In plane shear and bending for first gradient inextensible pantographic sheets: numerical study of deformed shapes and global constraint reactions

Abstract

The aim of the present paper is the analysis of a two-dimensional continuum with two families of inextensible fibers that are orthogonal in the initial configuration. In the first part of the work, a new formulation is presented, in which the problem is reduced to a standard nonlinear constrained minimization, while in the second part of the work several numerical investigations are presented considering different boundary conditions with respect to standard symmetric bias extensional tests. The conceptual framework can be recognized in the researches by Pipkin and Rivlin on inextensible nets. Furthermore, an implicit version of the Rivlin representation of the generic placement for a two-dimensional sheet with two families of inextensible fibers is provided by considering the angles of the fiber directors as degrees of the freedom of the formulation. In this way the first gradient formulation is given in terms of two angle fields only.

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

“…Numerical results [37][38][39][40][41] show that the final configurations obtained by using second gradient energies are smoother than those with the first gradient approach. In the following, numerical results will be reported for standard bias extension, shear and rotation tests, which confirm such a statement.…”

Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models

“…The mechanical systems which could be described by the introduced discrete model include those considered and studied in [2,4,5,10,11]. Indeed, (1) the pivots interconnecting (see Fig.…”

“…However, we believe that the potential scope of the presented model is wider: it very likely may include the most relevant deformation phenomena in the woven fabrics described in [12][13][14][15][16][17][18][19]. Indeed, the fibers constituting these fabrics intersect one with the others very similarly than in pantographic structures, and in their deformation, one can still distinguish extension, bending and shear: even if the hypothesis of elastic behavior in woven fabrics is applicable in a smaller range of deformation states, the model presented here seems capable of catching some relevant aspects of its phenomenology.…”

A 1D Continuum Model for Beams with Pantographic Microstructure: Asymptotic Micro-Macro Identification and Numerical Results

“…A comprehensive review on the modeling of pantographic structures can be found in [61,62]. Several results of numerical investigations can be found in [30,[63][64][65][66][67][68][69][70][71], while for an outline of recent experimental results we refer to [72,73]. The work is organized in the following way.…”