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An iso-parametric \(\pmb {\mathrm {G}^1}\)-conforming finite element for the nonlinear analysis of Kirchhoff rod. Part I: the 2D case

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Abstract

A geometrically exact nonlinear iso-parametric \(\mathrm {G}^1\)-conforming finite element formulation for the analysis of Kirchhoff rods, based on the cubic Bézier curve interpolation, is presented. In this work, the formulation is restricted to the planar 2D case. Introducing the \(\mathrm {G}^1\)-map, the interpolation preserves the continuity requirement during the deformation process of the rod. In this way, the \(\mathrm {G}^1\)-conformity is implicitly accounted at the element formulation level.

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References

  1. 1.

    Cottrell, A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley (2009)

  2. 2.

    Greco, L., Cuomo, M.: B-spline interpolation of Kirchhoff–Love space rods. Comput. Methods Appl. Mech. Eng. 256, 251–269 (2013)

  3. 3.

    Greco, L., Cuomo, M.: An implicit \(G^1\) multi patch B-spline interpolation for Kirchhoff–Love space rod. Comput. Methods Appl. Mech. Eng. 269, 173–197 (2014)

  4. 4.

    Bauer, A.M., Breitenberger, M., Philipp, B., Wüchner, R., Blatzinger, K.-U.: Nonlinear isogeometric spatial Bernoulli beam. Comput. Methods Appl. Mech. Eng. 303, 101–127 (2016)

  5. 5.

    Gerald, F.: Curves and Surfaces for CAGD: A Practical Guide. The Morgan Kaufmann Series in Computer Graphics. Morgan Kaufmann, 5 edn (2001)

  6. 6.

    Barsky, B.A., DeRose, T.D.: Geometric continuity of parametric curves: three equivalent characterizations. IEEE Comput. Graph. Appl. 9(6), 60–68 (1989)

  7. 7.

    Hohmeyer, M.E., Barsky, B.A.: Rational continuity: parametric, geometric, and Frenet frame continuity of rational curves. ACM Trans. Graph. 8(4), 335–359 (1989)

  8. 8.

    Armero, F., Valverde., J.: Invariant Hermitian finite element for thin Kirchhoff rods. I: the linear plane case. Comput. Methods Appl. Mech. Eng. 213–216, 427–457 (2012)

  9. 9.

    Generalized Continua from the Theory to Engineering Applications: volume 541 of CISM International Centre for Mechanical Sciences (Courses and Lectures), chapter Cosserat-Type Rods. Springer, Vienna (2013)

  10. 10.

    Altenbach, H., Bîrsan, M., Eremeyev, V.A.: On a thermodynamic theory of rods with two temperature fields. Acta Mech. 223(8), 1583–1596 (2012)

  11. 11.

    Simo, J.C., Vu-Quoc, L.: A three-dimensional finite-strain rod model. Part II: computational aspects. Comput. Methods Appl. Mech. Eng. 58, 79–116 (1986)

  12. 12.

    Simo, J.C.: A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput. Methods Appl. Mech. Eng. 49, 55–70 (1985)

  13. 13.

    Bouclier, R., Elguedj, T., Coumbescure, A.: Locking free isogeometric formulations of curved thick beams. Comput. Methods Appl. Mech. Eng. 245–246, 144–162 (2012)

  14. 14.

    Cazzani, A., Malagú, M., Turco, E.: Isogeometric analysis of plane curved beams. Math. Mech. Solids 21(5), 562–577 (2016)

  15. 15.

    Cazzani, A., Malagú, M., Turco, E., Stochino, F.: Constitutive models for strongly curved beams in the frame of isogeometric analysis. Math. Mech. Solids 21(2), 182–209 (2016)

  16. 16.

    Greco, L., Cuomo, M., Contrafatto, L., Gazzo, S.: An efficient blended mixed B-spline formulation for removing membrane locking in plane curved Kirchhoff rods. Comput. Methods Appl. Mech. Eng. 324, 476–511 (2017)

  17. 17.

    Maurin, F., Dedé, L., Spadoni, A.: Isogeometric rotation-free analysis of planar extensible-elastica for static and dynamic applications. Nonlinear Dyn. 81, 77–96 (2015)

  18. 18.

    Greco, L., Cuomo, M.: An isogeometric implicit \(G^{1}\) mixed finite element for Kirchhoff space rods. Comput. Methods Appl. Mech. Eng. 298, 325–349 (2016)

  19. 19.

    Weeger, O., Yeung, S.-K., Dunn, M.L.: Isogeometric collocation methods for cosserat rods and rod structures. Comput. Methods Appl. Mech. Eng. 316, 100–122 (2017)

  20. 20.

    Marino, E.: Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams. Comput. Methods Appl. Mech. Eng. 307, 383–410 (2016)

  21. 21.

    Marino, E.: Locking-free isogeometric collocation formulation for three-dimensional geometrically exact shear-deformable beams with arbitrary initial curvature. Comput. Methods Appl. Mech. Eng. 324, 546–572 (2017)

  22. 22.

    Maurin, F., Greco, F., Dedoncker, S., Desmet, W.: Isogeometric analysis for nonlinear planar Kirchhoff rods: weightedresidual formulation and collocation of the strong form. Comput. Methods Appl. Mech. Eng. 340, 1023–104 (2019)

  23. 23.

    Turco, E.: Discrete is it enough? The revival of Piola–Hencky keynotes to analyze three-dimensional elastica. Continuum Mech. Thermodyn. 30(5), 1039–1057 (2018)

  24. 24.

    Giorgio, I., Del Vescovo, D.: Energy-based trajectory tracking and vibration control for multi-link highly flexible manipulators. Math. Mech. Complex Syst. 7(2), 159–174 (2019)

  25. 25.

    Giorgio, I., Del Vescovo, D.: Non-linear lumped-parameter modeling of planar multi-link manipulators with highly flexible arms. Robotics (2018). https://doi.org/10.3390/robotics7040060

  26. 26.

    Baroudi, D., Giorgio, I., Battista, A., Turco, E., Igumnov, L.A.: Nonlinear dynamics of uniformly loaded elastica: experimental and numerical evidence of motion around curled stable equilibrium configurations. Zeitschrift für Angewandte Mathematik und Mechanik 99(7), e201800121 (2019)

  27. 27.

    Spagnuolo, M., Andreaus, U.: A targeted review on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling. Math. Mech. Solids 24(1), 258–280 (2019)

  28. 28.

    Grabovsky, Y., Truskinovsky, L.: The flip side of buckling. Continuum Mech. Thermodyn. 19, 211–243 (2007)

  29. 29.

    Cuomo, M.: Continuum model of microstructure induced softening for strain gradient materials. Math. Mech. Solids 24(8), 2374–2391 (2018)

  30. 30.

    Cuomo, M.: Continuum damage model for strain gradient materials with applications to 1D examples. Continuum Mech. Thermodyn. 31(4), 969–987 (2019)

  31. 31.

    Spagnuolo, M., Barcz, K., Pfaff, A., dell’Isola, F., Franciosi, P.: Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech. Res. Commun. 83, 47–52 (2017)

  32. 32.

    Altenbach, H., Eremeyev, V.A.: On the constitutive equations of viscoelastic micropolar plates and shells of differential type. Math. Mech. Complex Syst. 3(3), 273–283 (2015)

  33. 33.

    Greco, L., Cuomo, M.: Consistent tangent operator for an exact Kirchhoff rod model. Continuum Mech. Thermodyn. 27, 861–877 (2015)

  34. 34.

    Thomas, D.C., Scott, M.A., Evans, J.A., Tew, K., Evans, E.J.: Bézier projection: a unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis. Comput. Methods Appl. Mech. Eng. 284, 55–105 (2015)

  35. 35.

    Simo, J.C.: The (symmetric) Hessian for geometrically non linear models in solids mechanics: intrinsic definition and geometric interpretation. Comput. Methods Appl. Mech. Eng. 96, 189–200 (1992)

  36. 36.

    Lo, S.H.: Geometrically nonlinear formulation of 3d finite strain beam element with large rotations. Comput. Struct. 44(1–2), 147–157 (1992)

  37. 37.

    DaDeppo, D.A., Schmidt, R.: Instability of clamped-hinged circular arches subjected to a point load. J. Appl. Mech. 42(4), 894–896 (1975)

  38. 38.

    Li, W., Ma, H., Gao, W.: Geometrically exact curved beam element using internal force field defined in deformed configuration. Int. J. Nonlinear Mech. 89, 116–126 (2017)

  39. 39.

    Leahu-Aluas, I., Abed-Meraim, F.: A proposed set of popular limit-point buckling benchmark problems. Struct. Eng. Mech. 38(6), 767–802 (2011)

  40. 40.

    Barchiesi, E., dell’Isola, F., Laudato, M., Placidi, L., Seppecher, P.: A 1D continuum model for beams with pantographic microstructure: asymptotic micro-macro identification and numerical results. In: Porubov, A., dell’Isola, F., Eremeyev, V. (eds.) Advances in Mechanics of Microstructured Media and Structures, vol. 87, pp. 43–77. Springer, Cham (2018)

  41. 41.

    Alibert, J.J., Seppecher, P., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003)

  42. 42.

    Barchiesi, E., Eugster, S.R., Placidi, L., dell’Isola, F.: Pantographic beam: a complete second gradient 1D-continuum in plane. Zeitschrift für angewandte Mathematik und Physik (2018). https://doi.org/10.1007/s00033-019-1181-4

  43. 43.

    Barchiesi, E., Kakalo, S.: Variational asymptotic homogenization of beam-like square lattice structures. Math. Mech. Solids 20(10), 3295–3318 (2019)

  44. 44.

    Barchiesi, E., Laudato, M., Di Cosmo, F.: Wave dispersion in non-linear pantographic beams. Mech. Res. Commun. 94, 128–132 (2018)

  45. 45.

    Giorgio, I., Corte, A.D., dell’Isola, F.: Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn. 88(1), 21–31 (2017)

  46. 46.

    Maurin, F., Greco, F., Desmet, W.: Isogeometric analysis for nonlinear planar pantographic lattice: discrete and continuum models. Continuum Mech. Thermodyn. 31, 1051–1064 (2019)

  47. 47.

    Turco, E.: Numerically driven tuning of equilibrium paths for pantographic beams. Continuum Mech. Thermodyn. 31(6), 1941–1960 (2019)

  48. 48.

    Eremeyev, V.A., Turco, E.: Enriched buckling for beam-lattice metamaterials. Mech. Res. Commun. 103, 103458 (2020). https://doi.org/10.1016/j.mechrescom.2019.103458

  49. 49.

    dell’Isola, F., Seppecher, P., Alibert, J.J., et al.: Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Continuum Mech. Thermodyn. 31, 851–884 (2019). https://doi.org/10.1007/s00161-018-0689-8

  50. 50.

    dell’Isola, F., Seppecher, P., Spagnuolo, M., et al.: Advances in pantographic structures: design, manufacturing, models, experiments and image analyses. Continuum Mech. Thermodyn. 31, 1231–1282 (2019). https://doi.org/10.1007/s00161-019-00806-x

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Correspondence to L. Greco.

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Communicated by Holm Altenbach and Victor A. Eremeyev.

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Greco, L. An iso-parametric \(\pmb {\mathrm {G}^1}\)-conforming finite element for the nonlinear analysis of Kirchhoff rod. Part I: the 2D case. Continuum Mech. Thermodyn. (2020) doi:10.1007/s00161-020-00861-9

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Keywords

  • Conforming element
  • Kirchhoff rod
  • \(\mathrm {G}^1\) continuity
  • Isogeometric analysis
  • Finite element formulation