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

Abstract

To cite this version:I. Giorgio, N. Rizzi, E. Turco. Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2017The, , 473 (2207The, ), 10.1098The, /rspa.2017 Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis Giorgio, I. and Rizzi, N. L. and Turco, E.Abstract A nonlinear 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. In order 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 postbuckling behaviour of the same specimen, subjected to two di↵erent loading conditions are analysed as tests in the nonlinear range. The problems have been solved numerically by means of COMSOL Multiphysics FE software.

Pantographic metamaterials: an example of mathematically driven design and of its technological challenges

“…23) both show that, after a critical load, out-of-plane buckling occurs during bias extension, because the transverse (curved) beams in the middle of the specimen undergo buckling induced by the shortening of the middle width of the specimen. A 2D continuum model embedded in a 3D space has been also proposed [48] where, relying on a variational framework, the following strain energy density is proposed…”

Determination of metamaterial parameters by means of a homogenization approach based on asymptotic analysis

“…A feasible approach is to use first order at the microscale and second order at the macroscale leading to additional parameters at the macroscale to be determined, we refer to various formulations of a second order theory in [1][2][3][6][7][8]33,37,63,74,75,77,78,80,81,85,89]. Higher order theories are called generalized mechanics and homogenization in the framework of generalized mechanics is a challenging task endeavored by many scientists, among others by [17,31,40,49,55,71,79]. Mostly, it is agreed that homogenization of an RVE by involving so-called higher gradient terms of the macroscopic field is a natural way to include size effect [14,[41][42][43][44]58].…”

Nonlinear dynamics of uniformly loaded Elastica: Experimental and numerical evidence of motion around curled stable equilibrium configurations

“…Limiting our attention to mechanical problem and choosing, in the literature those works whose spirit is closer to ours, the application of Hencky's modeling concept can be found again in some more recent works which deal with single beams, see [], or with assemblages of beams, see []. Moreover Hencky‐type models can be used to conceive specially targeted lattice materials structured at different scales, see [].…”

“…This problem played a relevant role in the theory of metamaterials and in particular when dealing with pantographic microstructures with long beams. [12,29,34,35,40,48,51,61,104] In pantographic sheets [6,43,69] the single fiber, in a certain class of situations, can be modelled as a beam interacting with the other beams/fibers, in such a way that in the homogenized limit, a distributed load can be assumed to be applied. Actually in pantographic structures with long 'fibres' (see, e.g., Figure 1), we can observe the existence of 'long and soft beams' undergoing large deformations and subjected to 'distributed' loads.…”