Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Continuum Mechanics and Thermodynamics
Erschienen in: Continuum Mechanics and Thermodynamics 6/2019

22.10.2019 | Original Article

Numerically driven tuning of equilibrium paths for pantographic beams

verfasst von: Emilio Turco

Erschienen in: Continuum Mechanics and Thermodynamics | Ausgabe 6/2019

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

The nonlinear mechanical behavior of pantographic beams, mimicking the results of the Euler’s beam, is explored. In particular, using a very simple Lagrangian model already tested for pantographic lattices and an algorithm based on the Riks’ approach to reconstruct via a stepwise procedure the equilibrium path, the results of several numerical simulations, capable of determining the buckling load, the pre- and post-buckling paths and the imperfection analysis, are presented and discussed. The goal of this campaign of numerical simulations is double. Indeed, besides to make available to the scientific community some reference solutions for a kind of very popular metamaterial, the enriched capabilities of pantographic beams can also be exploited also for a fine-tuning of equilibrium paths.
Vorheriger Artikel Nonlinear travelling strain waves in a gradient-elastic medium
Nächster Artikel Phenomenon of skin effect in metals due to hydrogen absorption

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren
Fußnoten
1
If the cost related to the model construction and to the strategy for obtaining the problem solution is not considered.
 
2
The used bending measure is not usual but surely admissible; furthermore, it is computationally convenient since it avoids to compute the angle \(\beta \). However, it is also admissible to use as bending strain measure one similar to that used for the shearing strain measure.
 
3
It is obvious that this deformation is admissible only in the case of permitted overlapping for the pantographic beam elements.
 
4
See the exegesis of the work Fundamentals of the mechanics of continua by E. Hellinger reported in [23‐25] for an insightful analysis on the continuum idea.
 
Literatur
1.
Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability, 2nd edn. McGraw-Hill, Auckland (1963)
2.
Pignataro, M., Rizzi, N., Luongo, A.: Stability. Bifurcation and Postcritical Behaviour of Elastic Structures. Elsevier, Amsterdam (1991)
3.
Antman, S.S.: Nonlinear Problems of Elasticity, volume 107 of Applied Mathematical Sciences. Springer, New York, 2nd edn (2005)
4.
Bažant, Z.P., Cedolin, L.: Stability of Structures. Elastic, lnelastic, Fracture and Damage Theories. World Scientific Publishing, New Jersey (2010)MATHCrossRef
5.
Bertoldi, K., Vitelli, V., Christensen, J., van Hecke, M.: Flexible mechanical metamaterials. Nat. Rev. Mater. 2(17066) (2017)
6.
dell’Isola, F., Seppecher, P., Spagnuolo, M., Barchiesi, E., Hild, F., Lekszycki, T., Giorgio, I., Placidi, L., Andreaus, U., Cuomo, M., Eugster, S.R., 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., Emek Abali, B., Liebold, C., Ganzosch, G., Harrison, P., Drobnicki, R., Igumnov, L., Alzahrani, F., Hayat, T.: Advances in pantographic structures: design, manufacturing, models, experiments and image analyses. Continuum Mech. Thermodyn. (2019). https://​doi.​org/​10.​1007/​s00161-019-00806-x CrossRef
7.
dell’Isola, F., Seppecher, P., Alibert, J.J., Lekszycki, T., Grygoruk, R., Pawlikowski, M., Steigmann, D.J., Giorgio, I., Andreaus, U., Turco, E., Gołaszewski, M., Rizzi, N., Boutin, C., Eremeyev, V.A., 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, J.-F., Muller, W., Ganzosch, G., Spagnuolo, M., Pfaff, A., Barcz, K., Hoschke, K., Neggers, J., Hild, F.: Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Continuum Mech. Thermodyn. 31(4), 851–884 (2019)ADSMathSciNetCrossRef
8.
Barchiesi, E., Spagnuolo, M., Placidi, L.: Mechanical metamaterials: a state of the art. Math. Mech. Solids 24(1), 212–234 (2019)MathSciNetMATHCrossRef
9.
Golaszewski, R., Grygoruk, M., Giorgio, I., Laudato, M., di Cosmo, F.: Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions. Continuum Mech. Thermodyn. 31(4), 1015–1034 (2019)ADSCrossRef
10.
Turco, E., Barchiesi, E.: Equilibrium paths of Hencky pantographic beams in three-point bending problem. Math. Mech. Complex Syst. (submitted):1–20 (2019)
11.
Eremeyev, V.A., Turco, E.: Enriched buckling for beam-lattice metamaterials. Mech. Res. Commun. (submitted):1–7 (2019)
12.
dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.L.: Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 472(2185), 1–23 (2016)CrossRef
13.
Filipov, E.T., Tachi, T., Paulino, G.H.: Origami tubes assembled into stiff, yet reconfigurable structures and metamaterials. PNAS; Proc. Natl. Acad. Sci. 112(40), 12321–12326 (2015)ADSCrossRef
14.
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)MathSciNetMATHCrossRef
15.
Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für Angewandte Mathematik und Physik 67(85), 1–28 (2016)MathSciNetMATH
16.
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. Int. J. Solids Struct. 147, 94–109 (2018)CrossRef
17.
Riks, E.: The application of Newton’s method to the problem of elastic stability. Journal of Applied Mechanics, Transactions ASME 39 Ser E(4), 1060–1065 (1972)ADSMATHCrossRef
18.
Wriggers, P.: Nonlinear Finite Element Methods. Springer, Heidelberg (2008)MATH
19.
Clarke, M.J., Hancock, G.J.: A study of incremental-iterative strategies for non-linear analyses. Int. J. Numer. Meth. Eng. 29, 1365–1391 (1990)CrossRef
20.
Seppecher, P., Alibert, J.-J., dell’Isola, F.: Linear elastic trusses leading to continua with exotic mechanical interactions. In: Journal of Physics: Conference Series,Vol. 319(1), p. 012018. IOP Publishing (2011)
21.
Misra, A., Lekszycki, T., Giorgio, I., Ganzoch, G., Müller, W.H., dell’Isola, F.: Pantographic metamaterials show atypical Poynting effect reversal. Mech. Res. Commun. 89, 6–10 (2018)CrossRef
22.
De Angelo, M., Barchiesi, E., Giorgio, I., Emek Abali, B.: Numerical identification of constitutive parameters in reduced order bi-dimensional models for pantographic structures: application to out-of-plane buckling. Arch. Appl. Mech. 89(7), 1333–1358 (2019)CrossRef
23.
Eugster, S.R., dell’Isola, F.: Exegesis of the Introduction and Sect. I from “Fundamentals of the Mechanics of Continua” by E. Hellinger. ZAMM—J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 97(4), 477–506 (2017)
24.
Eugster, S.R., dell’Isola, F.: Exegesis of Sect. II and III. A from ”Fundamentals of the Mechanics of Continua ” by E. Hellinger. ZAMM—J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 98(1), 31–68 (2018)
25.
Eugster, S.R., dell’Isola, F.: Exegesis of Sect. III. B from “Fundamentals of the Mechanics of Continua” by E. Hellinger. ZAMM—J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 98(1), 69–105 (2018)
26.
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)MathSciNetMATHCrossRef
27.
Giorgio, I., dell’Isola, F., Steigmann, D.J.: Axisymmetric deformations of a 2nd grade elastic cylinder. Mech. Res. Commun. 94, 45–48 (2018)CrossRef
28.
Giorgio, I., dell’Isola, F., Steigmann, D.J.: Edge effects in Hypar nets. Comptes Rendus Mécanique 347, 114–123 (2019)ADSCrossRef
29.
Barchiesi, E., Ganzosch, G., Liebold, C., Placidi, L., Grygoruk, R., Müller, W.H.: Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation. Continuum Mech. Thermodyn. 31, 33–45 (2019)ADSMathSciNetCrossRef
30.
Placidi, L., dell’Isola, F., Barchiesi, E.: Mechanics of Fibrous Materials and Applications, chapter Heuristic homogenization of Euler and pantographic beams, pp. 123–155. Springer, Cham (2020)
31.
Barchiesi, E., Eugster, S.R., Placidi, L., dell’Isola, F.: Pantographic beam: a complete second gradient 1D-continuum in plane. Zeitschrift für angewandte Mathematik und Physik 70(135) (2019)
32.
Abdoul-Anziz, H., Seppecher, P.: Strain gradient and generalized continua obtained by homogenizing frame lattices. Math. Mech. Complex Syst. 6(3), 213–250 (2018)MathSciNetMATHCrossRef
33.
34.
Andreaus, U., Spagnuolo, M., Lekszycki, T., Eugster, S.R.: A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler–Bernoulli beams. Continuum Mech Thermodyn 1–21 (2018)
35.
Turco, E.: Discrete is it enough? The revival of Piola–Hencky keynotes to analyze three-dimensional Elastica. Continuum Mech. Thermodyn. 30(5), 1039–1057 (2018)ADSMathSciNetMATHCrossRef
36.
Greco, L., Cuomo, M., Contrafatto, L.: A reconstructed local B formulation for isogeometric Kirchhoff–Love shells. Comput. Method Appl. Mech. Eng. 332, 462–487 (2018)ADSMathSciNetCrossRef
37.
Greco, L., Cuomo, M., Contrafatto, L.: A quadrilateral G1-conforming finite element for the Kirchhoff plate model. Comput. Method Appl. Mech. Eng. 346, 913–951 (2019)ADSCrossRef
38.
Greco, L., Cuomo, M., Contrafatto, L.: Two new triangular G1-conforming finite elements with cubic edge rotation for the analysis of Kirchhoff plates. Comput. Method Appl. Mech. Eng. 356, 354–386 (2019)ADSCrossRef
39.
Ogden, R.W.: Non-linear Elastic Deformations. Dover, Mineola (1997)
40.
41.
Eremeyev, V.A., Zubov, L.M.: On the stability of elastic bodies with couple stresses. Mech. Solids 29(3), 172–181 (1994)
42.
Sheydakov, D.N., Altenbach, H.: Stability of inhomogeneous micropolar cylindrical tube subject to combined loads. Math. Mech. Solids 21(9), 1082–1094 (2016)MathSciNetMATHCrossRef
43.
Lakes, R.S.: Stability of Cosserat solids: size effects, ellipticity and waves. J. Mech. Mater. Struct. 13(1), 83–91 (2018)MathSciNetCrossRef
44.
45.
Balobanov, V., Niiranen, J.: Locking-free variational formulations and isogeometric analysis for the Timoshenko beam models of strain gradient and classical elasticity. Comput. Methods Appl. Mech. Eng. 339, 137–159 (2018)ADSMathSciNetCrossRef
46.
Misra, A., Poorsolhjouy, P.: Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math. Mech. Complex Syst. 3(3), 285–308 (2015)MathSciNetMATHCrossRef
47.
Pideri, C., Seppecher, P.: A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Continuum Mech. Thermodyn. 9(5), 241–257 (1997)ADSMathSciNetMATHCrossRef
48.
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)MathSciNetMATHCrossRef
49.
Niiranen, J., Balobanov, V., Kiendl, J., Hosseini, S.B.: Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models. Math. Mech. Solids 24(1), 312–335 (2019)MathSciNetMATHCrossRef
50.
Giorgio, I., Del Vescovo, D.: Energy-based trajectory tracking and vibration control for multi-link highly flexible manipulators. Math. Mech. Complex Syst. 7(2), 159–174 (2019)MathSciNetMATHCrossRef
51.
Giorgio, I., Del Vescovo, D.: Non-linear lumped-parameter modeling of planar multi-link manipulators with highly flexible arms. Robotics 7(4), 60 (2018) CrossRef
52.
Giorgio, I., Della Corte, A., dell’Isola, F.: Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn. 88(1), 21–31 (2017)CrossRef
53.
Chróścielewski, J., Schmidt, R., Eremeyev, V.A.: Nonlinear finite element modeling of vibration control of plane rod-type structural members with integrated piezoelectric patches. Continuum Mech. Thermodyn. 31(1), 147–188 (2019)ADSMathSciNetCrossRef
Metadaten
Titel
Numerically driven tuning of equilibrium paths for pantographic beams
verfasst von
Emilio Turco
Publikationsdatum
22.10.2019
Verlag
Springer Berlin Heidelberg
Erschienen in
Continuum Mechanics and Thermodynamics / Ausgabe 6/2019
Print ISSN: 0935-1175
Elektronische ISSN: 1432-0959
DOI
https://doi.org/10.1007/s00161-019-00840-9

Weitere Artikel der Ausgabe 6/2019

Continuum Mechanics and Thermodynamics 6/2019 Zur Ausgabe

Original Article

Nonlinear travelling strain waves in a gradient-elastic medium

Original Article

Topology optimization of cracked structures using peridynamics

Original Article

Stability of elastic cylindrical tube with surface stresses

Original Article

On the correspondence between two- and three-dimensional Eshelby tensors

Original Article

On 3D theory of acoustic metamaterials with a triple-periodic system of interior obstacles

Original Article

Singularity-free defect mechanics for polar media

Premium Partner

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen. 

    Zur Marktübersicht
    Bildnachweise