Issue 18

G. Del Piero et alii, Frattura ed Integrità Strutturale, 18 (2011) 5-13; DOI: 10.3221/IGF-ESIS.18.01 9 at the onset of the inelastic regime are summarized in Fig.1. The subsequent part of the response curve depends on the analytic expression of the cohesive energy  away from the origin. Anyway, the picture shows a major defect of the model, that is, the incapability of reproducing the regime of strain softening , in which the response curve exhibits a negative slope. Indeed, catastrophic failure occurs as soon as   (0) takes a negative value.  T HE NON - LOCAL MODEL he reproduction of strain softening becomes possible in the non-local model obtained by adding to the energy (3) a term proportional to the square of the gradient of the inelastic deformation    l dx x x w E 0 ))) (( )) (( ( ) ,(      +  l dx x 0 2 2 1 )('   (17) where  is a small positive constant. The addition of the new term brings additional terms to the yield condition (8)      (  ( x ))    ( x ) (18)  and to the Kuhn-Tucker conditions (11) 0 )(  x   , 0 )('' )) (('   x x     , 0 )( )) ('' )) ((' (   x x x       (19) and (13) 0 )(  x   , 0 )('' )( )) ((''   x x x          , 0 )( )) ('' )( )) (('' (   x x x x            ,  x (20) It also adds a positive term to the second variation. Then the necessary condition (14) is relaxed, and this is the reason why a description of strain-softening becomes possible. At the onset of the inelastic regime, due to the additional boundary conditions required by the supplementary gradient term, the inelastic deformation is not anymore constant all over the bar. The choice of the additional boundary conditions is a delicate point. Our choice of taking   ( l ) =  (0) = 0 (21)  is discussed in detail in the paper [1]. With these conditions, for the inelastic strain rate we get the expression xk x l x sinh 2 tanh 2 tanh )0('' )(                , (22) with  =(   (0) /  ) 1/2 , if   (0) > 0, and  ) ( 2 )( x lx x        (23) if   (0) = 0. In both cases, the added energy term has the effect of increasing the slope of the force-elongation response curve. In particular, for   (0) = 0 a hardening response takes the place of the perfectly plastic response predicted by the local model. For   (0) < 0, there are two types of solutions: the full-size solution xk lk xk x sin 2 tan 2 tan )0('' )(              , (24) with k = (    (0) /  ) 1/2 , if kl < 2  , and the localized solution  )) ( cos 1( )0('' )( axk x         , (25) T