Dynamic Behavior of Magnetically Affected Rod-Like Nanostructures with Multiple Defects via Nonlocal-Integral/Differential-Based Models

Through considering both nonlocality and surface energy effects, this paper suggests suitable mathematical-continuum-based models for free vibration of nanorods with multiple defects acted upon by a bidirectional-transverse magnetic field. By employing both theories of elasticity of Eringen and Gurtin–Murdoch, the equations of motion for the magnetically affected-damaged rod-like nanostructures are derived using the nonlocal-differential-based and the nonlocal-integral-based models. The local defects are modeled by a set of linearly appropriate axial springs at the interface of appropriately divided nanorods. Through constructing the nonlocal-differential equations of motion for sub-divided portions and by imposing the appropriate interface conditions, the natural frequencies as well as the vibrational modes are explicitly obtained for fixed–free and fixed–fixed nanorods with low numbers of defects. The extracted nonlocal-integral governing equations are also solved for natural frequencies using the finite-element technique. For a particular situation, the model’s results are successfully verified with those of another work. Subsequently, the effects of nonlocality, surface energy, defect’s location, nanorod’s diameter, magnetic field strength, and number of defects on the dominant free vibration response of the magnetically defected nanorods with various end conditions are displayed and discussed.