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An improved dynamic model for a silicone material beam with large deformation

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Abstract

Dynamic modeling for incompressible hyperelastic materials with large deformation is an important issue in biomimetic applications. The previously proposed lower-order fully parameterized absolute nodal coordinate formulation (ANCF) beam element employs cubic interpolation in the longitudinal direction and linear interpolation in the transverse direction, whereas it cannot accurately describe the large bending deformation. On this account, a novel modeling method for studying the dynamic behavior of nonlinear materials is proposed in this paper. In this formulation, a higher-order beam element characterized by quadratic interpolation in the transverse directions is used in this investigation. Based on the Yeoh model and volumetric energy penalty function, the nonlinear elastic force matrices are derived within the ANCF framework. The feasibility and availability of the Yeoh model are verified through static experiment of nonlinear incompressible materials. Furthermore, dynamic simulation of a silicone cantilever beam under the gravity force is implemented to validate the superiority of the higher-order beam element. The simulation results obtained based on the Yeoh model by employing three different ANCF beam elements are compared with the result achieved from a commercial finite element package as the reference result. It is found that the results acquired utilizing a higher-order beam element are in good agreement with the reference results, while the results obtained using a lower-order beam element are different from the reference results. In addition, the stiffening problem caused by volumetric locking can be resolved effectively by applying a higher-order beam element. It is concluded that the proposed higher-order beam element formulation has satisfying accuracy in simulating dynamic motion process of the silicone beam.

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References

  1. Fung, Y.C.: Biomechanics: Mechanical Properties of Living Tissues. Springer, New York (1993)

    Book  Google Scholar 

  2. Shabana, A.A.: Definition of the slopes and the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 1, 339–348 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Shabana, A.A.: Computational Continuum Mechanics, 1st edn. Cambridge University Press, Cambridge (2008)

    Book  MATH  Google Scholar 

  4. Shabana, A.A., Schwertassek, R.: Equivalence of the floating frame of reference approach and finite element formulations. Int. J. Non-Linear Mech. 33, 417–432 (1998)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. Maqueda, L.G., Mohamed, 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, 021003 (2010)

    Article  Google Scholar 

  8. Kerkkanen, K.S., Sopanen, J.T., Mikkola, A.M.: A linear beam finite element based on the absolute nodal coordinate formulation. J. Mech. Des. 127, 621–630 (2005)

    Article  Google Scholar 

  9. Gerstmayr, J., Shabana, A.A.: Analysis of thin beams and cables using the absolute nodal coordinate formulation. Nonlinear Dyn. 45, 109–130 (2006)

    Article  MATH  Google Scholar 

  10. 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, 461–487 (2008)

    Article  Google Scholar 

  11. Gerstmayr, J., Sugiyama, H., Mikkola, A.: Review on the absolute nodal coordinate formulation for large deformation analysis of multibody systems. J. Comput. Nonlinear Dyn. 8, 031016 (2013)

    Article  Google Scholar 

  12. Jung, S., Park, T., Chung, W.: Dynamic analysis of rubber-like material using absolute nodal coordinate formulation based on the non-linear constitutive law. Nonlinear Dyn. 63, 149–157 (2011)

    Article  Google Scholar 

  13. Hussein, B.A., Sugiyama, H., Shabana, A.A.: Coupled deformation model in the large deformation finite-element analysis: problem definition. J. Comput. Nonlinear Dyn. 2, 146–154 (2007)

    Article  Google Scholar 

  14. Shabana, A.A., Yakoub, R.Y.: Three dimensional absolute nodal coordinate formulation for beam elements: theory. J. Mech. Des. 123, 606–613 (2001)

    Article  Google Scholar 

  15. Yakoub, R.Y., Shabana, A.A.: Three dimensional absolute nodal coordinate formulation for beam elements: implementation and applications. J. Mech. Des. 123, 614–621 (2001)

    Article  Google Scholar 

  16. Schwab, A.L., Meijaard, J.P.: Comparison of three-dimensional flexible beam elements for dynamic analysis: classical finite element formulation and absolute nodal coordinate formulation. J. Comput. 5, 011010 (2010)

    Google Scholar 

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

    Article  Google Scholar 

  18. Bonet, J., Wood, R.D.: Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  19. Yeoh, O.H.: Characterization of elastic properties of carbon black filled rubber vulcanizates. Rubber Chem. Technol. 66, 754–771 (1990)

    Article  Google Scholar 

  20. Selvadurai, A.P.S.: Deflections of a rubber membrane. J. Mech. Phys. Solids 54, 1093–1119 (2006)

    Article  MATH  Google Scholar 

  21. Bathe, K.J.: Finite Element Procedures. Prentice Hall, Upper Saddle River (1996)

    MATH  Google Scholar 

  22. Li, P., Gantoi, F.M., Shabana, A.A.: Higher order representation of the beam cross section deformation in large displacement finite element analysis. J. Sound Vib. 330, 6495–6508 (2011)

    Article  Google Scholar 

  23. Orzechowski, G., Shabana, A.A.: Analysis of warping deformation modes using higher order ANCF beam element. J. Sound Vib. 363, 428–445 (2016)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (11772186 and 11272203).

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Authors and Affiliations

  1. Department of Engineering Mechanics, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Qiping Xu & Jinyang Liu

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  1. Qiping Xu
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  2. Jinyang Liu
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Correspondence to Jinyang Liu.

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Xu, Q., Liu, J. An improved dynamic model for a silicone material beam with large deformation. Acta Mech. Sin. 34, 744–753 (2018). https://doi.org/10.1007/s10409-018-0759-y

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  • DOI: https://doi.org/10.1007/s10409-018-0759-y

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