1999/06/09 by Youn-Sha Chan, Chan, Youn-Sha, Albert Fannjiang +3
Engineering · Materials Science · #Analysis of PDEs (math.AP) #Elasticity and Material Modeling #FOS: Mathematics #Nonlocal and gradient elasticity in micro/nano structures #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.math/9906058
openalex publication_date 1999/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Hypersingular integrals of the type Iα(Tn,m,r) = ∫-11 \hpsngAbs \fracTn(s)(1-s2)^m-1/2(s-r)αds |r|<1 and Iα(Un,m,r) = ∫-11 \hpsngAbs \fracUn(s)(1-s2)^m-1/2(s-r)αds |r|<1 are investigated for general integers α (positive) and m (non-negative), where Tn(s) and Un(s) are the Tchebyshev polynomials of the 1st and 2nd kinds, respectively. Exact formulas are derived for the cases α= 1, 2, 3, 4 and m = 0, 1, 2, 3; most of them corresponding to new solutions derived in this paper. Moreover, a systematic approach for evaluating these integrals when α> 4 and m>3 is provided. The integrals are also evaluated as |r|>1 in order to calculate stress intensity factors (SIFs). Examples involving crack problems are given and discussed with emphasis on the linkage between mathematics and mechanics of fracture. The examples include classical linear elastic fracture mechanics (LEFM), functionally graded materials (FGM), and gradient elasticity theory. An appendix, with closed form solutions for a broad class of integrals, supplements the paper.