R Desmorat

F Gatuingt

F Ragueneau

Non standard thermodynamics framework for robust computations with induced anisotropic damage

Many anisotropic damage models have been proposed for different materials, including concrete. The main drawback of the corresponding analyses is that a large number of material parameters is often introduced, leading to identification difficulties but also to model complexity and associated numerical difficulties. It is also sometimes difficult to ensure the continuity of the stresses if the quasi-unilateral effect of microcracks closure and the dissymmetry tension/compression are represented.

In order to solve those difficulties, one proposes to write the damage models in a specific non standard thermodynamics framework. The damage states are represented by a symmetric second order tensor and the damage rate is assumed governed by a positive second order tensor having a clear meanning: the absolute or the positive value of the plastic strain rate tensor for ductile materials, the positive part of the total strain tensor in quasi-brittle materials. Such a non standard feature makes the proof of the the positivity of the intrinsic dissipation necessary. This important proof is given in the considered framework for any damage law ensuring (anisotropic) damage increase and for any case, 3D, proportional or non proportional. This extends then to induced anisotropy the isotropic case property of a positive damage rate as a sufficient condition for the thermodynamics second principle to be fulfilled.

Altogether with the fact that the thermodynamics potential can be continuously differentiated, the anisotropic damage model for concrete (build in this framework) allows for robust Finite Element implementation. Both space (classical nonlocal with internal length, nonlocal with internal time) and time regularizations (visco-or delay-damage) are used and applied to quasi-static and dynamic cases. Examples on concrete and reinforced concrete structures are given.

INTRODUCTION

In standard thermodynamics framework [START_REF] Halphen | Sur les matériaux standards généralisés[END_REF] both the state and the evolution laws derive from potentials, Gibbs free enthalpy ρψ ⋆ written in terms of stress σ σ σ in the present work for the first one, a dissipation or pseudo-dissipation potential for the second one. The elasticity law and the strain energy release rate density Y Y Y , variable associated with D D D, are respectively gained as the derivative of the state potential with respect to the elastic strain tensor ǫ ǫ ǫ e and with respect to the damage tensor D D D. In the standard framework of thermodynamics a pseudo-dissipation potential quadratic function of Y Y Y is most often considered, the damage law taking the form

Ḋ D D = λ J J J : Y Y Y (1)
with λ a positive multiplier and J J J a positive fourth order tensor, eventually nonlinear function of the thermodynamics variables, so that the dissipation

D = Y Y Y : Ḋ D D = λ Y Y Y : J J J : Y Y Y (2)
due to the degradation mechanisms remains positive for any kind of loading. The choice of damage laws of the form (1) is very restrictive, for instance concerning the possibility to model induced anisotropy directly driven by the strains or the stresses. But, as we will illustrate on numerical examples, to ensure the positivity of the intrinsic dissipation also has good numerical properties.

NON STANDARD ANISOTROPIC FRAMEWORK

State potential

A general form for the strain energy coupled with anisotropic damage has been proposed by Ladevèze [START_REF] Ladevèze | On an anisotropic damage theory[END_REF][START_REF] Lemaitre | Engineering Damage Mechanics : Ductile, Creep, Fatigue and Brittle Failures[END_REF], as ρψ ⋆ = a 1 tr (H H Hσ σ σH H Hσ σ σ) + a 2 σ σ σ : σ σ σ + a 3 g(D H )(tr σ σ σ) 2 + a 4 (tr σ σ σ) 2 (3)

with

H H H = (1 1 1 -D D D) -1/2 (4)
a symmetric tensor and

D H = 1 3 tr D D D (5)
g(D H ) is a positive increasing function of D H , for example g(D H ) = 1/(1 -ηD H ). The a i (a 1 ≥ 0, a 3 ≥ 0) as well as η > 0 are material parameters. The first term of Eq. ( 3) can take different -non equivalentforms [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] Papa | Anisotropic damage model for the multiaxial static and fatigue behaviour of plain concrete[END_REF][START_REF] Lemaitre | Engineering Damage Mechanics : Ductile, Creep, Fatigue and Brittle Failures[END_REF],

a
allowing in the last case for the mechanical representation of the quasiunilateral effect of micro-defects closure, with σ σ σ + (resp. σ σ σ D + ) a special positive part [START_REF] Ladevèze | On an anisotropic damage theory[END_REF][START_REF] Desmorat | Quasi-unilateral conditions in anisotropic elasticity[END_REF][START_REF] Desmorat | Modeling micro-defects closure effect with isotropic/anisotropic damage[END_REF] built from the eigenvalues and the eigenvectors of H H Hσ σ σ (resp. of H H Hσ σ σ D ) and with . -the negative part in terms of principal components of a tensor. Two examples of strain energy densities are a first one based on a splitting deviatoric / hydrostatic quantities (with E and ν the Young's modulus and Poisson's ratio of the undamaged material and η a shearbulk coupling parameter [START_REF] Lemaitre | Anisotropic damage law of evolution[END_REF][START_REF] Delaplace | Discrete 3d model as complimentary numerical testing for anisotropic damage[END_REF]),

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 -ηD H (9)
a second one based on the feature of a constant ν/E ratio,

ρψ ⋆ = 1 2E tr (H H Hσ σ σH H Hσ σ σ) + ν 2E σ σ σ : σ σ σ -(tr σ σ σ) 2 (10)

Non standard damage evolution laws

Concerning the evolution laws, many non standard damage laws can be formulated for induced anisotropy [START_REF] Mazars | The unilateral behavior of damage concrete[END_REF][START_REF] Dragon | Modélisation de l'endommagement par mésofissuration : comportement unilatéral et anisotropie induite[END_REF][START_REF] Lemaitre | Anisotropic damage law of evolution[END_REF][START_REF] Billardon | Creep damage behaviour of a copper alloy on a large temperature range[END_REF], with second order tensorial damage rate proportional

• to the positive part of the strain tensor ǫ ǫ ǫ + or to ǫ ǫ ǫ α + with α a damage exponent,

• to the absolute value of the plastic strain tensor ǫ ǫ ǫ p , to the positive part of ǫ ǫ ǫ p , or to any linear combination

ζ |ǫ ǫ ǫ p | + (1 -ζ) ǫ ǫ ǫ p + ,
• to a power 2s of the stress tensor,

• to a linear combination ζ σ σ σ 2 + + (1 -ζ) σ σ σ 2 -(eventually at the power s) where to take ζ = 1 will lead to the modeling of the unilateral damage effect of no damage growth in compression, and to take 1/2 < ζ < 1 will lead to the modeling of the quasi-unilateral damage effect of a damage growth in compression smaller than in tension.

Induced damage anisotropy governed by the positive extensions is adapted to quasi-brittle materials as concrete. The other expressions will allow to generalize to induced anisotropy Lemaitre's damage law

Ḋ = Y S s ṗ ( 11
)
of a damage rate governed by the accumulated plastic strain rate ṗ and enhanced by the strain energy

Y = 1 2 ǫ ǫ ǫ e : E E E : ǫ ǫ ǫ e (12)
with E E E Hooke's tensor. As possible generalization, one has

Ḋ D D = Y S s ζ |ǫ ǫ ǫ p | + (1 -ζ) ǫ ǫ ǫ p + (13) Ḋ D D = ζ σ σ σ 2 + + (1 -ζ) σ σ σ 2 - 2ES s ṗ ( 14
)
where E denotes the Young's modulus and S and s the damage parameters.

For more details on ductile damage, refer to [START_REF] Lemaitre | Engineering Damage Mechanics : Ductile, Creep, Fatigue and Brittle Failures[END_REF].

The next question will be wether one automatically can ensure the positvity of the dissipation with the simple feature of a positive (tensorial) damage rate Ḋ D D.

Link with fourth order damage

The elasticity law corresponding to Gibbs potential (3) is

ǫ ǫ ǫ e = ρ ∂ψ ⋆ ∂σ σ σ = 2a 1 H H Hσ σ σH H H + 2a 2 σ σ σ + 2 (a 3 g(D H ) + a 4 ) tr σ σ σ 1 1 1 (15)
It can used to exhibit the fourth order effective elasticity tensor as ǫ ǫ ǫ e = Ẽ Ẽ Ẽ-1 : σ σ σ and then

Ẽ Ẽ Ẽ-1 (D D D) = 2a 1 H H H⊗H H H + 2a 2 I I I + 2 (a 3 g(D H ) + a 4 ) 1 1 1 ⊗ 1 1 1 (16)
with 1 1 1 and I I I respectively the second and fourth order unit tensors and where (H H H⊗H H H) ijkl = 1 2 (H ik H jl + H il H jk ) and with

a 1 + a 2 = 1 + ν 2E a 3 g(0) + a 4 = - ν 2E (17)
Different expressions corresponds to different choices of material parameters a i . Micromechanics and homogeneization calculations a of medium with cracks and cavities can be used as guidelines for such choices. A fourth order damage tensor D D D can then be built from (

I I I -D D D) : E E E = Ẽ E E as D D D = I I I -E E E -1 : Ẽ E E (18)
The a priori choice of a state potential is a limitation of the modelling, as one knows that the second order damage representation is only an approximation [START_REF] Leckie | Tensorial nature of damage measuring internal variables, chapter Physical Non-Linearities in Structural Analysis[END_REF][START_REF] Krajcinovic | Damage mechanics[END_REF]: systematic studies in non proportional loading cases are needed to fully validate any damage model built in present non standard framework (this applies to the damage model for concrete considered next). But note that the direct use of fourth order damage in phenomenological constitutive models often leads to a damage anisotropy difficult to visualize (6 principal damages instead of 3 for the second order framework) and to too many damage parameters.

POSITIVITY OF THE INTRINSIC DISSIPATION

Previous elastic energy densities can be continuously differentiated as

dρψ ⋆ = ǫ ǫ ǫ e : dσ σ σ + Y Y Y : dD D D (19)
or

dρψ ⋆ = [2a 1 (H H Hσ σ σH H H) + 2a 2 σ σ σ + 2a 3 g(D H ) tr σ σ σ 1 1 1 + 2a 4 tr σ σ σ 1 1 1] : dσ σ σ + 2a 1 (σ σ σH H Hσ σ σ) : dH H H + 1 3 a 3 g ′ (D H )(tr σ σ σ) 2 tr dD D D (20)
leading to a dissipation due to damage mechanisms expressed as

D = Y Y Y : Ḋ D D = 2a 1 (σ σ σH H Hσ σ σ) : Ḣ H H + 1 3 a 3 g ′ (D H )(tr σ σ σ) 2 tr Ḋ D D (21)
For the elastic energy densities written with the terms (6) or (8), the first term 2a With any damage law leading to a positive damage rate tensor, i.e. with positives eigenvalues ( Ḋ D D) J , the term

tr Ḋ D D = 3 J=1 ( Ḋ D D) J (23) is positive so that 1 3 a 3 g ′ (D H )(tr σ σ σ) 2 tr Ḋ D D ≥ 0 (24)
It is important to precise that the eigenvalues ( Ḋ

) : Ḣ H H, note that the expression H H H = (1 1 1 -D D D) -1/2 rewritten in terms of principal components H J = 1 √ 1 -D J (25)
gives positive increasing eigenvalues H J of tensor H H H which is then also positive and increasing during any damage process. The positivity of the symmetric matrix (s s sH H Hs s s) is gained by seeking the sign of its eigenvalues, denoted χ, solution of (s s sH H Hs s s) g = χ g (26)

with g the corresponding eigenvectors. The eigenvalues χ are equivalently solution of (H H Hs s s)

2 g = χ H H H g (27)
with obviously (H H Hs s s) 2 a positive matrix. These eigenvalues take the form

χ = g T (H H Hs s s) 2 g g T H H H g (28)
and, as ratio of positive terms, are positive. Last, the tensorial product (s s sH H Hs s s) : Ḣ H H of two symmetric positive tensors (s s sH H Hs s s) and Ḣ H H being positive, one can conclude to the positivity of the intrinsic dissipation D for any damaging loading, monotonic or not, uniaxial or multiaxial, proportional or non proportional... at the simple condition extended here to anisotropic damage that the damage rate Ḋ D D must remain a positive tensor.

Considering conversely the set of states represented by deviatoric tensors σ σ σ = σ σ σ D , the dissipation reduced to 2a 1 (s s sH H Hs s s) : Ḣ H H ≥ 0 ∀ (s s sH H Hs s s) ≥ 0 0 0 leads to Ḣ H H ≥ 0 0 0, therefore to the fact that Ḋ D D ≥ 0 0 0 is a necessary and sufficient condition for the positivity of the dissipation due to damage

D = Y Y Y : Ḋ D D.

ANISOTROPIC DAMAGE MODEL FOR CONCRETE

An anisotropic damage model has been proposed for concrete in previous non standard thermodynamics framework [START_REF] Desmorat | 5è journéees du Regroupement Francophone pour la Recherche et la Formation sur le Béton (RF2B)[END_REF][START_REF] Desmorat | Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials[END_REF] introducing a single damage variable, a second order tensor, as the representation of the damage state due to microcracking. Mainly due to induced anisotropy, the dissymmetric response of concrete in tension and in compression is obtained with a low number of material parameters (figure 1): 2 for elasticity, 1 as damage threshold, 2 for damage evolution.

Mazars strain damage criterion [START_REF] Mazars | Application de la mécanique de l'endommagement au comportement non linéaire et à la rupture du béton de structure[END_REF] is used in this initial model, with the advantage of simplicity for instance at the numerical level.

-50 -40 -30 -20 -10 0

! (x10 -4 ) -30 -20 -10 0 " (MPa)
Figure 1: Response of the model in tension and compression (E = 37 GPa, ν = 0.2, κ 0 = 5 • 10 -5 , a = 3 • 10 -4 , A = 5 • 10 3 ).

Induced anisotropic damage model

For concrete, the microcracks due to tension are mainly orthogonal to the loading direction, when the microcracks due to compression are mainly parallel to the loading direction. The damage state has then to be represented by a tensorial variable D D D either a fourth rank tensor or a second rank tensor. The use of a second order damage tensor is more convenient for practical applications (as well as for the material parameters identification) and this is the choice made here. The damage anisotropy induced by either tension or compression is simply modeled by the consideration of damage evolution laws ensuring a damage rate proportional to the positive part of the strain tensor, i.e. a damage governed by the principal extensions [START_REF] Mazars | The unilateral behavior of damage concrete[END_REF]. The full set of constitutive equations 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 (29)
• Effective stress ((.) D standing for the deviatoric part),

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 (30)
• Damage criterion f = ǫ -κ(tr D D D), so that the condition f < 0 -→ elastic loading or unloading, f = 0, ḟ = 0 -→ damage growth, where ǫ = ǫ ǫ ǫ + : ǫ ǫ ǫ + is Mazars equivalent strain and where

κ -1 (ǫ) = aA arctan ǫ a -arctan κ 0 a (31)
introducing κ 0 as damage threshold, A and a as damage parameters.

• Induced damage anisotropy governed by the positive extensions,

Ḋ D D = λ ǫ ǫ ǫ 2 + (32)
The damage multiplier λ is determined from the consistency condition f = 0, ḟ = 0.

The use of a damage criterion function f written in terms of strains instead of stresses allows for a simple implementation in a Finite Element computer code [START_REF] Desmorat | Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials[END_REF]. Even if Euler backward scheme is used, there is no need of an iterative process. Note that at the final stage of numerical implementation the elasticity law needs 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 (33)

Extension to nonlocal

Classical mesh dependency occurs when using previous local damage model in a Finite Element code. A nonlocal regularization can be used to gain the mesh independency. One just has to replace local Mazars strain ǫ in the damage criterion f by nonlocal Mazars strain ǫnl and to consider as nonlocal criterion,

f = ǫnl -κ (34)
where nonlocal equivalent strain can be defined using an integral form with W(ξ 2 ) a nonlocal weight function [START_REF] Pijaudier-Cabot | Nonlocal damage theory[END_REF], usually a gaussian function e -4ξ 2 or a bell-shaped function 1 -

ξ 2 2 + , ǫnl (x) = 1 V r Ω W x -s 2 ℓ 2 c ǫ(s) ds V r (x) = Ω W x -s 2 ℓ 2 c ds (35)
or using a second gradient form [START_REF] Aifantis | The physics of plastic deformation[END_REF][START_REF] Peerlings | Gradientenhanced damage model for quasi-brittle materials[END_REF],

ǫnl -γ ∇ 2 ǫnl = ǫ (36)
Both the integral form (through W) and the gradient form (through γ) introduce a characteristic length ℓ c . Nevertheless, and as pointed out by [START_REF] Pijaudier-Cabot | Non-local damage model with evolving internal length[END_REF]) from acoustic emission measurements during the damage process, the internal length is evolving with damage and is therfore probably better represented by a positive fourth order tensor w w w function of D D D instead of the scalar ℓ c . This can be done formally by replacing

W x -s 2 ℓ 2 c by W ((x -s) : w w w(D D D) : (x -s)) (37)
in previous integral equations. The isotropic scalar internal length theory is the particular case w w w = ℓ -2 c I I I. In the same spirit, the gradient term γ ∇ 2 ǫnl may be rewritten ∇(Γ Γ Γ∇ǫ nl ) with Γ Γ Γ = Γ Γ Γ(D D D) an evolving second order tensor.

An other possibility, naturally taking into account the effect of damage (therefore its anisotropy in terms of internal lengths, for second order damage as well as for fourth order damage), is to consider an internal time τ c instead of ℓ c and to replace the argument of the weight function W by the ratio τ xs /τ c , i.e.

W

xs 2 To take into account in finite elements computations an evolving internal length means to actualize at each time step and for each couple (x, s) the value of the weight function. This (very) costly choice is not made here. Even if induced anisotropy is considered next, the introduction of a single (constant) internal length will be considered as sufficient for practical applications.

VISCOUS REGULARIZATION FOR IMPACT

A regularization possibility for fast dynamics and impact consists in taking into account the strain rate effect on the dynamic response of concrete. For instance, introduce a characteristic time which, altogether with the consideration of the laws of dynamics, will indirectly defines a characteristic length. In the present case of elasticity coupled with damage this can simply be done by introducing a viscosity law ǫ v = ǫ v ( Ḋ) in Mazars criterion. The damage evolution occurs not anymore at f = 0 but at f = ǫ v > 0. A classical law for isotropic damage is Norton-Perzyna power law, ǫ v = k Ḋ1/n , with k and n the viscosity parameters. It leads to an unbounded damage rate often too high at high strain rates.

It is possible to bound the damage rate, for instance by the maximum rate Ḋ∞ = 1/τ c material dependent equal to the inverse of the characteristic time τ c [START_REF] Allix | Delay-damage modelling for fracture prediction of laminated composites under dynamic loading[END_REF][START_REF] Ladevèze | Computational damage mechanics approach for laminates: identification and comparison with experimental results[END_REF]. To gain this property, these authors rewrite the criterion surface as f = g(ǫ) -D (with g = κ -1 ) and define the viscosity law as

f = D v > 0 D v = - 1 b ln Ḋ∞ - Ḋ Ḋ∞ ( 39
)
from which derives the delay-damage law, saturating at high strain rates,

Ḋ = Ḋ∞ [1 -exp (-b(g(ǫ) -D))] (40)
The viscosity parameters, material dependent, are then Ḋ∞ and b. This regularization is defined locally (i.e. at a structure Gauss point) and is well adapted for dynamics computations. It can be extended to the case of induced anisotropic damage by setting (Desmorat et al., 2008):

tr Ḋ D D = Ḋ∞ 1 -exp -b κ -1 (ǫ) -tr D D D (41)
The full set of constitutive equations now 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 (42)
• 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 (43)
• Damage criterion (local) f = κ -1 (ǫ) -tr D D D, using the viscous regularization (39),

f ≤ 0 -→ elastic loading or unloading f > 0 with f = - 1 b ln Ḋ∞ -tr Ḋ D D Ḋ∞ -→ damage growth (44)
with Ḋ∞ and b the delay-damage parameters.

• Induced damage anisotropy governed by the positive extensions,

Ḋ D D = λ ǫ ǫ ǫ 2 + (45)
The damage multiplier λ is determined from the damage criterion expression for f > 0 (Eq. 44).

The delay-damage law ( 40) is recovered from previous equations.

Again, the numerical implementation in a Finite Elements computer code is quite simple as -once again -it does not need an iterative process (Desmorat et al., 2008).

STRUCTURAL EXAMPLES

The local and nonlocal integral anisotropic damage model has been implemented in CEA CAST3M Finite Element code. The anisotropic delaydamage model has been implemented in the explicit code LS-Dyna. The numerical scheme for the time integration is for both cases Euler's backward scheme but solved explicitly [START_REF] Desmorat | Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials[END_REF]Desmorat et al., 2008). Finite Element examples on 3D structures are given next to illustrate the model capabilities.

Plain concrete mixed-mode fracture

The double edge notched specimen tested by [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 is a symmetric 200 mm× 200 mm mortar square with two horizontal notches, 30 mm long and 5 mm thick.

The rotation of the external boundary of the plate is restricted around the vertical axis. The concrete specimen is first loaded by an increasing shear (lateral) 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.

The case study is here carried out for a lateral load F M ax = 10 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: 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 is ℓ c = 2 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: a coarse mesh with a total of 4936 elements, a medium mesh with 11294 elements, and a fine mesh with 14766 elements. The characteristic length corresponds then close to the notch to 2 elements of the coarse mesh, to 8 elements of the medium mesh, to 14 elements of the fine mesh.

Figure 2 shows the anisotropic damage patterns computed for the different meshes (at the figure top is the cracking pattern experimentally observed [START_REF] Nooru-Mohamed | Mixed-mode fracture of concrete: An experimental approach[END_REF]). The left column corresponds to the D 22 damage field, the right one to the D 11 damage field, both in nonlocal computations. They exhibit the now classical convergence and mesh independence of the results obtained with a nonlocal model, here in case of anisotropic damage. 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 to the crack patterns observed with the rotation of two main cracks represented. The cracks are not perfectly symmetric with respect to the center of the specimen due to the application of the experimental boundary conditions.

Reinforced concrete structure

The objectives of this section are to evaluate the robustness and 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 3 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]. 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, 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].

As one can see in the D 11 -damage maps given in Figure 4, due to the reinforcing steel bar implying a flexural rupture the effect of the characteristic length is important on the cracking pattern as it can be seen from damage maps. The choice ℓ c = 150 mm is the more appropriate here, as expected. The computation represents quite well the multiple cracks propagation for the characteristic length ℓ c = 150 mm.

3D dynamics tension test

In order to get tensile results at very high strain rates, tensile tests by scabbing were developed [START_REF] Klepaczko | An experimental method for dynamic tensile testing of concrete by spalling[END_REF][START_REF] Schuler | Spall experiments for the measurement of the tensile strength and fracture energy of concrete at high strain rates[END_REF].

Figure 5 shows the principle of the test. The setup consists of a striker (launched at the velocity V), an input bar and the tested specimen. The input bar of [START_REF] Klepaczko | An experimental method for dynamic tensile testing of concrete by spalling[END_REF] experiment has a diameter of 40 mm for a one meter length, while the concrete sample has the same diameter for a length of 120 mm. After the impact of the striker, an incident wave propagates in the input bar. One part of the wave is transmitted into the specimen and another one is reflected at the bar/specimen interface.

The transmitted compression wave is reflected at the free end and becomes a tensile wave. This leads to fracture in the spall plane. Three Finite Element meshes have been used. A coarse mesh is made of 24000 underintegrated 8-nodes brick elements, a medium mesh of 48000 elements and a fine mesh of 96000 elements. At time t = 0 the mesh boundaries are free and the experimental pressure wave is applied on the right face of the specimen. The simulation of the test must make it possible to find the rupture of the sample experimentally observed, i.e. a single main rupture crack at the distance X = 65.8 mm of the impacted face.

Figure 6 shows the damage field D 11 associated with the axial axe e 1 . When the material is subjected to compression the imposed strain is not sufficient to damage the material. To the opposite, when the state of tension becomes sufficiently large after the compressive wave reflection on the free surface, one obtains a damage D 11 close to 1 in a cross section. This cross section, which represents the location in which cracking will be initiated, is located at a distance X ranging between 64 and 69 mm from the impacted face. This measure is in good agreement with experimental one.

In order to illustrate the mesh independency (due to the viscous regularization of the delay-damage model), the results are presented on the coarse, medium and fine meshes. The damage maps obtained for the three meshes are shown in Figure 6. One can notice that the width of the localized damage band is the same for the three meshes and equal to approximately 5 × ℓ c , therefore of the order of magnitude of a characteristic length introduced from the knowledge of the wave celerity c L , ℓ c = c L × τ c = c L /D ∞ . Note that the times steps are different in the three simulations due to the Courant's condition based on the mesh size.

Impact on a reinforced concrete slab

In order to evaluate the ability of the anisotropic damage model to describe the concrete behavior in a case rather complex but representative of an Figure 7 shows the finite element mesh used for the simulations on a 4 meters width and 0.5 meters thick slab. The slab is meshed with 24000 3D underintegrated elements and the reinforcements are represented by 2300 truss-bars. The impacted area has a refined mesh whereas the other part of the slab has a coarse one.

Figure 8 shows the damages D 11 , D 22 and D 33 into the slab. One can notice that due to the symmetry condition, the damage D 11 and D 22 have a similar pattern. The damage D 33 represents the cracks in the slab thickness and is representative of the scabbing phenomenon. In our simulation, damages D 11 and D 22 are quite large exhibiting a shear rupture of the concrete slab with the apparition of a punch cone as experimentally observed in cases of thin slabs (more precisely of small ratio slab thickness / slab stiffness). In the same time, the damage D 33 remains small and does not exhibit scabbing.

CONCLUSION

A non standard thermodynamics framework for induced anisotropic damage guaranties the positivity of the intrinsic dissipation. A proof is given and the scalar feature Ḋ ≥ 0 of a positive damage rate for isotropic modeling is extended to anisotropy as the simple feature Ḋ D D ≥ 0 of a positive damage rate tensor. The convexity of the state potential with respect to the damage variable is not necessary.

An anisotropic damage model is described in the previous non stan- dard framework, with a quite low numbers of material parameters (5 including the elasticity parameters for the initial plus an internal length or an internal time for the nonlocal model or plus 2 viscosity parameters for the delay-damage model). The good mathematical properties of the model (differentiability of the state potential, positivity of the dissipation due to anisotropic damage even for complex non proportional loading) prove to be efficient for numerical computations of 3D structures, in both quasi-static and dynamic cases.

  1 tr H H Hσ σ σ D H H Hσ σ σ D (6) where (.) D denotes the deviatoric part, or a 1 tr (H H Hσ σ σ + H H Hσ σ σ + ) + σ σ σ -: σ σ σ - (7) or a 1 tr H H Hσ σ σ D + H H Hσ σ σ D + + σ σ σ D -: σ σ σ D -

  D) J of Ḋ D D are not the derivatives ḊJ of the eigenvalues of D D D (except in the particular case where D D D and Ḋ D D have the same principal directions), the positivity of the eigenvalues ( Ḋ D D) J nevertheless implies the increase of the eigenvalues of D D D. Concerning the term 2a 1 (s s sH H Hs s s

  τ xs is the time for the deformation information to go from point s to point x (calculated by wave propagation, wave speed affected by damage denoted c). When D D D = 0 this formulation is equivalent to Pijaudier-Cabot and Bazant nonlocal theory: then τ xs = xs /c and τ c = ℓ c /c =Cst is set (c = c(D D D = 0) is the wave speed of the virgin material).

Figure 2 :

 2 Figure 2: Damage maps for Nooru-Mohamed test at U = 3.5 10 -3 mm-(a) left column: D 22 fields, (b) right column: D 11 fields

Figure 3 :

 3 Figure 3: Reinforced concrete beam

Figure 4 :Figure 5 :

 45 Figure 4: Mesh and D 11 damage field obtained at the begining of steels yielding (left : ℓ c = 150 mm, right: ℓ c = 250 mm -left and right correspond to two different computations)

Figure 6 :

 6 Figure 6: Damage in the concrete sample (D 11 field, 1 ≡ horizontal)

Figure 7 :

 7 Figure 7: Finite Element mesh

  Figure 8: Damage map


Anisotropic damage modelling of biaxial behaviour and rupture of concrete structures

This paper deals with damage induced anisotropy modelling for concrete-like materials. A thermodynamics based constitutive relationship is presented coupling anisotropic damage and elasticity. The biaxial behaviour of such a model is analysed through the effects of yield surface modifications by the introduction of new equivalent strains.

INTRODUCTION

Regarding the ultimate behaviour of reinforced and prestressed concrete structures, the prevision of oriented micro-cracks openings is of main importance for the structural integrity. It is even a key issue for multi-physics analyses such as in diffusion problems. In that sense, the mesoscopic approach using Continuum Damage Mechanics at the Representative Element Volume scale is a relevant tool to deal with large scale structures if loading induced damage anisotropy is represented [START_REF] Lemaitre | Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures[END_REF]. It allows to represent the local loss of stiffness of the material and the strain localization zone representative of macroscopic cracks. Within the thermodynamics framework, the state damage variable may be a scalar or tensorial quantity. Dealing with induced anisotropy in cementitious materials and with non-symmetric tension/compression behaviour, the choice for a second order damage tensor [START_REF] Cordebois | Endommagement anisotrope en élasticité et plasticité[END_REF], Murakami, 1988) is a pragmatic approach with regard to robustness and numerical implementations. The 3D effects are not commonly taken into account with damage models. Plasticity is often required to gain some important features as of course permanent strains but also as confined features and responses.

A good approximation of the material response sustaining multiaxial states of stresses is required in Finite Elements, the states of stresses being naturally multiaxial at the structure Gauss points. Important works have then combined damage behaviour for tensile loadings and plasticity-like behaviour -eventually coupled with damage -for compressive or confined loadings [START_REF] Gatuingt | Coupled damage and plasticity modelling in transient dynamic analysis of concrete[END_REF]. The drawback of such approaches is the complexity of the models and of their numerical implementation: one often has to deal with multi-surfaces plasticity-damage modelling, robustness is difficult to ensure and making the models nonlocal also becomes a difficult task. Damage models (with no plasticity) usually do not represent permanent strains, but they represent properly the monotonic softening response of materials, at least under low confinement conditions, conditions often encountered in structural design when tension and shear are the main cause of structural failure. Not modelling plasticity reduces the numbers of materials parameters introduced. It allows also to consider more naturally criterion surfaces in the strains space, choices computationally efficient.

The choice is made in the present work just to address such monotonic failure cases and to see if damage models, anisotropic but with a limited numbers of material parameters, may prove sufficient in this task. Concerning modelling, the tension/compression coupled to shear response usually needs an adequate expression for the equivalent stresses or strains used in the threshold or criterion function. Based on the works of [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] Drucker | Soils mechanics and plastic analysis or limit design[END_REF][START_REF] De Vree | Comparison of nonlocal approaches in continuum damage mechanics[END_REF], different strain based damage criteria are proposed next and extended to nonlocal framework. Their influence both at the material and structural levels is studied when ultimate behaviour and rupture occur. In a first part, the initial thermodynamics damage modelling is recalled allowing for the rigorous expression of constitutive equations. The main numerical features of the new modelling are mentioned in part 2. The third part of the paper allows for dealing with the effects of different expression for the equivalent strains and for the elasticity domains regarding the biaxial behaviour of concrete at the material scale. At last, a structural example is presented.

ANISOTROPIC MODELLING OF CONCRETE

The necessity to account for micro-cracks orientation in the description of the mechanical behaviour of concrete naturally leads to the use of continuum anisotropic damage framework [START_REF] Chaboche | Le concept de contrainte effective appliqué à l'élasticité et à la viscoplasticité en présence d'un endommagement anisotrope[END_REF][START_REF] Cordebois | Endommagement anisotrope en élasticité et plasticité[END_REF][START_REF] Ladevèze | On an anisotropic damage theory[END_REF][START_REF] Chow | An anisotropic theory for continuum damage mechanics[END_REF], Murakami 1988, Dragon & Halm, 1988, Papa & Talercio, 1996[START_REF] Lemaitre | Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures[END_REF][START_REF] Badel | Application of some anisotropic damage model to the prediction of the failure of some complex industrial concrete structures[END_REF]. To be computationally efficient, the expression of constitutive relationships has to be coupled with a proper numerical algorithm allowing dealing with analyses at the structural scale. From the numerical point of view, considering both a hydrostatic / deviatoric splitting and a damage threshold based on an equivalent strain allows to avoid the Gauss point iterative resolution of the constitutive equations and related evolution laws, even if implicitly discretized [START_REF] Desmorat | Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials[END_REF]. The efficiency of such an approach for the use of anisotropic damage in reinforced and pre-stressed concrete has been pointed out treating large scale structures.

The mesh dependency induced by strain softening at the local level is avoided by adopting integral nonlocal type regularization [START_REF] Pijaudier-Cabot | Nonlocal Damage Theory[END_REF]. The transition between an homogeneous state of cracking and a macro-crack propagation is solved by adopting a scalar critical damage value Dc close to unity for which the principal damagesi.e. the eigenvalues of the damage tensor -can no more increase in the direction where this critical value is reached.

Elasticity coupled with anisotropic damage

Modelling crack initiation and growth at the representative volume element scale within a macroscopic phenomenological framework needs the introduction of a thermodynamic variable. A choice has to be made concerning the damage kinematics, between scalar or tensorial representations. The easiest one consists in using a scalar damage variable, representing an isotropic state of concrete degradation [START_REF] Mazars | Application de la mécanique de l'endommagement au comportement non linéaire et à la rupture du béton de structure[END_REF]. This approach allows for the expression of efficient material models dealing with robust stress integration algorithms. Large scale computation can be handled nevertheless without the information concerning the cracks orientation. If the description of the crack orientation is a major point, an anisotropic kinematics for the damage variable has to be introduced. Damage affecting directly the elasticity law, the most natural approach could use a fourth order tensor [START_REF] Chaboche | Le concept de contrainte effective appliqué à l'élasticité et à la viscoplasticité en présence d'un endommagement anisotrope[END_REF][START_REF] Leckie | Tensorial nature of damage measuring internal variables[END_REF]. It is possible to express a new thermodynamic variable or to directly use the elasticity operator as a variable being able to degrade [START_REF] Krajcinovic | Continuous damage mechanics revisited: basic concepts and definitions[END_REF][START_REF] Simo | Strain and stress based continuum damage models--I formulation[END_REF][START_REF] Ju | On energy-based coupled elastoplastic damage theories: constitutive modelling and computational aspects[END_REF][START_REF] Govindjee | Anisotropic modelling and numerical simulations of brittle damage in concrete[END_REF][START_REF] Meschke | An anisotropic elastoplastic-damage model for plain concrete[END_REF].

In such a case, this framework encounters strong limitations due to the difficulty in proceeding to robust numerical integration within a finite element code and to deal with adequate identification for the large number of material parameters involved.

An alternative approach consists in using a symmetric orthotropic second order damage tensor as thermodynamic variable [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], Chaw & Wang 1987, Murakami 1988). The resulting elasticity operator has also to be symmetric, depending on the strain, stress or energy used in the expression of effective stress (the real stress acting on the sane part of the material). Difficulty remains in dealing with effective stress, independent from material parameters (Lemaitre 2002[START_REF] Lemaitre | Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures[END_REF], or to account for complete stiffness recovery when passing from tension to compression [START_REF] Ladevèze | On an anisotropic damage theory[END_REF]).

In the present study, only partial stress recovery is introduced. A single but tensorial damage variable is used in describing the non-symmetric behaviour of concrete in tension and in compression. According to the general expression [START_REF] Ladevèze | On an anisotropic damage theory[END_REF][START_REF] Lemaitre | Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures[END_REF] insuring continuity of the stress-strain path whatever the loadings, the thermodynamics potential takes the following form

( )( ) [ ] ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + - - + - - + = - + - - 2 2 2 / 1 2 / 1 * 1 6 2 1 2 1 σ σ σ σ Tr Tr Tr E Tr E D D D D 1 D 1 ν ν ρψ (1)
where E and ν are the elasticity parameters, ρ the density, σ the Cauchy stress and D the damage tensor. The notation () D

. indicates the deviatoric part of a tensor and ( )

2 / 1 - -D 1 is gained from diagonal form of ( ) D 1 - as ( ) ( ) P D 1 P D 1 - = - -1 diag and ( ) ( ) -1 2 / 1 P D 1 P D 1 -1/2 diag - = --
. The notation ) , 0 max( x x = + stands for the positive part of the scalar x and ) , 0 min( x x = -stands for its negative part.

The state laws are obtained by derivation of the potential with respect to the thermodynamics variables. The elasticity law coupled with anisotropic damage reads:

( )( ) [ ] 1 D D 1 D 1 ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + - - + - - + = = - + - - σ σ σ σ ε Tr Tr Tr E E D D 1 3 2 1 1 2 / 1 2 / 1 * ν ν ∂ ∂ψ ρ (2)
And the strain energy release rate density Y (the thermodynamics force associated with D ) is:

D Y ∂ ∂ψ ρ * = .
The decoupling between the deviatoric member of the stress-strain relation and its volumic part induces only a partial stiffness recovery sufficient for monotonic applications. In compression, damage does not affect the hydrostatic response with a bulk modulus

) 2 1 ( 3 / ν - = E K
. In tension the damaged bulk modulus is

( )K Tr K D - = 1 ~.
Previous elasticity law rewritten ε σ

: E = ~
with E the undamaged Hooke tensor defines analytically the relationship between the Cauchy stress and the effective stress σ ~,

( )( ) [ ] 1 D D 1 D 1 ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - - - + - - = + + - - σ σ σ σ Tr Tr Tr D D 1 3 1 ~2 / 1 2 / 1
(3) so that the elasticity laws simply sums up as :

1 σ σ σ Ε ε 1 : 1 Tr E E ν ν - + = = - (4)

Damage threshold and evolution laws

Damage evolution is linked to the violation or not of a criterion. Depending on the materials, the damage criterion may be expressed, similarly to plasticity, thanks to the stresses [START_REF] Ortiz | A constitutive theory for the inelastic behavior of concrete[END_REF][START_REF] Warnke | Constitutive model for triaxial behaviour of concrete[END_REF][START_REF] Voyiadjis | Plasticity model for concrete using the bouding surface model[END_REF], using the 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, in Cracking and Damage[END_REF][START_REF] Ramtani | Contribution à la modélisation du comportement multiaxial du béton endommagé avec description du caractère unilatéral[END_REF][START_REF] De Vree | Comparison of nonlocal approaches in continuum damage mechanics[END_REF][START_REF] Geers | Damage and crack modelling in singleedge and double-edge notched concrete beams[END_REF] or using energy quantities like the damage energy release rate [START_REF] Marigo | Formulation d'une loi d'endommagement d'un matériau élastique[END_REF][START_REF] Laborderie | Crack closure effect in continuum damage mechanics: numerical implementation[END_REF]).

Most of the finite elements codes use displacements based interpolation functions. The most efficient way to express constitutive equations arguing for explicit numerical integration is to make use of a damage threshold based on the strains. For brittle materials like concrete, Mazars's criterion (1984) defining an equivalent strain is used for the anisotropic evolution of damage. The equivalent strain ε ) is constructed selecting the positive principal strain I ε :

∑ + + + = = 2 : ˆI ε ε ε ε (5)
The damage threshold takes the simple form:

( ) 0 ˆ≤ - = D tr f κ ε (6) ( ) D tr κ
is the consolidation function in the strains space, depending on the trace of the damage tensor. The initial value defines the damage threshold ( )

0 0 κ κ = .
The advantages are multiple. On the one hand, the use of equivalent quantities based on strains will make easier the explicit derivation of the subsequent numerical scheme for stress integration. On the other hand, dealing only with one scalar equivalent quantity based on strains, the classical drawbacks of spurious mesh dependency when using softening constitutive equations [START_REF] Bazant | Instability, ductility and size effect in strain-softening concrete[END_REF]

( ) ∫ - = V r nl dv x s s x V x ψ ε ε ) ( ) ( 1 ) ( ˆ with ( ) ∫ - = V r dv x s V ψ and ( ) ) / 2 exp( 2 2 c l x s x s - - = - ψ (7)
Only one damage tensorial variable is used to represent the micro-cracks pattern in concrete.

The non-symmetric tension/compression response is obtained thanks to the anisotropic feature of damage. The damage evolution in one direction is guided by the level of extension in this direction. Then the damage can be considered proportional to the positive part of the strain tensor

+ ε or to 2 + ε . The power 2 is used next as 2 + ∝ ε Tr TrD & simplifies in 2 ε ∝ D & Tr .
Within the thermodynamics framework, the corresponding choice for the non associated potential is:

2 : + = ε Y F (8)
The damage evolution law is obtained by derivation with regard to the strain energy release rate density:

2 + = ∂ ∂ = ε λ λ & & & Y D F (9)
The

Uniaxial responses

The responses of the model, subject to uniaxial states of stresses are presented in the next figures. The material parameters used in the analysis are E = 42 GPa, ν = 0.2, κ 0 = 5 10 -5 , A = 5 10 3 , a = 2.93 10 -4 .

One can observe that with only one thermodynamic variable (the damage tensor), the model is able to correctly describe the non symmetric uniaxial behaviour of concrete. Due to the decoupling between the hydrostatic part and the volumic part of the model, the volumic strain of the response is not affected in compression. Even if dilatancy is not represented, this is an improvement compared to the loss of bulk modulus obtained with isotropic damage models.

NUMERICAL IMPLEMENTATION

Euler backward scheme for numerical stresses and damage computations

One major advantage of this model is its easy numerical implementation within any finite element code and its corresponding robustness regarding computations at the structural scale. In fact, any evolution equation can be analytically (explicitly) calculated, even when using an implicit discretisation scheme. No internal iteration, very CPU time consuming, are needed at Gauss point level. The principal steps for stress computation are recall hereafter.

Knowing all the variables and stress at time n t as well as the final state of strain at time 1

+ n t , 1 + n
ε , the stress-damage algorithm aims at computing the internal variable (the damage tensor

+ + + + + + = = 1 1 1 1 , : ˆn n n n eq ε ε ε ε or others..)
If the elasticity criterion is not violated, i.e. 0 < f , the damage does not evolve (

n n D D = +1
) the stress increment is obtained thanks to a reversible elastic change of state (steps 6 and 7).

On the contrary, the internal variable has to be corrected, according to the nonlinear constitutive equations and the consolidation function.

2. Calculate the non local equivalent strain if regularisation procedure is adopted (eq. 7).

3. Discretize the damage evolution as : A critical value c D for damage is introduced, allowing defining the numerical transition between a micro-cracked medium range and the occurrence of a macroscopic crack, for which continuum damage mechanics looses sense and consistency, benefiting to nonlinear fracture mechanics or extended finite element formulation. Both handle strong discontinuities in the material [START_REF] Belytschko | Elastic crack growth in fininite elements with minimal remeshing[END_REF][START_REF] Jirasek | Comparative study on elements with embedded discontinuities[END_REF]. In first approximations for concrete, c D is taken equal to 0.99.

2 1 1 + + + ∆ = - = ∆
) ( ) ( ) ( ) ( ) [ ] 1 D D 1 D D 1 D 1 D 1 - + + + + + + + + + + + + + - + - - - - - - = 1 1 1 1 1 1 1 2 / 1 1 1 2 / 1 1 1 1 3 1 3 : ~n n n n n n n n n
Concerning the hydrostatic part of the constitutive equations, the limitations at high state of damage appears clearly when observing the reversible process equations (from eq (2)).

( )

K Tr Tr K Tr Tr 3 1 3 - + + - = σ σ ε D (11)
To keep positive the damaged bulk modulus, it is necessary to only limit in tensile loadings the evolution of the trace of D to c D . In such a way that, when trD reaches c D , the bulk modulus takes the critical value of

( )K D K c - = 1 ~ when 0 > ε Tr .
For the deviatoric case, to ensure that the damaged elasticity operator remains positive definite, one only has to impose that the eigenvalues of the second order damage tensor are bounded by 1, or by c D from a numerical point of view [START_REF] Lemaitre | Anisotropic damage law of evolution[END_REF][START_REF] Badel | Contributions à la simulation mécanique de structures en béton armé[END_REF]).

Under these conditions, the general evolution equations of the previous section have to be [ ]

- + + - + - = ε ε ε σ Tr Tr D K D G c D c ) 1 ( ) 1 ( 2 (13) with ) 1 ( 2 / ν + = E G
, the shear modulus.

BIAXIAL BEHAVIOUR AND EQUIVALENT STRAIN

The biaxial behaviour of brittle materials like concrete has to be handled with care, due to the complexity of the degradation modes and crack pattern [START_REF] Kupfer | Behaviour of concrete under biaxial stresses[END_REF]. The suitable representation of biaxial rupture needs to possess an adequate elasticity limit as well as evolution equations able to deal with confinement effects [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] Ramtani | Contribution à la modélisation du comportement multiaxial du béton endommagé avec description du caractère unilatéral[END_REF]).

This section is devoted to evaluate the biaxial response of the previous model using different equivalent strains. Although being within an anisotropic framework, such an analysis can easily be performed due to the high level of modularity of the constitutive model based on the use on equivalent strains.

Biaxial responses for an isotropic model and for the anisotropic model

The constitutive models based on continuum damage mechanics, isotropic or anisotropic, exhibit a high level of brittleness in biaxial compression. If the state of stress at rupture is plotted in the ( )

22
11 ,σ σ plane, one can observe a decrease of the load bearing capacity in biaxial compression (with regards to the uniaxial response in compression, rupt σ ) while the experimental results show an increase of the stress at rupture of about 20 % [START_REF] Kupfer | Behaviour of concrete under biaxial stresses[END_REF][START_REF] Van Mier | Strain-softening of concrete under multiaxial loading conditions[END_REF]. In figure 2a, one shows the elasticity limit, using the Mazars equivalent strain as well as the rupture envelop for the isotropic [START_REF] Mazars | Application de la mécanique de l'endommagement au comportement non linéaire et à la rupture du béton de structure[END_REF] damage model. [START_REF] Kupfer | Behaviour of concrete under biaxial stresses[END_REF][START_REF] Van Mier | Strain-softening of concrete under multiaxial loading conditions[END_REF] experimental data are drawn.

In figure 2b, the brittleness increase is even more clearly emphasized. The effects of damage in the non convexity of the rupture limit are even more pronounced for the anisotropic model, by localizing the crack in a unique plane, orthogonal to the loading plane. The lack of concavity in biaxial compression of the initial elasticity domain (flat end with Mazars's criterions) is the major source for this bad representation of biaxial responses. A possible remedy for compressive loadings is of course the consideration of both plasticity and damage mechanisms but the corresponding models becomes quite complex [START_REF] Meschke | An anisotropic elastoplastic-damage model for plain concrete[END_REF][START_REF] Ragueneau | Damage Model for Concrete Like Materials Coupling Cracking and Friction, Contribution towards Structural Damping: First Uniaxial Application[END_REF][START_REF] Huerta | An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with isotropic damage model[END_REF]) A second remedy, much simpler in terms of numerical efforts, consists in keeping the elasticity coupled to damage but in increasing the elasticity non-symmetry by adding new invariants in the expression of the equivalent strain.

Modification of the equivalent strain

Different solutions may be adopted to improve the biaxial responses of the models subject to complex state of stress. By only changing the definition and expression of the elasticity domain, the numerical integration is kept unchanged as well as the nonlocal formulation allowing proceeding to structural case study without mesh dependency. Different formulations can be adopted for concrete, based on the original expression of the Mazars equivalent strain

+ + = ε ε : ε
completed by terms function of the strain invariants :

) ( ) ( 1 1 ε ε Tr I I = = and ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - = = 2 1 2 2 3 1 : 2 1 ) ( I J J ε ε ε .
• Mazars-Drucker-Prager : Adding of the trace of the strain tensor. A complementary material parameter k has to be identified. Although it greatly improves the elasticity domain in bi-compression, the main disadvantage of this expression is to modify in the same way the tensile response making necessary a complete reidentification of all the material parameters.

) ( ) ( ) ( ) 2 2 2 1 2 2 1 1 12 2 1 1 2 1 2 1 2 1 J k I k k I k k eq ν ν ν ε + + - - + - - = (17)
Such a difference in the biaxial behaviour is more relevant when looking at the shear response of the different models. The isotropic Mazars model in shear is compared to the anisotropic one using the previous different expressions for the equivalent strain and elasticity surface. The shear behaviour of concrete, due to aggregate interlocks, roughness and frictional sliding along the crack surfaces should be much more ductile than the response in pure tension. This point is illustrated in figure 5, especially for the de Vree surface.

NOORU-MOHAMED'S STRUCTURAL CASE-STUDY

The classical experiment of Nooru-Hohamed (1992) is used to analyse, at the structural level, the effects of the changes in the expression of the equivalent strain. Due to the interaction between tension and shear following a non proportional path, this test is relevant for comparison in our case.

The specimen geometry and the experimental testing set up are shown in Fig. 6. It is a symmetric 200 mm • 200 mm mortar square with two notches, 30 mm long and 5 mm thick.

The case study is here carried out for a maximum shear load FMax = 10 kN exhibiting mixed mode fracure. The 3D Finite Element 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 mesh, the boundary conditions as well as the loading specifications are presented in the figure 6.

The model parameters used for the simulation are those of section 2 for concrete: E = 42000

MPa, ν = 0.2, 0 κ = 5.10 -5 , A = 5.10 3 , a = 2.9310 -4 .

In order to avoid any spurious mesh dependency due to strain localisation, nonlocal computations have been performed on a medium mesh. The characteristic length is set to 2 mm for all the analyse. The convergence of the computations has been numerically shown using three types of mesh in a previous work [START_REF] Desmorat | Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials[END_REF] One can observe that the crack path is well suited by the numerical computations with the non-symmetric boundary conditions of figure 6. The difference between the two criteria is emphasized in figure 8, for which the results obtained with the two different equivalent strains are plotted in the tensile load -vertical displacement diagram. The ductility as well as the peak displacement is better simulated using the de Vree equivalent strain. Although it allows for a better response modelling in the biaxial regime, the de Vree criterion overestimates the shear response. This point may explain the overload registered in the numerical computations in figure 8.

CONCLUSIONS

  are simply numerically avoided by implementing the law within the framework of nonlocal media through the use of a nonlocal weight function ψ (Pijaudier-Cabot & Bazant 1987). It only needs the identification of a new parameter c l , which is the internal length of the nonlocal medium. This length can be linked to the size of the heterogeneities of the material or to the resulting scale effects when dealing with structural analyses. A equivalent strain εˆ is the made nonlocal as

  damage multiplier λ & is determined from the consistency condition 0 identification of the behaviour of different concretes is performed by the definition of the consolidation function ( ) D tr κ . For concrete, a simple expression has been found allowing to fit the concrete responses in tension and in compression, introducing only two material parameters a and A in addition to the Young's modulus E, the Poisson's ratio ν and the damage threshold 0 κ .

  the Cauchy stresses by inverting eq (3).

(

  control procedureDescribing rupture of structural elements needs to deal with damage constitutive equations reaching the ultimate state of damage (eigenvalues close to unity). For isotropic damage, respecting the Clausius-Duhem inequality, it is quite clear that the maximum value of the damage variable should not exceed one. Within an anisotropic framework with partial stiffness recovery, such a criterion is no more so simple. The tensorial damage acts by its individual values on the deviatoric part of the behaviour and it acts by its trace on the volumic part. It is obvious to consider two different treatments for the damage evolution law, depending on which part of the behaviour is treated. Such a treatment simply just ensures the property of a positive effective damaged elasticity tensor.

Π

  adapted. If the maximum eigenvalue of damage I D reaches its critical value in the direction I n , damage growth in that direction is stopped, defining a first plane of fixed crack in the solid. Damage only goes on growing in the remaining ( ) damage evolution law is kept unchanged by only considering the terms of the strain tensor in the ( ) the projection operator. If the loading continues, a second direction for which damage reaches its critical value is detected, defining in the same way, orthogonally to the two first directions, the third one, and so the eigen base of the damage operator for the cracked medium. Only three families of cracks are introduced at the final stage and the fully broken behaviour is an elastic one (with different bulk modulus in tension and in compression),

•

  Modified Mazars-Drucker-Prager: Adding of the trace of the negative part of the strain tensor, without modifying the response in tension. This non-convex surface may generate, depending on the material parameters choice, instability of the mechanical response.

•

  Mazars-Mises-Drucker-Prager : : Adding of the trace of the strain tensor and of the second invariant of the deviatoric part of the strain tensor. This equivalent strain defines a convex elasticity domain making easier the constitutive equation parameters identification.

•

  De Vree criterion: a non-centered von Mises criterion in the strain space (de Vree et al. 1995). The ellipsoïdal shape, far from the previous one allows for a direct control of the non-symmetric behaviour of concrete through the introduction of the material parameter k.

(

  . Two types of computations are performed with the two models giving the most different responses in bicompression and in shear. The first one uses the original anisotropic model, i.e. with the damage threshold based on Mazars equivalent strain. The second one uses the de Vree equivalent strain. The results are given in the following using the medium mesh. The comparisons between the experimental crack pattern and the anisotropic D 11 and D 22 damage fields are plotted in the figure 7. Whatever the criterion, the damage fields obtained are quite close so that only the results obtained with the original Mazars equivalent strain are plotted in figure 7.

  Fig 1 : Uniaxial response of the anisotropic damage model. (a) Tension/compression non symmetric feature and (b) volumic behaviour in compression ( ε Tr V = ε

Fig 2 .

 2 Fig 2. Elasticity domain and ultimate state at rupture for (a) the isotropic model (Mazars 1984) and (b) the initial anisotropic model using the Mazars equivalent strain.

Fig 4 .

 4 Fig 4. Elasticity domain and ultimate state at rupture for (a) the anisotropic damage model using Mazars-Mises-Drucker-Prager equivalent strain and (b) the de Vree Criterion.

Fig 6 .

 6 Fig 6. Nooru-Mohamed test. Mesh, boundary conditions and loading history

Fig 8 .Fig 2 .Fig 3 .Fig 4 .Fig 5 .

 82345 Fig 8. Nooru-Mohamed test. Comparisons of two types of damage threshold during the loading.

  Fig 6. Nooru-Mohamed test. Mesh, boundary conditions and loading history