Issue 19

K. Pham et alii, Frattura ed Integrità Strutturale, 19 (2012) 5-19; DOI: 10.3221/IGF-ESIS.19.01 6 the body and the internal length of the material. The goal of the present paper is to extend a part of the results (the questions of stability will no be investigated) of [2] for a large class of elastic-softening material. Specifically, we propose a general method to construct localized solutions of the damage evolution problem and we study the influence of the constitutive parameters on the response. Several scenarii depending on the bar length and on the material parameters enlighten the size effects induced by the non local term. The paper is structured as follows. Section Setting of the gamage problem is devoted to the statement of the damage evolution problem. In Section Non homogeneous solutions of the damage problem we describe, perform and illustrate the method of construction of localized solutions and conclude by the different scenarii of responses. The following notation are used: the prime denotes either the spatial derivative or the derivative with respect to the damage parameter, the dot the time derivative, e.g. = /    u u x , ( ) = ( ) /     E dE d , = /      t . S ETTING OF THE DAMAGE PROBLEM The gradient damage model e consider a one-dimensional gradient damage model in which the damage variable  is a real number growing from 0 to 1, = 0  is the undamaged state and =1  is the full damaged state. The behavior of the material is characterized by the state function  W which gives the energy density at each point x . It depends on the local strain ( )  u x ( u denotes the displacement and the prime stands for the spatial derivative), the local damage value ( )  x and the local gradient ( )   x of the damage field at x . Specifically, we assume that  W takes the following form 2 2 2 0 1 1 ( , , ) = ( ) ( ) 2 2              W u E u w E (1) where 0 E represents the Young modulus of the undamaged material, ( )  E the Young modulus of the material in the damage state  and ( )  w can be interpreted as the density of the energy dissipated by the material during a homogeneous damage process (i.e. a process such that ( )   x = 0) where the damage variable of the material point grows from 0 to  . The last term in the right hand side of (1) is the ``non local" part of the energy which plays, as we will see later, a regularizing role by limiting the possibilities of localization of the damage field. For obvious reasons of physical dimension, it involves a material characteristic length  that will fix the size of the damage localization zone. The local model associated with the gradient model consists in setting = 0  and hence in replacing  W by 0 W : 2 0 1 ( , ) := ( ) ( ) 2       W u E u w (2) The qualitative properties of the (gradient or local) model, in particular its softening or hardening character, strongly depend on some properties of the stiffness function ( )    E , the dissipation function ( )    w , the compliance function ( ) =1 / ( )     S E and their derivatives. From now on we will adopt the following hypothesis, the importance of which will appear later: Hypothesis 1 (Constitutive assumptions) ( )    E and ( )    w are non negative and continuously differentiable with (1) = 0 E , (0) = 0 w , ( ) < 0   E and ( ) > 0   w for all [0,1)   . Moreover ( ) / ( )      w E is increasing to  while ( ) / ( )     w S is decreasing to 0 when  grows from 0 to 1. Let us comment this Hypothesis before to give an example 1. The interval of definition of  can always be taken as [0,1] after a change of the damage variable; 2. The condition < 0  E denotes the decrease of the material stiffness when the damage grows; 3. The condition (1) = 0 E ensures the total loss of stiffness when =1  ; 4. The positivity and the monotonicity of w is natural since ( )  w represents the energy dissipated during a damage process where the damage grows homogeneously in space from 0 to  ; 5. The boundedness of w is characteristic of strongly brittle materials with softening; this condition disappears in the case of weakly brittle materials with softening or in the case of brittle material with hardening; W