Issue 19

K. Pham et alii, Frattura ed Integrità Strutturale, 19 (2012) 5-19; DOI: 10.3221/IGF-ESIS.19.01 15 Figure 4 : Damage profile in the localization zone when the bar breaks, for = 4 q and different values of the parameter p ( = 1/ 2,1, 2, 4 p ) in the family of brittle materials of Example 1. Figure 5 : The damage profile for a given t and its evolution with t by assuming that   t t is decreasing in the case of the model of Example 1 with = 2 p and = 4 q . The rupture occurs when = 0  t and ( ) = 1   t . We check numerically that ( )    D is decreasing. It is easy to check that ( )    d  is decreasing with ( ) = 0  d c  while (0) d  represents the dissipated energy in a localization zone during to the process of damage up to the rupture. Let us call fracture energy and denote by c G this energy by reference to the Griffith surface energy density in Griffith theory of fracture. Since (0) =1  , we have Property 6 (Fracture energy) The dissipated energy in an inner localization zone during the damage process up to the rupture is a material constant c G which is given by 1 0 0 = 8 ( )     c G E w d (34) Because of the lack of constraint on the damage at the boundary, the dissipated energy in a boundary localization zone up to the rupture is / 2 c G . Example 6 In the case of the family of strongly materials of Example 1 the fracture energy is given by 1 0 = 2 , = 1     p c p c p q G J J v dv p (35) Thus c G is proportional to the product of the critical stress by the internal length, the coefficient of proportionality depending on the exponents p and q . The force-displacement relation The time is fixed and we still omit the index t . Let U be the prescribed displacement, = /  U L the average strain and  the stress in the bar which contains 1  n localization zones. Using (11), recalling that = 0  outside the localization zones and that all localization zones have the same size 2 ( )  D and the same profile, we get