A dynamical analysis of a spherically symmetric collapsing star surrounded by a locally anisotropic environment under an expansion-free condition is presented in \(f(R,T)\) gravity, where \(R\) corresponds to the Ricci scalar and \(T\) stands for the trace of the energy momentum tensor. The modified field equations and evolution equations are reconstructed in the framework of \(f(R,T)\) gravity. In order to acquire the collapse equation we implement the perturbation on all matter variables and dark source components comprising the viable \(f(R,T)\) model. The instability range is described in the Newtonian and post-Newtonian approximation. It is observed that the unequal stresses and density profile define the instability range rather than the adiabatic index. However, the physical quantities are constrained to maintain positivity of the energy density and a stable stellar configuration.

The present study explores the $$f(\mathcal {R},\mathcal {T})$$f(R,T) modified gravity on the basis of observational data for three different compact stars with matter profile as anisotropic fluid without electric charge. In this respect, we adopt the well-known Karmarker condition and assume a specific and interesting model for $$\mathrm {g}_{rr}$$grr metric potential component which is compatible with this condition. This choice further leads to a viable form of metric component $$\mathrm {g}_{tt}$$gtt by utilizing the Karmarkar condition. We also present the interior geometry in the reference of Schwarzschild interior and Kohler–Chao cosmological like solutions for $$f(\mathcal {R},\mathcal {T})$$f(R,T) theory. Moreover, we calculate the spacetime constants by using the masses and radii from the observational data of three different compact stars namely 4U 1538-52, LMC X-4 and PSR J1614-2230. In order to explore the viability and stability of the obtained solutions, some physical parameters and properties are presented graphically for all three different compact object models. It is noticed that the parameters c and $$\lambda $$λ have some important and considerable role for these solutions. It is concluded that our obtained solutions are physically acceptable, bearing a well-behave nature in $$f(\mathcal {R},\mathcal {T})$$f(R,T) modified gravity.

We discuss the dynamical analysis in f(R, T) gravity (where R is the Ricci scalar and T is the trace of the energy momentum tensor) for gravitating sources carrying axial symmetry. The self-gravitating system is taken to be anisotropic and the line element describes an axially symmetric geometry avoiding rotation about the symmetry axis and meridional motions (zero vorticity case). The modified field equations for axial symmetry in f(R, T) theory are formulated, together with the dynamical equations. Linearly perturbed dynamical equations lead to the evolution equation carrying the adiabatic index \(\Gamma \), which defines the impact of a non-minimal matter to geometry coupling on the range of instability for Newtonian and post-Newtonian approximations.