DOI QR코드

DOI QR Code

An analysis of an elastic solid incorporating a crack under the influences of surface effects in plane & anti-plane deformations

  • Kim, Chun Il (Department of Mechanical Engineering, University of Alberta)
  • Received : 2010.11.11
  • Accepted : 2011.02.18
  • Published : 2011.06.25

Abstract

We review a series of crack problems arising in the general deformations of a linearly elastic solid (Mode-I, Mode-II and Mode-III crack) and, perhaps more significantly, when the contribution of surface effects are taken into account. The surface mechanics are incorporated using the continuum based surface/interface model of Gurtin and Murdoch. We show that the deformations of an elastic solid containing a single crack can be decoupled into in-plane (Mode-I and Mode-II crack) and anti-plane (Mode-III crack) parts, even when the surface mechanics is introduced. In particular, it is shown that, in contrast to classical fracture mechanics (where surface effects are neglected), the incorporation of surface elasticity leads to the more accurate description of a finite stress at the crack tip. In addition, the corresponding stress fields exhibit strong dependency on the size of crack.

References

  1. Cammarata, R.C. (1997), "Surface and interface stress effects on interfacial and nanostructured materials", Mater. Sci. Eng. A, 237(2), 180-184. https://doi.org/10.1016/S0921-5093(97)00128-7
  2. Chakrabarti, A. (1999), "Numerical solution of a singular integro-differential equation", ZAMM Z. Agnew. Math. Mech., 79(4), 233-241.
  3. Duan, H.L., Wang, J., Huang, Z.P. and Karhaloo, B.L. (2005), "Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress", J. Mech. Phys. Solids, 53(7), 1574-1596. https://doi.org/10.1016/j.jmps.2005.02.009
  4. England, A.H. (1971), Complex variable methods in elasticity, John Wiley & Sons Ltd. London.
  5. Gurtin, M.E. and Murdoch, A.I. (1975), "A continuum theory of elastic material surfaces", Arch. Ration. Mech. An., 57(4), 291-323.
  6. Gurtin, M.E., Weissmuller, J. and Larche, F. (1998), "A general theory of curved deformable interface in solids at equilibrium", Philos. Mag. A., 78(5), 1093-1109. https://doi.org/10.1080/01418619808239977
  7. Kim, C.I., Schiavone, P. and Ru, C.Q. (2009), "Analysis of a mode-III crack in the presence of surface elasticity and a prescribed non-uniform surface traction", ZAMP. Z. Angew. Math. Phys. (DOI 10.1007/s00033-009-0021-3) https://doi.org/10.1007/s00033-009-0021-3
  8. Kim, C.I., Schiavone, P. and Ru, C.Q. (2010a), "Analysis of plane-strain crack problems (Mode-I & Mode-II) in the presence of surface elasticity", J. Elasticity. (DOI 10.1007/s10659-010-9287-0) https://doi.org/10.1007/s10659-010-9287-0
  9. Kim, C.I., Schiavone, P. and Ru, C.Q. (2010b), "The effects of surface elasticity on an elastic solid with mode- III crack: complete solution", J. Appl. Mech. - ASME, 77(2), 021011(1-7). https://doi.org/10.1115/1.3177000
  10. Miller, R.E. and Sheny, V.B. (2000), "Size-dependent elastic properties of nanosized structural elements", Nanotechnology, 11(3), 139-147. https://doi.org/10.1088/0957-4484/11/3/301
  11. Muskhelishvili, N.I. (1953), Some basic problems of the mathematical theory of elasticity, P. Noordhof, Groningen, The Netherlands.
  12. Ogden, R.W., Steigmann, D.J. and Haughton, D.M. (1997), "Effect of elastic surface coating on the finite deformation and bifurcation of a pressurized circular annulus", J. Elasticity, 47(2), 121-145. https://doi.org/10.1023/A:1007448209058
  13. Philip, P. (2009), "A quasistatic crack propagation model allowing for cohesive forces and crack reversibility", Interact. Multiscale Mech., 2(1), 31-44. https://doi.org/10.12989/imm.2009.2.1.031
  14. Ru, C.Q. (2010), "Simple geometrical explanation of Gurtin-Murdoch model of surface elasticity with clarification of its related versions", Sci. China, 53(3), 536-544.
  15. Sharma, P. and Ganti, S. (2004), "Size-dependent Eshelby's tensor for embedded nano-inclusions incorporating surface/interface engergies", J. Appl. Mech. - ASME, 71(5), 663-671. https://doi.org/10.1115/1.1781177
  16. Sih, G.C. (1965), "Boundary problems for longitudinal shear cracks", Develop. Theor. Appl. Mech., 2, 117-130.
  17. Sugiyama, A., Taguchi, Y., Nagaoka, S. and Nakajima, A. (2010), "Size-dependent magnetic properties of naked and ligand-capped nickel nanoparticles", Chem. Phys. Lett., 485, 129-132. https://doi.org/10.1016/j.cplett.2009.12.004
  18. Suzuki, T., Endo, H. and Shibayama, M. (2008), "Analysis of surface structure and hydrogen/deuterium exchange of colloidal silica suspension by contrast-variation small-angle neutron scattering", Langmuir, 24, 4537-4543. https://doi.org/10.1021/la7039515
  19. Tian, L. and Rajapakse, R.K.N.D. (2007), "Analytical solution of size-dependent elastic field of a nano-scale circular inhomogeneity", J. Appl. Mech. - ASME, 74(3), 568-574. https://doi.org/10.1115/1.2424242
  20. Wang, G.F. and Wang, T.J. (2006), "Deformation around nanosized elliptical hole with surface effect", Appl. Phys. lett., 89, 161901. https://doi.org/10.1063/1.2362988
  21. Wu, C.H. (1999), "The effect of surface stress on the configurational equilibrium of voids and cracks", J. Mech. Phys. Solids, 47, 2469-2492. https://doi.org/10.1016/S0022-5096(99)00021-6
  22. Wu, C.H. and Wang, M.L. (2000), "The effect of crack-tip point loads on fracture", J. Mech. Phys. Solids, 48, 2283-2296. https://doi.org/10.1016/S0022-5096(00)00011-9