Skip to main content
SpringerLink
Log in

Elastic Wave Propagation in a Class of Cracked, Functionally Graded Materials by BIEM

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

Elastic wave propagation in cracked, functionally graded materials (FGM) with elastic parameters that are exponential functions of a single spatial co-ordinate is studied in this work. Conditions of plane strain are assumed to hold as the material is swept by time-harmonic, incident waves. The FGM has a fixed Poisson’s ratio of 0.25, while both shear modulus and density profiles vary proportionally to each other. More specifically, the shear modulus of the FGM is given as μ (x)=μ 0 exp (2ax 2), where μ 0 is a reference value for what is considered to be the isotropic, homogeneous material background. The method of solution is the boundary integral equation method (BIEM), an essential component of which is the Green’s function for the infinite inhomogeneous plane. This solution is derived here in closed-form, along with its spatial derivatives and the asymptotic form for small argument, using functional transformation methods. Finally, a non-hypersingular, traction-type BIEM is developed employing quadratic boundary elements, supplemented with special edge-type elements for handling crack tips. The proposed methodology is first validated against benchmark problems and then used to study wave scattering around a crack in an infinitely extending FGM under incident, time-harmonic pressure (P) and vertically polarized shear (SV) waves. The parametric study demonstrates that both far field displacements and near field stress intensity factors at the crack-tips are sensitive to this type of inhomogeneity, as gauged against results obtained for the reference homogeneous material case

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

Similar content being viewed by others

References

  1. Ang WT, Clements D, Vahdati N (2003) A dual-reciprocity boundary element method for a class of elliptic boundary value problems for non-homogeneous anisotropic media. Eng Anal Boundary Elem 27:49–55

    Article  MATH  Google Scholar 

  2. Boore DM (1972) Finite difference methods for seismic wave propagation in heterogeneous materials. In: Bolt BA (eds) Methods in computational physics, vol 11. Academic Press, New York, pp 1–37

    Google Scholar 

  3. Chen YF, Erdogan F (1996) The interface crack problem for a nonhomogeneous coating bonded to a homogeneous substrate. J Mech Phys Solids 44:771–787

    Article  Google Scholar 

  4. Itagaki M (2000) Advanced dual-reciprocity method based on polynomial source and its application to eigenvalue problem for non-uniform media. Eng Anal Boundary Elem 24:169–176

    Article  MATH  Google Scholar 

  5. Manolis GD, Shaw RP (1996) Green’s function for the vector wave equation in a mildly heterogeneous continuum. Wave Motion 24:59–83

    Article  MATH  MathSciNet  Google Scholar 

  6. Rangelov TV, Manolis GD, Dineva PS (2005) Elastodynamic fundamental solutions for certain families of 2D inhomogeneous anisotropic domains: basic derivations. Eur J Mech/A Solids 24:820–836

    Article  MathSciNet  MATH  Google Scholar 

  7. Tanaka M, Matsumoto T, Suda Y (2001) A dual-reciprocity boundary element method applied to the steady-state heat conduction problem of functionally gradient materials. In: Brebbia CA (eds) Proceedings of BETEQ 2001, 2nd international conference on boundary element techniques. WIT Press, Southampton

    Google Scholar 

  8. Yue ZQ, Xiao HT, Tham LG (2003) Boundary element analysis of crack problems in functionally graded materials. Int J Solids Struct 40:3273–3291

    Article  MATH  Google Scholar 

  9. Ang WT, Clements DL (1987) On some crack problems for inhomogeneous elastic materials. Int J Solids Struct 23:1089–1104

    Article  MATH  Google Scholar 

  10. Ang WT, Clements DL, Cooke T (1999) A hypersingular boundary integral equation for anti-plane crack problems for a class of inhomogeneous anisotropic elastic materials. Eng Anal Boundary Elem 23:567–572

    Article  MATH  Google Scholar 

  11. Manolis GD (2003) Elastic wave scattering around cavities in inhomogeneous continua by the BEM. J Sound Vib 266:281–305

    Article  Google Scholar 

  12. Manolis GD, Dineva PS, Rangelov TV (2004) Wave scattering by cracks in inhomogeneous continua using BIEM. Int J Solids Struct 41:3905–3927

    Article  MATH  Google Scholar 

  13. Sladek J, Sladek V, Zhang C (2003) Dynamic response of a crack in a functionally graded material under anti-plane shear impact load. Key Eng Mater 251 & 252:123–129

    Google Scholar 

  14. Sladek J, Sladek V, Zhang C (2005) An advanced numerical method for computing elastodynamic fracture parameters in functionally graded materials. Comput Mater Sci 32: 532–543

    Article  Google Scholar 

  15. Sladek J, Sladek V, Zhang C (2005) Crack analysis of anisotropic functionally graded materials by meshless local Petrov-Galerkin method. In: Aliabadi MH, Buchholtz FG, Alfaiate J, Planas J, Abersek B, Nishida S (eds) Advances in fracture and damage mechanics IV. EC Press, London, pp 267–273

    Google Scholar 

  16. Sladek J, Sladek V, Zhang C (2005) Stress analysis in anisotropic functionally graded materials by the MLPG method. Eng Anal Boundary Elem 29:597–609

    Article  Google Scholar 

  17. Zhang C, Sladek J, Sladek V (2003) Effect of material gradients on transient dynamic mode-III: Stress intensity factor in FGM. J Solids Struct 40:5252–5270

    Google Scholar 

  18. Zhang C, Sladek J, Sladek V (2003) Numerical analysis of cracked functionally graded materials. Key Eng Mater 251 & 252:463–471

    Google Scholar 

  19. Zhang C, Sladek J, Sladek V (2004) Crack analysis in unidirectionally and bidimensionally functionally graded materials. Int J Fracture 129:385–406

    Article  Google Scholar 

  20. Zhang C, Sladek J, Sladek V (2004) 2-D elastodynamic crack analysis in FGMs by a time-domain BIEM. In: Leitao VMA, Aliabadi MH (eds) Advances in boundary element techniques. EC Press, London, pp 181–190

    Google Scholar 

  21. Zhang C, Sladek J, Sladek V (2004) A time-domain BIEM for crack analysis in FGMs under dynamic loading. In: Yao ZH, Yuan MW, Zhong WX (eds) Computational mechanics. Tsinghua University Press/Springer, Berlin Heidelberg New York, CD-ROM paper no. 447

    Google Scholar 

  22. Zhang C, Sladek J, Sladek V (2005) Transient dynamic analysis of cracked functionally graded materials. In: Aliabadi MH, Buchholtz FG, Alfaiate J, Planas J, Abersek B, Nishida S (eds) Advances in fracture and damage mechanics IV. EC Press, London, pp 301–308

    Google Scholar 

  23. Guo LC, Wu LZ, Zeng T, Ma, L (2004) Mode I crack problem for a functionally graded orthotropic strip. Eur J Mech/A Solids 23:219–234

    Article  MATH  Google Scholar 

  24. Rousseau CE, Tippur HV (2001) Dynamic fracture of compositionally graded materials with cracks along the elastic gradient: experiments and analysis. Mech Mater 33:403–421

    Article  Google Scholar 

  25. DeHoop AT (1995) Handbook of radiation and scattering of waves. Academic Press, London

    Google Scholar 

  26. Hook JF (1962) Green’s function for axially symmetric elastic waves in unbounded inhomogeneous media having constant velocity gradients. Trans ASME, J Appl Mech E-29:293–298

    MATH  MathSciNet  Google Scholar 

  27. Moodie TB, Barclay DW, Haddow JB (1979) The propagation and reflection of cylindrical symmetric waves in inhomogeneous anisotropic elastic materials. Int J Eng Sci 17:95–105

    Article  MATH  Google Scholar 

  28. Vrettos C (1990) Dispersive SH-surface waves in soil deposits of variable shear modulus. Soil Dyn Earthquake Eng 9:255–264

    Google Scholar 

  29. Vrettos C (1991) In-plane vibrations of soil deposits with variable shear modulus. II: line load. Int J Numerical Anal Methods Geomech 14:649–662

    Article  Google Scholar 

  30. Watanabe K (1982) Transient response of an inhomogeneous elastic solid to an impulsive SH-source (Variable SH-wave velocity). Bull JSME 25–201:315–320

    Google Scholar 

  31. Watanabe K (1982) Scattering of SH-wave by a cylindrical discontinuity in an inhomogeneous elastic medium. Bull JSME 25–205:1055–1060

    Google Scholar 

  32. Watanabe K, Takeuchi T (2002) Green’s function for two-dimensional waves in a radially inhomogeneous elastic solid. In: Watanabe K, Ziegler F (eds) IUTAM symposium on dynamics of advanced materials and smart structures, pp 459–468

  33. Wang CY, Achenbach JD (1994) Elastodynamic fundamental solutions for anisotropic solids. Geophys J Int 118:384–392

    Google Scholar 

  34. Vladimirov V (1984) Equations of mathematical physics. Mir Publishers, Moscow

    Google Scholar 

  35. Ludwig D (1966) The Radon transform in Euclidean space. Commun Pure Appl Math 29:49–81

    MathSciNet  Google Scholar 

  36. Mathematica 5.0 (2003) Wolfram Research. Champaign, Illinois

    Google Scholar 

  37. Zhang C, Gross D (1998) On wave propagation in elastic solids with cracks. Computational Mechanics Publications, Southampton

    MATH  Google Scholar 

  38. Rangelov TV, Dineva PS, Gross D (2003) A hypersingular traction boundary integral equation method for stress intensity factor computation in a finite cracked body. Eng Anal Boundary Elem 27:9–21

    Article  MATH  Google Scholar 

  39. Aliabadi M, Rooke D (1991) Numerical fracture mechanics. Computational Mechanics Publications, Southampton

    MATH  Google Scholar 

  40. Chen EP, Sih GC (1977) Scattering waves about stationary and moving cracks. In: Sih GC (eds) Mechanics of fracture: elastodynamic crack problems. Noordhoff Publishers, Leyden

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Continuum Mechanics, Institute of Mechanics, Bulgarian Academy of Sciences, 1113, Sofia, Bulgaria

    P. S. Dineva

  2. Department of Mathematical Physics, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113, Sofia, Bulgaria

    T. V. Rangelov

  3. Department of Civil Engineering, Aristotle University, Thessaloniki, 54006, Greece

    G. D. Manolis

Authors
  1. P. S. Dineva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. V. Rangelov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. D. Manolis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. D. Manolis.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dineva, P.S., Rangelov, T.V. & Manolis, G.D. Elastic Wave Propagation in a Class of Cracked, Functionally Graded Materials by BIEM. Comput Mech 39, 293–308 (2007). https://doi.org/10.1007/s00466-005-0027-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-005-0027-4

Keywords

Search

Navigation