# A Novel Finite Element Method Approach in the Modelling of Edge Trimming of CFRP Laminates
## 4 FEM Damage Algorithm Basics  
The composite damage model was implemented via VUMAT Fortran user-subroutine in Abaqus/Explicit. Four damage types were employed to reduce the mechanical properties linearly: (1) fibre traction, (2) fibre compression, (3) matrix traction, and (4) matrix compression. These damage types are included in the stiffness matrix to degrade the mechanical properties separately, as shown in Equation (1). The formulation of the stiffness matrix was first proposed by [PERSON] and [PERSON] [46] and later applied in composite machining by [PERSON] et al. [43]. \[\left[ \begin{array}{c}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{array} \right]=\frac{1}{D}\left[ \begin{array}{ccc}(1-d_{f})E _{11}&(1-d_{f})(1-d_{m})v_{21}E_{11}&0\\ (1-d_{f})(1-d_{m})v_{12}E_{22}&(1-d_{m})E_{22}&0\\ 0&0&D(1-d_{s})G_{12}\end{array} \right]\left[ \begin{array}{c}\varepsilon_{11 }\\ \varepsilon_{22}\\ \varepsilon_{12}\end{array} \right] \tag{1}\]  
where \(D=1-(1-d_{f})(1-d_{m})v_{12}v_{21}\); \(d_{s}=1-(1-d_{ft})(1-d_{fc})(1-d_{mt})(1-d_{mc})\)\(d_{f}=max\{d_{ft},d_{fc}\}\); \(d_{m}=max\{d_{mt},d_{mc}\}\); \(d_{l}e[0,1]\) and \(I=(ft,fc,mt,mc)\). A hybrid composite damage algorithm was used to predict the onset of the damage modes that were studied. To determine the damage initiation in fibre damage modes, a simple maximum stress failure criteria [47] detailed in the following equations was used. - Fibre traction \((\sigma_{11}>0)\)  
\[F_{ft}=\frac{\sigma_{11}}{X_{\rm T}}\geq 1 \tag{2}\]  
- Fibre compression \((\sigma_{11}<0)\)  
\[F_{fc}=\left|\frac{\sigma_{11}}{X_{\rm C}}\right|\geq 1 \tag{3}\]  
The accurate modelling of matrix damage modes is vital to predict the machining forces reliably [48]; therefore, the use of robust composite failure criteria is essential for excellent predictions. [PERSON]'s failure criterion was selected because of its high capability of determining the failure with a large number of off-axis loading scenarios [49]. For the purpose of brevity, [PERSON]'s criterion insights are not explained in detail here. This criterion is thoroughly explained in [50]. Three different matrix damage modes were distinguished: Mode A, Mode B, and Mode C. Mode A predicts matrix failure in tensile states, while Mode B and Mode C were utilised to predict matrix failure in compressive loads. These damage modes are formulated below. - Mode A \((\sigma_{22}\geq 0)\)  
Figure 6: Meshed elements distribution in the modelled laminate. \[F_{mmn}=\sqrt{\left(\frac{\sigma_{12}}{R_{\perp\parallel}^{A}}\right)^{2}+\left(1- \frac{p_{\perp\parallel}^{(+)}}{R_{\perp\parallel}^{A}}R_{\perp}^{(+)A}\right)^ {2}\left(\frac{\sigma_{22}}{R_{\perp}^{(+)A}}\right)^{2}}+\frac{p_{\perp \parallel}^{(+)}}{R_{\perp\parallel}^{A}}\sigma_{22}\geq 1\] (4) - Mode B (\(\sigma_{22}<0\)_and_ \(\sigma_{22}>-R_{\perp\perp}^{A}\))  
\[F_{mmb}=\sqrt{\left(\frac{\sigma_{12}}{R_{\perp\parallel}^{A}}\right)^{2}+ \left(\frac{p}{R}\right)^{2}\sigma_{22}^{2}+\left(\frac{p}{R}\right)\sigma_{2 2}\geq 1}\] (5) - Mode C (\(\sigma_{22}\leq-R_{\perp\perp}^{A}\))  
\[F_{mmc}=\frac{1}{2\left[1+\left(\frac{p}{R}\right)R_{\perp\perp}^{A}\right]} \left[\left(\frac{\sigma_{12}}{R_{\perp\parallel}^{A}}\right)^{2}+\left(\frac {\sigma_{22}}{R_{\perp\perp}^{A}}\right)^{2}\right]\frac{R_{\perp\perp}^{A}}{ -\sigma_{22}}\geq 1 \tag{6}\]  
In the above equations, the term \(F_{I}\) denotes the damage activation functions, where I is whichever of the four investigated damage modes (ft, fc, mt, and mc). All terms with \(R\) represent the strength components related to the fracture plane. Term \(R_{\perp}^{(+)A}\) is generally associated to the transverse matrix traction strength (\(Y_{T}\)), \(R_{\perp\parallel}^{A}\) is considered the composite shear strength (\(S_{12}\)). \(R_{\perp\perp}^{A}\) represents the transverse/transverse shear strength. In general terms, an accurate measurement of this variable presents a cumbersome venture. Therefore, this variable is usually inferred from Equation (7).