# Modelling of High-Velocity Impact on Woven Carbon Fibre-Reinforced Plastic Laminate
## 2 Constitutive Model
The function \(f(\sigma(t))\) was normalised with the corresponding member of the material stiffness matrix, so that evolution of the damage variables was defined as:  
\[\dot{\omega}=\dot{\omega}(\omega,\sigma)=\Omega_{\omega}\bigg{(}\frac{\sigma} {C_{m}(1-\omega)}-\frac{\sigma_{\text{cr}}}{C_{m}}\bigg{)}H\bigg{(}\frac{ \sigma}{C_{m}(1-\omega)}-\frac{\sigma_{\text{cr}}}{C_{m}}\bigg{)} \tag{11}\]  
\[\dot{\omega}_{\text{del}}=\dot{\omega}_{\text{del}}(\omega,\sigma)=\Omega_{ \text{del}}\bigg{(}\frac{\sigma_{33}}{C_{33}(1-\omega_{3})}-\frac{\sigma_{ \text{cr}}}{C_{33}}\bigg{)}H\bigg{(}\frac{\sigma_{33}}{C_{33}(1-\omega_{3})}- \frac{\sigma_{\text{cr}}}{C_{33}}\bigg{)} \tag{12}\]  
where \(\Omega_{\omega}\) and \(\Omega_{\text{del}}\) are material parameters determined by time to failure, \(\sigma\) and \(\sigma_{33}\) are effective stress and out-of-plane stress, respectively, \(\sigma_{\text{cr}}\) and \(\sigma_{\text{cr}}\)del are critical effective stress and critical out-of-plane stress, \(C_{m}\) and \(C_{33}\) are maximum in-plane stiffness member and through the thickness coefficient, respectively, \(H\) is Heaviside function. The integration of Equations (11) and (12) provides damage internal variables for principal directions of damage, \(\omega_{1}\), \(\omega_{2}\) and \(\omega_{3}\), which are further used for the update of the damaged stiffness tensor in Equation (4). Note that the first two variables correspond to the fibre directions in the woven material, whilst the third is a damage variable for the out-of-plane direction. The proposed damage model has been implemented into the LLNL code DYNA3D [6] and coupled with an orthotropic elastic constitutive model with vector equation of state (EOS), derived in [30] and outlined below. The vector EOS defines pressure as a state of stress induced by the volumetric strain only. This physically means that the pressure is a vector in the stress space, which is not collinear with hydrostat, and can be written as:  
\[-\dot{P}\dot{\psi}_{\text{jj}}=C_{\text{jjkl}}\delta_{\text{kl}}\epsilon_{ss} /3=C_{\text{jjkk}}\epsilon_{p} \tag{13}\]where the direction of the pressure in the stress space is defined by tensor \(\psi_{\rm ij}\) and volumetric strain is given as \(\varepsilon_{v}=\varepsilon_{\rm ss}/3\). Tensor \(\psi_{\rm ij}\) has diagonal members only, so indices in the brackets imply no summation. The governing equations of the decomposition of the stress tensor are summarised below. The first equation in (14) decomposes the stress onto the components \(\hat{P}\psi_{\rm ij}\) and \(\hat{S}_{\rm ij}\), which are not mutually orthogonal, whilst the second equation in (14) is used in the constitutive model, where the orthogonality between \(\widetilde{P}\psi_{\rm ij}\) and \(\widetilde{S}_{\rm ij}\) holds. The relationships between the components in Equation (14) are given below; for a detailed derivation, see [30]. \[\sigma_{\rm ij}=-\hat{P}\psi_{\rm ij}+\hat{S}_{\rm ij}=-\widetilde{P}\psi_{\rm ij }+\widetilde{S}_{\rm ij} \tag{14}\]  
\[\widetilde{P}=\hat{P}-\frac{\hat{S}_{\rm ij}\psi_{\rm ij}}{\psi_{\rm kl}\psi_{ \rm kl}}=\hat{P}-\hat{S}=-\frac{\sigma_{\rm ij}\psi_{\rm ij}}{3} \tag{15}\]  
\[\widetilde{S}_{\rm ij}=\hat{S}_{\rm ij}-\frac{\hat{S}_{\rm kl}\psi_{\rm kl}}{ \psi_{\rm sr}\psi_{\rm sr}}\psi_{\rm ij}=\hat{S}_{\rm ij}-\hat{S}\psi_{\rm ij} \tag{16}\]  
\[\widetilde{S}_{\rm ij}=\sigma_{\rm ij}-\frac{\sigma_{\rm kl}\psi_{\rm kl}}{ \psi_{\rm sr}\psi_{\rm sr}}\psi_{\rm ij}=\sigma_{\rm ij}+\widetilde{P}\psi_{ \rm ij} \tag{17}\]  
The implemented constitutive model was validated against the experiments conducted with woven CFRP, which was modelled with orthotropic material properties, as described in the subsequent sections. However, the model can be applied to the transversally isotropic and isotropic materials, by using appropriate material properties.