Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients

Placidi, Luca; Andreaus, Ugo; Corte, Alessandro Della; Lekszycki, Tomasz

doi:10.1007/s00033-015-0588-9

Cited by 132 publications

(78 citation statements)

References 66 publications

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“…[50][51][52], compatible identifications of the constitutive parameters appearing in Eq. (31) have been carried out, thus completely characterizing the set of constitutive parameters in terms of the fiber base material parameters (i.e., Young's modulus), of the fiber cross section parameters (i.e., area and moment of inertia), and of the distance between the nearest pivots.…”

Section: Analytical Identification Of Elastic Plate Modelsmentioning

confidence: 99%

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

Dell’Isola

Seppecher

Alibert

et al. 2018

Continuum Mech. Thermodyn.

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295

141

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Pantographic metamaterials: an example of mathematically driven design and of its technological challenges Abstract In this paper, we account for the research efforts that have been started, for some among us, already since 2003, and aimed to the design of a class of exotic architectured, optimized (meta) materials. At the first stage of these efforts, as it often happens, the research was based on the results of mathematical investiga-tions. The problem to be solved was stated as follows: determine the material (micro)structure governed by those equations that specify a desired behavior. Addressing this problem has led to the synthesis of second gradient materials. In the second stage, it has been necessary to develop numerical integration schemes andDipartimento di Ingegneria Strutturale e Geotecnica, Università degli Studi di Roma "La Sapienza.", Via Eudossiana 18, 00184 Rome, Italy E-mail: barchiesiemilio@gmail.com the corresponding codes for solving, in physically relevant cases, the chosen equations. Finally, it has been necessary to physically construct the theoretically synthesized microstructures. This has been possible by means of the recent developments in rapid prototyping technologies, which allow for the fabrication of some complex (micro)structures considered, up to now, to be simply some mathematical dreams. We show here a panorama of the results of our efforts (1) in designing pantographic metamaterials, (2) technology of rapid prototyping, and (3) in the mechanical testing of many real prototypes. Among the key findings that have been obtained, there are the following ones: pantographic metamaterials (1) undergo very large deformations while remaining in the elastic regime, (2) are very tough in resisting to damage phenomena, (3) exhibit robust macroscopic mechanical behavior with respect to minor changes in their microstructure and micromechanical properties, (4) have superior strength to weight ratio, (5) have predictable damage behavior, and (6) possess physical properties that are critically dictated by their geometry at the microlevel.

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Section: Analytical Identification Of Elastic Plate Modelsmentioning

confidence: 99%

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

Dell’Isola

Seppecher

Alibert

et al. 2018

Continuum Mech. Thermodyn.

Self Cite

295

141

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“…In [50,51] the integration by part procedure valid for boundaries with corner points is considered. The form of δU requires that only a class of external loads can be applied: indeed the virtual work of external loads must be consistent with it.…”

Section: Energy For Pantographic Sheets and Equilibrium Conditionsmentioning

confidence: 99%

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

Eremeyev

Dell’Isola

Boutin

et al. 2017

J Elast

125

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The well-posedness of the boundary value problems for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see, e.g., Chambon and Moullet in Comput. Methods Appl. Mech. Eng. 193:2771-2796 and Mareno and Healey in SIAM J. Math. Anal. 38:103-115, 2006.The study of the equilibrium of planar pantographic lattices has been approached in two different ways: in dell 'Isola et al. (Proc. R. Soc. Lond. Ser. A 472:20150, 2016) a discrete model was introduced involving extensional and rotational springs which is also valid in large deformations regimes while in Boutin et al. (Math. Mech. Complex Syst. 5:127-162, 2017) the lattice has been modelled as a set of beam elements interconnected by internal pivots, but the analysis was restricted to the linear case. In both papers a homogenized second gradient deformation energy, quadratic in the neighbourhood of non deformed configuration, is obtained via perturbative methods and the predictions obtained with the obtained continuum model are successfully compared with experiments. This energy is not strongly elliptic in its dependence on second gradients. We consider in this paper also the important particular case of pantographic lattices whose first gradient energy does not depend on shear deformation: this could be considered either a pathological case or an important exceptional case (see Stillwell et al. in Am. Math. Mon. 105:850-858, 1998 and Turro in Angew. Chem., Int. Ed. Engl. 39:2255Engl. 39: -2259Engl. 39: , 2000. In both cases we believe that such a particular case deserves some attention because of what we can understand by studying it (see Dyson in Science 200:677-678, 1978). This circumstance motivates the present paper, where we address the well-posedness of the planar linearized equilibrium problem for homogenized pantographic lattices. To do so: (i) we introduce a class of subsets of anisotropic Sobolev's space as the most suitable energy space E relative to assigned boundary conditions; (ii) we prove that the considered strain energy density is coercive and positive definite in E; (iii) we prove that the set of placements for which the strain energy is vanishing (the so-called floppy modes) must strictly include rigid motions; (iv) we determine the restrictions on displacement boundary conditions which assure existence and uniqueness of linear static problems. The presented results represent one of the first mechanical applications of the concept of Anisotropic Sobolev space, initially introduced only on the basis of purely abstract mathematical considerations.

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“…From the point of view of a homogenization approach [37][38][39][40][41][42], a procedure for a correct identification of the constitutive parameters is needed. In this regard, the authors in [43] present a strategy for a 2D second gradient material that can also be fruitfully adopted for a pantographic-like structure. Also, identification techniques based on the evaluation of dynamic properties can be properly employed for this purpose (see, e.g., [44]).…”

Section: Introductionmentioning

confidence: 99%

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

Greco

Giorgio

Battista

2016

Mathematics and Mechanics of Solids

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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.

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Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients

Cited by 132 publications

References 66 publications

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

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

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

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

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