  Collapsing solutions in $f(R)$ gravity are restricted due to junction
conditions that demand continuity of the Ricci scalar and its normal derivative
across the time-like collapsing hypersurface. These are obtained via the method
of $R$-matching, which is ubiquitous in $f(R)$ collapse scenarios. In this
paper, we study spherically symmetric collapse with the modification term
$\alpha R^2$, and use $R$-matching to exemplify a class of new solutions. After
discussing some mathematical preliminaries by which we obtain an algebraic
relation between the shear and the anisotropy in these theories, we consider
two metric ansatzes. In the first, the collapsing metric is considered to be a
separable function of the co-moving radius and time, and the collapse is
shear-free, and in the second, a non-separable interior solution is considered,
that represents gravitational collapse with non-zero shear viscosity. We arrive
at novel solutions that indicate the formation of black holes or locally naked
singularities, while obeying all the necessary energy conditions. The separable
case allows for a simple analytic expression of the energy-momentum tensor,
that indicates the positivity of the pressures throughout collapse, and is
further used to study the heat flux evolution of the collapsing matter, whose
analytic solutions are presented under certain approximations. These clearly
highlight the role of modified gravity in the examples that we consider.
