Computational Mechanics

, Volume 56, Issue 5, pp 739–752 | Cite as

Strict upper and lower bounds of stress intensity factors at 2D elastic notches based on constitutive relation error estimation

Original Paper
First Online:
Received:
Accepted:
  • 173 Downloads

Abstract

This paper aims to evaluate the stress intensity factors (SIFs) at 2D elastic notches that are of concerns in structural failure analysis. Strict upper and lower bounds of the SIFs are acquired by a unified approach within the framework of the constitutive relation error (CRE) estimation. The main ingredient is to gain a unified representation of the SIFs which is achieved by a path independent integral with the aid of the specialized auxiliary fields. With the unified approach, one can (1) Establish the dual problem based on the unified representation of the SIFs; (2) Perform dual error analysis, resulting in the mixture of errors in both primal and dual problems; (3) Acquire strict upper and lower bounds of the SIFs by utilizing the featured strict bounding property of the CRE estimation. Numerical examples are studied to illustrate the strict bounding properties of SIFs at cracks and notches.

Keywords

Stress intensity factors (SIFs) at notches Strict upper and lower bounds A unified approach CRE estimation  Path independent integral 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

The present investigation was performed with the support of the National Natural Foundation of China (No. 51178247, No. 51378294).

References

  1. 1.
    Brahtz JHA (1933) Stresses distribution in a reentrant corner. Trans ASME (J Appl Mech) 55:31–37Google Scholar
  2. 2.
    Williams ML (1952) Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J Appl Mech 19:526–528Google Scholar
  3. 3.
    Sinclair GB, Okajima M, Griffin JH (1984) Path independent integrals for computing stress intensity factors at sharp notches in elastic plates. Int J Numer Methods Eng 20:999–1008MATHCrossRefGoogle Scholar
  4. 4.
    Lazzarin P, Tovo R (1998) A notch intensity factor approach to the stress analysis of welds. Fatigue Fract Eng Mater Struct 21:1089–1103CrossRefGoogle Scholar
  5. 5.
    Babuška I, Miller A (1984) The post-processing approach in the finite element method–part 2: the calculation of stress intensity factors. Int J Numer Methods Eng 20:1111–1129MATHCrossRefGoogle Scholar
  6. 6.
    Freund LB (1978) Stress intensity factor calculations based on a conservation integral. Int J Solids Struct 14:241–250CrossRefGoogle Scholar
  7. 7.
    Stern M, Becker EB, Dunham RS (1976) A contour integral computation of mixed-mode stress intensity factors. Int J Fract 12:359–368Google Scholar
  8. 8.
    Gallimard L, Panetier J (2006) Error estimation of stress intensity factors for mixed-mode cracks. Int J Numer Methods Eng 68:299–316MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Babuška I, Miller A (1984) The post-processing approach in the finite element method–part 1: calculation of displacements, stresses and other higher order derivatives of the displacements. Int J Numer Methods Eng 20:1111–1129MATHCrossRefGoogle Scholar
  10. 10.
    Kim JH, Paulino GH (2003) T-stress, mixed-mode stress intensity factors and crack initiation angles in functionally graded materials: a unified approach using the interaction integral method. Comput Methods Appl Mech Eng 192:1463–1494MATHCrossRefGoogle Scholar
  11. 11.
    Rethore J, Gravouil A, Morestin F, Combescure A (2005) Estimation of mixed-mode stress intensity factors using digital image correlation and an interaction integral. Int J Fract 132:65–79CrossRefGoogle Scholar
  12. 12.
    Ladevèze P, Pelle JP (2004) Mastering calculations in linear and nonlinear mechanics. Springer, New YorkGoogle Scholar
  13. 13.
    Ainsworth A, Oden JT (2000) A posterior error estimation in finite element analysis. Wiley, New YorkCrossRefGoogle Scholar
  14. 14.
    Zienkiewicz OC, Zhu JZ (1987) A simple error estimator and adaptive procedure for practical engineering analysis. Int J Numer Methods Eng 24:337–357MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Babuška I, Rheinboldt WC (1978) A posterior estimators for the finite element method. Int J Numer Methods Eng 12:1597–1615MATHCrossRefGoogle Scholar
  16. 16.
    Ladevèze P, Leguillon D (1983) Error estimate procedure in the finite element method and application. SIAM J Numer Anal 20(3):485–509MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Almeida JPM, Pereira OJBA (2006) Upper bounds of the error in local quantities using equilibrated and compatible finite element solutions for linear elastic problems. Comput Methods Appl Mech Eng 195:279–296MATHCrossRefGoogle Scholar
  18. 18.
    Wang L, Zhong H (2014) A traction-based equilibrium finite element free from spurious kinematic modes for linear elasticity problems. Int J Numer Methods Eng 99:763–788CrossRefMathSciNetGoogle Scholar
  19. 19.
    Xuan ZC, Parés N, Peraire J (2006) Computing upper and lower bounds for the J -integral in two-dimensional linear elasticity. Comput Methods Appl Mech Eng 195:430–443MATHCrossRefGoogle Scholar
  20. 20.
    Patera AT, Peraire J (2002) A general Lagrangian formulation for the computation of a posteriori finite element bounds. Chapter in “Error estimation and adaptive discretization methods in CFD”. In: Barth T, Deconinck H (eds) Lecture notes in computational science and engineering, vol 25. Springer, HeidelbergGoogle Scholar
  21. 21.
    Ladevèze P (2006) Upper error bounds on calculated outputs of interest for linear and nonlinear structural problems. C R Mec 334:399–407MATHCrossRefGoogle Scholar
  22. 22.
    Fraeijs de Veubeke B (1964) Upper and lower bounds in matrix structural analysis. AGARDograf 72:165–201Google Scholar
  23. 23.
    Kempeneers M, Debongnie JF, Beckers P (2010) Pure equilibrium tetrahedral finite elements for global error estimation by dual analysis. Int J Numer Methods Eng 81:513–536MATHMathSciNetGoogle Scholar
  24. 24.
    Almeida JPM, Maunder EAW (2009) Recovery of equilibrium on star patches using a partition of unity technique. Int J Numer Methods Eng 79:1493–1516MATHCrossRefGoogle Scholar
  25. 25.
    Ladevèze P, Maunder EAW (1996) A general method for recovering equilibrating element tractions. Comput Methods Appl Mech Eng 137:111–151MATHCrossRefGoogle Scholar
  26. 26.
    Pled F, Chamoin L, Ladevèze P (2011) On the technique for constructing admissible stress fields in model verification: performances on engineering examples. Int J Numer Methods Eng 88:409–441MATHCrossRefGoogle Scholar
  27. 27.
    Ladevèze P, Chamoin L, Florentin É (2010) A new non-intrusive technique for the construction of admissible stress fields in model verification. Comput Methods Appl Mech Eng 199:766–777MATHCrossRefGoogle Scholar
  28. 28.
    Pares N, Diez P, Huerta A (2006) Subdomain-based flux-free a posterior error estimators. Comput Methods Appl Mech Eng 195(4–6):297–323MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Wang L, Zhong H (2015) A unified approach to strict upper and lower bounds of quantities in linear elasticity based on constitutive relation error estimation. Comput Methods Appl Mech Eng 286:332–353CrossRefMathSciNetGoogle Scholar
  30. 30.
    Giles MB, Süli E (2002) Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numer 11:145–236MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Pierce NA, Giles MB (2000) Adjoint recovery of superconvergent functionals from PDE approximations. SIAM Rev 42(2):247–264CrossRefMathSciNetGoogle Scholar
  32. 32.
    Oden JT, Prudhomme S (2001) Goal-oriented error estimation and adaptivity for the finite element method. Comput Math Appl 41:735–756MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Chamoin L, Ladevèze P (2009) Strict and practical bounds through a non-intrusive and goal-oriented error estimation method for linear viscoelasticity problems. Finite Elem Anal Des 45(4):251–262CrossRefGoogle Scholar
  34. 34.
    Panetier J, Ladevèze P, Chamoin L (2010) Strict and effective bounds in goal-oriented error estimation applied to fracture mechanics problems solved with XFEM. Int J Numer Methods Eng 81:671–700MATHGoogle Scholar
  35. 35.
    Ladevèze P, Chamoin L (2010) Calculation of strict error bounds for finite element approximations of nonlinear pointwise quantities of interest. Int J Numer Methods Eng 84:1638–1664MATHCrossRefGoogle Scholar
  36. 36.
    Ladevèze P, Blaysat B, Florentin É (2012) Strict upper bounds of the error in calculated outputs of interest for plasticity problems. Comput Methods Appl Mech Eng 245–246:194–205CrossRefGoogle Scholar
  37. 37.
    Waeytens J, Chamoin L, Ladevèze P (2012) Guaranteed error bounds on pointwise quantities of interest for transient viscodynamics problems. Comput Mech 49:291–307MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Ladevèze P (2008) Strict upper error bounds on computed outputs of interest in computational structural mechanics. Comput Mech 42:271–286Google Scholar
  39. 39.
    Seweryn A, Molski K (1996) Elastic stress singularities and corresponding generalized stress intensity factors for angular corners under various boundary conditions. Eng Fract Mech 55:529–556CrossRefGoogle Scholar
  40. 40.
    Gregory RD (1979) Green’s functions, bi-linear forms, and completeness of the eigenfunctions for the elastostatic strip and wedge. J Elast 9:283–309MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Filippi S, Lazzarin P, Tovo R (2002) Developments of some explicit formulas useful to describe elastic stress field ahead of notches in plates. Int J Solids Struct 39:4543–4565MATHCrossRefGoogle Scholar
  42. 42.
    Bui HD (1994) Inverse problems in the mechanics of materials, an introduction. CRC Press, Boca Raton, pp 20–21Google Scholar
  43. 43.
    Gross R, Mendelson A (1972) Plane elastostatic analysis of V-notched plates. Int J Fract Mech 8:267–276CrossRefGoogle Scholar
  44. 44.
    Gratsch T, Hartmann F (2003) Finite element recovery techniques for local quantities of linear problems using fundamental solutions. Comput Mech 33:15–21CrossRefMathSciNetGoogle Scholar
  45. 45.
    Hwu C, Kuo TL (2007) A unified definition for stress intensity factors of interface corners and cracks. Int J Solids Struct 44:6340–6359MATHCrossRefGoogle Scholar
  46. 46.
    Dunn ML, Suwito W, Cunningham S (1997) Stress intensities at notch singularities. Eng Fract Mech 57:417–430CrossRefGoogle Scholar
  47. 47.
    Chen DH (1995) Stress intensity factors for V-notched strip under tension or in-plane bending. Int J Fract 70:81–97CrossRefGoogle Scholar
  48. 48.
    Treifi M, Oyadiji SO, Tsang DKL (2009) Computation of the stress intensity factors of double-edge and centre V-notches plates under tension and anti-plane shear by the fractal like finite element method. Eng Fract Mech 76:2091–2108CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Civil EngineeringTsinghua UniversityBeijingChina

Personalised recommendations