Advertisement

Meccanica

, Volume 47, Issue 5, pp 1247–1260 | Cite as

An asymmetrical dynamic model for bridging fiber pull-out of unidirectional composite materials

  • N. C. Lü
  • Y. H. Cheng
  • X. G. Li
  • J. Cheng
Article

Abstract

An elastic analysis of an internal crack with bridging fibers parallel to the free surface in an infinite orthotropic elastic plane is studied. An asymmetrical dynamic model for bridging fiber pull-out of unidirectional composite materials is presented for analyzing the distributions of stress and displacement with the internal asymmetrical crack under the loading conditions of an applied non-homogenous stress and the traction forces on crack faces yielded by the bridging fiber pull-out model. Thus the fiber failure is determined by maximum tensile stress, resulting in fiber rupture and hence the crack propagation would occur in a self-similarity manner. The formulation involves the development of a Riemann-Hilbert problem. Analytical solution of an asymmetrical propagation crack of unidirectional composite materials under the conditions of two moving loads given is obtained, respectively. After those analytical solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be obtained.

Keywords

An asymmetrical dynamic model Bridging fiber pull-out Unidirectional composite materials Analytical solutions Crack 

Notes

Acknowledgements

Project supported by the postdoctoral foundation of China (No. 2005038199), HeiLongJiang provincial nature science foundation of China (ZJG04-08), HeiLongJiang provincial nature science foundation of China (E2007-36), HeiLongJiang provincial Education Cadre Teacher Foundation (1055G055).

References

  1. 1.
    Sorensen BF, Jacobsen TK (1998) Large-scale bridging in composites: R-curves and bridging laws. Composites, Part A 29A:1443–1451 CrossRefGoogle Scholar
  2. 2.
    Budiansky B, Hutchinson JW, Evens AG (1986) Matrix fracture in fiber-reinforced ceramics. J Mech Phys Solids 34:167–189 ADSMATHCrossRefGoogle Scholar
  3. 3.
    Ji M, Ishikawa H (1997) Analysis of an internal central crack with bridging fibers in a finite orthotropic plate. Int J Eng Sci 35:549–560 MATHCrossRefGoogle Scholar
  4. 4.
    Marshall DB, Cox BN, Evans AG (1985) The mechanics of matrix cracking in brittle-matrix fiber composites. Acta Metall 33:2013–2021 CrossRefGoogle Scholar
  5. 5.
    Marshall DB, Cox BN (1987) Tensile fracture of brittle matrix composites: influence of fiber strength. Acta Metall 35:2607–2619 CrossRefGoogle Scholar
  6. 6.
    Woo CW, Wang YH (1993) Analysis of an internal crack in a fine anisotropic plate. Int J Fract 62:203–208 ADSCrossRefGoogle Scholar
  7. 7.
    Lee JC (1990) Analysis of fiber bridged crack near a free surface in ceramic matrix composites. Eng Fract Mech 37:209–219 CrossRefGoogle Scholar
  8. 8.
    Tsai WT, Dharani IR (1993) Non self-similar fiber fracture in unidirectional composites. Eng Fract Mech 44:43–49 ADSCrossRefGoogle Scholar
  9. 9.
    Liu WN (1994) Stress ahead of the tip of a finite-width center-crack in fiber-reinforced composite specimens: subjected to non-linearly distributed bridging stresses. Int J Fract 70(4):L31–L35 Google Scholar
  10. 10.
    Liao K, Reifsnider K (2000) A tensile strength model for unidirectional fiber-reinforced brittle matrix composite. Int J Fract 106:95–115 CrossRefGoogle Scholar
  11. 11.
    Tamuzs V, Tarasovs S, Vilks U (2001) Progressive delamination and fibre bridging modeling in double cantilever beam composite specimens. Eng Fract Mech 68(5):513–525 CrossRefGoogle Scholar
  12. 12.
    Bao G, Suo Z (1992) Remarks on crack-bridging concepts. Appl Mech Rev 45(8):355–366 ADSCrossRefGoogle Scholar
  13. 13.
    Jacobsen TK, Sorensen BF (2001) Mode I intra-laminar crack growth in composites-modelling of R-curves from measured bridging laws. Composites, Part A 32:1–11 CrossRefGoogle Scholar
  14. 14.
    Piva A, Viola E (1988) Crack propagation in an orthotropic media. Eng Fract Mech, 29:535–547 CrossRefGoogle Scholar
  15. 15.
    De J, Patra B (1998) Elastodynamic crack problems in an orthotropic medium through complex variable approach. Eng Fract Mech 41:895–909 Google Scholar
  16. 16.
    Cheng J (1985) Problems on elastodynamics of some orthotropic anisotropic bodies. J Harbin Inst Tech 1985:8–21 (in Chinese) (Supplement of Engineering mechanics) Google Scholar
  17. 17.
    Lü NC, Cheng J (2000) Models of crack dynamics of bridge in composite materials. Eng Mech 17(6):117–120 (in Chinese) Google Scholar
  18. 18.
    Lü NC, Cheng J (2002) A model of crack dynamics problem on composite material. Chin Q Mech 23(4):504–508 (in Chinese) Google Scholar
  19. 19.
    Cox BN, Sridhar N, Beyerlein IJ (2001) Inertial effects in the pullout mechanism during dynamic loading of a bridged crack. Acta Mater 49:3863–3877 CrossRefGoogle Scholar
  20. 20.
    Lü NC, Cheng J, Cheng YH (2005) Models of fracture dynamics of bridging fiber pull-out of composite materials. Mech Res Commun 32(1):1–14 MATHCrossRefGoogle Scholar
  21. 21.
    Lü N-C, Cheng Y-H, Xu H-M, Cheng J, Tang L-Q (2004) Dynamic crack models on problems of bridging fiber pull-out of composite materials. App Math Mech 25(10):1194–1202 MATHCrossRefGoogle Scholar
  22. 22.
    Jiang M-Z, Lü N-C, Cheng J (2005) A fracture dynamics model of bridging fiber pull-out of composite materials. J Harbin Inst Tech 37(3):423–426 (in Chinese) Google Scholar
  23. 23.
    Sih GC (1968) Some elastodynamics problems of cracks. Int J Fract 1(1):51–68 Google Scholar
  24. 24.
    Sharon E, Gross SP, Fineberg J (1996) Energy dissipation in dynamic fracture. Phys Rev Lett 76(12):2117–2120 ADSCrossRefGoogle Scholar
  25. 25.
    Broberg KB (1960) The propagation of a brittle crack. Ark Fys 18:159–192 MathSciNetGoogle Scholar
  26. 26.
    Atkinson C (1965) The propagation of a brittle crack in anisotropic material. Int J Eng Sci 3:77–91 MATHCrossRefGoogle Scholar
  27. 27.
    Lü N-C, Cheng Y-H, Cheng J (2008) An asymmetrical self-similar dynamic crack model of bridging fiber pull-out in unidirectional composite materials. Int J Comput Methods Eng Sci Mech 9(3):171–179 MATHCrossRefGoogle Scholar
  28. 28.
    Lü N-C, Cheng YH, Cheng J (2006) Mode I crack tips propagating at different speeds under differential surface tractions. Theor Appl Fract Mech 46(3):262–275 CrossRefGoogle Scholar
  29. 29.
    Craggs YW (1966) The growth of a disk-shaped crack. Int J Eng Sci 4:113–124 MATHCrossRefGoogle Scholar
  30. 30.
    Goree JG, Gross RS (1979) Analysis of a unidirectional composite containing broken fibers and matrix damage. Eng Fract Mech 33:563–578 Google Scholar
  31. 31.
    Goree JG, Dharnt LR, Jones WF (1989) Crack growth and fracture of continuous fiber metal matrix composites: analysis and experiments. In: Metal matrix composites: testing, analysis, and failure modes. ASTM STP, vol 1032, pp 251–269 CrossRefGoogle Scholar
  32. 32.
    Charepanov GP, Afanasov EF (1970) Some dynamic problems of the theory of elasticity—a review. Int J Eng Sci 12:665–690 CrossRefGoogle Scholar
  33. 33.
    Charepanov GP (1973) Mechanics of brittle fracture. Nauka, Moscow, pp 732–792 Google Scholar
  34. 34.
    Erigen AC, Suhubi ES (1975) Elastodynamics. Linear theory, vol 2. Academic Press, New York Google Scholar
  35. 35.
    Muskhlishvili NI (1968) Singular integral equations. Nauka, Moscow Google Scholar
  36. 36.
    Muskhlishvili NI (1966) Some fundamental problems in the mathematical theory of elasticity. Nauka, Moscow Google Scholar
  37. 37.
    Gakhov FD (1963) Boundary-value problems. Fitzmatigiz, Moscow Google Scholar
  38. 38.
    Hoskins RF (1979) Generalized functions. Ellis Horwood, Chichester Google Scholar
  39. 39.
    Wang XS (1993) Singular functions and their applications in mechanics. Scientific Press, Beijing, pp 3–45 (in Chinese) Google Scholar
  40. 40.
    Sih GC (1977) Mechanics of fracture 4. Elastodynamics crack problems. Noordhoff, Leyden, pp 213–247 Google Scholar
  41. 41.
    Kanwal RP, Sharma DL (1976) Singularity methods for elastostatics. J Elast 6(4):405–418 MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Lü N-C, Li X-G, Cheng Y-H, Cheng J (2011) An asymmetrical dynamic crack model of bridging fiber pull-out of composite materials. Fiber Polym 12(1):79–88 CrossRefGoogle Scholar
  43. 43.
    Shen G-L (1996) Mechanics of composite materials. Tsinghua University Press, Beijing (in Chinese) Google Scholar
  44. 44.
    Editorial group of mathematics handbook (2002) Mathematics handbook. Advanced Education Press, Beijing, pp 244–300 (in Chinese) Google Scholar
  45. 45.
    Teaching office of mathematics of Tongji University (1994) Advanced mathematics, vol 1. Advanced Education Press, Beijing, pp 167–172 (in Chinese) Google Scholar
  46. 46.
    Wu KC (2000) Dynamic crack growth in anisotropic material. Int J Fract 106(1):1–12 CrossRefGoogle Scholar
  47. 47.
    Kalthof JF, Beinert J, Winkler S (1977) Measurements of dynamic stress intensity factors for fastrunning and arresting cracks in double-cantilever-beam specimens. In: Fast fracture and crack arrest. ASTM STP, vol 627. Pa Publisher, Philadelphia, pp 161–176 CrossRefGoogle Scholar
  48. 48.
    Kobayashi AS (1979) Dynamic fracture analysis by dynamic finite element method. Generation and prediction analyses. In: Nonlinear and dynamic fracture mechanics. AMD, vol 35. ASME, New York, pp 19–36 Google Scholar
  49. 49.
    Ravi-Chandar K, Knauss WG (1984) An experimental investigation into dynamic fracture: Part 1, crack initiation and arrest. Int J Fract 25(41):247–262 CrossRefGoogle Scholar
  50. 50.
    Ravi-Chandar K, Knauss WG (1984) An experimental investigation into dynamic fracture: Part 2, microstructural aspects. Int J Fract 26(11):65–80 CrossRefGoogle Scholar
  51. 51.
    Ravi-Chandar K, Knauss WG (1984) An experimental investigation into dynamic fracture: Part 3, on steady-state crack propagation and crack branching. Int J Fract 26(2):141–152 CrossRefGoogle Scholar
  52. 52.
    Ravi-Chandar K, Knauss WG (1984) An experimental investigation into dynamic fracture: Part 4, on the interaction of stress waves with propagation cracks. Int J Fract 26(3):189–200 CrossRefGoogle Scholar
  53. 53.
    Sneddon NI (1951) Fourier transform. McGraw-Hill, New York Google Scholar
  54. 54.
    Muskhelishvili NI (1953) Some basic problems from the mathematical theory of elasticity. Noordhoff, Groningen MATHGoogle Scholar
  55. 55.
    Galin LA (1953) Contact problems in elasticity theory. GITTL, Moscow Google Scholar
  56. 56.
    Lü NC, Cheng YH, Li XG, Cheng J (2010) Dynamic propagation problem of mode I semi-infinite crack subjected to superimpose loads. Fatigue Fract Eng Mater Struct 33(3):141–148 CrossRefGoogle Scholar
  57. 57.
    Wang YH, Cheung YK, Woo CW (1992) Anti-plane shear problem for an edge crack in a finite orthotropic plate. Eng Fract Mech 42(6):971–976 ADSCrossRefGoogle Scholar
  58. 58.
    Lü N-C, Cheng Y-H, Li X-G, Cheng J (2008) Dynamic propagation problems concerning the surfaces of asymmetrical mode III crack subjected to moving loads. Appl Math Mech 29(10):1279–1290 MATHCrossRefGoogle Scholar
  59. 59.
    Lü NC, Cheng YH, Wang YT, Cheng J (2010) Dynamic fracture of orthotropic solids under anti-plane shear loading. Mech Adv Mat Struct 17(3):215–224 CrossRefGoogle Scholar
  60. 60.
    Lü N-C, Cheng Y-H, Wang Y-T, Cheng J (2011) Dynamic extension problems concerning asymmetrical mode III crack. Appl Math Model 35:2499–2507 MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Lü N-C, Cheng Y-H, Wang Y-T, Cheng J (2011) Fracture dynamics problems of orthotropic solids under anti-plane shear loading. Nonlinear Dyn 63(4):793–806 CrossRefGoogle Scholar
  62. 62.
    Wang Y-S, Wang D (1996) Transient motion of an interface dislocation and self-similar propagation of an interface crack: anti-plane motion. Eng Fract Mech 55(5):717–725 CrossRefGoogle Scholar
  63. 63.
    Wu K-C (2003) Transient motion of an interfacial line force or dislocation in an anisotropic elastic material. Int J Solids Struct 40(8):1811–1823 MATHCrossRefGoogle Scholar
  64. 64.
    Atkinson C (1975) On the dynamic stress and displacement field associated with a crack propagating across the interface between two media. Int J Eng Sci 13(5):491–506 CrossRefGoogle Scholar
  65. 65.
    Lü N-C, Yang D-N, Cheng Y-H, Cheng J (2007) Asymmetrical dynamic propagation problems on mode III interface crack. Appl Math Mech 28(4):501–510 MATHCrossRefGoogle Scholar
  66. 66.
    Lü N-C, Cheng J, Tian X-B, Cheng Y-H (2005) Dynamic propagation problem on Dugdale model of mode III interface crack. Appl Math Mech 26(9):1212–1221 MATHCrossRefGoogle Scholar
  67. 67.
    Lü N-C, Cheng Y-H, Cheng J (2008) Dynamic propagation problems concerning asymmetrical mode III interface crack. Int J Comput Methods Eng Sci Mech 9(4):246–253 MATHCrossRefGoogle Scholar
  68. 68.
    Lü NC, Cheng YH, Li XG, Cheng J (2010) Asymmetrical dynamic propagation problems concerning mode III interface crack. Compos Interfaces 17(1):37–48 CrossRefGoogle Scholar
  69. 69.
    Lü N-C, Cheng J, Cheng Y-H (2001) Self-similar solutions of fracture dynamics problems on axially symmetry. Appl Math Mech 22(12):1429–1435 MATHCrossRefGoogle Scholar
  70. 70.
    Lü NC, Cheng YH, Si HL, Cheng J (2007) Dynamics of asymmetrical crack propagation in composite materials. Theor Appl Fract Mech 47(3):260–273 CrossRefGoogle Scholar
  71. 71.
    Lü NC, Cheng YH, Li XG, Cheng J (2011) Asymmetrical dynamic fracture model of bridging fiber pull-out of unidirectional composite materials. Nonlinear Dyn 66(1):1–14 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • N. C. Lü
    • 1
    • 3
  • Y. H. Cheng
    • 2
  • X. G. Li
    • 3
  • J. Cheng
    • 4
  1. 1.School of Material Science and EngineeringShenyang Ligong UniversityShenyangP.R. China
  2. 2.Department of Civil EngineeringNortheastern UniversityShenyangP.R. China
  3. 3.School of Civil EngineeringHarbin Engineering UniversityHarbinP.R. China
  4. 4.Department of Astronautics and MechanicsHarbin Institute of TechnologyHarbinP.R. China

Personalised recommendations