Non-linear continuum models for planar extensible beams and pantographic lattices of beams: Heuristic homogenization, experimental and numerical examples of equilibrium in large deformation

3 March 2016
Francesco dell'Isola
There are relatively few results in the literature of non-linear beam theory: we recall here the very first classical results by Euler–Bernoulli and the researches stemming from von Kármán for moderately large rotations but small strains. In this paper, we consider a discretized springs model for extensible beams and propose a heuristic homogenization technique of the kind first used by Piola. The homogenized energy obtained has some peculiar features which we start to describe by solving numerically some exemplary deformation problems. Furthermore we consider pantographic structures constituted by the introduced nonlinear beams and study some planar deformation problems. Numerical solutions for these 2D problems are obtained via minimization of energy and are compared via some experimental measurements, in which the importance of elongation phenomena are clearly pointed out. In the conclusions we indicate a list of some mathematical problems which seems worth of consideration. 
Indeed Already Piola in 1848 introduces for microscopically discrete systems to be described via a continuum model: i) the micro-macro kinematical map, ii) the identification of micro- macro work functional and iii) the consequent determination of macro-constitutive equations in terms of the micro properties of considered mechanical system.
Piola uses, following the standards of his time, a rigorous mathematical deduction process and considers separately one dimensional, two dimensional and three dimensional continua as continua whose reference configuration is a curve, a surface or a regular connected subset of Euclidean three dimensional space. This subdivision of the presented matter is also followed by Cosserat Brothers: how to detect the influence on their works exerted by Piola’s pioneering ones is a historical problem which deserves further in-depth studies.
In the present paper we follow the spirit of Piola while looking for Lagrange density functions for a class of non-linear one dimensional continua in planar motion: we focus on modeling phenomena in which both extensional and bending deformations are of relevance.
Usually in literature the simultaneous extension and bending deformation of a beam is not considered: however when considering two dimensional continua embedding families of fibers as a model of some specific microstructured mechanical systems (as fiber fabrics or pantographic sheets ) the assumption that the fibers cannot extend while bending is not phenomenologically well-grounded. Therefore, we are led in the second part of the present paper to present some two dimensional continua in which the second gradient of in plane displacement (involving so called geodesic bending) appears in the expression of deformation energy.
The modeling assumptions are, in both cases, based on a physically reasonable discrete microstructure of used beams: in engineering literature these microstructures, constituted by extensional and rotational springs and possibly rigid bars, were introduced in order to get discrete Lagrangian approximation of continuum models in linearized regimes.
A natural development, involving the study of spatial placements of one dimensional or two dimensional continua or the introduction of three dimensional continua embedding reinforcement fibers will be subject of further investigations.
The study of pantographic sheets by means of a micro model based on Cauchy first gradient continuum models involves the choice of relatively small length scale, implying the introduction of numerical models involving finite elements with several millions of degrees of freedom: the computational burden of such models makes their use, at least in the mid term horizon, absolutely inappropriate. The higher gradient reduced order model presented in this paper involves a rather more effective numerical modeling whose performances (as will be shown in a forthcoming paper Giorgio et al. in preparation) are however absolutely comparable.
However the problem of formulating intermediate meso modeling, involving a class of Generalised Beam Theories, will be necessarily to be confronted: for instance the deformation of beam sections involving warping, Poisson effects, elastic necking or large shear or twist deformation can definitively be studied via reduced order models not resorting to the most detailed micro Cauchy first gradient models.
One should also remark that higher gradient continuum models may require novel integration schemes, more suitable to their intrinsic structure: we expect that isogeometric methods may further increase the effectiveness of the reduced models we present here, especially when completely spatial models will be considered .
  • Industrial and Applied Mathematics Seminar