numero25

D. A. Hills et alii, Frattura ed Integrità Strutturale, 25 (2013) 27-35 ; DOI: 10.3221/IGF-ESIS.25.05 28       , ( , , ) zz rr r r r           (2) where r is the radial distance from the notch tip, the angle  is measured from the notch bisector and taken to be positive in the counter-clockwise direction, the eigenvalues I  , II  , are the lowest roots of the equations sin 2 sin 2 0 I I      (3) sin 2 sin 2 0 II II      (4) and the generalised stress intensity factors are defined along the bisector of the ‘notch’   0   as   1 0  lim , 0 I I r K r r       (5)   1 0 lim , 0 II II r r K r r       (6) Williams' solution may be written in an alternative form that is more suitable for frictional contacts. The punch has an included angle  , and is in contact with a half-plane, as shown in Fig. 1, so that the total included angle is 2      . The contact interface lies along the line   / 2      , and is denoted int  , while the distance from the notch tip along the interface line is x . The direct   p x , and shearing   q x , interfacial tractions may therefore be written as         1  1  0 1  0 1  , I II I II I II int I int II int I II p x x K x f K x f K x K x                      (7)         1  1  0 1  0 1  , I II I II I II I II r int I r int II r int I r II r q x x K x f K x f K x g K x g                       (8) where compression is taken to be positive, and the generalised stress intensity factors calibrated along the interface line are denoted 0 n K , where   , n I II  , and are found from   0 I I I int K K f    ,   0 II II II int K K f    (9) where     I r int I r I int f g f       ,     II r int II r II int f g f       (10) Figure 1 : A diagram of the idealised geometry considered, including the coordinate system. E DGE S EPARATION illiams’ solution displays a power law variation of the stress field with radial distance from the notch tip and expresses this as a series expansion, of which we consider only the first two terms; I K and II K . Because each term in the series is raised to a different power (except in the case of an edge crack when    ), the relative   x r  int Half-plane Punch W