Free vibration and large deformation characteristics of metamaterial thickness-deformable plates with initial geometrical imperfection

A geometrically imperfect third-order auxetic metamaterial thickness- and shear-deformable plate is considered, and both the non-linear bending as well as the free vibrations are analysed. Distribution of the effective material properties from plate’s top surface to the bottom one follows a functionally graded form; material properties are effectively approximated via genetic programming-assisted micromechanics models from previous studies. Without ignoring geometrical non-linearities, the fourfold coupled axial, transverse, rotational, and stretching non-linear motion equations are formulated for a geometrically imperfect third-order auxetic metamaterial thickness- and shear-deformable plate using Hamilton’s principle. The generalised differential quadrature method is implemented to discretise the motion equations; the discretised motion equations are then solved for both the non-linear bending as well as the linear free vibrations. For partial validation, the model is compared with available data from the open literature for simplified cases (i.e., single-layer homogeneous plates without metamaterial characteristics) and with a single-layered isotropic rectangular perfectly straight plate modelled in ANSYS. The complex non-linear mechanics and linear free vibration of the metamaterial system are analysed for different geometrical parameters, graphene origami contents, folding degrees, and geometrical imperfections, and also for both the symmetric and asymmetric distribution patterns of graphene origami. The results reveal that the geometric imperfections reduce the transverse deflection, and metamaterial plates with asymmetric graphene origami distributions consistently have the largest non-linear deflections among all the graphene origami distributions studied.