A new model for non-linear vibration of functionally graded porous nano-Beam based on non-local curvature and strain gradient tensors

In this paper, non-linear free vibration analysis of nano-beam has been studied. The non-local strain gradient theory and curvature tensor are used to show the size effect. The length scale parameter expresses the effect of strain gradient tensor in the non-local strain gradient theory. However, the aim of this article is to show the simultaneous effect of curvature and strain gradient tensors in non-linear vibration of functionally graded porous nano-beams. The effect of curvature tensor is demonstrated with the curvature tensor dependent parameter. Considering non-linear Von Karman strains and Euler-Bernoulli beam theory, the governing vibrational equation of FG porous nano-beams are derived using Hamilton's principle in the presence of strain gradient and curvature tensors simultaneously. The non-linear differential equation is extracted by using Galerkin's method and the non-linear natural frequency of nano-beam is obtained according to Hamiltonian approach. Results represent the simultaneous effects of the length scale and curvature tensor dependent parameters on dimensionless non-linear natural frequencies. Also effects of different parameters such as non-local parameter, length scale parameter, porosity volume index, and power-law index are discussed in the presence and absence of the curvature tensor dependent parameter. Also, the beginning points of stiffness-hardening and stiffness-softening of nano-beam are always constant values in the non-local strain gradient theory, whereas considering the curvature tensor changes the beginning points of stiffness-hardening and stiffness-softening. The results are also compared with previous researches for validation.