G Pijaudier-Cabot

MICRO-CRACK CLUSTERING, NON LOCAL AND GRADIENT DAMAGE MODELS

The aim of this chapter is to show how the results from statistical analyses of damage and micro mechanics can inspire phenomenological moddling. namely continuous damage models. Non local and gradient dependent damage models are introduced. The analysis of stain localisation shows the importance of incorporating an internal length in the constitutive relations. while this internal length can be regarded as a correlation length in statistical models. Finally. these continuum modds are shown to be consistent with fracture mechanics analyses of structural components.

Introduction

Continuous damage models are constitutive relations in which the mechanical effect of cracking and void growth is introduced with internal state variables which act on the elastic stiffness of the material (see e.g. Krajcinovic 1989 or Lemaitre and Chaboche 1989 for reviews on this topic). In most cases, continuous damage model ling is a phenomenological approach to the description of an elastic material containing voids or cracks. It is possible. however. to lay the basis of such macroscopic models on analyses of lattices and/or arrays of cracks.

From these analyses. we will see that two characteristics of damage models arise. The first one is that damage should enter in the macroscopic description as a variation of the material stiffness. The second one is that. in the course of progressive damage. initial disorder in the material tends to disappear and spatial correlation occurs.

In this chapter. we will start ti•om such observations and see how they can be incorporated in constitutive relations. We will arrive to non local and gradient damage models and investigate. at the macroscopic level. how failure can be described as a process of localisation of damage. This study will highlight the necessity l)f using non local model on physical grounds since, without them. failure is predicted to occur without energy dissipation.

Dama�e in a discrete system

). The lattice size is L x L where L is related to the total number cf bonds n == '2L2. Each bond behaves line a rly up to an assigned threshold where brittle failure is reached. The model does not aim at describing a specific material. It ought to be the simplest

model whose thermodynamic limit (lattice of infinite size) should be described by the damage theory, as we will see further. Yet, this description includes the essential ingredients: a two dimensional geometry, initial disorder, interactions and redistribution as the number of broken bonds increases. Instead of solving a mechanical problem, we use an electrical analogy which turns out to be strictly equivalent to the mechanical problem. The equations of equilibrium are similar but one dimensional instead of being two dimensional (same as in heat conduction problems). The results can be interpreted in a mechanical fashion provided the current is replaced by the stress, the voltage by the strain and the conductance by the Young's modulus. The scaling properties of the mechanical problem and those of the electrical problem are indistinguishable (de [START_REF] De Arcangelis | Scaling and Multiscaling Laws in Random Fuse Network[END_REF]. Figure 1-b shows the behaviour of one bond. The same conductance (stiffness) is assigned to all the bonds. The heterogeneity of the material properties is restricted to the variability of the maximum current at failure ( (which is equivalent to the peak stress). We have chosen here a distribution which is constant between 0 and I, hence it is representative of a large disorder which yields to diffuse damage which localises progressively.

The boundary conditions at the limit of the lattice are periodic so that the behaviour of an infinite system is represented and boundary effects are avoided. A constant jump of voltage is applied along the two boundaries which are perpendicular to the direction y. and symmetl)' conditions are applied along the two other boundaries. At each stage of loading. a unit current I (load) is applied on the lattice. The current in each bond i is computed (solution of a linear algebraic system of n I 2 equations according to Kirchhoff law) along with the overall V conductance of the lattice. The computational algorithm removes one bond at a time. In order to determine the next bond to be broken at a given step. \\ e look for the bond where the ratio i, 1 i is minimum. This bohd w i ll fail when the current applied to the laHice is:

. I.

I, = mm(�)

I (!)
The voltage V and the overall conductance of the lattice are obtained from this critical current.

I - c
Note that there are two competing effects in Eq.( I): On one side and because the material is disordered, bonds with a small threshold (. may fail first. On the other side. single crack propagation in the lattice produces a concentration of current (e.g. at the crack tip) which may cause a bond with a high threshold to fail. This equation illustrates how re d ist ribution and disorder may compete during the failure process. Initially all local currents are identical, and thus the first bond to break is the weakest. If the disorder is strong enough, the enhancement cf current in the vicinity of this first broken bond is not enough to induce a failure, and the second weakest bond may fail. This process yields a diffuse damage. However, the current distribution becomes broader progressively and thus bonds are expecte d to break in series c{ spatially localised clusters leading to bigger and bigger micro-cracks which are still distributed over the entire lattice. Eventually, at some stage. the current hete rogeneity due to the presence of a large density of broken bonds will dominate over the distribution of strength and lead to a macroscopic localisation. A more quantitative discussion along these lines will be proposed in the seque I. Each lattice possesses a unique distribution of bond thresholds. Obviously, analyses cannot rely on a single realisation of the random process which assigns the threshold distribution. Several computations \Vith different random seeds must be perfom1ed and analysed in a statistical fashion. The number of runs decreases as the size of the lattice increases in order to maintain computer times which are reasonable. For instance the computation of each lattice cf size 64 x 64 took 4 cpu hours on a medium size workstation. Figure 2 shows one of these plots for a lattice of size ( 32 x 32).

•

The envelope of the characteristic points yields a response which is similar to that cf concrete. The various snap-backs mean that under increasing voltage (strain), several bonds may break during an infinitesimal positive increment of voltage. Note that at any stage, the unloading path is linear. The equivalent mechanical lattice has a global response which agrees with continuous damage models, without damage induced inelastic strains. unless the equivalent of internal stresses are introduced in the model as in Schmittbuhl and Roux ( 1994).

Figure 3 shows a 64 x 64 lattice at the inception of complete failure, when only one bond remains unbroken (i.e. when the lattice is about to separate into two pieces). As the number cf broken bonds increases, their location localises over a rough I ine progressively -damage is diffuse at the beginning of the failure process and localises to fom1 a macro crack. It is important to note that this localisation concerns the incremental damage. The final aspect cf the distribution of mi�ro cracks will appear as diffuse since it incorporates micro cracks generated at all stages.

, r r,'\; .

. . ' . ,,, ,.

•'

'r '• :• . ..

• \ � • I' • �, • ,. ,. ,f'• l/. ': '•/ ' { . '' \ • . ... • • I , ) �\ ' , �I • .' o I ... .. �� \ o •.' I'
, .

'J� .w\.' .� � .,; . The numbers of broken bonds at peak and at failure are interesting because they show that damage is distributed. Would damage be localised along a single line (as for brittle materials),

• . . , ,• , .. .. , ,_ .. f • , •• •' �, �-' • '•.1 /"' j, 0 0 A•f • , j , I,;
Clustering, the number of broken bonds at failure would be twice the system size, i.e. the number of bonds a long a straight line in the x d irection, and the number of broken bonds at peak would be zero. This result would be achieved when there is no disorder in the lattice. Table I shows for each different size of lattice considered the number of runs performed, the averaged number of broken bonds at the peak current (load) n, and the averaged number of broken bonds at failure n,. It should be noted that the number of broken bonds at failure is always higher than the number c{ broken bonds at peak. This indicates that the failure process is rather progressive and that the behaviour of the lattice departs from a brittle response due to spatial corTelation and local ' disorder. First, we are interested in the evolution of damage in the lattice and its influence on the global properties. For this, we will look at the distribution of the current in the discrete model at several states of damage. Second, we expect to observe a correlation length which characterises the smallest representative volume element at a given state of damage. This correlation length appears in the spatial distribution of currents and consequently in the spatial distribution of broken bonds. Both aspects will be examined in the next sections.

Global properties of the discrete model

The global properties of the discrete model at any state of damage are. in this approach.

direct ly connected to the local distri bution of c urrent N(i). I nstead of characterising the distribution itself, valuable information can be obtained with the analysis of the moments cf the distribution of the current. The moment of order m is defined as:

'/ f ""'N( .) , . !Ym= I /(/ (2)
Here, we will limit the analysis to moment of order up to 4. This assum ption is equivalent to a truncature in a series cl eve lopment. These mom ents are of interest because of their physical meaning: the moment of order zero is the number of unbroken bonds. The first order moment is related to the average current, the second order moment is proportional to the overall conductance G:

(3)

where r is the local resi stance of the bond (unit resi stance here) and V is the global voltage jump applied to the lattice, equal to one in our case. Note that the fourth order moment is a measure of the dispersion of conductance.

It is natural to investi gate whether the number of broken bonds is a variable which cbaracterises correctly the evolut ion of damage in the continuum sense, that is a degradation of t �e global conductance or stiffness. If the number of broken bonds q I L2 (d ivided by the dimension of the lattice in order to have a quantity which is size independent) is the pert inent damage variable, the plots of the global conductance versus the non dimensional number of broken elements should be independent of the size of the discrete model. Delaplace et al.

( 1 996) have shown that th is is not true. Figure 4 shows a log-log plot of the moments as a function of the second order moment fo r different sizes of lattice. These plots do not depend on the size of the system, at least before the peak is reached. Hence, damage can be defined as the variation of the overall conductance during failure. This variable describes well the distribution of the current (stress) in a si;e independent fa shion. A similar observation has been reported by Krajcinovic and Bas ista (1991). As we will see fu rther, it will be used in the derivation of phenomenological damage models.

0.6 ......... _ -m=lL=l6 --m=2L=l6 -m=3L=l6 -m�L=l6 �-•••-••o m=l L=64 --m=2 L=64 o--.. •-•o m=J L=64

Existence of a correlation length

Assume that there is a length denoted as � wh ich defines the smallest size of the representative vol ume of the material (RVE). By construction, this length is smaller or equal to the size of the lattice. Th is quantity is a correlation length in a statistical sense. Above this size, the material can be regarded as homogeneous, i.e. without correlation. The local response of each bond in the discrete model is not independent for cells in the lattice bellow this size. In such cells, the response cannot be regarded as that of an homogeneous continuum. In order to �h i bit the existence of a correlation length as the number of broken bonds increases, one can analyse their distribution over the cell. This analysis is based on the variation of distance between two bonds in the lattice which are consecutively broken during the loading history accord ing to the fo regoing definitions: the distance between to neighbouring vertical bonds is I because there are L bonds on each column of the cell, and the distance between to neighbouring horizontal bonds is 0.5 because there are 2L bonds on each row of the cell. 0.00 �------�--------�--------�------� 0.0 2.0 4.0 6.0 8.0 Fioure 5 shows for S)'Stems of size 16x 16 the histograms of the distribution of horizontal � distance between two consecutive broken bonds denoted as h(d) at the beginning of damage (20 broken bonds). near the peak (60 broken bonds) and near failure (I 00 broken bonds). In the absence of a correlation length, these histograms should be a set of horizontal lines, wh ich means that the next broken bond can found anywhere in the lattice. These histograms can be approximated at least up to the peak: the distribution of the broken links is assumed to follow a power law fu nction of the distance up to � and then it is an horizontal line. The length � corresponds then exactly to the smallest size of the RYE:

h(d) = d-r for d :s; �. h(d) = h • ford E [ �. �] (4)
The value of exponent T is 0.67 for the normalised distributions shown on figure 5. The

.stance � is also the length at which the distribution changes fr om diffuse damage (constant distribution) to a progressive localisation of damage. Hence, it can be considered as a measure of the size of the zone in wh ich damage localises. The first order moment of the distribut ion h(d) can be obtained numerically. Substitution of Eq. (4) in the expression of the fi rst order moment yields an equation where the only remaining unknown is the correlation length. Figure 6 shows the evolution of this correlation length. It increases with increasing damage. The existence of a correlation length suggests that the damage process should be spatially correlated. In section 4, we will consider the deterministic case of a array of micro cracks and show a g ain that some spatial correlation exists in the develo p ment of damage.

Continuous Damage

As concluded in the previous section, the influence of damage on the response of the material is a degradation of its elastic stiffn ess. The stress-strain relation reads:

CJ -E " ' """ge J E ij-tj k l k l (5)
where a ii is the stress component, Ekl the strain component, and E :} �twgeJ is the stiffness coefficient of the damaged material. We will focus attention on the simple, isotropic, Scalar damage model because it is the most widely damage model employed and also because it has been proved to be an efficient model for the description of the tensile fa ilure of concrete and

i+ v0 CJ--- v0 a i' .. E,i = Eo(l-d) '' E o( l -d ) [ kk ulJ] (7)
According to this equation, the Poisson 's ratio of the material is not affected by damage. The elastic (i.e. fr ee) energy per unit mass of material is:

(8)
where E 0 is the stiffness tensor of the undamaged material. This energy is assumed to be the state potential, which means that the thermodynamic fo rces associated to the two variables describing the material, the strain tensor and the damage variable, derive fr om this potential:

(9)

Note that Eq. (9-a) is exactly the same as Eq. (7). Y is the damage energy release rate. The meaning of this variable is deduced from the calculation of the energy dissipation. The d i ssipation rate is the difference between the total (internal) variation of mechanical energy for any arbitrary rate of strain and the elastic (recoverable) energy rate:

(10)

Substitution of Eqs (7-9) yields the dissipation rate � as a fu nction of the damage variab le:

(11)

Therefore, Y is the energy released per unit rate of damage. In this constitutive relation (with a single damage variable), the second principle of thermodynamics constrains the time derivative of damage to be zero or positive. Damage cannot decrease because the rate of dissipation cannot decrease according to the Clausius-Duhem inequality. The evolution of damage requires the definition of a loading fu nction and an evolution law which can be similar to the evolution equation in associated plasticity. The constitutive relation is sometimes given in an integrated format. We will see in section 5 two examples cf such constitutive re lations.

Micro Cracking and Continuous Damage

In most experiments, say a direct tension test or a bending test [START_REF] Mazars | Application de la mecanique de l'endommagement au comportement non lineaire et a la rupture du beton de structure[END_REF], damage starts as a uniform distribution of propagating micro cracks. At the inception of fa ilure, crack propagation is localised into a narrow zone and only a few micro cracks propagate to fo rm the visible macro crack at fa ilure [START_REF] Chudnovsky | The Role of Microdefects in Fracture Propagation Process[END_REF]. In order to provide some justifications fo r damage and non local damage models, Pijaudier-Cabot and Berthaud ( 1990) have considered the two problems shown on figure 7 which represent : (I) the initial state cf damage, i.e. a regular array of cracks in an elastic matrix and (2) the situation in which one crack has propagated while all the others crack of the array remain stable. Th is last situation is meant to simulate the inception of fa ilure with damage localisation in a simplistic (qualitative), one-dimensional fa shion .

Fracture Mechanics Analysis

Consider the regular crack array shown on figure 7-a. This array is subjected to a remote tensile loading denoted as a"". The objective is to compute the stress intemity factor and energy release for each crack tip and to exh ibit the effect of crack interaction. For this, an approx imate method. originally proposed by [START_REF] Kachanov | Elastic Solids with Many Cracks. A Simple Method of Analysis[END_REF] is implemented. First, we use the superposition and replace the tensile loading a"" by an internal pressure -0'00 in each crack S i . Then, this equ i valent problem ofN interacting cracks is replaced by N problems in which each crack is considered alone in an infinite medium and is subjected to an unknown internal pressure P;(x;) given by Eq.( 12):

r=O

The maximum order of the expansion is denoted R and a� are the coeffic ients of the expansion •

where i refers to the cracks S i and r refers to the considered orde r of expansion. It is also natural to expand the interaction tenns Pji of the S j crack on the Sj crack: R pj i (x;) = "La/Jj;(x;)

r=O

In this equation, the fu nction ff; represents the stress field on the imaginary location of the crack S i due to the crack Sj loaded by a pressure distribution given by I!,.

The coefficients a� are the solution of the linear algebraic system which is obtained from Eq.( 13) for i=I, N using the orthogonality properties of Legendre polynomials. The stress intensity factor is approximated using the second term of Eq.( 13). In the present examples, we took R = 2 which provides a sufficient accuracy.

Homogenisation -Relationship with Continuous Damage

The aim is here to evaluate the macroscopic stiffuess of the micro cracked medium depicted above, in the direction which is orthogonal to the crack planes. Obviously, the micro cracked material wh ich is considered is not isotropic because the crack orientation is not random but periodic. Therefore, the results established in this section are only qualitative. The elastic energy in each cell is computed and a simple equivalence with an elastic continuum is assumed. The elastic energy in each cell. of the micro cracked body is:

h/2 112 W = � J J a(x, y ): E-1 : a(x,y)d y dx -h/2-//2 (15)
where E-1 is the compliance matrix of the elastic, uncracked, material. In the equivalent homogeneous material, the average elastic strain due to the crack opening in each cell is:

I JW r=- ht d.� ( 16
)
The total strain in each cell is then the sum of the deformation of the material and of the average strain due to crack opening. Th is deformation is totally reversible, provided the crack lengths in each cell remain constant. It fo llows the equalities (see Bazant,198 7):

a� iŞ=r + E' 0 ( 17
)
where E0 is the directional stiffuess of the material without cracks and E is the directional stiffness of the material with cracks. Th is stiffn ess is set proportional to E 0 and it is a function of the damage variable. Eqs.( 15-17) provide an expression of the damage variable d as a function of the crack density . and of the interaction between cracks:

d = __!!_L I + H.Ă' (18)
H is the interaction parameter which depends on the crack configuration. For the plane-strain mode I problem where the crack lengths are equal in each cell, this interaction parameter is:

0 u:l ....... 1.0 0.8 0.6 ' a) H = 2iĐ(l-v-)(a0 --=-) 8 •••---1/b = 1 •� ----1/b ... 0.5 '• t\ ---1/b = 0.4 \'� --S.C.M. •\ . '"•-without interact i on ' , , \',,
''\ \ ':'.... (

19).

In this case the coefficients a� (here a0,a2) are equal from one cell to another. They are solution of a 2x2 linear algebraic system of equations which is solved independently from the intensity of the remote stress (J"". Eq.( 1 8) highligh ts the effect of the interactions between cracks and of the crack density on the equivalent stiffness of the micro cracked medium. The evolution of the elastic modulus E is plotted on figure 8 for different crack spacings b and l.

The theoretical prediction obtained with a Self Consistent Method (SCM) [START_REF] Laws | The Effect of Micro crack Systems on the Loss of Stiffness of Brittle Solids[END_REF]) is also plotted. Compared to SCM, the present approach shows that interactions produce first a shielding effect for low crack density, and an am plification effect fià large crack densities. Th is amplification results in a stiffn ess which is much lower than the one calculated with SCM in the case of a square cell. For a dilute distribution of micro cracks (small crack density), the two models are similar. They are consistent with the lattice analysis in the regime controlled by the initial disorder. We will see next that interaction may play a central role when the geometry of distributed cracking changes, same as in lattice analysis when the damage process start to exhibits a non zero correlation length.

Crack Interaction and Non Local Damage

Consider again the elastic body with cracks shown on figure 7. When the crack density is low (i.e. bl c? 1), the interactions between collinear cracks can be neglected and we can consider only parallel cracks which represent the first stage of damage in a tensile loading test. The inception of localisation in that crack array is described by the case where one of the cracks has propagated symmetrically (Fig. 7-b). We are going to compute the new value of the damage variable corresponding to this crack configuration and to compare with the previous analysis qualitatively. For the purpose of this demonstration, the crack propagation conditions are not considered. We will compare two states of damage very close to each other.

Crack interaction appears naturally in the homogenisation as a non local effect. The crack opening in one cell is affected by the state of deform ation and cracking in the other cells. In the parallel crack system shown on figure 7-b, the crack length in cell 0 has been perturbed of a small quantity 8c . The strain energy in a given cell k can be approximated in plane strain by:

7 'V - O'""Pk (1-v-)ck � k- + Eo h/2 h/2 (20) �[ I O'(x,-1 I 2).ii.ii(x,-l I 2)dx + I O'(x,l I 2).ii.ii(x,l I 2)dx] -h/2 -h/2
In this expression, the second and third terms represent the flow of energy through the contour ofeach cell k due to the interaction. These are typically non local terms which appear in all cells, including those where the crack length does not change. According to the homogenisation method, the influence of this perturbation will produce a variation of damage in those cells although the crack length has not changed. Damage must not only be a fu nction ofthe crack density in each cell but also of the crack densities in neighbouring cells. Then, the elastic energy in an equivalent homogenised material should not be a fu nction of the overall strain and damage variable only. It should include quantities which refer to the crack or damage distribution. One possibility fo r this is to define damage, denoted now as D, in the equivalent, homogenised, material as a weighted volume average:

D(x) = J IJI(x-s)d(s)ds V (21)
D i s cal led the "non local damage variable" and d is the "local damage variable .. by reference to the term inology used in non local elasticity ( Eringen and Edelen, 1972). The weight function 1Jf is normalised so that

J IJ!(s)ds = I I' (22)
It fo llows from this definition that when the local damage is homogeneous over an infinite 11ody, it is equal to the non local damage. Therefore, d(x) is equal to the valu e of damage given in Eq.( l8) calculated in a fictitious medium which has a constant crack density equal to f3(x) . This is the reason why d(x) is still called a damage variable. It is a local quantity de fined at each m aterial poin t independently from the distribution of cracking in the structure considered.

.0.1. ,... With these definitions, the weight function in Eq. ( 21) can be calculated: the value of the mac roscop i c non local damage [)k in each cell k is computed using Eqs. (16-17) in which the non local damage is substituted to the local damage. The de finition of the local damage provides also the exprc:::ssion d111 in each cel l m. Its val ue in cell m is computed by considering that ce ll m is s ittin g in a fi c titious medium of constant crack density equal to th::l! in cell m. In each cell, the local and non local damage are ditTerent because the perturbation of the crack length produces a modification of the crack interactions. Eq. ( 21) is rewritten as a discrete sum: m=-too

1.1 �--------------------------, l=2 0.9 l=3 0.7 -eo •0 o.5 � 0.3 0.1 ---� -O . f�TTTO���TT,rrrrr����
D k = L ifi k m d m m=-oo
where .Àkm is the mean value of i over the cell m when D is calculated is the cell k.

(23)

Since the values of local and non local damage are known everywhere, the weight fu nction may easily be deduced. Figure 9 shows this weight fu nction for two crack spacings l = 2c and ( = 3c. This plot shows also that damage will grow in cells number 1 and -/, which are adjacent to the cell 0 in which the crack propagates, despite the length of the cracks in these cells remains constant.

Would the interactions remain similar during fa ilure, the variable which describes the effect of micro cracking on the elastic constants would be purely local. Non locality is caused by a change of interactions during progressive fa ilure. These results show also that the weight fu nction, which is the mathematical representation of the variations of crack interaction, is a fu nction of the micro crack length and of the state of strain (or stress). In the non local damage models, this weight function will be assumed to remain constant. Furthermore, the weight distribution will be assumed to be spherically symmetric. A constant weight fu nction is an approximation of the average variations of interactions during a specific failure mode (tension in the present applications). According to homogenisation, different weight fu nctions should be employed for diffe rent failure modes at least.

Non Local Damage Models

Fracture mechanics and crack interaction considerations have shown that defin ing damage as a weighted average seems more appropriate, especially with regard to the description of localisation of cracking in an elastic material containing a distribution of micro cracks. A non local variable enables the description of micro structural changes, (and more particularly determ inistic interactions) that a local variable cannot accommodate. This result is also consistent with lattice analysis, as observed in section I. Such a phenomenological model has been investigated by Bazant and Pijaudier-Cabot ( 1988), along with a constitutive relation where the variable which controls damage is averaged, instead of averaging damage itself.

Scalar non local damage models are presented in this section. Similar developments could be performed starting from a plasticity model or with an anisotropic damage model [START_REF] Valanis | A Global Damage Theory and the Hyperbo licity of the Wave Problem[END_REF].

Energy-based model

The constitutive relation is exactly the same as for the local damage model (Eq. 7). The evolution of damage is different. The growth of damage is defined by a loading function f:

\' f( y, D)= f F(z)dz-D 0 ( 24
)
where F is (in this chapter) a fu nction of the strain tensor which is deduced from experimental data. Instead of depending on the local energy re lease rate y(x), the loading function depends on y(x), the average energy release rate due to damage at point x of the solid:

y(x) = - 1 -J If!( X-s)y(s)ds and V,(x) = J If!( X-s)ds V, (x) \' \' (25)
V is the volume of the solid, and y(s) is the energy release rate due to damage at point s defined by:

I Y(s) = -E(s): E : E(s)
. 2

If!( X-s) is the weight fu nction : The evolution law is prescribed as in plasticity models:

lfl(x-s) = lfl o exp( _ll x -����) 2(� (
(28)

with the Kuhn-Tucker conditions 8 � 0, f :s; 0 and 8f = 0. The fu nction g is the evolution potential controlling the growth of damage and 8 is here the damage multiplier. In the following, we have set g = y. For concrete in tension, the evolution of damage is:

(29)

with the constants h1 = 605MPa-1• b2 = 5.24 104 /'.!Pa-1• Y0 = 60 10-6 MPa.

Strain-Based Model

This model is the non local extension of the constitutive relation due to [START_REF] Mazars | Application de la mecanique de l'endommagement au comportement non lineaire et a la rupture du beton de structure[END_REF]. The positive strains control the growth of damage which is mainly due to micro crack opening in mode I. The following norm called equivalent strain is defined :

E= (30) i=l
where (e;t = 0 if E; < 0 and (e;t = E ; if E; � 0; and E; (i e [I, 3]) are the principal strains.

The non local variable £, which represents the average of the equivalent strain over the representative volume surrounding each point in the material, is the variable that controls the growth of damage (Saouridis, 1988) -I f -

E(x) = --llf(x-s)E(s)ds V,(x) V (31 )
V is the volume of the structure, V,(x) is the representative volume at point x, and llf(x-s)

is the weight function, the same as for the previous model. The evolution of damage is specified accord ing to following conditions:

f(£) = £ -/( and ifj(E) = 0 andj(£) = 0 then D = F(£) (33) if j(E) < 0 or ifj(E) = 0 andj(£) < 0 then .¿ = 0
Equations (33) define the damage surface and the growth of damage. I( is the softening parameter and takes the largest value of £ ever reached during the previous loading history at a given time and at the considered point in the medium. Initially K = K 0 , where K o is the threshold of damage. Ko is the tensile strain at which damage is initiated, that is when the maximum tensile stress .y is reached in a uniaxial test.

The evolution law, denoted as F(E) in Eq.( 33) is a function of the strain, instead ofthe strain rate. It is integrated with respect to time so that damage can be directly computed at any state of deformat ion. Damage is decomposed into two parts, q for tension and D, for compression. D is computed from the relation (34) which combines these two types of damage:

{34)

f) = I -"'•o (I -A t ) - A t 1 [ cxp(-B1(f-i'0 )) f) = I -"'•o ( I -A, ) - A, ' [ exp(-B,I[ -1\•0 )) (35)
and constants A c , Be, A t , B t are model parameters. These coefficients characterise the hardening/softening response of concrete. The factors at and ac are expressed as non dimensional functions of the principal strains. Note that in this model, the condition that damage cannot decrease should be added to the loading function in Eq. (33) because the factors at and ac may vary in the reversible regime and could yield a decrease of damage which is inconsistent with the second principle of thermodynamics.

Gradient Damage Model

Non local con<>titutive relations can be considered as the point of departure of gradient models. Following the approach of section 5.2, the definition of the non local effective strain in Eq. (3 1) can be written as: (37) where a is a material parameter of the dimension of a length squared. Equation (3 7) is, in fact, obtained from a second order Tay lor development of the effective strain in the non local expression (Eq. 31 ). Hence, we assume that the support of the weight function is very small compared to the spatial variation of the effective strain over the structure analysed. This new expression of the non local effective strain can be substituted in the equations defining the evolution of damage in section 5.2. Numerically, this constitutive model can be very awkward to implement in a finite element model because Eq. ( 37) is a differential equation which involves the th ird order derivatives of the displacement functions. To obviate this problem, Eq. ( 37) can be approximated by (38) which is much easier to implement in a finite element code as it does not involve more than the first order derivatives of the displacement (see [START_REF] Peerlings | Computational Modelling of Gradient Enhanced Damage for Fracture and Fatigue Problems[END_REF].

In the non local and grad ient models discussed above, the internal length is postulated, as the result of interactions between growing defects which are, by definition, of a non local character. In some situations, however, it is possible to derive such a model with an internal length from existing, but more advanced, theories of elasticity. These theories are directly related to the multiple field theories described by Mariano and Trovalusci ( 1999) and recalled in the next chapter of this book.

Let us consider a material which contains voids with isotropic characteristics. In fa ct, we consider a porous isotropic material whose porosity is growing as damage, defined as a scalar quantity, develops. From the micromechanical point of view, damage will be characterised by the variation of volume fraction of material denoted as r(x, t). For a strain free material, the volume fraction is equal to I initially, and it decreases when damage grows. Failure is reached �hen the volume fraction is equal to zero. Starting from a reference configuration where the m aterial is strain-fr ee and the volume fraction is �"R(X), the variation of the volume fraction cf material is cp(x, t) = v(x, t)-�• R (x). This variation of volume fraction of material can be due to damage growth or straining.

For constant damage, the porous material is elastic and its behaviour is modelled using the theory of elastic material with voids [START_REF] Cowin | Linear Elastic Materials with Voids[END_REF]. The governing equations are (in the absence of body forces):

(J .. = 0 lj . .J h;.; + g = 0 (39a) (39b)
where h; is the equilibrated stress vector, g is the equilibrated body fo rce, and a is the overall stress in the porous material. Equation (39b) was first suggested in the case of granular materials by Goodman and Cow in ( 1972), and arises also in the microstructural theories cf elastic materials. Variables g and h; have been given spec ific interpretation: they are related to the stresses due to centres of dilation made of three couples of opposite fo rces without moments acting along three mutually orthogonal directions at material points. Such fo rces con•espond to the local pressures necessary to augment the size of an existing void, in a reversible or iiTeversible manner and they create a local stress distribution. These fo rces produce void growth and a variation of the overall volume fraction of the body.

The elastic (free) energy reads:

' I e e

{3

.ģ e

I I � 2 \l = -Ck1E E + u E cp + -acp 4J +-scfJ 2 I} •i tl IJ 'I 2 .I .I 2 (40)
where c,1 are the overall stiffness coefficients of the porous materiaL and (a. where gR and h t are the values of the equilibrated body forces and equilibrated stress vector in the reference state, respectively. Hence equilibrium in the reference (stress-free) state yields:

(42)

The constitutive relations are:

(43

)
where oij is the Kronecker symbol.

As pointed out by Cowin and Nunziato, the stiffness coefficients and the material parameters should depend on the reference volume fraction. In order to obtain an extension cf this model to the case of a damaged material, it is assumed that whenever damage grows, it modifies the volume fraction of the material in the reference configuration considered by the linear elastic theory. The variation of volume fraction is rewritten as:

dif;(.r, f)= d!f>.ae ( x, t) + d!f> i r (x, t), !!/> ' " ' (x, t) = v(x, t) -. v R (x, t),
with rp i r (x, t) = v R (x, t)v 0 (x), (44) where q/"' is the reversible variation of volume fr action, vR(x, t) is the volume fr action of the damaged material when it is free of loads, !/>" is the irreversi ble variation of volume fraction due to the growth of damage measured when the material is free of loads and v0 is the initial volume fraction of the material, when damage is equal to zero. The free energy of the material is now rewritten with the introduction of the degradation of the stiffness. Therefore, we assume that g R and ht defined in the elastic theory are related to 1/>ir [START_REF] Pijaudier-Cabot | Damage and Localisation in Elastic Materials with Voids[END_REF].

A "source term" A which controls the irreversible variation of volume fraction of the material from the initial state is defined with a loading function j(Eij , Z) and an evolution equation. It is added to the right hand-side term in Eq. (39b ). Z is the hardening -softening variable. For the sake of simplicity, we assume now that !/> ' " ' is very small compared to !/> ; , as damage grows. We will use, in the following, a linear relation between damage and the irreversible variation of volume fraction : d = -!/>;' . The constitutive relations which follow from these assumptions are similar to those of the gradient damage model. The governing equations and the stress -strain relation are now: Jai J. J =0 , 1w. ii -�d =A,

a = (1 -d)[At-e 8 • + 2f.1E e ]-f38 d. I} U IJ ij I) (45)
In this final fo rm , we may interpret the different material parameters. � is related to the evolution law of damage, in an integrated fo m1 and the coefficient f3 controls the amount c{ ' volumetric irrevers ible strain due to damage.

For the finite element implementation, consider now a finite body denoted as Q with boundary on. This body is initially at rest, the displacement and damage velocities being equal to zero. The two governing equations of motion are those defined in Eqs. (45a,b) and the boundary and initial conditions corresponding to damage equation are those defined in the linear elastic theory :

(46) a g rad(d(x, t)).ii = 0 on on, where ii is now the unit outward normal to the boundary iß. The meaning of the third (natural) boundary condition in Eq. ( 46) remains a problem to be solved.

In the absence of body fo rces and irrevers ible strains, the two govern ing equations, along with the boundary conditions, are equivalent to the fo llowing conditions: find a displacement field ii and a damage field d such that for any cinematically admissible virtual displacement ii *and damage fie Id d * :

J E* : iÝv = J ii * .f:ds, n ,;n J { agrad(d*). g rad(d) + �d * d}dv = J d * .Adv, n n (47)
where .° are external applied fo rces. For the finite element implementation, the displacement field components and damage field are discretised, same as in coupled thermo-mechanical problems. A similar variational principle can be obtained in an incremental fa shion for the purpose of implementing the model within a Newton -Raphson procedure.

Analysis of Strain and Damage Localisation

Throughout this section, we are going to investigate the propert ies of the non local damage model when strain and damage localisation occurs. For the sake of simplic ity, we will use the energy-based damage model. The strain-based model is certainly closer to the behaviour cf concrete and rock-like materials. lt is much more complex from a mathematical point of view (lfecause of the expression of the equivalent tensile strain which involves positive parts) and exhibits similar properties which have been verified numerically. The bifurcation analyses which are presented hereafter aim at understanding the properties of the non local model and the influence of the internal length.

7. I One Dimensional Approach to Localisation

For the sake of simplicity we will consider fi rst the one dimensional case of a bar which is in an initial state of constant strain E. The stress-strain curve of the material is shown in Fig. 11 a. It exhibits strain softening with a constant modulus Et and a secant modulus denoted as E11 • The length of the bar is L and its cross section is A .

(a)

... Assume that the loading is displacement controlled. If the strain E = u I L is lower than the strain at peak stress, the stress in the bar is CJ = E11E. For initial states of strain beyond this limit, the solution of the equation of equilibrium to small stress perturbation is not unique.

Assume now that fo r x E [O,h), the incremental stiffn ess is E11 and that fo r x E )h,L) the incremental stiffness is Et . It is then possible to find a possible perturbation, solution to th is problem fo r .mich the constitutive and compatibility equations are:

for x E [0, lz) for x E ]h, L] (48)
If the variation of the displacement at the extremity of the bar is fixed, the corresponding variation of tensile load is:

(49)

Hence, for any value of the u nknown h, it i s possi ble to find a solution to this problem. The number of solutions for this problem is infinite. It depends on the size of the zone where the strain is increasing incrementally, and also on its location.

Under displacement controlled conditions, the second order work corresponding to the perturbation is:

L ' I f d-U = -z dE.da dx -dFdu (50) 0
Upon substitution of the constitutive relations and of Eq. ( 49), the fo l lowing form of the second order work is:

, I '(hE ) d-U =--AE (dE t - 1 +(L-h) 2 I I E If ( 5 1 )
It fo llows that the state of equilibrium from which the perturbation is considered is stable if

-1 + (L-fz) >0 (hE ) E" ( 52
)
and stability is directly a fu nction of the size of the region in which the strain increases incrementally.

If this size is not controlled by the constitutive relations, the stable solution will be the one that minimises the second order work, according to Gibbs definition of stability. A simple calculation shows that the stable solution is obtained for h � L. In this case, all the bar unloads except fo r one point which fo llows the softening curve. This is the one dimensional localisation mode in which the admissible incremental strain is discontinuous. As a consequence, the energy dissipation is confined to a point (region of zero volume) and the total energy diss ipated b.v the bar during fa ilure is zero (the dissipation per unit volume iÞ material is finite). Obviously, this result is not realistic. It is also in contradiction with crack propagation criteria (such as Griffith criterion).

The localisation mode studied in this example is possible because of strain softening. It cannot occur in the hardening regime because it is not possible then to find a perturbation c:f the fo rm in Eq. ( 48) which satisfies equilibrium (the incremental strain has to be constant over the bar).

Let us consider now the case of a non local damage model. In order to avoid the difficulties inherent to the boundary conditions and the local averaging procedure near the boundaries, we t b ok at an infin ite bar which is in a homogeneous state of stress, strain, and damage initially denoted as (a0.E0 , D0 ). The rate equation of equilibrium is derived from Eqs. (8,9,24,25), assuming that the variation of stress is homogeneous over the bar:

(53) \vhere E0 is the elastic constant of the material. Note that the representative volume Vr (x) is constant for an infinite bar and that in this problem, the increments are replaced by rates, which is the same since the constitutive relations are rate independent.

Instead ofbeing a differential equation, the equation of equilibrium is now a second order linear integral equation in i,x . Thus, the mathematical nature of the problem is slightly changed. Nevertheless, it is possible to recover an algebraic equation by considering harmonic solutions of the velocity field ti(x) . This means that possible solutions are developed in Fourier series. Th is type of solution enables to separate, in the kernel of the integral, variable x fr om variable s. Hence, the kernel is degenerated in order to transform the integral equation into an algebraic equation. For details the reader should consult the book by Courant and Hilbert ( 1953).

ti(x)

= A exp( -i�x) (54)
where i is now the imaginary constant such that i 2 = -1 . Substitution in the equation c:f equilibrium yields: with

A � e x p ( -i.r )[ (I -D0 )E0 -F()' ).( � . .G )( E0e0 )2] = 0 � 2/ 2 Vt(�.()= exp(--" ) 2 (55) (56)
.(�. 0 is the Fourier transform of the weight fu nction. There is a trivial solution to this equation where � is zero and the strain remains homogeneous over the bar. However, another solution may exist:

(57)

The condition of existence of such a solution coincides with the condition that the tangent modulus of the material defom1ed homogeneously be negative or zero. Thus, loss of uniqueness is possible starting at the peak stress and beyond, in the strain-softening regime. The wave length of the periodic solution is proportional to the characteristic length of the material and depends on the initial state of strain (or tangent modulus) of the bar. In this simple case, it can be seen that the number of possible solutions of the rate equation of equilibrium is restricted to the trivial (homogeneous) solution and to a periodic solution. If the constitutive equations are local ( ( = 0 ), there is an infinite number of solutions to this problem, same as in the previous paragraph along with the same unrealistic fe ature of zero energy dissipation at fa ilure.

With the gradient damage model developed in section 6, the system of equations of equilibrium reduces to: We are using here a evolution of damage such that in the softening regime, the material behaves linearly with a tangent modulus denoted as h. Harmonic solutions are fo und upon softening (h < 0) on ly. The wave lengths e of these solutions are fu nction of the initial state of damage and of the material parameter a: f=2K

Eo (I -
(5 9 )

Bifurcation occurs when softening is encountered. Besides the trivial solution where the strain and damage rates remain homogeneous, there is again a second possible solution to the governing equations of equilibrium when the material softens (the term under the square root sign in Eq. (59) must be positive).

The square root of a is an internal length of the continuum which selects the wave length of the localised solution and scales the size of the localisation band. It plays exactly the same role as the internal length in other localisation limiters (see e.g. Sluys 1992, Pijaudier-Cabot and[START_REF] Pijaudier-Cabot | Strain Localisation and Bifurcation in a Non local Continuum[END_REF] . Figure 12 shows the variation of the wave length as a fu nction of the initial strain E0 about which bifurcation is considered. The material parameters are E=l MPa, h=-0.5MPa, ;=I, f3=0.1MPa, peak stress =IMPa with a= I mm2•

7.2 Possible Localisation Modes in Static's

We will consider now the case of an infinite solid on which adequate boundary conditions (rate of loading) are applied at infinity. It is assumed that the solid is initially deformed homogeneously and that it is subjected to a quasi-stat ic increment of deformation. For any variation about the initial equilibrium state, the rate equations of equilibrium are: The subscript 0 in this equation denotes the initial state of deformation and damage from which the perturbation are considered. Following Rudn icki and Rice analysis ( 1975). the velocity field corresponding this perturbation is assumed to be eo-continuous while the corresponding strain rate is C_ 1 -continuous, discontinuous across a surface denoted as r with nonnal vector ii (Fig. 13).

For equilibrium, the stress rate jump derived from the assumed strain distribution across r must be zero:

(6 1)
where [a] is the stress rate jump, that is the difference between the stress rate tensors across surface r. In order to proceed with the analysis, it is necessary to know what are the loading conditions in the two parts of the solid separated by r. A part from the trivial case in which the two parts unload and the strain rate jump is zero, it may be assumed that one part of the solid denoted as n-unloads (damage is constant according to Eq. 24) while damage keeps on growing in the other part denoted as n+. The average damage energy release rate at any point of the discontinuity surface is, however, a eo-continuous fu nction :

� I f . y(x) = --i(s)E0(.r + s): E: E(x + s)ds V, (x) 1, (62)
Thus, a discontinuity ofthe damage growth condition is impossible on r . the only remaining possibility for a discontinuity of the strain rate field to exist is that damage should grow in n and in n+ and the damage jump across the discontinuity surface r should be zero according to Eqs. (25,62). Therefore, the discontinuity of the nonnal stress across the interface is proport ional to the strain rate jump. It can be easily checked that this jump must be zero in order to satisfy equilibrium (Eq. 61 ). Consequently, a discontinuous strain rate ti eld cannot be an equilibrium solution according to the non local model. A similar property is trivial for gradient models, since the displacement field must be continuous enough so that second order gradients of the strain exist.

7.3 T h ree Dimensional Analysis with the Non Local Model

Strain localisation cannot be defined as the onset of a discontinuity of the velocity field. It remains however that strain softening may produce a loss of uniqueness of the solution to a bounda1y conditions problem. The detection of bifurcation points cannot be carried out analytically in the general context. This analysis can be perfonned in the case of an intinite body only. assuming that the boundary conditions are such that the de formation and damage variables are initially homogeneous over the solid. Hence. we will study the conditions of uniqueness and adm issibility of small perturbations which sat is!)' the rate constitutive equation (60) and the momentum equation:

In order to exhibit the localisation conditions in static's, the phase velocity in Eq.( 64) is taken equal to zero. Eq.( 69) is the bifurcation condition :

det[ii.H * Cs).ii] = o (69)
It has the same fo rm as Rudnicki and Rice condition (1975). In the limit of a vanishing internal length, Eq. ( 69) is exactly Rudnicki and Rice expression which corresponds to the occurrence of a surface of discontinuity of the strain rate in the considered solid. The acoustic t�nsor iiHii is singular.

The relation between the singularity of the acoustic tensor in a continuum description and the progressive concentration of fa ilure in discrete analysis is rather difficult to apprehend. A first reason is that it can be quite problematic to derive a tangent operator H in discrete analyses. De lap lace et al. (1999) have recalled that in some situations at least (e.g. fo r a discrete interface), the characteristic of the system (force vs. displacement) is not derivable. A second reason is that due to the initial disorder, bifurcation is very seldom to occur. The initial disorder serves as a perturbation which triggers damage localisation. According to the above analysis, the bifurcation condition is derived assuming that the system is initially in a homogeneous state of deformation. Would J bifurcation be perm itted in discrete analyses, responses of systems without localisation of fracturing events should be observed. It is from such results that an equivalent tangent operator of the underlying continuum approximation cf discrete system should be computed, before strain localisation, same as in Rudnicki and Rice analysis. This type of result has never been obtained to our knowledge. Hence, the determinant of the acoustic tensor, computed from an averaged response of several discrete analyses where a progressive concentration of damage occurs, should not vanish. At best, it should decrease without becoming singular. This type of result has been observed by [START_REF] Krajcinovic | Strain Localizat ion -Short to Long Correlation Length Transition[END_REF].

Finite Element Example

The non local model is re latively easy to implement since the equilibrium equations are standard. The integral relation due to the non local term is discretised according to the finite element mesh used fo r the analysis and an usual quadrature rule is employed fo r its evaluation. Notice that the weight fu nction does not vary during the loading process and is independent cf the boundary conditions. Therefore, the numerical evaluation of the non local terms is simplified: prior to initiate the calculations, the average weights are computed at each integration point and stored once and fo r all. In finite element calculations, the weight fu nction is chopped off: the weights that are less than 0.00 I are set to zero. The actual volume cf integration does not span over the entire volume of the solid and the calculation of the integrals requires less computer time and memory as the number of neighbouring integration points is reduced.

Figure 14 shows a schematic problem statement (see e.g. Pijaudier-Cabot and Bazant ( 1987) for details). A constant velocity is applied at the extremities of the bar such that a constant strain wave of amplitude 0.7t: *(where E * is the strain measured at the peak stress) is propagated. These waves meet at t*=ls in the middle of the bar where localisation occurs.

Relation with Fracture Mechanics

Fracture mechanics and its link with damage mechanics may possibly• help at understanding the e!lect of non locality of dam age and the intluence of the internal length on macroscopic fr acture properties. Our objectives in this section is to discuss a vel)• simplified lin k bet\veen the two theories which exhibit the intluence of the internal length of the material and may possibly be used for approxim at ing this length experimentally .

Consider an inf in ite body subjected to uniaxial tension in direction 2. a� 2 with a� = 0 for i :f. 2 and j :f. 2. Damage is denoted as [)0 and the strains are denoted as Eg . When small deviations fr om this equ ilibrium state arc analysed. harmonic displacement fi elds are solutions of the partial differential equations of equ ilibrium given in section 7.

We assume now that damage loc alisat ion is such that the band is perpendicular to the tensile load (Fig. 17). The wave length of the harmonic solution is entirely detenn ined from the evolution law of damage and the internal length of the continuum because the orientation of the band is fixed. The calculation of the approximated fracture energy is based on the assumption that at the onset of strain loc alisation. i.e. at the onset of localised cracking. the distributions of strain and damage jump sudde nly ti •om a homogeneous distribution to a harmonic solution with the smallest possible wave length . This is based on stability considerations which require that the width of the localisation zone should be as small as possibl e and on the assumption that intermediate states where maximum damage is neither equal to zero nor to I have a relatively small influence on the energy diss ipation. In an infinite body, localisation occurs suddenly and maximum damage jumps very rapidly to one. Therefore, this assumption does not seems to be too far away from the exact process of localisati0n in which the region where damage evolves shrinks with increasing damage as the loading progresses. Furthermore, this approx imat ion has been already tested successfully in finite element analyses of pre-damaged structures i n order to derive an equivalence between a cracked and a dam aged (in the continuum sense) structural component [START_REF] Bode | Failure Analysis of Initially Cracked Concrete Structures[END_REF]. Thi s equivalence was derived with the strain based damage model developed in section 5. W ith the minimum wave length (maximum value of s in Eq. ( 64)), the distribution of damage perpendicularly to the crack direction (the coordinate is denoted as x2 ) is: The energy consumption due to crack propagation is the integral of the energy dissipat ion at each material point of coordinate x2 in the fracture process zone which encountered damage up to D( x2 ). For propagation of a crack over a length 8a, the formula reads Therefore, the fracture energy is:

(7 1)

(72) ( 73
)
Note that the width of the fracture process zone is implicitly fixed by equation ( 70). The fracture energy tor an infinitely large specimen is related to the model parameters in the constitutive relations including the internal length. For a local modeL the wave length in Eq.( 70) would be zero and the calculation would yield a zero fracture energy too. This is not consistent with fracture mechanics where a ti nite amount of energy is required in order fo r a crack to propagate. The non local model provides a consistent result via the introduction of an internal length wh ich scales the localisation band (wave l ength of the localised m ode).

Conclusions

Continuous damage mechanics is a theory which aims at describing the m echanical efti:x:t cf cracking and void growth in an elastic material. Analyses on lattices show that in phcnomenological models, the variation of the material stifti1ess is the appropri ate damage variable. lt shows also that in the course of fa ilure spatial correlation develops which can be transposed in a non local fo rm at in macroscopic models. The description is essentially phcnomenological although it may be also motivated by several micro mechanics analyses.

Non locality introduces an internal length which scales the localisation process. In static's, the internal length contro ls the minimum wave length of the localisation modes. For a rate independent continuum. this property is central and insures a proper convergence of the finite clement calculations toward a unique solution or a ti nite number of solutions analogous to b uckl ing modes. Finally. the internal length makes it poss ible to relate damage models to fracture mechanics. The fi"acture energy is fu nction of this internal length. This last parameter could therefore be determ ined experimentally fr om rracture tests. using inverse analysis techniques.

  Figure I. (a) Lattice used for the anal y sis; (b) behaviour of one bond.

  figure 3. Lattic.: just befor.: compkte failure.

Figure 4 .

 4 Figure 4. Evolution of th� first four moments of the distribution of current as a function of the overall conductance.

Figure 5 .

 5 Figure 5. Histograms of th<! distribution of horizontal distance.

Figure 6 .

 6 Figure 6. Jnt.:rnal. correlation length vs. numba of broken bonds.

  reinforced concrete structural components (Mazars 1984, Mazars and Pijaudier-Cabot 1989). The stress strain relation of an isotropic damaged elastic material contains two damage variables d and 8are the Young's modulus and Poisson's ratio of the undamaged isotropic material respectively, and D;j is the kronecker symbol. The damage variables d and 8 are equal to 0 initially (for the undamaged material). At complete fa ilure, the elastic stiffn ess of the material vanishes and the two variables are equal to I. The subsequent assumption d = 8 yields the stress-strain relationship:

  r;) = -0'00 + L P p (.\' ;) j=l.j �i where pii(x;) represents the effect of the crack S j on the crack S i .

  Figure 7. l'vlicro cracked mode!� material: (a) uniform crack density; (b) ince p tion of localisation. The unknown pressure .C-.H) is expanded in a series of base functions which are Legendre polynomials:

Figure 8 .

 8 Figure 8. Evolution of the elastic modulus with crack density.

Figure 9 .

 9 Figure 9. Weight function at the inception of localisation.

  is the internal length of the non local continuum and If! 0 is a normalising factor. This fa ctor is such that for an infinite body . (x) = I . The internal length of the non local continuum depends on the size of the heterogeneities in the material. An usual approximation of this length is (. z 3((1 where d" is the maximum size of the aggregate in concrete[START_REF] Bazant | Measurement of Characteristic Length of Non local Continuum[END_REF].

Figure 10 :

 10 Figure 10: Response of the Strain-Based Model in Tension (a) and Compression (b). The factors CX 1 and a., are such that in uniaxial tension CX 1 =I. a., = 0. and D = D 1 , and in compression a. 1 = 0. a., = I. and [) = D,. The damage variables are functions of the average equivalent strain:

Figure I 0

 0 shows the response of this model in tension and in compression. The model parameters are:£ 0 = 23400MPa, v 0 = 0.2, K o = 2.610-4 , f3 = 1.05 At= 1., Bt = 15000, Ac = I .2, Be= 649 (36)More details on this constitutive relation, its finite element implementation, and comparisons with experimental data can be found in the works of[START_REF] Mazars | Application de la mecanique de l'endommagement au comportement non lineaire et a la rupture du beton de structure[END_REF], Saouridis ( 1988), Mazars and Pijaudier-Cabot (I 989).

  {3. �) are material IJ< parameters. The above fonn of the elastic energy is the result of the assumption that the material is isotropic with a centre of symmetry. The thermodynam ic forces associated to the variables ( E iJ , .. .o. ;) are defined as :(4 1) .

Figure 11 .

 11 Figure 11. Localisation in the one dimensional case

Figure 12 .

 12 Figure 12. Simplified one dimensional damage model: wave length of the localised mode as a fu nction of the strain at the onset of bifurcation.

  Figure13. Solid with a discontinuity of the strain rate field th rough the surface y.

  Figure 14. Dynamic bar problem.

Figure 15 Figure 15 .

 1515 Figure 15 shows the results of computations carried out with a local model (internal length set to zero) and several meshes w ith constant length elements. The damage profi les are plotted on th is figure at t ime 1.5t*.Damage local ises into the two elements at the middle of the bar. The size of the damag�d zone is controlled by the size of the finite elements: the smaller the element size, the smaller the damage zone will be. At the limit of a vanishing element size, which should correspond to the solution fo r a continuum, damage localises over a segment of zero length. Since the energy dissipation per unit volume of material is finite, the total energy dissipation fo r the entire bar, defined as the integration over time and over the bar of the dissipation rate, tends to zero.Failure occurs without energy dissipation, which is not physically realistic.

FigureFigure 16 .

 16 Figure16shows the same calculations perfom1ed with the non local model. Upon mesh refinement, the profiles of damage remain constant. The size of the damaged zone is proportional to the internal length of the continuum and fa ilure occurs with a finite. non zero energy dissipation because damage localises over a portion of the bar of fi nite, non zero length.

Figure 17 .

 17 Figure 17. Geometry of the local isation band assu med fo r the calculation of the fracture energy.

-J

  l{l(s)T](x2 +s)ds D( x2) = -'-�=--�----J lfi(S)T](s)ds with T](Z) = cos(�maxZ) If z E --,--

-

  c1iÜ = J W(D(x2 ))d.t2&L where W(D) is the energy dissipation due to damage up to the value D ll( r, ) f, (r, ) W( D(x2 )) = r -lEEi2(D)dD = -r -lEEi2F(_ v)dE 22 0 0

. cil �(x) di\'(.~(x)) = p--, ()r (63) where p is the mass density and ik is the unknown perturbation. We will consider here the non local damage model only. Similar results can be obtained with the gradient damage model.

With the non local model, Eq. ( 63) is an integro-diffe rential equation. Solutions of this equations are harmonic waves, propagating in direction ii, of amplitude A and phase velocity c:

Substitution of these solutions into the equation of motion yields the linear algebraic system:

where I is the 2x2 identity matrix, H * ( !;,) is the tangent operator govern ing the strain ratestress rate relation and:

ii.H * ( �).ii = (1-D0 )ii.E.ii -.Ë(�) F(y 0 )(E : E0 .ii) ® ( E: E0 .ii)

where ® denotes the tensorial product. ii.H * (.®).ii can be regarded as a pseudo acoustic tensor, fu nction of the wave number s entering in ljt(�) which is the Fourier transform of the weight fu nction:

(67)

The condition of bifurcation, i.e. the condition of admissibility and non uniqueness of harmonic waves is (a homogeneous deformation with constant velocity is already a trivial solution):

The major difference with the same analysis performed on a local continuum is that the pseudo-acoustic tensor is a fu nction of the wave number. In a local damage model, the weight fu nction is a Dirac fu nction. The Fourier transfom1 of such a function is constant. Whenever Eq.( 68) is satisfied fo r a local continuum, there is an infinity of possible solutions corresponding to any arbitrary wave number. On the contrary, Eq.( 68) may be satisfied in a non local continuum for a limited number of values of �.