Rodrigue Desmorat
email: desmorat@lmt.ens-cachan.fr

Fabrice Gatuingt

Frédéric Ragueneau

Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials

Keywords: concrete, damage, induced anisotropy, nonlocal, Finite Element computations

come

Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials

Introduction

To extend existing isotropic damage models to induced anisotropy is not an easy task as difficulties and questions specifically related to anisotropy arise. How to write the coupling damage/elasticity? What becomes then the effective stress concept associated with the principle of strain equivalence [START_REF] Lemaitre | Evaluation of dissipation, damage in metals submitted to dynamic loading[END_REF][START_REF] Cordebois | Endommagement anisotrope en élasticité et plasticité[END_REF] ? Which tensorial representation of damage shall be used ? If general damage anisotropy can be represented by a fourth order damage tensor [START_REF] Leckie | Tensorial nature of damage measuring internal variables[END_REF][START_REF] Krajcinovic | Continuous damage mechanics revisited: basic concepts and definitions[END_REF][START_REF] Lemaitre | Mécanique des matériaux solides, Dunod, english translation 1990[END_REF], this formally simple choice is difficult to work with, usually because of the high number of material parameters introduced. And how to model properly and efficiently (when numerical computations are in mind) the stiffness recovery due to the micro-defects closure effect [START_REF] Ladevèze | On an anisotropic damage theory[END_REF][START_REF] Ladevèze | Modeling and simulation of the mechanical behavior of cmcs, High Temperature Ceramic-Matrix Composite[END_REF][START_REF] Horii | Overall moduli of solids with microcracks: load induced anisotropy[END_REF][START_REF] Chaboche | Development of continuum damage mechanics for elastic solids sustaining anisotropic and unilateral damage[END_REF][START_REF] Dragon | Modélisation de l'endommagement par mésofissuration : com-portement unilatéral et anisotropie induite[END_REF][START_REF] Desmorat | Quasi-unilateral conditions in anisotropic elasticity[END_REF]? the damage growth higher in tension than in compression? In order to give a practical answers to these questions, the choice to represent the damage state by a second order damage tensor has been made by many authors [START_REF] Murakami | A constitutive equation of creep damage in pollicristalline metals[END_REF][START_REF] Cordebois | Endommagement anisotrope en élasticité et plasticité[END_REF][START_REF] Chow | An anisotropic theory for continuum damage mechanics[END_REF][START_REF] Murakami | Mechanical modeling of material damage[END_REF][START_REF] Ju | On energy-based coupled elasto-plastic damage theories: constitutive modeling and computational aspects[END_REF][START_REF] Meschke | An anisotropic elastoplastic-damage model for plain concrete[END_REF].

Concerning physical damage anisotropy, quasi-brittle materials such as concrete exhibit a micro-cracking pattern different in tensile and in compressive loadings [START_REF] Mazars | Application de la mécanique de l'endommagement au comportement non linéaire et à la rupture du béton de structure[END_REF][START_REF] Shah | Mechanical behavior of fiber-reinforced cement-based composites[END_REF]: the micro-cracks are mainly orthogonal to the loading direction in tension and parallel to it in compression. This induced anisotropy is responsible for the large dissymmetry tension/compression of concrete behavior and must be introduced to do so in constitutive modeling. For thermodynamics consistency and according to this last remark only one damage variable must be considered. As a state variable a damage variable represents the micro-cracks pattern, whatever the sign of the loading [START_REF] Ladevèze | Damage effective stress in quasi unilateral conditions[END_REF][START_REF] Lemaitre | A Course on Damage Mechanics[END_REF], and cannot be related to either tension or compression. For the sake of relative simplicity anisotropic damage is next represented by the second order tensor D D D of components D ij . If a tensile loading is applied in direction 1, induced anisotropic (diagonal) damage shall act as its component D 1 = D 11 instead of a damage variable "for tension". If the loading is compressive, damage shall act as its component D 2 = D 22 = D 33 instead of a damage variable "for compression". Written in the thermodynamics framework the model presented next follows these guidelines.

1 Coupling damage/elasticity using a 2nd order damage tensor

The thermodynamics framework proposed by Ladevèze leading to 3D continuous stressstrain responses is used [START_REF] Ladevèze | On an anisotropic damage theory[END_REF][START_REF] Papa | Anisotropic damage model for the multi-axial static and fatigue behaviour of plain concrete[END_REF][START_REF] Lemaitre | Engineering Damage Mechanics : Ductile, Creep, Fatigue and Brittle Failures[END_REF]. The damage state is represented by the 2nd order tensor D D D and there is one known thermodynamics potential ρψ ⋆ 0 from which derives a symmetric effective stress σ σ σ independent from the elasticity parameters [START_REF] Lemaitre | Anisotropic damage law of evolution[END_REF]:

ρψ ⋆ 0 = 1 + ν 2E tr H H Hσ σ σ D H H Hσ σ σ D + 1 -2ν 6E (tr σ σ σ) 2 1 -ηD H (1)
with E, ν the Young modulus and Poisson ratio of initially isotropic elasticity, η the hydrostatic sensitivity parameter (η ≈ 3 for most materials [START_REF] Lemaitre | Engineering Damage Mechanics : Ductile, Creep, Fatigue and Brittle Failures[END_REF]), ρ the density, where σ σ σ D = σ σ σ -1 3 tr σ σ σ 1 1 1 is the deviatoric stress and where H H H is the effective damage tensor, D H the hydrostatic damage,

H H H = (1 1 1 -D D D) -1/2 and D H = 1 3 tr D D D (2)
Quasi-brittle materials such as concrete exhibit a strong difference of behavior in tension and in compression due to damage. This micro-defects closure effect usually leads to complex models when damage anisotropy is considered [START_REF] Ortiz | A constitutive theory for the inelastic behavior of concrete[END_REF][START_REF] Chaboche | Development of continuum damage mechanics for elastic solids sustaining anisotropic and unilateral damage[END_REF][START_REF] Murakami | Constitutive and damage evolution equations of elasticbrittle materials based on irreversible thermodynamics[END_REF][START_REF] Halm | An anisotropic model of damage and frictional sliding for brittle materials[END_REF] and the purpose here is to show that for monotonic applications it is sufficient to consider damage anisotropy with a quasi-unilateral effect acting on the hydrostatic stress only, with a thermodynamics potential rewritten

ρψ ⋆ = 1 + ν 2E tr H H Hσ σ σ D H H Hσ σ σ D + 1 -2ν 6E tr σ σ σ 2 1 -tr D D D + -tr σ σ σ 2 (3)
so that the elasticity law reads

ǫ ǫ ǫ = ρ ∂ψ ⋆ ∂σ σ σ = 1 + ν E σ σ σ - ν E tr σ σ σ 1 1 1 (4)
and defines the symmetric effective stress σ σ σ independent from the elasticity parameters,

σ σ σ = (H H Hσ σ σ D H H H) D + 1 3
tr σ σ σ 1 -tr D D D --tr σ σ σ 1 1 1 [START_REF] Lemaitre | Mécanique des matériaux solides, Dunod, english translation 1990[END_REF] with (.) D = (.) -1 3 tr (.) 1 1 1 the deviatoric part. The notation . stands for the positive part of a scalar, x = max(x, 0).

Because of the splitting between deviatoric and hydrostatic contributions in the thermodynamics potential, the bulk modulus of the damaged material K remains constant for negative hydrostatic stresses, equal to the undamaged modulus K ( K = K = E 3(1-2ν) if tr σ σ σ < 0). In other words no volumetric damage is supposed to be associated with negative strains, and the collapse of micropores in the cement past is neglected under high hydrostatic compression. This simple and therefore first stage modeling choice has already been made by several authors [START_REF] Mazars | Application de la mécanique de l'endommagement au comportement non linéaire et à la rupture du béton de structure[END_REF][START_REF] Resende | A damage mechanics constitutive theory for the inelastic behaviour of concrete[END_REF][START_REF] Papa | Anisotropic damage model for the multi-axial static and fatigue behaviour of plain concrete[END_REF]. It is consistent with further consideration of Mazars damage criterion, an open criterion for the tricompression states. To consider a constant bulk modulus in compression is nevertheless a better choice for concrete-like materials than the feature of a constant (damage independent) apparent Poisson ratio encountered in most isotropic damage models. As often as long as the permanent strains due to micro-cracking and internal friction are not taken into account [START_REF] Herrmann | On the thermodynamics foundation of a damage theory in elastic solids[END_REF][START_REF] Laborderie | Crack closure effect in continuum damage mechanics: numerical implementation[END_REF][START_REF] Mazars | The unilateral behavior of damage concrete[END_REF][START_REF] Papa | Anisotropic damage model for the multi-axial static and fatigue behaviour of plain concrete[END_REF][START_REF] Dragon | Modélisation de l'endommagement par mésofissuration : com-portement unilatéral et anisotropie induite[END_REF][START_REF] Halm | An anisotropic model of damage and frictional sliding for brittle materials[END_REF][START_REF] Desmorat | 5è journéees du Regroupement Francophone pour la Recherche et la Formation sur le Béton (RF2B)[END_REF], dilatancy is not reproduced.

Once the coupling damage/elasticity is defined, it can be used to make quantitative damage measurements. For an uniaxial state of stress, the elasticity law reads

ǫ ǫ ǫ = σ E B B B(D 1 , D 2 ) (6)
with B B B a dimensionless tensor, diagonal, function of the principal damages D 1 and D 3 = D 2 only (the loading direction is denoted 1),

• in tension,

B 11 = (1 + ν) 4 9(1 -D 1 ) + 2 9(1 -D 2 ) + 1 -2ν 3(1 -D 1 -2D 2 ) B 22 = B 33 = -(1 + ν) 2 9(1 -D 1 ) + 1 9(1 -D 2 ) + 1 -2ν 3(1 -D 1 -2D 2 ) (7)
• in compression,

B 11 = (1 + ν) 4 9(1 -D 1 ) + 2 9(1 -D 2 ) + 1 -2ν 3
B 22 = B 33 = -(1 + ν) 2 9(1 -D 1 ) + 1 9(1 -D 2 ) + 1 -2ν 3 (8)
so that the measurement of the secant modulus E S = Ẽ = σ/ǫ 11 gives a possibility to measure the damage if either one of the principal damages D 1 or D 2 is zero (or constant) or if the ratio D 2 /D 1 is known for example from the damage evolution law (see section 3),

B 11 (D 1 , D 2 ) = E Ẽ -→ D 1 or D 2 (9)
Last, the state potential (3) defines the energy density release rate tensor Y Y Y as the thermodynamics variable associated with [START_REF] Lemaitre | Engineering Damage Mechanics : Ductile, Creep, Fatigue and Brittle Failures[END_REF].

D D D, Y Y Y = ρ ∂ψ ⋆ ∂D D D

Local and nonlocal damage criterion functions

As for plasticity, the elasticity domain can be defined through a criterion function f such as the domain f < 0 corresponds to elastic loading or unloading. Many criterion can be used, written in terms of stresses such as plasticity criteria [START_REF] Warnke | Constitutive model for triaxial behaviour of concrete[END_REF][START_REF] Ortiz | A constitutive theory for the inelastic behavior of concrete[END_REF][START_REF] Voyiadjis | Plasticity model for concrete using the bouding surface model[END_REF][START_REF] Gatuingt | Gurson's plasticity coupled to damage as a cap model for concrete compaction in dynamics[END_REF], strains [START_REF] Mazars | Application de la mécanique de l'endommagement au comportement non linéaire et à la rupture du béton de structure[END_REF][START_REF] Herrmann | On the thermodynamics foundation of a damage theory in elastic solids[END_REF][START_REF] Ramtani | Contribution à la modélisation du comportement multiaxial du béton endommagé avec description du caractère unilatéral[END_REF][START_REF] Geers | Damage and crack modelling in single-edge and double-edge notched concrete beams[END_REF], or strain energy release rate density [START_REF] Marigo | Formulation d'une loi d'endommagement d'un matériau élastique[END_REF][START_REF] Lemaitre | Mécanique des matériaux solides, Dunod, english translation 1990[END_REF][START_REF] Laborderie | Crack closure effect in continuum damage mechanics: numerical implementation[END_REF] The purpose here is to built a nonlocal constitutive model with a restricted number of material parameters, robust and easy to implement in Finite Element computer codes. Dilatancy will not be taken into account and one will accept an open criterion for the tricompression states.

These remarks lead us for the present work to the simple choice of Mazars criterion, function of the positive extensions ǫ I of the I th principal strain ǫ I ,

f = ǫ -κ ǫ = 3 I=1 ǫ I 2 = ǫ ǫ ǫ + : ǫ ǫ ǫ + ( 10
)
where ǫ is the equivalent strain for quasi-brittle materials and κ is the elastic strain limit in tension. The notation ǫ ǫ ǫ + stands for the positive part of the strain tensor in terms of principal values. Note that the choice of Mazars equivalent strain is the most efficient in terms of number of material parameters introduced (none!) but it leads to a tension/compression dissymmetry of the elasticity limit usually not large enough in the uniaxial case (case adjusted next by an adequate set of damage threshold and of damage parameters), feature to be even more emphasized for the bitension/bicompression dissymmetry case. One uses next Mazars criterion with these drawbacks in mind, i.e. with in mind that further studies on the confined multiaxial loading paths will need improvements concerning the definition of the equivalent strain ǫ as in [START_REF] De Vree | Comparison of nonlocal approaches in continuum damage mechanics[END_REF]. The proposed framework will nevertheless strictly apply with just such a change of definition. Note that Mazars strain present choice already allows to face 3D Finite Element computations with anisotropic damage of plain but also reinforced and pre-stressed concrete structures, as illustrated in section 8.

To build a nonlocal damage model, it will be sufficient to consider nonlocal Mazars criterion,

f = ǫnl -κ (11)
introducing the nonlocal equivalent strain ǫnl which can be defined using an integral form (W is the nonlocal weight function, [START_REF] Pijaudier-Cabot | Nonlocal damage theory[END_REF],

ǫnl = ǫnl (x) = 1 V r Ω W(x -s) ǫ(s) ds V r = V r (x) = Ω W(x -s)ds (12)
or using a second gradient form [START_REF] Aifantis | The physics of plastic deformation[END_REF][START_REF] Peerlings | Gradient-enhanced damage model for quasi-brittle materials[END_REF],

ǫnl -c ∇ 2 ǫnl = ǫ (13)
Both the integral form (through W) and the gradient form (through c) introduce a characteristic length l c . Even if induced anisotropy is considered next, the introduction of a single (isotropic) internal length will prove sufficient for practical applications.

Damage evolution laws for induced anisotropy

As already mentionned, a single (tensorial) damage variable is considered. This choice is thermodynamically consistent with the representation by an internal variable of the degradation mechanisms. It solves the problem of the artificial introduction of two scalar damage variables -one for tension, one for compression -in case of isotropic damage, as a damage variable represents a state of microcraking which remains the same at constant internal variables whatever the microcracks are positively or negatively loaded. Damage neverthess acts differently in tension and in compression and the induced anisotropy is then naturally responsible for the tension/compression dissymmetry. Damage is generally larger in the tensile direction denoted 1 in further developments with D 1 , D 2 , D 3 as principal damages.

For metals loaded in tension in direction 1 the damages D 2 and D 3 are of the order of D 1 /2 [START_REF] Lemaitre | Anisotropic damage law of evolution[END_REF]. This feature is modelled by a damage evolution law ensuring a damage rate proportional to the absolute value (in terms of principal components) of the plastic strain rate tensor:

Ḋ D D ∝ |ǫ ǫ ǫ p | (14)
For concrete, the damage pattern is different in tension and in compression [START_REF] Mazars | Application de la mécanique de l'endommagement au comportement non linéaire et à la rupture du béton de structure[END_REF][START_REF] Mazars | The unilateral behavior of damage concrete[END_REF][START_REF] Halm | An anisotropic model of damage and frictional sliding for brittle materials[END_REF]. For a tension loading applied in direction 1:

D D D ≈        D 1 0 0 0 0 0 0 0 0        (15)
when for a compression loading applied in direction 1:

D D D ≈        0 0 0 0 D 2 0 0 0 D 2        (16)
These two damage states can be modelled by damage evolution laws ensuring a damage tensor rate proportional to the positive part (in terms of principal components) of the total strain tensor:

Ḋ D D ∝ ǫ ǫ ǫ + (17)
but also by any rate law of the form

Ḋ D D ∝ ǫ ǫ ǫ α + ( 18
)
with α a damage exponent.

Non associated thermodynamics model

To propose a damage model written in the thermodynamics framework, consider as damage pseudo-potentiel :

F = F (Y Y Y ; ǫ ǫ ǫ) = Y Y Y : ǫ ǫ ǫ α + ( 19
)
where ǫ ǫ ǫ acts as a parameter so that the damage evolution law is derived from the normality rule as

Ḋ D D = λ ∂F ∂Y Y Y = λ ǫ ǫ ǫ α + ( 20
)
The damage multiplier λ is determined from the consistency condition f = 0, ḟ = 0. Making the simple choice κ = κ(tr D D D) [START_REF] Ramtani | Contribution à la modélisation du comportement multiaxial du béton endommagé avec description du caractère unilatéral[END_REF] gives for the local model

λ = dκ -1 dǫ ǫ tr ǫ ǫ ǫ α + ( 21
)
The anisotropic damage law takes then the general form:

Ḋ D D = dκ -1 dǫ ǫ ǫ ǫ α + tr ǫ ǫ ǫ α + ǫ ( 22
)
where the exponent α mainly plays a role in multiaxial states of stresses. Setting α = 2 (so that tr ǫ ǫ ǫ α + = ǫ2 ) makes once more the equivalent strain ǫ appear. This choice is then consistent with Mazars criterion, the damage law simplifying as

Ḋ D D = dκ -1 dǫ ǫ ǫ ǫ 2 + ǫ2 ǫ (23)
In tension performed in direction 1, ǫ = ǫ 1 > 0 and :

ǫ ǫ ǫ + =        ǫ 1 0 0 0 0 0 0 0 0        ǫ ǫ ǫ α + =        ǫ α 1 0 0 0 0 0 0 0 0        (24)
so that the form (15) of the damage tensor in tension is recovered.

In compression performed in direction 1, ǫ = √ 2ǫ 2 > 0, [START_REF] Murakami | Constitutive and damage evolution equations of elasticbrittle materials based on irreversible thermodynamics[END_REF] so that the form (16) of the damage tensor in compression is recovered.

ǫ ǫ ǫ + =        0 0 0 0 ǫ 2 0 0 0 ǫ 2        ǫ ǫ ǫ α + =        0 0 0 0 ǫ α 2 0 0 0 ǫ α 2       

Nonlocal damage law

For the nonlocal model, the equivalent strain ǫ is replaced by ǫnl in the criterion function. The only change is then in the expression for the damage multiplier with now:

λ = dκ -1 dǫ nl ǫnl tr ǫ ǫ ǫ α + ( 26
)
so that the damage evolution law becomes non local, i.e. Ḋ D D(x) at point x is (through ǫnl ) either function of the values of the surrounding points s or function of the strains gradient,

Ḋ D D = dκ -1 dǫ nl ǫ ǫ ǫ α + tr ǫ ǫ ǫ α + ǫnl or Ḋ D D = dκ -1 dǫ nl ǫ ǫ ǫ 2 + ǫ2 ǫnl (α = 2) ( 27
)
Of course, on the level of a single material point the nonlocal model will give the same response than the local one.

Positivity of the intrinsic dissipation

The thermodynamics framework has been used to derive both the state law and the damage evolution law but, due to the choices made for the potentials ρψ ⋆ and F , the model is not standard generalized [START_REF] Halphen | Sur les matériaux standards généralisés[END_REF]. The positivity of the intrinsic dissipation, here Y Y Y : Ḋ D D, is then not guaranteed and must be checked.

Even if the state potential ( 3) is not convex with respect to D D D, it can be differentiated as ρdψ ⋆ = ǫ ǫ ǫ : dσ σ σ + Y Y Y : dD D D and also as

ρdψ ⋆ = 1 + ν E (H H Hσ σ σ D H H H) D : dσ σ σ + 1 -2ν 3E tr σ σ σ 1 -tr D D D --tr σ σ σ tr dσ σ σ + 1 + ν E (σ σ σ D H H Hσ σ σ D ) : dH H H + 1 -2ν 6E tr σ σ σ 2 (1 -tr D D D) 2 tr dD D D (28) so that: Y Y Y : Ḋ D D = 1 + ν E (σ σ σ D H H Hσ σ σ D ) : Ḣ H H + 1 -2ν 6E tr σ σ σ 2 (1 -tr D D D) 2 tr Ḋ D D (29)
The term in tr Ḋ D D will be obviously positive for any monotonic increasing κ -1 -function so one has to concentrate on the term (σ σ σ D H H Hσ σ σ D ) : Ḣ H H.

Any damage law of the form [START_REF] Shah | Mechanical behavior of fiber-reinforced cement-based composites[END_REF] gives positive increasing eigenvalues D J for the damage tensor D D D. With the relation [START_REF] Lemaitre | Mécanique des matériaux solides, Dunod, english translation 1990[END_REF], the eigenvalues for H H H are

H J = 1 √ 1 -D J (30)
and are therefore also positive and increasing during any damage process. The positivity of the matrix (σ σ σ D H H Hσ σ σ D ) is last gained by checking the sign of its eigenvalues, denoted χ, solution of (σ σ σ D H H Hσ σ σ D ) g = χ g (with g standing for the corresponding eigenvectors), or equivalently solution of (H

H Hσ σ σ D ) 2 g = χ H H H g (31)
with (H H Hσ σ σ D ) 2 obviously a positive matrix. The eigenvalues χ read then:

χ = g T (H H Hσ σ σ D ) 2 g g T H H H g (32)
As ratios of positive terms they are positive as is then the second order tensor (σ σ σ D H H Hσ σ σ D ).

The tensorial product two positive tensors, (σ σ σ D H H Hσ σ σ D ) and Ḣ H H being positive, one can conclude that the positivity of the intrinsic dissipation is satisfied for any loading, eventually 3D, complex and/or non proportional.

Local and nonlocal anisotropic damage models

In order to derive and to implement the full set of constitutive equations for the anisotropic damage model one has last to define the function κ = κ(tr D D D). The simplest choice is to consider a linear function introducing two parameters only: the damage threshold κ 0 = κ(0) and a damage parameter A as [START_REF] Ramtani | Contribution à la modélisation du comportement multiaxial du béton endommagé avec description du caractère unilatéral[END_REF][START_REF] Halm | An anisotropic model of damage and frictional sliding for brittle materials[END_REF][START_REF] Chiarilli | Modelling of elasto-plastic damage behavior of claystone[END_REF]

κ(tr D D D) = 1 A tr D D D + κ 0 (33)
This leads to

κ -1 (ǫ) = A (ǫ -κ 0 ) (34)
and to the damage law (α = 2, nonlocal model):

Ḋ D D = A ǫ ǫ ǫ 2 + ǫ2 ǫnl (damage law 1) (35)
with for the local damage model the rate ǫnl replaced by local Mazars strain rate ǫ.

Damage anisotropy is different in tension and in compression. It affects differently the elasticity law and a strong difference in tension and in compression is finally obtained with the quite simple damage evolution law 1 (figure 2). Important point, this feature is gained with the consideration of one (tensorial) damage variable only, in accordance with the thermodynamics definition of a state variable: if one degradation mechanism is observed, only one damage variable shall represent the micro-cracks or micro-defects pattern, whatever the material is in tension or is in compression. The dissymmetry is nevertheless not sufficient with the linear κ-function with a too high damage rate in compression leading to a non physical snapback. One prefers then to consider as damage evolution law:

Ḋ D D = A   1 + ǫnl a 2   -1 ǫ ǫ ǫ 2 + ǫ2 ǫnl (damage law 2) ( 36
)
with a a material parameter of the order of magnitude the value of the strain reached in compression. In tension the term (ǫ/a) 2 << 1 is negligible so that the damage law 1 [START_REF] Ramtani | Contribution à la modélisation du comportement multiaxial du béton endommagé avec description du caractère unilatéral[END_REF] is recovered.

Comparing equations ( 27) and ( 36) allows to determine the κ-function,

dκ -1 dǫ nl = A   1 + ǫnl a 2   -1 (37)
This defines κ -1 and κ as

κ -1 (ǫ nl ) = aA arctan ǫnl a -arctan κ 0 a (38) κ(tr D D D) = a • tan tr D D D aA + arctan κ 0 a (39)
which have the same expression for the local damage model with also ǫnl (and ǫnl for the damage law) replaced by ǫ.

For accuracy reasons, the damage parameters A, a as well as the damage threshold must be identified by curve fitting on both the tension and compression curves of the material. Nevertheless, the damage measurement by eq. ( 9) with D 2 = 0 in tension and D 1 = 0 in compression allows to plot the trace of the damage tensor versus the equivalent strain curve. In tension,

tr D D D = D 1 , ǫ = ǫ 11 ; in compression, tr D D D = 2D 2 , ǫ = √ 2ǫ 22 = √ 2 B 22 (D 2 ) B 11 (D 2 ) ǫ 11 .
An example is given in figure 1 for concrete from which one can then justify to consider the function κ as a function of the trace of D D D: the experimental points from tension and from compression determine the same curve. Note also the characteristic shape of the damage curve obtained for models with no permanent strains: the damage seems to reach an asymptotic value. The concavity of the damage evolution will probably change if the permanent strains are modeled and if the damage is measured from the elastic unloading slope instead of from the secant modulus. In any case the principal damages D 1 , D 2 , D 3 are each bounded by 1 or most often by a critical damage D c . The trace tr D D D can then take values larger than 1 (and nevertheless bounded by 3 or 3D c ) but only in cases of negative states of stresses as tr D D D does not affect the compressibility modulus K kept equal to the initial modulus K in such cases. The full set of constitutive equations finally reads:

• Elasticity,

ǫ ǫ ǫ = 1 + ν E σ σ σ - ν E tr σ σ σ 1 1 1 or ǫ ǫ ǫ = E E E -1 : σ σ σ (40)
• Effective stress,

σ σ σ = (1 1 1 -D D D) -1/2 σ σ σ D (1 1 1 -D D D) -1/2 D + 1 3 tr σ σ σ 1 -tr D D D --tr σ σ σ 1 1 1 (41)
• Mazars equivalent strain, ǫ = ǫ ǫ ǫ + : ǫ ǫ ǫ + (42) • Nonlocal Mazars strain, defined either in gradient form [START_REF] Chow | An anisotropic theory for continuum damage mechanics[END_REF] or in integral form [START_REF] Murakami | A constitutive equation of creep damage in pollicristalline metals[END_REF].

With a Gaussian weight function,

ǫnl = 1 V r Ω exp - x -s 2 l 2 c ǫ(s) ds V r = Ω exp - x -s 2 l 2 c ds (43)
• Damage evolution law, local (if ǫnl = ǫ is set) or nonlocal, Ḋ D D = A   1 + ǫnl a 2   -1 ǫ ǫ ǫ 2 + ǫ2 ǫnl ( 45
)
where α = 2 is set.

It is a first order differential set of equations, time independent as no viscosity is introduced. The use of a damage criterion function f written in terms of strains will allow for a quite simple implementation in a Finite Elements computer code. Note that the elasticity law will then need to be inverted. This can be done in a closed form as:

σ σ σ = (1 1 1 -D D D) 1/2 σ σ σ (1 1 1 -D D D) 1/2 - (1 1 1 -D D D) : σ σ σ 3 -tr D D D (1 1 1 -D D D) + 1 3 [(1 -tr D D D) tr σ σ σ --tr σ σ σ ] 1 1 1 (46)
There is a total of only 5 material parameters for the final constitutive model with induced damage anisotropy (plus a characteristic length l c for the nonlocal model): E, ν for elasticity, κ 0 as damage threshold, A and a for damage evolution. A single set of material parameters is valid for both tension and compression. A critical damage D c may also be considered: when the maximum principal damage reaches D c there is rupture [START_REF] Lemaitre | Anisotropic damage law of evolution[END_REF][START_REF] Lemaitre | Isotropic and anisotropic damage law evolution[END_REF]. There is often no need to introduce a critical damage for nonlocal computations as they need the stress-strain response of the material up to vanishing stresses and as the mesoscopic cracks are represented by localized zones or shear bands. Figure 2 shows for both damage laws 1 and 2 the monotonic stress-strain curves for concrete obtained either in tension or in compression. Note that for the proposed anisotropic damage models the stress goes to zero in all cases for highly damaged materials. The material parameters describing well concrete behavior (using law 2) are: E = 42 GPa, ν = 0.2, κ 0 = 5 10 -5 , A = 5 10 3 , a = 2.93 10 -4 . The same value for the damage parameter A is used for law 1 illustrating the fact that there is an influence of parameter a on tension. An adequate choice for A will allow to make both tensile responses strictly match but not the compressive responses. Important point, one can see on such curves that damage anisotropy is truly responsible for the dissymmetry tension/compression, even if not in an adequate manner for the damage law 1. The snapback exhibited for law 1 is not physical and corresponds to a damage rate too high in compression, rate corrected through the consideration of parameter a. This kind of unstable compressive response was also obtained for Bažant and Gambarova initial microplane model [START_REF] Bazant | Crack shear in concrete: crack band microplane model[END_REF][START_REF] Bazant | Microplane model for progressive fracture of concrete and rock[END_REF] in which the local shear behavior was neglected (only local shear representation leading to correct concrete response in compression [START_REF] Bazant | Microplane model for brittle plastic material: I. theory and ii. verification[END_REF]). The property of shear/compression related behaviors is illustrated here for the anisotropic damage model by plotting the shear stress τ vs engineering shear strain γ monotonic model response (Fig. 3). To introduce the parameter a = 2.93 10 -4 in law 2 and then to model properly the concrete compressive response induce a less brittle response in shear. Note that this effect due to a lower damage growth will be even more pronounced if the quasi-unilateral conditions of microcracks closure are written on the deviatoric stresses also [START_REF] Lemaitre | Engineering Damage Mechanics : Ductile, Creep, Fatigue and Brittle Failures[END_REF][START_REF] Desmorat | Modeling micro-defects closure effect with isotropic/anisotropic damage[END_REF] when in the present case they are written on the hydrostatic stress only. Last, the fact that the thermodynamics potential can be continuously differentiated leads to the natural continuity of each component of the stress and strain tensors, even for non proportional loading paths. This property is essential for the ability of anisotropic damage models to deal with complex loading as encountered at Finite Element Gauss points.

Numerical implementation

The anisotropic damage model is in fact quite simple to implement in a Finite Element code: local iterations may be avoided even if Euler backward scheme is used, as proposed in present implementation procedure.

A global resolution of the equilibrium equations gives the displacements at time t n+1 with the internal damage variable D D D = D D D n kept unchanged from the last computed increment t n . The strains ǫ ǫ ǫ n+1 = ǫ ǫ ǫ(t n+1 ) at each Gauss point are calculated from the elements interpolation functions. To integrate the constitutive equations means to determine the stress σ σ σ n+1 and the damage D D D n+1 at time t n+1 . An iterative process, not described here, made of global equilibrium resolutions followed by local time integration of the constitutive equations often takes place [START_REF] Benallal | An integration algorithm and the corresponding consistent tangent operator for fully coupled elastoplastic and damage equations[END_REF][START_REF] Simo | Computational inelasticity[END_REF][START_REF] Jirasek | Embedded crack model: Part ii: Combination with smeared cracks[END_REF][START_REF] Jirasek | Inelastic Analysis of Structures[END_REF][START_REF] Besson | Mécanique non linéaire des matériaux[END_REF][START_REF] Besson | Numerical implementation of constitutive models, Local Approach to Fracture[END_REF]. One focus here on the numerical scheme for the local integration of the damage law.

Exact resolution of Euler backward discretization

The inputs of the time integration subroutine are the strain tensor ǫ ǫ ǫ n+1 and the damage tensor D D D n at a FE Gauss point; the ouputs are the damage and stress tensors D D D n+1 , σ σ σ n+1 at the same point. Instead of using the elastic stiffness, to calculate the secant operator E E E S helps the global resolutions, it helps a lot for the highly damaged parts subject to elastic unloadings. Computations will generally be more efficient if one uses the consistent tangent operator. But note that as one has to take the positive part of the strain tensor in previous constitutive equations, the stress-strain law cannot be continuously differentiated (the thermodynamics potential cannot be derivated twice) so that the tangent operator does not exist at some points, for instance for non proportional loading conditions with changes of sign of the strains. One has also in mind nonlocal integral computations for which it even more difficult to define a tangent operator [START_REF] Jirasek | Consistent tangent stiffness for nonlocal damage models[END_REF] and this key point for convergency acceleration with risk of loss of robustness is left to further developpments.

The damage multiplier is determined from the consistency condition numerically written f n+1 = ǫn+1 -κ(tr D D D n+1 ) = 0 so that:

tr D D D n+1 = κ -1 (ǫ n+1 ) (49
(3) Compute then the stresses using first the elasticity law written

σ σ σ n+1 = E E E : ǫ ǫ ǫ n+1 (52)
using then eq. ( 46),

σ σ σ n+1 = (1 1 1 -D D D n+1 ) 1/2 σ σ σ n+1 (1 1 1 -D D D n+1 ) 1/2 - (1 1 1 -D D D n+1 ) : σ σ σ n+1 3 -tr D D D n+1 (1 1 1 -D D D n+1 ) + 1 3 [(1 -tr D D D n+1 ) tr σ σ σ n+1 --tr σ σ σ n+1 ] 1 1 1 (53)
The numerical scheme is Euler backward scheme, therefore robust at the Gauss point level, and it has here the main advantage of the explicit schemes: there is no need of a local iterative process as the exact solution of the discretized constitutive equations can be gained.

Important feature, due to the use of a nonlocal damage criterion written in terms of strains, the nonlocal damage model is easily implemented in FE codes already working with nonlocal variables. For instance the existing subroutines computing nonlocal Mazars strain ǫnl can be used with no change. Once the strain ǫ ǫ ǫ n+1 is known from the global equilibrium, compute first ǫnl n+1 using for example the integral form [START_REF] Murakami | A constitutive equation of creep damage in pollicristalline metals[END_REF]. The only change in the integration of the constitutive law is then in equation ( 49) which has to be replaced by tr D D D n+1 = κ -1 ǫnl n+1 (54) The above procedure still applies.

Secant operator

The secant operator is in fact the effective (damaged) fourth order elasticity tensor Ẽ Ẽ Ẽ as when no permanent stresses are modelled,

σ σ σ = E E E S : ǫ ǫ ǫ = Ẽ Ẽ Ẽ : ǫ ǫ ǫ, Ẽ Ẽ Ẽ = Ẽ Ẽ Ẽ(D D D) (55)
In our case it has the same expression for both local and nonlocal anisotropic damage models.

Using eq. ( 46) allows to derive a closed form for Ẽ Ẽ Ẽ,

• if tr ǫ ǫ ǫ > 0, Ẽ Ẽ Ẽ = 2G (1 1 1 -D D D) ⊗(1 1 1 -D D D) 1/2 - (1 1 1 -D D D) ⊗ (1 1 1 -D D D) 3 -tr D D D + K(1 -tr D D D) 1 1 1 ⊗ 1 1 1 (56) • if tr ǫ ǫ ǫ < 0, Ẽ Ẽ Ẽ = 2G (1 1 1 -D D D) 1/2 ⊗(1 1 1 -D D D) 1/2 - (1 1 1 -D D D) ⊗ (1 1 1 -D D D) 3 -tr D D D + K 1 1 1 ⊗ 1 1 1 ( 57
)
where G = E/2(1 + ν) and K = E/3(1 -2ν) are respectively the shear and the bulk moduli and where the special tensorial product ⊗ is used: for two tensors A A A and B B B,

(A A A⊗B B B) ijkl = A ik B jl .
To know the expression for the secant tensor allows to built the global Finite Element secant matrix and to solve the equilibrium equations by use of quasi-Newton method. The secant method applies in the same manner to both local and nonlocal models. Note that with the gradient form (13) for nonlocal Mazars strain, a tangent operator can still be derived at Gauss points level [START_REF] Peerlings | Gradient-enhanced damage model for quasi-brittle materials[END_REF]. This is not the case anymore for the integral form [START_REF] Murakami | A constitutive equation of creep damage in pollicristalline metals[END_REF] and refer then to [START_REF] Jirasek | Consistent tangent stiffness for nonlocal damage models[END_REF] for the definition and use of a global tangent matrix.

Numerical control of rupture with anisotropic damage

Induced damage anisotropy often leads to numerical difficulties, more or less important, depending on the model, which can sometimes stop the computations very early (an example is given in next section for the present model, see figures 7 and 9).

As numerical difficulties one often thinks of a poorly estimated consistent tangent operator but for most models with no permanent strains, the use of the global secant stiffness proves sufficient [START_REF] Pijaudier-Cabot | Finite element analysis of bifurcation nonlocal strain softening solids[END_REF]. The main difficulties are in fact due to a bad numerical control of local rupture once a high level of damage, anisotropic, is reached. In order to continue the computation up to complete structural failure, one must ensure the local damaged Hooke operator to keep positive stiffnesses (i.e. positive Kelvin moduli) or in other words to remain positive definite [START_REF] Ladevèze | On an anisotropic damage theory[END_REF].

Broken bulk modulus

In the models proposed in previous sections, the trace of the damage tensor acts on the hydrostatic stress as:

tr ǫ ǫ ǫ = tr σ σ σ 3K(1 -tr D D D) - -tr σ σ σ 3K ( 58
)
with K the bulk modulus of the undamaged (isotropic) material. There are situations such as compression leading to tr D D D > 1. These are admissible as long as they are mainly compressive, more precisely as long as tr σ σ σ < 0, as the hydrostatic stiffness remains equal to 3K. But they become critical when a change in the loading sign occurs if the broken material behavior is not properly defined as then 3K(1 -tr D D D) is negative and cannot therefore act as a material stiffness. The physical meaning of a large damage trace due to compression is that the material, highly damaged, still resists for compressive states of stresses but will be broken in pieces as soon tensile stresses are applied on it. It is nevertheless possible to continue the computation by defining a fictitious hydrostatic broken behavior for the material such as

• a linear broken behavior,

tr σ σ σ = 3K broken tr ǫ ǫ ǫ -3K -tr ǫ ǫ ǫ (59)
where the "broken" bulk modulus K broken is very small and is related to the trace of the damage reached when the broken behavior is activated. If tr D D D ≥ D c at this time,

K broken = K(1 -D c ) (60)
• a constant broken behavior,

tr σ σ σ = σ broken if tr ǫ ǫ ǫ > σ broken 3K broken tr σ σ σ = 3K broken tr ǫ ǫ ǫ if 0 < tr ǫ ǫ ǫ ≤ σ broken 3K broken tr σ σ σ = 3K broken tr ǫ ǫ ǫ if tr ǫ ǫ ǫ ≤ 0 ( 61
)
with the residual stress σ broken as material parameter. • or even a softening broken behavior ensuring tr σ σ σ → 0 for "large" positive hydrostatic strains.

Principal damages bounded by the critical damage D c

Concerning the other (shear) moduli, the numerical difficulties can be avoided by ensuring the principal damages to remain bounded by 1 or, better, by the critical damage D c . Once the largest principal damage D I reaches D c , a solution is to keep the damage evolution law unchanged, here Ḋ D D = λ ǫ ǫ ǫ 2 + , but projected in such a manner that D I remains constant equal to D c . Such a projected damage law reads:

Ḋ D D = λ Π ⊥n I ǫ ǫ ǫ 2 + (62)
with Π ⊥n I the projection operator. It has the main advantage to consider the same damage law (with no need of additional parameters) as for the 3D case but in fact restricted to 2D as the material cannot damage anymore in the direction n I . The eigenvector n I corresponds to the eigenvalue D I , it is the normal of the main mesocrack initiated and is determined by solving

D D D n I = D I n I = D c n I , n I = 1 ( 63
)
The two other eigenvectors, also normalized, are denoted n II and n III and are associated with the principal damages D II , D III smaller than D c (consider here D I ≥ D II ≥ D III ) so that a less formal expression for the projected damage law is derived as:

Ḋ D D = λ ǫ ǫ ǫ 2 + -(n I • ǫ ǫ ǫ 2 + n I ) n I ⊗n I -(n II • ǫ ǫ ǫ 2 + n I ) n I ⊗n II sym -(n III • ǫ ǫ ǫ 2 + n I ) n I ⊗n III sym ( 64
σ σ σ = 2G(1 -D c )ǫ ǫ ǫ D + K [(1 -D c ) tr ǫ ǫ ǫ --tr ǫ ǫ ǫ ] 1 1 1 (65)
with ǫ ǫ ǫ D = ǫ ǫ ǫ -1 3 tr ǫ ǫ ǫ 1 1 1 the deviatoric strain.

The numerical implementation is similar to the one described in section 6.1 as the projected damage evolution law ( 64) is discretized in

∆D D D = D D D n+1 -D D D n = ∆λ Π ⊥n I ǫ ǫ ǫ n+1 2 +
(66) instead of Eq. ( 48) with a damage multiplier given by ∆λ

= tr D D D n+1 -tr D D D n tr (Π ⊥n I ǫ ǫ ǫ n+1 2 + ) (67)
instead of Eq. ( 50), so that the damage is updated as

D D D n+1 = D D D n + ∆λ Π ⊥n I ǫ ǫ ǫ n+1 2 + .
The risk of obtaining negative (non physical!) stiffnesses is in fact strong for structures computations with any anisotropic damage model introducing 1 1 1 -D D D (2nd order) terms or I I I -D D D (4th order) terms as one principal damage may reach 1 at some Gauss points for locally severe loading. This is usual for instance for reinforced concrete structures with perfect interfaces between steel and concrete so that severe shear easily occurs. The 1 1 1 -D D D and I I I -D D D terms must remain positive and the projection procedure just proposed can be applied to ensure this feature. Write the considered anisotropic damage evolution law in the general form Ḋ D D = λ Q Q Q (initial case of principal damages smaller than unity), with Q Q Q often a normal to a damage surface, i.e. a tensorial function of the stresses, strains and other variables. To replace the term ǫ ǫ ǫ 2 + by Q Q Q in equation ( 62) define in a general manner the projected damage law as Ḋ D D = λ Π ⊥n I Q Q Q. Note that for models based on compliance increase due to damage the problem just mentioned shall not occur, except once more if (1 1 1 -D D D) -1 or (I I I -D D D) -1 terms are introduced. For micromechanics based models such as the microplane damage model [START_REF] Bazant | Crack shear in concrete: crack band microplane model[END_REF][START_REF] Bazant | Microplane model for progressive fracture of concrete and rock[END_REF][START_REF] Bazant | Microplane model for brittle plastic material: I. theory and ii. verification[END_REF] this difficulty does not occur if the local scalar damages over each microcrack or over each microplane direction does not reach unity. This can be gained by using local uniaxial stress-strain relations having a softening part asymptotically equivalent to a monotonically decreasing function approaching 0 at large strains or by using local scalar damage laws monotonically increasing functions approaching 1 at large strains [START_REF] Bazant | Nonlocal microplane model for fracture, damage, and size effect in structures[END_REF][START_REF] Fichant | Isotropic and anisotropic descriptions of damage in concrete structures[END_REF]. This feature is more complex to gain for a tensorial writing of the damage evolution law.

Finite Element computations of structures

Once a constitutive model is developed it has to prove its efficiency in Finite Element computations of engineering structures, or at least on structural components. The difficulties encountered are numerous. The strain-damage localization due to stress softening is of course one of them, phenomenon usually regularized with use of nonlocal models, but at expensive computation cost. There are also important difficulties related to interfaces and steels yielding, difficulties preponderant in reinforced and pre-stressed concrete structures. Once illustrates here the ability of the anisotropic damage model to deal with such structures, starting first with Nooru-Mohamed test exhibiting rotations of the loading principal directions.

Plain concrete mixed-mode fracture

The double edge notched specimen tested by Nooru-Mohamed [START_REF] Nooru-Mohamed | Mixed-mode fracture of concrete: An experimental approach[END_REF] is analysed using the implementation of the model developed in the previous section. The specimen geometry and the experimental testing set up are shown in Figure 4. It is a symmetric 200 mm× 200 mm mortar square with two notches, 30 mm long and 5 mm thick.

The rotation of the external boundary of the plate is restricted around the Oz axis (thick borders). The concrete specimen is first loaded by an increasing shear force F (t) applied on the lateral surface. During the application of the shear force, the vertical displacement of the upper surface is totally free. In a second time, a vertical displacement U(t) is applied up to failure at constant F = F M ax , the higher F M ax the more curved the crack path [START_REF] Nooru-Mohamed | Mixed-mode fracture of concrete: An experimental approach[END_REF]. The case study is here carried out for F M ax = 22.5 kN. The FE discretization of the specimen is made by the use of four node tetrahedron elements with one integration point. In order to perform the computations in 3D at reasonable cost, a FE mesh with a 5 mm width is used when the real width of the specimen is 50 mm. The model parameters used for the simulation are those of section 5 for concrete: E = 42000 MPa, ν = 0.2, κ 0 = 5 10 -5 , A = 5 10 3 , a = 2.93 10 -4 . The nonlocal length used in the integral weight function (Gaussian type, Eq. 43) is l c = 8 mm, small value indeed justified by the fact that the material is a mortar with very small constituents [START_REF] Bazant | Measurement of characteristic length of nonlocal continuum[END_REF][START_REF] Ragueneau | Mechanical behavior related to continuum damage mechanics for concrete[END_REF]. Three meshes are used (Figure 5): a coarse mesh with a total of 862 elements, a medium mesh with 2428 elements, and a fine mesh with 4558 elements. The characteristic length corresponds then close to the notch to 2 elements of the coarse mesh, to 5 elements of the medium mesh, to 10 elements of the fine mesh. The above procedure for the control of rupture with anisotropic damage is considered with a linear broken behavior (D c = 0.99). The damage patterns computed corresponds well to the crack patterns experimentally observed [START_REF] Nooru-Mohamed | Mixed-mode fracture of concrete: An experimental approach[END_REF]. They are symmetric with respect to the center of the specimen (two main cracks obtained) so that the damage D 22 is drawn here in the upper part of the structure, D 11 in the lower part (most D 11 -damage occurs due to the control of rupture procedure). The right column computations exhibit the now classical convergence and mesh independence of the results obtained with a nonlocal model. Due to instabilities related to the too brittle response in local computations (induced by the mesh refinement), no convergence has been obtained for the local model with the fine mesh. Note that no specific procedure such as arclength methods has been used to overpass the numerical difficulties as the conceptually satisfactory introduction of a characteristic length proves to be efficient (convergence is obtained in nonlocal for the fine mesh). The importance of the procedure for the control of rupture must be emphasized: without its use the computations stop before the shear load F M ax is reached. sponse is of course mesh-dependent in case of local computations (not plotted), the finer the mesh the more brittle the structure behavior. This is classically not the case anymore for the nonlocal computations as convergence is reached for the nonlocal anisotropic damage model by mesh refinement.

To conclude, the nonlocal anisotropic damage model with an adequate control of rupture correctly represents the crack propagation for mixed mode fracture.

Reinforced concrete structure

The objectives of this section are to evaluate the ability of the anisotropic damage model to deal with a reinforced concrete element subject to flexion. The structure is a reinforced square cross section beam, subject to three point bend loading. Figure 8 shows geometric features for concrete and steel. During loading, multiple loading paths are encountered in different parts of the beam: tension on the lower part, compression on the upper part, shear near the edge and along the reinforcing bars. The corresponding different features of the constitutive equations are activated at the same time and the occurrence during loading of several competitive cracks usually makes difficult the global convergence scheme. For these reasons, this case-study was part of the international MECA benchmark, launched by E.D.F. to compare and discriminate different 3D constitutive models for concrete [START_REF] Delaplace | Modèles de fissuration de béton -projet MECA[END_REF]. For concrete, the material parameters used in the following computations are those of section 5 and used for previous computations of Nooru-Mohamed test. For steel, elastoplasticity with linear hardening is considered and the material parameters are imposed by the benchmark: Young's modulus E = 200000 MPa, Poisson ratio ν = 0.3, yield stress of 480 MPa, plastic modulus of 20000 MPa. For the computation, a 3D specimen has been meshed with 2 elements in the thickness for a total of 600 eight node parallelepipedic elements. Accounting for the different symmetries of the problem, only one reinforcing steel bar is modelled as shown in Figure 9. The mean dimension of the finite element size is 50 mm. The monotonic loading is applied up to failure. Two computations with two different characteristic lengths are performed in order to appreciate the effect of the nonlocal length (Gaussian weight function, Eq. ( 43), l c = 150 mm and l c = 250 mm). The choice l c = 150 mm is the more physical as it corresponds here to a characteristic length equal to 3 or 4 times the maximum aggregate size [START_REF] Bazant | Measurement of characteristic length of nonlocal continuum[END_REF].

Figure 9 shows comparisons between experiment and modeling in terms of global response, i.e. the load applied at the center of the beam versus the deflection. As often for numerical computations using the displacement based FE method, the initial beam stiffness is larger than the experimental one, phenomenon here enhanced by the high value of the concrete Young's modulus. As one also can see in Figure 9, due to the reinforcing steel bar implying a flexural rupture the effect of the characteristic length is small in terms of global response, but not of cracking pattern as it can be seen from damage maps. The choice l c = 150 mm is the more appropriate here, as expected. The maps of the damage variable D 11 are given in Figure 10. The computation represents quite well the multiple cracks propagation for the characteristic length l c = 150 mm. The three main stages of a reinforced concrete structure subject to flexion are then recovered: elasticity, cracking in tension, and yielding of the reinforcement.

Due to the high number of competitive cracks, the convergence of the local model stop very early during the analysis. Only the nonlocal approach allows for the calculations achievement without local instabilities. Last and again, the computations performed without the procedure for the rupture control stop too early when the computations with numerical rupture control (linear broken behavior) run up to complete failure of the beam.

Pre-stressed powerplant ring

Consider last the response under pressure of the concrete structure of Figure 11 representative of a ring of a powerplant containment vessel reinforced by half circular steel bars and pre-stressed by metallic cables. The diameter of the ring is 46.8 m, its thickness 0.9 m and only a height of 0.4 m is meshed. A vertical compressive stress of 8.5 MPa is applied to model the weight of the upper part of the structure (not meshed). There are two symmetric anchoring parts and the steel reinforcement is made of 88 rectangular stirrups (not considered in the computations), of 4 circular reinforcement bars and of 2 pre-stressed half-circular cables. A more detailled description of the studied structure can be found in [START_REF] Saouma | 3d nonlinear finite element/fracture mechanics analysis of pressurized nuclear reactor container ring[END_REF] and in [START_REF] Lemaitre | Engineering Damage Mechanics : Ductile, Creep, Fatigue and Brittle Failures[END_REF].

The FE analysis of the ring is made in 3D with the bars and the cables modelled with 812 two node bar elements and the concrete part with 4420 eight node bricks. The prestressing tension is applied through an equivalent loading allowing for the introduction of a constant tensile stress all along the cables. The equivalent loading of the steel cable on the concrete part of the structure corresponds to an initial tension of 4.4 MN for each cable.

Both the passive steel bars and the active metallic cables are assumed elastic-perfectly plastic. Concrete is modelled with the anisotropic damage model and the material parameters of previous sections. The material parameters for the steel and the cables are: E = 190000 MPa, ν= 0.3, yield stress of 500 MPa for the steel bars of diameter 25 mm ; E = 190000 MPa, ν= 0.3, yield stress of 1814 MPa for the active cables of cross section 5143 mm 2 . The deformed shape of the ring after the prestressing stage is shown in Figure 11a. An internal pressure P (t) is then applied. It corresponds to the every ten years experimental loading applied on powerplant containement vessels in order to measure the air leakage of the structure. The damage fields are computed up to P M ax = 2 MPa for which the average damage intensity corresponds to a structure already collapsed (of course real tests are performed with a much lower pressure level). Due to steel reinforcement a non uniformly deformed mesh is obtained (Figure 11b) in which the displacements have an amplification factor of 633). The anisotropic damage maps show an oriented microcracking pattern, mainly radial (truly represented by D θθ field, D 22 is the sectors θ ≈ 0 • and 180 • , D 11 in the sectors θ ≈ 90 • or -90 • ), which will be preferable paths for air leakage under pressure. The evolution of the principal damages at points A (external point of the anchoring part) and B (internal point of the anchoring) are plotted in Figure 13 under of the pre-stressing loading. Point A is the most loaded point after pre-stressing. During the pressure increase, the microcracks spread around the whole ring as the damage field tends to the uniform level D θθ ≈ 1 for P = P M ax = 2 MPa. The procedure for the rupture control of section 7 proves again efficient as without it one cannot numerically apply pre-stresses larger than 2 MN in the cables. The steel reinforcements prevent the localization modes, at least with the large elements used. In order to obtain large mesocracks initiation by strain-damage localization, nonlocal computations using smaller elements shall be performed but at a much expensive cost!

Conclusion

To conclude, an anisotropic damage model is proposed for quasi-brittle materials. The damage state corresponds to a given microcracks pattern and, accordingly to the definition of thermodynamics state variables, it is represented by a single 2nd order tensorial damage variable D D D whatever the sign of the loading. The strong dissymetry of concrete behavior is represented and is shown to be mainly due to induced damage anisotropy. A single set of damage parameters A and a is introduced and is valid for both tensile or compressive loading. The model is 3D and allows for computations of structures under either local and nonlocal assumptions. A proof of the positivity of the intrinsic dissipation is given.

From numerical examples up to structural failure, the anisotropic damage model shows its ability to be used in large scale engineering computations at reasonable computational cost. Reinforced and pre-stressed concrete structures can be studied up to a very high damage level and yielding of the reinforcement steels. Note that the numerical calculations are efficient due to three main features:

• a thermodynamics formulation (a state potential continuously differentiable, positive dissipation for any complex loading), • Euler backward scheme solved in an exact manner for the time integration of the constitutive equations (no local iterations), • the use of a procedure for the numerical control of rupture probably always necessary in case of anisotropic damage.

Last, even if anisotropic damage has not proven yet its superiority over the isotropic damage consideration for structural (mechanical) applications, it is nevertheless a more thermodynamically consistent framework. But probably the most promising point concerns the mechanical coupling with diffusion problems such as thermo-hydro-mechanical analyses. For instance, when dealing with the evaluation of the gas or liquid leakage of a cracked structure, one shall take advantage of the anisotropic description of the damage in the coupling with material permeability [START_REF] Bary | Poro-damage approach applied to hydrofracture analysis of concrete[END_REF].

Fig. 1 .

 1 Fig. 1. Trace of D D D function of Mazars equivalent strain (marks: experiments, continuous line: model)

•

  Damage criterion, local f = ǫ -κ(tr D D D) or nonlocal or f = ǫnl -κ(tr D D D), f < 0 or ḟ = 0 -→ elastic loading or unloading f = 0 and ḟ = 0 -→ damage growth (44)

Fig. 2 .

 2 Fig. 2. Stress-strain curves for concrete (tensile strength f t = 4 MPa, compressive strength f c = 38 MPa)

Fig. 3 .

 3 Fig. 3. Model response in shear for damage laws 1 and 2

( 1 ) 47 )( 2 )

 1472 Euler backward scheme is used, i.e. the variables are replaced by their value at time t n+1 in the constitutive equations when the damage rate Ḋ D D and the damage multiplier λ are replaced by ∆D D D = D D D n+1 -D D D n and ∆λ = λ n+1 -λ n in the damage law. In order to integrate the damage model proceed as follows: Compute the equivalent strain, ǫn+1 = ǫ ǫ ǫ n+1 + : ǫ ǫ ǫ n+1 + (Make a test on the criterion function f = ǫn+1 -κ(tr D D D n ) If f ≤ 0, D D D n+1 = D D D n , the material behaves elastically. If f > 0, the damage must be corrected by using the damage evolution law discretized as (α = 2 is set), ∆D D D = D D D n+1 -D D D n = ∆λ ǫ ǫ ǫ n+1 2 +

  ) which leads to the explicit expression of ∆λ even for Euler backward scheme, ∆λ = tr D D D n+1 -tr D D D n ǫ2 n+1 (50) and to the exact actualization of D D D, D D D n+1 = D D D n + ∆λ ǫ ǫ ǫ n+1 2 +

  ) and is then valid for fixed n I but for updated D II , D III , n II , n III as long as D II has not reached D c . Last, only D III will evolve as ḊIII = λ n III • ǫ ǫ ǫ 2 + n III at fixed n I and n II up to D III = D c so that one ends up to a diagonal (isotropic) damage D D D = D c 1 1 1. A 3D broken behavior is finally obtained, isotropic, dissymetric (linear case, D c constant close to 1),

Fig. 4 .

 4 Fig. 4. Nooru-Mohamed test

Fig. 5 .Figure 6

 56 Fig. 5. Different meshes of Nooru-Mohamed structure Figure 6 shows the anisotropic damage patterns computed. The left column computations correspond to local results, the right one to nonlocal results. The application of the shear load up to F M ax yields localized damage at the notch tip. The structural failure is then due to the application of the vertical displacement U(t) with mainly mode I cracks represented here in the Continuum Damage Mechanics framework by large D 22 values.The damage patterns computed corresponds well to the crack patterns experimentally observed[START_REF] Nooru-Mohamed | Mixed-mode fracture of concrete: An experimental approach[END_REF]. They are symmetric with respect to the center of the specimen (two main cracks obtained) so that the damage D 22 is drawn here in the upper part of the structure, D 11 in the lower part (most D 11 -damage occurs due to the control of rupture procedure). The right column computations exhibit the now classical convergence and mesh independence of the results obtained with a nonlocal model. Due to instabilities related to the too brittle response in local computations (induced by the mesh refinement), no convergence has been obtained for the local model with the fine mesh. Note that no specific procedure such as arclength methods has been used to overpass the numerical difficulties as the conceptually satisfactory introduction of a characteristic length proves to be efficient (convergence is obtained in nonlocal for the fine mesh). The importance of the procedure for the control of rupture must be emphasized: without its use the computations stop before the shear load F M ax is reached.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Damage maps for Nooru-Mohamed test at U = 3.5 10 -3 mm-(a) left column: results with local anisotropic damage model, (b) right column: results with nonlocal anisotropic damage model

Fig. 8 .

 8 Fig. 8. Reinforced concrete beam

Fig. 9 .Fig. 10 .

 910 Fig. 9. Comparisons experiment/computations

Fig. 11 .

 11 Fig. 11. Deformed shape of the ring (a) after prestressing and (b) after internal pressure -Points A and B of the anchoring part

Fig. 12 .

 12 Fig. 12. Anisotropic damage fields at P = 2 MPa

Fig. 13 .

 13 Fig. 13. Principal damages at points A and B during prestressing


Patrick Chevret
email: patrick.chevret@inrs.fr

Ange Ebissou
email: ange.ebissou@inrs.fr

Etienne Parizet
email: etienne.parizet@insa-lyon.fr

Evaluation of speech intelligibility in open-plan offices

In open-plan offices, ambient noise made of intelligible conversations is generally perceived as one of the most important annoyance for tasks requiring concentration efforts. This annoyance has been proved to lead to a decrease of task performance and to health troubles for people in the mean and long term (tiredness, stress, etc.) Consequently, the improvement of working conditions should pass by the evaluation of speech annoyance that could give rise to recommendations on the arrangement of the open-plan office and on acoustical treatment of walls, ceiling and furniture. In that context, a standard SRT (Speech Reception Threshold) measurement campaign was set up, in order to evaluate the effect of speech ambient noise on intelligibility. For the trial purpose, a large set of syllabically-balanced French sentences has been constituted from Combescure, Fourier and Hint sentences. The ambient noise was made of both synthetic (modulated white noise, speech like modulated white noise, inversed speech noise) and real recordings in open-plan offices. In parallel, a second measurement campaign with subjects was conducted whose aim was to study the influence of intelligibility on the performance and effort in the realization of a seriation task. Results showing some links between intelligibility, performance and effort are presented in the paper.

INTRODUCTION

Noise is reported in many studies as the most disturbing physical nuisance in open-plan offices. It has been noted that speech noise is particularly annoying for office workers [START_REF] Banbury | Office noise and employee concentration: Identifying causes of disruption and potential improvements[END_REF] . A need has thus emerged amongst occupational health specialists for a way to assess the acoustical quality of an office with regards to this problem. At INRS (French National Institute for Occupational Health), this concern was integrated in the research topics several years ago, starting with studies on the impact of continuous phone conversations in call centers. More recently, projects have been initiated on the effects of intelligibility on performance and workload in open-plan offices. This is the topic of the paper, which mainly focuses on the description of experimental campaigns and the associated procedures, the results and the orientation for the next years.

SPEECH INTELLIGIBILITY

Speech intelligibility is the main factor used to describe acoustic comfort in open-plan offices. It is commonly accepted, supported by that the speech intelligibility may be impacted by several factors:

1. The levels of the speech and the background noises (BGN), the latter being a mixture of equipment noise and speech noise. Indirectly, the ratio (SNR) between the two components (speech and BGN) is a main effect. 2. The arrangement of the offices and the quality of the materials used for the walls, the ceiling and the furniture inside the working site. These elements have an effect on the sound attenuation and the sound transmission between the speakers and the listener. 3. The type of activity and the number of persons that work in the space which determine the signal structure of the background noise, in particular its modulation rate.

In this section, we describe the set up of an experiment, whose objective is to evaluate the effect of the background noise modulation on sentence intelligibility. This is done by the determination of the Speech Reception Threshold (SRT) which is defined as the signal to noise SNR value for which 50 % of the sentences are 100% understood. Recently, such experiences have been conducted by Rhebergen 2 on simulated and natural signals (true voices). In this paper, we show that the use of the Plomp and Mimpen 3 procedure is adequate for the ranking of open-spaces in terms of intelligibility.

The experiment

The test procedure

The test is based on the SRT procedure developed by Plomp and Mimpen. Basically, it consists in determining the SNR value for which 50% of the sentences are totally understood by the listeners. The value also corresponds to an intelligibility score of 50%, and determines the inflection point of the psychometric function. SRT procedure consists in presenting sentences in noise to listeners with an adaptative SNR value which depends on the listener response (correct or incorrect). First, the SNR value is high (typically 8 dB), so that the listener has no doubt for its answer. Then, the SNR value is decreased by step of 4 dB until the response is incorrect. From this step, the SNR value is adapted with a value of +2dB when the response is wrong and -2dB when it is correct. For each series, 20 phonetically balanced sentences are presented to the listener. The lasts 15 values of SNR are averaged to give the SRT value.

The test conditions

For practical reasons, measurements have been conducted in two separate places so that two groups of people are constituted. The first group is installed in a small silent room (26 dBA of background noise) of dimensions 4m x 4m x 2.5m. This group is composed of 23 students aged from 21 to 37 years. They all had a normal hearing, as measured using an audiometric calibration. The second one was performed in a large semi-anechoic room (22 dBA of background noise) of dimensions 10m x 7m x 7m with 18 persons aging from 14 to 61 years. No hearing test was performed for this group but none of the people declared any hearing impairment.

Except for the room and the loudspeaker, the test conditions were exactly the same for the two groups. During the test, the listener is placed at a distance of about 1.5m from of a unique loudspeaker. Samples of sound, composed of a mixture of the target sentence and background noise, are presented.

The sound level of the sample is always held at a level of 60 dB at 1m from the loudspeaker. The SNR between the target sentence and the background level is adjusted according to the answer of the listener (repetition of the sentence), following the SRT procedure of Plomp and Mimpen.

Listening material

Lists of sentences in French (299 sentences in total), designed for audiometry purposes, were used as the target speech signal. The recordings 4 were provided by the Collège National d'Audiométrie (French Organization for Audiometry). The sentences are the so called Combescure, Fournier and HINTS sentences. They are pronounced by a male voice. Background noise was constituted either from the lists of sentences or synthetic signals according to the following description:

1. Stationary white noise (LTASS-WN), 2. White noise 100% sinusoidally intensity-modulated with 8Hz frequency modulation (LTASS-SIM8Hz), 3. Sentences pronounced by 1 female voice (LTASS-MOD1pers), 4. Mixture of sentences pronounced by 3 female voices (LTASS-MOD3pers), 5. White noise modulated by the envelop of sentences pronounced by 1 female voice (LTASS-WNMOD1pers), 6. White noise modulated by the envelop of a mixture of sentences pronounced by 3 female voice (LTASS-WNMOD3pers), 7. White noise modulated by the envelop of a mixture of sentences pronounced by 5 female voice (LTASS-WNMOD5pers), 8. Sentences pronounced by 1 female voice and time inversed (LTASS-MOD1pers-inv), 9. Ambient noise recorded in open-plan office 1 (LTASS-OPO1), 10. Ambient noise recorded in open-plan office 2 (LTASS-OPO2),

The reason for choosing a male voice for the target sentences and a female one for the background noise is to avoid confusion between the target sentences and the disturbing ones. The labeling LTASS means that all the samples are equalized according to the long term averaged speech spectrum; the male one for the target samples and the female one for the background noise.

All of these background noises are presented to the listeners in a randomized order. For each BGN and each listener, one SRT value is obtained which is used for the ANOVA described in the next section. The duration of the test is about 30min, this value mainly depending on the reactivity of the tested person.

The results

In this paper, two outputs of the test are analyzed through a standard Analysis of variance (ANOVA). First, the consistency is studied by presenting to the listeners the same background noise type at the beginning and at the end of the test (the type is different from one listener to the other). Second, the effect of background noise modulation is analyzed.

Test consistency

In order to analyze the test consistency, the same type of background noise is presented to the listeners (but different from one listener to the other) at the beginning (first series) and the end (last series) of the test. For each group, we analyze the consistency of the responses with the ANOVA in which we have two statistical populations; the first one is composed of the responses of the listeners for the first series and the second of the responses of the last series. The consistency is verified thanks to the H0 hypothesis. The respective probabilities for the equality of the mean values for the two groups are 97% and 68%. Therefore, consistency is obtained for group1, meaning that this group is less sensitive to a learning effect and to a fatigue effect that could be linked to the duration of the test. This is not true with the second group as far as there is a non negligible probability (32%) that either the learning phase is useful for the listener to get use to the test or that the effect of fatigue is noticeable. The difference of age between the two groups may be the more simple explanation. In the next section, the two groups are merged for the analysis of background noise effect.

Effects of background conditions

A 10[background condition BGN] × 41[subject] analysis of variance (ANOVA) was performed on the data. "Background condition" was found to be significant (F[9,400)] = 62.5, p < 0.001). The result of the ANOVA is presented on figure 1 which represents the SRT values for all the background noise configurations together with the corresponding confidence interval. The first remark is that 4 categories of background noise can be considered according to a SRT classification. These results confirm the literature in the sense that the more the BGN is modulated, the higher is the intelligibility. The first category seems to indicate that there is no effect of informational masking for this test. A major conclusion is that intelligibility is different between the two open-spaces, indicating that the procedure is relevant for establishing an intelligibility ranking of the open-spaces.

Confidence intervals are small for the largest values of SRT and increase with smaller values. This is due to a high variability of the listening conditions when only one speaker is making the background noise.

Figure 2 gives the SRT results for the two groups independently. We observe that the tendency is the same, except that a higher intelligibility is noticeable for the second group that performed the test in the semi-anechoic room (where the listening conditions are better).

EVALUATION OF PERFORMANCE

For certain office-related tasks, notably those involving short-term memory, a drop-off in performance can be observed. This phenomenon is usually referred to as Irrelevant Speech Effect. In laboratory settings, it is often explored through a seriation task, where seven to nine elements (digits, letters or words) are presented in rapid succession to participants. After a short retention period, they are required to report back the series in the exact order of presentation. The ISE then seems to be related to speech intelligibility: the more the ambient speech is comprehensible, the harsher the decrease in performance is [START_REF] Banbury | Auditory distraction and shortterm memory: Phenomena and practical implications[END_REF] . This is why an intelligibility index such as the STI has been used to propose target values for an open-plan office of good quality. In the same vein, Hongisto [START_REF] Hongisto | A model predicting the effect of speech of varying intelligibility on work performance[END_REF] proposed a STI-performance curve to predict the decrement of performance induced by ambient speech of a given intelligibility. According to this model, for unintelligible speech, performance is the same as in silence. A growing impairment, following a sigmoid curve, occurs for intermediate STI values ranging from 0.3 to 0.5, where the deterioration of performance is already approaching its maximum.

For other common tasks, no ISE can be noticed [START_REF] Schlittmeier | The impact of background speech varying in intelligibility: Effects on cognitive performance and perceived disturbance[END_REF] . As a consequence, when investigating the disturbing nature of ambient speech, one has to find alternative ways to estimate the difficulty of the task as perceived in a particular sound environment. In the ISE-related literature, questionnaires are the most frequently used method to obtain such an assessment. Their use is also relevant to complement performance measurements on short-term memory tasks. In that case, subjective assessments generally agree with objective ones. Participants express a higher level of disturbance when speech intelligibility increases. Nevertheless, both measurements do not seem to provide the same information with regard to the disturbance caused by the sound environment [START_REF] Schlittmeier | The impact of background speech varying in intelligibility: Effects on cognitive performance and perceived disturbance[END_REF] . It should also be noted that the questions asked to participants vary across studies, which can make comparisons difficult. The use of a widespread workload measure could be useful for future reference.

The first goal of this study is to improve our understanding of the variations of performance with speech intelligibility, as measured by the STI. By exploring the [0.3-0.5] range, it aims to offer some material to further the work undertaken by Hongisto. By using an objective measure of performance and a subjective report of difficulty, the following experiment also seeks to compare these two types of assessments. Their respective evolutions with regard to the ambient speech intelligibility will be observed in order to compare their ability to discriminate between ambient speeches of different intelligibility levels, as measured by the STI.

The experiment

The experimental group consisted of 57 participants, one half being students and the other half recruited through an outsourcing company specialized in clinical trials. There were 32 women and 25 men, aged from 22 to 73 years, 36 years on average. All participants reported normal hearing and were paid for their participation. The same lists of sentences as for the intelligibility tests were used as the speech signal as well as for the background noise rich with multi-talker babble. The total duration exceeded 12 minutes. Four values of the STI were implemented: 0.3, 0.4, 0.5 and 0.7. A model proposed by Hongisto 8 predicts the STI between two nearby workstations. For a given office setting, the model provides octave-band attenuations which are applied to the spectrums of both speech and noise signals in order to obtain the corresponding STI value.

Participants worked over two sessions on consecutive days. Each session included three 10minute blocks, separated by 5-minute pauses. One of them was carried out in silence, which served as a control condition for which STI = 0. For each of the other two, one of the four sound environments was used. At the end of the two sessions, each participant had been confronted to the four STI values. The order of the sentences was randomized before each block. The sound condition sequence was also balanced between subjects. The experiment took place in a soundtreated booth. Participants were alone in the booth, behind a desk. They faced a computer screen and were equipped with a mouse. Behind the screen was a loudspeaker, which created the sound environment. The A-weighted SPL was 46 dB(A) for all four noisy conditions. Participants had to achieve a seriation task. A permutation of integers from 1 to 9 was presented, one number at a time (700 ms on, 300 ms off; MS Sans Serif font, 3 cm high). After the last number had disappeared, a five-second pause was observed. The individual numbers then came into view in a 3x3 response array. Participants were asked to reproduce the series in the exact order of presentation by clicking on the buttons. After clicking, the number disappeared and could not be selected again. The next series began 3 seconds after the ninth number was proposed. The response phase was self paced, which means that participants carried out as many series as they could in the 10-minute block. Performance was measured by first reporting the number of figures incorrectly placed in any series. The average number of errors per series in a block was then calculated. After each block, participants were presented with a computer-based French version of the NASA-TLX questionnaire [START_REF] Hart | Development of NASA-TLX (Task Load Index): Results of empirical and theoretical research[END_REF] . This workload assessment method comprises six questions, each of them addressing a particular component of the mental workload as detailed in Table 1. For each question, participants gave out a score comprised between 0 and 100 using a 21-point scale. The lower the score, the more comfortable the situation was reported to be. A Raw Task Load Index (RTLX) was eventually calculated by averaging the six scores 10 .

Results

Statistical analyses were led using Stata v12.0. A high degree of heteroscedasticity for both performance and RTLX excluded the use of a classical ANOVA. Moreover, the data presented a hierarchical structure: blocks were carried out on a given session by a given participant. Therefore, within-subject and within-session correlations had to be taken into account. These observations led to the use of a three-level mixed-effects regression model, with a dummy variable for each condition. After modeling, it was then possible to assess separately betweensubject variability, the learning effect between sessions and the residual variance.

Performances

In the silent conditions, participants averaged 2.2 errors per series. A significant effect of the sound condition could be found (χ² = 22.84, d.f. = 4, p < 0.05). An agglomerative hierarchical clustering showed that the group of participants could be divided into two separate groups with distinctive sensitivities to the sound environment, as shown in Figure 3. Neither age nor sex had a significant influence on the classification. The first group is composed with high-performing subjects (32 members, 1.8 errors per series on average in the silent condition). Their behavior depended very little on the sound condition, as no significant effect of STI value could be found. The average performance in the second group was lower (2.8 errors per series on average in the silent condition, χ² = 7.89, d.f. = 1, p < 0.05). Most noticeably, its members displayed a higher sensitivity with regards to the sound environment. The decrease in performance reached 11% for STI = 0.7. The effect of STI value on DP was significant (χ² = 93.36, d.f. = 4, p < 0.05). Noise was significantly detrimental to performance when compared to the silence condition (χ² = 15.43, d.f. = 1, p < 0.05). More errors are made for the two highest levels of STI (0.5 and 0.7) than for the two lowest ones (0.3 and 0.4; χ² = 11.46, d.f. = 1, p < 0.05). The difference in DP was higher between the 0.4-and 0.7-levels (5.6 percentage points) than between the 0.3-and 0.5-levels (4.5 percentage points). This result does not comply with the shape of the curve in the prediction model.

Fig .3 -Decrease in performance for the two groups of participants.

Adding speech noise affected negatively the good execution of a seriation task. This phenomenon grew stronger as the ambient speech noise intelligibility increased. Nevertheless, any analysis should allow for the high level of between-subject variability. A sizable portion of the panel exhibited no sensitivity to noise. The relationship between the decrease in performance and the value of the STI did not conform to the curve proposed by Hongisto. This sigmoid function was chosen because it mimics the shape of the curve describing the dependence of subjective intelligibility of sentences on the STI. This assumption suggests that the level of performance in the task at hand depends on the meaning of the ambient speech. However, it does not appear to be the case for many tasks, such as simple seriation tasks [START_REF] Tremblay | Change of intensity fails to produce an irrelevant sound effect: Implications for the representation of unattended sound[END_REF] . A shape approaching the curve relative to CVC-syllables may be more adequate, as these elements do not carry any meaning.

Mental workload

There was a significant effect of the sound condition (χ² = 114.63, d.f. = 4, p < 0.05). The reported RTLX was much higher in a noisy environment than in silence, as shown in Figure 4. This difference should not, however, be attributed solely to a change in STI. It may be a known fact that performance in a seriation task is not sensitive to the noise level [START_REF] Tremblay | Change of intensity fails to produce an irrelevant sound effect: Implications for the representation of unattended sound[END_REF] , but it is not necessarily the case for mental workload assessments. Once again, no significant difference could be made between the intermediate STI values. In order to show whether both objective and subjective assessments of disturbance vary in the same way with the STI, separate analysis were led with the same groups that were presented in previous section. The results are shown in Figure 5. Subjects drawn from the first group reported lower RTLX scores than their counterparts from the second group (χ² = 16.96, d.f. = 1, p < 0.05). For both groups, the three intermediate conditions were not significantly different from one another. Although, the test is less discriminating for the first group (which is less sensitive to noise), it remains consistent with the second one (increase of workload with the intelligibility). Therefore, the test seems robust against a decreasing of the number of tested subjects. The NASA-TLX was conceived and validated as a global object, taking in consideration all questions. This makes any study of responses to an individual question theoretically unsound. Nevertheless, it can be a useful diagnostic tool to understand which questions bear the most importance in the variations of the global rating. The range of predicted scores for all 6 questions in noisy conditions, expressed as a relative increase from the lowest predicted score for each question, is presented in figure 6. It appears that questions 2 (related to the physical demand of the task) was the one for which the range of predicted scores was the widest. This is quite surprising because the task does not seem to be physically strenuous, which should make the question irrelevant to our experiment. The study of variance partition coefficients (VPC) shines a light on that fact: 70% of the total variance of the scores for question 2 could be explained by between-subject variability and only 2% by the sound condition. These numbers tend to put the previous observation regarding the high relative increase into perspective. As a matter of fact, despite the seemingly great differentiation, the residual error was too important for the effect of STI level to be significant. No significant differences between noisy conditions could either be noticed for questions 1 and 3. These results indicate an increase in STI did not change the way participants perceive the difficulty of the task. It should be noted that question 3 was treated very differently from subject to subject. While some considered primarily the time pressure they felt during the presentation of the numbers, others expressed that, as the response phase was self-paced, there was no particular temporal demand to report. This could account for the limited variations of the reported scores for this item. On a related issue, response time was also measured. No significant effect of the sound condition could be found. When considering the noisy conditions only, the effect of the STI value on reported scores reached significance for questions 4 (χ² = 9.61, d.f. = 3, p < 0.05) and 5 (χ² = 9.30, d.f. = 3, p < 0.05). These questions refer respectively to self-rated performance and effort level. Participants reported that an increase in STI forced them to work harder but still altered the quality of their output.

Adding speech noise increased the mental workload of a seriation task as reported by participants. The workload was stronger for ambient speech of high intelligibility. Noise affected less the perception of difficulty by the subjects than the appreciation of their work during the execution of the task. The information given by these subjective reports was not redundant with what the objective measurement of performance could bring by itself.

CONCLUSIONS

The objective of the paper was to report the experimental studies which have been done recently at INRS in order to evaluate the disturbance induced by ambient noise in open-plan offices.

The main goal was to create relevant set up and procedures in order to test both the intelligibility of speech, performances and mental workload in open-space conditions. This was done and the results presented indicate that, on the axis of intelligibility, the SRT procedure seems to be relevant for a ranking of the open-spaces: different open-spaces, different SRT values. The test has now to be extended to other in-situ measurements of ambient noise and an objective quantification of it (through a modulation rate for instance) has to be performed. As concern the evaluation of performance and workload, the measurement results are encouraging as far as they clearly highlight the detrimental effect of the ambient noise conditions with a good intelligibility. This result is consistent with the literature, but more has to be done regarding the shape of the general STI-performance curve. For instance, one should be aware of the high level of betweensubject variability. This problem may be mitigated by enlisting a greater number of participants. Further experiments are planned and a few remarks pertaining to the making up of an adequate protocol can be made.

Fig. 1 -

 1 Fig. 1 -SRT as a function of background noise

Fig. 4 -

 4 Fig. 4 -NASA-RTLX scores for the seriation task in sound environments of various STI values.

Fig. 5 -

 5 Fig. 5 -NASA-RTLX scores for the two groups of participants.

Fig. 6 -

 6 Fig. 6 -Increase in predicted score relative to the minimum for noisy conditions for the 6 questions of the NASA-TLX questionnaire.

  1. The category of BGN which are responsible of a very high intelligibility. In this category, are the sentences pronounced by one person normally or reversely. The SRT values are very low, which corresponds to very high level of BGN (60 +13.3 dB and 60 + 12.9 for, respectively, the normal and reverse signals). 2. The category of high intelligibility. It contains 2 BGN: the LTASS-WNMOD1pers (with noise modulated by the envelop of one person's voice) and the open-space2. 3. The category of moderate intelligibility which contains 3 BGN: LTASS-WNMOD3pers, LTASS-MOD3pers and LTASS-SIM8Hz which SRT value is very close to the one obtained by Rhebergen 4 . 4. The category of low intelligibility which contains 3 BGN: the white noise (LTASS-WN), the open-space 1 and LTASS-WNMOD5pers.

Table 1 :

 1 Overview of the NASA-TLX questionnaire

	Question 1	Mental demands of the task
	Question 2	Physical demands of the task
	Question 3	Temporal demands of the task
	Question 4	Self-rated performance
	Question 5	Effort level
	Question 6	Frustration

ACKNOWLEDGEMENTS

We acknowledge Professeur Joël Ducourneau, from the Laboratoire de Biomathématiques et Audioprothèse of the Université de Pharmacie of Nancy for his help in setting up the tests and providing the facilities and students.