Meccanica

, Volume 53, Issue 6, pp 1187–1201 | Cite as

Asymptotic homogenization of fibre-reinforced composites: a virtual element method approach

Novel Computational Approaches to Old and New Problems in Mechanics

Abstract

A virtual element method approach is presented for solving the unit cell problem, in application of the asymptotic homogenization method, and computing the antiplane shear homogenized material moduli of a composite material reinforced by cylindrical inclusions of arbitrary cross section. Validation of the proposed numerical method is proved by comparison with analytical and numerical reference solutions, for a number of micro-structural arrays and for different grading properties of the material constituents. A point on numerical efficiency is also made with respect to the possibility of local refinement granted by the innovative numerical procedure which relies on a mesh conformity concept ampler than the one of classical finite element method. The flexibility of the method allows for a large variety of microstructure shapes.

Keywords

VEM Virtual element method Homogenization Cell problem Periodic condition 

References

  1. 1.
    Antonietti PF, Beirão da Veiga L, Scacchi S, Verani M (2016) A \({C}^1\) virtual element method for the Cahn–Hilliard equation with polygonal meshes. SIAM J Numer Anal 54(1):34–56MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Artioli E, Beirão da Veiga L, Lovadina C, Sacco E (2017) Arbitrary order 2D virtual elements for polygonal meshes: part I, elastic problem. Comput Mech 60:355–377MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Artioli E, Beirão da Veiga L, Lovadina C, Sacco E (2017) Arbitrary order 2D virtual elements for polygonal meshes: part II, inelastic problem. Comput Mech 60:643–657MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Artioli E, Bisegna P (2013) Effective longitudinal shear moduli of periodic fibre-reinforced composites with functionally-graded fibre coatings. Int J Solids Struct 50:1154–1163CrossRefGoogle Scholar
  5. 5.
    Artioli E, Bisegna P, Maceri F (2010) Effective longitudinal shear moduli of periodic fibre-reinforced composites with radially-graded fibres. Int J Solids Struct 47:383–397CrossRefMATHGoogle Scholar
  6. 6.
    Artioli E, de Miranda S, Lovadina C, Patruno L (2017) A family of virtual element methods for plane elasticity problems based on the Hellinger–Reissner principle, submitted for publication, and online on: arxiv:1711.06168
  7. 7.
    Artioli E, de Miranda S, Lovadina C, Patruno L (2017) A stress/displacement virtual element method for plane elasticity problems. Comput Methods Appl Mech Eng 325:155–174ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Artioli E, Taylor RL (2018) Vem for inelastic solids. Comput Methods Appl Sci 46:381–394MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bathe KJ (1996) Finite element procedures. Prentice Hall, Upper Saddle RiverMATHGoogle Scholar
  10. 10.
    Beirão da Veiga L, Brezzi F, Cangiani A, Manzini G, Marini LD, Russo A (2013) Basic principles of virtual element methods. Math Models Methods Appl Sci 23(1):199–214MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Beirão da Veiga L, Brezzi F, Marini LD (2013) Virtual elements for linear elasticity problems. SIAM J Numer Anal 51(2):794–812MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Beirão da Veiga L, Brezzi F, Marini LD, Russo A (2014) The hitchhiker’s guide to the virtual element method. Math Models Methods Appl Sci 24(8):1541–1573MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Beirão da Veiga L, Lovadina C, Mora D (2015) A virtual element method for elastic and inelastic problems on polytope meshes. Comput Methods Appl Mech Eng 295:327–346ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Beirão Da Veiga L, Manzini M (2015) Residual a posteriori error estimation for the virtual element method for elliptic problems. ESAIM: M2AN 49:577–599MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Bensoussan A, Lions JL, Papanicolau G (1978) Asymptotic analysis for periodic structures. North-Holland, AmsterdamGoogle Scholar
  16. 16.
    Biabanaki S, Khoei A, Wriggers P (2014) Polygonal finite element methods for contact–impact problems on non-conformal meshes. Comput Methods Appl Mech Eng 269:198–221ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Bigoni D, Serkov SK, Valentini M, Movchan AB (1998) Asymptotic models of dilute composites with imperfectly bonded inclusions. Int J Solids Struct 35(24):3239–3258MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Brezzi F, Marini LD (2013) Virtual element methods for plate bending problems. Comput Methods Appl Mech Eng 253:455–462ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Cangiani A, Manzini G, Russo A, Sukumar N (2015) Hourglass stabilization and the virtual element method. Int J Numer Method Eng 102(3–4):404–436.  https://doi.org/10.1002/nme.4854 MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Chi H, Beirão da Veiga L, Paulino GH (2017) Some basic formulations of the virtual element method (VEM) for finite deformations. Comput Methods Appl Mech Eng 318:148–192ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Chi H, Talischi C, Lopez-Pamies O, Paulino GH (2015) Polygonal finite elements for finite elasticity. Int J Numer Methods Eng 101(4):305–328MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Duvaut G (1976) Homogeneization et materiaux composite. In: Ciarlet P, Rouseau M (eds) Theoretical and appliedMechanics. North-Holland, Amsterdam, pp 194–278Google Scholar
  23. 23.
    Engwirda D (2014) Locally-optimal Delaunay-refinement and optimisation-based mesh generation. Ph.D. thesis, The University of SydneyGoogle Scholar
  24. 24.
    Gain AL, Paulino GH, Leonardo SD, Menezes IFM (2015) Topology optimization using polytopes. Comput Methods Appl Mech Eng 293:411–430ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Gain AL, Talischi C, Paulino GH (2014) On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes. Comput Methods Appl Mech Eng 282:132–160ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Hashin Z (1983) Analysis of composite materials—a survey. J Appl Mech 50:481–505CrossRefMATHGoogle Scholar
  27. 27.
    Hashin Z (1991) The spherical inclusion with imperfect interface. J Appl Mech 58:444–449CrossRefGoogle Scholar
  28. 28.
    Hill R (1963) Elastic properties of reinforced solids: some theoretical pnnciples. J Mech Phys Solids 11:357–372ADSCrossRefMATHGoogle Scholar
  29. 29.
    Hollister SJ, Kikuchi N (1992) A comparison of homogenization and standard mechanics analyses for periodic porous composites. Comput Mech 10:73–95CrossRefMATHGoogle Scholar
  30. 30.
    Hughes TJR (2000) The finite element method linear static and dynamic finite element analysis, 2nd edn. Dover, Downers GroveMATHGoogle Scholar
  31. 31.
    Joyce D, Parnell WJ, Assier RC, Abrahams ID (2017) An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems. Proc R Soc 473:20170080MathSciNetCrossRefGoogle Scholar
  32. 32.
    Larsen EW (1975) Neutron transport and diffusion in inhomogeneous media. Int J Math Phys 16:1421–1427ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Lene F, Leguillon D (1982) Homogenized constitutive law for a partially cohesive composite material. Int J Solids Struct 18:443–458MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Leon SE, Spring D, Paulino GH (2014) Reduction in mesh bias for dynamic fracture using adaptive splitting of polygonal finite elements. Int J Numer Methods Eng 100:555–576MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Lions JL (1980) Asymptotic expansions in perforated media with a periodic structure. Rocky Mt J Math 10:125–140MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Lions JL (1981) Some methods in the mathematical analysls analysis of systems and their control. Gordon and Breach Science Publishers, New YorkGoogle Scholar
  37. 37.
    Michel JC, Moulinec H, Suquet P (1999) Effective properties of composite materials with periodic microstructure: a computational approach. Comput Methods Appl Mech Eng 172:109–143ADSMathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Moulinec H, Suquet P (1998) A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput Methods Appl Mech Eng 157:69–94ADSMathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Sanchez-Palencia E (1974) Comportements local et macroscopique d’un type de milieux physiques heterogenes. Int J Eng Sci 12:331–351CrossRefMATHGoogle Scholar
  40. 40.
    Sanchez-Palencia E (1980) Non-homogeneous media and vibration theory, lecture notes in physics. Springer, BerlinMATHGoogle Scholar
  41. 41.
    Shabana YM, Noda N (2008) Numerical evaluation of the thermomechanical effective properties of a functionally graded material using the homogenization method. Int J Solids Struct 45:3494–3506CrossRefMATHGoogle Scholar
  42. 42.
    Sukumar N, Tabarraei A (2004) Conforming polygonal finite elements. Int J Numer Methods Eng 61(12):2045–2066MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Suquet P (1987) Elements of homogenization theory for inelastic solid mechanics. In: Sanchez-Palencia E, Zaoui A (eds) Homogenizat!on techniques for composite media. Springer, Berlin, pp 194–278Google Scholar
  44. 44.
    Talischi C, Paulino GH, Pereira A, Menezes IFM (2010) Polygonal finite elements for topology optimization: a unifying paradigm. Int J Numer Methods Eng 82(6):671–698MATHGoogle Scholar
  45. 45.
    Willoughby N, Parnell WJ, Hazel AL, Abrahams ID (2012) Homogenization methods to approximate the effective response of random fibre-reinforced composites. Int J Solids Struct 49:1421–1433CrossRefGoogle Scholar
  46. 46.
    Wriggers P (2008) Nonlinear finite element methods. Springer, BerlinMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil Engineering and Computer Science (DICII)University of Rome - Tor VergataRomeItaly

Personalised recommendations