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
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
MTZ-Fachkongress

Antriebe und Energiesysteme von morgen


Austausch von Automobil- und Energiewirtschaft

Am 14./15. Mai geht es in Chemnitz um die Mobilitätswende durch Elektro- und Wasserstofffahrzeuge.
Erweiterte Suche
Anmelden
nach oben
Journal of Elasticity
Erschienen in: Journal of Elasticity 1/2021

26.08.2021

An Intrinsic Geometric Formulation of Hyper-Elasticity, Pressure Potential and Non-Holonomic Constraints

verfasst von: B. Kolev, R. Desmorat

Erschienen in: Journal of Elasticity | Ausgabe 1/2021

Einloggen

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

search-config
loading …

Abstract

Isotropic hyper-elasticity, altogether with the equilibrium equations and the usual boundary conditions, are formulated directly on the body ℬ, a three-dimensional compact and orientable manifold with boundary equipped with a mass measure. Pearson–Sewell–Beatty pressure potential on the boundary is recovered, using the Poincaré formula. The existence of such a potential requires conditions, which are formulated as non-holonomic constraints on the configuration space.
Vorheriger Artikel Large-Amplitude Elastic Free-Surface Waves: Geometric Nonlinearity and Peakons
Nächster Artikel On Hashin’s Hollow Cylinder and Sphere Assemblages in Anisotropic Nonlinear Elasticity

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
Anhänge
Nur mit Berechtigung zugänglich
Fußnoten
1
The term intrinsic can have additional meanings in geometry: for instance “coordinate free” or, in surface theory, “depending only on the metric and not on the embedding”.
 
Literatur
1.
Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications, 2nd edn. Applied Mathematical Sciences., vol. 75. Springer, New York (1988) MATHCrossRef
2.
Arnold, V.I.: On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid. Dokl. Akad. Nauk SSSR 162, 975–978 (1965) MathSciNet
3.
Arnold, V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16(fasc(1), 319–361 (1966) MATHCrossRef
4.
Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63(4), 337–403 (1976/77)
5.
Beatty, M.: Stability of hyperelastic bodies subject to hydrostatic loading. Non-Linear Mech. 5, 367–383 (1970) MATHCrossRef
6.
Bertram, A.: Elasticity and Plasticity of Large Deformations, 3rd edn. Springer Berlin Heidelberg, Berlin (2012). An introduction MATHCrossRef
7.
Cartan, H.: Differential Forms. Translated from the French. Houghton Mifflin Co., Boston, Mass (1970) MATH
8.
Ciarlet, P.G.: Mathematical Elasticity. Vol. I. Studies in Mathematics and Its Applications., vol. 20. North-Holland, Amsterdam (1988). Three-dimensional elasticity MATH
9.
Clarke, B.: The metric geometry of the manifold of Riemannian metrics over a closed manifold. Calc. Var. Partial Differ. Equ. 39(3–4), 533–545 (2010) MathSciNetMATHCrossRef
10.
Dimitrienko, Y.I.: Nonlinear Continuum Mechanics and Large Inelastic Deformations. Solid Mechanics and Its Applications, vol. 174. Springer, Netherlands (2011) MATH
11.
Doll, S., Schweizerhof, K.: On the development of volumetric strain energy functions. J. Appl. Mech. 67(1), 17–21 (1999) MATHCrossRef
12.
13.
Ebin, D.G., Marsden, J.E.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. (2) 92, 102–163 (1970) MathSciNetMATHCrossRef
14.
Einstein, A.: The Meaning of Relativity. Princeton University Press, Princeton (1988) MATH
15.
16.
Epstein, M., Segev, R.: Differentiable manifolds and the principle of virtual work in continuum mechanics. J. Math. Phys. 21(5), 1243–1245 (1980) ADSMathSciNetMATHCrossRef
17.
Eringen, A.C.: Nonlinear Theory of Continuous Media. McGraw-Hill, New York-Toronto-London (1962)
18.
Fiala, Z.: Time derivative obtained by applying the Riemannian manifold of Riemannian metrics to kinematics of continua. C. R. Mecanique 332, 97–102 (2004) ADSMATHCrossRef
19.
20.
Fiala, Z.: Geometry of finite deformations and time-incremental analysis. Int. J. Non-Linear Mech. 81, 230–244 (2016) ADSCrossRef
21.
Freed, D.S., Groisser, D.: The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group. Mich. Math. J. 36(3), 323–344 (1989) MathSciNetMATHCrossRef
22.
Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry, Universitext 3rd edn. Springer Berlin Heidelberg, Berlin (2004) MATHCrossRef
23.
Gil-Medrano, O., Michor, P.W.: The Riemannian manifold of all Riemannian metrics. Q. J. Math. Oxf. Ser. (2) 42(166), 183–202 (1991) MathSciNetMATHCrossRef
24.
Green, A.E., Zerna, W.: Theoretical Elasticity, 2nd edn. Clarendon Press, Oxford (1968) MATH
25.
Grubic, N., LeFloch, P.G., Mardare, C.: The equations of elastostatics in a Riemannian manifold. J. Math. Pures Appl. 102(6), 1121–1163 (2014) MathSciNetMATHCrossRef
26.
27.
Hart-Smith, L.: Elasticity parameters for finite deformations of rubber-like materials. J. Appl. Phys. 17, 608–625 (1966)
28.
Haupt, P.: Continuum Mechanics and Theory of Materials, second edn. Springer, Berlin (2002). Traduction de la quatrième édition allemande par, Juvet, G. and Leroy, R. MATHCrossRef
29.
30.
31.
Inci, H., Kappeler, T., Topalov, P.: On the Regularity of the Composition of Diffeomorphisms, 1st edn. Memoirs of the American Mathematical Society., vol. 226. Am. Math. Soc., Providence (2013) MATH
32.
Jiménez, V.M., de León, M., Epstein, M.: Characteristic distribution: an application to material bodies. J. Geom. Phys. 127, 19–31 (2018) ADSMathSciNetMATHCrossRef
33.
34.
Kriegl, A., Michor, P.W.: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, vol. 53. Am. Math. Soc., Providence (1997) MATH
35.
Kupferman, R., Olami, E., Segev, R.: Continuum dynamics on manifolds: application to elasticity of residually-stressed bodies. J. Elast. 128(1), 61–84 (2017) MathSciNetMATHCrossRef
36.
Lang, S.: Fundamentals of Differential Geometry. Graduate Texts in Mathematics, vol. 191. Springer, New York (1999) MATH
37.
Tensor-distributions, A.L.: In: Magnetohydrodynamics: Waves and Shock Waves in Curved Space-Time, pp. 1–17. Springer, Netherlands (1994)
38.
Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover, New York (1994). Corrected reprint of the 1983 original MATH
39.
Milnor, J.: Remarks on infinite-dimensional Lie groups. In: Relativity, Groups and Topology, II, Les Houches, 1983, pp. 1007–1057. North-Holland, Amsterdam (1984)
40.
Nanson, E.J.: Note on hydrodynamics. Messenger Math. 7, 182–185 (1878)
41.
42.
43.
Noll, W.: A General Framework for Problems in the Statics of Finite Elasticity. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proceedings of the International Symposium on Continuum Mechanics and Partial Differential Equations, pp. 363–387. Elsevier, Amsterdam (1978)
44.
Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Graduate Texts in Mathematics, vol. 107. Springer, New York (1993) MATHCrossRef
45.
46.
Penn, R.W.: Volume changes accompanying the extension of rubber. Trans. Soc. Rheol. 14(4), 509–517 (1970) CrossRef
47.
Podio-Guidugli, P., Caffarelli, G.V.: Surface interaction potentials in elasticity. In: Mechanics and Thermodynamics of Continua, pp. 345–385. Springer Berlin Heidelberg, Berlin (1991) CrossRef
48.
Podio-Guidugli, P., Vianello, M.: Constraint manifolds for isotropic solids. Arch. Ration. Mech. Anal. 105(2), 105–121 (1989) MathSciNetMATHCrossRef
49.
Rougée, P.: Formulation lagrangienne intrinsèque en mécanique des milieux continus. J. Méc. 19, 7–32 (1980) MathSciNetMATH
50.
Rougée, P.: The intrinsic Lagrangian metric and stress variables. In: Finite Inelastic Deformations - Theory and Applications, IUTAM Symposium Hannover/Germany 199, pp. 217–226. (1991)
51.
Rougée, P.: Mécanique des Grandes Transformations. Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 25, Springer-Verlag, Berlin (1997)
52.
Rougée, P.: An intrinsic Lagrangian statement of constitutive laws in large strain. Comput. Struct. 84(17–18), 1125–1133 (2006) MathSciNetCrossRef
53.
Rudin, W.: Functional Analysis, 2nd edn. International Series in Pure and Applied Mathematics. McGraw-Hill Inc., New York (1991) MATH
54.
55.
Segev, R., Epstein, M.: Geometric Continuum Mechanics. ACM. Springer International Publishing, vol. 42. Birkhäuser, Basel (2020) MATHCrossRef
56.
57.
58.
Simo, J.C., Marsden, J.E.: Stress tensors, Riemannian metrics and the alternative descriptions in elasticity. In: Trends and Applications of Pure Mathematics to Mechanics, Palaiseau, 1983. Lecture Notes in Phys., vol. 195, pp. 369–383. Springer, Berlin (1984) CrossRef
59.
Souriau, J.-M.: Géométrie et relativité. Enseignement des Sciences, VI. Hermann, Paris (1964)
60.
Steinmann, P.: Geometrical Foundations of Continuum Mechanics. Lecture Notes in Applied Mathematics and Mechanics., vol. 2. Springer, Heidelberg (2015). An application to first- and second- order elasticity and elasto-plasticity MATH
61.
Stumpf, H., Hoppe, U.: The application of tensor algebra on manifolds to nonlinear continuum mechanics—invited survey article. Z. Angew. Math. Mech. 77(5), 327–339 (1997) MathSciNetMATHCrossRef
62.
Truesdell, C., Noll, W.: The non-linear field theories of mechanics. In: Handbuch der Physik, Band III/3, pp. 1–602. Springer, Berlin (1965)
63.
64.
Wang, C.C., Truesdell, C.: Introduction to Rational Elasticity. Noordhoff International Publishing, Leyden (1973). Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics of Continua MATH
65.
Metadaten
Titel
An Intrinsic Geometric Formulation of Hyper-Elasticity, Pressure Potential and Non-Holonomic Constraints
verfasst von
B. Kolev
R. Desmorat
Publikationsdatum
26.08.2021
Verlag
Springer Netherlands
Erschienen in
Journal of Elasticity / Ausgabe 1/2021
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI
https://doi.org/10.1007/s10659-021-09853-5

Weitere Artikel der Ausgabe 1/2021

Journal of Elasticity 1/2021 Zur Ausgabe

OriginalPaper

Large-Amplitude Elastic Free-Surface Waves: Geometric Nonlinearity and Peakons

OriginalPaper

A Constrained Cosserat Shell Model up to Order : Modelling, Existence of Minimizers, Relations to Classical Shell Models and Scaling Invariance of the Bending Tensor

OriginalPaper

On Hashin’s Hollow Cylinder and Sphere Assemblages in Anisotropic Nonlinear Elasticity

OriginalPaper

Effective Model for Elastic Waves in a Substrate Supporting an Array of Plates/Beams with Flexural and Longitudinal Resonances

OriginalPaper

Representation of Solutions of Lamé–Navier System

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