If a sizeable fraction of the energy of supernova remnant shocks is channeled into energetic particles (commonly identified with Galactic cosmic rays), then the morphological evolution of the remnants must be distinctly modified. Evidence of such modifications has been recently obtained with the Chandra and XMM-Newton X-ray satellites. To investigate these effects, we coupled a semi-analytical kinetic model of shock acceleration with a 3D hydrodynamic code (by means of an effective adiabatic index). This enables us to study the time-dependent compression of the region between the forward and reverse shocks due to the back reaction of accelerated particles, concomitantly with the development of the Rayleigh-Taylor hydrodynamic instability at the contact discontinuity. Density profiles depend critically on the injection level η of particles: for η ≲ 10-4 modifications are weak and progressive, for η ˜ 10-3 modifications are strong and immediate. Nevertheless, the extension of the Rayleigh-Taylor unstable region does not depend on the injection rate. A first comparison of our simulations with observations of Tycho's remnant strengthens the case for efficient acceleration of protons at the forward shock.