Gul, UfukAydogdu, Metin2024-06-122024-06-1220221678-58781806-3691https://doi.org/10.1007/s40430-022-03635-3https://hdl.handle.net/20.500.14551/19166In this study, the longitudinal vibration of nanorods is investigated in the framework of nonlocal strain gradient theory and Bishop's rod model for the first time in the literature. Unlike simple rod theory, radial deformation and radial inertia are considered in Bishop rod theory. After deriving the kinematic relations of this Bishop nanorod model, Ritz method is used in order to determine the natural frequencies for different boundary conditions. It is obtained that the nonlocal strain gradient theory predicts the softening or hardening material behaviour compared to classical Bishop rod theory depending on the magnitude of the material length scale parameter and nonlocal parameter. This mechanical behaviour can provide a favourable design of nano/micro-scale structures. To show the validation of this study, natural frequencies calculated by using the present nonlocal strain gradient model are compared to results of other nonlocal strain gradient models in the literature and good agreement has been observed. The dimensionless frequencies of Bishop nanorod model are obtained by using the nonlocal strain gradient theory for different parameters such as material length scale parameter, nonlocal parameter, nanorod length, radius of gyration and mode number and a comprehensive review is executed in the numerical results.en10.1007/s40430-022-03635-3info:eu-repo/semantics/closedAccessBishop Nanorod ModelNonlocal Strain Gradient TheoryRitz MethodVibrationHetero-Junction CntsCarbon NanotubesTorsional VibrationWave-PropagationAxial VibrationElasticityRodsDeformationDislocationVelocityArticle448Q3WOS:0008362919000062-s2.0-85135204062Q2