Nicola Luigi Rizzi

Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenization, experimental and numerical examples of equilibrium

The aim of this paper is to find a computationally efficient and predictive model for the class of systems that we call ‘pantographic structures’. The interest in these materials was increased by the possibilities opened by the diffusion of technology of three-dimensional printing. They can be regarded, once choosing a suitable length scale, as families of beams (also called fibres) interconnected to each other by pivots and undergoing large displacements and large deformations. There are, however, relatively few ‘ready-to-use’ results in the literature of nonlinear beam theory. In this paper, we consider a discrete spring model for extensible beams and propose a heuristic homogenization technique of the kind first used by Piola to formulate a continuum fully nonlinear beam model. The homogenized energy which we obtain has some peculiar and interesting features which we start to describe by solving numerically some exemplary deformation problems. Furthermore, we consider pantographic structures, find the corresponding homogenized second gradient deformation energies and study some planar problems. Numerical solutions for these two-dimensional problems are obtained via minimization of energy and are compared with some experimental measurements, in which elongation phenomena cannot be neglected.

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.

A 1D higher gradient model derived from Koiter’s shell theory

A thin rectangular plate is modelled as an (initially flat) shell. Following Koiter, the two fundamental forms of the deformed middle surface are then used to define the strain measures of the body. On the middle surface of the plate two local coordinates are introduced: we will call them longitudinal and transversal, respectively. It is assumed that the components of the displacement field which characterize the middle surface kinematics can be expressed as a product of two functions: one defined along the longitudinal coordinate and one defined along the transversal coordinate. Given an explicit expression of the latter functions, the 2D plate fields are reduced to 1D ones. The functions of the transversal coordinate are chosen so that the stretch along it together with the membrane shear vanish. It is worth noting that the linearization of these constraints leads to the well-known Vlasov’s assumptions. It is shown that by modelling each side of a thin walled beam as a 1D continuum, the entire assembly can be reduced to a 1D model as well. This procedure gives rise to an hyperelastic 1D beam model in which at least the warping effect is taken into account. The main features of the model are shown by means of some simple applications.

Frequency shifts in natural vibrations in pantographic metamaterials under biaxial tests

In this paper a 2D continuum model, thought as the homogenized limit of a microstructured pantographic sheet, is studied. The microstructure is characterized by two families of parallel fibers, whose deformation measures account for bending, elongation and relative rotation of the fibers. The deformation energy density of the homogenized model depends on both first and second gradients of the displacement. Modal analysis is performed in order to assess the peculiarities of the dynamic behavior of higher gradient models, and in particular the difference, with respect to classical laminae, in the dependence of the eigenfrequencies on the stiffness.

Modal analysis of laminates by a mixed assumed-strain finite element model:

Fibre-reinforced plates and shells are finding an increasing interest in engineering applications; in most cases dynamic phenomena need to be taken into account. Consequently, effective and robust computational tools are sought in order to provide reliable results for the analysis of such structural models. In this paper the mixed assumed-strain laminated plate element, previously used for static analyses, has been extended to the dynamic realm. This model is derived within the framework of the so-called First-order Shear Deformation Theory (FSDT). What is peculiar in this assumed-strain finite element is that in-plane strain components are modeled directly; the corresponding stress components are deduced via constitutive law. By enforcing the equilibrium equations for each lamina, and taking continuity requirements into account, the out-of-plane shear stresses are computed and, finally, constitutive law provides the corresponding strains. The resulting global strain field depends only on a fixed number of parameters, regardless of the total number of layers. Since the proposed element is not locking-prone, even in the thin plate limit, and provides an accurate description of inter-laminar stresses, an extension to the dynamic range seems to be particularly attractive. The same kinematic assumptions will lead to the formulation of a consistent mass matrix. The element, developed in this way, has been extensively tested for several symmetric lamination sequences; comparison with available analytical solutions and with numerical results obtained by refined 3-D models are also presented.

Leo: a Multimedia Tale of Structural Mechanics

Abstract.The authors have devised a method for teaching structural mechanics articulated in three phases: observations (the description of mechanical phenomena, increasingly complex, selected with regards to their pertinence of the problem that one wants to affront, and their efficiency); modeling (the construction of a physical-mathematical model that takes into account its formal content and stresses its importance as an instrument and has the potential for other applications); design (suggestion of cues for applications stimulate the student to exercise his creative imitation). What is proposed to the student is not so much a set of notions, as a method and set of instruments for selecting experiences (for example previous design solutions) to the end of evaluating their repeatability in diverse situations, by means of a physical-mechanical reading which comes from phenomena which one finds in daily life. “Leo” was created as a teaching instrument which is presented as a tale in the form of a hypertext