\[[K]=\begin{bmatrix}l_{1}^{2}&m_{1}^{2}&n_{1}^{2}&2l_{1}m_{1}&2m_{1}n_{1}&2n_{1}l_{ 1}\\ l_{2}^{2}&m_{2}^{2}&n_{2}^{2}&2l_{2}m_{2}&2m_{2}n_{2}&2n_{2}l_{2}\\ l_{3}^{2}&m_{3}^{2}&n_{3}^{2}&2l_{3}m_{3}&2m_{3}n_{3}&2n_{3}l_{3}\\ l_{4}^{2}&m_{4}^{2}&n_{4}^{2}&2l_{4}m_{4}&2m_{4}n_{4}&2n_{4}l_{4}\\ l_{5}^{2}&m_{5}^{2}&n_{5}^{2}&2l_{5}m_{5}&2m_{5}n_{5}&2n_{5}l_{5}\\ l_{6}^{2}&m_{6}^{2}&n_{6}^{2}&2l_{6}m_{6}&2m_{6}n_{6}&2n_{6}l_{6}\end{bmatrix} \tag{7}\]  
where \(l_{i}\), \(m_{i}\), \(n_{i}\) (\(i\) = 1,2,3,... 6) are the direction cosines of the angle between \(\sigma_{ni}\) direction and the \(x\), \(y\), and \(z\) axis, respectively. It is worth emphasizing here that since we have to compute \([K]\)\(-\)1, the determinant of \([K]\) cannot be zero. If \(|K|\) = 0, it indicates that the six normal stresses measured by the sensor are not independent, and the stress state of the measuring point cannot be calculated. For example, the determinant of \([K]\) is zero when two sensing units measure in opposite directions. As the normal stress measured in opposite directions should be equal, they are not independent. In addition, if three of the six measurement directions are orthogonal to each other, and the other three are also orthogonal to each other, then the determinant is also zero. According to the elastic theory, the sum of the normal stress in the three orthogonal directions is the first principal stress invariant, that is, the following relationship exists:  
\[I_{1}=\sigma_{n1}+\sigma_{n2}+\sigma_{n3}=\sigma_{n4}+\sigma_{n5}+\sigma_{n6} \tag{8}\]  
Equation (8) means that at least one of the six measuring directions is not independent, and the stress state of the measuring point cannot be completely determined.