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Non-material Finite Elements for Spatial Deformations of Belts

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Contributions to Advanced Dynamics and Continuum Mechanics

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 114))

Abstract

We present a novel mixed Eulerian–Lagrangian beam finite element formulation. Large spatial deformations of shear-rigid, but extensible rods with natural curvature are considered. The three-dimensional deformation of a thin strip clamped at both ends is computed with this novel method and compared with semi-analytic solutions of the boundary value problem of the incremental rod theory as well as with the finite element solution for an equivalent shell model. Stability of the straight clamped beam in the absence of gravity is considered analytically for the sake of comparison and the critical value of the natural curvature is found. Finally, the contact problem of a belt spanned between two pulleys is discussed.

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References

  1. Antman, S.S.: Nonlinear Problems of Elasticity. Applied Mathematical Sciences. Springer, Berlin (1994)

    Google Scholar 

  2. Belyaev, A., Eliseev, V., Irschik, H., Oborin, E.: Contact of two equal rigid pulleys with a belt modelled as Cosserat nonlinear elastic rod. Acta Mechanica 228, 4425–4434 (2017)

    Article  MathSciNet  Google Scholar 

  3. Belyaev, A.K., Eliseev, V.V.: Flexible rod model for the rotation of a drill string in an arbitrary borehole. Acta Mechanica 229(2), 841–848 (2017)

    Article  MathSciNet  Google Scholar 

  4. Eliseev, V.: Mechanics of Deformable Solid Bodies. St Petersburg State Polytechnical University Publishing House, St Petersburg (2006)

    Google Scholar 

  5. Gruber, P.G., Nachbagauer, K., Vetyukov, Y., Gerstmayr, J.: A novel director-based Bernoulli-Euler beam finite element in absolute nodal coordinate formulation free of geometric singularities. Mech. Sci. 4(2), 279–289 (2013)

    Article  Google Scholar 

  6. Hong, D., Ren, G.: A modeling of sliding joint on one-dimensional flexible medium. Multibody Syst. Dyn. 26, 91–106 (2011)

    Article  MathSciNet  Google Scholar 

  7. Kong, L., Parker, R.: Steady mechanics of belt-pulley systems. ASME J. Appl. Mech. 72, 25–34 (2005)

    Article  ADS  Google Scholar 

  8. Liu, J., Cheng, Z., Ren, G.: An arbitrary Lagrangian Eulerian formulation of a geometrically exact Timoshenko beam running through a tube. Acta Mechanica 229, 3161–3188 (2018)

    Article  MathSciNet  Google Scholar 

  9. Simitses, G., Hodges, D.H.: Fundamentals of Structural Stability. Butterworth Heinemann, London (2005)

    MATH  Google Scholar 

  10. Vetyukov, Y.: Nonlinear Mechanics of Thin-Walled Structures, Asymptotics. Direct Approach and Numerical Analysis. Foundation of Engineering Mechanics. Springer, Vienna (2014)

    Book  Google Scholar 

  11. Vetyukov, Y.: Non-material finite element modelling of large vibrations of axially moving strings and beams. J. Sound Vib. 414, 299–317 (2018)

    Article  ADS  Google Scholar 

  12. Vetyukov, Y., Eliseev, V.V.: Modeling of building frames as spatial rod structures with geometric and physical nonlinearities. Comput. Cont. Mech. 3(3), 32–45 (2010)

    Article  Google Scholar 

  13. Vetyukov, Y., Schmidrathner, C.: A rod model for large bending and torsion of an elastic strip with a geometrical imperfection. Acta Mechanica (submit.) 1–15, (2019)

    Google Scholar 

  14. Vetyukov, Y., Gruber, P.G., Krommer, M.: Nonlinear model of an axially moving plate in a mixed Eulerian-Lagrangian framework. Acta Mechanica 227(10), 2831–2842 (2016)

    Article  MathSciNet  Google Scholar 

  15. Vetyukov, Y., Gruber, P.G., Krommer, M., Gerstmayr, J., Gafur, I., Winter, G.: Mixed eulerian-lagrangian description in materials processing: deformation of a metal sheet in a rolling mill. Int. J. Numer. Methods Eng. 109(10), 1371–1390 (2016)

    Article  MathSciNet  Google Scholar 

  16. Vetyukov, Y., Oborin, E., Scheidl, J., Krommer, M., Schmidrathner, C.: Flexible belt hanging on two pulleys: contact problem at non-material kinematic description. Int. J. Solids Struct. (submit.)

    Google Scholar 

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Acknowledgements

The authors wish to thank for the support of the Austrian Research Promotion Agency (FFG), project number 861493.

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Authors and Affiliations

  1. Vienna University of Technology, Getreidemarkt 9, 1060, Vienna, Austria

    Christian Schmidrathner & Yury Vetyukov

Authors
  1. Christian Schmidrathner
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  2. Yury Vetyukov
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Corresponding author

Correspondence to Yury Vetyukov .

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Editors and Affiliations

  1. Institut für Mechanik, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

    Holm Altenbach

  2. Institut für Technische Mechanik, Johannes Kepler Universität Linz, Linz, Austria

    Hans Irschik

  3. Institute of Continuous Media Mechanics, Russian Academy of Sciences, Perm, Russia

    Valery P. Matveenko

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Schmidrathner, C., Vetyukov, Y. (2019). Non-material Finite Elements for Spatial Deformations of Belts. In: Altenbach, H., Irschik, H., Matveenko, V. (eds) Contributions to Advanced Dynamics and Continuum Mechanics. Advanced Structured Materials, vol 114. Springer, Cham. https://doi.org/10.1007/978-3-030-21251-3_13

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