The ceramic and ceramic-metal composite plates upon high velocity impact by a kinetic energy rod may undergo spalling, fragmentation, interface debonding, perforation, bending, and stretching. A challenge in studying these problems is using a material model that realistically simulates the material response at high plastic strains, plastic strain rates and hydrostatic pressures. The choice of the material model and the computational framework influence predictions of deformations of the projectile-target system. Three constitutive relations generally employed to study deformations of a ceramic are due to Johnson and Holmquist, generally abbreviated as JH1, JH2 and JHB. We have implemented these in a computational algorithm using Smoothed Particle Hydrodynamics (SPH) basis functions and a pseudo-spring technique to simulate the initiation and propagation of material failure. After ensuring that the computed results for two flyer plate impact experiments agree well with the test findings, we have studied penetration of monolithic ceramic and ceramic/aluminium targets by kinetic energy rods having blunt, hemispherical and conical nose shapes. It is shown that the present approach can successfully predict spalling, formation of conoid, fragmentation and crack branching in the ceramic, and bending/stretching of metal backing plates. Whereas results from the JH1 and the JHB material models are qualitatively similar, those using the JHB constitutive relation are closest to the test observations. This work suggests that one should use the JHB model for analyzing impact and penetration of a ceramic plate.