Parameter identification of a second-gradient model for the description of pantographic structures in dynamic regime

Pantographic structures are examples of metamaterials with such a microstructure that higher-gradient terms’ role is increased in the mechanical response. In this work, we aim for validating parameters of a reduced-order model for a pantographic structure. Experimental tests are carried out by applying forced oscillation to 3D-printed specimens for a range of frequencies. A second-gradient coarse-grained nonlinear model is utilized for obtaining a homogenized 2D description of the pantographic structure. By inverse analysis and through an automatized optimization algorithm, the parameters of the model are identified for the corresponding pantographic structure. By comparing the displacement plots, the performance of the model and the identified parameters are assessed for dynamic regime. Qualitative and quantitative analyses for different frequency ranges are performed. A good agreement is present far away from the eigenfrequencies. The discrepancies near the eigenfrequencies are a possible indication of the significance of higher-order inertia in the model.

A 3D extension of pantographic geometries to obtain metamaterial with semi-auxetic properties

In this work we present a three-dimensional extension of pantographic structures and describe its properties after homogenization of the unit cell. Here we rely on a description involving only the first gradient of displacement, as the semi-auxetic property is effectively described by first-order stiffness terms. For a homogenization technique, discrete asymptotic expansion is used. The material shows two positive ([Formula: see text]) and one negative Poisson’s ratios ([Formula: see text]). If, on the other hand, we assume inextensible Bernoulli beams and perfect pivots, we find a vanishing stiffness matrix, suggesting a purely higher gradient material.

Stiffness optimization in nonlinear pantographic structures

Mechanical metamaterials are microstructured mechanical systems showing an overall macroscopic behaviour that depends mainly on their microgeometry and microconstitutive properties. Moreover, their exotic properties are very often extremely sensitive to small variations of mechanical and geometrical properties in their microstructure. Clearly, the methods of structural optimization, once combined with the techniques used to describe multiscale systems, are expected to determine a dramatic improvement in the quality of newly designed metamaterials. In this paper, we consider, only as a demonstrative example, planar pantographic structures which have proved to be extremely tough in extension, To describe pantographic structure behaviour in an efficient way, it has been proposed to use Piola–Hencky-type Lagrangian models, in which the understanding of the mechanics of involved microdeformation processes allows for the formulation of efficient numerical codes. In this paper, we prove that it is possible, via a suitable choice of the macroscopic shear stiffness, to increase the maximal elongation of pantographic structures, in the standard bias test, before the occurrence of rupture phenomena. The basic tool employed to this aim is a constrained optimization algorithm, which uses the numerical tool, previously developed for determining equilibrium shapes, as a subroutine. Actually, one looks for the shear stiffness distribution, which, given the imposed elongation of the pantographic structure and the force applied to it by the used hard device, minimizes the total elongation energy. The so-optimized shear stiffness distribution does prove able to extend the range of imposed elongations that the specimen can experience while remaining undamaged.

Are higher-gradient models also capable of predicting mechanical behavior in the case of wide-knit pantographic structures?

The central theme of this study is to investigate a remarkable capability of a second-gradient continuum model developed for pantographic structures. The model is applied to a particular type of this metamaterial, namely the wide-knit pantograph. As this type of structure has low fiber density, the applicability of such a continuum model may be questionable. To address this uncertainty, numerical simulations are conducted to analyze the behavior of a wide-knit pantographic structure, and the predicted results are compared with those measured experimentally under bias extension testing. The results presented in this study show that the numerical predictions and experimental measurements are in good agreement; therefore, in some useful circumstances, this model is applicable for the analysis of wide-knit pantographic structures.