Skip to main content
SpringerLink
Log in

Dynamical Vector Fields on Pantographic Sheet: Experimental Observations

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

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

Abstract

In this work, we will present and discuss some experimental observations of the dynamical displacement vector field on a pantographic sheet. We will sketch the experimental setup and we will qualitatively describe the observed behavior for a set of relevant frequencies.

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, Müller WH, dell’Isola F (2017) Theory and computation of higher gradient elasticity theories based on action principles. Archive of Applied Mechanics 87(9):1495–1510

    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

    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, dell’Isola F, Giorgio I, Placidi L, Lekszycki T, Rizzi N (2016) Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity. International Journal of Engineering Science 108:34–50

    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 

  • Auffray N, dell’Isola F, Eremeyev V, Madeo A, Rosi G (2015) Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids. Mathematics and Mechanics of Solids 20(4):375–417

    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: Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials, Springer, 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: Advances in Mechanics of Microstructured Media and Structures, Springer, pp 43–74

    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 pp 1–13

    Google Scholar 

  • Barchiesi E, Spagnuolo M, Placidi L (2018c) Mechanical metamaterials: a state of the art. Mathematics and Mechanics of Solids p 1081286517735695

    Google Scholar 

  • Battista A, Rosa L, dell’Erba R, Greco L (2016) Numerical investigation of a particle system compared with first and second gradient continua: Deformation and fracture phenomena. Mathematics and Mechanics of Solids p 1081286516657889

    Google Scholar 

  • Berezovski A, Giorgio I, Della Corte A (2015) Interfaces in micromorphic materials: Wave transmission and reflection with numerical simulations. Mathematics and Mechanics of Solids p 1081286515572244

    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 

  • Caprino S, Esposito R, Marra R, Pulvirenti M (1993) Hydrodynamic limits of the vlasov equation. Communications in partial differential equations 18(5):805–820

    Google Scholar 

  • Carcaterra A, dell’Isola F, Esposito R, Pulvirenti M (2015) Macroscopic description of microscopically strongly inhomogenous systems: A mathematical basis for the synthesis of higher gradients metamaterials. Archive for Rational Mechanics and Analysis pp 1–24

    Google Scholar 

  • Carinci G, De Masi A, Giardinà C, Presutti E (2014a) Hydrodynamic limit in a particle system with topological interactions. Arabian Journal of Mathematics 3(4):381–417

    Google Scholar 

  • Carinci G, De Masi A, Giardinà C, Presutti E (2014b) Super-hydrodynamic limit in interacting particle systems. Journal of Statistical Physics 155(5):867–887

    Google Scholar 

  • Cecchi A, Rizzi N (2001) Heterogeneous elastic solids: a mixed homogenization-rigidification technique. International journal of solids and structures 38(1):29–36

    Google Scholar 

  • Chatzigeorgiou G, Javili A, Steinmann P (2014) Unified magnetomechanical homogenization framework with application to magnetorheological elastomers. Mathematics and Mechanics of Solids 19(2):193–211

    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 

  • Cuomo M, dell’Isola F, Greco L, Rizzi N (2017) First versus second gradient energies for planar sheets with two families of inextensible fibres: investigation on deformation boundary layers, discontinuities and geometrical instabilities. Composites Part B: Engineering 115:423–448

    Google Scholar 

  • De Masi A, Olla S (2015) Quasi-static hydrodynamic limits. Journal of Statistical Physics 161(5):1037–1058

    Google Scholar 

  • De Masi A, Galves A, Löcherbach E, Presutti E (2015) Hydrodynamic limit for interacting neurons. Journal of Statistical Physics 158(4):866–902

    Google Scholar 

  • Del Vescovo D, Giorgio I (2014) 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, Andreaus U, Placidi L (2014a) 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 p 1081286513509811

    Google Scholar 

  • dell’Isola F, Maier G, Perego U, Andreaus U, Esposito R, Forest S (2014b) The complete works of Gabrio Piola: volume I. Cham, Switzerland: Springer

    Google Scholar 

  • dell’Isola F, Giorgio I, Andreaus U (2015a) Elastic pantographic 2d lattices: a numerical analysis on static response and wave propagation. Proceedings of the Estonian Academy of Sciences 64(3):219–225

    Google Scholar 

  • dell’Isola F, Seppecher P, Della Corte A (2015b) The postulations á la D’alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc R Soc A 471(2183):20150,415

    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, Cuomo M, Greco L, Della Corte A (2016b) Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies. Journal of Engineering Mathematics pp 1–31

    Google Scholar 

  • dell’Isola F, Giorgio I, Pawlikowski M, Rizzi N (2016c) Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenization, experimental and numerical examples of equilibrium. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472(2185)

    Google Scholar 

  • dell’Isola F, Steigmann D, Della Corte A (2016d) Synthesis of fibrous complex structures: Designing microstructure to deliver targeted macroscale response. Applied Mechanics Reviews 67(6):21–pages

    Google Scholar 

  • dell’Isola F, Seppecher P, Alibert JJ, et al (2018) Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Continuum Mechanics and Thermodynamics pp 1–34

    Google Scholar 

  • dell’Isola F, Maier G, Perego U, Andreaus U, Esposito R, Forest S (2019) The complete works of gabrio piola: volume ii. Cham, Switzerland: Springer

    Google Scholar 

  • Deymier P (2013) Acoustic Metamaterials and Phononic Crystals. Springer Series in Solid-State Sciences, Springer Berlin Heidelberg

    Google Scholar 

  • Eremeyev VA, Altenbach H (2014) Equilibrium of a second-gradient fluid and an elastic solid with surface stresses. Meccanica 49(11):2635–2643

    Google Scholar 

  • Eremeyev VA, dell’Isola F, Boutin C, Steigmann D (2017) Linear pantographic sheets: existence and uniqueness of weak solutions. Journal of Elasticity pp 1–22

    Google Scholar 

  • Esposito R, Pulvirenti M (2004) From particles to fluids. Handbook of mathematical fluid dynamics 3:1–82

    Google Scholar 

  • Ferretti M, Madeo A, dell’Isola F, Boisse P (2014) Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory. Zeitschrift für angewandte Mathematik und Physik 65(3):587–612

    Google Scholar 

  • Ganzosch G, dell’Isola F, Turco e, Lekszycki T, Müller W (2016) Shearing tests applied to pantographic structures. Acta Polytechnica CTU Proceedings 7:1–6

    Google Scholar 

  • Ganzosch G, Hoschke K, Lekszycki T, Giorgio I, Turco E, Müller W (2018) 3d-measurements of 3d-deformations of pantographic structures. Technische Mechanick 38(3):233–245

    Google Scholar 

  • Giorgio I (2016) Numerical identification procedure between a micro-Cauchy model and a macrosecond gradient model for planar pantographic structures. Zeitschrift fur Angewandte Mathematik und Physik 67(4)

    Google Scholar 

  • Giorgio I, Della Corte A, dell’Isola F, Steigmann DJ (2016) Buckling modes in pantographic lattices. Comptes rendus Mecanique 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 N, Turco E (2017b) Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proceedings of the Royal Society of London Series A 473:20170,636

    Google Scholar 

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

    Google Scholar 

  • Hild F, Roux S (2006) Digital image correlation: from displacement measurement to identification of elastic properties–a review. Strain 42(2):69–80

    Google Scholar 

  • Javili A, McBride A, Steinmann P (2012) Thermomechanics of solids with lower-dimensional energetics: On the importance of surface, interface, and curve structures at the nanoscale. A unifying review. Applied Mechanics Reviews 65:010,802–1–31

    Google Scholar 

  • Javili A, Chatzigeorgiou G, Steinmann P (2013a) Computational homogenization in magnetomechanics. International Journal of Solids and Structures 50(25):4197–4216

    Google Scholar 

  • Javili A, McBride A, Mergheim J, Steinmann P, Schmidt U (2013b) Micro-to-macro transitions for continua with surface structure at the microscale. International Journal of Solids and Structures 50(16):2561–2572

    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

    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

    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

    Google Scholar 

  • Madeo A, Neff P, Ghiba ID, Placidi L, Rosi G (2013) Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps. Continuum Mechanics and Thermodynamics pp 1–20

    Google Scholar 

  • Madeo A, Della Corte A, Greco L, Neff P (2014) Wave propagation in pantographic 2D lattices with internal discontinuities. Accepted for publication in Proceedings of the Estonian Academy of Sciences

    Google Scholar 

  • Misra A, Poorsolhjouy P (2015) Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math Mech Complex Syst 3(3):285–308

    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 

  • Pideri C, Seppecher P (1997) A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Continuum Mechanics and Thermodynamics 9(5):241–257

    Google Scholar 

  • Placidi L (2014) A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model. Continuum Mechanics and Thermodynamics

    Google Scholar 

  • Placidi L, Barchiesi E (2018) Energy approach to brittle fracture in strain-gradient modelling. Proc R Soc A 474(2210):20170,878

    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, 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 66(6):3699–3725

    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) Simulation results for damage with evolving microstructure and growing strain gradient moduli. Continuum Mechanics and Thermodynamics pp 1–21

    Google Scholar 

  • Pulvirenti M (1996) Kinetic limits for stochastic particle systems. Lecture Notes in Mathematics-Springer-Verlag

    Google Scholar 

  • Rahali Y, Giorgio I, Ganghoffer J, dell’Isola F (2015) Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices. International Journal of Engineering Science 97:148–172

    Google Scholar 

  • Rinaldi A, Placidi L (2013) A microscale second gradient approximation of the damage parameter of quasi-brittle heterogeneous lattices. ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik/Journal of Applied Mathematics and Mechanics

    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(5):050,801

    Google Scholar 

  • Scerrato D, Giorgio I, Rizzi N (2016a) Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Zeitschrift fur Angewandte Mathematik und Physik 67(3)

    Google Scholar 

  • Scerrato D, Zhurba Eremeeva I, Lekszycki T, Rizzi N (2016b) On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik

    Google Scholar 

  • Seppecher P (2002) Second-gradient theory: application to cahn-hilliard fluids. In: Continuum Thermomechanics, Springer, pp 379–388

    Google Scholar 

  • Trinh DK, Janicke R, Auffray N, Diebels S, Forest S (2012) Evaluation of generalized continuum substitution models for heterogeneous materials. International Journal for Multiscale Computational Engineering 10(6)

    Google Scholar 

  • Turco E (2018) Discrete is it enough? the revival of Piola–Hencky keynotes to analyze threedimensional elastica. Continuum Mechanics and Thermodynamics pp 1–19

    Google Scholar 

  • Turco E, Barcz K, Rizzi N (2016a) Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part II: comparison with experimental evidence. Zeitschrift fur Angewandte Mathematik und Physik 67(5)

    Google Scholar 

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

    Google Scholar 

  • Turco E, dell’Isola F, Rizzi N, Grygoruk R, Müller W, Liebold C (2016c) 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 (2016d) 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, Golaszewski M, Giorgio I, D’Annibale F (2017a) Pantographic lattices with nonorthogonal fibres: experiments and their numerical simulations. Composites Part B: Engineering 118:1–14

    Google Scholar 

  • Turco E, Golaszewski M, Giorgio I, Placidi L (2017b) Can a hencky-type model predict the mechanical behaviour of pantographic lattices? In: Mathematical Modelling in Solid Mechanics, Springer, pp 285–311

    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 

  • 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 

  • Yang Y, Misra A (2012) Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity. International Journal of Solids and Structures 49(18):2500–2514

    Google Scholar 

  • Yang Y, ChingW, Misra A (2011) Higher-order continuum theory applied to fracture simulation of nanoscale intergranular glassy film. Journal of Nanomechanics and Micromechanics 1(2):60–71

    Google Scholar 

Download references

Acknowledgements

The authors want to thanks Luca Manzari for the precious help in the production of the figures.

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Via Vetoio, Coppito, 67100, L’Aquila, Italy

    Marco Laudato & Rafał Drobnicki

  2. Dipartimento di Ingegneria Civile, Edile, Ambientale e Architettura, Università degli Studi dell’Aquila, Via Giovanni Gronchi 18 - Zona industriale di Pile, 67100, L’Aquila, Italy

    Fabio Di Cosmo

  3. International Research Center M&MoCS, Università degli Studi dell’Aquila, Via Giovanni Gronchi 18 - Zona industriale di Pile, 67100, L’Aquila, Italy

    Fabio Di Cosmo

  4. Department of Machinery Design and Biomedical Engineering, Warsaw University of Technology, Warsaw, Poland

    Rafał Drobnicki

  5. KTH Royal Institute of Technology, Marcus Wallenberg Laboratory for Sound and Vibration Research, SE-100 44, Stockholm, Sweden

    Peter Göransson

Authors
  1. Marco Laudato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabio Di Cosmo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rafał Drobnicki
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Peter Göransson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Laudato .

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

Laudato, M., Di Cosmo, F., Drobnicki, R., Göransson, P. (2019). Dynamical Vector Fields on Pantographic Sheet: Experimental Observations. 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_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-13307-8_19

  • 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