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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
  • 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
Original Article
  • 40 Downloads

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.

Keywords

Pantographic fabrics Metamaterials Scientific design Higher gradient materials 

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Notes

Acknowledgements

This work was supported by a grant from the Government of the Russian Federation (contract No. 14.Y26.31.0031).

References

  1. 1.
    dell’Isola, F., Steigmann, D., Della Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev. 67(6), 060804 (2015)CrossRefGoogle Scholar
  2. 2.
    Milton, G., Briane, M., Harutyunyan, D.: On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials. Math. Mech. Complex Syst. 5(1), 41–94 (2017)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Eremeyev, V.A., Pietraszkiewicz, W.: Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Math. Mech. Solids 21(2), 210–221 (2016)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bertram, A., Glüge, R.: Gradient materials with internal constraints. Math. Mech. Complex Syst. 4(1), 1–15 (2016)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Russo, L.: The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn. Springer, Berlin (2013)Google Scholar
  6. 6.
    Stigler, S.M.: Stigler’s law of eponymy. Trans. N. Y. Acad. Sci. 39(1 Series II), 147–157 (1980)CrossRefGoogle Scholar
  7. 7.
    dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math. Mech. Solids 20(8), 887–928 (2015)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    dell’Isola, F., Corte, A.D., Giorgio, I.: Higher-gradient continua: the legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math. Mech. Solids 22(4), 852–872 (2017)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Del Vescovo, D., Giorgio, I.: Dynamic problems for metamaterials: review of existing models and ideas for further research. Int. J. Eng. Sci. 80, 153–172 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    dell’Isola, F., Lekszycki, T., Pawlikowski, M., Grygoruk, R., Greco, L.: Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Z. angew. Math. Phys. 66, 3473–3498 (2015)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Alibert, J.-J., Seppecher, P., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Pideri, C., Seppecher, P.: A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Contin. Mech. Thermodyn. 9(5), 241–257 (1997)ADSMathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc. R. Soc. A 472(2185), 23 (2016)Google Scholar
  14. 14.
    dell Isola, F., Seppecher, P., Della Corte, A.: The postulations á la d alembert and á la cauchy for higher gradient continuum theories are equivalent: a review of existing results. In: Proceedings of the Royal Society A, Volume 471, p. 20150415. The Royal Society (2015)Google Scholar
  15. 15.
    Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., Rosi, G.: Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids. Math. Mech. Solids 20(4), 375–417 (2015)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Altenbach, H., Eremeyev, V.: On the linear theory of micropolar plates. ZAMM 89(4), 242–256 (2009)ADSMathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Pietraszkiewicz, W., Eremeyev, V.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46(3), 774–787 (2009)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Rahali, Y., Giorgio, I., Ganghoffer, J.F., dell’Isola, F.: Homogenization à la piola produces second gradient continuum models for linear pantographic lattices. Int. J. Eng. Sci. 97, 148–172 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bilotta, A., Formica, G., Turco, E.: Performance of a high-continuity finite element in three-dimensional elasticity. Int. J. Numer. Methods Biomed. Eng. 26(9), 1155–1175 (2010)MATHCrossRefGoogle Scholar
  20. 20.
    Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches. Contin. Mech. Thermodyn. 28(1–2), 139–156 (2016)ADSMathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    de Saint-Venant, M.: Mémoire sur la torsion des prismes: avec des considérations sur leur flexion ainsi que sur l’équilibre intérieur des solides élastiques en général: et des formules pratiques pour le calcul de leur résistance à divers efforts s’ exerçant simultanément. Imprimerie nationale (1856)Google Scholar
  22. 22.
    Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(1), 415–448 (1962)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Dillon, O.W., Perzyna, P.: Gradient theory of materials with memory and internal changes. Arch. Mech. 24(5–6), 727–747 (1972)MathSciNetMATHGoogle Scholar
  24. 24.
    Abdoul-Anziz, H., Seppecher, P.: Strain gradient and generalized continua obtained by homogenizing frame lattices (2017) . \(<\)hal-01672898\(>\) Google Scholar
  25. 25.
    Turco, E., Giorgio, I., Misra, A., dell’Isola, F.: King post truss as a motif for internal structure of (meta) material with controlled elastic properties. Open Sci. 4(10), 171153 (2017)Google Scholar
  26. 26.
    Everstine, G.C., Pipkin, A.C.: Boundary layers in fiber-reinforced materials. J. Appl. Mech. 40, 518–522 (1973)ADSMATHCrossRefGoogle Scholar
  27. 27.
    Hilgers, M.G., Pipkin, A.C.: Elastic sheets with bending stiffness. Q. J. Mech. Appl. Math. 45, 57–75 (1992)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Hilgers, M.G., Pipkin, A.C.: Energy-minimizing deformations of elastic sheets with bending stiffness. J. Elast. 31, 125–139 (1993)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Hilgers, M.G., Pipkin, A.C.: Bending energy of highly elastic membranes ii. Q. Appl. Math. 54, 307–316 (1996)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Hu, M.Z., Kolsky, H., Pipkin, A.C.: Bending theory for fiber-reinforced beams. J. Compos. Mater. 19, 235–249 (1985)ADSCrossRefGoogle Scholar
  31. 31.
    Pipkin, A.C.: Generalized plane deformations of ideal fiber-reinforced materials. Q. Appl. Math. 32, 253–263 (1974)MATHCrossRefGoogle Scholar
  32. 32.
    Pipkin, A.C.: Energy changes in ideal fiber-reinforced composites. Q. Appl. Math. 35, 455–463 (1978)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Pipkin, A.C.: Some developments in the theory of inextensible networks. Q. Appl. Math. 38, 343–355 (1980)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    dell’Isola, F., d’Agostino, M.V., Madeo, A., Boisse, P., Steigmann, D.: Minimization of shear energy in two dimensional continua with two orthogonal families of inextensible fibers: the case of standard bias extension test. J. Elast. 122(2), 131–155 (2016)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Placidi, L., Greco, L., Bucci, S., Turco, E., Rizzi, N.L.: A second gradient formulation for a 2d fabric sheet with inextensible fibres. Z. angew. Math. Phys. 67(5), 114 (2016)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Rivlin, R.S.: Plane strain of a net formed by inextensible cords. In: Collected Papers of RS Rivlin, pp. 511–534. Springer (1997)Google Scholar
  37. 37.
    Greco, L., Giorgio, I., Battista, A.: In plane shear and bending for first gradient inextesible pantographic sheets: numerical study of deformed shapes and global constraint reactions. Math. Mech. Solids, p. 1081286516651324 (2016)Google Scholar
  38. 38.
    Cuomo, M., Dell’Isola, F., Greco, L., Rizzi, N.L.: First versus second gradient energies for planar sheets with two families of inextensible fibres: investigation on deformation boundary layers, discontinuities and geometrical instabilities. Compos. B Eng. 115, 423–448 (2017)CrossRefGoogle Scholar
  39. 39.
    Cuomo, M., dell’Isola, F., Greco, L.: Simplified analysis of a generalized bias test for fabrics with two families of inextensible fibres. Z. angew. Math. Phys. 67(3), 1–23 (2016)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    dell’Isola, F., Cuomo, M., Greco, L., Della Corte, A.: Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies. J. Eng. Math. 103(1), 127–157 (2017)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Greco, L., Giorgio, I., Battista, A.: In plane shear and bending for first gradient inextensible pantographic sheets: numerical study of deformed shapes and global constraint reactions. Math. Mech. Solids 22(10), 1950–1975 (2017)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Giorgio, I.: Numerical identification procedure between a micro-cauchy model and a macro-second gradient model for planar pantographic structures. Z. angew. Math. Phys. 67(4), 95 (2016)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z. angew. Math. Phys. 67, 28 (2016)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Eremeyev, V.A., dell’Isola, F., Boutin, C., Steigmann, D.: Linear pantographic sheets: existence and uniqueness of weak solutions (2017).  https://doi.org/10.1007/s10659-017-9660-3
  45. 45.
    Placidi, L., Barchiesi, E., Turco, E., Rizzi, N.L.: A review on 2D models for the description of pantographic fabrics. Z. angew. Math. Phys. 67(5), 121 (2016)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    dell’Isola, F., Steigmann, D.J.: A two-dimensional gradient-elasticity theory for woven fabrics. J. Elast. 18, 113–125 (2015)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Giorgio, I., Grygoruk, R., dell’Isola, F., Steigmann, D.J.: Pattern formation in the three-dimensional deformations of fibered sheets. Mech. Res. Commun. 69, 164–171 (2015)CrossRefGoogle Scholar
  48. 48.
    Giorgio, I., Rizzi, N.L., Turco, E.: Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc. R. Soc. A, p. 21 (2017).  https://doi.org/10.1098/rspa.2017.0636
  49. 49.
    Auffray, N., Dirrenberger, J., Rosi, G.: A complete description of bi-dimensional anisotropic strain-gradient elasticity. Int. J. Solids Struct. 69, 195–206 (2015)CrossRefGoogle Scholar
  50. 50.
    Boutin, C., dell’Isola, F., Giorgio, I., Placidi, L.: Linear pantographic sheets: asymptotic micro–macro models identification. Math. Mech. Complex Syst. 5(2), 127–162 (2017)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Placidi, L., Andreaus, U., Corte, A.D., Lekszycki, T.: Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Z. angew. Math. Phys. 66(6), 3699–3725 (2015)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Placidi, L., Barchiesi, E., Battista, A.: An inverse method to get further analytical solutions for a class of metamaterials aimed to validate numerical integrations. In: Mathematical Modelling in Solid Mechanics, pp. 193–210. Springer (2017)Google Scholar
  53. 53.
    Scerrato, D., Zhurba Eremeeva, I.A., Lekszycki, T., Rizzi, N.L., Rizzi, N.L.: On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets. ZAMM 96(11), 1268–1279 (2016)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    dell’Isola, F., Giorgio, I., Andreaus, U.: Elastic pantographic 2d lattices: a numerical analysis on static response and wave propagation. Proc. Est. Acad. Sci. 64, 219–225 (2015)CrossRefGoogle Scholar
  55. 55.
    dell’Isola, F., Della Corte, A., Giorgio, I., Scerrato, D.: Pantographic 2D sheets. Int. J. Non Linear Mech. 80, 200–208 (2016)ADSCrossRefGoogle Scholar
  56. 56.
    Madeo, A., Della Corte, A., Greco, L., Neff, P.: Wave propagation in pantographic 2d lattices with internal discontinuities. arXiv preprint arXiv:1412.3926 (2014)
  57. 57.
    Turco, E., dell’Isola, F., Rizzi, N.L., Grygoruk, R., Müller, W.H., Liebold, C.: Fiber rupture in sheared planar pantographic sheets: numerical and experimental evidence. Mech. Res. Commun. 76, 86–90 (2016)CrossRefGoogle Scholar
  58. 58.
    Spagnuolo, M., Barcz, K., Pfaff, A., Dell’Isola, F., Franciosi, P.: Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech. Res. Commun. 83, 47–52 (2017)CrossRefGoogle Scholar
  59. 59.
    Ganzosch, G., dell’Isola, F., Turco, E., Lekszycki, T., Müller, W.H.: Shearing tests applied to pantographic structures. Acta Polytech. CTU Proc. 7, 1–6 (2016)CrossRefGoogle Scholar
  60. 60.
    Turco, E., Golaszewski, M., Giorgio, I., D’Annibale, F.: Pantographic lattices with non-orthogonal fibres: experiments and their numerical simulations. Compos. B Eng. 118, 1–14 (2017)CrossRefGoogle Scholar
  61. 61.
    Sutton, M.A., Orteu, J.J., Schreier, H.: Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications. Springer, Berlin (2009)Google Scholar
  62. 62.
    Hild, F., Roux, S.: Digital Image Correlation, pp. 183–228. Wiley-VCH, Weinheim (2012)Google Scholar
  63. 63.
    Turco, E., Misra, A., Pawlikowski, M., dell’Isola, F., Hild, F.: Enhanced piola-hencky discrete models for pantographic sheets with pivots without deformation energy: numerics and experiments (submitted for publication) (2018)Google Scholar
  64. 64.
    Tomičevć, Z., Hild, F., Roux, S.: Mechanics-aided digital image correlation. J. Strain Anal. Eng. Des. 48(5), 330–343 (2013)CrossRefGoogle Scholar
  65. 65.
    Hild, F., Roux, S., Gras, R., Guerrero, N., Marante, M.E., Flórez-López, J.: Displacement measurement technique for beam kinematics. Opt. Lasers Eng. 47(3), 495–503 (2009)CrossRefGoogle Scholar
  66. 66.
    Leclerc, H., Périé, J.-N., Roux, S., Hild, F.: Integrated digital image correlation for the identification of mechanical properties. In: Gagalowicz, A., Philips, W. (eds.) International Conference on Computer Vision/Computer Graphics Collaboration Techniques and Applications, Volume LNCS 5496, pp. 161–171. Springer, Berlin (2009)Google Scholar
  67. 67.
    Lindner, D., Mathieu, F., Hild, F., Allix, O., Ha Minh, C., Paulien-Camy, O.: On the evaluation of stress triaxiality fields in a notched titanium alloy sample via integrated DIC. J. Appl. Mech. 82(7), 071014 (2015)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Francesco dell’Isola
    • 1
    • 2
    • 3
    • 4
  • Pierre Seppecher
    • 2
    • 5
  • Jean Jacques Alibert
    • 5
  • Tomasz Lekszycki
    • 2
    • 6
    • 7
  • Roman Grygoruk
    • 6
  • Marek Pawlikowski
    • 6
  • David Steigmann
    • 8
  • Ivan Giorgio
    • 2
    • 4
  • Ugo Andreaus
    • 1
  • Emilio Turco
    • 2
    • 9
  • Maciej Gołaszewski
    • 6
  • Nicola Rizzi
    • 10
  • Claude Boutin
    • 11
  • Victor A. Eremeyev
    • 2
    • 12
    • 13
  • Anil Misra
    • 2
    • 14
  • Luca Placidi
    • 2
    • 4
    • 15
  • Emilio Barchiesi
    • 1
    • 2
    • 4
  • Leopoldo Greco
    • 2
    • 16
  • Massimo Cuomo
    • 2
    • 16
  • Antonio Cazzani
    • 17
  • Alessandro Della Corte
    • 2
    • 3
    • 4
  • Antonio Battista
    • 2
    • 4
    • 18
  • Daria Scerrato
    • 2
    • 4
  • Inna Zurba Eremeeva
    • 2
  • Yosra Rahali
    • 19
  • Jean-François Ganghoffer
    • 2
    • 19
  • Wolfgang Müller
    • 20
  • Gregor Ganzosch
    • 20
  • Mario Spagnuolo
    • 4
    • 21
  • Aron Pfaff
    • 22
  • Katarzyna Barcz
    • 6
  • Klaus Hoschke
    • 22
  • Jan Neggers
    • 23
  • François Hild
    • 23
  1. 1.Dipartimento di Ingegneria Strutturale e GeotecnicaUniversità degli Studi di Roma “La Sapienza.”RomeItaly
  2. 2.International Research Center M&MoCSUniversità degli Studi dell’AquilaL’AquilaItaly
  3. 3.Dipartimento di Ingegneria Civile, Edile-Architettura e AmbientaleUniversità degli Studi dell’AquilaL’AquilaItaly
  4. 4.Research Institute for MechanicsNational Research Lobachevsky State University of Nizhni NovgorodNizhny NovgorodRussia
  5. 5.Institut de Mathématiques de ToulonUniversité de Toulon et du VarLa Garde CedexFrance
  6. 6.Institute of Mechanics and PrintingWarsaw University of TechnologyWarsawPoland
  7. 7.Department of Experimental Physiology and PathophysiologyMedical University of WarsawWarsawPoland
  8. 8.Department of Mechanical EngineeringUniversity of California at BerkeleyBerkeleyUSA
  9. 9.Dipartimento di Architettura, Design, UrbanisticaUniversità degli Studi di SassariAlgheroItaly
  10. 10.Dipartimento di ArchitetturaUniversità degli studi Roma TreRomeItaly
  11. 11.Ecole Nationale des Travaux Publics de l’Etat, LGCB CNRS 5513 - CeLyAUniversité de LyonVaulx-en-Velin CedexFrance
  12. 12.Faculty of Civil and Environmental EngineeringGdańsk University of TechnologyGdańskPoland
  13. 13.Mathematics, Mechanics and Computer Science DepartmentSouth Federal UniversityRostov-on-DonRussia
  14. 14.Civil, Environmental and Architectural Engineering DepartmentThe University of KansasLawrenceUSA
  15. 15.Engineering FacultyInternational Telematic University UninettunoRomeItaly
  16. 16.Dipartimento di Ingegneria Civile ed Ambientale (sezione di Ingegneria Strutturale)Università di CataniaCataniaItaly
  17. 17.Dipartimento di Ingegneria Civile, Ambientale e ArchitetturaUniversità degli studi di CagliariCagliariItaly
  18. 18.Laboratoire des Sciences de l’Ingénieur pour l’ EnvironnementUniversité de La RochelleLa RochelleItaly
  19. 19.Laboratoire d’Energétique et de Mécanique Théorique et AppliquéeUniversity of LorraineVandoeuvre-lés-Nancy cedexFrance
  20. 20.Faculty of MechanicsBerlin University of TechnologyBerlinGermany
  21. 21.Laboratoire des Sciences des Procédés et des MatériauxUniversité Paris 13VilletaneuseFrance
  22. 22.Fraunhofer Institute for High-Speed DynamicsErnst-Mach-InstitutFreiburgGermany
  23. 23.Laboratoire de Mécanique et Technologie (LMT)ENS Paris-Saclay/CNRS/Université Paris-SaclayCachan CedexFrance

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