Journal of Materials Science

, Volume 43, Issue 15, pp 5157–5167 | Cite as

Design of graded two-phase microstructures for tailored elasticity gradients

  • Shiwei Zhou
  • Qing LiEmail author


Being one of new generation of composites, functionally graded materials (FGMs) possess gradually changed physical properties due to their compositional and/or microstructural gradients. In literature, exhaustive studies have been carried out in compositional modeling and design, while limited reports are available for microstructural optimization. This article presents an inverse homogenization method for the design of two-phase (solid/void) FGM microstructures, whose periodic base cells (PBCs) vary in a direction parallel to the property gradient but periodically repeat themselves in the perpendicular direction. The effective elasticity tensor at each PBC is estimated in terms of the homogenization theory. The overall difference between the effective tensor and their target is minimized by seeking for an optimal PBC material topology. To preserve the connectivity between adjacent PBCs, three methods, namely connective constraint, pseudo load, and unified formulation with nonlinear diffusion are proposed herein. A number of two-dimensional examples possessing graded volume fraction and Young’s modulus but constant positive or negative Poisson’s ratios are presented to demonstrate this computational design procedure.


Topology Optimization Unify Formulation Selective Laser Melting Nonlinear Diffusion Solid Isotropic Material With Penalization 



The financial supports from Australian Research Council (Nos. DP0558497 and DP0773726) are acknowledged.


  1. 1.
    Niion M, Maeda S (1990) J Iron Steel Inst Jpn 30:699CrossRefGoogle Scholar
  2. 2.
    Miyamoto Y (1999) Functionally graded materials: design, processing and applications. Kluwer Academic Publishers, Boston, LondonCrossRefGoogle Scholar
  3. 3.
    Paul W (2001) 21st Century manufacturing. Prentice-Hall Inc., New JerseyGoogle Scholar
  4. 4.
    Amada S, Untao S (2001) Compos B Eng 32:449CrossRefGoogle Scholar
  5. 5.
    Nogata F, Takahashi H (1995) Compos Eng 5:743CrossRefGoogle Scholar
  6. 6.
    Silva ECN, Walters MC, Paulino GH (2006) J Mater Sci 41:6991. doi: CrossRefGoogle Scholar
  7. 7.
    Ray AK, Das SK, Mondal S, Ramachandrarao P (2004) J Mater Sci 39:1055. doi: CrossRefGoogle Scholar
  8. 8.
    Bendsøe MP, Sigmund O (2003) Topology, optimisation: theory, methods, and applications. Springer, Berlin, New YorkGoogle Scholar
  9. 9.
    Sigmund O (1994) Int J Solids Struct 31:2313. doi: CrossRefGoogle Scholar
  10. 10.
    Sigmund O (1994) Design of material structures using topology optimization. Technical University of DenmarkGoogle Scholar
  11. 11.
    Bensoussan A, Papanicolaou G, Lions JL (1978) Asymptotic analysis for periodic structures. North Holland Pub Co., AmsterdamGoogle Scholar
  12. 12.
    Sanchez-Palencia E (1980) Non-homogeneous media and vibration theory. Springer-Verlag, BerlinGoogle Scholar
  13. 13.
    Sigmund O, Torquato S (1996) Appl Phys Lett 69:3203. doi: CrossRefGoogle Scholar
  14. 14.
    Zhou SW, Li Q (2007) J Phys D Appl Phys 40:6083. doi: CrossRefGoogle Scholar
  15. 15.
    Zhou SW, Li Q (2008) J Mater Res 23:798. doi: CrossRefGoogle Scholar
  16. 16.
    Guest JK, Prévost JH (2007) Comput Methods Appl Mech Eng 196:1006. doi: CrossRefGoogle Scholar
  17. 17.
    Mumtaz KA, Hopkinson N (2007) J Mater Sci 42:7647. doi: CrossRefGoogle Scholar
  18. 18.
    Dimitrov D, Schreve K, de Beer N (2006) Rapid Prototyping J 12:136. doi: CrossRefGoogle Scholar
  19. 19.
    Wang JW, Shaw LL (2006) J Am Ceram Soc 89:3285. doi: CrossRefGoogle Scholar
  20. 20.
    Lin CY, Hsiao CC, Chen PQ, Hollister SJ (2004) Spine 29:1747. doi: CrossRefGoogle Scholar
  21. 21.
    Lin CY, Schek RM, Mistry AS, Shi XF, Mikos AG, Krebsbach PH et al (2005) Tissue Eng 11:1589. doi: CrossRefGoogle Scholar
  22. 22.
    Hollister SJ (2005) Nat Mater 4:518. doi: CrossRefGoogle Scholar
  23. 23.
    Chen KZ, Feng XA (2004) Comput Aid Des 36:51. doi: CrossRefGoogle Scholar
  24. 24.
    Zhu F, Chen KZ, Feng XA (2006) Adv Eng Software 37:20. doi: CrossRefGoogle Scholar
  25. 25.
    Seepersad CC, Kumar RS, Allen JK, Mistree F, McDowell DL (2004) J Comput Aided Mater Des 11:163. doi: CrossRefGoogle Scholar
  26. 26.
    Schramm U, Zhou M (eds) (2006) Recent developments in the commercial implementation of topology optimization. Springer, NetherlandsGoogle Scholar
  27. 27.
    Bendsøe MP, Kikuchi N (1988) Comput Methods Appl M 71:197. doi: CrossRefGoogle Scholar
  28. 28.
    Sigmund O (1994) J Intell Mater Syst Struct 5:736. doi: CrossRefGoogle Scholar
  29. 29.
    Cheng KT, Olhoff N (1981) Int J Solids Struct 17:305. doi: CrossRefGoogle Scholar
  30. 30.
    Markworth AJ, Ramesh KS, Parks WP (1995) J Mater Sci 30:2183. doi: CrossRefGoogle Scholar
  31. 31.
    Mori T, Tanaka K (1973) Acta Metall 21:571. doi: CrossRefGoogle Scholar
  32. 32.
    Hill R (1965) J Mech Phys Solids 13:213. doi: CrossRefGoogle Scholar
  33. 33.
    Reiter T, Dvorak GJ, Tvergaard V (1997) J Mech Phys Solids 45:1281. doi: CrossRefGoogle Scholar
  34. 34.
    Hashin Z, Shtrikman S (1962) J Appl Phys 33:3125. doi: CrossRefGoogle Scholar
  35. 35.
    Zhou SW, Li Q (2008) Compu Mater Sci in progressGoogle Scholar
  36. 36.
    Zhou M, Rozvany GIN (1991) Comput Methods Appl Mech Eng 89:309. doi: CrossRefGoogle Scholar
  37. 37.
    Stolpe M, Svanberg K (2001) Struct Multidiscip Optim 22:116. doi: CrossRefGoogle Scholar
  38. 38.
    Bendsoe MP, Sigmund O (1999) Arch Appl Mech 69:635. doi: CrossRefGoogle Scholar
  39. 39.
    Svanberg K (1987) Int J Numer Methods Eng 24:359. doi: CrossRefGoogle Scholar
  40. 40.
    Haug EJ, Choi KK, Komkov V (1986) Design sensitivity analysis of structural systems. Academic Press, OrlandoGoogle Scholar
  41. 41.
    Aubert G, Kornprobst P (2006) Mathematical problems in image processing: partial differential equations and the calculus of variations. Springer, New YorkGoogle Scholar
  42. 42.
    Wang MY, Zhou S, Ding H (2004) Struct Multidiscip Optim 28:262. doi: CrossRefGoogle Scholar
  43. 43.
    Bourdin B (2001) Int J Numer Methods Eng 50:2143. doi: CrossRefGoogle Scholar
  44. 44.
    Brandel B, Lakes RS (2001) J Mater Sci 36:5885. doi: CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.School of Aerospace, Mechanical and Mechatronic EngineeringThe University of SydneySydneyAustralia

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