Issue 19

K. Pham et alii, Frattura ed Integrità Strutturale, 19 (2012) 5-19; DOI: 10.3221/IGF-ESIS.19.01 18 It remains to study under which condition this relation between t and  is invertible. In other words, we have to find when the overall curve  --  does not contain snap-backs. Specifically, in order that there is no snap-back, we must have / ( ) 0, d d       . By virtue of (36), that gives an upper bound for L : 1 0 (0, ]0 ( ) := inf d M d L n E nL d               (41) Of course, this condition is never satisfied when 1 ( ) d     is not decreasing. (For example, in the case of the family of models of Example 2, it is never satisfied if < 2 q , cf Example 7.) Depending on the properties of the model, the length of the bar and the number of localization zones, we can obtain different scenarii. Let us consider only the situation with one localization zone ( =1 n ), to simplify the presentation. We can distinguish two cases: 1. Case 0 > 2 ( ) m M L D L   . In such a case we have three situations (a) For very short bars, i.e. 0 2 ( ) L D   , no solution with localization is possible. The homogeneous solution is the unique solution; (b) For short bars, i.e. 0 2 ( ) < m D L L   , a solution with localization is possible just after the elastic phase, but the localization zone will progressively cover all the bar and a snap-back is possible; (c) For long bars, i.e. > m L L , a solution with localization is possible, but it is necessarily discontinuous in time because of the presence of a snap-back in the overall stress-strain response. 2. Case < m M L L . We have then four possibilities, the first two (a) and (b) are the same as before (a) For intermediate bars, i.e. < m M L L L  , a continuous solution with localization is possible, the damaged zone does not reach the boundary, there is no snap-back; (b) For long bars, i.e. > M L L , a solution with localization is possible, but it is necessarily discontinuous in time because of the presence of a snap-back. Example 8 In the case of the family of models of Example 2, if < 2 q , then < 0 M L . It is not possible to find a non homogeneous solution without snap-back. If = 2 q and = 4 p , then 0 0 4 72 = < = 6 17 17 m M L L      . Therefore, for bars with an intermediate length we can find a continuous in time localized solution, cf Fig. 7 (left), while for long bars a localized solution is necessarily discontinuous in time just after the elastic phase, cf Fig. 7 (right). Figure 7 : Overall stress-strain relations for a law of Example 2 with = 4 p and = 2 q (Thin curve=homogeneous response; Thick curve=localized response). Left: For a bar of intermediate length; Right: For a long bar. C ONCLUSION AND PERSPECTIVES e have proposed a method of construction of non homogeneous solutions for the one-dimensional damage evolution problem of a bar under traction. We have shown that the properties of such a localized solution is very sensitive to the parameters of the model. This strong dependence could be very interesting from an W