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

TLDR: P pantographic metamaterials undergo very large deformations while remaining in the elastic regime, are very tough in resisting to damage phenomena, and exhibit robust macroscopic mechanical behavior with respect to minor changes in their microstructure and micromechanical properties.

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 investigations. 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 and 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) in exploiting the modern 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.

Figures

Fig. 34 Force (N ) versus non-dimensional displacement for the shear test of a pantographic sheet up to the onset of fiber breakage. The black curve is the experimental data, and the red curve has been obtained via numerical simulation (color figure online)

Fig. 35 Dependence of the resistance to sliding Kint on δ

Fig. 31 Force versus prescribed displacement for a uniaxial bias extension test. a Sample before first beam breakage (i.e., breakdown onset); b upper-left corner beam breakage; c–f further fiber breakage

Full text

HAL Id: hal-01829943
https://hal.archives-ouvertes.fr/hal-01829943
Submitted on 22 Jul 2018
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Pantographic metamaterials: an example of
mathematically driven design and of its technological
challenges
Francesco Dell’Isola, Pierre Seppecher, Jean-Jacques Alibert, Tomasz
Lekszycki, Roman Grygoruk, Marek Pawlikowski, David Steigmann, Ivan
Giorgio, Ugo Andreaus, Emilio Turco, et al.
To cite this version:
Francesco Dell’Isola, Pierre Seppecher, Jean-Jacques Alibert, Tomasz Lekszycki, Roman Grygoruk, et
al.. Pantographic metamaterials: an example of mathematically driven design and of its technological
challenges. Continuum Mechanics and Thermodynamics, Springer Verlag, 2019, 31 (4), pp.851-884.
�10.1007/s00161-018-0689-8�. �hal-01829943�

Continuum Mech. Thermodyn.
https://doi.org/10.1007/s00161-018-0689-8
ORIGINAL ARTICLE
Francesco dell’Isola · Pierre Seppecher · Jean Jacques Alibert · Tomasz Lekszycki ·
Roman Grygoruk · Marek Pawl
ikowski · David Steigmann · Ivan Giorgio · Ugo Andreaus ·
Emilio Turco · Maciej Gołaszewski · Nicola Rizzi · Claude Boutin · Victor A. Eremeyev ·
Anil Misra · Luca Placidi · Emilio Barchiesi
· Leopoldo Greco · Massimo Cuomo ·
Antonio Cazzani · Alessandro Della Corte · Antonio Battista · Daria Scerrato ·
Inna Zurba Eremeeva · Yosra Rahali · Jean-François Ganghoffer · Wolfgang Müller ·
Gregor Ganzosch · Mario Spagnuolo · Aron Pfaff · Katarzyna Barcz · Klaus Hoschke ·
Jan Neggers · François Hild
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 and
F. dell’Isola · U. Andreaus · E. Barchiesi (
B
)
Dipartimento di Ingegneria Strutturale e Geotecnica, Università degli Studi di Roma “La Sapienza.”, Via Eudossiana 18,
00184 Rome, Italy
E-mail: barchiesiemilio@gmail.com
F. dell’Isola · P. Seppecher · T. Lekszycki · I. Giorgio · E. Turco · V. A . E r e meyev · A. Misra · L. Placidi · E. Barchiesi ·
L. Greco · M. Cuomo · A. D. Corte · A. Battista · D. Scerrato · I. Z. Eremeeva · J.-F. Ganghoffer
International Research Center M&MoCS, Università degli Studi dell’Aquila, Via Giovanni Gronchi 18 - Zona industriale di Pile,
67100 L’Aquila, Italy
F. dell’Isola · A. D. Corte
Dipartimento di Ingegneria Civile, Edile-Architettura e Ambientale, Università degli Studi dell’Aquila, Via Giovanni Gronchi
18 - Zona industriale di Pile, 67100 L’Aquila, Italy
F. dell’Isola · I. Giorgio · L. Placidi · E. Barchiesi · A. D. Corte · A. Battista · D. Scerrato · M. Spagnuolo
Research Institute for Mechanics, National Research Lobachevsky State University of Nizhni Novgorod, Nizhny Novgorod,
Russia
P. Seppecher · J. J. Alibert
Institut de Mathématiques de Toulon, Université de Toulon et du Var, Avenue de l’ Université, BP 132, 83957 La Garde Cedex,
France
T. Lekszycki · R. Grygoruk · M. Pawlikowski · M. Gołaszewski · K. Barcz
Institute of Mechanics and Printing, Warsaw University of Technology, 85 Narbutta Street, 02-524 Warsaw, Poland
T. Lekszycki
Department of Experimental Physiology and Pathophysiology, Medical University of Warsaw, 1b Banacha Street, 02-097
Warsaw, Poland

F. dell’Isola et al.
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) in exploiting the modern
D. Steigmann
Department of Mechanical Engineering, University of California at Berkeley, 6133 Etcheverry Hall, Mailstop 1740, Berkeley,
CA, USA
E. Turco
Dipartimento di Architettura, Design, Urbanistica, Università degli Studi di Sassari, Asilo Sella, Via Garibaldi 35 (I piano),
07041 Alghero, SS, Italy
N. Rizzi
Dipartimento di Architettura, Università degli studi Roma Tre, Via della Madonna dei Monti 40, 00184 Rome, Italy
C. Boutin
Ecole Nationale des Travaux Publics de l’Etat, LGCB CNRS 5513 - CeLyA, Université de Lyon, 69518 Vaulx-en-Velin Cedex,
France
V. A. Erem eyev
Faculty of Civil and Environmental Engineering, Gda´nsk University of Technology, ul. Gabriela Narutowicza 11/12,
80-233 Gda´nsk, Poland
V. A. Erem eyev
Mathematics, Mechanics and Computer Science Department, South Federal University, Milchakova, str., 8a, Rostov-on-Don,
Russia 344090
A. Misra
Civil, Environmental and Architectural Engineering Department, The University of Kansas, 1530 W. 15th Street, Lawrence,
KS 66045-7609, USA
L. Placidi
Engineering Faculty, International Telematic University Uninettuno, C.so Vittorio Emanuele II 39, 00186 Rome, Italy
L. Greco · M. Cuomo
Dipartimento di Ingegneria Civile ed Ambientale (sezione di Ingegneria Strutturale), Università di Catania, Edificio Polifun-
zionale, IV piano, Viale Andrea Doria, 6 I, 95125 Catania, Italy
A. Cazzani
Dipartimento di Ingegneria Civile, Ambientale e Architettura, Università degli studi di Cagliari, Via Marengo 2, Cagliari, Italy
A. Battista
Laboratoire des Sciences de l’Ingénieur pour l’ Environnement, Université de La Rochelle, 23 avenue Albert Einstein, BP 33060,
17031 La Rochelle, Italy
Y. Raha l i · J.-F. Ganghoffer
Laboratoire d’Energétique et de Mécanique Théorique et Appliquée, University of Lorraine, 2 Avenue de la Foret de Haye, BP
90161, 54505 Vandoeuvre-lés-Nancy cedex, France
W. Müller · G. Ganzosch
Faculty of Mechanics, Berlin University of Technology, Einsteinufer 5, 10587 Berlin, Germany
M. Spagnuolo
Laboratoire des Sciences des Procédés et des Matériaux, Université Paris 13, Campus de Villetaneuse 99 avenue Jean-baptiste
Clément, 93430 Villetaneuse, France
A. Pfaff · K. Hoschke
Fraunhofer Institute for High-Speed Dynamics, Ernst-Mach-Institut, Eckerstraße 4, 79104 Freiburg, Germany

Pantographic metamaterials
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.
Keywords Pantographic fabrics · Metamaterials · Scientific design · Higher gradient materials
Introduction
Like every other human activity, the design, manufacturing and testing of prototypes of novel materials having
a complex and purpose-tailored (micro)structure need the organized efforts of many specialists having a large
scope of competence. Therefore, the present work needed the collaboration of many scientists, each one with
his/her own specific competences. The order of the authors of this paper has been formed with a simple
criterion: it is related to the length of the time period that has seen their involvement in the described joint
research efforts and, therefore, does not express any evaluation of the importance of each contribution.
Let us note that, in this paper, no specific length scale is attached to the word “micro.” Specifically, with
its use it is meant that at one or at multiple smaller (with respect to the unique macroscale corresponding to
that at which phenomena are observed) length scales the material is made of complex microstructures: they
consist in the organization of the distribution of matter and its (possibly varying) physical properties.
The aim of this paper is to account, in a unique panoramic view, for the research efforts that we have started
(at least the first ones among us) since 2003 and that has produced, in our opinion, some interesting results.
The aim of the investigations was more specifically (1) to design novel and exotic architectured metamaterials
based on a mathematical understanding of the related mechanical problems and on suitably designed numerical
simulations, (2) to build the designed prototypes by using 3D printing technology, (3) to test with sensitive
apparatuses the so-builtprototypes, (4) toelaborate the obtaineddata withmodern image correlationtechniques,
(5) to produce a careful model fitting of the experimental data by means of the systematic use of numerical
simulations, and (6) to compare the proposed models with experimental evidence.
At the first stage of the research effort, as it often happens, the problem was approached from a theoretical
point of view. The mathematical models, which were initially introduced, belong to the class of generalized
continua: the introduced independent kinematic fields include not only the displacement field but, eventually,
also microstretch and/or microrotation fields. The particular class of second gradient continua was more
specifically considered: in these media, the strain energy depends on the displacement gradient and on its
second gradient. The reasons of their name are therefore clear: in second gradient continua the strain energy may
depend on the second gradient of displacement. Second gradient continua can be regarded as media endowed
with a tensorial microstructure in which a constraint is applied, namely it requires that the microstructure
tensor is equal to the placement gradient. The problem to be solved was: given a desired behavior, to find at
first the evolution equations modeling such a behavior and then to characterize the material (micro)structure
governed by the chosen equations.
In the second stage, it was necessary to develop numerical integration schemes and the corresponding
codes for solving, in physically relevant cases, the equations chosen to describe the desired behavior. Finally,
it was necessary to build the microstructures. This was possible b y means of the recent developments of rapid
prototyping technologies, which allow for the fabrication of those which, up to now, were simply mathematical
dreams.
In this paper, we show the results of our efforts in designing pantographic metamaterials, in the mechanical
testing of real prototypes, and evidence is provided on their exotic behavior. With the latest advancements (e.g.,
3D-printing technology and, more generally, of rapid prototyping techniques), the small-scale production of
materials with complex geometries has become more affordable than ever [1–4]. The exploitation of these
new technologies has made possible the development in the last few years of materials with very different
substructures.
J. Neggers · F. Hild
Laboratoire de Mécanique et Technologie (LMT), ENS Paris-Saclay/CNRS/Université Paris-Saclay, 61 avenue du Président
Wilson, 94235 Cachan Cedex, France

F. dell’Isola et al.
Fig. 1 Example of pantographic structure [13]
One of the research goals whose achievement has been sped up by rapid prototyping is to determine and
study new microstructures that, at a well-specified macroscopic scale, exhibit a behavior that can be described
by nonstandard mathematical models like generalized continuum theories. Many of these theories, that today
are being called “generalized” (as opposed to “classical” theories), were formulated before or together with
so-called “classical” theories and then lost [5,6]. It is possible to state that some of these theories were
already known at least two centuries ago [7,8]. Pantographic structures (Fig. 1) have been proposed as a
metamaterial [9], which is well described by second gradient continuum theories [10–12].
1 Modeling and experiments in elastic regime
The theoretical interest in pantographic structures derives from the fact that, in order to describe their exotic
phenomenology, one has to utilize higher gradient continuum theories [14,15] or micromophic theories [16,17]
with the related problem of homogenization [18] and of different strategies for numerical integration [19,20].
1.1 Homogenization of periodic truss modular structures
Throughout the history of mechanics, several multiscale procedures have been developed in order to relate
macromodels with micromodels, the first attempts tracing back to Maxwell and Saint-Venant [21]. An approach
that has proven to be effective is based on the postulate of a macroscopic and a microscopic model and of
a kinematic correspondence between the deformations defined within the two models. Successively, it is
postulated that the power expended in corresponding motions coincides. In this way, it is possible to obtain
the coefficients of the constitutive equations of the macromodel in terms of properties of the building blocks
constituting the microscopic model. The macromodel is not the result of the homogenization process but is,
instead, assumed a priori. Formal asymptotic expansion can help to encompass this difficulty, and a microscopic
model made up of linear Euler beams leads to a simple macroscopic second gradient model of a 1D planar
beam [11].
The structure that is considered at the microlevel is the so-called pantographic structure (Fig. 2). It is
assumed that the considered pantographic microstructure is made up of a very large number of small modules
and the limit behaviorwhen such a number tends to infinity,i.e., the homogenized macromodel, is studied. Using
Gamma-convergence technique, it is proven that the homogenized model is the postulated second gradient
model [11]. Successively, a modified (Warren-type) pantographic structure is proposed as micromodel in order
to get for the first time a third gradient planar beam model (Fig. 3), whose general properties were already
studied by Mindlin and Tiersten [22], and Dillon and Perzyna [23]. Such structures possess other floppy modes
(i.e., placements for which the strain energy vanishes) than, trivially, rigid motions. The pantographic beam
does not store any energy when undergoing uniform extension, while the Warren-type pantographic beam does
not store any energy when undergoing uniform flexure.
In Ref. [24], formal asymptotic expansion procedures, already employed [11,13], are systematically con-
sidered in the framework of linear elasticity in order to determine the effective behavior of periodic structures

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Pantographic metamaterials: an example of mathematically driven design and of its technological challenges" ?

In this paper, the authors 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. The problem to be solved was stated as follows: determine the material ( micro ) structure governed by those equations that specify a desired behavior. The authors show here a panorama of the results of their efforts ( 1 ) in designing pantographic metamaterials, ( 2 ) in exploiting the modern D. Steigmann Department of Mechanical Engineering, University of California at Berkeley, 6133 Etcheverry Hall, Mailstop 1740, Berkeley, CA, USA E. Turco Dipartimento di Architettura, Design, Urbanistica, Università degli Studi di Sassari, Asilo Sella, Via Garibaldi 35 ( I piano ), 07041 Alghero, SS, Italy N. Rizzi Dipartimento di Architettura, Università degli studi Roma Tre, Via della Madonna dei Monti 40, 00184 Rome, Italy C. Boutin Ecole Nationale des Travaux Publics de l ’ Etat, LGCB CNRS 5513 CeLyA, Université de Lyon, 69518 Vaulx-en-Velin Cedex, France V. A. Eremeyev Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, ul. 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. 

Q2. What is the definition of anisotropic sobolev space?

It is observed that the energy space of linear pantographic sheets, i.e., the space of functions fulfilling boundary conditions for which the strain energy is meaningful, is included in a special class of Sobolev spaces, the so-called Anisotropic Sobolev Space. 

Q3. What was the morphological operation used to construct the mesh?

Thanks to the uniform background, simple morphological operations were performed in order to construct this mesoscale mesh from a mask. 

Q4. What is the coordinate system of the euclidean space?

A Cartesian coordinate system (O, (ê1, ê2)) is introduced, with X = (X1, X2) the coordinates of the generic point in the Euclidean space R2. 

Q5. What is the criterion for rupture of a spring?

In particular, the criterion for rupture of a spring at iteration t , which discriminates whether that spring has to be removed from the computations at iteration t + 1 or not, is based on (constant) thresholds for the relative elongation of extensional springs, e.g., (‖pi+1, j − pi, j‖ − ) (upper and lower thresholds are employed for this deformation measure). 

Q6. What is the definition of the push-forward vectors in the current configuration of the vectors?

As customary, D1 and D2 are defined as the push-forward vectors in the current configuration of the vectors D1 and D2, respectively, i.e., Dα = FDα, α = 1, 2. 

Q7. How often is the strain a standard relation between the width and height of a fabric specimen?

Very often, it is assumed that N = 3M , which is the standard relation between the width and height of a fabric specimen for experimental and numerical tests. 

Q8. What is the buckling of the fibers?

Many fiber reference curvatures have been considered (e.g., sinusoidal, spiral, parabolic fibers), and for parabolic fibers, experiments (Fig. 22) and model (Fig. 23) both show that, after a critical load, out-of-plane buckling occurs during bias extension, because the transverse (curved) beams in the middle of the specimen undergo buckling induced by the shortening of the middle width of the specimen. 

Q9. What is the strain energy density of a 2D continuummodel?

A 2D continuummodel embedded in a 3D space has been also proposed [48] where, relying on a variational framework, the following strain energy density is proposedπ 

Q10. What is the simplest way to address the homogenization of pantographic fabrics?

Reference [18] has first addressed the homogenization à la Piola of pantographic fabrics in a linear setting, proving that the homogenization of pantographic fabrics gives rise to second gradient continua. 

Q11. What is the simplest explanation of the hencky-type micromodel?

1.4 À la Piola homogenized elastic plate modelConsidering the discrete Hencky-type micromodel presented above, a 2D continuum macromodel has been derived by means of micro–macro transitions. 

Q12. What is the space of admissible placements for the Pipkin continuum under study?

the space of admissible placements for the Pipkin continuum under study is uniquely determined by the continuous piecewise twice continuously differentiable fields μ1(ξ1) and μ2(ξ2).