  We consider a gravitational model in which matter is non-minimally coupled to
geometry, with the effective Lagrangian of the gravitational field being given
by an arbitrary function of the Ricci scalar, the trace of the matter
energy-momentum tensor, and the contraction of the Ricci tensor with the matter
energy-momentum tensor. The field equations of the model are obtained in the
metric formalism, and the equation of motion of a massive test particle is
derived. In this type of models the matter energy-momentum tensor is generally
not conserved, and this non-conservation determines the appearance of an
extra-force acting on the particles in motion in the gravitational field. The
Newtonian limit of the model is also considered, and an explicit expression for
the extra-acceleration which depends on the matter density is obtained in the
small velocity limit for dust particles. We also analyze in detail the
so-called Dolgov-Kawasaki instability, and obtain the stability conditions of
the model with respect to local perturbations. A particular class of
gravitational field equations can be obtained by imposing the conservation of
the energy-momentum tensor. We derive the corresponding field equations for the
conservative case by using a Lagrange multiplier method, from a gravitational
action that explicitly contains an independent parameter multiplying the
divergence of the energy-momentum tensor. The cosmological implications of the
model are investigated for both the conservative and non-conservative cases,
and several classes of analytical solutions are obtained.
