Surface singularity and crack propagation

Zdeněk P. Bažant*, Luis F. Estenssoro

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

397 Scopus citations


The three-dimensional singular stress field near the terminal point 0 of the crack front edge at the surface of an elastic body is investigated, using spherical coordinates r, θ, φ and assuming all three displacements to be of the form rλppF(θ, λ) where p = distance from the singularity line (crack front edge or notch edge) and p = given constant. The variational principle governing the displacement distribution on a unit sphere about point 0, which has previously been obtained from the differential equations of equilibrium, is now derived more directly from potential energy. The previously developed finite element method on the unit sphere is used to reduce the problem to the form k(λ)X = 0 where X = column matrix of the nodal values of displacements on the unit sphere and k(λ) = square matrix, all coefficients of which are quadratic polynomials in λ. It is proven that the variational principle as well as matrix k must be nonsymmetric, which means that complex eigenvalues A are possible. The dependence of A upon Poisson's ratio v for Mode I cracks whose front edge is normal to the surface is solved numerically and it closely agrees with the analytical solution of Benthem. Previously unavailable solutions for Modes II and HI and for cracks (of all modes) with inclined front edge and inclined crack plane are also obtained. By energy flux argument, it is found that the front edge of a propagating crack must terminate at the surface obliquely, at a certain angle whose dependence upon the inclination of the crack plane is also solved. The angle is the same for Modes II and III, but different for Mode I. For this mode, the surface point trails behind the interior of the propagating crack, while for Modes II and III it moves ahead. Consequently, a combination of Mode I with Modes II and III is impossible at the surface terminal point of a propagating crack whose plane is orthogonal. When the plane is inclined, the three stress intensity factors can combine only in certain fixed ratios. The angle of crack edge is a function of the angle of crack plane. Some results with complex λ for two-material interfaces are also given.

Original languageEnglish (US)
Pages (from-to)405-426
Number of pages22
JournalInternational Journal of Solids and Structures
Issue number5
StatePublished - 1979

ASJC Scopus subject areas

  • Modeling and Simulation
  • Materials Science(all)
  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics


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