## 2 Numerical Investigations
### Constitutive Models for Composite Materials  
It is generally considered that the volume unit of the composite single-layer plate structure is composed of two parts: the matrix unit and the fiber unit. Based on this, the composite material single-layer plate structure can be regarded as an orthotropic elastomer with three mutual Vertical elastic planes of symmetry. According to the relevant theory of composite mechanics, for anisotropic composite materials, the constitutive relation is shown in Formula (1):\[\left[\begin{array}{c}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\\ \tau_{23}\\ \tau_{31}\\ \tau_{12}\end{array}\right]=\left[\begin{array}{cccccc}C_{11}&C_{12}&C_{13}&C_{ 14}&C_{15}&C_{16}\\ C_{21}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\ C_{31}&C_{32}&C_{33}&C_{34}&C_{35}&C_{36}\\ C_{41}&C_{42}&C_{43}&C_{44}&C_{45}&C_{46}\\ C_{51}&C_{52}&C_{53}&C_{54}&C_{55}&C_{56}\\ C_{61}&C_{62}&C_{63}&C_{64}&C_{65}&C_{66}.\end{array}\right]\left[\begin{array}[] {c}\varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{3}\\ \tau_{723}\\ \tau_{731}\\ \tau_{712}\end{array}\right] \tag{1}\]  
Formula (1) can be abbreviated as:  
\[\sigma=C\varepsilon \tag{2}\]  
In Formulas (1) and (2), the subscripts 1, 2, and 3 respectively represent the main directions of the three elastic symmetry planes; \(\sigma_{i}\) is the normal stress, \(\tau_{ij}\) is the shear stress, \(\gamma_{ij}\) is the shear strain, \(\varepsilon_{i}\) is the line strain, \(C\) is the stiffness matrix and \(C_{ij}\) is the stiffness coefficient.  
If the strain energy is considered, it can be seen by a derivation that there are only 21 independent stiffness coefficients in the elastic material stiffness matrix; for an orthotropic single-layer plate, because there are 3 orthogonal elastic symmetry planes, there are only 21 independent stiffness coefficients in the stiffness matrix. There are 9 independent stiffness coefficients, then Equation (1) can be simplified as:  
\[\left[\begin{array}{c}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\\ \tau_{23}\\ \tau_{31}\\ \tau_{12}\end{array}\right]=\left[\begin{array}{cccccc}C_{11}&C_{12}&C_{13} &0&0&0\\ C_{12}&C_{22}&C_{23}&0&0&0\\ C_{13}&C_{23}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{55}&0\\ 0&0&0&0&0&C_{66}.\end{array}\right]\left[\begin{array}{c}\varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{3}\\ \tau_{723}\\ \tau_{712}\end{array}\right] \tag{3}\]  
When the composite laminate structure is subjected to fatigue load, different forms of damage will appear inside the structure. With the increase in fatigue loading times, the damage inside the laminate structure will gradually expand until the composite structure finally fails. In the process of fatigue failure of composite structure, its bearing capacity will also gradually decrease. When analyzing the fatigue problem of composite material structure, it is necessary to update the stiffness matrix of composite material to the stiffness matrix after performance attenuation in real time. Assuming that the stiffness reduction matrix is K, the stiffness matrix after attenuation can be expressed as:  
\[C^{\prime}=KC \tag{4}\]  
By substituting Equation (4) into Equation (3), the stress update at the next moment can be obtained.