Issue 19

K. Pham et alii, Frattura ed Integrità Strutturale, 19 (2012) 5-19; DOI: 10.3221/IGF-ESIS.19.01 8 with the initial condition 0 0 ( ) = ( )   x x . In (8), '( , )( , )   u v  denotes the derivative of  at ( , )  u in the direction ( , )  v and is given by 2 2 0 0 1 '( , )( , ) = ( ) ( ) ( ) 2                                 L u v E Su v E Su w S E S dx  The set of admissible displacement rates  u can be identified with ( )  U t  , while the set of admissible damage rates   can be identified with  because the damage can only increase for irreversibility reasons. Inserting in (8) =    t and =   t v u w with 0  w  , we obtain the variational formulation of the equilibrium of the bar, 0 0 ( ( )) ( ) ( ) = 0,       L t t E x u x w x dx w  (9) From (9), we deduce that the stress along the bar is homogeneous and is only a function of time = 0, = ( ( )) ( ), (0, )        t t t t E x u x x L (10) Dividing (10) by ( )  t E , integrating over (0, ) L and using boundary conditions (5), we find 0 ( ( )) =    L t t t S x dx U (11) The damage problem is obtained after inserting (9)-(11) into (8). That leads to the variational inequality governing the evolution of the damage 2 2 0 0 0 0 ( ) 2 ( ) 2 0                    L L L t t t t S dx w dx E dx (12) where the inequality must hold for all    and becomes an equality when =    t . After an integration by parts and using classical tools of the calculus of variations, we find the strong formulation for the damage evolution problem: For (almost) all 0  t , Irreversibility condition : 0    t (13) Damage criterion : 2 2 0 ( ) 2 ( ) 2 0             t t t t S w E (14) Loading/unloading condition : 2 2 0 ( ( ) 2 ( ) 2 ) = 0              t t t t t S w E (15) Remark 1 We can deduce also from the variational approach natural boundary conditions and regularity properties for the damage field. In particular, we obtain that   t must be continuous everywhere. As boundary conditions at = 0 x and = x L we will simply take (0) = ( ) = 0     t t L although the more general ones induced by the variational principle correspond to a combination of inequalities and equalities like (14)-(15). These regularity properties of the damage field (and consequently the boundary conditions) hold only for the gradient model ( 0   ) and disappear for the local model ( = 0  ). As long as the regularity in time is concerned, we will only consider evolution such that   t t is at least continuous. In terms of energy, we have the following property Property 1 (Balance of energy) Let us assume that the bar is undamaged and unstretched at time 0, i.e. 0 = 0  and 0 = 0 U . By definition, the work done by the external loads up to time t is given by 0 ( ) =    t e s s t U ds  (16) the total dissipated energy in the bar during the damage process up to time t is given by 2 2 0 0 0 1 ( ) = ( ) ( ( )) 2        L L d t t t E x dx w x dx  (17) while the elastic energy which remains stored in the bar at time t is equal to