Physics

Finite element method

Quadrilateral

Mathematics

Mechanics

Porosity

Structural engineering

Length scale

Engineering

Mathematical analysis

Boundary value problem

Composite material

Vibration

Materials science

Geometry

Quantum mechanics

Volume fraction

Shear (geology)

Advances in nano research

In the present paper, a refined trigonometric higher-order shear deformation theory has been presented with the conjunction of nonlocal theory for the vibrational response of functionally graded (FG) porous nanoplate. The displacement field is chosen based on assumptions that the out of the plane displacement consists of bending and shear components whereas the transverse shear-strain has nonlinear variation along the thickness direction. The number of unknown variables is four, as against five in other renowned shear deformation theories. The governing equations have been derived using Hamilton's principle. A generalized porosity model has also been developed to accommodate both even and uneven type of distribution of porosity in the FG nanoplates. The closed-form solution of simply-supported FG porous nanoplates is obtained and the results are compared with the available reported results. In finite element solution, a C0 continuous isoparametric quadrilateral element has been used with various conventional and unconventional boundary conditions. The effects of various parameters like small-scale effect, aspect ratio, volume fraction index, porosity volume fraction, and thickness ratio have been investigated. The significant influence of small-scale effects and porosity inclusions have been observed in the reported results. It has been reported that both closed-form and finite element solutions with the present theory can make accurate predictions of the free vibration response.

Size-dependent vibration response of porous graded nanostructure with FEM and nonlocal continuum model