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A Multi-disciplinary Approach for Mechanical Metamaterial Synthesis: A Hierarchical Modular Multiscale Cellular Structure Paradigm

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State of the Art and Future Trends in Material Modeling

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 100))

Abstract

Recent advanced manufacturing techniques such as 3D printing have prompted the need for designing new multiscale architectured materials for various industrial applications. These multiscale architectures are designed to obtain the desired macroscale behavior by activating interactions between different length scales and coupling different physical mechanisms. Although promising results have been recently obtained, the design of such systems still represents a challenge in terms of mathematical modeling, experimentation, and manufacturing. In this paper, some research perspectives are discussed aiming to determine the most efficient methodology needed to design novel metamaterials. A multidisciplinary approach based on Digital Image Correlation (DIC) techniques may be very effective. The main feature of the described DIC-based approach consists of the integration of different methodologies to create a synergistic relationship among the different steps from design to fabrication and validation. Experimental techniques and modeling approaches are envisioned to be combined in feedback loops whose objective is to determine the required multiscale architectures of newly designed metamaterials. Moreover, it is necessary to develop appropriate mathematical models to estimate the behavior of such metamaterials. Within this new design approach, the manufacturing process can be effectively guided by a precise theoretical and experimental framework. In order to show the applicability of the proposed approach, some preliminary results are provided for a particular type of mechanical metamaterial, namely, pantographic metamaterials. Lastly, the most relevant challenges are highlighted among those that must be addressed for future applications.

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References

  • Abdoul-Anziz H, Seppecher P (2018) Strain gradient and generalized continua obtained by homogenizing frame lattices. Mathematics and mechanics of complex systems 6(3):213–250

    Google Scholar 

  • Alibert JJ, Seppecher P, dell’Isola F (2003) Truss modular beams with deformation energy depending on higher displacement gradients. Mathematics and Mechanics of Solids 8(1):51–73

    Google Scholar 

  • Altenbach H, Eremeyev VA, Naumenko K (2015) On the use of the first order shear deformation plate theory for the analysis of three-layer plates with thin soft core layer. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 95(10):1004–1011

    Google Scholar 

  • Ambati M, Fang N, Sun C, Zhang X (2007) Surface resonant states and superlensing in acoustic metamaterials. Physical Review B 75(19):195,447

    Google Scholar 

  • Andreau O, Koutiri I, Peyre P, Penot JD, Saintier N, Pessard E, De Terris T, Dupuy C, Baudin T (2019) Texture control of 316l parts by modulation of the melt pool morphology in selective laser melting. Journal of Materials Processing Technology 264:21–31

    CAS  Google Scholar 

  • Andrews E, Sanders W, Gibson LJ (1999) Compressive and tensile behaviour of aluminum foams. Materials Science and Engineering: A 270(2):113–124

    Google Scholar 

  • Andreykiv A, Prendergast P, Van Keulen F, Swieszkowski W, Rozing P (2005) Bone ingrowth simulation for a concept glenoid component design. Journal of biomechanics 38(5):1023–1033

    CAS  Google Scholar 

  • Ao X, Chan C (2008) Far-field image magnification for acoustic waves using anisotropic acoustic metamaterials. Physical Review E 77(2):025,601

    Google Scholar 

  • Auffray N, dell’Isola F, Eremeyev V, Madeo A, Placidi L, Rosi G (2014) Least action principle for second gradient continua and capillary fluids: A lagrangian approach following piola’s point of view. In: dell’Isola F, Maier G, Perego U, Andreaus U, Esposito R, Forest S (eds) The Complete Works of Gabrio Piola: Volume I, Springer, Cham, Advanced Structured Materials, vol 38, pp 606–694

    Google Scholar 

  • Barchiesi E, Placidi L (2017) A review on models for the 3d statics and 2d dynamics of pantographic fabrics. In: Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials, Springer, pp 239–258

    Google Scholar 

  • Barchiesi E, Spagnuolo M, Placidi L (2018) Mechanical metamaterials: a state of the art. Mathematics and Mechanics of Solids 24(1):212–234, DOI:https://doi.org/10.1177/1081286517735695

    Google Scholar 

  • Barchiesi E, Spagnuolo M, Placidi L (2019) Mechanical metamaterials: a state of the art. Mathematics and Mechanics of Solids 24(1):212–234, DOI:https://doi.org/10.1177/1081286517735695

    Google Scholar 

  • Bergamini A, Christen R, Maag B, Motavalli M (2006) A sandwich beam with electrostatically tunable bending stiffness. Smart materials and structures 15(3):678

    CAS  Google Scholar 

  • Bergamini AE, Zündel M, Flores Parra EA, Delpero T, Ruzzene M, Ermanni P (2015) Hybrid dispersive media with controllable wave propagation: A new take on smart materials. Journal of Applied Physics 118(15):154,310

    Google Scholar 

  • Boutin C, Giorgio I, Placidi L, et al (2017) Linear pantographic sheets: Asymptotic micro-macro models identification. Mathematics and Mechanics of Complex Systems 5(2):127–162

    Google Scholar 

  • Carcaterra A, dell’Isola F, Esposito R, Pulvirenti M (2015) Macroscopic description of micro- scopically strongly inhomogeneous systems: A mathematical basis for the synthesis of higher gradients metamaterials. Archive for Rational Mechanics and Analysis 218(3):1239–1262

    Google Scholar 

  • Casadei F, Delpero T, Bergamini A, Ermanni P, Ruzzene M (2012) Piezoelectric resonator arrays for tunable acoustic waveguides and metamaterials. Journal of Applied Physics 112(6):064,902

    Google Scholar 

  • Cazzani A, Ruge P (2013) Rotor platforms on pile-groups running through resonance: A comparison between unbounded soil and soil-layers resting on a rigid bedrock. Soil Dynamics and Earthquake Engineering 50:151–161, DOI:https://doi.org/10.1016/j.soildyn.2013.02.022

    Google Scholar 

  • Cazzani A, Ruge P (2016) Stabilization by deflation for sparse dynamical systems without loss of sparsity. Mechanical Systems and Signal Processing 70–71:664–681, DOI:https://doi.org/10.1016/j.ymssp.2015.09.027

    Google Scholar 

  • Cazzani A, Malagù M, Turco E (2016a) Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches. Continuum Mechanics and thermodynamics 28(1-2):139–156

    Google Scholar 

  • Cazzani A, Malagù M, Turco E (2016b) Isogeometric analysis of plane-curved beams. Mathematics and Mechanics of Solids 21(5):562–577

    Google Scholar 

  • Cazzani A, Malagù M, Turco E, Stochino F (2016c) Constitutive models for strongly curved beams in the frame of isogeometric analysis. Mathematics and Mechanics of Solids 21(2):182–209

    Google Scholar 

  • Cazzani A, Stochino F, Turco E (2016d) An analytical assessment of finite element and isogeometric analyses of the whole spectrum of timoshenko beams. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 96(10):1220–1244

    Google Scholar 

  • Cazzani A, Rizzi N, Stochino F, Turco E (2018a) Modal analysis of laminates by a mixed assumed-strain finite element model. Mathematics and Mechanics of Solids 23(1):99–119, DOI:https://doi.org/10.1177/1081286516666405

    Google Scholar 

  • Cazzani A, Serra M, Stochino F, Turco E (2018b) A refined assumed strain finite element model for statics and dynamics of laminated plates. Continuum Mechanics and Thermodynamics pp 1–28, DOI:https://doi.org/10.1007/s00161-018-0707-x

  • Chevalier L, Calloch S, Hild F, Marco Y (2001) Digital image correlation used to analyze the multiaxial behavior of rubber-like materials. Eur J Mech A/Solids 20:169–187

    Google Scholar 

  • Chia HN, Wu BM (2015) Recent advances in 3d printing of biomaterials. Journal of Biological Bngineering 9(1):4

    Google Scholar 

  • Cima M, Sachs E, Cima L, Yoo J, Khanuja S, Borland S, Wu B, Giordano R (1994) Computer-derived microstructures by 3d printing: bio-and structural materials. In: Solid Freeform Fabr Symp Proc: DTIC Document, pp 181–90

    Google Scholar 

  • di Cosmo F, Laudato M, Spagnuolo M (2018) Acoustic metamaterials based on local resonances: homogenization, optimization and applications. In: Generalized Models and Non-classical Approaches in Complex Materials 1, Springer, pp 247–274

    Google Scholar 

  • Del Vescovo D, Giorgio I (2014a) Dynamic problems for metamaterials: review of existing models and ideas for further research. International Journal of Engineering Science 80:153–172

    Google Scholar 

  • Del Vescovo D, Giorgio I (2014b) Dynamic problems for metamaterials: review of existing models and ideas for further research. International Journal of Engineering Science 80:153–172

    Google Scholar 

  • dell’Isola F, Placidi L (2011) Variational principles are a powerful tool also for formulating field theories. In: Variational models and methods in solid and fluid mechanics, Springer, pp 1–15

    Google Scholar 

  • dell’Isola F, Andreaus U, Placidi L (2015a) 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. Mathematics and Mechanics of Solids 20(8):887–928

    Google Scholar 

  • dell’Isola F, Lekszycki T, Pawlikowski M, Grygoruk R, Greco L (2015b) Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Zeitschrift für angewandte Mathematik und Physik 66(6):3473–3498

    Google Scholar 

  • dell’Isola F, Steigmann D, Della Corte A (2015c) Synthesis of fibrous complex structures: de- signing microstructure to deliver targeted macroscale response. Applied Mechanics Reviews 67(6):060,804

    Google Scholar 

  • dell’Isola F, Bucci S, Battista A (2016a) Against the fragmentation of knowledge: The power of multidisciplinary research for the design of metamaterials. In: Advanced Methods of Continuum Mechanics for Materials and Structures, Springer, pp 523–545

    Google Scholar 

  • dell’Isola F, Giorgio I, Pawlikowski M, Rizzi N (2016b) Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc R Soc A 472(2185):20150,790

    Google Scholar 

  • dell’Isola F, Della Corte A, Giorgio I (2017) Higher-gradient continua: The legacy of piola, mindlin, sedov and toupin and some future research perspectives. Mathematics and Mechanics of Solids 22(4):852–872

    Google Scholar 

  • dell’Isola F, Seppecher P, Alibert JJ, Lekszycki T, Grygoruk R, Pawlikowski M, Steigmann D, Giorgio I, Andreaus U, Turco E, Golaszewski M, Rizzi N, Boutin C, Eremeyev VA, Misra A, Placidi L, Barchiesi E, Greco L, Cuomo M, Cazzani A, Corte AD, Battista A, Scerrato D, Zurba Eremeeva I, Rahali Y, Ganghoffer JF, Müller W, Ganzosch G, Spagnuolo M, Pfaff A, Barcz K, Hoschke K, Neggers J, Hild F (2019a) Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Continuum Mechanics and Thermodynamics DOI:https://doi.org/10.1007/s00161-018-0689-8

    Google Scholar 

  • dell’Isola F, Seppecher P, Spagnuolo M, Barchiesi E, Hild F, Lekszycki T, Giorgio I, Placidi L, Andreaus U, Cuomo M, Eugster SR, Pfaff A, Hoschke K, Langkemper R, Turco E, Sarikaya R, MISRA A, DE ANGELO M, D'Annibale F, Bouterf A, Pinelli X, Misra A, Desmorat B, Pawlikowski M, Dupuy C, Scerrato D, Peyre P, Laudato M, Manzari L, Göransson P, Hesch C, Hesch S, Franciosi P, Dirrenberger J, Maurin F, Vangelatos Z, Grigoropoulos C, Melissinaki V, Farsari M, Muller W, Abali E, Liebold C, Ganzosch G, Harrison P, Drobnicki R, Igumnov LA, Alzahrani F, Hayat T (2019b) Advances in Pantographic Structures: Design, Manufactur- ing, Models, Experiments and Image Analyses. Continuum Mechanics and Thermodynamics DOI:https://doi.org/10.1007/s00161-019-00806-x

    Google Scholar 

  • dell’Isola F, Seppecher P, Spagnuolo M, Barchiesi E, Hild F, Lekszycki T, Giorgio I, Placidi L, Andreaus U, Cuomo M, et al (2019c) Advances in pantographic structures: design, manufacturing, models, experiments and image analyses. Continuum Mechanics and Thermodynamics pp 1–52, DOI:https://doi.org/10.1007/s00161-019-00806-x

    Google Scholar 

  • Deng K, Ding Y, He Z, Zhao H, Shi J, Liu Z (2009) Theoretical study of subwavelength imaging by acoustic metamaterial slabs. Journal of Applied Physics 105(12):124,909

    Google Scholar 

  • Engheta N, Ziolkowski RW (2006) Metamaterials: physics and engineering explorations. John Wiley & Sons

    Google Scholar 

  • Eremeyev V, Zubov L (2007) On constitutive inequalities in nonlinear theory of elastic shells. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik: Applied Mathematics and Mechanics 87(2):94–101

    Google Scholar 

  • Eremeyev VA, Lebedev LP (2011) Existence theorems in the linear theory of micropolar shells. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 91(6):468–476

    Google Scholar 

  • Eremeyev VA, Lebedev LP (2016) Mathematical study of boundary-value problems within the framework of steigmann–ogden model of surface elasticity. Continuum Mechanics and Thermo- dynamics 28(1-2):407–422

    Google Scholar 

  • Eremeyev VA, Pietraszkiewicz W (2016) Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Mathematics and Mechanics of Solids 21(2):210–221

    Google Scholar 

  • Eugster S, Hesch C, Betsch P, Glocker C (2014) Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates. International Journal for Numerical Methods in Engineering 97(2):111–129

    Google Scholar 

  • Eugster SR, Glocker C (2013) Constraints in structural and rigid body mechanics: a frictional contact problem. Annals of solid and structural mechanics 5(1-2):1–13

    Google Scholar 

  • Eugster SR, Glocker C (2017) On the notion of stress in classical continuum mechanics. Mathematics and Mechanics p 299

    Google Scholar 

  • Ferretti M, Piccardo G (2013) Dynamic modeling of taut strings carrying a traveling mass. Continuum Mechanics and Thermodynamics 25(2-4):469–488

    Google Scholar 

  • Francfort GA, Murat F (1986) Homogenization and optimal bounds in linear elasticity. Archive for Rational mechanics and Analysis 94(4):307–334

    Google Scholar 

  • Gatt R, Mizzi L, Azzopardi JI, Azzopardi KM, Attard D, Casha A, Briffa J, Grima JN (2015) Hierarchical auxetic mechanical metamaterials. Scientific reports 5:8395

    Google Scholar 

  • Geers M, Kouznetsova VG, Brekelmans W (2003) Multiscale first-order and second-order computa- tional homogenization of microstructures towards continua. International Journal for Multiscale Computational Engineering 1(4)

    Google Scholar 

  • Germain P (1973) The method of virtual power in continuum mechanics. part 2: Microstructure. SIAM Journal on Applied Mathematics 25:556–575, DOI:https://doi.org/10.1137/0125053

    Google Scholar 

  • Gibson LJ, Ashby MF, Karam G, Wegst U, Shercliff H (1995) The mechanical properties of natural materials. ii. microstructures for mechanical efficiency. Proceedings of the Royal Society of London Series A: Mathematical and Physical Sciences 450(1938):141–162

    Google Scholar 

  • Giorgio I, Galantucci L, Della Corte A, Del Vescovo D (2015) Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: current and upcoming applications. International Journal of Applied Electromagnetics and Mechanics 47(4):1051–1084

    Google Scholar 

  • Giorgio I, Andreaus U, Madeo A (2016a) The influence of different loads on the remodeling process of a bone and bioresorbable material mixture with voids. Continuum Mechanics and Thermodynamics 28(1-2):21–40

    Google Scholar 

  • Giorgio I, Andreaus U, Scerrato D, dell’Isola F (2016b) A visco-poroelastic model of functional adaptation in bones reconstructed with bio-resorbable materials. Biomechanics and modeling in mechanobiology 15(5):1325–1343

    Google Scholar 

  • Giorgio I, Andreaus U, Lekszycki T, Corte AD (2017) The influence of different geometries of matrix/scaffold on the remodeling process of a bone and bioresorbable material mixture with voids. Mathematics and Mechanics of Solids 22(5):969–987

    Google Scholar 

  • Golaszewski M, Grygoruk R, Giorgio I, Laudato M, Di Cosmo F (2018) Metamaterials with relative displacements in their microstructure: technological challenges in 3d printing, experiments and numerical predictions. Continuum Mechanics and Thermodynamics pp 1–20, DOI:https://doi.org/10.1007/s00161-018-0692-0

    Google Scholar 

  • Grédiac M, Hild F (2012) Full-field measurements and identification in solid mechanics. John Wiley & Sons

    Google Scholar 

  • Grillanda N, Chiozzi A, Bondi F, Tralli A, Manconi F, Stochino F, Cazzani A (2019) Numerical insights on the structural assessment of historical masonry stellar vaults: the case of santa maria del monte in cagliari. Continuum Mechanics and Thermodynamics pp 1–24, DOI:https://doi.org/10.1007/s00161-019-00752-8

  • Gunenthiram V, Peyre P, Schneider M, Dal M, Coste F, Fabbro R (2017) Analysis of laser–melt pool–powder bed interaction during the selective laser melting of a stainless steel. Journal of Laser Applications 29(2):022,303

    Google Scholar 

  • Haboudou A, Peyre P, Vannes A (2003) Study of keyhole and melt pool oscillations in dual beam welding of aluminum alloys: effect on porosity formation. In: First International Symposium on High-Power Laser Macroprocessing, International Society for Optics and Photonics, vol 4831, pp 295–301

    Google Scholar 

  • Heckele M, Schomburg W (2003) Review on micro molding of thermoplastic polymers. Journal of Micromechanics and Microengineering 14(3):R1

    Google Scholar 

  • Hild F, Roux S (2012a) Comparison of local and global approaches to digital image correlation. Experimental Mechanics 52(9):1503–1519

    Google Scholar 

  • Hild F, Roux S (2012b) Digital image correlation. In: Rastogi P, Hack E (eds) Optical Methods for Solid Mechanics. A Full-Field Approach, Wiley-VCH, Weinheim (Germany), pp 183–228

    Google Scholar 

  • Hild F, Raka B, Baudequin M, Roux S, Cantelaube F (2002) Multi-scale displacement field measurements of compressed mineral wool samples by digital image correlation. Appl Optics IP 41(32):6815–6828

    Google Scholar 

  • Jia H, Ke M, Hao R, Ye Y, Liu F, Liu Z (2010) Subwavelength imaging by a simple planar acoustic superlens. Applied Physics Letters 97(17):173,507

    Google Scholar 

  • Kadic M, Bückmann T, Stenger N, Thiel M, Wegener M (2012) On the practicability of pentamode mechanical metamaterials. Applied Physics Letters 100(19):191,901

    Google Scholar 

  • Khakalo S, Niiranen J (2017) Isogeometric analysis of higher-order gradient elasticity by user elements of a commercial finite element software. Computer-Aided Design 82:154–169

    Google Scholar 

  • Khakalo S, Niiranen J (2018) Form ii of mindlin’s second strain gradient theory of elasticity with a simplification: For materials and structures from nano-to macro-scales. European Journal of Mechanics-A/Solids 71:292–319

    Google Scholar 

  • Lakes R (1987) Foam structures with a negative poisson’s ratio. Science 235:1038–1041

    CAS  Google Scholar 

  • Lakes R (1993) Materials with structural hierarchy. Nature 361(6412):511

    Google Scholar 

  • Laudato M, Manzari L, Barchiesi E, Di Cosmo F, Göransson P (2018) First experimental observation of the dynamical behavior of a pantographic metamaterial. Mechanics Research Communications 94:125–127, DOI:https://doi.org/10.1016/j.mechrescom.2018.11.003

    Google Scholar 

  • Leclerc H, Périé J, Roux S, Hild F (2009) Integrated digital image correlation for the identification of mechanical properties. In: Gagalowicz A, Philips W (eds) Computer Vision/Computer Graphics CollaborationTechniques. MIRAGE 2009, Springer, Berlin, Heidelberg, Lecture Notes in Computer Science, vol 5496, pp 161–171

    Google Scholar 

  • Lee JH, Singer JP, Thomas EL (2012) Micro-/nanostructured mechanical metamaterials. Advanced materials 24(36):4782–4810

    CAS  Google Scholar 

  • Lekszycki T, dell’Isola F (2012) A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 92(6):426–444

    Google Scholar 

  • Liu Y, Greene MS, Chen W, Dikin DA, Liu WK (2013) Computational microstructure characterization and reconstruction for stochastic multiscale material design. Computer-Aided Design 45(1):65–76

    Google Scholar 

  • Liu Z, Zhang X, Mao Y, Zhu Y, Yang Z, Chan CT, Sheng P (2000) Locally resonant sonic materials. science 289(5485):1734–1736

    CAS  Google Scholar 

  • Liu Z, Lee H, Xiong Y, Sun C, Zhang X (2007) Far-field optical hyperlens magnifying sub-diffraction-limited objects. science 315(5819):1686–1686

    CAS  Google Scholar 

  • Luongo A, Zulli D (2012) Dynamic instability of inclined cables under combined wind flow and support motion. Nonlinear Dynamics 67(1):71–87

    Google Scholar 

  • Luongo A, Zulli D, Piccardo G (2008) Analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables. Journal of Sound and Vibration 315(3):375–393

    Google Scholar 

  • Madeo A, Lekszycki T, dell’Isola F (2011) A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery. Comptes Rendus Mécanique 339(10):625–640

    CAS  Google Scholar 

  • Madeo A, George D, Lekszycki T, Nierenberger M, Rémond Y (2012) A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodelling. Comptes Rendus Mécanique 340(8):575–589

    CAS  Google Scholar 

  • Madou MJ (2011) Manufacturing techniques for microfabrication and nanotechnology, vol 2. CRC press

    Google Scholar 

  • Maggi A, Li H, Greer JR (2017) Three-dimensional nano-architected scaffolds with tunable stiffness for efficient bone tissue growth. Acta Biomaterialia 63:294–305

    CAS  Google Scholar 

  • Masuzawa T, Fujino M, Kobayashi K, Suzuki T, Kinoshita N (1985) Wire electro-discharge grinding for micro-machining. CIRP Annals 34(1):431–434

    Google Scholar 

  • Mathieu F, Leclerc H, Hild F, Roux S (2015) Estimation of elastoplastic parameters via weighted FEMU and integrated-DIC. Exp Mech 55(1):105–119

    Google Scholar 

  • Mindlin RD (1965) Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures 1(4):417–438

    Google Scholar 

  • Misra A, Lekszycki T, Giorgio I, Ganzosch G, Müller WH, dell’Isola F (2018) Pantographic metamaterials show atypical Poynting effect reversal. Mechanics Research Communications 89:6–10

    Google Scholar 

  • Niiranen J, Khakalo S, Balobanov V, Niemi AH (2016) Variational formulation and isogeometric analysis for fourth-order boundary value problems of gradient-elastic bar and plane strain/stress problems. Computer Methods in Applied Mechanics and Engineering 308:182–211

    Google Scholar 

  • Niiranen J, Kiendl J, Niemi AH, Reali A (2017) Isogeometric analysis for sixth-order boundary value problems of gradient-elastic kirchhoff plates. Computer Methods in Applied Mechanics and Engineering 316:328–348

    Google Scholar 

  • Norris AN (2009) Acoustic metafluids. The Journal of the Acoustical Society of America 125(2):839–849

    Google Scholar 

  • Ok JG, Seok Youn H, Kyu Kwak M, Lee KT, Jae Shin Y, Jay Guo L, Greenwald A, Liu Y (2012) Continuous and scalable fabrication of flexible metamaterial films via roll-to-roll nanoimprint process for broadband plasmonic infrared filters. Applied Physics Letters 101(22):223,102

    Google Scholar 

  • Pearton S, Abernathy C, Ren F, Lothian J, Wisk P, Katz A (1993) Dry and wet etching characteristics of inn, aln, and gan deposited by electron cyclotron resonance metalorganic molecular beam epitaxy. Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films 11(4):1772–1775

    Google Scholar 

  • Pendry JB, Holden A, Stewart W, Youngs I (1996) Extremely low frequency plasmons in metallic mesostructures. Physical review letters 76(25):4773

    CAS  Google Scholar 

  • Piccardo G, Ranzi G, Luongo A (2014) A complete dynamic approach to the generalized beam theory cross-section analysis including extension and shear modes. Mathematics and Mechanics of Solids 19(8):900–924

    Google Scholar 

  • Placidi L, Rosi G, Giorgio I, Madeo A (2014) Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials. Mathematics and Mechanics of Solids 19(5):555–578

    Google Scholar 

  • Placidi L, Andreaus U, Giorgio I (2017a) Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. Journal of Engineering Mathematics 103(1):1–21

    Google Scholar 

  • Placidi L, Barchiesi E, Battista A (2017b) An inverse method to get further analytical solutions for a class of metamaterials aimed to validate numerical integrations. In: Mathematical Modelling in Solid Mechanics, Springer, pp 193–210

    Google Scholar 

  • Quiligotti S, Maugin GA, dell’Isola F (2002) Wave motions in unbounded poroelastic solids infused with compressible fluids. Zeitschrift für angewandte Mathematik und Physik ZAMP 53(6):1110–1138

    Google Scholar 

  • Rosi G, Giorgio I, Eremeyev VA (2013) Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 93(12):914–927

    Google Scholar 

  • Rumpf RC, Pazos J, Garcia CR, Ochoa L, Wicker R (2013) 3d printed lattices with spatially variant self-collimation. Progress In Electromagnetics Research 139:1–14

    Google Scholar 

  • Scerrato D, Giorgio I, Rizzi NL (2016) Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Zeitschrift für angewandte Mathematik und Physik 67(3):53

    Google Scholar 

  • Seppecher P, Alibert JJ, dell’Isola F (2011) Linear elastic trusses leading to continua with exotic mechanical interactions. Journal of Physics: Conference Series 319(1):012,018

    Google Scholar 

  • Sutton M (2013) Computer vision-based, noncontacting deformation measurements in mechanics: A generational transformation. Appl Mech Rev 65(AMR-13-1009):050,802

    Google Scholar 

  • Sutton M, Orteu J, Schreier H (2009) Image correlation for shape, motion and deformation measurements: basic concepts, theory and applications. Springer Science & Business Media

    Google Scholar 

  • Teo WE, Ramakrishna S (2006) A review on electrospinning design and nanofibre assemblies. Nanotechnology 17(14):R89

    CAS  Google Scholar 

  • Tomičević Z, Hild F, Roux S (2013) Mechanics-aided digital image correlation. The Journal of Strain Analysis for Engineering Design 48(5):330–343

    Google Scholar 

  • Turco E, dell’Isola F, Cazzani A, Rizzi NL (2016a) Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für angewandte Mathematik und Physik 67(4):85

    Google Scholar 

  • Turco E, Golaszewski M, Cazzani A, Rizzi NL (2016b) Large deformations induced in planar pantographic sheets by loads applied on fibers: experimental validation of a discrete lagrangian model. Mechanics Research Communications 76:51–56

    Google Scholar 

  • Turco E, Giorgio I, Misra A, dell’Isola F (2017a) King post truss as a motif for internal structure of (meta) material with controlled elastic properties. Royal Society open science 4(10):171,153

    Google Scholar 

  • Turco E, Golaszewski M, Giorgio I, D’Annibale F (2017b) Pantographic lattices with non-orthogonal fibres: Experiments and their numerical simulations. Composites Part B: Engineering 118:1–14

    Google Scholar 

  • Turco E, Misra A, Pawlikowski M, dell’Isola F, Hild F (2018) Enhanced piola–hencky discrete models for pantographic sheets with pivots without deformation energy: Numerics and experiments. International Journal of Solids and Structures

    Google Scholar 

  • Vangelatos Z, Komvopoulos K, Grigoropoulos C (2018) Vacancies for controlling the behavior of microstructured three-dimensional mechanical metamaterials. Mathematics and Mechanics of Solids p 1081286518810739

    Google Scholar 

  • Vangelatos Z, Melissinaki V, Farsari M, Komvopoulos K, Grigoropoulos C (2019) Intertwined microlattices greatly enhance the performance of mechanical metamaterials. Mathematics and Mechanics of Solids p 1081286519848041

    Google Scholar 

  • Veselago V (1967) Properties of materials having simultaneously negative values of the dielectric and magnetic susceptibilities. Soviet Physics Solid State USSR 8:2854–2856

    Google Scholar 

  • Veselago VG (1968) The electrodynamics of substances with simultaneously negative values of and µ. Soviet physics uspekhi 10(4):509

    Google Scholar 

  • Wegst UG, Bai H, Saiz E, Tomsia AP, Ritchie RO (2015) Bioinspired structural materials. Nature materials 14(1):23

    Google Scholar 

  • Zhang S, Yin L, Fang N (2009) Focusing ultrasound with an acoustic metamaterial network. Physical review letters 102(19):194,301

    Google Scholar 

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Authors and Affiliations

  1. Department of Naval Architecture and Ocean Engineering, Istanbul Technical University, 34469, Maslak, Istanbul, Turkey

    Mustafa Erden Yildizdag

  2. International Research Center for the Mathematics and Mechanics of Complex Systems, University of L’Aquila, L’Aquila, Italy

    Mustafa Erden Yildizdag, Chuong Anthony Tran, Emilio Barchiesi, Mario Spagnuolo & Francesco dell’Isola

  3. Dipartimento di Ingegneria Civile, Edile-Architettura e Ambientale, Università degli Studi dell’Aquila, L’Aquila, Italy

    Francesco dell’Isola

  4. Laboratoire de Mécanique et Technologie (LMT), ENS Paris-Saclay, CNRS, Université Paris-Saclay, 94235, Cachan Cedex, France

    François Hild

Authors
  1. Mustafa Erden Yildizdag
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  2. Chuong Anthony Tran
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  3. Emilio Barchiesi
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  4. Mario Spagnuolo
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  5. Francesco dell’Isola
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  6. François Hild
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Corresponding author

Correspondence to Mario Spagnuolo .

Editor information

Editors and Affiliations

  1. Faculty of Mechanical Engineering, Otto von Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

    Holm Altenbach

  2. Faculty of Mechanical Engineering, Esslingen University of Applied Sciences, Esslingen, Germany

    Andreas Öchsner

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Yildizdag, M.E., Tran, C.A., Barchiesi, E., Spagnuolo, M., dell’Isola, F., Hild, F. (2019). A Multi-disciplinary Approach for Mechanical Metamaterial Synthesis: A Hierarchical Modular Multiscale Cellular Structure Paradigm. In: Altenbach, H., Öchsner, A. (eds) State of the Art and Future Trends in Material Modeling . Advanced Structured Materials, vol 100. Springer, Cham. https://doi.org/10.1007/978-3-030-30355-6_20

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