Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions

Gołaszewski, Maciej; Grygoruk, Roman; Giorgio, Ivan; Laudato, Marco; Cosmo, Fabio Di

doi:10.1007/s00161-018-0692-0

Cited by 73 publications

(36 citation statements)

References 52 publications

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“…In order to make such disclosure evident, and to experimentally investigate the special role that perfect pivots played in synthesized microstructures, the following technological demand arose: "to find an innovative design and production process to fabricate, via additive manufacturing, pantographic sheets made for instance of polyamide, whose pivots actually twist without deformation and with negligible dissipation" [5]. The specimens could be printed in a monolith and required no post-assembly but only an easily standardized run-in procedure.…”

Section: 2mentioning

confidence: 99%

Enhanced Piola–Hencky discrete models for pantographic sheets with pivots without deformation energy: Numerics and experiments

Turco

Misra

Pawlikowski

et al. 2018

International Journal of Solids and Structures

108

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The problem of the synthesis of second gradient (meta)materials, via architectured microstructures made of micro-lattices, has been solved [1, 2] by choosing ideal pivots as preferred constraints. The obtained homogenized macro-equations [3, 4]) show some pathologies that reflect the exotic behavior of the considered metamaterials, even if they are of interest by themselves [4]. The theoretical issues that they raise not only represent an intellectual challenge but also means for disclosing potentially interesting new phenomena. To make such disclosure evident, the related technological demand arose, namely, to find an innovative design and production process to construct [5], by using additive manufacturing, some pantographic sheets (made in this instance of polyamide but hopefully later also using metals or alloys) whose pivots do twist practically without deformation and with negligible dissipation. Remarkably the specimen could be printed in a monolith and required no post-assembly but only an easily standardized run-in procedure. In this paper, in order to introduce a mathematical description for pantographic sheets with perfect pivots and to avoid to face the aforementioned pathologies, a discrete, finite dimensional, Lagrangian model is formulated. Moreover, in order to include the case in which the beams interconnecting the pivots are long enough to store non negligible bending energy between the closest pairs of pivots, an enhanced Piola-Hencky discrete model is introduced. Two types of nodes are distinguished, the first one interconnects two pantographic fibers, the second one simply interconnects two different segments of the same fiber. The Vietnam long neck peculiar deformed shape experimentally observed in standard extension bias test, is obtained with very short computing time, so that the innovative code which has been elaborated can be used as subroutine in more complex computation schemes. A preliminary digital image correlation analysis, see [6, 7], is performed and shows that a remarkable agreement between theoretical predictions and experimental evidence can be obtained. This circumstance is easily explained by observing that said numerical code is based on a discrete model directly inspired by the mechanical properties of pantographic sheets and that, therefore, the passages to a continuum model via homogenization [8] and then to the subsequent re-discretization, via the introduction of more or less suitable finite elements, are avoided. In our opinion a theory driven formulation of a directly discrete numerical model presents many advantages and it seems suitable for attacking future structural optimization problems.

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Section: 2mentioning

confidence: 99%

Enhanced Piola–Hencky discrete models for pantographic sheets with pivots without deformation energy: Numerics and experiments

Turco

Misra

Pawlikowski

et al. 2018

International Journal of Solids and Structures

108

View full text Add to dashboard Cite

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“…Another source of inelastic behavior is the contact of the beams in the lattice and the related adhesion interactions [56,57]. For current state of the pantographic metamaterials, we refer to [58][59][60][61][62][63]. Beam lattices can be used as "meso-models" of cellular solids or regular open-cell foams which are widely used in engineering and tissue engineering: we therefore believe that the homogenized models introduced in the present paper can have a wider range of application.…”

Section: Introductionmentioning

confidence: 98%

On existence and uniqueness of weak solutions for linear pantographic beam lattices models

Eremeyev

Alzahrani

Cazzani

et al. 2019

Continuum Mech. Thermodyn.

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In this paper, we discuss well-posedness of the boundary-value problems arising in some “gradient-incomplete” strain-gradient elasticity models, which appear in the study of homogenized models for a large class of metamaterials whose microstructures can be regarded as beam lattices constrained with internal pivots. We use the attribute “gradient-incomplete” strain-gradient elasticity for a model in which the considered strain energy density depends on displacements and only on some specific partial derivatives among those constituting displacements first and second gradients. So, unlike to the models of strain-gradient elasticity considered up-to-now, the strain energy density which we consider here is in a sense degenerated, since it does not contain the full set of second derivatives of the displacement field. Such mathematical problem was motivated by a recently introduced new class of metamaterials (whose microstructure is constituted by the so-called pantographic beam lattices) and by woven fabrics. Indeed, as from the physical point of view such materials are strongly anisotropic, it is not surprising that the mathematical models to be introduced must reflect such property also by considering an expression for deformation energy involving only some among the higher partial derivatives of displacement fields. As a consequence, the differential operators considered here, in the framework of introduced models, are neither elliptic nor strong elliptic as, in general, they belong to the class so-called hypoelliptic operators. Following (Eremeyev et al. in J Elast 132:175–196, 2018. 10.1007/s10659-017-9660-3) we present well-posedness results in the case of the boundary-value problems for small (linearized) spatial deformations of pantographic sheets, i.e., 2D continua, when deforming in 3D space. In order to prove the existence and uniqueness of weak solutions, we introduce a class of subsets of anisotropic Sobolev’s space defined as the energy space E relative to specifically assigned boundary conditions. As introduced by Sergey M. Nikolskii, an anisotropic Sobolev space consists of functions having different differential properties in different coordinate directions.

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“…Moreover, they are becoming fashionable as an elementary constituent of microstructured objects manufactured with computer-aided techniques (Atai and Steigmann, 1997;dell'Isola et al, 2016b;Ravari and Kadkhodaei, 2015;Ravari et al, 2014;dell'Isola et al, 2016a;Turco et al, 2016Turco et al, , 2017Milton et al, 2017;Spagnuolo et al, 2017). These objects are potentially advantageous for their mechanical characteristics, as shown in recent literature (Boutin et al, 2017;Giorgio et al, 2017;Scerrato et al, 2016;Eremeyev, 2017;Golaszewski et al, 2019;Turco et al, 2019;Franciosi et al, 2019) and their theoretical study requires tools from homogenization theory (Boubaker et al, 2007;Ravari et al, 2016;Dos Reis and Ganghoffer, 2012;Reda et al, 2016), discrete mechanics (Turco, 2018;Jawed et al, 2018;Andreaus et al, 2018), theory of generalized continua (Misra et al, 2016;Placidi et al, 2017Placidi et al, , 2014Placidi et al, , 2016 as well as developments in nonlinear elasticity of beams. In this last regard, certainly a full understanding of the onset and the characteristics of multiple solutions for the static problem under distributed load would be an important step forward, and possibly more general beam models will also have to be considered (Diyaroglu et al, 2015;Challamel, 2013;Challamel et al, 2013) because of the exotic properties of microstructured continua (Misra et al, 2018;dell'Isola et al, 2018;Diyaroglu et al, 2017).…”

Section: Introductionmentioning

confidence: 99%

Higher Gradient Materials and Related Generalized Continua

Dell’Isola

Corte

Battista

et al. 2019

Advanced Structured Materials

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In this paper we present numerical solutions to a geometrically nonlinear version of the extensible Timoshenko beam model under distributed load. The particular cases in which: i) extensional stiffness is infinite (inextensible Timoshenko model), ii) shear stiffness is infinite (extensible Euler model) and iii) extensional and shear stiffnesses are infinite (inextensible Euler model) will be numerically explored. Parametric studies on the axial stiffness in both the Euler and Timoshenko cases will also be shown and discussed.

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Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions

Cited by 73 publications

References 52 publications

Enhanced Piola–Hencky discrete models for pantographic sheets with pivots without deformation energy: Numerics and experiments

Enhanced Piola–Hencky discrete models for pantographic sheets with pivots without deformation energy: Numerics and experiments

On existence and uniqueness of weak solutions for linear pantographic beam lattices models

Higher Gradient Materials and Related Generalized Continua

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