Nonlinear Dynamics

, Volume 73, Issue 1–2, pp 183–198 | Cite as

A thick anisotropic plate element in the framework of an absolute nodal coordinate formulation

  • M. Langerholc
  • J. Slavič
  • M. Boltežar
Original Paper


In this research, the incorporation of material anisotropy is proposed for the large-deformation analyses of highly flexible dynamical systems. The anisotropic effects are studied in terms of a generalized elastic forces (GEFs) derivation for a continuum-based, thick, and fully parameterized absolute nodal coordinate formulation plate element, of which the membrane and bending deformation effects are coupled. The GEFs are first derived for a fully anisotropic, linearly elastic material, characterized by 21 independent material parameters. Using the same approach, the GEFs are obtained for an orthotropic material, characterized by nine material parameters. Furthermore, the analysis is extended to the case of nonlinear elasticity; the GEFs are introduced for a nonlinear Cauchy-elastic material, characterized by four in-plane orthotropic material parameters. Numerical simulations are performed to validate the theory for statics and dynamics and to observe the anisotropic responses in terms of displacements, stresses, and strains. The presented formulations are suitable for studying the nonlinear dynamical behavior of advanced elastic materials of an arbitrary degree of anisotropy.


Absolute nodal coordinate formulation Thick plate Anisotropy Nonlinear elasticity Nonlinear dynamics Large deformation 



The operation was partially financed by the European Union, European Social Fund.


  1. 1.
    Abbas, L.K., Rui, X., Hammoudi, Z.S.: Plate/shell element of variable thickness based on the absolute nodal coordinate formulation. Nonlinear Dyn. 224, 127–141 (2010) Google Scholar
  2. 2.
    Abbas, L., Rui, X., Marzocca, P.: Panel flutter analysis of plate element based on the absolute nodal coordinate formulation. Multibody Syst. Dyn. 27, 135–152 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Avril, S., Bonnet, M., Bretelle, A., Grédiac, M., Hild, F., Ienny, P., Latourte, F., Lemosse, D., Pagano, S., Pagnacco, E., Pierron, F.: Overview of identification methods of mechanical parameters based on full-field measurements. Exp. Mech. 48, 381–402 (2008) CrossRefGoogle Scholar
  4. 4.
    Bronstein, I.N., Semendyayev, M.G.: Handbook of Mathematics, 4th edn. Springer, Berlin (2003) Google Scholar
  5. 5.
    Burden, R.L., Faires, D.J.: Numerical Analysis, 9th edn. Brooks Cole, California (2010) Google Scholar
  6. 6.
    Čepon, G., Boltežar, M.: Dynamics of a belt-drive system using a linear complementarity problem for the belt-pulley contact description. J. Sound Vib. 319(3–5), 1019–1035 (2009) Google Scholar
  7. 7.
    Čepon, G., Manin, L., Boltežar, M.: Introduction of damping into the flexible multibody belt-drive model: a numerical and experimental investigation. J. Sound Vib. 324(1–2), 283–296 (2009) Google Scholar
  8. 8.
    Cho, S.H., Kim, G., McCarthy, T.J., Farris, R.J.: Orthotropic elastic constants for polyimide film. Polym. Eng. Sci. 41(2), 301–307 (2001) CrossRefGoogle Scholar
  9. 9.
    Dmitrochenko, O.: Finite elements using absolute nodal coordinates for large-deformation flexible multibody dynamics. J. Comput. Appl. Math. 215(2), 368–377 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dmitrochenko, O., Mikkola, A.: A formal procedure and invariants of a transition from conventional finite elements to the absolute nodal coordinate formulation. Multibody Syst. Dyn. 22, 323–339 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dufva, K., Kerkkänen, K., Maqueda, L.G., Shabana, A.A.: Nonlinear dynamics of three-dimensional belt drives using the finite-element method. Nonlinear Dyn. 48(4), 449–466 (2007) zbMATHCrossRefGoogle Scholar
  12. 12.
    Hussein, B., Negrut, D., Shabana, A.A.: Implicit and explicit integration in the solution of the absolute nodal coordinate differential/algebraic equations. Nonlinear Dyn. 54, 283–296 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kerkkänen, K., GarcíaVallejo, D., Mikkola, A.: Modeling of belt-drives using a large deformation finite element formulation. Nonlinear Dyn. 43, 239–256 (2006) zbMATHCrossRefGoogle Scholar
  14. 14.
    Langerholc, M., Česnik, M., Slavič, J., Boltežar, M.: Experimental validation of a complex, large-scale, rigid-body mechanism. Eng. Struct. 36, 220–227 (2012) CrossRefGoogle Scholar
  15. 15.
    Langerholc, M., Slavič, J., Boltežar, M.: Absolute nodal coordinates in digital image correlation. Exp. Mech. (2012). doi: 10.1007/s11340-012-9691-4 Google Scholar
  16. 16.
    Maqueda, L.G., Mohamed, A.N.A., Shabana, A.A.: Use of general nonlinear material models in beam problems: application to belts and rubber chains. J. Comput. Nonlinear Dyn. 5(2), 021003 (2010) CrossRefGoogle Scholar
  17. 17.
    Maqueda, L.G., Shabana, A.A.: Poisson modes and general nonlinear constitutive models in the large displacement analysis of beams. Multibody Syst. Dyn. 18(3), 375–396 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Mikkola, A., Shabana, A.A.: A non-incremental finite element procedure for the analysis of large deformation of plates and shells in mechanical system applications. Multibody Syst. Dyn. 9, 283–309 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Nada, A., El-Assal, A.: Absolute nodal coordinate formulation of large-deformation piezoelectric laminated plates. Nonlinear Dyn. 67, 2441–2454 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Newnham, R.E.: Properties of Materials: Anisotropy, Symmetry, Structure. Oxford University Press, London (2005) Google Scholar
  21. 21.
    Pogorelov, D., Dmitrochenko, O.: Generalization of plate finite elements for absolute nodal coordinate formulation. Multibody Syst. Dyn. 10(1), 17–43 (2003) zbMATHCrossRefGoogle Scholar
  22. 22.
    Shabana, A.A.: Computational Dynamics, 3rd edn. Wiley, New York (2010) zbMATHCrossRefGoogle Scholar
  23. 23.
    Shabana, A.A.: Computational Continuum Mechanics. Cambridge University Press, Cambridge (2008) zbMATHCrossRefGoogle Scholar
  24. 24.
    Schwab, A.L., Gerstmayr, J., Meijaard, J.P.: Comparison of three-dimensional flexible thin plate elements for multibody dynamic analysis: finite element formulation and absolute nodal coordinate formulation. ASME Conf. Proc. 4806X, 1059–1070 (2007) Google Scholar
  25. 25.
    Sugiyama, H., Shabana, A.A.: Application of plasticity theory and absolute nodal coordinate formulation to flexible multibody system dynamics. J. Mech. Des. 126(3), 478–487 (2004) CrossRefGoogle Scholar
  26. 26.
    Sugiyama, H., Shabana, A.A.: On the use of implicit integration methods and the absolute nodal coordinate formulation in the analysis of elasto-plastic deformation problems. Nonlinear Dyn. 37, 245–270 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Tian, Q., Liu, C., Machado, M., Flores, P.: A new model for dry and lubricated cylindrical joints with clearance in spatial flexible multibody systems. Nonlinear Dyn. 64, 25–47 (2011) CrossRefGoogle Scholar
  28. 28.
    Yoo, P.S., Pogorelov, D., Dmitrochenko, O.: Large deflection analysis of a thin plate: computer simulations and experiments. Multibody Syst. Dyn. 11, 185–208 (2004) zbMATHCrossRefGoogle Scholar
  29. 29.
    Yu, L., Ren, G.T.: Integration of absolute nodal elements into multibody system. Nonlinear Dyn. 62, 931–943 (2010) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Knauf Insulation d.o.o.Central Engineering EuropeŠkofja LokaSlovenia
  2. 2.Faculty of mechanical engineeringLjubljanaSlovenia

Personalised recommendations