STABILITY OF THE NUMERICAL PROCEDURE FOR SOLUTION OF SINGULAR INTEGRAL-EQUATIONS ON SEMIINFINITE INTERVAL - APPLICATION TO FRACTURE-MECHANICS

Singular integral equations (Cauchy-type singularity) play a key role in computational fracture mechanics. In modeling the micromechanical aspects of fracture, one examines effects produced by defects with dimensions much smaller than the crack size. Thus, to focus attention on these effects, one has to consider a crack as infinitely large and to bring the problem to a defect scale. The integral equations resulting from these problems are formulated on a semi-infinite interval, and they are of the first kind with Cauchy-type singular term. On a finite interval, the numerical procedure for these equations is well developed and analyzed. On a semi-infinite interval, this procedure cannot be directly applied without a stabilization of the numerical scheme. The existence of a solution to a homogeneous equation creates a stability problem in forming the numerical scheme. Depending on the mesh size, different magnitudes of the homogeneous solutions can be sensed by the numerical scheme. A numerical procedure with a stabilizing term and its implementations in fracture mechanics on microscale is discussed.