Issue 19

K. Pham et alii, Frattura ed Integrità Strutturale, 19 (2012) 5-19; DOI: 10.3221/IGF-ESIS.19.01 7 6. The condition of monotonicity of /   w E is introduced by sake of simplicity; it is unessential and denotes that the strain does not decrease when the damage grows; 7. The condition of monotonicity of /   w S is essential; it denotes the softening property, i.e. the decreasing of the stress when the damage grows. 8. The condition 1 ( ) / ( ) = 0 lim       w S ensures that the material cannot sustain any stress when its damage state is 1. Example 1 A family of models which satisfy the assumptions above is the following one, when > > 0 q p : 2 0 0 ( ) = (1 ) , ( ) = (1 (1 ) ) 2         q p c q E E w pE (3) It contains five material parameters: the sound Young modulus 0 > 0 E , the dimensionless parameters p and q , the critical stress > 0  c and the internal length > 0  whose physical interpretation will be given in Section The homogeneous solution and the issue of uniqueness. The condition > 0 q is necessary and sufficient in order that ( )    E be decreasing from 0 E to 0 while the condition > 0 p is necessary and sufficient in order that ( )    w be increasing from 0 to a finite value. If > 0 p and > 0 q , then the condition > q p is necessary and sufficient in order that ( ) / ( )        w E be increasing to  while ( ) / ( )       w S is automatically decreasing to 0 . Example 2 Another interesting family of models which satisfy the assumptions above is the following one 2 0 0 0 (1 ) ( ) = , ( ) = ( ) (1 ) 2          q p E E w p q E (4) where 1  p and 1  q are two constants playing the role of constitutive parameters and 0  represents the critical stress of the material. The damage problem of a bar under traction Let us consider a homogeneous bar whose natural reference configuration is the interval (0, ) L and whose cross-sectional area is S . The bar is made of the nonlocal damaging material characterized by the state function  W given by (1). The end = 0 x of the bar is fixed, while the displacement of the end = x L is prescribed to a non negative value t U (0) = 0, ( ) = 0, 0   t t t u u L U t (5) where, in this quasi-static setting, t denotes the loading parameter or shortly the ``time", t u is the displacement field of the bar at time t . The evolution of the displacement and of the damage in the bar is obtained via a variational formulation, the main ingredients of which are recalled hereafter, see [2] for details. Let U t  and  be respectively the kinematically admissible displacement fields at time t and the convex cone of admissible damage fields:       0 = : (0) = 0, ( ) = , = : (0) = 0, ( ) = 0 , = : ( ) 0,     U t t v v v L U v v v L x x    (6) where 0  is the linear space associated with U t  . The precise regularity of these fields is not specified here, we will simply assume that there are at least continuous and differentiable everywhere. Then with any admissible pair ( , )  u at time t , we associate the total energy of the bar 2 2 2 0 0 0 1 1 ( , ) := ( ( ), ( ), ( )) = ( ( )) ( ) ( ( )) ( ) 2 2                       L L u W u x x x Sdx E x Su x w x S E S x dx  (7) For a given initial damage field 0  , the damage evolution problem reads as: For each > 0 t find ( , )  t t u in  U t   such that For all ( ) ( , ) , '( , )( , ) 0             t t t t U t v u v u    (8)