Issue 19

K. Pham et alii, Frattura ed Integrità Strutturale, 19 (2012) 5-19; DOI: 10.3221/IGF-ESIS.19.01 12 ( , ( )) = 0, 0 < ( ) <1      H (27) ( )   corresponds to the maximal value of the damage (at the given time), taken at the center of the localization zone. Therefore, we have obtained the Property 2 (Profile of a localized damage field) For a given stress (0, )    c , the damage field in an inner localized damage zone ( ( ), ( ))     i i x D x D is given by (31) while the half-length ( )  D of the localized damage zone is finite, proportional to the internal length  and given by (28). The damage profile is symmetric with respect to the center i x of the localized damage zone, maximal at the center, the maximal value ( )   being given by (27). The damage profile is a continuously differentiable function of x , decreasing from ( )   at the center to 0 at the boundary of the localized damage zone. The matching with the undamaged part of the bar is smooth, the damage and the gradient of damage vanishing at the boundary of the localized damage zone, see Fig. 2. Figure 2 : A typical damage profile in an inner localization zone and in a boundary localization zone when 0 < <   c Concerning the dependence of ( )   on  , we have: Property 3 (The dependence on the stress of the maximal value of the damage in the localization zone.) When  decreases from 0  to 0, ( )   increases from 0 to 1. Proof. Indeed, let 1 2 0 0 < <     . Since 1 1 2 2 1 2 0 = ( , ( )) = ( , ( )) < ( , ( ))          H H H , and, since 1 ( , ) < 0   H when 1 ( ) < <1    , we have 1 2 ( ) > ( )     . Hence ( )     is decreasing. Since 0 / ( , ) < 0      H for > 0  , we have 0 ( ) = 0   . Let us prove that 0 ( ) =1 lim     . Let 0 = ( ) lim      m (the limit exists and is positive since ( )   is decreasing). If <1  m , passing to the limit in (27) when  goes to 0 gives 0 = (0, ) = 2 ( )   m m H w , a contradiction. Hence =1  m . The size of the localization zone is deduced from (25) by integration. It depends also on  and is given by ( ) 0 ( ) = ( , )         d D H (28) ( )  D is proportional to the internal length and is finite because the integral is convergent. (Indeed, ( , )   H behaves like ( , 0)      H near = 0  and like ( , ( ))( ( ))           H near = ( )    . Since ( , 0) > 0     H and ( , ( )) < 0       H , the integral is convergent.) Provided that 2 ( )   L D , it is really possible to insert a localization zone of size 2 ( )  D inside the bar. Concerning the dependence of ( )  D on  , we obtain the following fundamental property the proof of which is not given here (it is based on a careful study of the behavior of the integral giving ( )  D ): Property 4 (Dependence on the stress of the size of the localization zone.) The size ( )  D of the localization zone varies continuously with  , 0 ( )  D and 0 0 = ( ) lim    D D are finite and given by 1 0 0 0 0 2 0 0 2 ( ) = , = (0) 2 (0) 2 ( )            E E D D d S w w (29)