Abstract
Nonlinear free vibrations of functionally graded porous Bernoulli–Euler nano-beams resting on an elastic foundation through a stress-driven nonlocal elasticity model are studied taking into account von Kármán type nonlinearity and initial geometric imperfection. By using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural flexural frequency is then established using the Hamiltonian approach to nonlinear oscillators. Several comparisons with existing models in the literature are performed to validate the accuracy and reliability of the proposed approach. Finally, a numerical investigation is developed in order to analyze the effects of the gradient index coefficient, porosity volume fraction, initial geometric imperfection, and the Winkler elastic foundation coefficient, on the nonlinear flexural vibrations of metal–ceramic FG porous Bernoulli–Euler nano-beams.
Highlights
Graded materials (FGMs) are advanced composites designed and fabricated in a way that their physical and mechanical properties spatially vary in their structures
Functionally graded materials are made of a combination of metal and ceramic to offer a wide range of applications for various equipment subject to extreme thermo-mechanical stresses [5,6,7,8,9,10,11,12,13]
Nano-beam, resting on a Winkler elastic foundation, whose material properties are listed in Table 1, is considered as a case study
Summary
Graded materials (FGMs) are advanced composites designed and fabricated in a way that their physical and mechanical properties spatially vary in their structures. Some recent applications of the theories mentioned above are addressed in [42,43,44] These difficulties can be overcome by adopting the stress-driven nonlocal integral model (SDM) recently proposed by Romano and Barretta [45], in which the roles of stress and elastic strain fields are swapped with respect to the strain-driven model. In this case, for a class of bi-exponential kernels, the integral form of the constitutive equations is shown to be mathematically equivalent to differential equations subjected to some higher order constitutive boundary conditions. Small scale effects are imperfection, the gradient index coefficient, the porosity volume fraction and the Winkler elastic foundation coefficient on nonlinear fundamental frequencies of porous FG supported nanobeams are presented and discussed
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