  In the present work, we study about highly dense compact stars which are made
of quarks, specially strange quarks, adopting the Krori-Barua
(KB)~\cite{Krori1975} metric in the framework of $f(R,T)$ gravity. The equation
of state (EOS) of a strange star can be represented by the MIT bag model as
$p_r(r)=\frac{1}{3}[\rho(r)-4B_g]$ where $B_g$ is the bag constant, arises due
to the quark pressure. Main motive behind our study is to find out singularity
free and physically acceptable solutions for different features of strange
stars. Here we also investigate the effect of alternative gravity in the
formation of strange stars. We find that our model is consistent with various
energy conditions and also satisfies Herrera's cracking condition, TOV
equation, static stability criteria of Harrison-Zel$'$dovich-Novikov etc. The
value of the adiabatic indices as well as the EOS parameters re-establish the
acceptability of our model. Here in detail we have studied specifically three
different strange star candidates, viz. $PSRJ~1614~2230, Vela~X-1$ and
$Cen~X-3$. As a whole, present model fulfils all the criteria for stability.
Another fascinating point we have discussed is the value of the bag constant
which lies in the range $(40-45)$~MeV/fm$^{3}$. This is quite smaller than the
predicted range, i.e., $(55-75)$~MeV/fm$^{3}$ ~\cite{Farhi1984,Alcock1986}. The
presence of the constant ($\chi$), arises due to the coupling between matter
and geometry, is responsible behind this reduction in $B_g$ value. For
$\chi=0$, we get the higher value for $B_g$ as the above mentioned predicted
range.
