Issue 19

K. Pham et alii, Frattura ed Integrità Strutturale, 19 (2012) 5-19; DOI: 10.3221/IGF-ESIS.19.01 16 0 1 0 2 ( ) = ( ( )) = ( ( ))                L L nD U S x dx n S x dx E  The integral 1 ( ( ))   S x dx  can be transformed into an integral over the range of  by using (25). Indeed, by symmetry it can reads as 1 ( ) 1 2 ( ( ))     x x D S x dx . Making the change of variables   x , since = ( , )     d H dx , we obtain ( ) 0 1 ( ) ( ( )) = 2 ( , )           S d S x dx H  Recalling (28), we finally obtain the overall stress-strain relation 1 = ( ) ( )        e d n L (36) with ( ) 1 0 0 0 1 ( ) = , ( ) = 2 ( ) ( , )                     e d d S E E H (37) Remark 3 For a given n , (36) gives the average strain in term of the stress. That corresponds to a curve in the    plane, parametrized by  varying from 0 to 0  . The curve = 0 n is the segment corresponding to the elastic phase. Thus  can be decomposed into two terms, one associated with the elastic part of bar, the other with the localization zones. Note that 1 ( )     d depends neither on the length of the bar nor on the internal length of the material. The properties of monotonicity of the function 1 ( )     d play an important role on the presence of snap-backs in the overall response of the bar, see the next subsection. Since 1 0 ( ) = 0   d and since 1 ( ) > 0   d for 0 <   , 1 ( )     d is necessarily decreasing in the neighborhood of 0  . We have in particular Property 7 (Behaviour of 1 ( )     d near 0  ). 1 0 ( ) = 0   d and 5/2 2 2 1/2 0 0 1 0 2 3/2 0 2 (0) ( ) = ( (0) 2 (0))            d S E d d S w (38) On the other hand, the behavior of 1 ( )   d when 0 /   is small is very sensitive to the constitutive parameters as it is shown in the following example and on Fig. 6. Figure 6 : Graph of the function 1 ( )     d giving the contribution of a localized zone on the overall strain in the case of the model of Example 2 with = 4 p and different values of q (dashed: =1 q , thick: = 2 q , thin: = 3 q ). Example 7 In the case of the family of models of Example 2, we have