An explicit elastic solution for a brittle film with periodic cracks

  • H. M. Yin
  • G. H. Paulino
  • W. G. Buttlar
Original Paper


A two-dimensional explicit elastic solution is derived for a brittle film bonded to a ductile substrate through either a frictional interface or a fully bonded interface, in which periodically distributed discontinuities are formed within the film due to the applied tensile stress in the substrate and consideration of a “weak form stress boundary condition” at the crack surface. This solution is applied to calculate the energy release rate of three-dimensional channeling cracks. Fracture toughness and nominal tensile strength of the film are obtained through the relation between crack spacing and tensile strain in the substrate. Comparisons of this solution with finite element simulations show that the proposed model provides an accurate solution for the film/substrate system with a frictional interface; whereas for a fully bonded interface it produces a good prediction only when the substrate is not overly compliant or when the crack spacing is large compared with the thickness of the film. If the section is idealized as infinitely long, this solution in terms of the energy release rate recovers Beuth’s exact solution for a fully cracked film bonded to a semi-infinite substrate. Interfacial shear stress and the edge effect on the energy release rate of an asymmetric crack are analyzed. Fracture toughness and crack spacing are calculated and are in good agreement with available experiments.


Energy release rate Elastic materials Fracture toughness Thin film Channeling crack Periodic cracks 


  1. Agrawal DC, Raj R (1989) Measurement of the ultimate shear-strength of a metal ceramic interface. Acta Metall 37: 1265–1270. doi: 10.1016/0001-6160(89)90120-X CrossRefGoogle Scholar
  2. Alaca BE, Saif MTA, Sehitoglu H (2002) On the interface debond at the edge of a thin film on a thick substrate. Acta Mater 50:1197–1209. doi: 10.1016/S1359-6454(01)00421-9 CrossRefGoogle Scholar
  3. Bažant ZP (1999) Size effect on structural strength: a review. Arch Appl Mech 69:703–725. doi: 10.1007/s004190050252 MATHCrossRefGoogle Scholar
  4. Bell RO, Rupprecht G (1963) Elastic constants of Strontium Titanate. Phys Rev 129:90–94. doi: 10.1103/PhysRev.129.90 CrossRefADSGoogle Scholar
  5. Beuth JL (1992) Cracking of thin bonded films in residual tension. Int J Solids Struct 29:1657–1675. doi: 10.1016/0020-7683(92)90015-L CrossRefGoogle Scholar
  6. Beuth JL, Klingbeil NW (1996) Cracking of thin films bonded to elastic-plastic substrates. J Mech Phys Solids 44:1411–1428. doi: 10.1016/0022-5096(96)00042-7 CrossRefADSGoogle Scholar
  7. Bordet H, Ignat M, Dupeux M (1998) Analysis of the mechanical response of film on substrate systems presenting rough interfaces. Thin Solid Films 315:207–213. doi: 10.1016/S0040-6090(97)00755-4 CrossRefADSGoogle Scholar
  8. Chen BF, Hwang J, Chen IF, Yu GP, Huang J-H (2000) A tensil-film-cracking model for evaluating interfacial shear strength of elastic film on ductile substrate. Surf Coat Technol 126:91–95. doi: 10.1016/S0257-8972(99)00669-6 CrossRefGoogle Scholar
  9. Etzkorn EV, Clarke DR (2001) Cracking of GaN films. J Appl Phys 89:1025–1034. doi: 10.1063/1.1330243 CrossRefADSGoogle Scholar
  10. Fleck NA, Qiu XM (2007) The damage tolerance of elastic–brittle, two-dimensional isotropic lattices. J Mech Phys Solids 55:562–588. doi: 10.1016/j.jmps.2006.08.004 CrossRefADSMathSciNetMATHGoogle Scholar
  11. Hu MS, Evans AG (1989) The cracking and decohesion of thin films on ductile substrates. Acta Metall 37:917–925. doi: 10.1016/0001-6160(89)90018-7 CrossRefGoogle Scholar
  12. Hutchinson JW, Suo Z (1992) Mixed mode cracking in layered materials. Adv Appl Mech 29: 63–191MATHCrossRefGoogle Scholar
  13. Leevers PS, Godart M-A (2008) Adiabatic decohesion in a thermoplastic craze thickening at constant or increasing rate. J Mech Phys Solids 56:2149–2170. doi: 10.1016/j.jmps.2008.02.001 CrossRefADSMATHGoogle Scholar
  14. Liu XH, Suo Z, Ma Q (1999) Split singularities: stress field near the edge of a Silicon die on a polymer substrate. Acta Mater 47:67–76. doi: 10.1016/S1359-6454(98)00345-0 CrossRefGoogle Scholar
  15. Malzbender J (2004) Stress profile and thermal expansion of layered materials determined from surface stresses. Appl Phys Lett 84:4661–4662. doi: 10.1063/1.1759773 CrossRefADSGoogle Scholar
  16. Nakamura T, Kamath SM (1992) Three-dimensional effects in thin film fracture mechanics. Mech Mater 13:67–77. doi: 10.1016/0167-6636(92)90037-E CrossRefGoogle Scholar
  17. Parmigiani JP, Thouless MD (2006) The roles of toughness and cohesive strength on crack deflection at interfaces. J Mech Phys Solids 54:266–287. doi: 10.1016/j.jmps.2005.09.002 MATHCrossRefADSGoogle Scholar
  18. Saif MTA, Hui CY, Zehnder AT (1993) Interface shear stresses induced by non-uniform heating of a film on a substrate. Thin Solid Films 224:159–167. doi: 10.1016/0040-6090(93)90427-Q CrossRefADSGoogle Scholar
  19. Shenoy VB, Schwartzman AF, Freund LB (2001) Crack patterns in brittle thin films. Int J Fract 109:29–45. doi: 10.1023/A:1010973729754 CrossRefGoogle Scholar
  20. Tadepalli R, Turner KT, Thompson CV (2008) Mixed-mode interface toughness of wafer-level Cu–Cu bonds using asymmetric chevron test. Mech Phys Solids 56:707–718. doi: 10.1016/j.jmps.2007.07.016 MATHCrossRefADSGoogle Scholar
  21. Thouless MD, Olsson E, Gupta A (1992) Cracking of brittle films on elastic substrates. Acta Metall Mater 40: 1287–1292. doi: 10.1016/0956-7151(92)90429-I CrossRefGoogle Scholar
  22. Timm DH, Guzina BB, Voller VR (2003) Prediction of thermal crack spacing. Int J Solids Struct 40:125–142. doi: 10.1016/S0020-7683(02)00496-1 MATHCrossRefGoogle Scholar
  23. Vlassak JJ (2003) Channel cracking in thin films on substrates of finite thickness. Int J Fract 119/120:299–323. doi: 10.1023/A:1024962825938 CrossRefGoogle Scholar
  24. Wang J, Qiao P (2004) Interface crack between two shear deformable elastic layers. J Mech Phys Solids 52:891–905. doi: 10.1016/S0022-5096(03)00121-2 MATHCrossRefADSGoogle Scholar
  25. Wang JS, Sugimura Y, Evans AG, Tredway WK (1998) The mechanical performance of DLC films on steel substrates. Thin Solid Films 325:163–174. doi: 10.1016/S0040-6090(98)00418-0 CrossRefADSGoogle Scholar
  26. Xia ZC, Hutchinson JW (2000) Crack patterns in thin films. J Mech Phys Solids 48:1107–1131. doi: 10.1016/S0022-5096(99)00081-2 MATHCrossRefADSGoogle Scholar
  27. Yin HM, Buttlar WG, Paulino GH (2007) Periodic thermal cracking in an asphalt overlay bonded to a rigid pavement. J Transp Eng 133:39–46. doi: 10.1061/(ASCE)0733-947X(2007)133:1(39) CrossRefGoogle Scholar
  28. Yu HH, He MY, Hutchinson JW (2001) Edge effects in thin film delamination. Acta Mater 49:93–107. doi: 10.1016/S1359-6454(00)00293-7 CrossRefGoogle Scholar
  29. Zhao M-H, Fu R, Lu D, Zhang T-Y (2002) Critical thickness for cracking of Pb(Zr 0.53 Ti 0.47)O 3 thin films deposited on Pt/Ti/Si(100) substrates. Acta Mater 50:4241–4254. doi: 10.1016/S1359-6454(02)00254-9 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Civil Engineering and Engineering MechanicsColumbia UniversityNew YorkUSA
  2. 2.Department of Civil and Environmental EngineeringUniversity of Illinois at Urbana-Champaign, Newmark LaboratoryUrbanaUSA

Personalised recommendations