Issue 32

I. Telichev, Frattura ed Integrità Strutturale, 32 (2015) 24-34; DOI: 10.3221/IGF-ESIS.32.03 30 Using the expansion of the function u 2 (ξ) in terms of Lagrange interpolation polynomials over the Chebyshev’s nodes we obtain the expression for the function * * 2 2 ( ) g x :         * 2 1 1 * * * 2 2 2 2 2 1 1 1 1 ( ) (1) 1 2 1 N N k r k r k r x g x g u T T d N                             (6) After integration   * 2 1 * 2 2 1 x d arccos x      and     * 2 1 * 2 2 1  1 r x T d sin r arccos x r           we get           1 * * * * * 2 2 2 2 2 2 1 1 1 1 ( ) (1)    2 N N k r k k r g x g u arccos x T sin r arccos x N r                            (7) Analogously we obtain the expressions for * * 0 0 ( ) g x and * * 1 1 ( ) g x at * 0 0 0 0 0 / ,   x x l x L   and * 1 1 1 1 1 / ,     x x l x L   :               1 * * * * * 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 ( ) (1) / 2 N N k r k k r g x g g x g l l u arccos x T sin r arccos x N r                                  (8)     * * * 1 1 1 1 1 1 1 1 ( ) (1) / g x g g x g l l                   1 * * 1 1 1 1 1 1 1 2 N N k r k k r u arccos x T sin r arccos x N r                          (9) From Eq. (2) in the symmetric case we have           ' ' ' ' 1 æ v v v 4 g x x x x G                (10) Integrating we obtain the relation: (1 ) ( ) v v v( ) 2 4 n n æ g x x C G        , where n is a segment number. The constants of integration C n are determined by displacement at the end of the segment: 2 0 C    2 2 1 (1 ) 4 æ g l C G          1 1 0 1 1 1 2 2 (1 ) (1 ) 4 4 æ g l æ C C g l g l G G              Thus the crack opening displacement (COD) for the segment L n is defined as following * * * * (1 ) ( ) ( ) 2v( ) 2 2 n n n n n n æ l g x COD x x C G     (11) Since for the plane stress (1 ) 2 4 æ G E   , the expression for * ( ) n COD x takes the form                                  1 * * * 1 1 4 1 ( ) 2 [ 2 ] ,    0,1,2 N N n k n Y n n n r k n k r Y u l S COD x C arccos x T sin r arccos x n EN S r (12)