Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis

A nonlinear two-dimensional (2D) continuum with a latent internal structure is introduced as a coarse model of a plane network of beams which, in turn, is assumed as a model of a pantographic structure made up by two families of equispaced beams, superimposed and connected by pivots. The deformation measures of the beams of the network and that of the 2D body are introduced and the former are expressed in terms of the latter by making some kinematical assumptions. The expressions for the strain and kinetic energy densities of the network are then introduced and given in terms of the kinematic quantities of the 2D continuum. To account for the modelling abilities of the 2D continuum in the linear range, the eigenmode and eigenfrequencies of a given specimen are determined. The buckling and post- buckling behaviour of the same specimen, subjected to two different loading conditions are analysed as tests in the nonlinear range. The problems have been solved numerically by means of the COMSOL Multiphysics finite element software.