Skip to main content
SpringerLink
Log in

Two-dimensional modeling of heterogeneous structures using graded finite element and boundary element methods

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

In the present work, graded finite element and boundary element methods capable of modeling behaviors of structures made of nonhomogeneous functionally graded materials (FGMs) composed of two constituent phases are presented. A numerical implementation of Somigliana’s identity in two-dimensional displacement fields of the isotropic nonhomogeneous problems is presented using the graded elements. Based on the constitutive and governing equations and the weighted residual technique, effective boundary element formulations are implemented for elastic nonhomogeneous isotropic solid models. Results of the finite element method are derived based on a Rayleigh–Ritz energy formulation. The heterogeneous structures are made of combined ceramic–metal materials, in which the material properties vary continuously along the in-plane or thickness directions according to a power law. To verify the present work, three numerical examples are provided in the paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

References

  1. Suresh S, Mortensen A (1998) Functionally graded materials. Institute of Materials. IOM Communications, London

    Google Scholar 

  2. Miyamoto Y, Kaysser WA, Rabin BH (1999) Functionally graded materials: design, processing and applications. Kluwer, Netherlands, Dordrecht

    Book  Google Scholar 

  3. Tornabene F (2009) Vibration analysis of functionally graded conical, cylindrical and annular shell structures with a four-parameter power-law distribution. Comput Methods Appl Mech Eng 198:2911–2935

    Article  ADS  MATH  Google Scholar 

  4. Tornabene F, Viola E (2009) Free vibration analysis of functionally graded panels and shells of revolution. Meccanica 44:255–281

    Article  MATH  Google Scholar 

  5. Tornabene F, Viola E (2009) Free vibrations of four-parameter functionally graded parabolic panels and shell of revolution. Eur J Mech A, Solids 28:991–1013

    Article  MATH  Google Scholar 

  6. Viola E, Tornabene F (2009) Free vibrations of three–parameter functionally graded parabolic panels of revolution. Mech Res Commun 36:587–594

    Article  Google Scholar 

  7. Malekzadeh P, Golbahar Haghighi MR, Atashi MM (2011) Free vibration analysis of elastically supported functionally graded annular plates subjected to thermal environment. Meccanica 46:893–913

    Article  MathSciNet  Google Scholar 

  8. Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics, 6th edn. Elsevier Butterworth-Heinemann, Oxford

    MATH  Google Scholar 

  9. Santare MH, Lambros J (2000) Use of graded finite elements to model the behavior of nonhomogeneous materials. J Appl Mech 67:819–822

    Article  MATH  Google Scholar 

  10. Chinosi C, Della Croce L (2007) Approximation of functionally graded plates with non-conforming finite elements. J Comput Appl Math 210:106–115

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Azadi M, Shariyat M (2010) Nonlinear transient transfinite element thermal analysis of thick-walled FGM cylinders with temperature-dependent material properties. Meccanica 45:305–318

    Article  MathSciNet  Google Scholar 

  12. Shariyat M (2012) A general nonlinear global-local theory for bending and buckling analyses of imperfect cylindrical laminated and sandwich shells under thermomechanical loads. Meccanica 47:301–319

    Article  MathSciNet  Google Scholar 

  13. Asemi K, Akhlaghi M, Salehi M (2012) Dynamic analysis of thick short length FGM cylinders. Meccanica 47:1441–1453

    Article  MathSciNet  Google Scholar 

  14. Nguyen–Xuan H, Tran LV, Thai CH, Nguyen-Thoi T (2012) Analysis of functionally graded plates by an efficient finite element method with node-based strain smoothing. Thin-Walled Struct 54:1–18

    Article  Google Scholar 

  15. Paris F, Canas J (1997) Boundary element method. Oxford University Press, New York

    MATH  Google Scholar 

  16. Aliabadi MH (2002) The boundary element method: applications in solids and structures. Wiley, New York

    MATH  Google Scholar 

  17. Aliabadi MH, Wen PH (2011) Boundary element methods in engineering and sciences. Imperial College Press, London

    Google Scholar 

  18. Kim J-H, Paulino GH (2002) Isoparametric graded finite elements for nonhomogeneous isotropic and orthotropic materials. J Appl Mech 69:502–514

    Article  MATH  Google Scholar 

  19. Tornabene F, Viola E, Inman DJ (2009) 2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical and annular shell structures. J Sound Vib 328:259–290

    Article  ADS  Google Scholar 

  20. Tornabene F, Liverani A, Caligiana G (2011) FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations. Int J Mech Sci 53:446–470

    Article  Google Scholar 

  21. Dave EV, Paulino GH, Buttlar WG (2011) Viscoelastic functionally graded finite element method using correspondence principle. J Mater Civ Eng 23:39–48

    Article  Google Scholar 

  22. Singh IV, Mishra BK, Bhattacharya S (2011) XFEM simulation of cracks, holes and inclusions in functionally graded materials. Int J Mech Mater Des 7:199–218

    Article  Google Scholar 

  23. Sladek J, Sladek V, Zhang C, Schanz M (2006) Meshless local Petrov-Galerkin method for continuously nonhomogeneous linear viscoelastic solids. Comput Mech 37:279–289

    Article  MATH  Google Scholar 

  24. Gray LJ, Kaplan T, Richardson JD, Paulino GH (2003) Green’s function and boundary integral analysis for exponentially graded materials: heat conduction. J Appl Mech 40:543–549

    Article  Google Scholar 

  25. Kuo HY, Chen T (2005) Steady and transient green’s functions for anisotropic conduction in an exponentially graded solid. Int J Solids Struct 42:1111–1128

    Article  MathSciNet  MATH  Google Scholar 

  26. Chan YS, Gray LJ, Kaplan T, Paulino GH (2004) Green’s function for a two-dimensional exponentially graded elastic medium. Proc R Soc A 460:1689–1706

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Criado R, Ortiz JE, Mantic V (2007) Boundary element analysis of three-dimensional exponentially graded isotropic elastic solids. Comput Model Eng Sci 22:151–164

    MathSciNet  MATH  Google Scholar 

  28. Criado R, Gray LJ, Mantic V, Paris F (2008) Green’s function evaluation for three-dimensional exponentially graded elasticity. Int J Numer Methods Eng 74:1560–1591

    Article  MathSciNet  MATH  Google Scholar 

  29. Gao XW, Zhang C, Sladek J, Sladek V (2008) Fracture analysis of functionally graded materials by a BEM. Compos Sci Technol 68:1209–1215

    Article  Google Scholar 

  30. Ashrafi H, Bahadori MR, Shariyat M (2012) Two-dimensional modeling of functionally graded viscoelastic materials using a boundary element approach. Adv Mater Res 463–464:570–574

    Article  Google Scholar 

  31. Li XF, Peng XL (2009) A pressurized functionally graded hollow cylinder with arbitrarily varying material properties. J Elast 96:81–95

    Article  MathSciNet  MATH  Google Scholar 

  32. Chen YZ, Lin XY (2008) Elastic analysis for thick cylinders and spherical pressure vessels made of functionally graded materials. Comput Mater Sci 44:581–587

    Article  Google Scholar 

  33. Hongjun X, Zhifei S, Taotao Z (2006) Elastic analyses of heterogeneous hollow cylinders. Mech Res Commun 33:681–691

    Article  MATH  Google Scholar 

  34. Tutuncu N, Ozturk M (2001) Exact solutions for stresses in functionally graded pressure vessels. Composites, Part B, Eng 32:683–686

    Article  Google Scholar 

  35. Horgan CO (1999) The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials. J Elast 55:43–59

    Article  MathSciNet  MATH  Google Scholar 

  36. Nie GJ, Batra RC (2010) Exact solutions and material tailoring for functionally graded hollow circular cylinders. J Elast 99:179–201

    Article  MathSciNet  MATH  Google Scholar 

  37. Gao XW (2002) The radial integration method for evaluation of domain integrals with boundary-only discretization. Eng Anal Bound Elem 26:905–916

    Article  MATH  Google Scholar 

  38. Ashrafi H, Farid M (2009) A mathematical boundary integral equation analysis of standard viscoelastic solid polymers. Comput Math Model 20:397–415

    Article  MATH  Google Scholar 

  39. Ashrafi H, Bahadori MR, Shariyat M (2012) Modeling of viscoelastic solid polymers using a boundary element formulation with considering a body load. Adv Mater Res 463:499–504

    Article  Google Scholar 

  40. Wang BL, Mai YW (2005) Transient one-dimensional heat conduction problems solved by finite element. Int J Mech Sci 47:303–317

    Article  MATH  Google Scholar 

  41. Shariyat M, Khaghani M, Lavasani SMH (2010) Nonlinear thermoelasticity, vibration, and stress wave propagation analyses of thick FGM cylinders with temperature-dependent material properties. Eur J Mech A, Solids 29:378–391

    Article  Google Scholar 

  42. Shariyat M, Lavasani SMH, Khaghani M (2010) Nonlinear transient thermal stress and elastic wave propagation analyses of thick temperature-dependent FGM cylinders, using a second-order point-collocation method. Appl Math Model 34:898–918

    Article  MathSciNet  MATH  Google Scholar 

  43. Hosseini SM, Sladek J, Sladek V (2011) Meshless local Petrov–Galerkin method for coupled thermoelasticity analysis of a functionally graded thick hollow cylinder. Eng Anal Bound Elem 35:827–835

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran

    H. Ashrafi & M. Shariyat

  2. Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran

    K. Asemi & M. Salehi

Authors
  1. H. Ashrafi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Asemi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Shariyat
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Salehi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Shariyat.

Appendix

Appendix

For the simplex linear triangular element, the following formulation is used.

(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
(A.14)

In absence of the body forces, the total potential energy of the element may be expressed as [8]:

(A.15)

Therefore, employing principle of minimum total potential energy gives the governing equations as:

(A.16)

Equation (A.16) may be represented by the following standard form:

(A.17)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ashrafi, H., Asemi, K., Shariyat, M. et al. Two-dimensional modeling of heterogeneous structures using graded finite element and boundary element methods. Meccanica 48, 663–680 (2013). https://doi.org/10.1007/s11012-012-9623-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-012-9623-5

Keywords

Search

Navigation