Static analysis of porous Bishop nanorods embedded in an elastic medium

This work presents a novel analytical formulation for the static response of porous, thick circular nanorods embedded in an elastic medium, developed within the framework of Bishop's rod theory. The model incorporates two distinct sources of non-locality: an internal one, governed by a stress-driven integral law describing the mechanical behavior of the nanorod material, and an external one, modeled through a displacement-based integral formulation accounting for the nonlocal response of the surrounding elastic matrix. The resulting variational problem is first established in integral form and subsequently recast into an equivalent well-posed differential representation through the adoption of suitable convolution kernels. The analytical framework enables an efficient evaluation of the coupled effects of porosity, non-locality, and foundation stiffness on the mechanical response. Parametric studies, supported by analytical–numerical simulations, reveal the fundamental influence of the governing length scales on the global stiffness and deformation patterns of the system. The proposed approach provides a consistent theoretical basis for the design and optimization of nanostructures, resonators, and sensing elements, and can be readily extended to the modeling of nanostructured composites and innovative sustainable materials.