International Journal of Engineering Science, Vol.41, No.7, pp.683-720, 2003
Y.S. Chan and A.C.
Fannjiang
Department of Mathematics and Graduate Group in Applied Mathematics, University of
California, Davis, CA 95616-5294, U.S.A.
G.H. Paulino,
Department of Civil and Environmental Engineering, University of Illinois
at Urbana-Champaign,
Newmark Laboratory, 205 North Mathews Avenue,
IL 61801, U.S.A.
Abstract
Hypersingular integrals of the type
and
are investigated for general integers a (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 a = 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 a > 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.
Key words: hypersingular integrals, Tchebyshev polynomials, stress intensity factors, functionally graded materials, gradient elasticity theory, linear elastic fracture mechanics.
Representative Results
 |
Closed form solutions |