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There is a class of planar 1D-continua which can be described exclusively by their placement functions which in turn are curves in a two-dimensional space. In contrast to the Elastica for which the deformation energy depends on the projection of the second gradient to the normal vector of the placement function, i.e. the material curvature, the pro...
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Over the past decade, kirigami--the Japanese art of paper cutting--has been playing an increasing role in the emerging field of mechanical metamatertials and a myriad of other mechanical applications. Nonetheless, a deep understanding of the mathematics and mechanics of kirigami structures is yet to be achieved in order to unlock their full potenti...
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... This occurs because, to meet the higher-order end boundary conditions (strains), higher-order forces (or double forces) from Eq. (27a) generate within the rod. Similar findings have been explored in Barchiesi et al. (2019) for pantographic beams. ...
Micro-and nano-rods have been identified for various applications in actuators, sensors, and energy harvesters. This paper develops a general framework for micro-and nano-rods based on the one-dimensional strain-gradient theory for special Cosserat rods that considers large displacement and rotation of the cross section, chirality, and size effects. Initially, we obtain the linear momentum balance and angular momentum balance equations for rods utilizing the three-dimensional strain-gradient elasticity theory. Subsequently, we derive the constitutive relations for the strain-gradient elastic rods while considering material objectivity. We further identify the strain-gradient measures and their corresponding higher-order forces and higher-order moment terms. Using these constitutive relations, we show the applicability of our theory by deriving several closed-form solutions for geometrically nonlinear thin rods undergoing extension, shear, torsion, and bending deformation. Finally, we rigorously examine the effect of the length scale parameter on all these deformations of the strain-gradient elastic special Cosserat rod under classical and higher-order boundary conditions. Our analysis through these interesting examples can help in developing next-generation architected metamaterials using micro-and nano-rods.
... These structures consist of (at least) two families of beams periodically arranged in parallel planes with an offset connected by a pivot. Constraining the periodicity of the arrangement of beams to only one dimension, we can describe a pantographic beam [1][2][3][4] and for two dimensions a pantographic sheet [5][6][7][8]. By adding additional parallel pantographic sheets, which are all interconnected by pivots, pantographic blocks are formed, and a periodicity is added to the third dimension [9][10][11]. ...
... Using the SE(3)-logarithm map 1 Log SE (3) , we obtain the fibers' curved initial configurations by solving the nonlinear least squares problem ...
... For Log SE (3) : SE(3) → R 6 we abused notation in accordance with Harsch et al. [47], Eugster and Harsch [48]. ...
In this paper, we apply a numerical integration strategy recently developed for determining the deformation shapes of structures constituted by Cosserat rods, to predict the behavior of panto-cylinders. Panto-cylinders have, as microstructure, a set of two families of helicoidal beams interconnected by perfect or elastic joints. The pivot’s free rotation axis is, in the reference configuration, orthogonal to the cylindrical surfaces spanned by the beams. We perform a series of numerical simulations looking for the mechanical parameters which exalt the chirality effects in the structure. For the performed compression, extensions, shear, and torsion tests, we find chiral deformation patterns with a dependence on the type of joint and its length.
... For a more wide showcase, see, e.g. [42][43][44][45][46][47][48][49]. Finally, we talked about dynamics and dissipation issues. ...
... Pantographic lattices have been extensively studied in the recent literature due to their promising mechanical behavior. To investigate their mechanical behavior, different continuum [3][4][5][6] and discrete [7][8][9][10] models have been proposed. Specifically, a lot of effort has been put into the development of continuum models which are based on second-gradient modeling approaches. ...
In this paper, linear wave propagation in pantographic lattices is investigated. It is assumed that the pantographic lattice is attached to a material modeled by the classical first-gradient theory with a structured interface having its own material properties. By using a variational principle, governing equations and jump conditions at the structured interface are obtained. To this end, the pantographic lattice is modeled by a well-known second-gradient continuum model. Transmis- sion and reflection characteristics are investigated considering four different types of constraints at the structured interface, namely, generalized internal clamp, generalized internal hinge, gen- eralized internal roller, and generalized internal free ends. The effects of elastic moduli and material properties of both continua and the structured interface are analyzed by conducting a parameter study for each considered constraint.
... The study of deformations at the level of the mesostructure of the pantographic metamaterial is crucial. From the point of view of continuum models describing their mechanical response, the related flexural energy determines the part of macrostrain energy depending on the second gradient of macroplacement fields (Yang and Misra, 2012;Auffray, dell'Isola, Eremeyev, Madeo and Rosi, 2015;Placidi, Andreaus and Giorgio, 2017;Barchiesi, Eugster, Placidi and dell'Isola, 2019). The aim is then to observe experimentally the effects due to second gradient energy terms, in a mechanical test that reduces the strains relative to the elongation of the fibers (contributing to first gradient strain energy). ...
... Pantographic structures [1,2,3] are metamaterials that received their name from the characteristic of their fundamental principle of two beams connected by a mechanical link or compliant pivot, allowing large relative rotations with small deformation energies. When considered as continuous media, like 1D pantographic beams [4,5,6], 2D sheets [7,8,9,10] and 3D blocks [11,12,13], they show properties that can not be reproduced by Cauchy-Born continuum mechanics, but more general continuum theories are needed. Commonly, the homogenized models fall within the framework of second-gradient elasticity, which involves the second-gradient of displacement in the deformation energy density [14,15,16,17,18,19]. ...
... In the second gradient theory, the Lagrangian function also depends on the second derivatives of the displacement field and not only on the first ones. The origin of this model lies in the fact that higher-order derivatives have sometimes been included to obtain more general models; Lagrangian functions dependent on higher-order derivatives are often called non-local in the literature [37][38][39][40][41][42][43][44][45][46][47][48][49][50]. This terminology can be explained by considering that, while for a standard material, in a discrete framework with a lattice of points, a simple interaction between a given point and the immediately neighbouring points, in the continuum limit involves an energy function based on the first derivative of the placement field according to the Piola's ansatz. ...
In this paper, we consider a deformable continuous medium and its discrete representation realized by a lattice of points. The former is solved using the classical variational formulation with the finite element method. The latter, a 2D discrete “kinematic” model, instead is conceived to determine the displacements of the lattice points depending on interaction rules among them and thus provides the final configuration of the system. The kinematic model assigns the displacements of some points, so-called leaders, by solving Newton’s law; the other points, namely followers, are left to rearrange themselves according to the lattice structure and the flocking rules. These rules are derived from the effort to describe the behaviour of a robot swarm as a single whole organism. The advantage of the kinematic model lies in reducing computational cost and the easiness of managing complicated structures and fracture phenomena. In addition, generalizing the discrete model to non-local interactions, such as for second gradient materials, is easier than solving partial differential equations. This paper aims to compare and discuss the deformed configurations obtained by these two approaches. The comparison between FEM and the kinematic model shows a reasonable agreement even in the case of large deformations for the standard case of the first gradient continuum.
... The crucial point is to conjecture the right mathematical model that is faithful to a certain degree of the physical system under consideration and allows for the evaluation up to a pre-established accuracy of some aspects of interest. For pantographic (i.e., mesostructured) materials, the following three approaches have been developed for describing their mechanical behavior [20,21,22], each having its limitations and strengths: 25 • continuum approach [23, 24,25,26]; ...
... The crucial point is to conjecture the right mathematical model that is faithful to a certain degree of the physical system under consideration and allows for the evaluation up to a pre-established accuracy of some aspects of interest. For pantographic (i.e., mesostructured) materials, the following three approaches have been developed for describing their mechanical behavior [20,21,22], each having its limitations and strengths: 25 • continuum approach [23, 24,25,26]; ...
... Further, a macro-scale model that can describe large displacements and deformations of pantographic beams can be given by introducing a meso-length scale using Hencky discrete springs that are equivalent to Euler beams, such that macro-deformation energy is expressed in terms of the extensional and rotational kinematic quantities ρ and ϑ as [61], ...
Novel theories are needed for the discovery of innovative and exotic metamaterial and for their
rational design. The current practice of mechanical analyses based upon moribund classical theories and
experimental trial-error campaigns is caught in an inescapable vortex and illusion of inductive reasoning.
The needed novel research paradigm is one in which the formulation of theoretical concepts precede their
experimental validation. In the absence of theoretical understanding, the design experiments and collection
of experimental evidence will remain unavoidably circumscribed. History of science can provide us guidance
in the search for the needed powerful tools required for discovery. The principle of virtual work provides the
necessary framework for development of theories that can lead to novelmetamaterials, as it was the unifying
principle which allowed the French-Italian School, headed by D’Alembert, Lagrange and Gabrio Piola,
to found modern continuum mechanics. Based upon this framework we have conceived a metamaterial
synthesis schema that exploits micro-macro identification traceable to the early days of the formulation of
continuum theories for deformable solids. The schema is illustrated with application to metamaterials with
pantographic and granularmotifs based upon higher-gradient and higher-order theories.
... The metamaterials synthesized by using peculiar mesostructures (Turco et al., 2017a;Ciallella, 2020;Aydin et al., 2022;Barchiesi et al., 2023) exhibit some unusual properties whose exploitation may lead to interesting engineering applications (dell'Isola et al., 2019a,b;Barchiesi et al., 2021;Spagnuolo et al., 2022b;Spagnuolo, 2022a;Eremeyev et al., 2023). Moreover, on the basis of available theoretical results (Barchiesi et al., 2019;Eremeyev et al., 2019Eremeyev et al., , 2020dell'Isola and Steigmann, 2020;Spagnuolo and Barchiesi, 2021;Eremeyev and Reccia, 2022), when using pantographic meso-architectures as fundamental substructures at multiple length scales, one may synthesize a large class of generalized continua that include at least the set of th gradient continua (Alibert et al., 2003;Seppecher et al., 2011). Very similar pantographic substructures may also be found in fiber-reinforced composites (Cuomo et al., 2016;Spagnuolo et al., 2022c). ...
