numero25

D. A. Hills et aliii, Frattura ed Integrità Strutturale, 25 (2013) 27-35; DOI: 10.3221/IGF-ESIS.25.05 33 Figure 4 : A plot showing the size of the plastic zone p 0 x / d , as a function of strength of the applied load 0 G / k , for example punch angles of   60 , 90 ,120      , shown in red, black, and blue, respectively. D ETERMINATION OF THE D OMINANT S LIP P ROCESS : P LASTIC VS . F RICTIONAL e have described how to calculate the relative sizes of the implied regions of adhesion, slip, and separation, and also how the size of the plastic zone specified along the interface line scales with remote load, so we are in a position to address the question that motivated this analysis; "when is the frictional slip enveloped by plastic slip, so that the corner is effectively a notch in a monolithic material?" The first thing to be said on this issue is that, if frictional slip is implied to extend out past the mode II region of the solution and into the region where bounded terms control the behaviour of the contact, then whether or not the plastic zone fully envelops the zone of frictional slip at the edge of contact cannot be determined through examination of the implications Williams' asymptotic solution. To find out when this condition obtains we consider the value of the implied traction ratio in the mode II region, which, for the example punch angles of   60 ,90 ,120      , is   0.3224, 0.2189, 0.8500 II r g     , respectively. Thus, if the coefficient of friction is less than the absolute value of this implied traction ratio, i.e. if II r f g   , then the question of whether or not the frictional slip zone is enveloped by plasticity cannot be determined by consideration of Williams' asymptote. If, however, slip is not implied to extend past the mode II region of the solution, and is contained within the asymptote, i.e. if II r f g   , then further examination of the implications of Williams' solution is merited. For cases when II r f g   and thus adhesion is implied in the mode II region of the solution, we can calculate the strength of load required to imply a plastic zone of equal size to the implied zone of frictional slip. This is achieved by setting     0 0 / / p s x d x d  in Eq. (25), and solving for the value of 0 / G k . The result of this calculation is plotted against the coefficient of friction f , for punch angles of   60 ,90 ,120      , in Fig. 5. Also plotted in Fig. 5, are three horizontal dashed lines showing the values of II r g  for the three punch angles considered. This information is added because, as stated above, if the coefficient of friction is less than this value, i.e. if II r f g   , then slip is implied to extend outside the region controlled by the asymptote, and the magnitude of load required to imply a process zone of equal size to the frictional slip zone cannot be calculated within the asymptote. The solid lines in Fig. 5 plot the magnitude of load 0 / G k , that results in the conditions     0 0 / / p s x d x d  and II r f g   both being met, while the dotted lines plot the value of 0 / G k that result in only the first condition being satisfied. So, for loads greater in strength than the solid line in Fig. 5, slip is implied only in the mode I dominated region (because II r f g   ), and the process zone is implied to be larger than the slip zone in this region. Thus, in this case the contact is expected to behave like a notch. For loads greater than the dotted lines in Fig. 5, which 60° 90° 120° 1 2 3 4 5 x p d 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 G 0 k W