Transverse vibration of an embedded nanotube made of functionally graded porous material based on NSGT

The present work focuses on examining the transverse vibration behavior of a functionally graded porous nanotube resting on a Winkler–Pasternak elastic foundation, incorporating size-dependent effects and deformable boundary conditions. An analytical approach based on nonlocal strain gradient theory is adopted, which accounts for small-scale effects through two distinct size parameters—one with a strengthening effect and the other with a weakening effect. For the first time, deformable boundary conditions are considered the functionally graded porous nanotube in this context. An eigenvalue problem is formulated using the Stokes' transform, with constant coefficients assigned to the springs, along with Winkler–Pasternak foundation coefficients, material properties, and scale parameters. The resulting characteristic equation is solved analytically for various values of foundation stiffness, elastic spring parameters, strain gradient, and nonlocal effects to determine the system’s vibration frequencies. A comparative assessment with earlier research is conducted to validate the findings, confirming their accuracy. Then, the effects of small-scale parameters, volume fraction index, deformable boundaries, elastic foundation, rotary inertia and porosity density are discussed.