Stress intensity factors for 3-D axisymmetric bodies containing cracks by p-version of F.E.M.

This paper presents an extensive investigation of bi-directional functionally graded (BFG) deep circular nanobeam’s free vibration behaviour. BFG deep circular nanobeam’s displacement field is expressed in accordance with Timoshenko beam theory. To obtain the governing equations in canonical form, Lagrangian equations are first implemented, and the resulting expressions are then converted into a canonical equation set utilizing the conjugate momentums and their associated derivatives. In addition, nanoscale impacts are incorporated into the analyses utilizing Eringen’s approach. In accordance with this procedure, an algorithm is established and its accuracy is verified by solving a benchmark free vibration example from literature and comparing the corresponding results. Finally, an extensive parametric study is realized, in which the impacts of nanoscale, boundary condition, bidirectional material gradation, and different geometric characteristics on BFG deep circular nanobeams’ fundamental frequencies are evaluated.