Theoretical model of rigid projectile penetration into metal targets considering strain hardening, strain rate, and temperature effects

In this study, we present a theoretical penetration model that incorporates the coupled effects of strain hardening, strain rate, and temperature, applicable to rigid ogive-nose long rod steel projectiles that penetrate aluminium targets under a normal impact within a specific range of striking velocities. Firstly, we employ an explicit stress–plastic strain relationship to accurately characterise the flow stress in the plastic region. This helps derive the approximate solution for the dynamic expansion of the incompressible spherical cavity, and analyse the effects of the strain rate and temperature softening on the radial stress of the spherical cavity. Subsequently, we derived the closed-form penetration equation using the spherical cavity expansion (SCE) approximation and by incorporating time-dependent velocity boundary conditions. The proposed method is then employed to validate the model by comparing the results obtained from the proposed SCE approximation penetration model with the existing experimental data derived from the final depth of penetration of three caliber-radius-head (CRH) ogive-nose long rod projectiles used to strike aluminium targets. Furthermore, we performed nonlinear finite element simulations to determine the importance of different components of the Johnson–Cook constitutive relationship in affecting the penetration resistance of the target. Lastly, we analysed the effect of thermal softening on the penetration resistance based on the variations in CRH and velocities. The results indicate that the strain rate effect enhances the resistance of the target, particularly for the strain-rate sensitive materials. Alternatively, the thermal softening effect reduces the resistance of the target. The thermal softening significantly affects the target plate within the plastic region, where $1<r\left(x\right)/a\left(x\right)<2$; however, its influence in other plastic regions is relatively minimal. The normal component of velocity at the projectile surface decreases progressively with the increase in the CRH, thereby attenuating the effect of thermal softening. Conversely, the strain rate increases with the increase in the initial velocity. This presents a further increase in the temperature, which exacerbates thermal softening, thereby causing a reduction in the penetration resistance by approximately 5.4% to 6.1% and diminishing the cavitation resistance of the target plate material.