Postbuckling and nonlinear free vibration of postbuckled porous functionally graded micro/nanotubes via nonlocal strain and velocity gradient theory

Background

Vibration response analysis serves as a critical tool in investigating the behavior of micro/nanoscale structures operating in dynamic environments, offering valuable insights into their performance and ultimately refining the design of devices. Particularly, when these structures are deliberately engineered to function near or within the postbuckling regime, understanding their vibratory behavior in this state becomes essential. This study focuses on exploring the postbuckling behavior and nonlinear frequencies of simply supported buckled porous functionally graded (PFG) size-dependent tubes. Internal resonances are not considered in this analysis.

Method

The nonlocal strain and velocity gradient theory, within the framework of the Euler-Bernoulli beam hypothesis, is employed to derive the nonlinear partial differential equations of motion. It is assumed that the material properties are gradually graded in the radial direction. Additionally, two different porosity distribution patterns are used in the radial direction. The method of multiple scales is used to solve the system of nonlinear ordinary differential equations obtained by applying the Galerkin method.

Results

The closed expression for the i-th nonlinear frequency of buckled porous functionally graded size-dependent tubes is determined based on the amplitude of the vibration modes involved. The findings indicate that porous M/NTs exhibit a loss of static stability at lower compressive axial loads compared to their nonporous counterparts. Furthermore, the softening effects resulting from a uniform porosity distribution are more pronounced than those from an uneven porosity distribution. Interestingly, nonporous M/NTs display the lowest nonlinear postbuckling frequency among the studied configurations. Moreover, it is observed that the nonlinear frequency tends to increase with a rise in the compressive axial load, while it decreases with an increase in the excitation amplitude.