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International Journal of Fracture

, Volume 147, Issue 1–4, pp 269–283 | Cite as

Material forces for crack analysis of functionally graded materials in adaptively refined FE-meshes

  • Rolf Mahnken
Original Paper

Abstract

This work describes the computation of fracture parameters in functionally graded materials (FGMs) with stationary cracks. To this end the continuum concept of material forces is employed, such that the corresponding balance equation can be discretized with a standard Galerkin finite element procedure. A domain-type formulation is used for evaluation of a vectorial J-integral, where in the practical implementation the material nodal forces of the finite element discretization are summed up in a finite region of the crack-tip. In this way the numerical calculation is completely independent from the alignment of the finite element mesh or any selected integration contour, which is most attractive for adaptively refined finite element meshes. For illustrative purpose the accuracy of the method is discussed for two examples based on comparison with available theoretical and numerical solutions.

Keywords

Material forces Finite elements Stress intensity factor J-integral 

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References

  1. Ainsworth M, Zhu JZ, Craig AW, Zienkiewicz OC (1989) Analysis of the Zienkiewicz-Zhu a-posteriori error estimator in the finite element method. Int J Numer Meth Eng 28: 2161–2174CrossRefGoogle Scholar
  2. Anlas G, Santare MH, Lambros J (2000) Numerical calculation of stress intensity factors in functionally graded materials. Int J Fract 104: 131–143CrossRefGoogle Scholar
  3. Bendsoe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer-Verlag, BerlinGoogle Scholar
  4. Braun M (1997) Configurational forces induced by finite-element discretization. Proc Estonian Acad Sci Phys Math 46: 24–31Google Scholar
  5. Chadwick P (1975) Applications of an energy-momentum tensor in non-linear elastostatics. J Elast 5: 249–258CrossRefGoogle Scholar
  6. Cherepanov GP (1967) Crack propagation in continuous media. J Appl Math Mech 31: 503–512CrossRefGoogle Scholar
  7. Cherepanov GP (1968) Cracks in solids. Int J Solids Struct 4: 811–831CrossRefGoogle Scholar
  8. Delale F, Erdogan F (1983) Fracture mechanics of functionally graded materials. J Appl Mech 50: 609–614CrossRefGoogle Scholar
  9. Denzer R (2006) Computational configurational forces. Dissertation, Technical University of Kaiserslautern, Report No.: UKL/LTM T 06-04, Lehrstuhl für Technische MechanikGoogle Scholar
  10. Denzer R, Barth FJ, Steinmann P (2003) Studies in elastic fracture mechanics based on the material force method. Int J Num Meth Eng 58: 1817–1835CrossRefGoogle Scholar
  11. Eischen JW (1987a) Fracture of non-homogeneous materials. Int J Fract 34: 3–22Google Scholar
  12. Eischen JW (1987b) An improved method for computing the J 2 integral. Eng Fract Mech 26: 691–700CrossRefGoogle Scholar
  13. Erdogan F (1995) Fracture mechanics of functionally graded materials. Composit Eng 5: 753–770Google Scholar
  14. Erdogan F, Wu BH (1997) The surface crack problem for a plate with functionally graded materials. ASME J Appl Mech 64: 449–456Google Scholar
  15. Eshelby JD (1951) The force on an elastic singularity. Philos Trans Roy Soc, Math Phys Sci A 244: 87–112CrossRefGoogle Scholar
  16. Eshelby JD (1965) The continuum theory of lattice defects. Prog Solid States Phys 3: 79–114CrossRefGoogle Scholar
  17. Giannakopoulos AE, Surush S, Finot M, Olsson M (1995) Elastoplastic analysis of thermal cycling: layered materials with compositional gradients. Acta Metall Mater 43(4): 1335–1354CrossRefGoogle Scholar
  18. Gu P, Dao M, Asaro RJ (1999) A simplified method for calculating the crack-tip field of functionally graded materials using the domain integral. ASME J Appl Mech 66(1): 101–108CrossRefGoogle Scholar
  19. Gurtin ME (2000) Configurational forces as basic concepts of continuum physics. Springer-VerlagGoogle Scholar
  20. Haddi A, Weichert D (1995) On the computation of the J-integral for three-dimensional geometries in inhomogeneous materials. Comput Mat Sci 5: 143–150CrossRefGoogle Scholar
  21. Hirano T, Teraki J, Yamada T (1990) On the design of functionally graded materials. In: Yamanouchi M, Koizumi M, Hirai T, Shiota I (eds) Proceedings of the 1st International Symposium on Functionally Gradient Materials, Sendai, Japan, 1990Google Scholar
  22. Honein T, Herrmann G (1997) Conservation laws in non-homogenous plane elastostatics. J Mech Phys Solids 45: 789–805CrossRefGoogle Scholar
  23. Hughes TJR (1987) The Finite Element Method: linear static and dynamic analysis. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  24. Kienzler R, Herrmann G (2000) Mechanics in material space with application to defect and fracture mechanics. Springer-VerlagGoogle Scholar
  25. Kim JH, Paulino GH (2002) Finite element evaluation of mixed mode stress intensity factors in functionally graded materials. Int J Num Meth Eng 53: 1903–1935CrossRefGoogle Scholar
  26. Konda N, Erdogan F (1994) The mixed mode crack problem in a non-homogeneous elastic plane. Eng Fract Mech 47: 533–545CrossRefGoogle Scholar
  27. Li FZ, Shih CF, Needleman A (1985) A comparison of methods for calculating energy release rates. Eng Frac Mech 21: 405–421CrossRefGoogle Scholar
  28. Makowski J, Stumpf H, Hackl K (2006) The fundamental role of nonlocal and local balance laws of material forces in finite elastoplasticity and damage mechanics. Int J Solids Struct 43: 3940–3959CrossRefGoogle Scholar
  29. Marur PR, Tippur H (1998) Evaluation of mechanical properties of functionally graded materials. J Test Eval JTEVA 26(6): 539–545Google Scholar
  30. Marur PR, Tippur H (2000) Numerical analysis of crack-tip fields in functionally graded materials with a crack normal to the elastic gradient. Int J Solids Struct 37: 5353–5370CrossRefGoogle Scholar
  31. Maugin GA (1993) Material inhomogeneities in elasticity. Chapman & Hall, LondonGoogle Scholar
  32. Müller R, Maugin GA (2002) On material forces and finite element discretizations. Comput Mech 29(1): 52–60CrossRefGoogle Scholar
  33. Müller R, Kolling S, Gross D (2001) On configurational forces in the context of the finite-element method. Int J Num Meth Eng 53: 1557–1574CrossRefGoogle Scholar
  34. Müller R, Gross D, Maugin GA (2004) Use of material forces in adaptive finite element methods. Comput Mech 33: 421–434CrossRefGoogle Scholar
  35. Nguyen TD, Govindjee S, Klein PA (2005) A material force method for inelastic fracture mechanics. J Mech Phys Solids 53: 91–121CrossRefGoogle Scholar
  36. Parameswaran V, Shukla A (2002) Asymptotic stress fields for dynamic stationary cracks along the gradient in functionally gradient materials. ASME J Appl Mech 69: 240–243Google Scholar
  37. Paulino GH, Jin ZH, Dodds RH (2003) Failure of functionally graded materials. In: Karihaloo B, Knauss WG (eds) Comprehensive structural integrity, vol 2. Elsevier Science, pp 607–644Google Scholar
  38. Rajagopal A, Sivakumar SM (2007) A combined r-h adaptive strategy based on material forces and error assessment for plane problems and bimaterial interfaces. Comput Mech 41: 49–72CrossRefGoogle Scholar
  39. Rice JR (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 35: 379–386Google Scholar
  40. Shield RT (1967) Inverse deformation results in finite elasticity. ZAMP 18: 490–500CrossRefGoogle Scholar
  41. Silva ECN, Paulino GH (2005) Topology optimization design of functionally graded structures. Proc of 6th World Congress of Structural and Multidisciplinary Optimization, Ria de Janeiro, 30 May-03 June 2005, BrazilGoogle Scholar
  42. Spiegel MR (1959) Vector analysis and an introduction to tensor analysis. Schaum’s outline of theory and problems. McGraw-Hill, New YorkGoogle Scholar
  43. Steinmann P (2001a) Application of material forces to hyperelastic fracture mechanics, Part I: continuum mechanical setting. Int J Solids Struct 37: 7371–7391CrossRefGoogle Scholar
  44. Steinmann P (2001b) A view on the theory and computation of hyperelastic defect mechanics. In: Proc. of European Conference on Computational Mechanics, ECCM, Crakow, PolandGoogle Scholar
  45. Steinmann P, Ackermann D, Barth FJ (2001) Application of material forces to hyperelastic fracture mechanics, Part II: computational setting. Int J Solids Struct 38: 5509–5526CrossRefGoogle Scholar
  46. Surush S, Mortensen A (1998) Fundamentals of functionally graded materials. Institute of Materials, LondonGoogle Scholar
  47. Williams ML (1957) On the stress distribution at the base of a stationary crack. ASME J Appl Mech 24: 109–114Google Scholar
  48. Zienkiewicz OC, Taylor RL (2005) The finite element method, 6th edn. vol 1. Mc Graw-Hill, LondonGoogle Scholar
  49. Zienkiewicz OC, Zhu JZ (1987) A simple error estimator and adaptive procedure for practical engineering analysis. Int J Num Meth Eng 24: 337–357CrossRefGoogle Scholar
  50. Zuiker J, Dvorak G (1994) The effective properties of functionally graded composites- I. Extension of the Mori-Tanaka method to linearly varying fields. Composit Eng 4: 19–35Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Engineering Mechanics (LTM)University of PaderbornPaderbornGermany

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