  Very recently, the authors of [PRL {\bf 118} (2017) 021102] have shown that
violation of energy-momentum tensor (EMT) could result in an accelerated
expansion state via appearing an effective cosmological constant, in the
context of unimodular gravity. Inspired by this outcome, in this paper we
investigate cosmological consequences of violation of the EMT conservation in a
particular class of $f(R,T)$ gravity when only the pressure-less fluid is
present. In this respect, we focus on the late time solutions of models of the
type $f(R,T)=R+\beta \Lambda(-T)$. As the first task, we study the solutions
when the conservation of EMT is respected and then we proceed with those in
which violation occurs. We have found, provided that the EMT conservation is
violated, there generally exist two accelerated expansion solutions which their
stability properties depend on the underlying model. More exactly, we obtain a
dark energy solution for which the effective equation of state (EoS) depend on
model parameters and a de Sitter solution. We present a method to parametrize
$\Lambda(-T)$ function which is useful in dynamical system approach and has
been employed in the herein model. Also, we discuss the cosmological solutions
for models with $\Lambda(-T)=8\pi G(-T)^{\alpha}$ in the presence of the ultra
relativistic matter.
