  We show that in modified $f(R)$ type gravity models with non-minimal coupling
between matter and geometry, both the matter Lagrangian, and the
energy-momentum tensor, are completely and uniquely determined by the form of
the coupling. This result is obtained by using the variational formulation for
the derivation of the equations of motion in the modified gravity models with
geometry-matter coupling, and the Newtonian limit for a fluid obeying a
barotropic equation of state. The corresponding energy-momentum tensor of the
matter in modified gravity models with non-minimal coupling is more general
than the usual general-relativistic energy-momentum tensor for perfect fluids,
and it contains a supplementary, equation of state dependent term, which could
be related to the elastic stresses in the body, or to other forms of internal
energy. Therefore, the extra-force induced by the coupling between matter and
geometry never vanishes as a consequence of the thermodynamic properties of the
system, or for a specific choice of the matter Lagrangian, and it is non-zero
in the case of a fluid of dust particles.
