  We propose a new model of gravity where the Ricci scalar (R) in
Einstein-Hilbert action is replaced by an arbitrary function of R and of the
norm of energy-momentum tensor i.e., $f(R,T_{\mu\nu}T^{\mu\nu})$. Field
equations are derived in the metric formalism. We find that the equation of
motion of massive test particles is non-geodesic and these test particles are
acted upon by a force which is orthogonal to the four-velocity of the
particles. We also find the Newtonian limit of the model to calculate the extra
acceleration which can affect the perihelion of Mercury. There is a deviation
from the general relativistic(GR) result unless the energy density of fluid is
constant. Arranging $\alpha$ parameter gives an opportunity to cure the
inconsistency between the observational values for the abundance of light
elements and the standard Big Bang Nucleosynthesis results. Even the dust
dominated universe undergoes an accelerated expansion without using a
cosmological constant in Model II. With this specific choice of
$f(R,T_{\mu\nu}T^{\mu\nu})$, we get the a Cardassian-like expansion.
