Skip to main content
SpringerLink
Log in

How the Properties of Pantographic Elementary Lattices Determine the Properties of Pantographic Metamaterials

  • Chapter
  • First Online:
New Achievements in Continuum Mechanics and Thermodynamics

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 108))

Abstract

In this paper we describe a three-scales homogenization process which we use to determine a macroscopic model for pantographic metamaterials. The smallest scale refers to the length at which the considered deformable mechanical system can be modeled as a Cauchy’s continuum. Of course, at this scale, its geometry is rather complex. The meso-scale refers to a length at which the system can be modeled as a Hencky-type discrete system constituted by masses interconnected by extensional and rotational springs. At macro-scale the model to be used is a generalized plate whose deformation energy depends on geodesic curvature. While the direct identification from the smallest scale to the macro-scale seems rather diffcult, the identification from smallest scale to meso-scale can be successfully obtained. The geometrical properties, along with Young and Poisson coeffcients of the used isotropic material, at the smallest scale determine the extensional and rotational stiffnesses to be used at the meso-scale. On the other hand, the Piola-type identification process allows us to determine the stiffnesses of the macroscopic generalized plate model, via a simple asymptotic expansion. We have observed that this process is valid in both cases when the smallest scale is of the order of microns and when the smallest scale is of the order of tenth of millimeters. Some experimental and numerical results supporting this statement are exhibited.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Abali BE, Wu CC, Müller WH (2016) An energy-based method to determine material constants in nonlinear rheology with applications. Continuum Mechanics and Thermodynamics 28(5):1221–1246

    Google Scholar 

  • Abd-alla AN, Alshaikh F, Del Vescovo D, Spagnuolo M (2017) Plane waves and eigenfrequency study in a transversely isotropic magneto-thermoelastic medium under the effect of a constant angular velocity. Journal of Thermal Stresses 40(9):1079–1092

    Google Scholar 

  • Alibert JJ, Della Corte A (2015) Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Zeitschrift für Angewandte Mathematik und Physik 66(5):2855–2870

    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 

  • Andreaus U, Spagnuolo M, Lekszycki T, Eugster SR (2018) A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear euler–bernoulli beams. Continuum Mechanics and Thermodynamics pp 1–21

    Google Scholar 

  • Atai A, Steigmann DJ (1997) On the nonlinear mechanics of discrete networks. Archive of Applied Mechanics 67(5):303–319

    Google Scholar 

  • Balobanov V, Niiranen J (2018) Locking-free variational formulations and isogeometric analysis for the Timoshenko beam models of strain gradient and classical elasticity. Computer Methods in Applied Mechanics and Engineering 339:137–159

    Google Scholar 

  • Barchiesi E, Placidi L (2017) A review on models for the 3d statics and 2d dynamics of pantographic fabrics. In: Sumbatyan MA (ed) Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials, Advanced Structured Materials, vol 59, Springer, Singapore, pp 239–258

    Google Scholar 

  • Barchiesi E, dell’Isola F, Laudato M, Placidi L, Seppecher P (2018a) A 1D continuum model for beams with pantographic microstructure: asymptotic micro-macro identification and numerical results. In: dell’Isola F, Eremeyev V, Porubov A (eds) Advances in Mechanics of Microstructured Media and Structures. Advanced Structured Materials, Springer

    Google Scholar 

  • Barchiesi E, Ganzosch G, Liebold C, Placidi L, Grygoruk R, Müller WH (2018b) Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation. Continuum Mechanics and Thermodynamics

    Google Scholar 

  • Barchiesi E, Spagnuolo M, Placidi L (2018c) Mechanical metamaterials: a state of the art. Mathematics and Mechanics of Solids 24(1):212–234

    Google Scholar 

  • Berezovski A, Giorgio I, Della Corte A (2016) Interfaces in micromorphic materials: wave transmission and reflection with numerical simulations. Mathematics and Mechanics of Solids 21(1):37–51

    Google Scholar 

  • Birsan M, Altenbach H, Sadowski T, Eremeyev VA, Pietras D (2012) Deformation analysis of functionally graded beams by the direct approach. Composites Part B: Engineering 43(3):1315–1328

    Google Scholar 

  • Boutin C, dell’Isola F, Giorgio I, Placidi L (2017) Linear pantographic sheets. Asymptotic micromacro models identification. Mathematics and Mechanics of Complex Systems 5(2):127–162

    Google Scholar 

  • Braides A, Causin A, Solci M (2018) A homogenization result for interacting elastic and brittle media. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 474(2218)

    Google Scholar 

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

    Google Scholar 

  • Challamel N, Lerbet J, Wang CM (2014) On buckling of granular columns with shear interaction: Discrete versus nonlocal approaches. Journal of Applied Physics 115(23):234,902

    Google Scholar 

  • Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley

    Google Scholar 

  • Cuomo M, Contraffatto L, Greco L (2014) A variational model based on isogeometric interpolation for the analysis of cracked bodies. International Journal of Engineering Science 80:173–188

    Google Scholar 

  • De Masi A, Merola I, Presutti E, Vignaud Y (2008) Potts models in the continuum uniqueness and exponential decay in the restricted ensembles. Journal of Statistical Physics 133(2):281–345

    Google Scholar 

  • De Masi A, Merola I, Presutti E, Vignaud Y (2009) Coexistence of ordered and disordered phases in Potts models in the continuum. Journal of Statistical Physics 134(2):243–306

    Google Scholar 

  • dell’Isola F, Maier G, Perego U, Andreaus U, Esposito R, Forest S (2014) The complete works of Gabrio Piola: Volume I - Commented English Translation. Springer International Publishing

    Google Scholar 

  • dell’Isola F, Andreaus U, Placidi L (2015) 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, Giorgio I, Pawlikowski M, Rizzi NL (2016) Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 472(2185):1–23

    Google Scholar 

  • dell’Isola F, Seppecher P, Alibert JJ, Lekszycki T, Grygoruk R, Pawlikowski M, Steigmann DJ, Giorgio I, Andreaus U, Turco E, Gołaszewski M, Rizzi N, Boutin C, Eremeyev VA, Misra A, Placidi L, Barchiesi E, Greco L, Cuomo M, Cazzani A, Della Corte A, Battista A, Scerrato D, Zurba Eremeeva I, Rahali Y, Ganghoffer JF, Muller W, Ganzosch G, Spagnuolo M, Pfaff A, Barcz K, Hoschke K, Neggers J, Hild F (2018) Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Continuum Mechanics and Thermodynamics 10.1007/s00161-018-0689-8

    Google Scholar 

  • dell’Isola F, Maier G, Perego U, Andreaus U, Esposito R, Forest S (2019) The complete works of Gabrio Piola: Volume II - Commented English Translation. Springer International Publishing

    Google Scholar 

  • di Cosmo F, Laudato M, Spagnuolo M (2018) Acoustic metamaterials based on local resonances: Homogenization, optimization and applications. In: Altenbach H, Pouget J, Rousseau M, Collet B, Michelitsch T (eds) Generalized Models and Non-classical Approaches in Complex Materials 1, Advanced Structured Materials, vol 89, Springer International Publishing, Cham, pp 247–274

    Google Scholar 

  • Diyaroglu C, Oterkus E, Oterkus S, Madenci E (2015) Peridynamics for bending of beams and plates with transverse shear deformation. International Journal of Solids and Structures 69:152–168

    Google Scholar 

  • Engelbrecht J, Berezovski A (2015) Reflections on mathematical models of deformation waves in elastic microstructured solids. Mathematics and Mechanics of Complex Systems 3(1):43–82

    Google Scholar 

  • Eremeyev V, dell’Isola F (2018) A note on reduced strain gradient elasticity. In: Altenbach H, Pouget J, Rousseau M, Collet B, Michelitsch T (eds) Generalized Models and Non-classical Approaches in Complex Materials 1, Advanced Structured Materiasl, vol 89, Springer International Publishing, Cham, pp 301–310

    Google Scholar 

  • Eremeyev VA, dell’Isola F, Boutin C, Steigmann D (2018) Linear pantographic sheets: Existence and uniqueness of weak solutions. Journal of Elasticity 132(2):175–196

    Google Scholar 

  • Eugster SR, 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 

  • Franciosi P, Spagnuolo M, Salman OU (2018) Mean Green operators of deformable fiber networks embedded in a compliant matrix and property estimates. Continuum Mechanics and Thermodynamics pp 1–32

    Google Scholar 

  • Giorgio I, Grygoruk R, dell’Isola F, Steigmann DJ (2015) Pattern formation in the threedimensional deformations of fibered sheets. Mechanics Research Communications 69:164–171

    Google Scholar 

  • Giorgio I, Della Corte A, dell’Isola F, Steigmann DJ (2016) Buckling modes in pantographic lattices. Comptes Rendus - Mécanique 344(7):487–501

    Google Scholar 

  • Giorgio I, Della Corte A, dell’Isola F (2017a) Dynamics of 1D nonlinear pantographic continua. Nonlinear Dynamics 88(1):21–31

    Google Scholar 

  • Giorgio I, Rizzi NL, Turco E (2017b) Continuum modelling of pantographic sheets for out-ofplane bifurcation and vibrational analysis. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473(20170636):1–21

    Google Scholar 

  • Giorgio I, Harrison P, dell’Isola F, Alsayednoor J, Turco E (2018) Wrinkling in engineering fabrics: a comparison between two different comprehensive modelling approaches. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474(20180063):1–20

    Google Scholar 

  • Golaszewski R Mand Grygoruk, 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

    Google Scholar 

  • Greco L, Cuomo M (2013) B-Spline interpolation of Kirchhoff–Love space rods. Computer Methods in Applied Mechanics and Engineering 256:251–269

    Google Scholar 

  • Greco L, Cuomo M, Contraffatto L, Gazzo S (2017) An effcient blended mixed B-spline formulation for removing membrane locking in plane curved Kirchhoff rods. Computer Methods in Applied Mechanics and Engineering 324:476–511

    Google Scholar 

  • Grimmett GR (2016) Correlation inequalities for the Potts model. Mathematics and Mechanics of Complex Systems 4(3-4):327–334

    Google Scholar 

  • Hencky H (1921) Über die angenäherte lösung von stabilitätsproblemen im raum mittels der elastischen gelenkkette. PhD thesis, Engelmann

    Google Scholar 

  • Javili A, Dortdivanlioglu B, Kuhl E, Linder C (2015) Computational aspects of growth-induced instabilities through eigenvalue analysis. Computational Mechanics 56(3):405–420

    Google Scholar 

  • Karamooz MR, Kadkhodaei M (2015) A computationally effcient modeling approach for predicting mechanical behavior of cellular lattice structures. Journal of Materials Engineering and Performance 24(1):245–252

    Google Scholar 

  • Karamooz MR, Kadkhodaei M, Badrossamay M, Rezaei R (2014) Numerical investigation on mechanical properties of cellular lattice structures fabricated by fused deposition modeling. International Journal of Mechanical Sciences 88:154–161

    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 (to appear)

    Google Scholar 

  • Khakalo S, Balobanov V, Niiranen J (2018) Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: applications to sandwich beams and auxetics. Journal of Engineering Science 127:33–52

    Google Scholar 

  • Laudato M, Di Cosmo F (2018) Euromech 579 Arpino 3-8 April 2017: Generalized and microstructured continua: new ideas in modeling and/or applications to structures with (nearly) inextensible fibers - a review of presentations and discussions. Continuum Mechanics and Thermodynamics pp 1–15

    Google Scholar 

  • Lekszycki T, Di Cosmo F, Laudato M, Vardar O (2018) Application of energy measures in detection of local deviations in mechanical properties of structural elements. Continuum Mechanics and Thermodynamics

    Google Scholar 

  • Meo DD, Diyaroglu C, Zhu N, Oterkus E, Siddiq MA (2016) Modelling of stress-corrosion cracking by using peridynamics. International Journal of Hydrogen Energy 41(15):6593–6609

    Google Scholar 

  • Piegl L, Tiller W (1997) The NURBS Book, 2nd edn. Springer-Verlag Berlin Heidelberg

    Google Scholar 

  • Placidi L, Barchiesi E (2018) Energy approach to brittle fracture in strain-gradient modelling. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474(20170878):1–19

    Google Scholar 

  • Placidi L, El Dhaba AR (2017) Semi-inverse method à la Saint-Venant for two-dimensional linear isotropic homogeneous second-gradient elasticity. Mathematics and Mechanics of Solids 22(5):919–937

    Google Scholar 

  • Placidi L, Andreaus U, Della Corte A, Lekszycki T (2015) Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coeffcients. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 66(6):3699–3725

    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: dell’Isola F, Sofonea M, Steigmann D (eds) Mathematical Modelling in Solid Mechanics, Springer Singapore, Singapore, pp 193–210

    Google Scholar 

  • Placidi L, Barchiesi E, Della Corte A (2017c) Identification of two-dimensional pantographic structures with a linear D4 orthotropic second gradient elastic model accounting for external bulk double forces. In: dell’Isola F, Sofonea M, Steigmann D (eds) Mathematical Modelling in Solid Mechanics, Springer Singapore, Singapore, pp 211–232

    Google Scholar 

  • Placidi L, Barchiesi E, Misra A (2018a) A strain gradient variational approach to damage: a comparison with damage gradient models and numerical results. Mathematics and Mechanics of Complex Systems 6(2):77–100

    Google Scholar 

  • Placidi L, Misra A, Barchiesi E (2018b) Two-dimensional strain gradient damage modeling: a variational approach. Zeitschrift für angewandte Mathematik und Physik 69(3):56

    Google Scholar 

  • Saeb S, Steinmann P, Javili A (2016) Aspects of computational homogenization at finite deformations: A unifying review from Reuss’ to Voigt’s bound. Applied Mechanics Reviews 68:050,801

    Google Scholar 

  • Scerrato D, Giorgio I, Rizzi NL (2016) Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. ZAMP - Journal of Applied Mathematics and Physics 67(53):1–19

    Google Scholar 

  • Seppecher P, Alibert JJ, dell’Isola F (2011) Linear elastic trusses leading to continua with exotic mechanical interactions. In: Journal of Physics: Conference Series, IOP Publishing, vol 319(1), p 012018

    Google Scholar 

  • Shirani M, Luo C, Steigmann DJ (2018) Cosserat elasticity of lattice shells with kinematically independent flexure and twist. Continuum Mechanics and Thermodynamics pp 1–11

    Google Scholar 

  • Spagnuolo M, Andreaus U (2018) A targeted review on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling. Mathematics and Mechanics of Solids p 108128651773700

    Google Scholar 

  • Steigmann DJ, dell’Isola F (2015) Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. Acta Mechanica Sinica 31(3):373–382

    Google Scholar 

  • Turco E (2017) Tools for the numerical solution of inverse problems in structural mechanics: review and research perspectives. European Journal of Environmental and Civil Engineering 21(5):509–554

    Google Scholar 

  • Turco E (2018) Discrete is it enough? the revival of Piola–Hencky keynotes to analyze threedimensional elastica. Continuum Mechanics and Thermodynamics 30(5):1039–1057

    Google Scholar 

  • Turco E, Rizzi NL (2016) Pantographic structures presenting statistically distributed defects: numerical investigations of the effects on deformation fields. Mechanics Research Communications 77:65–69

    Google Scholar 

  • Turco E, Barcz K, Pawlikowski M, Rizzi NL (2016a) Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part I: numerical simulations. Zeitschrift für Angewandte Mathematik und Physik 67(122):1–16

    Google Scholar 

  • Turco E, Barcz K, Rizzi NL (2016b) Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part II: comparison with experimental evidence. Zeitschrift für Angewandte Mathematik und Physik 67(123):1–16

    Google Scholar 

  • Turco E, dell’Isola F, Cazzani A, Rizzi NL (2016c) Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für Angewandte Mathematik und Physik 67(4):1–28

    Google Scholar 

  • Turco E, dell’Isola F, Rizzi NL, Grygoruk R, Müller WH, Liebold C (2016d) Fiber rupture in sheared planar pantographic sheets: numerical and experimental evidence. Mechanics Research Communications 76:86–90

    Google Scholar 

  • Turco E, Golaszewski M, Cazzani A, Rizzi NL (2016e) 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 properties. Royal Society Open Science 4(171153)

    Google Scholar 

  • Turco E, Golaszewski M, Giorgio I, D’Annibale F (2017b) Pantographic lattices with nonorthogonal 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 (2018a) Enhanced Piola–Hencky discrete models for pantographic sheets with pivots without deformation energy: numerics and experiments. International Journal of Solids and Structures 147:94–109

    Google Scholar 

  • Turco E, Misra A, Sarikaya R, Lekszycki T (2018b) Quantitative analysis of deformation mechanisms in pantographic substructures: experiments and modeling. Continuum Mechanics and Thermodynamics pp 1–15

    Google Scholar 

  • Yang H, Ganzosch G, Giorgio I, Abali BE (2018) Material characterization and computations of a polymeric metamaterial with a pantographic substructure. Zeitschrift für angewandte Mathematik und Physik 69(4):105

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Architecture, Design and Urban planning (DADU), University of Sassari and M&MOCS International Research Center, Sassari, Italy

    Emilio Turco

Authors
  1. Emilio Turco
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emilio Turco .

Editor information

Editors and Affiliations

  1. Institute of Mechanics, Technische Universität Berlin, Berlin, Germany

    Bilen Emek Abali

  2. Institut für Mechanik, Otto-von-Guericke-University Magdeburg, Magdeburg, Germany

    Holm Altenbach

  3. Dipartimento di Ingegneria Strutturale e Geotecnica, Università degli Studi di Roma “La Sapienza”, Roma, Italy

    Francesco dell'Isola

  4. Faculty of Civil and Environmental Engineering, Gdansk University of Technology, Gdansk, Poland

    Victor A. Eremeyev

  5. Fakultät Maschinenbau, Esslingen University of Applied Sciences, Esslingen, Germany

    Andreas Öchsner

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Turco, E. (2019). How the Properties of Pantographic Elementary Lattices Determine the Properties of Pantographic Metamaterials. In: Abali, B., Altenbach, H., dell'Isola, F., Eremeyev, V., Öchsner, A. (eds) New Achievements in Continuum Mechanics and Thermodynamics. Advanced Structured Materials, vol 108. Springer, Cham. https://doi.org/10.1007/978-3-030-13307-8_33

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-13307-8_33

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-13306-1

  • Online ISBN: 978-3-030-13307-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation