Advertisement

Generalized strain space formulations and eigenprojection tensors

  • Klaus HeiduschkeEmail author
Original Paper
  • 32 Downloads

Abstract

The relations between the symmetric second-order tensors of deformation rate and of {qr}-generalized strain rate are given by transformations with fourth-order tensors, which are determined by the eigenprojection algorithm and summed up from functions of the distinct eigenvalues multiplied with dyadic products of the corresponding eigenprojections of the stretch tensors. These fourth-order transformation tensors also define the tensors of {qr}-generalized stress which are work-conjugate to the corresponding {qr}-generalized strain (rate). For finite deformations, every tensor pair of {qr}-generalized stress and strain constitutes a distinct material model and forms a unique element of the generalized strain space formulations. For \(q=r=0\), the tensors of logarithmic strain and stress emerge from the {qr}-generalized strains and stresses together with the corresponding fourth-order logarithmic transformation tensors. The resulting logarithmic strain space formulation has proven as most accurate, stable, and efficient by its implementation into the special-purpose finite element simulation tools AutoForm, Pafix, and Urmel.

Notes

Compliance with ethical standards

Financial disclosure

None reported.

Conflict of interest

The author declares no potential conflict of interest.

References

  1. 1.
    Anderheggen, E.: On the design of a new program for simulating thin sheet metal forming processes. In: FE-simulation of 3-D sheet metal forming processes in automotive industry, VDI-Bericht, vol. 894, pp. 231–245. Düsseldorf (1991)Google Scholar
  2. 2.
    Anderheggen, E., Ekchian, D., Heiduschke, K., Bartelt, P.: A contact algorithm for explicit dynamic FEM analysis. In: Brebbia, C.A., Aliabadi, M.H. (eds.) Contact Mechanics, Proceedings 1st International Conference, pp. 271–283. Southhampton, 13–15 July (1993)Google Scholar
  3. 3.
    Cardano, G.: Ars magna [Hieronymi Cardani, praestantissimi Mathematici, Philosophi, ac Medici, Artis magnae, sive de regulis Algebraicis, Lib.unus. Qui & totius operis de Arithmetica, quod opus perfectum inscripsit,est in ordine Decimus], Norimbergae per Ioh. Petreium excusum. Anno M. D. XLV, Nürnberg (1545). http://gateway-bayern.de/VD16+C+877
  4. 4.
    Cauchy, A.L.: Sur l’équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques. Bulletin de la Société philomatique, 300–304, Paris (1823). http://gallica.bnf.fr/ark:/12148/bpt6k901948/f308
  5. 5.
    Cauchy, A.L.: De la pression ou tension dans un corps solide. In: Exercices de mathématiques, vol. 2, pp. 42–56, Paris (1827). http://www.google.ch/books?id=NBMOAAAAQAAJ
  6. 6.
    Cauchy, A.L.: Sur la condensation et la dilatation des corps solides. In: Exercices de mathématiques, vol. 2, pp. 60–69, Paris (1827). http://www.google.ch/books?id=NBMOAAAAQAAJ
  7. 7.
    Darijani, H., Naghdabadi, R.: Constitutive modeling of solids at finite deformation using a second-order stress–strain relation. Int. J. Eng. Sci. 48, 223–236 (2010)CrossRefGoogle Scholar
  8. 8.
    Doyle, T.C., Ericksen, J.L.: Nonlinear elasticity. In: Kuerti, G. (ed.) Advances in Applied Mechanics, vol. 4, pp. 53–115. Academic Press, New York (1956) Google Scholar
  9. 9.
    Finger, J.: Über die gegenseitigen Beziehungen gewisser in der Mechanik mit Vortheil anwendbaren Flächen zweiter Ordnung nebst Anwendungen auf Probleme der Astatik. In: Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, mathematisch-naturwissenschaftliche Classe, Band 101, Abtheilung II a, 1105–1142 (1892). https://www.zobodat.at/pdf/SBAWW_101_2a_1105-1142.pdf
  10. 10.
    Flory, P.J.: Thermodynamic relations for high elastic materials. Trans. Farad. Soc. 57, 829–838 (1961)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Green, G.: On the propagation of light in crystallized media, Trans. Cambr. Philos. Soc. 7(Part II), 121–140 (1839). http://ia600300.us.archive.org/ 20/items/transactionsofca07camb/transactionsofca07camb.pdf
  12. 12.
    Green, A.E., Naghdi, P.M.: A general theory of an elastic–plastic continuum. Arch. Ration. Mech. Anal. 18, 251–281 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Heiduschke, K., Anderheggen, E., Reissner, J.: Constitutive equations for sheet metal forming. In: FE-simulation of 3-D sheet metal forming processes in automotive industry, VDI-Bericht 894, 17–37, Düsseldorf (1991)Google Scholar
  14. 14.
    Heiduschke, K.: The logarithmic strain space description. Int. J. Solids Struct. 32, 1047–1062 (1995)CrossRefzbMATHGoogle Scholar
  15. 15.
    Heiduschke, K.: Axisymmetric three- and four-node finite elements for large strain elasto-plasticity. Int. J. Numer. Methods Eng. 38, 2303–2324 (1995)CrossRefzbMATHGoogle Scholar
  16. 16.
    Heiduschke, K.: Computational aspects of the logarithmic strain space description. Int. J. Solids Struct. 33, 747–760 (1996)CrossRefzbMATHGoogle Scholar
  17. 17.
    Heiduschke, K.: An elastic isotropic, plastic orthotropic constitutive model based on deviator transformations. Int. J. Solids Struct. 34, 2339–2356 (1997)CrossRefzbMATHGoogle Scholar
  18. 18.
    Heiduschke, K.: Solder joint lifetime assessment of electronic devices. Int. J. Numer. Methods Eng. 41, 211–231 (1998)CrossRefzbMATHGoogle Scholar
  19. 19.
    Heiduschke, K.: On the multiplicative logarithmic strain space formulation. Tech. Mech. 38, 22–40 (2018).  https://doi.org/10.24352/UB.OVGU-2018-004 Google Scholar
  20. 20.
    Heiduschke, K.: A commutative-symmetrical multiplicative decomposition of left and right stretch tensors. Int. J. Solids Struct. 144–145, 59–65 (2018).  https://doi.org/10.1016/j.ijsolstr.2018.04.013 CrossRefGoogle Scholar
  21. 21.
    Hencky, H.: Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen, Zeitschrift für technische Physik 9, 215–220, 457 (1928)Google Scholar
  22. 22.
    Hill, R.: On constitutive inequalities for simple materials. J. Mech. Phys. Solids 16, 229–242 (1968)CrossRefzbMATHGoogle Scholar
  23. 23.
    Hoger, A.: The stress conjugate to logarithmic strain. Int. J. Solids Struct. 23, 1645–1656 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Itskov, M.: On the application of the additive decomposition of the generalized strain measures in large strain plasticity. Mech. Res. Commun. 31, 507–517 (2004)CrossRefzbMATHGoogle Scholar
  25. 25.
    Kernighan, B., Ritchie, D.: The C Programming Language, 2nd edn. Prentice Hall, Englewood Cliffs, NJ (1988). ISBN 0-13-110362-8zbMATHGoogle Scholar
  26. 26.
    Kirchhoff, G.R.: Über die Gleichungen des Gleichgewichtes eines elastischen Körpers bei nicht unendlich kleinen Verschiebungen seiner Theile, 762–773. In: Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, mathematisch-naturwissenschaftliche Classe 9 (1852). http://biodiversitylibrary.org/page/6482116
  27. 27.
    Kubli, W.: Prozessoptimierte implizite FEM-Formulierung für die Umformsimulation grossflächiger Blechbauteile, Diss. ETH Nr. 11175. In: Fortschrittberichte VDI Reihe 20, Nr. 204, Düsseldorf (1996)Google Scholar
  28. 28.
    Levi-Civita, T.: Lezioni di calcolo differenziale assoluto. In: Stock, A. (ed.), Roma (1925). http://mathematica.sns.it/opere/411/
  29. 29.
    Luehr, C.P., Rubin, M.B.: The significance of projection operators in the spectral representation of symmetric second order tensors. Comput. Methods Appl. Mech. Eng. 84, 243–246 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Macvean, D.B.: Die Elementararbeit in einem Kontinuum und die Zuordnung von Spannungs- und Verzerrungstensoren, Diss. ETH Nr. 4062. J. Appl. Math. Phys. [ZAMP] 19, 157–185 (1968)Google Scholar
  31. 31.
    Meyers, A., Xiao, H., Bruhns, O.T.: Elastic stress ratchetting and corotational stress rates. Tech. Mech. 23, 92–102 (2003)Google Scholar
  32. 32.
    Ogden, R.W.: Non-linear Elastic Deformations. Ellis Horwood Limited, Chichester (1984)zbMATHGoogle Scholar
  33. 33.
    Piola, G.: Nuova analisi per tutte le questioni della meccanica molecolare. In: Memorie di matematica e di fisica della Società Italiana delle Scienze, vol. 21, pp. 155–321, Modena (1825). http://books.google.ch/books?id=cv6ngCOYD7MC
  34. 34.
    Richter, H.: Zur Elastizitätstheorie endlicher Verformungen. Math. Nachr. 8, 65–73 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Sayir, M.: Zur Fließbedingung der Plastizitätstheorie. Ing. Arch. 39, 414–432 (1970)CrossRefzbMATHGoogle Scholar
  36. 36.
    Seth, B.R.: Generalized strain measure with applications to physical problems. In: Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics, Proceedings IUTAM 23–27 April 1962, Haifa, pp. 162–172. Pergamon Press, New York (1964)Google Scholar
  37. 37.
    Sylvester, J.J.: Sur l’équations en matrices px = xq. C. R. Acad. Sci. 99, 67–71, 115, 116 (1884)Google Scholar
  38. 38.
    Thomson, W., Tait, P.G.: Treatise on Natural Philosophy, Part I. Cambridge University Press, Cambridge (1879)zbMATHGoogle Scholar
  39. 39.
    Truesdell, C.A., Noll, W.: The non-linear field theories of mechanics. In: Flügge, S. (ed.) Handbuch der Physik III/3. Springer, Berlin (1965) + Corrigenda and addenda = Second edition (1992); Third edition by S.S. Antman (2004)Google Scholar
  40. 40.
    Voigt, W.: Lehrbuch der Kristallphysik. Teubner-Verlag, Leipzig (1910)zbMATHGoogle Scholar
  41. 41.
    Xiao, H., Bruhns, O.T., Meyers, A.: Thermodynamic laws and consistent Eulerian formulation of finite elastoplasticity with thermal effects. J. Mech. Phys. Solids 55, 338–365 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Alumnus of Institut für MechanikETH ZürichSchaffhausenSwitzerland

Personalised recommendations