A thermodynamically consistent strain-gradient model for functionally graded nanobeams with composite damage mechanics on elastic substrate medium

This study develops a functionally graded (FG) nanobeam model incorporating composite damage mechanics within a thermodynamically consistent strain-gradient framework. The formulation is based on Euler–Bernoulli beam theory, while the nanobeam–substrate interaction is represented by a Winkler foundation. Damage is introduced through a partially degraded ceramic phase, whereas the metal phase remains intact. Three effective-property models, namely the Voigt, Reuss, and proposed formulations, are used to describe the damaged FG material response. The governing equations are derived using the principle of virtual work, and analytical solutions are obtained for the static bending response. The results show that damage reduces effective stiffness and increases bending deflection, while higher ceramic content and substrate stiffness enhances the overall rigidity. The proposed model yields physically admissible predictions bounded by the Voigt and Reuss limits, demonstrating its consistency for damaged FG nanobeam–substrate systems.