Skip to main content
SpringerLink
Log in

Stress analysis for multilayered coating systems using semi-analytical BEM with geometric non-linearities

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

Abstract

For a long time, most of the current numerical methods, including the finite element method, have not been efficient to analyze stress fields of very thin structures, such as the problems of thin coatings and their interfacial/internal mechanics. In this paper, the boundary element method for 2-D elastostatic problems is studied for the analysis of multi-coating systems. The nearly singular integrals, which is the primary obstacle associated with the BEM formulations, are dealt with efficiently by using a semi-analytical algorithm. The proposed semi-analytical integral formulas, compared with current analytical methods in the BEM literature, are suitable for high-order geometry elements when nearly singular integrals need to be calculated. Owing to the employment of the curved surface elements, only a small number of elements need to be divided along the boundary, and high accuracy can be achieved without increasing more computational efforts. For the test problems studied, very promising results are obtained when the thickness of coated layers is in the orders of 10−6–10−9, which is sufficient for modeling most coated systems in the micro- or nano-scales.

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

Abbreviations

BEM:

Boundary element method

FEM:

Finite element method

CBEM:

Conventional boundary element method

BIEs:

Boundary integral equations

MDBEM:

Multi-domain boundary element method

References

  1. Subramanian C, Strafford KN (1993) Review of multicomponent and multilayer coatings for tribological applications. Wear 165(1): 85–95

    Article  Google Scholar 

  2. Luo JF, Liu YJ, Berger EJ (2000) Interfacial stress analysis for multi-coating systems using an advanced boundary element method. Comput Mech 24(6): 448–455

    Article  MATH  Google Scholar 

  3. Djabella H, Arnell RD (1994) Finite element analysis of elastic stresses in multilayered systems. Thin Solid Films 245(1–2): 27–33

    Article  Google Scholar 

  4. Bouzakis KD, Vidakis N (1997) Prediction of the fatigue behaviour of physically vapour deposited coatings in the ball-on-rod rolling contact fatigue test, using an elastic-plastic finite elements method simulation. Wear 206(1): 197–203

    Article  Google Scholar 

  5. Basar Y, Ding Y (1996) Finite-element analysis of hyperelastic thin shells with large strains. Comput Mech 18(3): 200–214

    Article  MATH  Google Scholar 

  6. Igic PM, Mawby PA (1999) An advanced finite element strategy for thermal stress field investigation in aluminium interconnections during processing of very large scale integration multilevel structures. Microelectron J 30(12): 1207–1212

    Article  Google Scholar 

  7. Bennani HH, Takadoum J (1999) Finite element model of elastic stresses in thin coatings submitted to applied forces. Surf Coatings Technol 111(1): 80–85

    Article  Google Scholar 

  8. Providas E, Kattis MA (1999) A simple finite element model for the geometrically nonlinear analysis of thin shells. Comput Mech 24(2): 127–137

    Article  MATH  Google Scholar 

  9. Genes MC, Kocak S (2005) Dynamic soil–structure interaction analysis of layered unbounded media via a coupled finite element/boundary element/scaled boundary finite element model. Int J Numer Methods Eng 62(6): 798–823

    Article  MATH  Google Scholar 

  10. Moreira R, Rodrigues J, Ferreira A (2006) A generalized layerwise finite element for multi-layer damping treatments. Comput Mech 37(5): 426–444

    Article  MATH  Google Scholar 

  11. Bialas M (2008) Finite element analysis of stress distribution in thermal barrier coatings. Surf Coatings Technol 202(24): 6002–6010

    Article  Google Scholar 

  12. Ranjbar-Far M, Absi J, Mariaux G, Dubois F (2010) Simulation of the effect of material properties and interface roughness on the stress distribution in thermal barrier coatings using finite element method. Mater Des 31(2): 772–781

    Article  Google Scholar 

  13. Liu GR, Tani J, Watanabe K, Ohyoshi T (1990) A semi-exact method for the propagation of harmonic waves in anisotropic laminated bars of rectangular cross section. Wave Motion 12(4): 361–371

    Article  MATH  Google Scholar 

  14. Liu GR, Tani J, Ohyoshi T, Watanabe K (1991) Transient waves in anisotropic laminated plates, part 2: application. J Vib Acoust 113(2): 235–239

    Article  Google Scholar 

  15. Liu GR, Tani J, Watanabe K, Ohyoshi T (1990) Harmonic wave propagation in anisotropic laminated strips. J Sound Vib 139(2): 313–324

    Article  Google Scholar 

  16. Lam KY, Liu GR, Wang YY (1997) Time-harmonic response of a vertical crack in plates. Theor Appl Fract Mech 27(1): 21–28

    Article  Google Scholar 

  17. Xi ZC, Liu GR, Lam KY, Shang HM (2000) Dispersion and characteristic surfaces of waves in laminated composite circular cylindrical shells. J Acoust Soc Am 108(5): 2179–2186

    Article  Google Scholar 

  18. Liu GR, Wang XJ, Xi ZC (2000) Elastodynamic responses of an immersed to a gaussian beam pressure. J Sound Vib 233(5): 813–833

    Article  Google Scholar 

  19. Liu GR, Achenbach JD, Kim JO, Li ZL (1992) A combined finite element method/boundary element method technique for V(z) curves of anisotropic-layer/substrate configurations. J Acoust Soc Am 92(5): 2734–2740

    Article  Google Scholar 

  20. Brebbia CA, Telles JCF, Wrobel LC (1984) Boundary element techniques: theory and applications in engineering. Springer, New York

    MATH  Google Scholar 

  21. Cruse TA (1988) Boundary element analysis in computational fracture mechanics. Boston, Dordrecht

    MATH  Google Scholar 

  22. Cheng AHD, Cheng DT (2005) Heritage and early history of the boundary element method. Eng Analysis Boundary Elements 29(3): 268–302

    Article  MATH  Google Scholar 

  23. Cruse TA, Aithal R (1993) Non-singular boundary integral equation implementation. Int J Numer Methods Eng 36(2): 237–254

    Article  Google Scholar 

  24. Sladek V, Sladek J (1998) Singular integrals and boundary elements. Comput Methods Appl Mech Eng 157(3–4): 251–266

    Article  MATH  MathSciNet  Google Scholar 

  25. Luo JF, Liu YJ, Berger EJ (1998) Analysis of two-dimensional thin structures (from micro- to nano-scales) using the boundary element method. Comput Mech 22(5): 404–412

    Article  MATH  Google Scholar 

  26. Zhang YM, Gu Y (2010) An effective method in BEM for potential problems of thin bodies. J Mar Sci Technol 18(1): 137–144

    Google Scholar 

  27. Jun L, Beer G, Meek JL (1985) Efficient evaluation of integrals of order 1/r 1/r2, 1/r3 using Gauss quadrature. Eng Analysis 2(3): 118–123

    Article  Google Scholar 

  28. Gao XW, Yang K, Wang J (2008) An adaptive element subdivision technique for evaluation of various 2D singular boundary integrals. Eng Analysis Boundary Elements 32(8): 692–696

    Article  Google Scholar 

  29. Lutz E (1992) Exact Gaussian quadrature methods for near-singular integrals in the boundary element method. Eng Analysis Boundary Elements 9(3): 233–245

    Article  Google Scholar 

  30. Telles JCF (1987) A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals. Int J Numer Methods Eng 24(5): 959–973

    Article  MATH  Google Scholar 

  31. Nagarajan A, Mukherjee S (1993) A mapping method for numerical evaluation of two-dimensional integrals with 1/r singularity. Comput Mech 12(1): 19–26

    Article  MATH  MathSciNet  Google Scholar 

  32. Huang Q, Cruse TA (1993) Some notes on singular integral techniques in boundary element analysis. Int J Numer Methods Eng 36(15): 2643–2659

    Article  MATH  MathSciNet  Google Scholar 

  33. Sladek V, Sladek J, Tanaka M (2000) Optimal transformations of the integration variables in computation of singular integrals in BEM. Int J Numer Methods Eng 47(7): 1263–1283

    Article  MATH  Google Scholar 

  34. Ma H, Kamiya N (2002) A general algorithm for the numerical evaluation of nearly singular boundary integrals of various orders for two- and three-dimensional elasticity. Comput Mech 29(4): 277–288

    Article  MATH  Google Scholar 

  35. Johnston BM, Johnston PR, Elliott D (2007) A sinh transformation for evaluating two-dimensional nearly singular boundary element integrals. Int J Numer Methods Eng 69(7): 1460–1479

    Article  MATH  MathSciNet  Google Scholar 

  36. Ye WJ (2008) A new transformation technique for evaluating nearly singular integrals. Comput Mech 42(3): 457–466

    Article  MATH  Google Scholar 

  37. Zhang YM, Gu Y, Chen JT (2009) Boundary layer effect in BEM with high order geometry elements using transformation. Comput Model Eng Sci 45(3): 227–247

    MathSciNet  Google Scholar 

  38. Zhang YM, Gu Y, Chen JT (2010) Boundary element analysis of the thermal behaviour in thin-coated cutting tools. Eng Analysis Boundary Elements 34(9): 775–784

    Article  MathSciNet  Google Scholar 

  39. Yoon SS, Heister SD (2000) Analytic solution for fluxes at interior points for the 2D Laplace equation. Eng Analysis Boundary Elements 24(2): 155–160

    Article  MATH  Google Scholar 

  40. Fratantonio M, Rencis JJ (2000) Exact boundary element integrations for two-dimensional Laplace equation. Eng Analysis Boundary Elements 24(4): 325–342

    Article  MATH  Google Scholar 

  41. Liu YJ, Rizzo FJ (1992) A weakly singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems. Comput Methods Appl Mech Eng 96(2): 271–287

    Article  MATH  MathSciNet  Google Scholar 

  42. Niu Z, Wendland WL, Wang X, Zhou H (2005) A semi-analytical algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods. Comput Methods Appl Mech Eng 194(9–11): 1057–1074

    Article  MATH  Google Scholar 

  43. Schulz H, Schwab C, Wendland WL (1998) The computation of potentials near and on the boundary by an extraction technique for boundary element methods. Comput Methods Appl Mech Eng 157(3–4): 225–238

    Article  MATH  MathSciNet  Google Scholar 

  44. Zhou HL, Niu ZR, Cheng CZ, Guan ZW (2008) Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems. Comput Struct 86(15): 1656–1671

    Article  Google Scholar 

  45. Zhang YM, Gu Y, Chen JT (2009) Analysis of 2D thin walled structures in BEM with high-order geometry elements using exact integration. Comput Model Eng Sci 50(1): 1–20

    MATH  MathSciNet  Google Scholar 

  46. Sladek V, Sladek J, Tanaka M (1993) Nonsingular BEM formulations for thin-walled structures and elastostatic crack problems. Acta Mech 99(1): 173–190

    Article  MATH  MathSciNet  Google Scholar 

  47. Liu YJ (1998) Analysis of shell-like structures by the Boundary Element Method based on 3-D elasticity: formulation and verification. Int J Numer Methods Eng 41(3): 541–558

    Article  MATH  Google Scholar 

  48. Du F, Lovell MR, Wu TW (2000) Boundary element method analysis of temperature fields in coated cutting tools. Int J Solids Struct 38: 4557–4570

    Article  Google Scholar 

  49. Chen XL, Liu YJ (2001) Thermal stress analysis of multi-layer thin films and coatings by an advanced boundary element method. Comput Model Eng Sci 2(3): 337–349

    MATH  Google Scholar 

  50. Fairweather G, Karageorghis A (1998) The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 9(1): 69–95

    Article  MATH  MathSciNet  Google Scholar 

  51. Chen CS (1995) The method of fundamental solutions for non-linear thermal explosions. Commun Numer Methods Eng 11(8): 675–681

    Article  MATH  Google Scholar 

  52. Chen JT, Chang MH, Chen KH, Lin SR (2002) The boundary collocation method with meshless concept for acoustic eigenanalysis of two-dimensional cavities using radial basis function. J Sound Vib 257(4): 667–711

    Article  Google Scholar 

  53. Chen W, Tanaka M (2002) A meshless, integration-free, and boundary-only RBF technique. Comput Math Appl 43(3–5): 379–391

    Article  MATH  MathSciNet  Google Scholar 

  54. Young DL, Chen KH, Lee CW (2005) Novel meshless method for solving the potential problems with arbitrary domain. J Comput Phys 209(1): 290–321

    Article  MATH  Google Scholar 

  55. Chen W (2002) Meshfree boundary particle method applied to Helmholtz problems. Eng Analysis Boundary Elements 26(7): 577–581

    Article  MATH  Google Scholar 

  56. Chen W, Wang FZ (2010) A method of fundamental solutions without fictitious boundary. Eng Analysis Boundary Elements 34(5): 530–532

    Article  Google Scholar 

  57. Fu Z, Chen W, Yang W (2009) Winkler plate bending problems by a truly boundary-only boundary particle method. Comput Mech 44(6): 757–763

    Article  MATH  MathSciNet  Google Scholar 

  58. Liu YJ (2010) A new boundary meshfree method with distributed sources. Eng Analysis Boundary Elements 34(11): 914–919

    Article  Google Scholar 

  59. Berger JR, Karageorghis A (1999) The method of fundamental solutions for heat conduction in layered materials. Int J Numer Methods Eng 45(11): 1681–1694

    Article  MATH  Google Scholar 

  60. Berger JR, Karageorghis A (2001) The method of fundamental solutions for layered elastic materials. Eng Analysis Boundary Elements 25(10): 877–886

    Article  MATH  Google Scholar 

  61. Zhang YM, Wen WD, Wang LM, Zhao XQ (2004) A kind of new nonsingular boundary integral equations for elastic plane problems. Chin J Theor Appl Mech 36(3): 311–321

    Google Scholar 

  62. Gao XW, Yang K (2009) Interface integral BEM for solving multi-medium elasticity problems. Comput Methods Appl Mech Eng 198(15–16): 1429–1436

    Article  Google Scholar 

  63. Gao XW, Guo L, Zhang C (2007) Three-step multi-domain BEM solver for nonhomogeneous material problems. Eng Analysis Boundary Elements 31(12): 965–973

    Article  Google Scholar 

  64. Johnson KL (1985) Contact mechanics. Cambridge University Press, London

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, Shandong University of Technology, Zibo, 255049, Shandong Province, China

    Yao-Ming Zhang & Yan Gu

  2. Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, 20224, Taiwan

    Jeng-Tzong Chen

Authors
  1. Yao-Ming Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jeng-Tzong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yao-Ming Zhang.

Additional information

The support of the National Natural Science Foundation of China (10571110), the National Natural Science Foundation of Shangdong Province of China (ZR2010AZ003), and the Opening Fund of the State Key Laboratory of Structural Analysis for Industrial Equipment (GZ1017) are gratefully acknowledged.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, YM., Gu, Y. & Chen, JT. Stress analysis for multilayered coating systems using semi-analytical BEM with geometric non-linearities. Comput Mech 47, 493–504 (2011). https://doi.org/10.1007/s00466-010-0559-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-010-0559-0

Keywords

Search

Navigation