Finite element method for transient response of viscoelastic multi-directional FGP skew-nanoplate resting on visco-Pasternak foundation taking into account surface effect using nonlocal strain gradient theory

The finite element approach is used for the first time to simulate and examine the free oscillation and transient response of a visco-elastic multi-directional functionally graded porous (MFGP) skew-nanoplate, taking into account surface effects using nonlocal strain gradient hypothesis. The mechanical characteristics of the materials vary in all three directions of length, width, and thickness of the plate according to the exponential law. Additionally, it has viscoelastic behavior according to the Kelvin-Voigt model. The novelty of this paper lies in the incorporation of the spatial variability of nonlocal and length-scale factors as additional mechanical characteristics of the material. The overall equation of motion for the plate is derived by including the classical plate hypothesis and Hamilton’s principle. A quadrilateral plate element with four nodes and six degrees of freedom is created using a non-conforming C2-level Hermitian function. This function offers precise results and rapid convergence for various forms and boundary conditions (BCs) that low-order elements cannot accomplish. The Newmark-beta direct integration technique is used to calculate the transient responses of the visco-elastic MFGP skew-nanoplate under various BCs. Furthermore, a thorough assessment of the impacts of several factors such as residual surface stress, grading indices, elastic foundation stiffness, skew angle, other geometrical parameters, and BCs on the transient responses of the viscoelastic MFGP skew-nanoplate has been uncovered.