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Scale-dependent vibration and stability of three-dimensional functionally graded (FG) porous Rayleigh nanobeams with axial motion in varying environmental conditions are investigated by considering scale effects in the thickness orientation. It is supposed that the effective material characteristics of the nanobeam in the longitudinal, width, and thickness orientations are graded according to exponential and power-law functions. Different patterns involving uniform, non-uniform, and logarithmic distributions are considered to model the porosity impacts. Based on the nonlocal strain gradient Rayleigh beam model and incorporating thickness effects, the dynamical equation of the system is derived by considering linear, parabolic, and sinusoidal elastic foundations. The Galerkin discretization technique and Laplace transform are adopted to solve the eigenvalue problem and accomplish the stability analysis. Several comparative studies are performed with the available data in the open literature to validate the solution approach and results. Also, detailed parametric investigations are accomplished to interpret the importance of scale factor, rotary inertia factor, porosity characteristics, magneto-hygro-thermal loads, foundation parameters, gradient indices, and boundary conditions on the dynamical behavior, critical divergence, and flutter axial speeds. The results revealed that, contrary to the effect of the material gradient in thickness orientation, the longitudinal and width gradation of materials can lead to the increment of vibrational frequencies. The outcomes declared that the instability threshold improves significantly by considering the scale effects along the thickness orientation. Also, it is demonstrated that when the middle surface of the cross-section has the highest porosity, the stability regions expand by increasing the porosity coefficient. The present research outcomes can offer valuable insights into the optimum design of high-speed moving multi-directional graded porous nanosystems.
Xiaolan Chen (Tue,) studied this question.