# Effect of Cohesive Properties on Low-Velocity Impact Simulations of Woven Composite Shells
## 2 Finite Element Models
### Intralaminar Damage Modelling  
To capture the progression of damage at the intralaminar level, a constitutive model specifically designed for fabric-reinforced composites, was used in ABAQUS/Explicit [17]. This constitutive model, initially developed by [PERSON] et al. [19] and based on the work of [PERSON] and [PERSON] [20], offers comprehensive capabilities for capturing the intralaminar damage phenomena. It can be seamlessly integrated into the analysis workflow as a VUMAT user subroutine, with the user-defined material identified by the text \"ABQ_PLY_FABRIC\" [18]. This integration enables effective modelling and simulation of the intralaminar damage process in fabric-reinforced composites. In this damage model, each woven fabric-reinforced layer is modelled as a homogeneous orthotropic elastic material with the capability of sustaining progressive stiffness degradation due to fibre failure and matrix cracking, and plastic deformation under shear loading [16; 18]. Figure 1: FE modelling approach based on continuum shell elements and cohesive surfaces: (**a**) Representation of the complete FE model. (**b**) Schematic representation of the shell in the through-thickness direction. The parameter \(t\) refers to the thickness of a layer. Notice that in the absence of specific experimental data for the tested material, reference values from similar studies/materials were employed as a starting point. However, recognising the need for a more refined and comprehensive analysis, a parametric study was conducted to thoroughly investigate and evaluate the influence of fracture toughness on the numerical-experimental correlation. The intralaminar damage model utilised in this study adopts the maximum stress failure criterion to determine the initiation of damage in the fibres. Additionally, a damage evolution model based on fracture energies was employed to govern the progressive reduction in stiffness. To describe the mechanical behaviour of the degraded orthotropic material, the Hooke's law was applied, and its expression is as follows:  
\[\varepsilon= \begin{bmatrix}\frac{S_{1}}{(1-d_{1})}&S_{12}&0\\ S_{21}&\frac{S_{2}}{(1-d_{2})}&0\\ 0&0&\frac{S_{6}}{(1-d_{12})}\end{bmatrix} \sigma \tag{1}\]  
where \(S_{ij}\) represents the components of the compliance matrix for the undamaged orthotropic material. The nominal Cauchy stress tensor is denoted as \(\sigma\), while \(\varepsilon\) represents the elastic strain tensor. The damage coefficients, \(d_{1}\), \(d_{2}\), and \(d_{12}\), have specific associations: \(d_{1}\) and \(d_{2}\) relate to fibre fracture along directions 1 and 2, respectively, whereas \(d_{12}\) corresponds to matrix microcracking caused by shear deformation. The elastic domain, at any given time, is calculated based on the damage activation functions, \(F_{\alpha}\), expressed as:  
\[F_{\alpha}=\frac{\tilde{\sigma}_{\alpha}}{X_{\alpha}}-r_{\alpha}\leq 0\quad \text{with}\quad\alpha=1\pm,2\pm,12 \tag{2}\]  
where \(\tilde{\sigma}_{\alpha}\) and \(X_{\alpha}\) represent the effective stresses and strengths, respectively, and \(r_{\alpha}\) denotes the damage thresholds, initially set to 1. Once the damage onset is reached, indicated by \(\frac{\tilde{\sigma}_{\alpha}}{X_{\alpha}}=1\), the evolution of the damage coefficients \(d_{1}\) and \(d_{2}\) (associated with tensile and compressive loading) is determined using Equations (3) and (4). Similarly, the damage coefficient \(d_{12}\) (related to shear loading) is obtained from Equation (5).