Extraction formulas of stress intensity factors for the biharmonic equations containing crack singularities

• Published in: Computers & mathematics with applications (1987) 2020-09, Vol.80 (5), p.1142-1163
• Main Authors: Kim, Seokchan, Palta, Birce, Oh, Hae-Soo
• Format: Article
• Language: English
• Subjects: Biharmonic equations; Boundary conditions; Crack singularity; Enriched Galerkin Method; Galerkin method; Harmonic functions; Hermite elements and B-splines; Integrals; Isotopes; Iterative Method; Mathematical analysis; Singularity (mathematics); Stress intensity factor; Stress intensity factors
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2020.05.026

We derive formulas extracting stress intensity factors of the biharmonic equations on cracked domains with clamped (or simply supported or free) boundary conditions along the crack faces. Each of these formulas can be written in terms of the integral of the given source function multiplied by the cut-off dual singularity and the integral of the unknown true solution multiplied by the cut-off dual singularity over the unit disk. The unknown true solution in the extraction formulas is calculated by either the Implicitly Enriched Galerkin Method or the Iteration Methods. The former was developed by the authors (Kim, Oh, Palta, Kim) and the latter proposed in this paper is the sum of the solution of a regular biharmonic equation and singular functions with iteratively estimated stress intensity factors as coefficients. We show the Iteration Methods quickly converge and the proposed Enrichment Method yields highly accurate stress intensity factors. We also demonstrate that for a known true solution, the extraction formulas yield exact stress intensity factors.