Advertisement

International Journal of Fracture

, Volume 189, Issue 1, pp 103–110 | Cite as

Validity of linear elasticity in the crack-tip region of ideal brittle solids

  • Gaurav SinghEmail author
  • James R. Kermode
  • Alessandro De Vita
  • Robert W. Zimmerman
Brief Note

Abstract

It is a well known that, according to classical elasticity, the stress in the crack-tip region is singular, which has led to a debate over the validity of linear elasticity in this region. In this work, comparisons of finite and small strain theories have been made in the crack-tip region of a brittle crystal to comment on the validity of linear elasticity in the crack tip region. We find that linear elasticity is capable of accurately defining the state of stress very close (\(\sim \)1 nm) to a static crack tip.

Keywords

Crack Singularity Elasticity Brittle 

Notes

Acknowledgments

We acknowledge funding from the Rio Tinto Centre for Advanced Mineral Recovery based at Imperial College, London, and useful discussions with Gert van Hout throughout the project. J.R.K and A.D.V. acknowledge funding from the EPSRC HEmS Grant EP/L014742/1 and from the European Commission ADGLASS FP7 project.

References

  1. Barenblatt GI (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7:55–129CrossRefGoogle Scholar
  2. Bernstein N, Hess DW (2003) Lattice trapping barriers to brittle fracture. Phys Rev Lett 91:25501–25504CrossRefGoogle Scholar
  3. Broberg KB (1971) Crack-growth criteria and non-linear fracture mechanics. J Mech Phys Solids 19(6):407–418CrossRefGoogle Scholar
  4. Buehler M, van Duin A, Goddard W (2006) Multiparadigm modeling of dynamical crack propagation in silicon using a reactive force field. Phys Rev Lett 96:95505Google Scholar
  5. Buehler M (2008) Atomistic modeling of materials failure. Springer, BostonCrossRefGoogle Scholar
  6. Chaudhuri RA (2014) Three-dimensional mixed mode I+II+III singular stress field at the front of a (111) [112] \(\times \) [110] crack weakening a diamond cubic mono-crystalline plate with crack turning and step/ridge formation. Int J Fract 187:15–49. doi: 10.1007/s10704-013-9891-7
  7. Cherepanov GP (1967) Crack propagation in continuous media. J Appl Math Mech 31(3):503–512CrossRefGoogle Scholar
  8. Cramer T, Wanner A, Gumbsch P (2000) Energy dissipation and path instabilities in dynamic fracture of silicon single crystals. Phys Rev Lett 85(4):788–791CrossRefGoogle Scholar
  9. Csányi G, Winfield S, Kermode JR, De Vita A, Comisso A, Bernstein N, Payne MC (2007) Expressive programming for computational physics in Fortran 95+. IoP Computational Physics Newsletter. p 27Google Scholar
  10. Freund LB (1998) Dynamic fracture mechanics. Cambridge University Press, CambridgeGoogle Scholar
  11. Gerberich WW, Oriani RA, Lji MJ, Chen X, Foecke T (1991) The necessity of both plasticity and brittleness in the fracture thresholds of iron. Philos Mag A 63(2):363–376CrossRefGoogle Scholar
  12. Geubelle PH, Knauss WG (1994) Finite strains at the tip of a crack in a sheet of hyperelastic material: I. Homogeneous case. J Elast 35:61–98CrossRefGoogle Scholar
  13. Gleizer A, Peralta G, Kermode JR, De Vita A, Sherman D (2014) Dissociative chemisorption of O\(_{2}\) inducing stress corrosion cracking in silicon crystals. Phys Rev Lett 112:115501Google Scholar
  14. Gol’dstein RV, Salganik RL (1974) Brittle fracture of solids with arbitrary cracks. Int J Fract 10(4):507–523CrossRefGoogle Scholar
  15. Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond Ser A 221:163–198CrossRefGoogle Scholar
  16. Holland D, Marder M (1998) Ideal brittle fracture of silicon studied with molecular dynamics. Phys Rev Lett 80(4):746– 749Google Scholar
  17. Irwin GR (1948) Fracturing of metals. Trans Am Soc Met 40:147Google Scholar
  18. Kermode JR, Albaret T, Sherman D, Bernstein N, Gumbsch P, Payne MC, Csanyi G, De Vita A (2008) Low-speed fracture instabilities in a brittle crystal. Nature 455(7217):1224–1227CrossRefGoogle Scholar
  19. Kermode JR, Ben-Bashat L, Atrash F, Cilliers JJ, Sherman D, De Vita A (2013) Macroscopic scattering of cracks initiated at single impurity atoms. Nat Commun 4:2441–2448Google Scholar
  20. Knauss WG (1966) Stresses in an infinite strip containing a semi-infinite crack. J Appl Mech 33(2):356–362CrossRefGoogle Scholar
  21. Lawn BR (1993) Fracture of brittle solids. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  22. Love AEH (1920) A treatise on the mathematical theory of elasticity. Cambridge University Press, CambridgeGoogle Scholar
  23. Mal AK, Singh SJ (1991) Deformation of elastic solids. Prentice Hall, New JerseyGoogle Scholar
  24. Maranganti R, Sharma P (2007) Length scales at which classical elasticity breaks down for various materials. Phys Rev Lett 98:195504Google Scholar
  25. Marder MP, Liu X (1993) Instability in lattice fracture. Phys Rev Lett 71:2417Google Scholar
  26. Marder MP (2004) Effects of atoms on brittle fracture. Int J Fract 130(2):517–555CrossRefGoogle Scholar
  27. Moras G, Choudhury R, Kermode JR, Csányi G, Payne MC, De Vita A (2010) Hybrid quantum/classical modeling of material systems: the “Learn on the Fly” molecular dynamics scheme. In: Dumitrica T (ed) Trends in computational nanomechanics transcending length and time scales. Springer, Berlin, pp 1–23Google Scholar
  28. Nair AK, Warner DH, Hennig RG, Curtin WA (2010) Coupling quantum and continuum scales to predict crack tip dislocation nucleation. Scr Mater 63:1212Google Scholar
  29. Rhee YW, Kim HW, Deng Y, Lawn BR (2001) Brittle fracture versus quasi plasticity in ceramics: a simple predictive index. J Am Ceram Soc 84(3):561–565CrossRefGoogle Scholar
  30. Rivlin RS, Thomas AG (1953) Rupture of rubber. I. Characteristicenergy for tearing. J Polym Sci 10(3):291–318CrossRefGoogle Scholar
  31. Slepyan LI (2002) Models and phenomena in fracture mechanics. Springer, BerlinCrossRefGoogle Scholar
  32. Stillinger FH, Weber TA (1985) Computer simulation of local order in condensed phases of silicon. Phys Rev B 31(8):5262CrossRefGoogle Scholar
  33. Swadener JG, Baskes MI, Nastasi M (2002) Molecular dynamics simulation of brittle fracture in silicon. Phys Rev Lett 89(8):85503–85504CrossRefGoogle Scholar
  34. Tadmor EB, Phillips R, Ortiz M (1996) Mixed atomistic and continuum models of deformation in solids. Langmuir 12:4529–4534CrossRefGoogle Scholar
  35. Thomson R, Hsieh C, Rana V (1971) Lattice trapping of fracture cracks. J Appl Phys 42(8):3154–3160CrossRefGoogle Scholar
  36. Wong FS, Shield RT (1969) Large plane deformations of thin elastic sheets of neo-hookean material. Zeitschrift f\({\ddot{u}}\)r Angewandte Math Phys (ZAMP) 20(2):176199Google Scholar
  37. Xi XK, Zhao DQ, Pan MX, Wang WH, Wu Y, Lewandowski JJ (2005) Fracture of brittle metallic glasses: brittleness or plasticity. Phys Rev Lett 94(12):125510–125513 Google Scholar
  38. Zimmerman JA, Webb EB, Hoyt JJ, Jones RE, Klein PA, Bammann DJ (2004) Calculation of stress in atomistic simulation. Model Simul Mater Sci Eng 12:S319–S332CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Gaurav Singh
    • 1
    Email author
  • James R. Kermode
    • 2
  • Alessandro De Vita
    • 2
    • 3
  • Robert W. Zimmerman
    • 1
  1. 1.Department of Earth Science and EngineeringImperial College LondonLondonUK
  2. 2.Department of PhysicsKing’s College LondonLondonUK
  3. 3.CENMAT-UTSTriesteItaly

Personalised recommendations