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Multi-scale modelling and simulation of effective properties of perforated sheets with periodic patterns

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Abstract

The elastic properties of a perforated sheet having hexagonal periodic patterns are evaluated using a representative volume element (RVE). The finite element method (FEM) is employed to evaluate the response of the RVE under periodic boundary conditions and numerical homogenisation technique is applied using the FEM solution to estimate the effective properties of the perforated sheet. Numerical homogenisation results are compared to analytical solutions from literature and experimental results. The RVE used in this work considers the radius formed between attached edges and investigates the effect of radius on the overall properties, which has not been sufficiently investigated in literature. Furthermore, the influence of enforcing periodic boundary conditions on the RVE along the thickness direction for such thin perforated structure is investigated. It is found that the elastic constants are overestimated on enforcing periodicity in the thickness direction.

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References

  1. Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11(5):357–372

    Article  Google Scholar 

  2. Nemat-Nasser S, Hori M (1999) Micromechanics: overall properties of heterogeneous materials. Elsevier

  3. Hassani B, Hinton E (1999) Homogenization and structural topology optimization? Theory, practice and software. Springer

  4. Charalambakis N (2010) Homogenization techniques and micromechanics. A survey and perspectives. Appl Mech Rev 63(3):030803. https://doi.org/10.1115/1.4001911

  5. Barbero EJ (2013) Finite element analysis of composite materials using Abaqus. CRC Press

  6. Utzig L, Karch C, Rehra J, Hannemann B, Schmeer S (2018) Modeling and simulation of the effective strength of hybrid polymer composites reinforced by carbon and steel fibers. J Mater Sci 53(1):667–677

    Article  Google Scholar 

  7. Würkner M, Berger H, Gabbert U (2011) On numerical evaluation of effective material properties for composite structures with rhombic fiber arrangements. Int J Eng Sci 49(4):322–332

    Article  Google Scholar 

  8. Karch C (2014) Micromechanical analysis of thermal expansion coefficients. Model Numer Simul Mater Sci 4:104–118

    Google Scholar 

  9. Karadeniz ZH, Kumlutas D (2007) A numerical study on the coefficients of thermal expansion of fiber reinforced composite materials. Compos Struct 78(1):1–10

    Article  Google Scholar 

  10. Dasgupta A, Agarwal RK, Bhandarkar SM (1996) Three-dimensional modeling of woven-fabric composites for effective thermo-mechanical and thermal properties. Compos Sci Technol 56(3):209–223

    Article  Google Scholar 

  11. Drougkas A, Roca P, Molins C (2015) Analytical micro-modeling of masonry periodic unit cells—elastic properties. Int J Solids Struct 69–70:169–188

    Article  Google Scholar 

  12. Omairey SL, Dunning PD, Sriramula S (2019) Development of an ABAQUS plugin tool for periodic RVE homogenisation. Eng Comput 35(2):567–577

    Article  Google Scholar 

  13. Pahr DH, Zysset PK (2008) Influence of boundary conditions on computed apparent elastic properties of cancellous bone. Biomech Model Mechanobiol 7(6):463–476

    Article  Google Scholar 

  14. Martínez-Ayuso G, Friswell MI, Adhikari S, Khodaparast HH, Berger H (2017) Homogenization of porous piezoelectric materials. Int J Solids Struct 113–114:218–229

    Article  Google Scholar 

  15. Deraemaeker A, Nasser H (2010) Numerical evaluation of the equivalent properties of macro fiber composite (MFC) transducers using periodic homogenization. Int J Solids Struct 47(24):3272–3285

    Article  Google Scholar 

  16. Berger H, Kari S, Gabbert U, Rodriguez-Ramos R, Bravo-Castillero J, Guinovart-Diaz R, Sabina FJ, Maugin GA (2006) Unit cell models of piezoelectric fiber composites for numerical and analytical calculation of effective properties. Smart Mater Struct 15(2):451–458

    Article  Google Scholar 

  17. Alzebdeh K (2012) Evaluation of the in-plane effective elastic moduli of single-layered graphene sheet. Int J Mech Mater Des 8(3):269–278

    Article  Google Scholar 

  18. Wang Y, Guan B, Mu L, Zang Y (2018) Equivalent tensile properties analysis of the dimpled sheet. J Fail Anal Prev 18(4):791–798

    Article  Google Scholar 

  19. Forskitt M, Moon JR, Brook PA (1991) Elastic properties of plates perforated by elliptical holes. Appl Math Model 15(4):182–190

    Article  Google Scholar 

  20. van Rens BJE, Brekelmans WAM, Baaijens FPT (1998) Homogenization of the elastoplastic behavior of perforated plates. Comput Struct 69(4):537–545

    Article  MathSciNet  Google Scholar 

  21. Sayed AM (2019) Numerical analysis of the perforated steel sheets under uni-axial tensile force. Metals 9(6):632

    Article  Google Scholar 

  22. Zhu JC, Ben Bettaieb M, Abed-Meraim F (2020) Numerical investigation of necking in perforated sheets using the periodic homogenization approach. Int J Mech Sci 166:105209

    Article  Google Scholar 

  23. Zhu JC, Bettaieb MB, Abed-Meraim F (2020) Comparative study of three techniques for the computation of the macroscopic tangent moduli by periodic homogenization scheme. Eng Comput. https://doi.org/10.1007/s00366-020-01091-y

    Article  Google Scholar 

  24. 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 

  25. Schmitz A, Horst P (2014) A finite element unit-cell method for homogenised mechanical properties of heterogeneous plates. Compos Part A Appl Sci Manuf 61:23–32

    Article  Google Scholar 

  26. Cai Y, Xu L, Cheng G (2014) Novel numerical implementation of asymptotic homogenization method for periodic plate structures. Int J Solids Struct 51(1):284–292

    Article  Google Scholar 

  27. Catapano A, Montemurro MA (2014) Homogenisation of core properties multi-scale approach for the optimum design of sandwich plates with honeycomb core. Part I. Compos. Struct. 118:664–676

    Article  Google Scholar 

  28. Gibson LJ, Ashby MF (2010) Cellular solids: structure and properties. Cambridge University Press

  29. Malek S, Gibson L (2015) Effective elastic properties of periodic hexagonal honeycombs. Mech Mater 91:226–240

    Article  Google Scholar 

  30. Balawi S, Abot JL (2008) A refined model for the effective in-plane elastic moduli of hexagonal honeycombs. Compos Struct 84(2):147–158

    Article  Google Scholar 

  31. Sun CT, Vaidya RS (1996) Prediction of composite properties from a representative volume element. Compos Sci Technol 56(2):171–179

    Article  Google Scholar 

  32. Xia Z, Zhang Y, Ellyin F (2003) A unified periodical boundary conditions for representative volume elements of composites and applications. Int J Solids Struct 40(8):1907–1921

    Article  Google Scholar 

  33. Cheng Q, Zhidong G, Siyuan J, Zengshan L (2017) A method of determining effective elastic properties of honeycomb cores based on equal strain energy. Chin J Aeronaut 30(2):766–779

    Article  Google Scholar 

  34. Varadharajan S (2019) Mechanical characterisation of speaker plate using a representative volume element approach. Master’s thesis, Technical University of Munich, Germany

  35. Aboudi J (1991) Mechanics of composite materials: a unified micromechanics approach. Elsevier

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Authors and Affiliations

  1. Department of Civil, Geo and Environmental Engineering, Technical University of Munich, Arcisstraße 21, 80333, Munich, Germany

    Srikkanth Varadharajan & Fabian Duddeck

  2. Simulation Interior, BMW Group, Knorrstraße 147, 80788, Munich, Germany

    Srikkanth Varadharajan & Lukas Utzig

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  1. Srikkanth Varadharajan
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  2. Lukas Utzig
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  3. Fabian Duddeck
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Correspondence to Lukas Utzig.

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Appendix

Appendix

Table 7 gives the constraint equations to enforce PBC for nodes located on edges and at the corners of the RVE.

Table 7 PBC constraints on corner and edge nodes
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Varadharajan, S., Utzig, L. & Duddeck, F. Multi-scale modelling and simulation of effective properties of perforated sheets with periodic patterns. Meccanica 57, 707–722 (2022). https://doi.org/10.1007/s11012-021-01463-8

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  • DOI: https://doi.org/10.1007/s11012-021-01463-8

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