We investigate the chemical and kinematic properties of the diffuse stellar haloes of six simulated Milky Way-like galaxies from the Aquarius Project. Binding energy criteria are adopted to defined two dynamically distinct stellar populations: the diffuse inner and outer haloes, which comprise different stellar sub-populations with particular chemical and kinematic characteristics. Our simulated inner- and outer-halo stellar populations have received contributions from debris stars (formed in sub-galactic systems while they were outside the virial radius of the main progenitor galaxies) and endo-debris stars (those formed in gas-rich sub-galactic systems inside the dark matter haloes). The inner haloes possess an additional contribution from disc-heated stars in the range $sim 3 - 30 %$, with a mean of $sim 20% $. Disc-heated stars might exhibit signatures of kinematical support, in particular among the youngest ones. Endo-debris plus disc-heated stars define the so-called insitu stellar populations. In both the inner- and outer-halo stellar populations, we detect contributions from stars with moderate to low [$alpha$/Fe] ratios, mainly associated with the endo-debris or disc-heated sub-populations. The observed abundance gradients in the inner-halo regions are influenced by both the level of chemical enrichment and the relative contributions from each stellar sub-population. Steeper abundance gradients in the inner-halo regions are related to contributions from the disc-heated and endo-debris stars, which tend to be found at lower binding energies than debris stars. (Abridged).
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