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Stress, deformation and damage fields near the tip of a crack in a damaged nonlinear material

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Abstract

The stress, strain, displacement and damage fields near the tip of a crack in a power-law hardening material with continuous damage formation under antiplane longitudinal shear loading are investigated analytically. The interaction between a major crack and distributed microscopic damage is considered by describing the effect of damage in terms of a damage variable D. A deformation plasticity theory coupled with damage and a damage evolution law are formulated. A hodograph transformation is employed to determine the singularity and angular distribution of the crack-tip quantities. Consequently, analytical solutions for the antiplane shear crack-tip fields are obtained. Effects of the hardening exponent n and the damage exponent m on the crack-tip fields are discussed. It is found that the present crack-tip stress and strain solutions for damaged nonlinear material are similar to the well-known HRR fields for virgin materials. However, damage leads to a weaker singularity of stress, and to a stronger singularity of strain compared to that for virgin materials, respectively. The stress associated with damage always falls below the HRR field for virgin material; but the distribution of strain associated with damage lies slightly above the HRR field for r/(J/τ0) > 1.5 while the difference becomes negligible when r/(J/τ0) > 2. The limiting distributions of stress and strain may indeed be given by the HRR field.

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References

  1. A.S. Argon, J. Im and R. Safogln, Cavity formation from inclusion in ductile fracture. Metallurgical Transactions 6A (1975) 825–837.

    Google Scholar 

  2. J.W. Hancock and A.C. Machenzie, On the mechanisms of ductile failure in high-strength steels subiected to multi-axial stress-state. Journal of the Mechanics and Physics of Solids 24 (1976) 147–169.

    Google Scholar 

  3. J.W. Hancock and M.J. Cowling, Role of state of stress in crack-tip failure processes. Metal Science 14 (1980) 293–304.

    Google Scholar 

  4. F.B. Beremin, Cavity formation from inclusions in ductile fracture of A508 steel. Metallurgical Transactions 12A (1981) 723–732.

    Google Scholar 

  5. B. Marini, F. Mudry and A. Pineau, Study of cavity growth in ductile rupture of A508 steel under nonradial loading. Engineering Fracture Mechanics 22 (1985) 375–386.

    Google Scholar 

  6. B. Marini, F. Mudry and A. Pineau, Study of cavity growth in ductile rupture of A508 steel under nonradial loading. Engineering Fracture Mechanics, 22 (1985) 989–996.

    Google Scholar 

  7. R.O. Ritchie and A.W. Thompson, On macroscopic and microscopic analyses for crack initiation and crack growth toughness in ductile alloys. Metallurgical Transactions 16A (1985) 233–248.

    Google Scholar 

  8. H. Miyamoto, K. Machida, H. Okaysu and T. Kawazoe, Study of the process zone at the crack tip (behavior of the voids at the crack tip of aluminum alloy specimen) in, Mechanical Behaviour of Materials-V, ICM-5, M.G. Yan et al. (eds.), Vol. 1, Pergamon Press, Oxford (1987) 39–50.

    Google Scholar 

  9. J.F. Knott, Effects of microstructure and stress-state in ductile fracture research in, Advances in Fracture Research, ICF7, K. Salama et al. (eds.), Vol. 1, Pergamon Press, Oxford (1989) 125–138.

    Google Scholar 

  10. S.-X. Wu, B. Cotterell and Y.-W. Mai, On the relationship between crack tip opening displacement at the initiation of a ductile tear in low alloy steel, hydrostatic stress, and void growth. International Journal of Fracture 51 (1991) 207–218.

    Google Scholar 

  11. T.J. Wang, Thermal and mechanical load induced damage behaviour of a low alloy steel: mechanisms and modelling. Engineering Fracture Mechanics 44 (1993) 971–980.

    Google Scholar 

  12. T.J. Wang, Thermal and mechanical load induced damage behaviour of a low alloy steel: mechanisms and modelling. Engineering Fracture Mechanics 45 (1993) 799–812.

    Google Scholar 

  13. J.R. Rice and G.F. Rosengren, Plane strain deformation near a crack tip in power-law hardening material. Journal of the Mechanics and Physics of Solids 16 (1968) 1–12.

    Google Scholar 

  14. J.W. Hutchinson, Singular behaviour at the end of a tensile crack in hardening material. Journal of the Mechanics and Physics of Solids 16 (1968) 13–31.

    Google Scholar 

  15. J.R. Rice, Stresses due to a sharp notch in a work-hardening elastic-plastic material loaded by longitudinal shear. ASME Journal of Applied Mechanics 34 (1967) 287–298.

    Google Scholar 

  16. J.C. Amazigo, Fully plastic crack in an infinite body under anti-plane shear. International Journal of Solids and Structures 10 (1974) 1003–1015.

    Google Scholar 

  17. C.F. Shih, Journal of the Mechanics and Physics of Solids 29 (1981) 305–326.

    Google Scholar 

  18. J.W. Huchinson, Crack tip singularity fields in nonlinear fracture mechanics: a survey of current status in, Advances in Fracture Research. ICF5, D. Francois et al. (eds.), Vol. 6, Pergamon Press, Oxford (1981) 2669–2684.

    Google Scholar 

  19. K.C. Hwang, S.W. Yu and W. Yang, Theoretical study of crack-tip singularity fields in China. Applied Mechanics Review 13 (1990) 19–33.

    Google Scholar 

  20. Y.C. Li and T.C. Wang, High order asymptotic field of tensile plane strain nonlinear crack problems. Scientia Sinica 29A (1986) 941–955.

    Google Scholar 

  21. F.Z. Li, A. Needleman and C.F. Shih, Characterization of crack tip stress and deformation fields in creeping solids. International Journal of Fracture 36 (1988) 163–186.

    Google Scholar 

  22. N.P. O'Dowd and C.F. Shih, Family of crack-tip fields characterized by a triaxiality parameter-1. Structure of fields. Journal of the Mechanics and Physics of Solids 39 (1991) 989–1015.

    Google Scholar 

  23. N.P. O'Dowd and C.F. Shih, Family of crack-tip fields characterized by a triaxiality parameter-1. Structure of fields. Journal of the Mechanics and Physics of Solids II 40 (1992) 939–963.

    Google Scholar 

  24. A.M. Al-Ani and J.W. Hancock, J-dominance of short cracks in tension and bending. Journal of the Mechanics and Physics of Solids 39 (1991) 23–43.

    Google Scholar 

  25. C. Betegon and J.W. Hancock, Two-parameter characterization of elastic-plastic crack-tip fields. ASME Journal of Applied Mechanics 58 (1991) 104–113.

    Google Scholar 

  26. F.S. Ma and Z.B. Kuang, Elastic-plastic fracture analysis of finite bodies-I. Description of the stress field. Engineering Fracture Mechanics 48 (1994) 227–737.

    Google Scholar 

  27. F.S. Ma and Z.B. Kuang, Elastic-plastic fracture analysis of finite bodies-I. Description of the stress field. Engineering Fracture Mechanics II 48 (1994) 739–744.

    Google Scholar 

  28. S.M. Sharma and N. Aravas, Determination of higher-order terms in asymptotic elastoplastic crack tip solutions. Journal of the Mechanics and Physics of Solids 39 (1991) 1043–1072.

    Google Scholar 

  29. L.M. Kachanov, Time of the rupture process under creep conditions. Izvestiya Akademii Nauk. SSR. Otd. Tekh. 8 (1958) 26–31.

    Google Scholar 

  30. Y.N. Robotnov, Creep Problems in Structural Members. North-Holland, Amsterdam (1969).

    Google Scholar 

  31. J.L. Chaboche, Continuum damage mechanics: a tool to describe phenomena before crack initiation. Nuclear Engineering and Design 64 (1981) 233–247.

    Google Scholar 

  32. J. Lemaitre, How to use damage mechanics. Nuclear Engineering and Design 80 (1984) 233–246.

    Google Scholar 

  33. J. Lemaitre, A continuum damage mechanics model for ductile fracture. ASME Journal of Engineering Materials and Technology 107 (1985) 83–89.

    Google Scholar 

  34. J. Lemaitre, Micromechanics of crack initiation. International Journal of Fracture 42 (1990) 87–99.

    Google Scholar 

  35. D. Krajcinovic, Continuum damage mechanics. Applied Mechanics Review 37 (1984) 1–6.

    Google Scholar 

  36. D. Krajcinovic, Damage mechanics. Mechanics of Materials 8 (1989) 117–197.

    Google Scholar 

  37. S. Murakami, Notion of continuum damage mechanics and its application to anisotropic creep damage theory. ASME Journal of Engineering Materials and Technology 105 (1983) 99–105.

    Google Scholar 

  38. J.L. Chaboche, Continum damage mechanics, Part I. General concepts. ASME Journal of Applied Mechanics 55 (1988) 59–65.

    Google Scholar 

  39. J.L. Chaboche, Continum damage mechanics, Part I. General concepts. ASME Journal of Applied Mechanics II 55 (1988) 66–72.

    Google Scholar 

  40. J. Lemaitre and J.L. Chaboche, Mechanics of Solid Materials. Cambridge University Press, Cambridge (1990).

    Google Scholar 

  41. J. Hult, Introduction and general overview in, Continum Damage Mechanics: Theory and Applications, D. Krajcinovic and J. Lemaitre (eds.), Springer-Verlag, Berlin (1987) 1–36.

    Google Scholar 

  42. J.L. Chaboche, Fracture mechanics and damage mechanics in, Numerical Methods in Fracture Mechanics, Proceedings, 4th International Conference, A.R. Luxmore et al. (eds.), Pineridge Press, Swansea (1987) 309–324.

    Google Scholar 

  43. C.L. Chow and J. Wang, Ductile fracture characterization with an anisotropic continuum damage theory. Engineering Fracture Mechanics (1988) 547–563.

  44. C.L. Chow and J. Wang, An anisotropic theory of elasticity for continuum damage mechanics. International Journal of Fracture 33 (1987) 3–16.

    Google Scholar 

  45. T.J. Wang, Unified CDM model and local criterion for ductile fracture-I. Engineering Fracture Mechanics 42 (1992) 177–182.

    Google Scholar 

  46. T.J. Wang, Unified CDM model and local criterion for ductile fracture-I. Engineering Fracture Mechanics II 42 (1992) 185–192.

    Google Scholar 

  47. T.J. Wang, Unified CDM model and local criterion for ductile fracture-I. Engineering Fracture Mechanics 48 (1994) 217–230.

    Google Scholar 

  48. J. Janson, Dugdale-crack in material with continuous damage formation. Engineering Fracture Mechanics 9 (1973) 891–899.

    Google Scholar 

  49. J. Janson, Dugdale-crack in material with continuous damage formation. Engineering Fracture Mechanics 10 (1978) 795–806.

    Google Scholar 

  50. H.D. Bui and Eh Ehrlacher, Propagation of damage in elastic and plastic solids. in, Advances in Fracture Research, ICF5, D. Francois et al. (eds.), Vol. 1, Pergamon Press, Oxford (1981) 533–551.

    Google Scholar 

  51. Q.H. Tang and T.C. Wang, The stress field near a crack-tip for damaged materials. Acta Mechanica Sinica 19 (1987) 333–341 (in Chinese).

    Google Scholar 

  52. G. Rousselier, Finite deformation constitutive relations including ductile fracture damage in, Proceedings of IUTAM Symposium on Three-Dimensional Constitutive Relations and Ductile Fracture (S. Nemat-Nasser (ed.), North-Holland, Amsterdam (1981) 331–355.

    Google Scholar 

  53. T.H. Hao, X.T. Zhang and K.C. Hwang, The antiplane shear field for crack in infinite slab of an elasto-damaged material. Acta Mechanica Sinica 23 (1991) 571–580 (in Chinese).

    Google Scholar 

  54. Z.H. Jin, S.W. Yu and K.C. Hwang, The near crack-tip fields of brittle damaged materials. Acta Mechanica Sinica (English edition) 5 (1989) 227–236.

    Google Scholar 

  55. J. Masars and J. Lemaitre, Application of continuous damage mechanics to strain and fracture behaviour of concrete. NATO ARW, Proceedings NATO Advanced Research Workshop on Application of Fracture Mechanics to Cementitious Composites, S.P. Shah (ed.), Nijhoff (1985).

  56. Z.H. Jin and S.W. Yu, The near crack-tip field of stress and damage for concrete. Acta Mechanica Solida Sinica 11 (1990) 209–216 (in Chinese).

    Google Scholar 

  57. J. Wang and C.L. Chow, HRR fields for damaged materials. International Journal of Fracture 54 (1992) 165–183.

    Google Scholar 

  58. X.T. Zhang, K.C. Huang and T.H. Hao, Asymptotic solution of mode-III crack in damaged softening materials. International Journal of Fracture 62 (1993) 269–281.

    Google Scholar 

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Authors and Affiliations

  1. Department of Engineering Mechanics, Xi'an Jiaotong University, 710049, Xi'an, People's Republic of China

    Tie-Jun Wang & Zhen-Bang Kuang

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  1. Tie-Jun Wang
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  2. Zhen-Bang Kuang
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Wang, TJ., Kuang, ZB. Stress, deformation and damage fields near the tip of a crack in a damaged nonlinear material. Int J Fract 79, 1–26 (1996). https://doi.org/10.1007/BF00017710

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  • DOI: https://doi.org/10.1007/BF00017710

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