# Hybrid Shielding for Hypervelocity Impact of Orbital Debris on Unmanned Spacecraft
## 2 Numerical Modelling
### Constitutive Modelling  
Due to the high energy involved in the hypervelocity impact, all Whipple shield components, including the honeycomb core and debris particle, were modelled with the Johnson-Cook constitutive law to ensure adequate plasticity, strain-rate and temperature material behaviour is captured. These three terms are explicitly and separately described in [PERSON]'s constitutive equation  
\[\tilde{\sigma}=\left[A+B\left(\tilde{\sigma}^{\text{pl}}\right)^{n}\right] \left[1+C\ln\left(\frac{\tilde{\sigma}^{\text{pl}}}{\tilde{\sigma}^{\text{pl} }_{0}}\right)\right]\left[1-\left(T^{*}\right)^{m}\right] \tag{1}\]  
where \(\tilde{\sigma}\) is the flow stress, \(\tilde{\sigma}^{\text{pl}}\) is the equivalent plastic strain, \(n\) is the strain hardening exponent, \(A\), \(B\), \(C\) and \(m\) are material constants that can be determined experimentally, \(\tilde{\sigma}^{\text{pl}}\) is the equivalent plastic strain rate [18], and \(T^{*}\) is the non-dimensional homologous temperature, defined as  
\[T^{*}=\frac{T-T_{\text{t}}}{T_{\text{m}}-T_{\text{t}}} \tag{2}\]  
where \(T\) is the current temperature, \(T_{\text{m}}\) is the melting temperature and \(T_{\text{t}}\) is the transition temperature, at or below which there is no temperature dependence for the yield stress.  
Damage caused by the impact is modelled using the associated Johnson-Cook damage model, which similarly considers the effects of plasticity, strain-rate and temperature, and can be described by the equation  
\[\tilde{\sigma}^{\text{pl}}_{\text{D}}=\left[D_{1}+D_{2}\exp(-D_{3}\eta)\right] \left[1+D_{4}\ln\left(\frac{\tilde{\sigma}^{\text{pl}}}{\tilde{\epsilon}_{0} }\right)\right](1+D_{5}T^{*}) \tag{3}\]  
where \(D_{i}\) (\(i=1,\ldots,5\)) are the damage parameters measured at or below the transition temperature and \(\eta\) is the stress triaxiality, which represents the ratio of pressure to von Mises stress. The damage parameter is then calculated as  
\[D=\sum\frac{\tilde{\epsilon}_{\text{D}}}{\tilde{\epsilon}^{\text{pl}}_{\text {D}}} \tag{4}\]  
and damage occurs when the damage parameter \(D\) reaches a value of 1.0. After damage initiation, the material stiffness is progressively degraded according to the damage evolution relationship [18].  
The developed models of the Whipple shield honeycomb core were tested with two different lightweight materials: an aluminium alloy (AL2024-T3) and a titanium alloy (Ti-6 Al-4V). The impact energy absorption efficiency of both materials is analysed and compared. The front plate, the back plate and the debris particle (projectile) are aluminium (AL2024-T3) throughout. The material parameters for all materials in the models are listed in Table 1[19; 20].  
Due to the high energy involved in the hypervelocity impact, the compressibility of the materials is modelled using the non-linear Mie-Gruneisen equation of state  
\[p=\frac{\rho_{0}G^{2}\mu\left[1+\left(1-\frac{\gamma_{0}}{2}\right)\mu-\frac{ \alpha}{2}\mu^{2}\right]}{\left[1-\left(S_{1}-1\right)\mu\right]^{2}}+\left( \gamma_{0}+a\mu\right)E \tag{5}\]  
where \(E\) is the internal energy, \(\mu=\rho/\rho_{0}-1\) is the relative density, \(\gamma_{0}\) is the non-dimensional Gruneisen constant, \(a\) is the volume correction for \(\gamma_{0}\) and \(G,S_{1}\) are material constants that depend on the shock wave and particle velocities. The corresponding parameters are listed in Table 1. This simplified version of the non-linear Mie-Gruneisen equation of state in Equation (5) only considers the first-order dependency of the pressure on the material density (\(S_{2}=S_{3}=0\)).