We study strange stars in the framework of $fleft(R,mathcal{T}right)$ theory of gravity where the strange quark matter distribution inside the stellar system is governed by the phenomenological MIT Bag model equation of state (EOS). Further, for a specific value of $B$ and observed values of mass of the strange star candidates we obtain the exact solution of the modified Tolman-Oppenheimer-Volkoff (TOV) equation in the framework of $fleft(R,mathcal{T}right)$ gravity and have studied in detail the dependence of the different physical parameters due to the chosen different values of $chi$. To check the physical acceptability and stability of the stellar system based on the obtained solutions we have performed different physical tests, viz., the energy conditions, Herrera cracking concept, adiabatic index etc. In this work, we also have explained the effects, those are arising due to the interaction between the matter and the curvature terms in $fleft(R,mathcal{T}right)$ gravity, on the anisotropic compact stellar system. It is interesting to note that as the values of $chi$ increase the strange stars become more massive and their radius increase gradually so that eventually they gradually turn into less dense compact objects. The present study reveals that the modified $fleft(R,mathcal{T}right)$ gravity is a suitable theory to explain massive stellar systems like recent magnetars, massive pulsars and super-Chandrasekhar stars, which can not be explained in the framework of GR. However, for $chi=0$ the standard results of Einsteinian gravity are retrieved.
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