numero25

D. A. Hills et alii, Frattura ed Integrità Strutturale, 25 (2013) 27-35; DOI: 10.3221/IGF-ESIS.25.05 32 Plots of the implied slip regions (from Eq. (21)) as a function of the coefficient of friction f , for the case when I K is negative and II K is positive, are shown in Fig. 3, for three sample punch angles   60 ,90 ,120      . Finally, for cases in which both I K and II K are negative, separation is implied to extend from the interior of the contact, with closure and a slip region at the edge of contact whose extent is controlled by the next term in a series expansion. We do not consider this possibility here or in any subsequent analysis. T HE P LASTIC Z ONE he implied elastic state of stress near a sharp feature is singular, but in practice the singular behaviour is truncated by the presence of a plastic zone incorporating a process region, and whose extent is determined by the strength of the applied load and the yield strength of the material. In order to determine the size of the plastic zone, implied by violations of the yield condition, we use the second invariant of deviatoric stress   2 2 2 2 2 3 rr zz rr zz zz rr r J                      (22) and note that, with this scaling von Mises yield condition is 2 2 3 J k  (23) where k is the yield stress of the material in pure shear. We then modify Eq. (12) to be specifically along the interface of the contact, i.e.       1  1  0 0 0 , I II ij int I II ij int ij int x x x f f G d d                       (24) and use this expression (Eq. (24)) together with Eq. (22), and then simplify the result, to give             2 1  2 1  2  2 2 0 0 0 0 I II I II I int II int I II int x x x J G p p p d d d                                            (25) where   I int p  ,   II int p  , and   I II int p   represent the  -dependence of the mode I , mode II , and mixed mode terms in the solution, respectively. We now impose the yield condition given in Eq. (23), denote the size of the plastic zone along the interface line p x , and solve for 0 / G k , which gives             1  2 1  2 1  2  2 0 0 0 0 3 I II I II p p p I int II int I II int x x x G p p p k d d d                                             (25) If the punch angle  , is specified, Eq. (25) can be used to determine the size of the plastic zone 0 / p x d , as a function of strength of the applied load 0 / G k , and this is plotted for punch angles of   60 , 90 ,120      in Fig. 4. This figure illustrates the perhaps surprising property that the relationship between the strength of the applied load 0 / G k , and the normalised size of the plastic zone specified along the interface line 0 / p x d , is not monotonically increasing. This is because, as described in Hills et al. [1], as strength of the remote load is increased, the plastic front rotates and changes from being very mode I like when loaded lightly to being very mode II like in character when loaded heavily. Therefore, although the maximum radius of the plastic zone does increase monotonically in size with stronger loading, when specified along the interface line, the plastic radius takes on its maximum value when the remote loading causes the maximum plastic radius to coincide with the interface line. The plastic length along the interface then decreases for stronger loads, because the maximum plastic radius rotates away from the line corresponding to the frictional interface. T