F Delduc

E Ivanov
email: eivanov@theor.jinr.ru

New Model of N =8 Superconformal Mechanics

Keywords: 11.30.Pb, 11.15.-q, 11.10.Kk, 03.65.-w Keywords: Supersymmetry, mechanics, superfield

. This action is found to be invariant under the exceptional N =8, d=1 superconformal group F (4) with the R-symmetry subgroup SO(7). The general N =8 supersymmetric (1, 8, 7) action is a sum of the superconformal action and the previously known free bilinear action. We show that the general action is also superconformal, but with respect to redefined superfield transformation laws. The scalar potential can be generated by two Fayet-Iliopoulos N =4 superfield terms which preserve N =8 supersymmetry but break the superconformal and SO(7) symmetries.

Introduction

Superconformal mechanics (SCM) plays an important role in a wide circle of intertwining problems related to black holes, AdS 2 /CFT 1 , super Calogero-Moser models, branes, etc [1]- [14]. While the N =2 and N =4 SCM models were constructed long ago [1]- [10], much less is known about the higher N >4 cases, in particular, the N =8 one. This is related to the fact that the number of admissible non-equivalent d=1 superconformal groups is growing with N . For instance, there is only one choice for the N =2 superconformal group, SU (1, 1|1) ∼ OSp(2|2), whereas there are three possibilities, D(2, 1; α), OSp(4 ⋆ |2) and SU (1, 1|2), in the N =4 case 1 and four in the N =8 case: OSp(8|2) , OSp(4 ⋆ |4) , F (4) and SU (1, 1|4) [START_REF] Van Proeyen | Tools for supersymmetry[END_REF][START_REF] Frappat | Dictionary on Lie superalgebras[END_REF].

The N =8 superconformal actions of the off-shell multiplets (5,8,3) and (3,8,5) were explicitly given in [14] in terms of the properly constrained N =4, d=1 superfields. It was found that in both cases the underlying superconformal symmetry is OSp(4 ⋆ |4) . It is interesting to construct superconformal models associated with other N =8, d=1 superconformal groups. An example of such a system is presented here. It is the N =8 supersymmetric mechanics model associated with the off-shell multiplet (1,8,7) from the list of [START_REF] Bellucci | [END_REF]. The underlying N =8 superconformal symmetry is the exceptional supergroup F (4) with the R-symmetry subgroup SO (7). In the N =4 superfield approach which we use throughout the paper, the "manifest" superconformal group is D(2, 1; -1/3) ⊂ F (4) .

In terms of N =4 superfields, the multiplet in question amounts to a sum (1, 8, 7) = (1, 4, 3) ⊕ (0, 4, 4).

(1.1)

While for the multiplet (1, 4, 3) one can write manifestly N =4 supersymmetric actions in the ordinary N =4 superspace [3,18], actions of the fermionic N =4 multiplet (0, 4, 4) are naturally written in the analytic harmonic N =4 superspace [19,14,[START_REF] Bellucci | [END_REF]. In order to construct the N =8 supersymmetric actions of the multiplet (1, 8, 7) we use the N =4, d=1 harmonic superspace description for both N =4 multiplets in (1.1). A sum of the free (1, 8, 7) action [START_REF] Bellucci | [END_REF] and the superconformal action constructed here yields the most general (1, 8, 7) action. We present the relevant component off-shell action and show that it agrees with that found in [20] by a different method. Surprisingly, the general action is also superconformal, though with respect to redefined superfield transformation laws. For both N =4 multiplets one can construct N =8 supersymmetric Fayet-Iliopoulos terms which, however, break superconformal symmetry.

2 Preliminaries: N =4, d=1 harmonic superspace

The harmonic analytic N =4 superspace [START_REF] Galperin | [END_REF]19,[START_REF] Delduc | [END_REF]23] is parametrized by the coordinates

(ζ, u) = (t A , θ + , θ+ , u ± i ) , u +i u - i = 1 . (2.2)
They are related to the standard N =4 superspace (central basis) coordinates z = (t, θ i , θi ) as

t A = t -i(θ + θ-+ θ -θ+ ), θ ± = θ i u ± i , θ± = θi u ± i . (2.
3)

The N =4 covariant spinor derivatives and their harmonic projections are defined by

D i = ∂ ∂θ i + i θi ∂ t , Di = ∂ ∂ θi + iθ i ∂ t , (D i ) = -Di , {D i , Dk } = 2i δ i k ∂ t , (2.4)
D ± = u ± i D i , D± = u ± i Di , {D + , D-} = -{D -, D+ } = 2i ∂ t A .
(2.5)

1 Actually, the supergroups OSp(4 ⋆ |2) and SU (1, 1|2) can be treated as special cases of D(2, 1; α) .

1

In the analytic basis, the derivatives D + and D+ are short,

D + = ∂ ∂θ -, D+ = - ∂ ∂ θ-.
(2.6)

The analyticity-preserving harmonic derivative D ++ and its conjugate D --are given by

D ++ = ∂ ++ -2iθ + θ+ ∂ t A + θ + ∂ ∂θ -+ θ+ ∂ ∂ θ-, D --= ∂ ---2iθ -θ-∂ t A + θ -∂ ∂θ + + θ-∂ ∂ θ+ , ∂ ±± = u ± i ∂ ∂u ∓ i , (2.7)
and become the pure partial derivatives ∂ ±± in the central basis. They satisfy the relations

[D ++ , D --] = D 0 , [D 0 , D ±± ] = ±2D ±± , (2.8)
where D 0 is the operator counting external harmonic U (1) charges. The integration measures in the full harmonic superspace (HSS) and its analytic subspace are defined as

µ H = dudtd 4 θ = dudt A (D -D-)(D + D+ ) = µ (-2) A (D + D+ ) , µ (-2) A = dudζ (-2) = dudt A dθ + d θ+ = dudt A (D -D-) .
(2.9)

3 The multiplets (1,4,3) and (0, 4,

The off-shell multiplet (1, 4, 3) is described by a real N =4 superfield v(z) obeying the constraints [3]

D i D i v = Di Di v = 0 , [D i , Di ]v = 0 . (3.1)
The same constraints in HSS read [17]

D ++ v = 0 , D + D -v = D+ D-v = 0 , D + D-+ D+ D -v = 0 . (3.2)
The extra harmonic constraint guarantees the harmonic independence of v in the central basis.

Recently, it was shown [23] that this multiplet can be also described in terms of the real analytic gauge superfield V(ζ, u) subjected to the abelian gauge transformation

V ⇒ V ′ = V + D ++ Λ --, Λ --= Λ --(ζ, u) . (3.3)
In the Wess-Zumino gauge just the irreducible (1, 4, 3) content remains

V W Z (ζ, u) = x(t A ) + θ + ω i (t A )u - i + θ+ ωi (t A )u - i + 3iθ + θ+ A (ik) (t A )u - i u - k . (3.4) No residual gauge freedom is left. The original superfield v(z) is related to V(ζ, u) by v(t, θ i , θk ) = du V t -2iθ i θk u + (i u - k) , θ i u + i , θk u + k , u ± l . (3.5)
The constraints (3.1) are recovered as a consequence of the harmonic analyticity of V

D + V = D+ V = 0 . (3.6) 2
We shall need a "bridge" representation of V through the superfields v(z) and V --(z, u)

V = v + D ++ V --, v ′ = v , V --′ = V --+ Λ --. (3.7)
The term v(z) is just given by the expression (3.5). The analyticity conditions (3.6) imply

D -v + D + V --= D-v + D+ V --= 0 . (3.8)
Below are some useful corollaries of (3.8), (3.6) and (2.5)

D + D--D+ D -v = -2D + D+ V --, (3.9)
D + D+ V --= iD ++ V --+ i 2 D -D-v -i V , (3.10)
D + V --+ i 2 D -D-v = D+ V --+ i 2 D -D-v = 0 . (3.11)
The general invariant action of the multiplet (1, 4, 3) reads

S (v) gen = dtd 4 θ L gen (v) . (3.12)
The free action corresponds to the quadratic Lagrangian

S (v) f ree = -dtd 4 θ v 2 , (3.13)
while the action invariant under the most general N =4, d=1 superconformal group

D(2, 1; α) (except for the special values of α=0, -1) is [9] S (v) sc = -dtd 4 θ (v) -1 α , (3.14)
where, for the correct d=1 field theory interpretation, one must assume that v develops a nonzero background value, v = 1 + . . . . The transformation properties of some relevant objects under the conformal N =4 supersymmetry ⊂ D(2, 1; α) are as follows [19,23] 2

δ sc D ++ = -Λ ++ sc D 0 , δ sc D 0 = 0 , (3.15)
δ sc µ (-2) A = 0 δ sc µ H = µ H 2Λ sc - 1 + α α Λ 0 , δ sc (dtd 4 θ) = - 1 α (dtd 4 θ) Λ 0 , δ sc du = du D --Λ ++ sc (3.16
)

δ sc v = -Λ 0 v , δ sc V = -2Λ sc V , (3.17)
where

Λ ++ sc = 2iα(ε + θ + -ε + θ+ ) ≡ D ++ Λ sc , Λ sc = 2iα(ε -θ + -ε -θ+ ) , (D ++ ) 2 Λ sc = 0 , (3.18) Λ 0 = 2Λ sc -D --Λ ++ sc = 2iα θ i εi + θi ε i , D ++ Λ 0 = 0 , (3.19)
and ε ± = ε i u ± i , ε± = εi u ± i , ε i , εi being mutually conjugated Grassmann transformation parameters. Using (3.16), (3.17) and (3.18), (3.19), it is easy to check the D(2, 1; α) invariance of the action (3.14) and the covariance of the relation (3.5).

One can also construct an N =4 supersymmetric Fayet-Iliopoulos (FI) term

S (v) F I = i dudζ (-2) c +2 V c +2 = c ik u + i u + k , [c] = cm -1 , (3.20)
which produces a scalar potential after elimination of the auxiliary field A (ik) in the sum of (3.12) and (3.20). This term is superconformal only for the special choice α=0 [23].

3.2 (0, 4, 4)

The multiplet (0, 4, 4) comprises 4 fermionic fields and 4 bosonic auxiliary fields. It is described off shell by the fermionic analytic superfield Ψ +A , (Ψ +A ) = Ψ + A , obeying the constraint [19]:

D ++ Ψ +A = 0 ⇒ Ψ +A = ψ iA u + i + θ + a A + θ+ āA + 2iθ + θ+ ψiA u - i . (3.21)
With respect to the doublet index A (A = 1, 2), it is transformed by some extra ("Pauli-Gürsey") group SU (2) P G which commutes with N =4 supersymmetry. The requirement of superconformal covariance of the constraint (3.21) uniquely fixes the superconformal D(2, 1; α) transformation rule of Ψ +A , for any α, as

δ sc Ψ +A = Λ sc Ψ +A . (3.22)
In the central basis, the constraint (3.21) implies

Ψ +A (z, u) = Ψ iA (z)u + i , (3.23)
and the analyticity conditions D + Ψ +A = D+ Ψ +A = 0 amount to

D (i Ψ k)A (z) = D(i Ψ k)A (z) = 0 . (3.24)
The free action of Ψ +A , S

f ree = dudζ (-2) Ψ +A Ψ + A , (ψ)
is not invariant under D(2, 1; α) (except for the special case of α=0). However, we can construct a superconformal invariant by coupling Ψ +A to the (1, 4, 3) multiplet [23]

S (ψ) (sc) = dudζ (-2) V Ψ +A Ψ + A . (3.26)
This action is superconformal at any α, and it also respects the gauge invariance (3.3) as a consequence of the constraint (3.21). Assuming that V = 1 + V, v = 1 + v, (3.26) can be treated as a superconformal generalization of the free action (3.25). A simple analysis based on dimensionality and on the Grassmann character of the superfields Ψ +A , Ψ -A = D --Ψ +A shows that no self-interaction of the multiplet (0, 4, 4) can be constructed. Also, the coupling (3.26) is the only possible coupling of this fermionic multiplet to the multiplet (1, 4, 3) preserving the canonical number of fields with time derivative in the component action (no more than two for bosons and no more than one for fermions).

The only additional N =4 invariant one can construct is the appropriate FI-type term S (ψ)

F I = dudζ (-2) θ + ξ A Ψ +A + θ+ ξA Ψ + A , (3.27)
ξ A , ξA being SU (2) P G breaking constants. This term is superconformal at α= -1 [23].

N =8 supersymmetry

As shown in [START_REF] Bellucci | [END_REF], one can define the hidden N =4 supersymmetry3

δ η v = -η iA Ψ iA , δ η Ψ iA = 1 2 η A k D i Dk -Di D k v . (4.1)
It commutes with the explicit N =4 supersymmetry and so forms, together with the latter, N =8, d=1 Poincaré supersymmetry. It is easy to check the compatibility of (4.1) with the constraints (3.1), (3.24). The same transformations, being rewritten in HSS, read

δ η v = η -A Ψ + A -η +A Ψ - A , δ η Ψ +A = η -A D + D+ v - 1 2 η +A D + D--D+ D -v , (4.2)
where η ±A = η iA u ± i . The appropriate transformation of the analytic prepotential V is

δ η V(ζ, u) = 2η -A Ψ + A (ζ, u) . (4.3)
As expected, (4.3) closes on the time derivative of V only modulo a gauge transformation:

[δ η , δ η ′ ]V = 2i(η ′ iA η iA ) V -D ++ Λ --, Λ --= V --+ i 2 D -D-v , (4.4)
where we used the identities (3.9)-(3.11) and anticommutation relations (2.5). Now we wish to construct the most general action of the multiplets (1, 4, 3) and (0, 4, 4) which would enjoy the hidden supersymmetry (4.1), (4.2) and so present the N =4 superfield form of the general N =8 supersymmetric action of the multiplet (1,8,7) .

A convenient starting point of such a construction is offered by the Ψ-actions (3.25) and (3.26) in view of their uniqueness. The N =8 completion of the free action (3.25) was found in [START_REF] Bellucci | [END_REF]. The variation of (3.25) under (4.2) can be written as

δ η S (ψ) f ree = 4 µ (-2) A D + D+ v (η -A Ψ + A ) = 4 µ H v(η -A Ψ + A ) = -2 dtd 4 θ v(η iA Ψ iA ) , (4.5)
where we used the relation (2.9), constraint (3.21) and the harmonic independence of v in the central basis. This variation is cancelled by that of the free v action (3.13), so the action

S (N =8) f ree = 1 2 µ (-2) A Ψ +A Ψ + A -dtd 4 θ v 2 (4.6)
is N =8 supersymmetric. It breaks superconformal symmetry, since its first term is invariant under D(2, 1; α= -1/2) (see (3.14)), while the second one is invariant under D(2, 1; α=0) . Now let us promote the interaction action (3.26) to an N =8 invariant. To calculate the variation δ η S (ψ) sc , we firstly note that it is fully specified by the variation δ η Ψ +A , since δ η V does not contribute because of the nilpotency property (Ψ +A ) 3 = 0 . Then, using (3.9), we represent

δ η Ψ +A = D + D+ η -A v + η +A V --, (4.7)
restore the full superspace integration measure in δ η S (ψ) sc and rewrite this variation as

δ η S (ψ) sc = 2 µ H v 2 (η -A Ψ + A ) + vD ++ V --(η -A Ψ + A ) + vV --(η +A Ψ + A ) + V --D ++ V --(η +A Ψ + A ) , (4.8)
where the bridge representation (3.7) for V was used. Taking into account the harmonic constraint (3.21) and the properties D ++ η -A = η +A and D ++ η +A = 0 , we observe that all terms in (4.8) except for the first one are reduced to a total harmonic derivative, whence

δ η S (ψ) sc = 2 µ H v 2 (η -A Ψ + A ) = -dtd 4 θ(η iA Ψ iA )v 2 . (4.9)
This is cancelled out by the variation of -1/3 dtd 4 θ v 3 , so the second N =8 supersymmetric action is given by

S (N =8) sc = 1 2 µ (-2) A V Ψ +A Ψ + A - 1 3 dtd 4 θ v 3 . (4.10)
Since the first term in (4.10) is D(2, 1; α) invariant at any α, while the second one is invariant under D(2, 1; α=-1/3) (see eq. (3.14)), we conclude that the full action (4.10) is invariant under the N =4 superconformal group D(2, 1; α= -1/3) . Since it is also invariant under the rigid N =8, d=1 supersymmetry, it is invariant under some N =8 superconformal group. Therefore the action (4.10), provided that V and v start with a constant, V = 1 + Ṽ , v = 1 + ṽ , defines a new model of N =8 superconformal mechanics associated with the N =8 multiplet (1,8,7) . It is easy to recognize which N =8, d=1 superconformal group we are facing in the present case. As follows from [START_REF] Frappat | Dictionary on Lie superalgebras[END_REF], the only such supergroup in which one can embed D(2, 1; α= -1/3) (to be more exact, an equivalent supergroup D(2, 1; β) , β= -1+α α =2) is the exceptional N =8, d=1 superconformal group F (4) , with the R-symmetry subgroup SO (7).

As already mentioned, the actions (3.25) and (3.26) are the unique d=1 sigma model type actions simultaneously involving both the (0, 4, 4) and (1, 4, 3) multiplets. Hence the sum of their N =8 completions, (4.6) and (4.10), yields the most general N =8 supersymmetric sigma-model type off-shell action of the multiplet (1, 8, 7) in the N =4 superfield formulation:

S (N =8) gen = γ S (N =8) f ree + γ ′ S (N =8) sc , (4.11)
γ and γ ′ being two independent renormalization constants. Surprisingly, it is also F (4) invariant, though with respect to modified superfield transformation laws. Choosing, for simplicity, γ = 1, and making the redefinitions 1 +γ ′ V = Ṽ , 1 + γ ′ v = ṽ , γ ′ Ψ +A = Ψ+A , we observe that (4.11) is reduced, up to a constant renormalization factor, to (4.10) where all superfields are replaced by those with "tilde". Assuming for the new superfields the same transformation laws (3.17), (3.22) as for the original ones, we see that (4.11) is also F (4) invariant. The F (4) transformations of the original superfields v and V are of course modified, e.g.

δ ′ sc v = -Λ 0 [v+(γ ′ ) -1
] . The transformations of the hidden N =4 supersymmetry remain unaltered. There is no way to make superconformal the free action (4.6), while (4.11) is N =8 superconformal at any γ ′ = 0 .

One can also check that each of the two FI terms, (3.20) and (3.27), is invariant under N =8 supersymmetry (4.2), (4.3). However, they both break superconformal symmetry and SO (7) . The same symmetry properties are exhibited by the on-shell potential terms arising upon eliminating the auxiliary fields A ik , a m and ām from the sum of (4.11) and (3.20), (3.27).

Component actions

Using the explicit component expansions (3.4) and (3.21), it is straightforward to find the component form of the superfield actions (3.13) and (3.14)

S (N =8) f ree = dt ẋ ẋ - i 4 ω i ωi -ωi ωi + iψ iA ψiA + 1 2 A ik A ik + a A āA ≡ dt L (N =8)
f ree , (5.12)

S (N =8) sc = dt xL (N =8) f ree + i 2 A ik ψ A i ψ kA + 1 2 ω i ωk - 1 2 ωk ψ kA a A + ω k ψ A k āA . (5.13)
To interpret (5.13) as the superconformal action, one needs to make the shift x = 1 + x . We observe that the general off-shell action (4.11) is specified by the linear function f (x) = γ +γ ′ x , and its derivative f x = γ ′ , in agreement with the result of [20] where the component off-shell action of the multiplet (1, 8, 7) was constructed by a different method. Here we reproduced this result in a manifestly N =4 supersymmetric off-shell superfield formalism. We took advantage of the latter to show the F (4) superconformal invariance of the action (5.13), as well as of a sum of (5.12) and (5.13) (with respect to modified F (4) transformations). Also, we showed that the FI terms (3.20), (3.27) preserve N =8 supersymmetry (although they break F (4) ).

The component expressions for (3.20), (3.27) can be easily found

S (v) F I = -dt c ik A ik , S (ψ)
F I = dt(ξ A āA -ξA a A ) . (5.14)
After eliminating the auxiliary fields A ik , a A , āA in the sum

S (N =8) gen +S (v) F I +S (ψ)
F I by their algebraic equations of motion, there appears a scalar potential ∼ (γ + γ ′ x) -1 (plus some accompanying fermionic terms) which is not conformal. Perhaps, conformal potentials could be generated by coupling the multiplet (1,8,7) to some additional N =8 multiplets.

For completeness, we present the component form of the transformations (4.1)

δ η x = -η iA ψ iA , δ η ω i = -2η iA a A , δ η ωi = -2η iA āA , δ η A ik = 2η (iA ψk) A , δ η ψ iA = iη A k A ki -iη iA ẋ , δ η a A = iη iA ωi , δ η āA = iη iA ωi . (5.15)
It is straightforward to check the invariance of (5.12) and (5.13) under these transformations. Let us also give the R-symmetry transformations belonging to the coset SO(7)/[SU (2)] 3 , where [SU (2)] 3 is the product of three SU (2) symmetries: SU (2) P G acting on the indices A and commuting with the manifest N =4 supersymmetry, manifest R-symmetry SU (2) R acting on the doublet indices i, k, . . . , and one more hidden SU (2) R ′ which rotates θ i through θi , D i through Di , ω i through ωi and a A through āA . These transformations read

δ λ A ik = λ (ik)B a B - λ(ik) B āB , δ λ a B = -λ(ik)B A ik , δ λ āB = λ (ik)B A ik δ λ ψ iA = - i 2
λ (ik)A ω k + λ(ik)A ωk , δ λ ω i = -2i λ(ik)A ψ kA , δ λ ωi = 2i λ (ik)A ψ kA . (5.16)

Here λ (ik)B , λ(ik)B are 6 complex parameters, which, together with 9 real parameters of [SU (2)] 3 , are the 21 real parameters of the group SO (7) . The bosonic field x is an SO(7) singlet.

Conclusions

In this article, using the manifestly N =4 supersymmetric language of the N =4, d=1 harmonic superspace, we constructed a new N =8 superconformal model associated with the off-shell multiplet (1,8,7) and showed that the corresponding N =8 superconformal group is the exceptional supergroup F (4) . We also found that the generic sigma-model type off-shell action of this multiplet is given by a sum of the superconformal action which is trilinear in the involved N =4 superfields and the free bilinear action. The generic action is also superconformal, but with respect to modified F (4) transformations. The component action is in agreement with the one derived in [20]. The N =8 supersymmetric potential terms can be generated by two superfield FI terms which break both superconformal and SO (7) symmetries. An interesting problem for further study is to see whether superconformal potentials could be generated by coupling this model to some other known N =8 multiplets. Also it would be of interest to find out possible implications of this new superconformal model in the brane, black holes and AdS 2 /CFT 1 domains, e.g. along the lines of refs. [4]- [6], [10]- [START_REF] Gomis | [END_REF].

Invariance under these transformations is sufficient to check D(2, 1; α) invariance since the rest of the D(2, 1; α) transformations is contained in the closure of the conformal and manifest Poincaré N =4, d=1 supersymmetries.

The relative sign between these two variations was chosen so as to have the same closure for the hidden supersymmetry as for the manifest one.

Acknowledgements

The work of E.I. was supported in part by the RFBR grant 06-02-16684, the RFBR-DFG grant 06-02-04012-a, the grant DFG, project 436 RUS 113/669/0-3, the grant INTAS 05-7928 and a grant of Heisenberg-Landau program. He thanks Laboratoire de Physique, UMR5672 of CNRS and ENS Lyon, for the warm hospitality extended to him during the course of this work.


F Gatuingt
email: fabrice.gatuingt@ens-cachan.fr

L Snozzi

J F Molinari

Determination of the dynamic tensile response and dissipated fracture energy of concrete with a cohesive element model

Introduction

Modelling of concrete structures subjected to dynamic loading such as explosions, impacts and perforations is very demanding, both from the point of view of computer codes and from the point of view of material modelling. Several macroscopic models for concrete have been developed [START_REF] Ottosen | Constitutive model for short-time loading of concrete[END_REF][START_REF] Sluys | Wave propagation, localisation and dispersion in softening solids[END_REF][START_REF] Jf Dubé | Rate dependent damage model for concrete in dynamics[END_REF][START_REF] Sercombe | Viscous hardening plasticity for concrete in high-rate dynamics[END_REF][START_REF] Zp Bazant | Fracturing rate effect and creep in microplane model for dynamics[END_REF][START_REF] Gatuingt | Coupled damage and plasticity modelling in transient dynamic analysis of concrete[END_REF][START_REF] Hervé | On numerical implementation of a coupled rate dependent damage-plasticity constitutive model for concrete in application to high-rate dynamics[END_REF][START_REF] Pedersen | An analysis of dynamic fracture in concrete with a continuum visco-elastic visco-plastic damage model[END_REF][START_REF] Gatuingt | Anisotropic 3d delay-damage model to stimulate concrete structures[END_REF][START_REF] Pontiroli | Predicting concrete behaviour from quasi-static loading to hypervelocity impact An overview of the PRM model[END_REF] to perform non-linear numerical analysis of such problems. In these models a rate effect in tension has often been introduced to represent the experimental data for loading strain rates exceeding 1 /s [START_REF] Weerheijm | The rate dependency of concrete in tension -New data for wet, normal and dry conditions[END_REF][START_REF] Erzar | An Experimental Method to Determine the Tensile Strength of Concrete at High Rates of Strain[END_REF]. The influence of the internal microstructure of concrete and the mechanisms that lead to different crack patterns when varying the loading rate, remain open questions. In order to give some answers to these questions, concrete has to be considered as a heterogeneous material where the nature of the heterogeneity depends essentially on the scale of observation. If one takes the scale of sand's grain, concrete can be considered as a biphasic material made of aggregates of different sizes randomly distributed in a mortar matrix. Thus, the failure of a concrete sample in tension is related to processes that take place at the so-called mesoscale. In order to investigate the role of the meso-structure on the dynamic tensile response of concrete, one can use a numerical approach. This is achieved in this paper using a 2D finite element mesoscopic description of concrete (aggregates + matrix) with cohesive capability. This method has proven its efficiency on numerical simulations of fracture of brittle materials [START_REF] Levy | Dynamic fragmentation of a ring: Predictable fragment mass distributions[END_REF] and has been extended to concrete like materials [START_REF] Snozzi | Influence of the meso-structure in dynamic fracture simulation of concrete under tensile loading[END_REF].

Meso-scale model

Concrete is an heterogeneous material made of various components, which are present in different proportions. This produces a quasi-brittle material, whose mechanical behavior is defined by the wide range of the ingredients in the mixture. Since every component has an influence according to its characteristic length, considering concrete at a meso-scale level of observation allow to represent it as a biphasic material: aggregates embedded in a mortar paste matrix. Meso-scale continuum models have been tested by several authors [START_REF] Roelfstra | Le béton numerique[END_REF][START_REF] Stankowski | Numerical simulation of progressive failure in particle composites[END_REF][START_REF] Lopez | Microstructural analysis of concrete fracture using interface elements[END_REF][START_REF] Caballero | 3D meso-mechanical numerical analysis of concrete using interface elements[END_REF]. From those works it results that the meso-scale level of material observation is suitable to capture the main characteristics of the overall mechanical behavior of concrete: micro-cracking, fracture initiation and propagation, coalescence and localization. Moreover the constitutive equations might be relaxed when comparing with a macro-scale formulation and thus the number of parameters may be reduced.

In our model only medium and large aggregates are represented explicitly. Besides those two components the interfaces between the two phases, called interfacial transition zone (ITZ), are represented by dynamically inserted elements with the presented cohesive-frictional capability. In order to study the specific effect of the meso-structure on the dynamic tensile response, no random fields have been introduced on the properties. A finite element approach with cohesive capability is then used to simulate the cracks opening and propagation.

Meso-structure modelisation

Real concrete is based on randomly distributed aggregates in a cement paste matrix. In our description, we chose to describe explicitly only the aggregates with a diameter larger than 4 mm and smaller than 25 mm with six classes to capture roughly the aggregates size distribution. The smallest aggregates are then taken into account in the homogeneous matrix of mortar. The size of the specimens generated is 100×100 mm 2 . As in [START_REF] N'guyen | A mesoscopic approach for a better understanding of the transition from diffuse damage to localized damage[END_REF], knowing the aggregate size distribution, the number of perfect circular (in 2D) aggregates of each class can be calculated. To avoid an artificial increase of the volume matrix due to the boundary effect, the heterogeneities are placed in a larger sample with a final cut of all parts outside of the concrete specimen. Figure 1 shows different images of the meso-structures obtained using this method with the same aggregate size distribution.

Meso 1

Meso 2 Meso 3 Meso 4 A 2D finite element mesh is then generated from the meso-structure images using the PPM2OOF [20] public domain software created at the National Institute of Standards and Technology (NIST). Notice that this mesh generation is possible since the cohesive element methodology here does not require an a priori definition of the possible fracture planes [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements[END_REF][START_REF] Roelfstra | Le béton numerique[END_REF][START_REF] Stankowski | Numerical simulation of progressive failure in particle composites[END_REF][START_REF] Camacho | Computational modelling of impact damage in brittle materials[END_REF][START_REF] Carol | Micromechanical analysis of quasi-brittle materials using fracture-based interface elements[END_REF][START_REF] Caballero | 3D meso-mechanical numerical analysis of concrete using interface elements[END_REF].

The cohesive element method

A well-known method to model the onset of fracture is to having recourse to cohesive zone modeling, which has been introduced by Dugdale [START_REF] Dugdale | Yielding of steel sheets containing slits[END_REF] and Barenblatt [25] in the 1960's. This method describes fracture as a separation process by relating the displacement jump, which occurs at the crack tip, with tractions. The debonding is assumed to be confined in a small region of material called the cohesive zone, which represents the region where atomistic separation occurs. Within the computational framework this region is represented by interface elements with zero thickness. While damage is concentrated in these elements, the surrounding bulk material behaves linear elastically.

In our case the crack path is not known a priori and all lines in the mesh are considered as a potential crack path. During the simulation, the stress on the interface between two adjacent continuum elements is computed and compared to the fracture criterion at the end of every time step. The interfacial stress, σ, is calculated averaging stresses of the adjacent Gauss points of the two continuum elements. If the inter-element stress exceeds the critical stress value, the nodes located at the inter-element boundary are doubled, the two elements are topologically disconnected and a cohesive element is inserted (see Figure 2). After the nodal disconnection, the interfacial stress starts being controlled by the traction separation law implemented in the cohesive element (Figure 2

). n s s t t δ δ c δ max G c f ct t max
Figure 2: Schematics of the nodal disconnection between two continuum elements in which the interfacial stress has exceeded the critical stress and the traction/separation law where t is the effective cohesive traction, f ct represents the local material strength and δ c represents the effective relative displacement beyond which complete decohesion occurs. Note that the definition of f ct and δ c implicitly establish the existence of an effective fracture energy G c , which corresponds to the area under the curve of Figure 2:

G c = 1 2 f ct δ c (1)
Partially damaged cohesive elements have dissipated an energy W < G c . As the present work is focused on tensile uniaxial loading, the fracture energy of the different phases of our concrete is identified to the experimental one for Mode I, i.e. G c = G I c .

Numerical simulation of direct tensile test in dynamics 3.1 Initial and boundary conditions

The specimen is loaded under displacement control with an imposed strain rate ε. To avoid stress wave propagation and an early fracture near the boundaries [START_REF] Miller | Modeling and simulation of dynamic fragmentation in brittle materials[END_REF], all the nodes of the finite element mesh are prescribed an initial velocity in accordance to their vertical position y as illustrated in Figure 3:

V y (y) = 2V 0 h y (2)
To obtain the stress-strain curves, we define the macroscopic stress as the boundary reaction force F y divided by the initial width of the specimen and the macroscopic strain as the displacement U y of the boundary divided by the initial height h of the specimen.

Material parameters

As shown in section 2 our numerical model is based on three phases: the mortar paste, the aggregates and an ITZ. For these phases we have to identify the elastic and rupture parameters of equation 2.2. We used the parameters given in Table 1. The relationship between the cohesive element law and G c introduces a length scale -called the cohesive zone length l z -into the material description. This length has an important influence around the crack tip in numerical simulations and we shown that in our case it is of the order of 4 mm for the aggregates (see [START_REF] Gatuingt | Numerical determination of the tensile response and the dissipated fracture energy of concrete: role of the meso-structure and influence of the loading rate[END_REF]). So the finite elements have to be smaller than 1 mm to have at least four elements in the cohesive zone.

The dynamic tensile response

As previously mentioned, the numerical model does not contain any random field to represent the scattering often observed on the macroscopic response of concrete. This scattering can be important in experimental tests for the same initial concrete mix but tested on several specimens. The main differences are then due to the different aggregate arrangements and to the macro-porosity distribution. In this section, we study only the influence of spatial distribution of aggregates and not the role of the macroporosity. This porosity has probably a great influence on the crack initiation and ultimately on the crack path but in order to clearly separate the influence of aggregates, it will be omitted in our simulations.

The results of the strain-stress curves obtained for five numerical concrete specimens in tension for several strain rates are presented in Figure 4. One can see on this figure that the microstructure has a slight influence (less than 4 %) on the peak strength value of our concrete. The macroscopic tensile strength is mainly governed in our case by the ITZ strength and not by the meso-structure. This can be best explained by examining the failure process in quasi-static for which we can observe a gradual transition from diffuse micro-cracking to strain localization and finally to a macroscopic crack. Figure 5 shows this evolution to a unique macroscopic crack for ε = 0.1 /s. In the upper row of Figure 5, which was obtained at the peak stress just before strain localization, one can remark that for the five meso-structures, the micro-cracks are diffuse and that their numbers are almost independent of the aggregates spatial distribution. This explains the slight influence on the peak strength value observed at the macroscopic scale. Conversely, on the lower row of Figure 5, which is extracted from the softening part, we notice that the crack path is strongly dependent on the meso-structure. This is why one obtains a stronger difference (50 %) on the stress value for ε = 3.10 -4 at the end of the loading for ε = 0.1 /s.

Loading rate effect

It is well-known that experimental results on dynamic tension tests show a rate sensitivity of tensile strength [START_REF] Vegt | Modelling of impact behaviour of concrete -an experimental approach[END_REF][START_REF] Erzar | An Experimental Method to Determine the Tensile Strength of Concrete at High Rates of Strain[END_REF]. In quasi-statics, the macroscopic tensile strength is mainly governed in our case by the Interfacial Transition Zone strength (ITZ between the aggregates and the mortar paste) and not by the meso-structure [START_REF] Rr Pedersen | Mesoscopic modelling of concrete under impact[END_REF][START_REF] Gatuingt | Numerical determination of the tensile response and the dissipated fracture energy of concrete: role of the meso-structure and influence of the loading rate[END_REF]. For low strain rate -ε < 1 /s -the dynamic resistance increase is mainly due to the presence of water in the material [START_REF] Rossi | The dynamic behaviour of concrete: influence of free water[END_REF] and we have a slight Dynamic Increase Factor -equal to the ratio of the static versus the dynamic strengths. For higher strain rate -ε > 1 /s -the usual explanation of a more important DIF is the transition between single cracking in quasi-statics to diffuse cracks in dynamics.

The results of the strain-stress curves obtained for our numerical simulations in tension for several strain rates are presented in Figure 4. We can see on this figure the numerical rate effect in tension obtained. We can notice that in our case the dynamic increase factor is equal to almost 2 for ε = 100 /s which can be to slight compared to experimental one (∼ 3). We may have to take into account rate dependency at the material level, for example by linking the cohesive strength f ct to the rate of deformation of the surrounding material and to the crack opening rate, in order to achieve better agreement with experimental results. Figure 6 shows the numerical rate effect in tension obtained for three meso-structures with our assumptions. One can see on this figure that the computed increase with strain rate is slight concerning the tensile strength. We obtain here a dynamic increase factor -equal to the ratio of the static versus the dynamic strengths -of 1.3 for ε = 100 /s while it is equal to approximately 3 in the experiments. This result shows that the assumption of no rate effect in the cohesive traction law is not totally realistic. The micro-inertia effects in the fracture process zone are not sufficient to explain the rate dependency of concrete in tension even at the highest loading rate velocities. One has to take into account a rate dependency at the material level, for example by linking the cohesive strength f ct to the strain rate. 7 depicts the evolution of the dissipated fracture energy as a function of the macroscopic strain of the specimen for different loading rates and heterogeneities. One can see on this figure that the dissipated fracture energy strongly depends on the loading rate even with a rate independent local fracture energy. We note that even if the dynamic increase factor for the tensile strength is not large enough, it is less the case for the dissipated fracture energy of the specimen. According to the structural effect hypothesis, the explanation of higher dissipated fracture energy for high loading rates resides in a more diffuse micro-cracking. This effect can be seen on Figure 8 where the final aspect of the concrete specimen based on the meso-structure Meso 1 are compared for low and high loading rates. This behavior only explains that the differences between low and high loading rates are not important for the peak stress but only for the total dissipated fracture energy, which is in contradiction with experimental observations. To improve our simulations and the representativeness of our numerical concrete, the challenge will be to find a rate dependent cohesive traction law (at least for the f ct value) that will not increase too much the total dissipated fracture energy. But an open question remains. Should we use a rate dependent function for only the ITZ, mortar, or aggregates or for all the parts of the meso-structure? This point will be difficult to answer without some specific experiments on the different constituents of a real concrete.

Conclusion

In this paper we proposed a mesoscopic model for the analysis of dynamic tensile failure of concrete. This model is based on a 2D finite element description with cohesive capability of a mix of aggregates larger than 4 mm in a mortar paste matrix. The influence of the heterogeneous meso-structure of concrete and the loading rate on the tensile response and the dissipated fracture energy are studied. With our specimen size we observe a small impact of the aggregates arrangement on the tensile strength. This is of course based on the strong assumption of a perfect mix with no macro-porosity. Nevertheless when the post-peak response and then the dissipated fracture energy are considered, we show that the mesostructure has an influence.

The second main interest of the paper is the study of the dynamic loading rate effects on the tensile strength and on the dissipated fracture energy with our approach. The cohesive law used is independent of the local or global strain rate to see if the dynamic increase factor observed experimentally is due to a material effect or a structural (inertial) effect. The conclusions are not the same for the tensile strength and for the dissipated fracture energy. In the first case, the numerical results exhibit a small rate effect that is not coherent with the experiments. One the other hand, it is very interesting to see that for the second case we are very close to the experimental results. This means that the tensile strength increase is mostly due to a material effect (moisture, visco-elasticity, . . . ) while the dissipated fracture energy observed in the post-peak part is more due to the increase of the number of microcracks with the loading rate. With a small change in the cohesive law it will then be possible to have a realistic virtual material to test the influence of other experimental configurations.

Figure 1 :

 1 Figure 1: Different images of the meso-structures generated with the same aggregates size distribution.

Figure 3 :

 3 Figure 3: Velocity initial condition

Figure 4 :

 4 Figure 4: Influence of the meso-structure on the stress-strain curves for different strain rates: a/ ε = 0.1 /s, b/ ε = 1 /s, c/ ε = 10 /s Meso 1 Meso 2 Meso 3 Meso 4

Figure 5 :

 5 Figure 5: Influence of the meso-structure on the crack path for different meso-structures at ε = 0.1 /s: upper row at ε = 7.10 -5 for ≈ the peak stress (disp. × 2000), lower row at ε = 3.10 -4 (disp. × 100)

Figure 6 :

 6 Figure 6: Numerical rate effect in tension for three meso-structures

Figure 7 :

 7 Figure 7: Influence of the meso-structure on the dissipated fracture energy for different strain rates

Figure

  Figure 7 depicts the evolution of the dissipated fracture energy as a function of the macroscopic strain of the specimen for different loading rates and heterogeneities. One can see on this figure that the dissipated fracture energy strongly depends on the loading rate even with a rate independent local fracture energy. We note that even if the dynamic increase factor for the tensile strength is not large

Figure 8 :

 8 Figure 8: Influence of the loading rate on the final cracking for the meso-structure Meso 1 (disp. × 100): a/ ε = 0.1 /s, b/ ε = 100 /s,

Table 1 :

 1 Material properties used in the simulations

		Aggregate Mortar paste ITZ
	Density -ρ (kg/m 3 )	2700	2200	-
	Young's modulus -E (GPa)	75	25	-
	Poisson's ratio -ν	0.2	0.2	-
	Fracture Energy -G c (J/m 2 )	60	50	30
	Tensile Strength -f ct (MPa)	10	4	2.4 1
	Cohesive zone length -l z (mm)	41	72	-

This value is estimated from G c of the ITZ with δ c equal to the value obtained for the mortar. With this assumption we obtain a f ct smaller in the ITZ than in the mortar, which is commonly accepted for a classical concrete