  In this paper, we adopt minimal gravitational decoupling scheme to extend a
non-static spherically symmetric isotropic composition to anisotropic interior
in
$f(\mathcal{R},\mathcal{T},\mathcal{R}_{\mathrm{a}\mathrm{b}}\mathcal{T}^{\mathrm{a}\mathrm{b}})$
theory. A geometric deformation is applied only on $g_{rr}$ metric component
through which the modified field equations are separated into two sets, each of
them correspond to their parent (seed and newly added) source. An isotropic
model suggested by the Friedmann-Lemaitre-Robertson-Walker metric is adopted to
reduce the unknowns in the first set. We then obtain an isotropic solution by
making use of a linear equation of state and a particular form of the scale
factor. A density-like constraint is chosen to solve the other sector
containing the deformation function and multiple components of an additional
matter source. Further, the graphical interpretation of the developed model is
carried out to analyze how a decoupling parameter and modified gravity
influence the evolutionary phases of the universe. It is concluded that only
the radiation-dominated era meets stability criteria everywhere in this
matter-geometry coupled theory.
