## 2 Theoretical Modelling
### Derivation of the Damage Evolution States  
According to the conclusions drawn in a study by [PERSON] [29], the Hashin criterion is able to distinguish the physical properties of fibre and matrix damage. Therefore, the Hashin criterion was used to determine the damage initiation during the progressive failure analysis herein; the progressive failure flow, combined with the computational model in  
Figure 3: The 2-D periodic structure of the RVEs. this paper, is illustrated in Figure 4. The following five different damage modes were considered in the model: fibre tensile failure, fibre compression failure, matrix tensile failure, matrix compression failure, and matrix fibre shear failure [30; 31]. As far as the fibre-reinforced metal matrix composite is concerned, the stress-strain relationship in the principal coordinate system can be expressed as follows:  
\[ \begin{cases}\sigma_{11}\\ \sigma_{22}\\ \tau_{12}\end{cases} = \begin{bmatrix}E_{1}&\gamma_{21}E_{1}&0\\ \gamma_{12}E_{2}&E_{2}&0\\ 0&0&G_{12}\end{bmatrix}  \tag{18}\]  
where \(E_{1}\) and \(E_{2}\) are the longitudinal and transverse moduli of elasticity, respectively; \(\gamma_{21}\) and \(\gamma_{12}\) denote Poisson's ratios; and \(G_{12}\) is the shear modulus. According to the classical laminate theory, the following expression can be derived:  
\[\sigma^{\prime}=C_{0}\epsilon \tag{19}\]  
where \(C_{0}\) is the composite stiffness matrix and the principal coordinate system of the fibre-reinforced composite monolayer structure shown in Figure 1.  
When local damage is generated in the material, the stiffness degradation due to the damage is considered. Based on the principle of material property degradation (MPDG), the constitutive equation of the material is defined as follows:  
\[\sigma=C_{d}\epsilon \tag{20}\]  
where \(C_{d}\) denotes the damage stiffness matrix. The damage stiffness matrix of the transversely isotropic material \(C_{d}\) for the plane stress state can be expressed as follows:  
\[C_{d}=(1-d_{i})C_{0}=\frac{1}{D} \begin{bmatrix}(1-d_{f})E_{1}&(1-d_{f})(1-d_{ m})\gamma_{12}E_{1}&0\\ (1-d_{f})(1-d_{m})\gamma_{12}E_{2}&(1-d_{m})E_{2}&0\\ 0&0&(1-d_{s})G_{12}D\end{bmatrix}  \tag{21}\]  
where \(D=1-(1-d_{f})(1-d_{m})\gamma_{12}\gamma_{21}\), \(d_{f}\), \(d_{m}\), and \(d_{s}\) are the fibre, matrix, and shear damage state variables, respectively. The damage state variables have values between 0 and 1. When the damage state variable is 0, it means that the element has experienced  
Figure 4: Progressive failure calculation flow. no damage and that the material stiffness has not changed. On the other hand, when the damage state variable is 1, it implies that the element has completely failed and that the material stiffness is 0. Where the superscript \(t\) indicates stretching and specifies \(c\) compression for \(d\). After the damage has occurred, the effective stress of the material is \(\hat{\sigma}=M\sigma\), where the damage matrix \(M\) is defined as follows:  
\[M= \begin{bmatrix}\dfrac{1}{1-d_{f}}&0&0\\ 0&\dfrac{1}{1-d_{m}}&0\\ 0&0&\dfrac{1}{1-d_{s}}\end{bmatrix}  \tag{22}\]  
The local damage within the composite degrades the material properties, the stiffness of the damaged element is lowered, and the stress values are redistributed [31]. The stress increase near the damage site is also larger, and failure is more likely to occur when the load increases, causing the gradual expansion of damage. Figure 5 depicts the damage evolution pattern of the linear elastic material. In Figure 5, section OA indicates the state of the composite material without damage, whereas section AC shows the degradation of the material after the damage has occurred.