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Contact between shear-deformable beams with elliptical cross sections

  • M. Magliulo
  • A. Zilian
  • L. A. A. BeexEmail author
Original Paper

Abstract

Slender constituents are present in many structures and materials. In associated mechanical models, each slender constituent is often described as a beam. Contact between beams is essential to incorporate in mechanical models, but associated contact frameworks are only demonstrated to work for beams with circular cross sections. Only two studies have shown the ability to treat contact between beams with elliptical cross sections, but those frameworks are limited to point-wise contact, which narrows their applicability. This contribution presents initial results of a framework for shear-deformable beams with elliptical cross sections if contact occurs along a line or at an area (instead of at a point). This is achieved by integrating a penalty potential over one of the beams’ surfaces. Simo–Reissner geometrically exact beam elements are employed to discretize each beam. As the surface of an assembly of such beam elements is discontinuous, a smoothed surface is introduced to formulate the contact kinematics. This enables the treatment of contact for large sliding displacements and substantial deformations.

Notes

Acknowledgements

The authors gratefully acknowledge the financial support of the University of Luxembourg for project TEXTOOL.

References

  1. 1.
    Beex, L.A.A., Peerlings, R.: On the influence of delamination on laminated paperboard creasing and folding. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 370(1965), 1912–1924 (2012)CrossRefGoogle Scholar
  2. 2.
    Mäkelä, P., Östlund, S.: Orthotropic elastic-plastic material model for paper materials. Int. J. Solids Struct. 40, 5599–5620 (2003)CrossRefGoogle Scholar
  3. 3.
    Thakkar, B., Gooren, L., Peerlings, R., Geers, M.: Experimental and numerical investigation of creasing in corrugated paperboard. Philos. Mag. 88, 3299–3310 (2008)CrossRefGoogle Scholar
  4. 4.
    Lee, K.-Y., Aitomäki, Y., Berglund, L.A., Oksman, K., Bismarck, A.: On the use of nanocellulose as reinforcement in polymer matrix composites. Compos. Sci. Technol. 105, 15–27 (2014)CrossRefGoogle Scholar
  5. 5.
    Kulachenko, A., Uesaka, T.: Direct simulations of fiber network deformation and failure. Mech. Mater. 51, 1–14 (2012)CrossRefGoogle Scholar
  6. 6.
    Beex, L.A.A., Peerlings, R., van Os, K., Geers, M.: The mechanical reliability of an electronic textile investigated using the virtual-power-based quasicontinuum method. Mech. Mater. 80, 52–66 (2015)CrossRefGoogle Scholar
  7. 7.
    Boubaker, B.Ben, Haussy, B., Ganghoffer, J.: Discrete models of woven structures. Macroscopic approach. Compos. Part B Eng. 38, 498–505 (2007)CrossRefGoogle Scholar
  8. 8.
    Boisse, P., Gasser, A., Hivet, G.: Analyses of fabric tensile behaviour: determination of the biaxial tension-strain surfaces and their use in forming simulations. Compos. Part A Appl. Sci. Manuf. 32, 1395–1414 (2001)CrossRefGoogle Scholar
  9. 9.
    Peng, X.Q., Cao, J.: A continuum mechanics-based non-orthogonal constitutive model for woven composite fabrics. Compos. Part A Appl. Sci. Manuf. 36, 859–874 (2005)CrossRefGoogle Scholar
  10. 10.
    Miao, Y., Zhou, E., Wang, Y., Cheeseman, B.A.: Mechanics of textile composites: micro-geometry. Compos. Sci. Technol. 68, 1671–1678 (2008)CrossRefGoogle Scholar
  11. 11.
    Jung, A., Lach, E., Diebels, S.: New hybrid foam materials for impact protection. Int. J. Impact Eng. 64, 30–38 (2014)CrossRefGoogle Scholar
  12. 12.
    Jung, A., Beex, L.A.A., Diebels, S., Bordas, S.P.A.: Open-cell aluminium foams with graded coatings as passively controllable energy absorbers. Mater. Des. 87, 36–41 (2015)CrossRefGoogle Scholar
  13. 13.
    Sun, Y., Burgueño, R., Wang, W., Lee, I.: Modeling and simulation of the quasi-static compressive behavior of Al/Cu hybrid open-cell foams. Int. J. Solids Struct. 54, 135–146 (2015)CrossRefGoogle Scholar
  14. 14.
    Onck, P.R., Van Merkerk, R., De Hosson, J.T.M., Schmidt, I.: Fracture of metal foams: In-situ testing and numerical modeling. In: Advanced Engineering Materials, vol. 6, pp. 429–431, Wiley, Hoboken (2004)Google Scholar
  15. 15.
    Ashby, M.F.: The properties of foams and lattices. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 364, 15–30 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Beex, L.A.A., Kerfriden, P., Rabczuk, T., Bordas, S.P.A.: Quasicontinuum-based multiscale approaches for plate-like beam lattices experiencing in-plane and out-of-plane deformation. Comput. Methods Appl. Mech. Eng. 279, 348–378 (2014)CrossRefGoogle Scholar
  17. 17.
    Beex, L.A.A., Rokoš, O., Zeman, J., Bordas, S.P.A.: Higher-order quasicontinuum methods for elastic and dissipative lattice models: uniaxial deformation and pure bending. GAMM-Mitteilungen 38, 344–368 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mozafary, V., Payvandy, P., Rezaeian, M.: A novel approach for simulation of curling behavior of knitted fabric based on mass spring model. J. Text. Inst. 109, 1620–1641 (2018)CrossRefGoogle Scholar
  19. 19.
    Neto, A.Gay, Pimenta, P.M., Wriggers, P.: A master-surface to master-surface formulation for beam to beam contact. Part I: frictionless interaction. Comput. Methods Appl. Mech. Eng. 303, 400–429 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Neto, A.Gay, Pimenta, P.M., Wriggers, P.: A master-surface to master-surface formulation for beam to beam contact. Part II: frictional interaction. Comput. Methods Appl. Mech. Eng. 319, 146–174 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zavarise, G., De Lorenzis, L.: The node-to-segment algorithm for 2D frictionless contact: classical formulation and special cases. Comput. Methods Appl. Mech. Eng. 198, 3428–3451 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Simo, J.: A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput. Methods Appl. Mech. Eng. 49, 55–70 (1985)CrossRefGoogle Scholar
  23. 23.
    Simo, J., Vu-Quoc, L.: A three-dimensional finite-strain rod model. Part II: computational aspects. Comput. Methods Appl. Mech. Eng. 58, 79–116 (1986)CrossRefGoogle Scholar
  24. 24.
    Simo, J., Vu-Quoc, L.: A Geometrically-exact rod model incorporating shear and torsion-warping deformation. Int. J. Solids Struct. 27(3), 371–393 (1991)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Jelenić, G., Crisfield, M.: Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics. Comput. Methods Appl. Mech. Eng. 171, 141–171 (1999)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ibrahimbegović, A.: On finite element implementation of geometrically nonlinear Reissner’s beam theory: three-dimensional curved beam elements. Comput. Methods Appl. Mech. Eng. 122, 11–26 (1995)CrossRefGoogle Scholar
  27. 27.
    Romero, I.: The interpolation of rotations and its application to finite element models of geometrically exact rods. Comput. Mech. 34, 121–133 (2004)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Meier, C., Popp, A., Wall, W.A.: Geometrically exact finite element formulations for slender beams: Kirchhoff–Love theory versus Simo–Reissner theory. Arch. Comput. Methods Eng. 26, 163–243 (2019)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Wriggers, P.: Computational Contact Mechanics. Springer, Berlin, Heidelberg (2006)CrossRefGoogle Scholar
  30. 30.
    Zavarise, G., Wriggers, P.: A segment-to-segment contact strategy. Math. Comput. Model. 28, 497–515 (1998)CrossRefGoogle Scholar
  31. 31.
    Zavarise, G., Wriggers, P.: Contact with friction between beams in 3-D space. Int. J. Numer. Methods Eng. 49, 977–1006 (2000)CrossRefGoogle Scholar
  32. 32.
    Konyukhov, A., Mrenes, O., Schweizerhof, K.: Consistent development of a beam-to-beam contact algorithm via the curve-to-solid beam contact: analysis for the nonfrictional case. Int. J. Numer. Methods Eng. 113, 1108–1144 (2018)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Durville, D.: Contact modelling in entangled fibrous materials. In: Trends in Computational Contact Mechanics, pp. 1–22, Springer, Berlin (2011)Google Scholar
  34. 34.
    Vu, T., Durville, D., Davies, P.: Finite element simulation of the mechanical behavior of synthetic braided ropes and validation on a tensile test. Int. J. Solids Struct. 58, 106–116 (2015)CrossRefGoogle Scholar
  35. 35.
    Lengiewicz, J., Korelc, J., Stupkiewicz, S.: Automation of finite element formulations for large deformation contact problems. Int. J. Numer. Methods Eng. 85(10), 1252–1279 (2011)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Lengiewicz, J., Stupkiewicz, S.: Efficient model of evolution of wear in quasi-steady-state sliding contacts. Wear 303, 611–621 (2013)CrossRefGoogle Scholar
  37. 37.
    Korelc, J.: Automation of primal and sensitivity analysis of transient coupled problems. Comput. Mech. 44, 631–649 (2009)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Korelc, J.: Multi-language and multi-environment generation of nonlinear finite element codes. Eng. Comput. 18(4), 312–327 (2002)CrossRefGoogle Scholar
  39. 39.
    Wriggers, P., Krstulovic-Opara, L., Korelc, J.: Smooth C1-interpolations for two-dimensional frictional contact problems. Int. J. Numer. Methods Eng. 51, 1469–1495 (2001)CrossRefGoogle Scholar
  40. 40.
    Popov, V.L.: Contact Mechanics and Friction, vol. 52. Springer, Berlin (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Science, Technology and Communication, Institute of Computational EngineeringUniversity of LuxembourgEsch-sur-AlzetteLuxembourg

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