Fractal nonlocal thermoelastic dynamics of imperfect curved functionally graded nanobeam with memory effect

This study presents an exact analytical investigation into the vibrational behavior of imperfect curved nanobeams under uniform thermal conditions, incorporating fractal geometry to model material heterogeneity at multiple scales. The analysis employs a combined nonlocal–strain gradient model to account for scale-dependent behavior and introduces a memory-dependent constitutive model to reflect time-history influences. Material properties vary through the thickness according to a temperature-dependent modified power-law, accounting for porosity induced during manufacturing. The governing equations, derived via Timoshenko beam theory with memory and fractal effects, are solved using the Navier method. Results are validated against existing literature and emphasize the influence of nonlocality, porosity, thermal loading, gradient index, geometry, fractal scaling, and memory effects on natural frequencies. The study confirms that memory-dependent and fractal behaviors significantly affect dynamic responses, offering benchmark solutions for future research on functionally graded nanostructures with porosity and time-dependent effects.