A semi-infinite crack in front of a circular, thermally mismatched heterogeneity

stress intensity factors (SIFs) of a semi-infinite crack in front of a thermally and elastically mismatched, circular heterogeneity are studied based on a singular integral equation technique and on a self-consistent method. It is shown that the solution resulting from the selfconsistent method is equivalent to the one from the Cauchy-type singular integral equation if the kernel function in the integral equation is completely ignored. The self-consistent solution is then compared with the numerical solution of the integral equation for the full range of elastic mismatch using various discretization techniques. For Dundurs’ parameters within the range Ial < 0.6 and #I = a/4, the SIF’s predicted by the self-consistent formula agree within 7% or better when compared with the numerical results, provided that the crack tip is not situated extremely close to the heterogeneity. Finally, it is analyzed how the convergence of the SIFs of crack tips which are extremely close to the heterogeneity is influenced by the choice of discretization scheme : to generate computer codes which are easy to implement, time-efficient and numerically accurate, it is advantageous to use techniques which operate on a finite interval [ - 1, + l] (i.e. Gauss-Chebyshev, Lobatto-Chebyshev) as compared with those which cover the positive x-axis [0, co) (i.e. Radau-Chebyshev or Gauss- Hermite). Consequently, it is advisable to map the semi-infinite crack into a crack of finite size by using suitable transforms. It will be shown that among the discretizations for a finite interval the fastest to converge are those which explicitly use the end points - 1 and + 1 (LobattoChebyshev) followed closely by polynomial extrapolation of discrete solution data for (- 1, + 1) into the crack tips.