Skip to main content
SpringerLink
Log in

Relative orientation constraints in the nonlinear large displacement analysis: application to soft materials

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

A new simple formulation of the orientation constraints for very flexible bodies is introduced in this investigation. This constraint formulation avoids using the tangent or cross-section frames previously used in the literature by directly using two position-gradient vectors to define the orientation constraint equations that eliminate the relative rotations between two bodies. The orientation constraints are commonly used to define clamped (rigid) joint between very flexible bodies, and rigid and less-flexible bodies. The very flexible bodies are modeled using the absolute nodal coordinate formulation (ANCF) which employs position gradients as nodal coordinates, while other rigid and less-flexible bodies have kinematics described in terms of orientation parameters. The less-flexible bodies can be modeled using the floating frame of reference formulation. Conventional clamped-end conditions eliminate all rigid body and deformation degrees of freedom because of the low-order of the finite element interpolation. When using a higher interpolation order, distinction is made between fully- and partially-clamped joints; the former eliminates all the degrees of freedom, while the latter eliminates only the relative translations and rotations, allowing for local deformations at the joint definition point. The singularity problem that arises, in the formulation of the orientation constraints, as the result of using ANCF orientation vectors derived using one position-gradient vector is demonstrated using simple, but common, example of a cantilever beam. The applicability of Saint–Venant principle, which states that the effect of the load diminishes away from the point of application of the load, to constraint forces that eliminate degrees of freedom is examined. Numerical results obtained in this study demonstrate that elimination of degrees of freedom, which leads to different kinematic structures and deformation basis-vectors, can be more significant in the case of soft materials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24

Similar content being viewed by others

References

  1. Obrezkov, L.P., Matikainen, M.K., Harish, A.B.: A finite element for soft tissue deformation based on the absolute nodal coordinate formulation. Acta Mech. (2020). https://doi.org/10.1007/s00707-019-02607-4

    Article  MathSciNet  MATH  Google Scholar 

  2. Shabana, A.A., Zhang, D.: ANCF curvature continuity: application to soft and fluid materials. Nonlinear Dyn. 100, 1497–1517 (2020)

    Google Scholar 

  3. Fotland, G., Haskins, C., Rølvåg, T.: Trade study to select best alternative for cable and pulley simulation for cranes on offshore vessels. Syst. Eng. (2019). https://doi.org/10.1002/sys.21503

    Article  Google Scholar 

  4. Chen, Y., Zhang, D.G., Li, L.: Dynamic analysis of rotating curved beams by using absolute nodal coordinate formulation based on radial point interpolation method. Sound Vib. 441, 63–83 (2019)

    Google Scholar 

  5. Ebel, H., Matikainen, M.K., Hurskainen, V.V., Mikkola, A.: Higher-order beam elements based on the absolute nodal coordinate formulation for three-dimensional elasticity. Nonlinear Dyn. 88, 1075–1091 (2017)

    Google Scholar 

  6. Yang, S., Deng, Z., Sun, J., Zhao, Y., Jiang, S.: An improved variable-length beam element with a torsion effect based on the absolute nodal coordinate formulation. IMechE J. Multibody Dyn. 232, 69–83 (2018)

    Google Scholar 

  7. Dmitrochenko, O.N., Pogorelov, D.Y.: Generalization of plate finite elements for absolute nodal coordinate formulation. Multibody Syst. Dyn. 10, 17–43 (2003)

    MATH  Google Scholar 

  8. Gerstmayr, J., Irschik, H.: On the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach. J. Sound Vib. 318(3), 461–487 (2008)

    Google Scholar 

  9. Hu, W., Tian, Q., Hu, H.Y.: Dynamics simulation of the liquid-filled flexible multibody system via the absolute nodal coordinate formulation and SPH method. Nonlinear Dyn. 75, 653–671 (2014)

    MathSciNet  Google Scholar 

  10. Nachbagauer, K.: State of the art of ANCF elements regarding geometric description, interpolation strategies, definition of elastic forces, validation and locking phenomenon in comparison with proposed beam finite elements. Arch. Comput. Methods Eng. 21(3), 293–319 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Olshevskiy, A., Dmitrochenko, O., Kim, C.W.: Three-dimensional solid brick element using slopes in the absolute nodal coordinate formulation. ASME J. Comput. Nonlinear Dyn. 9(2), 1–10 (2014)

    Google Scholar 

  12. Orzechowski, G., Fraczek, J.: Nearly incompressible nonlinear material models in the large deformation analysis of beams using ANCF. Nonlinear Dyn. 82(1), 451–464 (2015)

    MathSciNet  Google Scholar 

  13. Tian, Q., Chen, L.P., Zhang, Y.Q., Yang, J.Z.: An efficient hybrid method for multibody dynamics simulation based on absolute nodal coordinate formulation. ASME J. Comput. Nonlinear Dyn. 4, 021009-1–021009-14 (2009)

    Google Scholar 

  14. Tian, Q., Sun, Y.L., Liu, C., Hu, H.Y., Paulo, F.: Elasto-hydro-dynamic lubricated cylindrical joints for rigid-flexible multibody dynamics. Comput. Struct. 114–115, 106–120 (2013)

    Google Scholar 

  15. Shen, Z., Li, P., Liu, C., Hu, G.: A finite element beam model including cross-section distortion in the absolute nodal coordinate formulation. Nonlinear Dyn. 77(3), 1019–1033 (2014)

    Google Scholar 

  16. Shabana, A.A.: Computational Continuum Mechanics, 3rd edn. Cambridge University Press, Cambridge (2018)

    MATH  Google Scholar 

  17. Hussein, B.A., Weed, D., Shabana, A.A.: Clamped end conditions and cross section deformation in the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 21(4), 375–393 (2009)

    MATH  Google Scholar 

  18. Gantoi, F.M., Brown, M.A., Shabana, A.A.: ANCF finite element/multibody system formulation of the ligament/bone insertion site constraints. ASME J. Comput. Nonlinear Dyn. 5(3), 031006-1–031006-9 (2010)

    Google Scholar 

  19. Goldstein, H.: Classical Mechanics. Addison-Wesley, Boston (1950)

    MATH  Google Scholar 

  20. Greenwood, D.T.: Principles of Dynamics. Prentice Hall, Upper Saddle River (1988)

    Google Scholar 

  21. Roberson, R.E., Schwertassek, R.: Dynamics of Multibody Systems. Springrt, Berlin (1988)

    MATH  Google Scholar 

  22. Shabana, A.A.: Dynamics of Multibody Systems, 5th edn. Cambridge University Press, Cambridge (2020)

    MATH  Google Scholar 

  23. Patel, M., Shabana, A.A.: Locking alleviation in the large displacement analysis of beam elements: the strain split method. Acta Mech. 229(7), 2923–2946 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Calisti, M., Giorelli, M., Levy, G., Mazzolai, G., Hochner, B., Laschi, C., Dario, P.: An octopus-bioinspired solution to movement and manipulation for soft robots. Bioinspir. Biomim. 6(3), 036002 (2011)

    Google Scholar 

  25. Ma, L., Wei, C., Zhao, Y.: Modeling and verification of a RANCF fluid element based on cubic rational bezier volume. ASME J. Comput. Nonlinear Dyn. 15, 041005 (2020)

    Google Scholar 

  26. Jing, L., Li, K., Yang, H., Chen, P.Y.: Recent advances in integration of 2D materials with soft matter for multifunctional robotic materials. Mater. Horiz. 7(1), 54–70 (2020)

    Google Scholar 

  27. Maqueda, L.G., Shabana, A.A.: Poisson modes and general nonlinear constitutive models in the large displacement analysis of beams. J. Multibody Syst. Dyn. 18(3), 375–396 (2007)

    MATH  Google Scholar 

Download references

Acknowledgments

This research was supported by the National Science Foundation (Projects # 1632302 and 1852510).

Author information

Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Ahmed A. Shabana & Ahmed E. Eldeeb

Authors
  1. Ahmed A. Shabana
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ahmed E. Eldeeb
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmed A. Shabana.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shabana, A.A., Eldeeb, A.E. Relative orientation constraints in the nonlinear large displacement analysis: application to soft materials. Nonlinear Dyn 101, 2551–2575 (2020). https://doi.org/10.1007/s11071-020-05839-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-05839-5

Keywords

Search

Navigation