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Homogenization assumptions for coupled multiscale analysis of structural elements: beam kinematics

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Abstract

This contribution proposes a multiscale scheme for structural elements considering beam kinematics. The scheme is based on a first-order homogenization approach fulfilling the Hill–Mandel condition. Within this paper, special focus is given to the transverse shear stiffness. Using basic boundary conditions, the transverse shear stiffness drastically depends on the size of the representative volume element (RVE). The reason for this size dependency is identified. As a consequence, additional internal constraints are proposed. With these new constraints, the homogenization scheme leads to cross-sectional values independent of the size of the RVE. As they are based on the beam assumptions, a homogeneous material distribution in the length direction yields optimal results. Furthermore, outcomes of the scheme are verified with simple linear elastic benchmark tests as well as nonlinear computations involving plasticity and cross-sectional deformations.

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References

  1. Buannic N, Cartraud P (2001) Higher-order effective modeling of periodic heterogeneous beams. II. Derivation of the proper boundary conditions for the interior asymptotic solution. Int J Solids Struct 38(40–41):7163–7180

    Article  MathSciNet  Google Scholar 

  2. Carrera E, Giunta G, Nali P, Petrolo M (2010) Refined beam elements with arbitrary cross-section geometries. Comput Struct 88(5–6):283–293

    Article  Google Scholar 

  3. Cartraud P, Messager T (2006) Computational homogenization of periodic beam-like structures. Int J Solids Struct 43(3):686–696

    Article  MathSciNet  Google Scholar 

  4. Coenen EWC, Kouznetsova VG, Geers MGD (2010) Computational homogenization for heterogeneous thin sheets. Int J Numer Methods Eng 83(8–9):1180–1205

    Article  Google Scholar 

  5. Feyel F (1998) Application du calcul parallèle aux modéles á grand nombre de variables internes, PhD Thesis. ONERA

  6. Feyel F, Chaboche JL (2000) FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Comput Methods Appl Mech Eng 183(3–4):309–330

    Article  Google Scholar 

  7. Geers MGD, Coenen EWC, Kouznetsova VG (2007) Multi-scale computational homogenization of structured thin sheets. Model Simul Mater Sci Eng 15(4):S393–S404

    Article  Google Scholar 

  8. Gruttmann F, Wagner W (2001) Shear correction factors in Timoshenko’s beam theory for arbitrary shaped cross-sections. Comput Mech 27(3):199–207

    Article  Google Scholar 

  9. Gruttmann F, Wagner W (2013) A coupled two-scale shell model with applications to layered structures. Int J Numer Methods Eng 94(13):1233–1254

    Article  MathSciNet  Google Scholar 

  10. Gruttmann F, Wagner W (2017) Shear correction factors for layered plates and shells. Comput Mech 59(1):129–146

    Article  MathSciNet  Google Scholar 

  11. Gruttmann F, Sauer R, Wagner W (1998) A geometrical nonlinear eccentric 3D-beam element with arbitrary cross-sections. Comput Methods Appl Mech Eng 160(3–4):383–400

    Article  Google Scholar 

  12. Gruttmann F, Wagner W, Sauer R (1998) Zur Berechnung von Wölbfunktion und Torsionskennwerten beliebiger Stabquerschnitte mit der Methode der finiten Elemente. Bauingenieur 73:138–143

    Google Scholar 

  13. Gruttmann F, Knust G, Wagner W (2017) Theory and numerics of layered shells with variationally embedded interlaminar stresses. Comput Methods Appl Mech Eng 326:713–738

    Article  MathSciNet  Google Scholar 

  14. Heller D, Gruttmann F (2016) Nonlinear two-scale shell modeling of sandwiches with a comb-like core. Compos Struct 144:147–155

    Article  Google Scholar 

  15. Hill R (1952) The elastic behaviour of a crystalline aggregate. Proc Phys Soc London Sect A 65(5):349–354

    Article  Google Scholar 

  16. Hodges DH (2006) Nonlinear composite beam theory, vol 213. Progress in astronautics and aeronautics. American Institute of Aeronautics and Astronautics, Reston

    Book  Google Scholar 

  17. Klinkel S, Govindjee S (2002) Using finite strain 3D-material models in beam and shell elements. Eng Comput 19(3):254–271

    Article  Google Scholar 

  18. Kollbrunner CF, Meister M (1961) Knicken, Biegedrillknicken, Kippen. Springer, Berlin

    Book  Google Scholar 

  19. Markovič D, Ibrahimbegović A (2004) On micro-macro interface conditions for micro scale based fem for inelastic behavior of heterogeneous materials. Comput Methods Appl Mech Eng 193(48–51):5503–5523

    Article  Google Scholar 

  20. Saeb S, Steinmann P, Javili A (2016) Aspects of computational homogenization at finite deformations: a unifying review from Reuss’ to Voigt’s bound. Appl Mech Rev 68(5):050801

    Article  Google Scholar 

  21. Sayyad AS (2011) Comparison of various refined beam theories for the bending and free vibration analysis of thick beams. Appl Comput Mech 5:217–230

    Google Scholar 

  22. Schröder J (2000) Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Stabilitätsproblemen. Habilitation, Bericht Nr I-7 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart

  23. Simo JC, Vu-Quoc L (1991) A geometrically-exact rod model incorporating shear and torsion-warping deformation. Int J Solids Struct 27(3):371–393

    Article  MathSciNet  Google Scholar 

  24. Timoshenko SP, Goodier JN (1951) Theory of elasticity. McGraw-Hill, New York

    MATH  Google Scholar 

  25. Vlachoutsis S (1992) Shear correction factors for plates and shells. Int J Numer Methods Eng 33(7):1537–1552

    Article  Google Scholar 

  26. Wackerfuß J, Gruttmann F (2009) A mixed hybrid finite beam element with an interface to arbitrary three-dimensional material models. Comput Methods Appl Mech Eng 198(27):2053–2066

    Article  MathSciNet  Google Scholar 

  27. Wackerfuß J, Gruttmann F (2011) A nonlinear Hu-Washizu variational formulation and related finite-element implementation for spatial beams with arbitrary moderate thick cross-sections. Comput Methods Appl Mech Eng 200(17–20):1671–1690

    Article  MathSciNet  Google Scholar 

  28. Wagner W, Gruttmann F (2001) Finite element analysis of Saint–Venant torsion problem with exact integration of the elastic–plastic constitutive equations. Comput Methods Appl Mech Eng 190(29–30):3831–3848

    Article  Google Scholar 

  29. Wagner W, Gruttmann F (2013) A consistently linearized multiscale model for shell structures. In: Pietraszkiewicz W (ed) Shell structures. CRC Press, Boca Raton

    MATH  Google Scholar 

  30. Xu L, Cheng G, Yi S (2016) A new method of shear stiffness prediction of periodic Timoshenko beams. Mech Adv Mater Struct 23(6):670–680

    Article  Google Scholar 

  31. Yu W, Hodges DH (2005) Generalized Timoshenko theory of the variational asymptotic beam sectional analysis. J Am Helicopter Soc 50(1):46–55

    Article  Google Scholar 

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Authors and Affiliations

  1. Lehrstuhl für Baustatik und Baudynamik, RWTH Aachen University, Mies-van-der-Rohe-Str. 1, 52074, Aachen, Germany

    Simon Klarmann & Sven Klinkel

  2. Fachgebiet Festkörpermechanik, Technische Universität Darmstadt, Franziska-Braun-Str. 7, 64287, Darmstadt, Germany

    Friedrich Gruttmann

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  1. Simon Klarmann
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  2. Friedrich Gruttmann
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  3. Sven Klinkel
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Correspondence to Simon Klarmann.

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Klarmann, S., Gruttmann, F. & Klinkel, S. Homogenization assumptions for coupled multiscale analysis of structural elements: beam kinematics. Comput Mech 65, 635–661 (2020). https://doi.org/10.1007/s00466-019-01787-z

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