  We assume cylindrically symmetric stars which begin collapsing by dissipating
energy in the form of heat flux. We wish to study the effects of
Carroll-Duvvuri-Trodden-Turner (CDTT) model, $f(R)=R+\sigma\frac{\mu^4}{R}$, on
the range of dynamical instability. For this purpose, perturbation scheme is
applied to all the metric functions, material functions and $f(R)$ model to
obtain the full set of dynamical equation which control the evolution of the
physical variables at the surface of a star. It is found that instability limit
involves adiabatic index $\Gamma$ which depends on the density profile and
immense terms of perturbed CDTT model. In addition, model is constrained by
some requirement, e.g. positivity of physical quantities. We also reduce our
results asymptotically as $\mu\rightarrow0$, being the GR results in both the
Newtonian and post Newtonian regimes.
