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

Abstract

In the standard asymptotic micro-macro identification theory, starting from a De Saint-Venant cylinder, it is possible to prove that, in the asymptotic limit, only flexible, inextensible, beams can be obtained at the macro-level. In the present paper we address the following problem: is it possible to find a microstructure producing in the limit, after an asymptotic micro-macro identification procedure, a continuum macro-model of a beam which can be both extensible and flexible? We prove that under certain hypotheses, exploiting the peculiar features of a pantographic microstructure, this is possible. Among the most remarkable features of the resulting model we find that the deformation energy is not of second gradient type only because it depends, like in the Euler beam model, upon the Lagrangian curvature, i.e. the projection of the second gradient of the placement function upon the normal vector to the deformed line, but also because it depends upon the projection of the second gradient of the placement on the tangent vector to the deformed line, which is the elongation gradient. Thus, a richer set of boundary conditions can be prescribed E. Barchiesi ⋅ F. dell'Isola

Large in-plane elastic deformations of bi-pantographic fabrics: asymptotic homogenization and experimental validation

“…Pantographic beam discrete model. The assembly and kinematics of a discrete pantographic beam slightly generalizing that presented in [33,36] are sketched in Fig. 1.…”

“…1 depict hinge constraints, requiring the end points of the corresponding springs to have the same position in space. We note that the assembly considered herein is a generalization of that studied in [33], as the angle γ ∈ (0, π) between springs concurring at point P i from the right in Fig. [33] is generally dierent from π /4. Moreover, further rotational springs, which are coloured in green in Fig.…”

“…In [33], the model of [34,35] has been generalized to the nite strain regime. Remarkably, the deformation energy density of such a 1D-continuum, [33], does not only depend on the material curvature but also on the stretch gradient.…”

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

“…We are sure of the greater efficiency of Hencky‐type models also when more complicated systems are considered (see []). Therefore we are confident that codes similar to the simple ones which were used in the present paper can be developed, and will be extremely useful, also in the dynamics study of i) Timoshenko beams, ii) lattice systems including many beam structural elements both in the case of 1D systems, in small or large deformations regimes, or in the case 2D and 3D structures, iii) more complex micro‐structures which arise in the theory of metamaterials (see []). We believe that the alternative approach, based on the formulation of continuous models and on their subsequent discretization although is substantially equivalent, may present some difficulties, when the discretization process of the continuous model is obtained without taking into account the physical nature of the modeled mechanical systems.…”

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

“…This method decomposes variables to their global variations and local fluctuations. Such a decomposition is used to generate closed form equations to determine constitutive parameters as applied in one-dimensional problems, for example in the analysis of composites [16,21], while 2D problems [13,15,22,29,72] have been investigated numerically. FEM is employed in [68] demonstrating that higher-order terms start dominating as the difference between parameters of composite materials increases.…”