  Supernova remnants (SNRs) are believed to be the major contributors to
Galactic cosmic rays. The detection of non-thermal emission from SNRs
demonstrates the presence of energetic particles, but direct signatures of
protons and other ions remain elusive. If these particles receive a sizeable
fraction of the explosion energy, the morphological and spectral evolution of
the SNR must be modified. To assess this, we run 3D hydrodynamic simulations of
a remnant coupled with a non-linear acceleration model. We obtain the
time-dependent evolution of the shocked structure, impacted by the
Rayleigh-Taylor hydrodynamic instabilities at the contact discontinuity and by
the back-reaction of particles at the forward shock. We then compute the
progressive temperature equilibration and non-equilibrium ionization state of
the plasma, and its thermal emission in each cell. This allows us to produce
the first realistic synthetic maps of the projected X-ray emission from the
SNR. Plasma conditions (temperature, ionization age) can vary widely over the
projected surface of the SNR, especially between the ejecta and the ambient
medium owing to their different composition. This demonstrates the need for
spatially-resolved spectroscopy. We find that the integrated emission is
reduced with particle back-reaction, with the effect being more significant for
the highest photon energies. Therefore different energy bands, corresponding to
different emitting elements, probe different levels of the impact of particle
acceleration. Our work provides a framework for the interpretation of SNR
observations with current X-ray missions (Chandra, XMM-Newton, Suzaku) and with
upcoming X-ray missions (such as Astro-H).
