Annie Raoult
email: annie.raoult@parisdescartes.fr

Quasiconvex Envelopes in Nonlinear Elasticity

We give several examples of modeling in nonlinear elasticity where a quasiconvexification procedure is needed. We first recall that the three-dimensional Saint Venant-Kirchhoff energy fails to be quasiconvex and that its quasiconvex envelope can be obtained by means of careful computations. Second, we turn to the mathematical derivation of slender structure models: an asymptotic procedure using Γ-convergence tools leads to models whose energy is quasiconvex by construction. Third, we construct an homogenized quasiconvex energy for square lattices.

1 The Saint Venant-Kirchhoff stored energy function 1.1 Non quasiconvexity of the Saint Venant-Kirchhoff stored energy function

This section is based on [START_REF] Raoult | Non-polyconvexity of the stored energy function of a Saint Venant-Kirchhoff material[END_REF] from which it is immediately derived that the Saint Venant-Kirchhoff stored energy function is not rankone convex, and as a consequence not polyconvex, nor quasiconvex.

The internal energy of an elastic material reads J(φ) = Ω W (∇φ(x))dx where Ω ⊂ R 3 is a reference configuration (here assumed to be homogeneous), W : M 3×3 → R is the stored energy function that is most of the time assumed to be continuous and the deformation φ : Ω → R 3 is sufficiently regular. This is the energy due to the deformation φ. Actually, the domain of W should be restricted to the set M + 3×3 of matrices with positive determinant and φ should satisfy in some sense det∇φ(x) > 0 in order to express that orientation is preserved by realistic deformations and to prevent matter interpenetration. This restriction leads to mathematical difficulties and is quite often left aside. The total energy is the sum of the internal energy and of the external energy which takes into account the action of external loads (body forces such as gravity, surface forces such as pressure . . .). The equilibrium problem in an energy form reads:

Find φ ∈ Φ such that I(φ) = min Φ I, ( 1
)
where I is the total energy and Φ is a functional space that takes into account placement conditions. A basic hypothesis for proving the existence of a minimizer of a given functional is that this functional be lower semicontinuous. We are dealing here with functionals defined on infinite dimensional spaces, namely W 1,p (Ω; R 3 ) spaces (for simple growth conditions on W , J is well defined on W 1,p (Ω; R 3 )). Therefore, the appropriate lower semicontinuity is lower semicontinuity with respect to the weak topology. Then, it is classical (see Morrey (1995) and [START_REF] Dacorogna | Direct Methods in the Calculus of Variations[END_REF] for a survey) that J is weakly lower semicontinuous if and only if W is quasiconvex. Recall that the quasiconvexity condition reads

∀F ∈ M 3×3 , ∀φ ∈ W 1,∞ 0 (Ω; R 3 ), W (F ) ≤ 1 |Ω| Ω W (F + ∇φ(x)) dx. (2)
Quasiconvexity is a nonlocal notion which makes it difficult to check or to contradict. Two pointwise notions make a lower-upper frame for quasiconvexity. Indeed, the implications polyconvexity ⇒ quasiconvexity ⇒ rank-one convexity (3) are valid. This chain of implications can be easily remembered by noticing that the three notions are listed in alphabetical order. The convexity notion can even be added at the left extremity of the chain. But for applications in nonlinear elasticity, this is not useful since convexity of the energy density has to be ruled out for modeling reasons. Rank-one convexity is simply convexity along the straight line generated by two matrices whose difference is of rank 1. It reads

∀F, G ∈ M 3,3 such that rank(F -G) ≤ 1, ∀λ ∈ [0, 1], W (λF + (1 -λ)G) ≤ λW (F ) + (1 -λ)W (G). (4)
Let us mention that rank-one convexity does not imply quasiconvexity as was proved by [START_REF] Sverak | Rank-one convexity does not imply quasiconvexity[END_REF]. Polyconvexity is a more complex notion that was introduced in [START_REF] Ball | Convexity conditions and existence theorems in nonlinear elasticity[END_REF]: an energy is polyconvex if one can find a convex function w such that ∀F ∈ M 3×3 , W (F ) = w(F, adjF, detF ).

(5)

The function w in the above equation is defined on M 3×3 × M 3×3 × R. Note that this definition may be restricted to matrices with positive determinant in which case w is defined on M 3×3 × M 3×3 × R + * which is the convex hull of {(F, adjF, detF ), detF > 0}. From implications (3), it is obvious that a way of proving that an energy is quasiconvex -the important notion -is to prove that it is polyconvex. This is why the possible polyconvexity of the Saint Venant-Kirchhoff energy was examined two decades ago. In fact, the proof provided the non rank-one convexity.

Theorem 1.1. The Saint Venant-Kirchhoff energy is not rank-one convex.

The following statements immediately follow.

Corollary 1.2. The Saint Venant-Kirchhoff energy is not polyconvex, nor quasiconvex.

Proof of Theorem 1.1. -The Saint Venant Kirchhoff energy reads

W (F ) = µ 4 F T F -I 2 + λ 8 ( F 2 -3) 2 , ( 6
)
where F 2 = tr F T F . Letting C(F ) = F T F , we have equivalently

W (F ) = µ 4 tr(C(F )) 2 + λ 8 F 4 - 3λ + 2µ 4 F 2 + 9λ + 6µ 8 . (7)
The first term reads µ 4 (v 4 1 + v 4 2 + v 4 3 ) where v 1 , v 2 , v 3 are the singular values of F . Since g defined by g(v 1 , v 2 , v 3 ) = (v 4 1 + v 4 2 + v 4 3 ) is a convex, symmetric function that is not decreasing with respect to each of its variable, it is known that this first term is convex in F . Proofs of such results can be found in [START_REF] Ball | Convexity conditions and existence theorems in nonlinear elasticity[END_REF], [START_REF] Ciarlet | Mathematical Elasticity. Vol I: Three-dimensional Elasticity[END_REF], Le [START_REF] Dret | Sur les fonctions de matrices convexes et isotropes[END_REF], [START_REF] Thompson | Eigenvalues of sums of hermitian matrices[END_REF]. The second term is obviously convex in F . But, the minus sign in the third term prevents the whole of the expression of being rank-one convex as shown by the following counter-example which uses the fact that for F small, this term is greater than the two previous ones. Let F = ε Id and G = ε diag(1, 1, 3), so that F + G 2 = ε diag(1, 1, 2). Notice that F -G is of rank 1. Matrices F and G are such that adj (

F + G 2 ) = adj F + adj G 2 , det( F + G 2 ) = det F + det G 2 .
If W were rank-one convex, we would get W (

F + G 2 ) ≤ 1 2 (W (F ) + W (G)), i.e., µ tr(C( F + G 2 )) 2 + λ 2 F + G 2 4 -(3λ + 2µ) F + G 2 2 ≤ 1 2 [µ tr(C(F )) 2 + λ 2 F 4 -(3λ + 2µ) F 2 ] + 1 2 [µ tr(C(G)) 2 + λ 2 G 4 -(3λ + 2µ) G 2 ].
Since terms tr(C(F )) 2 , tr(C(G)) 2 , tr(C( F +G 2 )) 2 and terms F 4 , G 4 , F +G 2 4 are of order 4 with respect to ε while the remaining ones are of order 2, this amounts to

- 1 ε 2 F + G 2 2 ≤ - 1 ε 2 ( 1 2 F 2 + 1 2 G 2 ).
The contradiction follows from the equalities

F 2 = 3ε 2 , G 2 = 11ε 2 , F +G 2 2 = 6ε 2 .
1.2 The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function

The quasiconvex envelope of the three-dimensional Saint Venant-Kirchhoff stored energy function was first computed in Le Dret and Raoult (1995b). Computations were made in a systematic but somewhat tedious way, and inspired by a preliminary work by the authors in a 2d×3d setting, see Le Dret and Raoult (1995a). Extending some results by Pipkin allows to simplify the proofs, see [START_REF] Pipkin | Relaxed energies for large deformations of membranes[END_REF], Le Dret and Raoult (1995c).

We denote by S + m the set of symmetric, positive semidefinite matrices.

Lemma 1.3. Let m ≤ n and Y : M n×m → R be a left O(n)-invariant, rank- one convex mapping. Then the mapping Ỹ : S + m → R such that Y (F ) = Ỹ (F T F ) for all F in M n×m satisfies Ỹ (C) ≤ Ỹ (C + S) for all C, S ∈ S + m . (8)
Remark 1.4. In the case when m < n, this result is due to [START_REF] Pipkin | Relaxed energies for large deformations of membranes[END_REF]. However, the argument does not apply to square matrices. In Pipkin's terminology, Ỹ is said to be increasing.

Proof -Following [START_REF] Pipkin | Convexity conditions for strain-dependent energy functions for membranes[END_REF], we first remark that proving (8) amounts to proving that

Ỹ (C) ≤ Ỹ (C + µv ⊗ v) for all C ∈ S + m and for all µ ≥ 0, v ∈ R m \ {0}. (9)
Indeed, (8) clearly implies (9). Conversely, any S in S + m admits a spectral decomposition S = i=1,m µ i v i ⊗ v i where µ i ≥ 0, and v i , i = 1, • • • , m are orthonormal eigenvectors of S. Applying inequality (9) m times, we obtain (8).

Let us now prove (9). Let C ∈ S + m and v ∈ R m \ {0} be given. Without loss of generality, we assume

v 2 := v T v = 1.
We first consider the case when either m < n or m = n and C is not invertible. In both cases, C can be written as C = F T F where F T is a noninjective m × n matrix. Therefore, there exists u in ker F T with u = 1. From the rank-one convexity of Y , we know that the function

y : t ∈ R → y(t) = Y (F + t u ⊗ v) ∈ R is convex. Moreover, since F T u = 0, y(t) = Ỹ (C + t 2 v ⊗ v)
. Therefore, y is even. It follows that y is monotone increasing on R + ; in particular, y(0) ≤ y(t) for all t ≥ 0. Choosing t = √ µ, we obtain (9). We now turn to the case when m = n and C is invertible. For all µ ≥ 0, the matrix

C µ = C + µ v ⊗ v is symmetric, positive definite. Hence, F µ = C 1/2 µ is invertible. We define a function h on R by h : t ∈ R → h(t) = Y (F µ + t F -1 µ v ⊗ v). It follows from the rank-one convexity of Y that h is convex. Moreover, h(t) = Ỹ (C + (µ + 2 t) v ⊗ v + t 2 v ⊗ v(C + µv ⊗ v) -1 v ⊗ v).
An easy computation shows that the function

t ∈ R → 2 t v ⊗ v + t 2 v ⊗ v(C + µ v ⊗ v) -1 v ⊗ v is symmetric with respect to t = -(v T (C+µ v⊗v) -1 v) -1 < 0.
The function h, in turn, is symmetric with respect to t. Therefore, h attains its minimum at t and is monotone increasing on [ t, +∞[. Obviously, h(0) = Ỹ (C+µ v⊗v). If we can find t such that t ≤ t ≤ 0 and

h(t) = Ỹ (C), (10)
then inequality (9) is proved. A sufficient condition for a real number t to solve (10) is

t 2 v T (C + µv ⊗ v) -1 v + 2 t + µ = 0. ( 11
)
The discriminant of equation ( 11) is positive if and only if

µv T (C + µv ⊗ v) -1 v ≤ 1. ( 12
)
Let us check that this is indeed the case. Let z = µ 1/2 C -1/2 v. Then we have

µv T (C + µv ⊗ v) -1 v = z T (I + z ⊗ z) -1 z = z 2 1 + z 2 ≤ 1,
hence the roots of equation ( 11) are real. Moreover, they are nonpositive and symmetric with respect to t. The largest root thus satisfies t ≤ t ≤ 0 and (10), which proves our claim.

With this lemma at hand, we can now state our main result.

Theorem 1.5. Let m ≤ n and let W : F ∈ M n×m → R be a left O(n)invariant, bounded from below, stored energy function such that the associated function W :

C → W (C) is convex on S + m . Then, QW (F ) = inf S∈S + m W (F T F + S). (13)
Proof -Since W is bounded from below, we can define, with Pipkin's notation, a function Wr on S + m by Wr (C) = inf

S∈S + m W (C + S).
It is easy to check that Wr (C) is convex. Indeed, given C 1 and C 2 ∈ S + m and an arbitrary ε > 0, there exists S 1 and

S 2 in S + m such that Wr (C i ) ≤ W (C i + S i ) and W (C i + S i ) ≤ Wr (C i ) + ε for i = 1, 2. Let t ∈ [0, 1] and S = t S 1 + (1 -t)S 2 . Then, Wr (t C 1 + (1 -t)C 2 ) ≤ W (t C 1 + (1 -t) C 2 + S) ≤ t W (C 1 + S 1 ) + (1 -t) W (C 2 + S 2 ) ≤ t Wr (C 1 ) + (1 -t) Wr (C 2 ) + ε.
The convexity of Wr follows at once.

Let us now remark that Wr obviously satisfies Wr (C) ≤ Wr (C + S) for all C and S in S + m . This implies that the function

Z := F ∈ M n×m → Wr (F T F ) is convex. Indeed, for all F and G in M n×m and for all t ∈ [0, 1], (tF + (1 -t)G) T (tF + (1 -t)G) = t F T F + (1 -t)G T G -t(1 -t)(F -G) T (F -G). Therefore, since t(1 -t)(F -G) T (F -G) is positive semidefinite, Z(t F + (1 -t)G) ≤ Wr (t F T F + (1 -t)G T G) ≤ t Z(F ) + (1 -t)Z(G),
by the convexity of Wr . Consequently, since Z is convex and below W , we see that Z ≤ QW . The reverse inequality is obtained as follows. From Le Dret and Raoult (1995a), we know that QW is also left O(n)-invariant. Applying Lemma 1 to Y = QW , which is rank-one convex, we obtain

QW (F ) = QW (F T F ) ≤ QW (F T F + S) = QW (F T F + S) 1/2 ≤ W (F T F + S)
for all S ∈ S + m . Therefore, QW (F ) ≤ Z(F ) and the proof is complete.

Remark 1.6. i) It follows clearly from the proof that the quasiconvex envelope is also in this case the convex and rank-one convex envelope of the stored energy function.

ii) We now proceed to show by means of a simple counterexample that Pipkin's formula fails for m > n. Let m = 2, n = 1. Consider the function W : M 1×2 → R, W (F ) = F T F -I 2 . We thus let W (C) = C -I 2 . This function is clearly convex with respect to C. If we denote F = (z 1 , z 2 ) with z i ∈ R, then we have [START_REF] Dacorogna | Direct Methods in the Calculus of Variations[END_REF]. Let us now take F = (1, -1) so that

W (F ) = (z 2 1 -1) 2 + (z 2 2 -1) 2 + 2z 2 1 z 2 2 = F 2 -1 2 + 1. Therefore, QW (F ) = CW (F ) and QW (F ) = W (F ) if F ≥ 1, 1 if F ≤ 1, see
C = 1 -1 -1 1 and W (C) = 2. With the choice S = 1 2 1 1 1 1 , we obtain Wr (C) ≤ W (C + S) = 1 < 2 = W (C) = W (F ) = QW (F ).
Application to an explicit computation: The Saint Venant-Kirchhoff stored energy function defined in (6) can equivalently be written under the form W (F ) = W (F T F ) where

W (C) = µ 4 C -I 2 + λ 8 (tr C -3) 2 (14)
for all C in S + 3 . The mapping W is clearly convex with respect to C. Therefore, Theorem 1.5 applies. Let us briefly show how computations can be organized. For any C in S + 3 , let J C : S ∈ S + 3 → W (C + S) ∈ R. This is a strictly convex, coercive mapping. Consequently, J C admits one and only one minimizer on S + 3 . By (13), we have to evaluate inf

S∈S + 3 J C (S) = min S∈S + 3 J C (S).
Assume first that C is diagonal. We deduce from (14) that J C (S) ≥ J C (diag (s 11 , s 22 , s 33 )). Minimizing J C (S) among semidefinite positive matrices thus amounts to minimizing J C (S) among diagonal positive matrices. Equivalently, we have to minimize on (R + ) 3 the mapping j C such that

j C (s 1 , s 2 , s 3 ) = µ 4 3 i=1 (c ii -1 + s i ) 2 + λ 8 ( 3 i=1 (c ii -1 + s i )) 2 .
Without loss of generality, we assume that c 11 ≤ c 22 ≤ c 33 . The optimality conditions for j C on (R + ) 3 read

Dj C (s 1 , s 2 , s 3 )(t 1 , t 2 , t 3 ) ≥ 0 for all (t 1 , t 2 , t 3 ) ∈ (R + ) 3 , Dj C (s 1 , s 2 , s 3 )(s 1 , s 2 , s 3 ) = 0, (s 1 , s 2 , s 3 ) ∈ (R + ) 3 .
They are equivalent to

∂ i j C (s 1 , s 2 , s 3 ) ≥ 0 for i = 1, 2, 3, ∂ i j C (s 1 , s 2 , s 3 ) s i = 0, s i ≥ 0 for i = 1, 2, 3, that is to say (2 µ + λ)(c ii -1 + s i ) + λ k =i (c kk -1 + s k ) ≥ 0, ((2 µ + λ)(c ii -1 + s i ) + λ k =i (c kk -1 + s k )) s i = 0 , s i ≥ 0.
We distinguish four different cases:

1) If c 33 ≤ 1, then we can choose s i = 1 -c ii and min (R + ) 3 j C = 0.
2) If c 33 ≥ 1 and 2 (λ + µ) c 22 + λ c 33 ≤ 3 λ + 2 µ, then we can choose

s 3 = 0, s j = -c jj - λ 2(λ + µ) c 33 + 3λ + 2µ 2(λ + µ) ≥ 0, j = 1, 2 and min (R + ) 3 j C = µ (3λ + 2µ) 8(λ + µ) (c 33 -1) 2 .
3) If 2(λ+µ) c 22 +λ c 33 ≥ 3 λ+2 µ and (λ+2µ) c 11 +λ (c 22 +c 33 ) ≤ 3λ+2µ,

then s 2 = s 3 = 0, s 1 = -c 11 - λ (λ + 2 µ) (c 22 + c 33 ) + 3λ + 2µ (λ + 2 µ) ≥ 0 and min (R + ) 3 j C = µ 4 ((c 22 -1) 2 + (c 33 -1) 2 ) + λµ 4(λ + 2 µ) (c 22 + c 33 -2) 2 .
4) If (λ + 2µ) c 11 + λ (c 22 + c 33 ) ≥ 3λ + 2µ, then we can choose s 1 = s 2 = s 3 = 0 and min

(R + ) 3 j C = W (C).
So far, we have determined QW (F ) when C = F T F is diagonal. To extend the result to arbitrary matrices C in S + 3 , we make use of the right O(3)-invariance of the Saint Venant-Kirchhoff density W which is inherited by QW . Therefore, QW (F ) only depends on the singular values of F . We denote by v 1 (F ) ≤ v 2 (F ) ≤ v 3 (F ) the singular values arranged in increasing order. It suffices to replace c ii by v i (F ) 2 in the above formulas to obtain an explicit expression for QW (F ).

The expressions thus obtained are the same as those obtained in Le Dret and Raoult (1995a) that we recall below. First, for making comparisons between the energy and its quasiconvex envelop easier, we express the Saint Venant-Kirchhoff energy in terms of the singular values and we obtain

W (F ) = E 8(1 + ν) 3 i=1 (v i (F ) 2 -1) 2 + Eν 8(1 + ν)(1 -2ν) 3 i=1 v i (F ) 2 -3 2 , ( 15
)
where the Young modulus and the Poisson ratio are given by

E = µ(3λ + 2µ) λ + µ and ν = λ 2(λ + µ)
.

Let T = {v ∈ R 3 ; 0 ≤ v 1 ≤ v 2 ≤ v 3 } be the convex tetrahedral cone of R 3 + delimited by the planes v 1 = 0, v 1 = v 2 and v 2 = v 3 .
We define a mapping q on T by

q(v) = E 8 [v 2 3 -1] 2 + + E 8(1 -ν 2 ) [v 2 2 + νv 2 3 -(1 + ν)] 2 + + E 8(1 -ν 2 )(1 -2ν) [(1 -ν)v 2 1 + ν(v 2 2 + v 2 3 ) -(1 + ν)] 2 + , (16)
where

[x] 2 + = x 2 if x ≥ 0, [x] 2 + = 0 if x < 0.
Previous computations allow to write the following theorem.

Theorem 1.7. The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function W is given by

∀F ∈ M 3 , QW (F ) = q (v 1 (F ), v 2 (F ), v 3 (F )) . ( 17
)
Let us examine more deeply the values taken by QW . We introduce three subsets of T :

H = {v ∈ T ; v 3 ≤ 1}, C = {v ∈ T ; v 2 2 + νv 2 3 -(1 + ν) ≤ 0}, E = {v ∈ T ; (1 -ν)v 2 1 + ν(v 2 2 + v 2 3 ) -(1 + ν) ≤ 0},
which correspond to the various positive parts that appear in formula ( 16).

It is easily checked that

H ⊂ C ⊂ E and that i) if v ∈ H, then q(v) = 0, ii) if v ∈ C \ H, then q(v) = E 8 (v 2 3 -1) 2 , iii) if v ∈ E \ C, then q(v) = E 8(1 + ν) 3 i=2 (v 2 i -1) 2 + Eν 8(1 -ν 2 ) 3 i=2 v 2 i -2 2 , iv) if v ∈ E, then q(v) = E 8(1 + ν) 3 i=1 (v 2 i -1) 2 + Eν 8(1 + ν)(1 -2ν) 3 i=1 v 2 i -3 2 .
For singular values outside of E, the energy and its quasiconvex envelope coincide. At the other end of the scale, for sufficiently small singular values, the quasiconvex energy is equal to 0. It is indeed a general fact that the quasiconvex envelope of a material indifferent and isotropic material vanishes on the set of singular values less than 1, see Le Dret and Raoult (1994).

2 Quasiconvexity in the derivation of slender structure models

In this section, we turn to the rigorous derivation of models for bidimensional structures from three-dimensional models. The idea of deriving simplified models from complete models goes back at least to the 50s with the works by R.D. Mindlin and by E.Reissner among others. In the linear case, correct bidimensional models can be obtained by quick, but mathematically frightening ways: assuming for instance that the 33 component of the linearized strain tensor is equal to 0 in some equations of the three-dimensional model, but not in other ones. Later on, came the idea of considering a sequence of structures with thickness 2ε, of writing elasticity models for each of these structures and of studying the asymptotic behavior of the solutions of the models. Many researchers in applied mathematics worked on this subject and most of them consider that this procedure was first formalized by [START_REF] Ciarlet | A justification of the two-dimensional plate structure[END_REF].

The first result obtained by this method was of no surprise: the usual linear plate model is recovered. In the linear case, this line of work was pursued by identifying more precise models (i.e. not only identifying the limit of the three-dimensional solutions, but identifying a higher-order term as well), studying dynamical cases, or considering more general materials such that piezo-electric materials, visco-elastic materials. Things get trickier when dealing with nonlinear models: existence results are not always available and convergence proofs are much harder when products of terms have to be considered. Identifying limit models was first obtained by formal asymptotic methods on the system of partial differential equations describing finite elasticity written under variational form. Then came the realm of rigorous Γ-convergence arguments and their escort of quasiconvexification tools. The work we present here is taken from Le Dret and Raoult (1995a).

We mention that a previous work by [START_REF] Acerbi | A variational definition for the strain energy of an elastic string[END_REF] existed and, although in a one-dimesional setting that only requires convexity arguments, gave a path to follow.

The three-dimensional and rescaled problems

For all ε > 0, let Ω ε = {x ∈ R 3 ; (x 1 , x 2 ) ∈ ω, |x 3 | < ε}, where ω is an open, bounded subset of R 2 with Lipschitz boundary. For all z i ∈ R 3 , i = 1, 2, 3, we note (z 1 |z 2 |z 3 ) the 3 × 3 matrix whose i-th column is z i . Let W : M 3×3 →
R be a continuous function that satisfies the following growth and coercivity hypotheses:

∃C > 0, ∃p ∈ ]1, +∞[, ∀F ∈ M 3×3 , |W (F )| ≤ C(1 + F p ), ( 18
) ∃α > 0, ∃β ≥ 0, ∀F ∈ M 3×3 , W (F ) ≥ α F p -β. ( 19
)
We assume that Ω ε is the reference configuration of a hyperelastic homogeneous three-dimensional body whose stored energy function is W . We assume for simplicity that the bodies are submitted to the action of dead loading body force densities f ε ∈ L q (Ω ε ; R 3 ) and surface traction densities g ε ∈ L r (S ε ; R 3 ) on S ε = ω × {±ε}, the top and bottom surfaces of Ω ε . For the sake of definiteness, we assume that q = r and 1/p + 1/q = 1, but other choices are indeed possible at no extra cost. Let Γ ε = ∂ω × ]-ε, ε[ be the lateral surface of Ω ε . We assume that the deformations of the bodies satisfy a boundary condition of place on Γ ε . The equilibrium problem may be formulated as a minimization problem:

Find φ ε ∈ Φ ε such that I ε (φ ε ) = inf ψ∈Φε I ε (ψ), (20)
where the total energy I ε is

I ε (ψ) = Ωε W (∇ψ) dx - Ωε f ε • ψ dx - Sε g ε • ψ dσ, (21)
and the set of admissible deformations is

Φ ε = {ψ ∈ W 1,p (Ω ε ; R 3 ); ψ(x) = x on Γ ε }. (22)
We do not assume that W is quasiconvex and problem (20) may well not possess any solutions. Naturally, if it does have solutions which are thus actual equilibrium deformations of the bodies, our results apply to these deformations.

Let us thus be given a diagonal minimizing sequence φ ε for the sequence of energies I ε over the sets Φ ε . More specifically, we assume that

φ ε ∈ Φ ε , I ε (φ ε ) ≤ inf ψ∈Φε I ε (ψ) + εs(ε), ( 23
)
where s is a positive function such that s(ε) → 0 when ε → 0. Such a sequence always exists and if the minimization problems have solutions, φ ε may be chosen to be such a solution.

In order to obtain a membrane model in the limit, it is of crucial importance to specify the order of magnitude of the applied loads. In effect, it is always possible to stretch all thin cylinders Ω ε into the same block, say Ω 1 , by applying sufficiently large forces. For such forces, the limit behavior is obviously not that of a membrane.

It turns out that the right order of magnitude is given by f ε L q (Ωε;R 3 ) ≤ Cε 1/q and g ε L q (Sε;R 3 ) ≤ Cε where the constant C does not depend on ε. For example, the weight of the material, f ε (x) = (0, 0, -ρg) T , is allowed.

In order to rescale the problem, we let Ω = Ω 1 , Γ = Γ 1 and S = S 1 and define a rescaling operator Θ ε by (

Θ ε ψ)(x 1 , x 2 , x 3 ) = ψ(x 1 , x 2 , εx 3 ). Let φ(ε) = Θ ε φ ε and φ 0 (ε)(x) = (x 1 , x 2 , εx 3 ) T .
Note that all components are treated in the same way: we only transport φ ε on the fixed domain Ω. This is the same rescaling as that used in [START_REF] Fox | A justification of nonlinear properly invariant plate theories[END_REF]. The rescaled displacement u(ε) = φ(ε) -φ 0 (ε) belongs to V = W 1,p Γ (Ω; R 3 ). We accordingly rescale the energies by setting

I(ε)(ψ) = ε -1 I ε (Θ -1 ε ψ), i.e., I(ε)(ψ) = Ω W ∂ 1 ψ ∂ 2 ψ ∂ 3 ψ ε dx - Ω f (ε) • ψ dx - S ε -1 g(ε) • ψ dσ, (24)
or in terms of the rescaled displacements

J(ε)(v) = Ω W e 1 + ∂ 1 v e 2 + ∂ 2 v e 3 + ∂ 3 v ε dx - Ω f (ε) • (φ 0 (ε) + v) dx - S ε -1 g(ε) • (φ 0 (ε) + v) dσ,
where

f (ε) = Θ ε f ε and g(ε) = Θ ε g ε . It is immediate that J(ε)(u(ε)) ≤ inf v∈V J(ε)(v) + s(ε). (25)
For simplicity, we assume that f (ε) = f and ε -1 g(ε) = g are independent of ε.

Computation of the Γ-limit of the rescaled energies

We use Γ-convergence theory to determine the asymptotic behavior of the rescaled displacements u(ε) when ε → 0. In the sequel, the thickness parameter ε will take its values in a sequence ε n → 0. Since the results do not depend on the sequence in question, and for notational brevity, we will simply use the notation ε. Let us recall that a sequence of functions G ε from a metric space X into R is said to Γ-converge toward G 0 for the topology of X if the following two conditions are satisfied for all x ∈ X:

∀x ε → x, lim inf G ε (x ε ) ≥ G 0 (x), ∃y ε → x, G ε (y ε ) → G 0 (x).
If the sequence G ε Γ-converges, its Γ-limit is lower semicontinuous and is alternatively given by

G 0 (x) = min{lim inf G ε (x ε ); x ε → x}.
In addition, the set of functions from X into R has a sequential compactness property with respect to Γ-convergence in the sense that any sequence G ε : X → R admits a Γ-convergent subsequence. The main interest of Γ-convergence is that if the minimizers of G ε stay in a compact set of X for all ε, then their limit points are minimizers of G 0 , see De [START_REF] Giorgi | Su un tipo di convergenza variazionale[END_REF], [START_REF] Attouch | Variational Convergence for Functions and Operators[END_REF], Dal [START_REF] Maso | An Introduction to Γ-convergence[END_REF].

We do not use J(ε) directly, since this would imply working with the weak topology of W 1,p (Ω; R 3 ), which is non metrizable. Instead, we extend the energies to L p (Ω; R 3 ) by setting

∀v ∈ L p (Ω; R 3 ), J(ε)(v) = J(ε)(v) if v ∈ V, +∞ otherwise. ( 26
)
This is a classical trick used in the applications of Γ-convergence: obviously, this does not change the minimization problem. It has the additional virtue of incorporating the boundary conditions in the energy functional.

Let us now proceed to compute the Γ-limit of the sequence J(ε) for the strong topology of L p (Ω; R 3 ). Let M 3×2 be the space of 3 × 2 real matrices endowed with the usual Euclidean norm F = tr ( F T F ). We note (z 1 |z 2 ) the matrix of M 3×2 whose α-th column is z α ∈ R 3 . For all F = (z 1 |z 2 ) ∈ M 3×2 and z ∈ R 3 , we also note ( F |z) the matrix whose first two columns are z 1 and z 2 and whose third column is z.

As in [START_REF] Acerbi | A variational definition for the strain energy of an elastic string[END_REF] for elastic strings, we define

W 0 : M 3×2 → R by W 0 ( F ) = inf z∈R 3 W (( F |z)). ( 27
)
Due to the coercivity assumption on W , it is clear that this function is well defined. Besides, since W is continuous, the infimum is attained.

Proposition 2.1. The function W 0 is continuous and satisfies the growth and coercivity estimates:

∃C ′ > 0, ∀ F ∈ M 3×2 , |W 0 ( F )| ≤ C ′ (1 + F p ), ( 28
) ∀ F ∈ M 3×2 , W 0 ( F ) ≥ α F p -β. (29)
Proof -Since W 0 is an infimum of continuous functions, it is upper semicontinuous. Let F ∈ M 3×2 and consider a sequence F n ∈ M 3×2 such that F n → F as n → +∞. Because of the coercivity assumption on W , there exists a compact set K such that for all F n the infimum in definition ( 27) is attained at a point z n ∈ K. Consider a subsequence, still denoted n, such that W 0 ( F n ) converges when n → +∞. We extract a further subsequence such that

z n → z ∈ K. By continuity of W , W 0 ( F n ) = W (( F n |z n )) → W (( F |z)) ≥ W 0 ( F ). As this is true for all subsequences such that W 0 ( F n ) converges, it follows that lim inf W 0 ( F n ) ≥ W 0 ( F ), hence W 0 is lower semicontinuous.
For all F ∈ M 3×2 , let z 0 be a point where the infimum in definition ( 27) is attained. Thus,

W 0 ( F ) = W (( F |z 0 )) ≥ α ( F |z 0 ) p -β ≥ α F p -β. Hence W 0 is coercive. Therefore, W 0 is nonnegative outside of a compact set K ′ . Since |W 0 | is continuous, it is bounded on K ′ and for F ∈ K ′ , |W 0 ( F )| = W 0 ( F ) ≤ W (( F |0)) ≤ C(1 + ( F |0) p ) = C(1 + F p ), which proves the growth estimate. Let QW 0 = sup{Z : M 3×2 → R, Z quasiconvex, Z ≤ W 0 } be the quasi- convex envelope of W 0 . Let us introduce the space V M = {v ∈ V ; ∂ 3 v = 0}, ( 30
)
which we call the space of membrane displacements. It is canonically isomorphic to W 1,p 0 (ω; R 3 ) and we let v denote the element of W 1,p 0 (ω; R 3 ) that is associated with v ∈ V M through this isomorphism. The expression of the Γ-limit of the sequence J(ε) is given in the following theorem.

Theorem 2.2. The sequence J(ε) Γ-converges for the strong topology of L p (Ω; R 3 ) when ε → 0. Let J(0) be its

Γ-limit. For all v ∈ L p (Ω; R 3 ) ∩ V M , J(0)(v) = 2 ω QW 0 ((e 1 +∂ 1 v|e 2 +∂ 2 v)) dx 1 dx 2 - ω F • (φ 0 (0)+v) dx 1 dx 2 (31) where F(x 1 , x 2 ) = 1 -1 f (x 1 , x 2 , x 3 ) dx 3 + g(x 1 , x 2 , 1) + g(x 1 , x 2 , -1), and J(0)(v) = +∞ if v ∈ L p (Ω; R 3 ) \ V M .
For clarity, we break the proof of Theorem 2.2 into a series of lemmas and propositions.

We begin by extracting a Γ-convergent subsequence and call J(0) its Γlimit. The uniqueness of J(0) will make the extraction of this subsequence superfluous a posteriori .

Lemma 2.3. Let v(ε) ∈ L p (Ω; R 3 ) be a sequence such that J(ε)(v(ε)) ≤ C < +∞ where C does not depend on ε. Then v(ε) is uniformly bounded in V and its limit points for the weak topology of V belong to V M .

Proof-Let v(ε) ∈ L p (Ω; R 3 ) be such that J(ε)(v(ε)) ≤ C < +∞.
Then, the definition (26) of the function J(ε) implies first of all that v(ε) ∈ V for all ε > 0. Let us call ψ(ε) = v(ε) + φ 0 (ε) the deformation that is associated with the displacement v(ε). The coercivity of the function W and the assumed uniform bound for the energies imply that

α Ω (∂ 1 ψ(ε)|∂ 2 ψ(ε)|ε -1 ∂ 3 ψ(ε)) p dx ≤ C ′ 1 + ψ(ε) W 1,p (Ω;R 3 ) ( 32
)
where C ′ does not depend on ε. It is clear that for all ε ≤ 1, (z

1 |z 2 |ε -1 z 3 ) ≥ (z 1 |z 2 |z 3 ) . Therefore, (32) implies that α ∇ψ(ε) p L p (Ω;R 3 ) ≤ C ′ 1 + ψ(ε) W 1,p (Ω;R 3 ) , (33)
which, together with the boundary condition of place ψ(ε) = φ 0 (ε) on Γ, yields the desired uniform bound for ψ(ε) in W 1,p (Ω; R 3 ) by Poincaré's inequality. Since φ 0 (ε) is obviously uniformly bounded in W 1,p (Ω; R 3 ), the same holds true for v(ε).

On the other hand, since (z 1 |z 2 |ε -1 z 3 ) ≥ ε -1 |z 3 |, where | • | denotes the Euclidean norm into R 3 , upon using the bound just established above in inequality (33) we obtain that ∂ 3 ψ(ε) L p (Ω;R 3 ) ≤ C ′′ ε, so that ∂ 3 ψ(ε) → 0 strongly in L p (Ω; R 3 ). If we let ψ denote any limit point of the sequence ψ(ε) for the weak topology of W 1,p (Ω; R 3 ), it follows at once that ∂ 3 ψ = 0. If v denotes the corresponding limit point of the sequence v(ε), since v = ψ -φ 0 (0) and ∂ 3 φ 0 (0) = 0, we obtain that v belongs to V M .

Corollary 2.4. If v ∈ L p (Ω; R 3 ) but v ∈ V M , then J(0)(v) = +∞.
Proof -Indeed, if J(0)(v) < +∞, there exists a sequence v(ε) that converges strongly to v in L p (Ω; R 3 ) and such that J(ε

)(v(ε)) → J(0)(v). Therefore, by Lemma 2.3, v ∈ V M .
We thus only have to compute the value of the Γ-limit for displacements in V M .

Proposition 2.5. For all v ∈ V M , we have that

J(0)(v) ≥ 2 ω QW 0 ((e 1 +∂ 1 v|e 2 +∂ 2 v)) dx 1 dx 2 - ω F • (φ 0 (0)+v) dx 1 dx 2 . (34)
Proof -Consider any v ∈ V M . Since J(ε)(v) is obviously bounded from above independently of ε, it follows that J(0)(v) < +∞. By the definition of Γ-convergence, there exists a sequence v(ε)

∈ V such that v(ε) → v strongly in L p (Ω; R 3 ) and J(ε)(v(ε)) → J(0)(v). Then, by Lemma 2.3, v(ε) ⇀ v weakly in V .
First of all, it is clear that, when ε → 0,

Ω f • (φ 0 (ε) + v(ε)) dx + S g • (φ 0 (ε) + v(ε)) dσ → ω F • (φ 0 (0)+v) dx 1 dx 2 .
(35) For the elastic energy, we have that (with

ψ(ε) = v(ε) + φ 0 (ε) as usual) Ω W ∂ 1 ψ(ε) ∂ 2 ψ(ε) ∂ 3 ψ(ε) ε dx ≥ Ω W 0 ((∂ 1 ψ(ε)|∂ 2 ψ(ε))) dx ≥ Ω QW 0 ((∂ 1 ψ(ε)|∂ 2 ψ(ε))) dx. Let G : W 1,p (Ω; R 3 ) → R be defined by G(ψ) = Ω QW 0 ((∂ 1 ψ|∂ 2 ψ)) dx. (36)
Let us define a function Z :

M 3 → R by Z((z 1 |z 2 |z 3 )) = QW 0 ((z 1 |z 2 )).
Since QW 0 is quasiconvex, Z is also quasiconvex. Indeed, let d be the unit square in R 2 and D = d × ]0, 1[. Consider any function φ ∈ D(D; R 3 ). For all y ∈ ]0, 1[, the function φ y defined by φ y (x 1 , x 2 ) = φ(x 1 , x 2 , y) belongs to D(d; R 3 ). Hence, for all

F ∈ M 3 , D Z(F + ∇φ) dx = D QW 0 ((z 1 + ∂ 1 φ|z 2 + ∂ 2 φ)) dx = 1 0 d QW 0 ((z 1 + ∂ 1 φ y |z 2 + ∂ 2 φ y )) dx 1 dx 2 dy ≥ 1 0 QW 0 ((z 1 |z 2 )) dy = Z(F ).
This implies that Z is quasiconvex. We now remark that Z is quasiconvex, bounded below by -β and satisfies the growth condition (18) since QW 0 satisfies (28). Therefore, the function G is sequentially weakly lower semicontinuous on W 1,p (Ω; R 3 ), Acerbi and Fusco (1984). Consequently, since ψ(ε

) ⇀ ψ = v + φ 0 (0) in W 1,p (Ω; R 3 ), lim inf ε→0 Ω W ∂ 1 ψ(ε) ∂ 2 ψ(ε) ∂ 3 ψ(ε) ε dx ≥ lim inf ε→0 G(ψ(ε)) ≥ G(ψ) = 2 ω QW 0 ((e 1 + ∂ 1 v|e 2 + ∂ 2 v)) dx, (37)
and the proof is complete.

Let us now turn to proving the reverse inequality.

Proposition 2.6. For all v ∈ V M , we have that

J(0)(v) ≤ 2 ω QW 0 ((e 1 +∂ 1 v|e 2 +∂ 2 v)) dx 1 dx 2 - ω F • (φ 0 (0)+v) dx 1 dx 2 . (38) Proof -Let us consider v ∈ V M . For all w ∈ W 1,p 0 (ω; R 3 ), we define a displacement v(ε)(x) = v(x 1 , x 2 ) + εx 3 w(x). (39)
Obviously, v(ε) → v strongly in W 1,p (Ω; R 3 ). Let us examine the limit behavior of the sequence J(ε)(v(ε)). By the dominated convergence theorem and the growth estimate, it is clear that

Ω W ((∂ 1 ( ψ+εx 3 w)|∂ 2 ( ψ+εx 3 w)|e 3 +w)) dx → Ω W ((∂ 1 ψ|∂ 2 ψ|e 3 +w)) dx when ε → 0. Consequently, → Ω W ((∂ 1 ψ|∂ 2 ψ|e 3 + w)) dx - ω F • (φ 0 (0)+v) dx 1 dx 2 .
As this is true for all w ∈ W 1,p 0 (ω; R 3 ), it follows from the definition of Γ-convergence that

J(0)(v) ≤ inf w∈W 1,p 0 (ω;R 3 ) Ω W ((∂ 1 ψ|∂ 2 ψ|e 3 +w)) dx- ω F •(φ 0 (0)+v) dx 1 dx 2 .
We remark that inf

W 1,p 0 (ω;R 3 ) Ω W ((∂ 1 ψ|∂ 2 ψ|e 3 +w)) dx = inf L p (ω;R 3 ) Ω W ((∂ 1 ψ|∂ 2 ψ|e 3 +w)) dx,
by the density of W 1,p 0 (ω; R 3 ) in L p (ω; R 3 ) and by the dominated convergence theorem. The function g : ω × R 3 → R defined by g(x, z) = W ((∂ 1 ψ(x)|∂ 2 ψ(x)|e 3 + z)) is a Carathéodory function. Hence, the measurable selection lemma, [START_REF] Ekeland | Analyse Convexe et Problèmes Variationnels[END_REF], shows that there exists a measurable function w 0 such that

W 0 ((∂ 1 ψ(x)|∂ 2 ψ(x))) = W ((∂ 1 ψ(x)|∂ 2 ψ(x)|e 3 + w 0 (x)))
for almost all x ∈ ω. Due to the coercivity estimate, w 0 ∈ p (ω; R 3 ) and thus inf

w∈L p (ω;R 3 ) Ω W ((∂ 1 ψ|∂ 2 ψ|e 3 + w)) dx ≤ Ω W 0 ((∂ 1 ψ|∂ 2 ψ)) dx. ( 40
)
Let G : W 1,p 0 (ω; R 3 ) → R be defined by

G(v) = 2 ω W 0 ((∂ 1 ψ|∂ 2 ψ)) dx 1 dx 2 - ω F • (φ 0 (0)+v) dx 1 dx 2 .
It follows from ( 40) that for all

v ∈ V M J(0)(v) ≤ G(v). (41)
Since J(0) is lower semicontinuous on L p (Ω; R 3 ), its restriction to V M is sequentially weakly lower semicontinuous on V M equipped with the W 1,p (ω; R 3 ) topology. Therefore, J(0

) | V M ≤ Γ-G
where Γ-G is the sequential weak lower semicontinuous envelope of G on W 1,p 0 (ω; R 3 ). It is known, see Acerbi and Fusco (1984), that Γ-G is given by

Γ-G(v) = 2 ω QW 0 ((∂ 1 ψ|∂ 2 ψ)) dx 1 dx 2 - ω F • (φ 0 (0)+v) dx 1 dx 2 .
which proves proposition 2.6. Theorem 2.2 is proved by gathering all previous results.

Convergence of the rescaled deformations and the nonlinear membrane model

We now use Theorem 2.2 to characterize the asymptotic behavior of diagonal minimizing sequences of rescaled deformations φ(ε) for the sequence of rescaled energies I(ε), which are such that

I(ε)(φ(ε)) ≤ inf ψ∈Φ(ε) I(ε)(ψ)+h(ε)
where h is a positive function such that h(ε) → 0 when ε → 0 and the sets of admissible deformations are

Φ(ε) = {ψ ∈ W 1,p (Ω; R 3 ); ψ(x) = φ 0 (ε)(x) on Γ}.
We introduce the space of membrane deformations as

Φ M = {ψ ∈ W 1,p (Ω; R 3 ), ∂ 3 ψ = 0, ψ(x) = (x 1 , x 2 , 0) T on Γ}, which is isomorphic to ΦM = { ψ ∈ W 1,p (ω; R 3 ), ψ(x 1 , x 2 ) = (x 1 , x 2 , 0) T on ∂ω}.
Theorem 2.7. The sequence φ(ε) is relatively weakly compact in W 1,p (Ω; R 3 ). Its limit points φ belong to Φ M and are identified with elements φ of ΦM , solutions of the minimization problem Ī(0)( φ) = inf ψ∈ ΦM Ī(0)( ψ), where the membrane energy Ī(0) is given by

Ī(0)( ψ) = 2 ω QW 0 (∇ ψ) dx 1 dx 2 - ω F • ψ dx 1 dx 2 . ( 42
)
Proof -We follow an argument due to De Giorgi. With the rescaled deformation φ(ε), we associate a rescaled displacement u(ε)

∈ V by u(ε) = φ(ε) -φ 0 (ε). Obviously, J(ε)(u(ε)) ≤ inf v∈V J(ε)(v) + h(ε). In particular, J(ε)(u(ε)
) is uniformly bounded and Lemma 2.3 shows that the sequence u(ε) is relatively weakly compact in V and that its limit points belong to V M . This implies in particular that u(ε) is relatively compact in L p (Ω; R 3 ).

On the other hand, it is clear that ≤ inf v∈L p (Ω;R 3 ) J(ε)(v) + h(ε). Let u be a limit point of u(ε) for the strong topology of L p (Ω; R 3 ) (such a u is also a limit point of u(ε) for the weak topology of V ). Without loss of generality, we may assume that u(ε) → u strongly in L p (Ω; R 3 ). Let v be an arbitrary element of L p (Ω; R 3 ) and consider a sequence v(ε)

∈ L p (Ω; R 3 ) such that v(ε) → v, J(ε)(v(ε)) → J(0)(v).
Such a sequence exists by the very definition of Γ-convergence. Since J

(ε)(u(ε)) ≤ J(ε)(v(ε)) + h(ε), it follows that J(0)(u) ≤ lim inf J(ε)(u(ε)) ≤ lim inf( J(ε)(v(ε)) + h(ε)) = J(0)(v).
Therefore, u is a minimizer of J(0). Rewriting this statement in terms of φ, we obtain Theorem 2.7.

Remark 2.8. i) It is classical in Γ-convergence theory that the energies converge as well, in the sense that

I(ε)(φ(ε)) → Ī(0)( φ).
ii) If the three-dimensional elasticity problem has solutions that minimize the elastic energy, Theorem 2.7 applies to these minimizers. Such is the case for the polyconvex stored energy functions. Examples of realistic stored energy functions that satisfy the above hypotheses include Ogden's stored energy functions with appropriate exponents. For a derivation of the membrane model under the constraint det(∇φ) = 0, see Anza [START_REF] Hafsa | The nonlinear membrane model: Variational derivation under the constraint det∇u = 0[END_REF].

iii) The weak limits φ of weakly convergent subsequences of φ(ε) do not depend on x 3 , in which sense the problem becomes two-dimensional in the limit. If regular enough, the associated function describes a deformation of ω into a surface of R 3 . The elastic energy of such a deformation, 2 ω QW 0 (∇ φ) dx 1 dx 2 , only depends on its first derivatives. There are thus no bending effects associated with curvature, in which sense the resulting model is a nonlinear membrane model. iv) If the function QW 0 is smooth enough, the Euler-Lagrange equations for the membrane problem assume the form

-∂ β ∂ ∂ F QW 0 (∇ φ) iβ = 1 2 F i in ω, φ(x 1 , x 2 ) = (x 1 , x 2 , 0) T on ∂ω, ( 43
)
where Greek indices take their values in the set {1, 2}, Latin indices in the set {1, 2, 3} and the summation convention is understood. System (43) is a system of three quasilinear partial differential equations in the three unknowns φi . If the applied load is the weight, assuming the membrane is horizontal in its reference configuration, the limit load is of the form 1/2F = (0, 0, -ρg) T . The function F → TR ( F ) = ∂ ∂ F QW 0 ( F ) appears as the constitutive law for the analogue of the first Piola-Kirchoff stress tensor in the membrane. It gives the Lagrangian description of the tensile stresses in the membrane. It appears that the limit membrane problem retains the full quasilinear structure of three-dimensional elasticity, in contrast with such nonlinear plate models as the von Kármán equations that are only semilinear.

v) The explicit computation of QW 0 when W is the three-dimensional Saint Venant Kirchhoff energy is given in Le Dret and Raoult (1995a).

vi) The formal derivation of two-dimensional bending models was obtained in [START_REF] Fox | A justification of nonlinear properly invariant plate theories[END_REF] by asymptotic expansions. The bending model was there shown to be part of a hierarchy of models starting with the membrane model and ending with the linear plate model. Preliminary results for a rigorous derivation of the bending model by Γ-convergence arguments were given in [START_REF] Pantz | Quelques Problèmes de Modélisation en Élasticité non Linéaire[END_REF]. The issue was completely settled in [START_REF] Friesecke | A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity[END_REF] where an important rigidity lemma is proved. See also [START_REF] Friesecke | A hierarchy of plate models derived from nonlinear elasticity by Γ-convergence[END_REF].

vii) The previous method can be adapted to the study of the consistency of director models (Cosserat theory). In such models, the deformation is assumed to have a specific form. Namely, the deformation of each point (x 1 , x 2 , x 3 ) of the three-dimensional body is assumed to be the sum of the deformation of point (x 1 , x 2 ) and of a term colinear to a vector depending on (x 1 , x 2 ) only, called the director vector. Some models assume the director to be of norm 1, or to remain normal to the deformed configuration. We proved in Le [START_REF] Dret | Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results[END_REF] that the inextensibility constraint is inadequate in the membrane regime. On the contrary, the unconstrained Cosserat hypothesis provides a correct framework to account for the nonlinear membrane behavior. In order to obtain information on the asymptotic behavior of the director field, we had to study the weak lower semicontinuity of integral functionals of the form

I(ψ, δ) = ω W (∇ψ, δ) dx where ψ : ω ⊂ R 2 → R 3 , δ : ω → R 3
. This led us to introduce the notion of cross-quasiconvexity which is related to A-convexity, see [START_REF] Fonseca | A-quasiconvexity, lower semicontinuity and Young measures[END_REF].

Quasiconvexity in the derivation of lattice homogenized models

This section deals with lattices. For the sake of definiteness, we study bidimensional lattices that can deform into R 3 . Given a lattice made of bars and nodes, or given an atom lattice, we aim at deriving a continuous model whose macroscopic behavior is similar to the discrete lattice behavior. The discrete model we start from takes into account the changes of lengths as well as the changes of angles under deformation. Formal results in this setting were first given in [START_REF] Caillerie | Cell-to-muscle homogenization. Application to a constitutive law for the myocardium[END_REF] and in [START_REF] Mourad | Description Topologique de l'Architecture Fibreuse et Modélisation Mécanique du Myocarde[END_REF] and they are valid for any lattice. They were detailed later on in [START_REF] Caillerie | Discrete homogenization in graphene sheet modeling[END_REF] for graphene sheets whose carbon atoms make hexagonal networks. The rigorous results we present here are valid for a restricted class of lattices, square lattices are a good example of. An extended version will be published in a joint work with Nicolas Meunier and Olivier Pantz.

The lattice energy

We consider a discrete two-dimensional lattice.

Let Ω =]0, 1[ 2 , N ∈ N, r = 1
N and let | • | denote the Euclidean norm into R 3 . We assume that the lattice nodes coincide at rest with the points M i,j = (ir, jr) ∈ Ω, (i, j) ∈ {0, 1, . . . , N } 2 . For the sake of simplicity, we assume that the lattice is fixed at points located on the boundary. All arguments below extend easily to other cases, for instance a lattice whose left edge nodes are fixed. Under a loading G acting on the internal nodes, the lattice undergoes a deformation ψ from {M i,j , (i, j) ∈ {0, 1, . . . , N } 2 } into R 3 that minimizes, under the placement constraint, the energy

k 1 2 0≤i≤N -1,1≤j≤N -1 (|ψ(M i+1,j ) -ψ(M i,j )| -r) 2 + k 1 2 1≤i≤N -1,0≤j≤N -1 (|ψ(M i,j+1 ) -ψ(M i,j )| -r) 2 + k 2 2 0≤i,j≤N ((k,l),(k ′ ,l ′ ))∈Cij ψ(M k ′ ,l ′ ) -ψ(M i,j ) |ψ(M k ′ ,l ′ ) -ψ(M i,j )| • ψ(M k,l ) -ψ(M i,j ) |ψ(M k,l ) -ψ(M i,j )| 2 - N -1 i,j=1 G(M i,j ) • ψ(M i,j ) (44)
where k 1 and k 2 are stiffness coefficients and where C ij allows to number the pairs of bars sharing one end identified by (i, j): for (i, j) ∈ {0, 1, . . . , N -1} 2 , C ij contains four elements; otherwise it contains two elements. Obviously, in such a model segments between nodes remain straight under any deformation. Therefore, when the deformation of nodes in known, the lattice deformation is known. This is the simplest example of an energy taking into account the length changes and the angle changes due to the deformation. The internal energy is equal to 0 when all segments are of length r and when all angles are equal to π/2. Such an energy applies to mechanical networks and to atom lattices as well, in which case segments are atomic bonds. Adjacent nodes cannot be set on a single point because bar lengths cannot be equal to 0. We do not assume ψ to be globally one-to-one thus allowing for folding along the bars. In other words, ψ must satisfy the constraint ψ(M k,l ) = ψ(M i,j ) for adjacent pairs of indices. As a consequence, denominators in (44) are not equal to 0 and the energy is well defined. We emphasize the fact that the geometry of the lattice we study here is extremely simple: at rest, translating a reference pattern made of one node and two segments along two orthogonal directions generate the whole of the structure. The derivation of a continous model has been performed by formal asymptotic methods in a more general setting. Namely, the reference pattern contains several nodes and an arbitrary number of bars, see [START_REF] Mourad | Description Topologique de l'Architecture Fibreuse et Modélisation Mécanique du Myocarde[END_REF] and [START_REF] Caillerie | Cell-to-muscle homogenization. Application to a constitutive law for the myocardium[END_REF], and [START_REF] Caillerie | Discrete homogenization in graphene sheet modeling[END_REF] for results specifc to hexagonal networks. Here, we aim at obtaining rigorous convergence proofs and our reasonings are in the spirit of those by [START_REF] Braides | Continuum limits of discrete sytems without convexity hypotheses[END_REF], [START_REF] Alicandro | A general integral representation result for continuum limits of discrete energies with superlinear growth[END_REF]. We consider that the square lattice behavior is close to the limit behavior of a sequence of lattices whose grid step εr goes to 0. Data for these lattices are given by assuming that k ε 1 = k 1 , k ε 2 = ε 2 k 2 and that there exists a continuous function g defined on Ω such that for all i, j = 1, . . . , 1 εr , G ε (i εr, j εr) = ε 2 r 2 g(i εr, j εr).

In order to write convergence results, we need the minimization problems associated with all lattices in this sequence to be defined on a single functional space. To this aim, we first subdivide squares with edge length h = εr into triangles. In other words, we construct a triangulation denoted by T 1 h a way reminiscent of finite element subdivisions (hence the choice of the notation h), see Fig. 1. With each mapping ψ defined on the nodes (ih, jh) we can associate a mapping defined on Ω that is affine on each triangle, globally continuous and coincides with ψ at each node. This new mapping can still be denoted by ψ. Obviously,

∀x 1 ∈]ih, (i + 1)h[, ψ((i + 1)h, jh) -ψ(ih, jh) = h ∂ 1 ψ(x 1 , jh), ∀x 2 ∈]jh, (j + 1)h[, ψ(ih, (j + 1)h) -ψ(ih, jh) = h ∂ 2 ψ(ih, x 2 ),
and ∂ 1 ψ, ∂ 2 ψ are R 3 -valued functions that remain constant over each triangle. This typically allows to write terms due to the length changes in the energy (44) corresponding to the h-lattice as integrals of functions depending on the gradient of ψ only. Inner products in the third line that correspond to two segments belonging to a single triangle of T 1 h can be written as integrals of functions depending on the gradient of ψ, as well. This trick does not work for inner products corresponding to pairs of bars that are not the edges of a single triangle in T 1 h . Therefore we introduce a second triangulation denoted by T 2 h that is orthogonal to T 1 h , see Fig. 1, and with each mapping ψ defined on the nodes (ih, jh) we associate a mapping ψ ⊥ defined on Ω that is affine on each triangle of T 2 h , globally continuous and coincides with ψ at each node. Upon some computation and dividing by the first stiffness coefficient, we arrive at the family of scaled energies

I h (ψ) = Ω (|∂ 1 ψ| -1) 2 + (|∂ 2 ψ| -1) 2 + K ∂ 1 ψ |∂ 1 ψ| • ∂ 2 ψ |∂ 2 ψ| 2 dx + Ω (|∂ 1 ψ ⊥ | -1) 2 + (|∂ 2 ψ ⊥ | -1) 2 + K ∂ 1 ψ ⊥ |∂ 1 ψ ⊥ | • ∂ 2 ψ ⊥ |∂ 2 ψ ⊥ | 2 dx -L h (ψ) (45)
where

K = 4 r 2 k p k l , L h (ψ) = h 2 1 h i,j=1 f (ih, jh) • ψ(ih, jh) with f = 4 k l g.
defined on the sets

A * h = {ψ ∈ C 0 ( Ω; R 3 ), ∀ T ∈ T 1 h , ψ |T ∈ P 1 (T ), ψ |∂Ω = id, ∀(k, l), (k ′ , l ′ )s.t.|k ′ -k| + |l ′ -l| = 1, ψ(k ′ h, l ′ h) = ψ(kh, lh)}.(46)
Functions satisfying the final assertion in (46) will be said to be pairwise one-to-one. In the sequel, we will also use the set A h which does not require the deformations to be pairwise one-to-one:

A h = {ψ ∈ C 0 ( Ω; R 3 ), ∀ T ∈ T 1 h , ψ |T ∈ P 1 (T ), ψ |∂Ω = id}. (47)

Convergence results

Minimizing I h on A * h amounts to minimize a functional that is defined on a subset of a finite-dimensional space. Nevertheless, the existence of a minimizer is not obvious since A * h is not closed. Therefore, we concentrate on the limit behavior of almost minimizers. More specifically, we consider functions φ h such that

φ h ∈ A * h , ∀ψ ∈ A * h , I h (φ h ) ≤ I h (ψ) + s(h), (48)
where s(h) ≥ 0, s(h) → 0 when h → 0. Our arguments follow the lines of reasoning that were described in Section 2. Let us first introduce some notation: let id : (

x 1 , x 2 ) ∈ Ω → (x 1 , x 2 , 0) ∈ R 3 and let W 1 : M 3×2 → R be defined by ∀F = [z 1 |z 2 ] ∈ M 3×2 , W 1 (F ) = (|z 1 | -1) 2 + (|z 2 | -1) 2 . ( 49
)
This function satisfies the coerciveness inequality

∀F = [z 1 |z 2 ] ∈ M 3×2 , W 1 (F ) ≥ 1 2 ||F || 2 -1. ( 50
)
Lemma 3.1. Let φ h a sequence in A * h satisfying (48). Then, there exists C > 0 such that for all h, I h (φ h ) ≤ C. Let ψ h ∈ A * h be a sequence such that I h (ψ h ) ≤ C < +∞ for all h. Then ψ h is bounded in H 1 (Ω; R 3 ).

Proof-Choosing ψ = id in (48), we have ψ ⊥ = id and we immediately obtain that I h (φ h ) ≤ I h (id)+s(h) = -L h (id)+s(h). Since for any h, L h (id) is a Riemann sum associated with f • id, the sequence L h (id) converges and we know by construction that s(h) goes to 0. Therefore, the first point in Lemma 3.1 is proved. Let us turn to the second point. We first establish that

∃C > 0, ∀h, ∀ψ h ∈ A h , ( h h 2 |ψ h (ih, jh)| 2 ) 1/2 ≤ ||ψ h || L 2 (Ω;R 3 ) . ( 51
)
This is obtained by sendind the restrictions of functions ψ h to all triangles in T 1 h onto a single triangle, say the unit rectangular triangle , as is customary in the finite element theory, and by using the fact that all norms are equivalent on P 1 ( T ). Now let ψ h ∈ A * h be a sequence such that I h (ψ h ) ≤ C < +∞. Then, by (51), ∀h,

Ω W 1 (∇ψ h (x)) dx ≤ C + L h (ψ h ) ≤ C(1 + ||ψ h || L 2 (Ω;R 3 ) ).
We conclude by (50) and by Poincaré inequality.

As a consequence, the following corollary holds true.

Corollary 3.2. For any sequence φ h satisfying (48), there exists a subsequence that we still label by h and there exists φ ∈ H 1 (Ω; R 3 ) such that φ h → φ in L 2 (Ω; R 3 ) and φ h ⇀ φ in H 1 (Ω; R 3 ).

In order to identify a problem whose φ is a minimizer, we use Γ-convergence tools. To this end, we extend energies I h to L 2 (Ω; R 3 ) in the classical way seen in Section 2. Namely, we set

∀ψ ∈ A * h , Ĩh (ψ) = I h (ψ), ∀ψ ∈ L 2 (Ω; R 3 ) \ A * h , Ĩh (ψ) = +∞. (52)
Obviously, φ h solves (48) if and only if φ h satisfies

φ h ∈ L 2 (Ω; R 3 ), ∀ψ ∈ L 2 (Ω; R 3 ), Ĩh (φ h ) ≤ Ĩh (ψ) + hs(h). (53)
Moreover, we need to work on vector spaces rather than on affine spaces. Therefore, we define the sets

V * h = A * h -id = {v ∈ C 0 ( Ω; R 3 ), ∀ T ∈ T 1 h , v |T ∈ P 1 (T ), v |∂Ω = 0, id + v pairwise one-to-one}, ( 54
)
with the associated functionals Jh given by

∀v ∈ V * h , Jh (v) = J h (v) := I h (id + v), ∀v ∈ L 2 (Ω; R 3 ) \ V * h , Jh (v) = +∞ (55) or equivalently ∀v ∈ L 2 (Ω; R 3 ), Jh (v) = Ĩh (id + v). (56)
Finally, we extract a Γ-convergent subsequence for the L 2 (Ω; R 3 )-topology and call J0 its Γ-limit. As usual, the uniqueness of J0 will make the extraction of this subsequence superfluous a posteriori .

Proposition 3.3. u := φ -id minimizes J0 on L 2 (Ω; R 3 ).

Proof-We recall the argument that was already given in Proposition 2.7. From Corollary 3.2, we know that u h := φ h -id converges to u in L 2 (Ω; R 3 ). Therefore, on the one hand, J0 (u) ≤ lim inf Jh (u h ). On the other hand, for any v in L 2 (Ω; R 3 ), there exists v h in L 2 (Ω; R 3 ) such that Jh (v h ) converges to J0 (v). Therefore, for any v ∈ L 2 (Ω; R 3 ),

J0 (u) ≤ lim inf Jh (u h ) ≤ lim inf( Jh (v h ) + hs(h)) = J0 (v) (57)
Let us now proceed to the identification of J0 (v) for any v in L 2 (Ω; R 3 ). The case when v does not belong to H 1 0 (Ω; R 3 ) is immediate.

Proposition 3.4. For all v in L 2 (Ω, R 3 ) \ H 1 0 (Ω, R 3 ), J0 (v) = +∞.
We proceed by contradiction. Suppose J0 (v) < +∞. Since Jh Γ-converges to J0 for the L 2 (Ω; R 3 )-topology, there exists a sequence v h in L 2 (Ω; R 3 ) such that v h → v in L 2 (Ω; R 3 ) and Jh (v h ) → J0 (v) < +∞. Obviously Jh (v h ) is bounded from above. Therefore, v h belongs to V * h and from Lemma 3.1, we deduce that v h converges weakly to v in H 1 (Ω; R 3 ) which states in particular that v belongs to H 1 0 (Ω; R 3 ).

This result implies that u := φ -id belongs to H 1 0 (Ω; R 3 ) and minimizes J0 on H 1 0 (Ω; R 3 ). Before looking for a precise expression of J0 (v) when v in H 1 0 (Ω; R 3 ), let us check that it is a finite number. This step was obvious in the framework of Section 2. Here, it requires some work. Actually, the result will not be obtained by simply defining a constant sequence v h = v and saying that (v) ≤ lim inf Jh (v h ) = lim inf J h (v h ) < +∞. Indeed, v is not necessarily affine not to mention not necessarily such that id + v is pairwise one-to-one. What we need to do is finding a sequence v h in V * h such that v h converges to v in L 2 (Ω; R 3 ). The following lemma gives stronger results. In the first assertion, we ignore the injectivity condition and we make use of the notation

V h = A h -id = {v ∈ C 0 ( Ω; R 3 ), ∀ T ∈ T 1 h , v |T ∈ P 1 (T ), v |∂Ω = 0}. ( 58
)
In the second assertion, we take the injectivity condition into account.

Lemma 3.5. For any v in H 1 0 (Ω; R 3 ), there exists a sequence v h such that

v h ∈ V h and v h → v in H 1 (Ω; R 3 ). Moreover elements v h can be taken in V * h .
Proof-Let Π h be the P 1 -interpolation operator associated with T 1 h . Standard results show that, for any w in H 2 (Ω; R 3 ), Π h (w) converges to w. If w belongs to (H 2 ∩ H 1 0 )(Ω; R 3 ), then Π h w is equal to 0 on the boundary. In other words, Π h w belongs to A h . Therefore, any w in (H 2 ∩ H 1 0 )(Ω; R 3 ) is the limit of a sequence w h , h → 0 with w h in A h . Now, well known density results state that any v in H 1 0 (Ω; R 3 ) is the limit of a sequence w n , n → +∞ such that w n belongs to (H 2 ∩ H 1 0 )(Ω; R 3 ). Upon extracting a diagonal sequence (recall that h is the inverse of an integer), we prove the first assertion of Lemma 3.5.

For obtaining the second assertion, we show that functions v h that were just obtained can be slightly modified into functions v * h such that the functions ψ * h := (id + v * h ) are pairwise one-to-one. We actually prove a stronger result, since we obtain functions ψ * h whose restrictions to the set of nodes are one-to-one. Define ψ h = id + v h and

N h = { (k, l), (k ′ , l ′ ) , k, l, k ′ , l ′ ∈ {0, . . . , 1 h } 2 ; ψ h (kh, lh) = ψ h (k ′ h, l ′ h)}.
This set is nonempty since ψ h = id on the boundary. Therefore,

C h := min N h |(ψ h (k ′ h, l ′ h) -ψ(kh, lh)| is strictly positive. Finally, let λ h = C h 2 √
2 . We define v * h as follows:

∀(x 1 , x 2 ) s.t. h ≤ x 1 , x 2 ≤ 1 -h, v * h = v h + λ h id, v * h|∂Ω = 0
and v * h is globally continuous and affine on each triangle in T 1 h . Note that since v h and id are affine per triangle, there is no contradiction in the previous requirements. Let us check that the restriction of ψ

* h = id + v * h to the set of nodes is one-to-one. Suppose that ψ * h (kh, lh) = ψ * h (k ′ h, l ′ h) with (k, l), (k ′ , l ′ ) ∈ {0, . . . , 1 h } 2 . Let us show that (k, l) = (k ′ , l ′ ).
We distinguish several cases: -Case 1: (kh, lh) and (k ′ h, l ′ h) belong to ∂Ω. Since ψ h coincides with id on ∂Ω, obviously (k, l) = (k ′ , l ′ ).

-Case 2: (kh, lh) and (k ′ h, l ′ h) belong to Ω. Then -either ψ h (kh, lh) = ψ h (k ′ h, l ′ h) and the assumption on ψ * h implies that λ

h (kh, lh, 0) = λ h (k ′ h, l ′ h, 0), whence (k, l) = (k ′ , l ′ ) because λ h = 0, -or ψ h (kh, lh) = ψ h (k ′ h, l ′ h) and the assumption on ψ * h implies that ψ h (kh, lh)-ψ h (k ′ h, l ′ h) = λ h ((k ′ -k)h, (l ′ -l)h, 0). Therefore, |ψ h (kh, lh)- ψ h (k ′ h, l ′ h)| ≤ λ h √ 2 < C h 2 which is contradictory with the definition of C h . -Case 3: (kh, lh) ∈ ∂Ω and (k ′ h, l ′ h) ∈ Ω. Similar to case 2. We still have to prove that v * h converges to v in H 1 (Ω; R 3 ). Clearly, C h ≤ h and λ h ≤ h 2 √ 2 . Therefore, v h +λ h id converges to v in H 1 (Ω; R 3 ). The result remains true for v * h which differs from v h + λ h id on of width h: L ∞ bounds for v *
h and its partial derivatives on these bands are easily obtained.

Corollary 3.6. J0 is finite on

H 1 0 (Ω; R 3 ). Proof-For any v in H 1 0 (Ω; R 3 ), there exists v h in V * h such that v h → v in L 2 (Ω; R 3 ). Therefore, J0 (v) ≤ lim inf Jh (v h ) = lim inf J h (v h ) < +∞.
Finding an expression for J0 (v), v in H 1 0 (Ω; R 3 ) is now divided into two steps. First, we exhibit a functional that is smaller than J0 ; then, we prove that it is greater as well. Let us recall the definition of J h . For any v h in V * h and

ψ h = id + v h , J h (v h ) = I h (ψ h ) = Ω (W 1 +W 2 )(∇ψ h (x))+(W 1 +W 2 )(∇ψ ⊥ h (x)) dx-L h (ψ h ), (59)
where

∀F = [z 1 |z 2 ] ∈ M 3×2 s.t. z 1 = 0 and z 2 = 0, W 2 (F ) = K z 1 |z 1 | • z 2 |z 2 | 2 . (60) Let v in H 1 0 (Ω; R 3 ). By the definition of Γ-convergence, there exists v h in L 2 (Ω; R 3 ) such that v h → v in L 2 (Ω; R 3 ) and Jh (v h ) → J0 (v) < +∞. Equivalently, ψ h = id + v h converges to ψ = id + v in L 2 (Ω; R 3
) and Ĩh (ψ h ) converges to Ĩ0 (ψ) < +∞. From Lemma 3.1, we derive that ψ h belongs to A * h and converges weakly to ψ in H 1 (Ω; R 3 ). Let us show that ψ ⊥ h exhibits similar properties.

Lemma 3.7. For any sequence ψ h in A h such that ψ h converges to ψ strongly in L 2 (Ω; R 3 ) and weakly in H 1 (Ω; R 3 ), the sequence ψ ⊥ h converges to ψ strongly in L 2 (Ω; R 3 ) and weakly in H 1 (Ω; R 3 ) as well. Moreover,

∇ψ ⊥ h L 2 (Ω;M3×2) = ∇ψ h L 2 (Ω;M3×2) .
Proof-Let Q be the square with vertices (ih, jh), ((i + 1)h, jh), ((i + 1)h, (j + 1)h), (ih, (j + 1)h). We divide Q into two triangles T 1 and T 3 where the vertices of T 1 (resp. T 3 ) are (ih, jh), ((i + 1)h, jh), (ih, (j + 1)h) (resp. ((i+1)h, jh), ((i+1)h, (j+1)h), (ih, (j+1)h). We divide Q into two triangles T 2 and T 4 as well where the vertices of T 2 (resp. T 4 ) are (ih, jh), ((i + 1)h, jh), ((i + 1)h, (j + 1)h) (resp. (ih, jh), ((i + 1)h, (j + 1)h), (ih, (j + 1)h). Restricted to T 1 (resp. T 3 ), ∂ 1 ψ h is a constant vector that is equal to

∂ 1 ψ ⊥ h restricted to T 2 (resp. T 4 ). Therefore, Q |∂ 1 ψ h | 2 dx = T1∪T3 |∂ 1 ψ h | 2 dx = T2∪T4 |∂ 1 ψ ⊥ h | 2 dx = Q |∂ 1 ψ ⊥ h | 2 dx.
Similar equalities hold for the derivatives with respect to x 2 . Upon adding the equalities for all squares Q, we obtain

∇ψ ⊥ h L 2 (Ω;M3×2) = ∇ψ h L 2 (Ω;M3×2) .
Hence, ∇ψ ⊥ h L 2 (Ω;M3×2) is bounded. As ψ ⊥ h coincides with the identity on ∂Ω, Poincaré inequality proves that ψ ⊥ h is bounded in H 1 (Ω; R 3 ). Therefore ψ ⊥ h converges weakly to some function in H 1 0 (Ω; R 3 ). Let us now prove that χ h := ψ ⊥ h -ψ h converges to 0 in L 2 (Ω; R 3 ). Since ψ h and ψ ⊥ h coincide on the vertices on any Q defined above, they coincide on the edges of Q. In other words, χ h is equal to 0 on ∂Q, and rewriting the classical proof of Poincaré inequality reads

χ h (x 1 , x 2 ) = x1 ih ∂ 1 χ h (ξ 1 , x 2 ) dξ 1 ,
which provides by Cauchy-Schwarz inequality,

|χ h (x 1 , x 2 )| 2 ≤ h (i+1)h ih (∂ 1 χ h ) 2 (ξ 1 , x 2 ) dξ 1 ,
and upon integrating with respect to x 2 (j+1)h

jh |χ h (x 1 , x 2 )| 2 dx 2 ≤ h Q |∂ 1 χ h (x)| 2 dx. Finally Q |χ h (x)| 2 dx ≤ h 2 Q |∂ 1 χ h (x)| 2 dx, Ω |χ h (x)| 2 dx ≤ h 2 Ω |∂ 1 χ h (x)| 2 dx,
which implies that χ h L 2 (Ω;R 3 ≤ h ∇χ h L 2 (Ω;M3×2) . Using the first part of proof, we obtain that ψ ⊥ h -ψ h converges to 0 in L 2 (Ω; R 3 ) from which Lemma 3.7 follows.

Let us now proceed to study the limit behavior of I h (ψ h ). To this aim, we extend W 2 to the whole of M 3×2 by letting: for all F = [z 1 |z 2 ] in M 3×2 , W e 2 (F ) = W 2 (F ) if z 1 = 0 and z 2 = 0, W e 2 (F ) = K if not.

(61)

This new function W e 2 is not continuous on M 3×2 , but it is Borel measurable and bounded by K. As a consequence, Z defined by

Z : M 3×2 → R, Z = W 1 + W e 2 (62)
is Borel measurable and satisfies the coerciveness and growth inequalities

∀F ∈ M 3×2 , α||F || 2 -β ≤ Z(F ) ≤ α ′ ||F || 2 + β ′ , α, α ′ , β, β ′ > 0. (63)
Since W e 2 extends W 2 , we can see (59),

I h (ψ h ) = Ω Z(∇ψ h (x)) + Z(∇ψ ⊥ h (x)) dx -L h (ψ h ). ( 64
)
Let QZ be the quasiconvex envelope of Z which is classically defined, see [START_REF] Dacorogna | Direct Methods in the Calculus of Variations[END_REF], by QZ(F ) = sup{g(F ); g : M 3×2 → R, g quasiconvex, g ≤ Z}.

(65)

Since Z takes finite values only, all functions g in ( 65) are continuous: indeed, rank-one convex functions that are finite valued are continuous. Therefore, QZ is upper semicontinuous, hence Borel measurable and quasiconvex. Let us note finally that

∀F ∈ M 3×2 , 0 ≤ QZ(F ) ≤ α ′ ||F || 2 + β ′ , (66)
and let us define W on M 3×2 by

W = 2Z = 2(W 1 + W e 2 ). ( 67
)
Proposition 3.8. For all ψ in (id+H 1 0 (Ω; R 3 )), Ĩ0 (ψ) ≥ Ω QW (∇ψ(x)) dx-Ω f (x) • ψ(x) dx.

Proving this proposition requires to deal in particular with the term due to the external loadings. As this is not central to our analysis, we do not detail the proof of the following lemma although it is not a totally trivial exercise in approximation theory. Lemma 3.9. For any sequence ψ h in A h such that ψ h converges weakly to ψ in H 1 (Ω, R 3 ), L h (ψ h ) converges to Ω f (x) • ψ(x) dx.

Proof of Proposition 3.8 -Obviously,

I h (ψ h ) ≥ Ω QZ(∇ψ h (x)) + QZ(∇ψ ⊥ h (x)) dx -L h (ψ h ),
where both integrands are in L 1 (Ω; R) since QZ is continuous and satisfies (66). For the same reasons, the functional

H : ψ ∈ H 1 (Ω; R 3 ) → H(ψ) = Ω QZ(∇ψ(x)) dx ∈ R
is well defined. It has been proved by Acerbi and Fusco (1984), see also (2007), that under properties (66), H is sequentially weakly lower semicontinuous on H 1 (Ω; R 3 ). Therefore,

Ĩ0 (ψ) = lim I h (ψ h ) ≥ lim inf(H(ψ h ) + H(ψ ⊥ h )) -lim L h (ψ h ) ≥ lim inf H(ψ h ) + lim inf H(ψ ⊥ h ) - Ω f (x) • ψ(x) dx ≥ 2H(ψ) - Ω f (x) • ψ(x) dx,
since by Lemma 3.7 both sequences ψ h and ψ ⊥ h converge weakly to ψ.

The remaining part of this section is devoted to showing that the inequality in Proposition 3.8 is actually an identity. Proposition 3.10. For all ψ in (id+H 1 0 (Ω; R 3 )), Ĩ0 (ψ) ≤ Ω QW (∇ψ(x)) dx-Ω f (x) ψ(x) dx.

Proof -By the definition of Γ-convergence, Ĩ0 (ψ) = J0 (v) ≤ lim inf I h (ψ h ) for any sequence ψ h ∈ A * h that converges to ψ in L 2 (Ω; R 3 ). From Lemma 3.5, we can choose a sequence ψ h ∈ A * h that converges to ψ in H 1 (Ω; R 3 ). Let us check that ψ ⊥ h converges to ψ in H 1 (Ω; R 3 ) as well. From Lemma 3.7, we already know that ψ ⊥ h converges to ψ strongly in L 2 (Ω; R 3 ) and weakly in H 1 (Ω; R 3 ). Therefore, it suffices to show that ψ ⊥ h H 1 (Ω;R 3 ) → ψ H 1 (Ω;R 3 ) . Actually, from Lemma 3.5 again,

ψ ⊥ h H 1 (Ω;R 3 ) = ψ ⊥ h L 2 (Ω;R 3 ) + ∇ψ ⊥ h L 2 (Ω;M3×2) = ψ ⊥ h L 2 (Ω;R 3 ) + ∇ψ h L 2 (Ω;M3×2) .
Hence,

ψ ⊥ h H 1 (Ω;R 3 ) → ψ L 2 (Ω;R 3 ) + ∇ψ L 2 (Ω;M3×2) = ψ H 1 (Ω;R 3 ) .
Let us examine the limit behavior of

I h (ψ h ) = Ω Z(∇ψ h (x)) + Z(∇ψ ⊥ h (x)) dx -L h (ψ h )
where we know that ψ h → ψ, ψ ⊥ h → ψ in H 1 (Ω; R 3 ), and where we recall that Z is not continuous on the set of matrices whose a column is equal to 0. We separate I h (ψ h ) in two parts by writing I h (ψ h ) = X h + Y h , with

X h = Ω W 1 (∇ψ h (x)) + W 1 (∇ψ ⊥ h (x)) dx -L h (ψ h ), Y h = Ω W 2 (∇ψ h (x)) + W 2 (∇ψ ⊥ h (x)) dx.
Since W 1 is continuous on M 3×2 , by the dominated convergence theorem and by Lemma 3.9,

X h → Ω 2 W 1 (∇ψ(x)) dx - Ω f (x) • ψ(x) dx.
In order to deal with Y h , we choose an element δ 1 (resp. δ 2 ) in the L 2 class of ∂ 1 ψ (resp. ∂ 2 ψ) and we decompose Ω in two measurable subsets defined by Ω 1 = {x ∈ Ω; δ 1 (x) = 0 and δ 2 (x) = 0}, Ω 2 = Ω \ Ω 1 .

Clearly,

Y h = Ω1 W 2 (

Figure 1 .

 1 Figure 1. Left: triangulation T 1 h , right: triangulation T 2 h

  The dominated convergence theorem allows to deal with the first term and we get For the second term we simply remark thatΩ2 W 2 (∇ψ h (x))+W 2 (∇ψ ⊥ h (x) dx ≤ 2 K meas(Ω 2 ) = 2

∇ψ h (x)) + W 2 (∇ψ ⊥ h (x) dx + Ω2 W 2 (∇ψ h (x)) + W 2 (∇ψ ⊥ h (x) dx. Ω1 W 2 (∇ψ h (x)) + W 2 (∇ψ ⊥ h (x) dx → 2 Ω1 W 2 (∇ψ(x)) dx = 2 Ω1 W e 2 (∇ψ(x)) dx.

Ω2 W e 2 (∇ψ(x)) dx.

Therefore,

2 (∇ψ(x)) dx.

At this point, we can say that ∀ψ ∈ (id + H 1 0 (Ω; R 3 )), Ĩ0 (ψ)

Since J0 is sequentially weakly lower semicontinuous on H 1 0 (Ω; R 3 ), it follows that J0 ≤ Γ -G where Γ -G is the sequential weak lower semicontinuous envelope of G on H 1 0 (Ω; R 3 ). We made use several times in this manuscript of the fact for Z : M 3×2 → R continuous, positive and satisfying coerciveness and growth hypotheses (63), the sequential weak lower semicontinuous envelope Γ -F of F defined by

Ω QZ(∇v(x)) dx. Although less known, the result remains true when Z is no longer continuous, but Borel measurable, see the relaxation theorems proved in [START_REF] Dacorogna | Direct Methods in the Calculus of Variations[END_REF]. This ends the proof of Proposition 3.10.

To conclude this section, we can state our final theorem.

Theorem 3.11. For all ψ in (id+H

Comments. The homogenized lattice energy QW inherits the invariance properties of W . On the one hand, W is obviously frame indifferent and it is immediate that QW is frame indifferent as well. Therefore, QW can be expressed as a function of C = F T F ; in other words, with obvious notation, there exists a function w : {(c 11 , c 22 , c 12 ) ∈ R 3 ; c 11 ≥ 0, c 2 12 ≤ c 11 c 22 } such that for all F in M 3×2 , W (F ) = w(c 11 , c 22 , c 12 ). On the other hand, the planar rotation of angle π/2 belongs to the symmetry group of W and this remains true for QW . As a consequence, it is easily shown that w is symmetric with respect to its first two variables. For more general results on the invariance of envelopes, see [START_REF] Bousselsal | Relaxation of functionals involving homogeneous functions and invariance of envelopes[END_REF]. Finally, let us mention that QW (F ) = 0 for matrices F such that c 11 ≤ 1, c 22 ≤ 1, c 2 12 ≤ (1 -c 11 )(1 -c 22 ). Therefore, QW does not coincide with W .

Bibliography

E. Acerbi and N. Fusco. Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal., 86:125-145, 1984.


Piotr Dworakowski
email: piotr.dworakowski@supergrid-institute.com

Michał Michna
email: michal.michna@pg.edu.pl

Andrzej Wilk
email: andrzej.wilk@pg.edu.pl

Thermal simulations and measurements of 3-phase medium frequency transformer in 100 kW DC-DC converter with 99.2% efficiency

This article deals with a 100 kW dual active bridge (DAB) DC-DC converter and a 20 kHz 3-phase medium frequency transformer (MFT). A computational fluid dynamics (CFD) model of the MFT is presented and compared with thermal measurements

showing a good fit. The core hotspot is close to the thermal runaway limit. The maximum winding temperature in the normal operating mode results in a comfortable headroom to enable the operation in the degraded mode. Moreover, it is likely that the transformer may be loaded with higher power. The efficiency of the DC-DC converter was measured 99.2% at 100 kW.

I. INTRODUCTION

The isolated DC-DC converter is the enabling technology in modern electrical power systems and transportation [START_REF] Dworakowski | Modelling and analysis of medium frequency transformers for power converters[END_REF]. Dual active bridge (DAB) is a prominent topology for high power applications requiring bidirectional power flow [START_REF] De Doncker | A threephase soft-switched high-power-density DC/DC converter for highpower applications[END_REF]. The 3-phase DAB (DAB3) is considered for the applications where high power density and high efficiency are required [START_REF] Soltau | Development and demonstration of a medium-voltage high-power DC-DC converter for DC distribution systems[END_REF], [START_REF] Xue | Single-phase vs. threephase high density power transformers[END_REF].

A medium frequency transformer (MFT) is one of the key components of the isolated DC-DC converters. The MFT is still quite a novel technology with lots of research interest [START_REF] Mogorovic | 100 kW, 10 kHz Medium-Frequency Transformer Design Optimization and Experimental Verification[END_REF]- [START_REF] Tomczuk | Electromagnetic and Temperature 3-D Fields for the Modular Transformers Heating Under High-Frequency Operation[END_REF]. In particular, only a few 3-phase MFT have been reported. A 10 kVA 1 kHz 3-phase MFT prototype was reported in [START_REF] Lee | Geometry optimization and characterization of three-phase medium frequency transformer for 10kVA Isolated DC-DC converter[END_REF] and a 2 kVA 100 kHz 3-phase MFT was reported in [START_REF] Noah | A novel three-phase LLC resonant converter with integrated magnetics for lower turn-off losses and higher power density[END_REF]. A 5 MW DAB3 was presented in [START_REF] Soltau | Development and demonstration of a medium-voltage high-power DC-DC converter for DC distribution systems[END_REF] but using three single-phase MFTs. The 3-phase MFT at high power (100 kW) and high frequency (20 kHz) was reported by the authors in [START_REF] Dworakowski | 3phase medium frequency transformer for a 100kW 1.2kV 20kHz Dual Active Bridge converter[END_REF]. The design of MFT must consider winding, core, insulation and cooling design constraints.

The choice of magnetic core material shall be done according to the material properties and cost [START_REF] Villar | Proposal and validation of medium-frequency power transformer design methodology[END_REF], [START_REF] Hurley | The Performance Factor for Magnetic Materials Revisited: The Effect of Core Losses on the Selection of Core Size in Transformers[END_REF]. The amorphous and especially nanocrystalline materials are preferred in low and medium frequencies due to the high flux density. The main advantage of ferrite cores is their low power loss, which makes them an attractive material for the construction of medium and high frequency transformers [START_REF] Bahmani | Design considerations of medium-frequency power transformers in HVDC applications[END_REF], [START_REF] Stojadinović | Modelling and Design of a Medium Frequency Transformer for High Power DC-DC Converters[END_REF]. The magnetic circuit can be realised with different geometrical structures: core type [START_REF] Villar | Optimal design and experimental validation of a Medium-Frequency 400kVA power transformer for railway traction applications[END_REF], shell type [START_REF] Mogorovic | 100 kW, 10 kHz Medium-Frequency Transformer Design Optimization and Experimental Verification[END_REF], matrix [START_REF] Ortiz | 1 Megawatt, 20 kHz, isolated, bidirectional 12kV to 1.2kV DC-DC converter for renewable energy applications[END_REF] or coaxial [START_REF] Kheraluwala | Coaxially wound transformers for high-power high-frequency applications[END_REF].

The design of MFT winding must take into account the skin and proximity effects. The foil winding is the cheapest but it causes a large parasitic capacitance of the winding [START_REF] Dworakowski | Hysteresis modelling of a medium frequency single-phase transformer[END_REF]. The tube conductor requires a specific manufacturing process and results with the significant conductor diameter [START_REF] Heinemann | An actively cooled high power, high frequency transformer with high insulation capability[END_REF]. The litz wire offers a good performance but it comes with a relatively high cost.

The insulation of MFT may be ensured by solid or liquid material. The solid insulation offers the dielectric strength of some tens of kV/mm. Usually it is realised by the impregnation of a material placed between the windings. In case of a breakdown the insulation material is definitively damaged. The advantage of the solid insulation is a limited maintenance. The liquid insulation, in addition to its primary function offers good cooling. A liquid-insulated transformer would be generally smaller than a solid-insulated one. However, the liquid insulation requires more maintenance effort and depending on the liquid used it may be more or less environmentally friendly. In MFT design, the high frequency effect on insulation material should be taken into account [START_REF] Birle | Breakdown of polymer dielectrics at high direct and alternating voltages superimposed by high frequency high voltages[END_REF], [START_REF] Anand | Electrical and optical partial discharge assessment of dielectric barriers in mineral oil and synthetic ester[END_REF].

The cooling of the MFT may be ensured by the natural or forced air convection or liquid cooling [START_REF] Ortiz | Optimized design of medium frequency transformers with high isolation requirements[END_REF]. The liquid cooling offers the best performance but it requires more complex cooling system and more important maintenance effort. The liquid cooling may be considered if a very high-power density is required or if the liquid is required for insulation. The thermal design of MFT is challenging due to the high-power loss density of a small size MFT.

There is a research competition in designing the highest efficiency DC-DC converter and MFT while keeping the small size. Mastering the power losses and simple cooling is the key to the successful design. This article presents the thermal design and efficiency measurement of a 3-phase MFT within a 100 kW DAB DC-DC converter. The measured efficiency of the isolated DC-DC converter is 99.2%. This article is organised as follows. Section II introduces the 3-phase DAB DC-DC converter and the MFT. Section III presents a computational fluid dynamics (CFD) model of the MFT. Section IV details the thermal measurements of the MFT including a comparison with the CFD simulation. Section V shows the efficiency measurement of the 100 kW DC-DC converter.

II. 3-PHASE DAB AND MFT

The 3-phase DAB (DAB3) is composed of two 3-phase bridges connected with a 3-phase transformer. The DAB3 circuit diagram is presented in Fig. 1. The operation of the DAB3 with the rectangular modulation is based on the phase shift δ between two bridges. The power flow is inversely proportional to the MFT leakage inductance Lac. A 100 kW DC-DC converter prototype was developed as detailed in [START_REF] Lagier | A 100 kW 1.2 kV 20 kHz DC-DC converter prototype based on the Dual Active Bridge topology[END_REF]. Both bridges are equipped with 1.7 kV SiC MOSFET [START_REF] Wolfspeed | CAS300M17BM2 1700V 225A 8.0mΩ SiC Half-Bridge[END_REF]. The normal operation mode is defined as Udc1 = Udc2 = 1200 V and the degraded operation mode as Udc1 = 1200 V and Udc2 = 0.8Udc1.

The design of the 3-phase MFT was detailed by the authors in [START_REF] Dworakowski | 3phase medium frequency transformer for a 100kW 1.2kV 20kHz Dual Active Bridge converter[END_REF]. The main data are summarized in Table I. The winding losses were calculated according to [START_REF] Tourkhani | Accurate analytical model of winding losses in round Litz wire windings[END_REF], which is derived from [START_REF] Dowell | Effects of eddy currents in transformer windings[END_REF]. The core losses were calculated according to [START_REF] Venkatachalam | Accurate prediction of ferrite core loss with nonsinusoidal waveforms using only Steinmetz parameters[END_REF], which is derived from [START_REF] Steinmetz | On the Law of Hysteresis[END_REF].

The 3-phase MFT power density is approximately 4 kVA/kg. A comparable 50 Hz transformer, at 400 V and natural air cooling, has the power density of less than 0.4 kVA/kg [START_REF] Abb | Dry-type low voltage transformers[END_REF]. The 3-phase MFT at 20 kHz is more than 10 times lighter than the 3-phase 50 Hz transformer.

III. CFD SIMULATION MODEL OF MFT

A computational fluid dynamics (CFD) model of the 3-phase MFT was developed in Ansys Icepak. Ansys Icepak is based on the Fluent solver which solves Navier-Stokes equations. The equation for conservation of mass can be written as follows:

𝜕𝜌 𝜕𝑡 + ∇ • (𝜌𝐯) = 0 , (1)
where ρ(x,t) is the fluid density which is a function of time t and position vector x and v is the fluid velocity vector. For an incompressible fluid (1) reduces to:

∇ • 𝐯 = 0 (2)
The law of conservation of momentum is:

𝜌 𝜕(𝜌𝐯) 𝜕𝑡 + ∇ • (𝜌𝐯𝐯) = -∇𝑝 + ∇ • 𝛕 + 𝜌𝐠 + 𝐅 , (3)
where p is the static pressure, τ is the stress tensor, g is the standard gravity vector and F is the external force vector. The stress tensor describes the momentum transfer on the surface of the fluid. For the Newtonian fluid, the stress tensor is proportional to the velocity gradient tensor:

𝛕 = 𝜇 [∇𝐯 + ∇𝐯 𝐓 - 2 3 ∇𝐯𝐈] , (4)
where μ is the molecular viscosity, I is the unit tensor. The energy equation considers the flow of mass and heat:

𝜕 𝜕𝑡 (𝜌ℎ) + ∇ • (𝜌ℎ𝐯) = ∇ • [(𝑘 + 𝑘 𝑡 )∇𝑇] + 𝑆 ℎ , (5)
where h is the sensible enthalpy, k he molecular conductivity, kt is the conductivity due to turbulent transport, T is the temperature, Sh includes any volumetric heat sources. In conducting solid regions, Ansys Icepak solves a simple conduction equation that includes the heat flux due to conduction and volumetric heat sources within the solid:

𝜕 𝜕𝑡 (𝜌ℎ) = ∇ • [𝑘∇𝑇] + 𝑆 ℎ (6)
The 3D model of the MFT presented in Fig. 2 was divided into four computational domains and the material properties were set as follows:

• Ω1 is the homogenized ferrite core with the thermal conductivity λ = 4.25 W/mK, specific heat c = 750 J/kgK, density ρ = 4800 kg/m 3 [START_REF] Ferroxcube | Soft ferrites and accessories, data handbook[END_REF],

• Ω2 is the litz wire winding; in [START_REF] Kyaw | Analytical Thermal Model for Inductor and Transformer Windings and Litz Wire[END_REF] a thermal model of a litz wire was proposed showing that the transverse thermal conductivity is below 1 W/mK; in the Icepak library, the material which has similar properties is epoxy resin with λ = 0.2 W/mK; hence, this homogenized material was used, • Ω3 is the background region with the air temperature 40°C, • Ω4 is the volume of fans with the same material properties as Ω3.

The emissivity coefficient of the core and winding was set to ε = 0.8. A turbulent air flow was considered. The bottom face of the background region is a wall with zero heat flux and all other faces are openings with constant temperature at 40°C. Three fans EBM papst 5214 NHH [START_REF]DC axial compact fan 5214 NHH[END_REF] were implemented with the linearized pressure-flow curve as presented in Fig. 3. The heat sources were assigned according to the winding and core power loss presented in Table I for the normal operating mode. The simulation result is presented in Fig. 4 showing the air velocity vectors displayed on a side view plane passing vertically across the middle of transformer and in Fig. 9 showing the temperature distribution on the core surface. It is seen that the windings are well cooled thanks to the air passing between the windings. The core is not cooled so well.

IV. THERMAL TEST OF MFT A. Test bench

A dedicated power converter test bench was developed allowing the experimental validation of the 3-phase DAB and MFT. The circuit diagram is presented in Fig. 5 and the test bench implementation in Fig. 6. The power circuit is arranged in back-to-back in order to test the DC-DC converter at full power with minimum energy consumption. The converter output is connected to its input and the whole is supplied from a DC power supply which sets the voltage reference and supplies the test circuit power loss. The DC-DC converter runs in the power regulation mode. The current transducers in the AC link are necessary for the internal converter protection and the control of DC bias [START_REF] Wattenberg | Impact of DC-Bias on Dual Active Bridge Control and How to Avoid it[END_REF].

B. Thermal test

The thermal test was performed at Udc1 = Udc2 = 1200 V and 100 kW corresponding to the normal operating mode. The MFT temperature was measured with some thermocouples placed around the transformer during the manufacturing. An additional measurement was done using a thermal camera. In Fig. 7 the thermal test result is presented showing the values of temperature rise ΔT. The test lasted for 6 hours. Some disturbances are present in the waveforms due to unintentional converter stops related to gate-drive EMI problems [START_REF] Geramirad | Experimental EMI study of a 3-phase 100kW 1200V Dual Active Bridge Converter using SiC MOSFETs[END_REF], [START_REF] Geramirad | Conducted EMI reduction in a 100kW 1.2kV Dual Active Bridge converter[END_REF]. At the end of the record the temperatures were stabilized and it seems fair to consider them as final values. The ambient temperature varied during the test between 23 and 37°C. At the end of the test the ambient temperature was close to 40°C set in CFD simulation.

As expected, the temperature of the central column (measured by thermocouples Tc2, Tc3, Tc4) is the highest, with Tc3 being the hotspot equal to 127°C (ΔT = 90 K). The core temperature is probably close to the thermal runaway limit. The Curie temperature of 3C90 is 220°C. The time constant of the hotspot is approximately 0.6 hours. The temperature of the side column (Tc10) is equal to 93°C.

The primary winding temperature (Tc5) is 56°C and the secondary winding temperature (Tc6) is nearly the same as ambient. The temperature of the insulation class H is respected. Some temperature margin is required to enable the DC-DC converter operation in degraded mode where the winding power loss is almost doubled when compared to the normal operation mode. Moreover, it is likely that the transformer may be loaded with higher power. In Fig. 8 there is presented the measured temperature distribution on the MFT surface using the thermal camera Flir A325 [START_REF] Flir | Thermal camera A325sc[END_REF]. The values should be considered approximate as the default thermal camera emissivity coefficients where used. The temperature measurement could be more precise if the emissivity coefficients were set. The general trend of temperature distribution is confirmed. The core central column is the hottest. The core yoke is cooler than the core side columns thanks to better heat exchange. The external winding is nearly at the ambient temperature.

C. Comparison between simulation and experimental results

In Fig. 9 there is presented the comparison between the CFD simulation and measurement. The temperature on the core surface is plotted and the values at the thermocouple locations are displayed. The red text gives the simulation result and the black text gives the measurement result. The measured temperature of the side column at 93°C is very similar to the simulated value at 90°C. The temperature of the central column is quite similar to the surface hotspot temperature simulated at 120°C and measured at around 127°C. The distribution of the temperature along the column height is not the same. The

V. EFFICIENCY MEASUREMENT OF DC-DC CONVERTER

The measurement of the DC-DC converter efficiency was performed using the precision power analyser ZES Zimmer LMG670 [START_REF] Zimmer | LMG670 -1 to 7 Channel Power Analyzer[END_REF]. The power of the DC supply was measured according to Fig. 6. This measurement technique was proven to be the most effective and rather overestimating the DC-DC converter power loss. This measurement takes into account the power of the complete test bench, including the DC-DC converter but also DC inductors and all wiring. The authors highlight that the measurement of power losses in highly efficient power converters is a metrological challenge. In this article, the DC-DC converter efficiency was measured accurately thanks to the converter ratio equal to 1.

The efficiency measurement result is presented in Fig. 10. At Udc1 = Udc2 = 1200 V and 100 kW, the DC-DC efficiency is 99.2%. A maximum efficiency of 99.25% is seen at 70 kW. Considering the transformer power loss contribution, the shape of the curve is as expected. Low efficiency at low power is due to the loss of zero voltage switching (ZVS) [START_REF] Lagier | Experimental validation and comparison of a SiC MOSFET based 100 kW 1.2 kV 20 kHz three-phase dual active bridge converter using two vector groups[END_REF], [START_REF] Lagier | Theoretical and experimental analysis of the soft switching process for SiC MOSFETs based Dual Active Bridge converters[END_REF]. 99.2% at 100 kW Assuming that the half of the power loss accounts for MFT then the half for the SiC MOSFETs [START_REF] Akagi | Power-loss breakdown of a 750-V, 100-kW, 20-kHz bidirectional isolated DC-DC converter using SiC-MOSFET/SBD dual modules[END_REF], the 3-phase MFT efficiency is expected approximately 99.6%. This corresponds to the MFT power loss at 400 W what is slightly lower than calculated in Table I (206+255=461 W). This is in line with the investigations presented in [START_REF] Michna | Modified Preisach model of hysteresis in multi air gap ferrite core medium frequency transformer[END_REF] where a new set of Steinmetz equation coefficients was proposed. However, if the power loss calculations are not valid, then it is surprising to see the good fit between the thermal simulation and measurements presented in section IV. Further analysis is required. In order to measure the MFT efficiency, the calorimetric methods should be considered [START_REF] Rothmund | 99% Efficient 10 kV SiC-Based 7 kV/400 V DC Transformer for Future Data Centers[END_REF]. The measurement of MFT efficiency using ZES Zimmer LMG670 was not successful.

VI. CONCLUSION

The 100 kW dual active bridge (DAB) converter and the 20 kHz 3-phase medium frequency transformer (MFT) were presented. The computational fluid dynamics (CFD) model of the 3-phase MFT was introduced. The thermal measurements of the MFT were detailed and compared with the CFD simulation. The efficiency of the DC-DC converter was measured.

The CFD simulation and experimental measurements show a good fit. The MFT surface hotspot is in the central column at 127°C. This high temperature is probably close to the thermal runaway limit but still well below the Curie temperature at 220°C. The maximum winding temperature in the normal operating mode is at the comfortable 56°C. There is enough headroom to enable the operation in degraded mode where the winding power loss is almost doubled. Moreover, it is likely that the transformer may be loaded with higher power.

The efficiency of the DC-DC converter was measured 99.2% at 100 kW. This is an accurate measurement which was only possible thanks to the DC-DC conversion ratio equal to 1. The 3-phase MFT efficiency is expected at approximately 99.6%. Any direct MFT efficiency measurement was not successful. Further research is required in order to identify an appropriate efficiency measurement method for MFT.

Fig. 1 .

 1 Fig. 1. Circuit model of the 3-phase dual active bridge (DAB3)

Fig. 2 .

 2 Fig. 2. CFD model of MFT including the domains: Ω1 core, Ω2 winding, Ω3 background and Ω4 fans

Fig. 3 .

 3 Fig. 3. EBM papst 5214 NHH fan curve [35] (blue) and linearized curve for CFD simulation (red)

Fig. 4 .

 4 Fig. 4. Thermal simulation result in steady state showing the air velocity vectors on a plane passing vertically across the middle of transformer, side view

Fig. 5 .Fig. 6 .

 56 Fig. 5. Circuit diagram of the test bench with 3-phase DAB DC-DC converter

Fig. 7 .

 7 Fig. 7. Thermal test result of the 3-phase MFT: core (Tc2, Tc3, Tc4 and Tc10) and winding (Tc5 and Tc6) temperature increase ΔT at the ambient temperature of 37°C maximum

Fig. 8 .

 8 Fig. 8. Thermal image of the 3-phase MFT

Fig. 9 .

 9 Fig. 9. Comparison between the thermal simulation (surface plot) and thermal measurement (text in black)

Fig. 10 .

 10 Fig. 10. Efficiency measurement of the 3-phase DAB. At 100 kW the DC-DC efficiency is 99.2%. Low efficiency at low power is due to the loss of zero voltage switching (ZVS)

TABLE I

 I

	3-PHASE MFT DATASHEET
	Rated active power	100 kW
	Primary and secondary phase voltage	566 V RMS
	Primary and secondary current (normal/degraded)	65 / 92 A RMS
	Frequency	20 kHz
	Vector group and core structure	Yy core type
	Winding material	copper litz wire
	Number of turns	20 (single layer)
	Winding wire cross section	30.39 mm 2
	Winding current density (normal/degraded)	2.1 / 3 A/mm 2
	Winding wire strand diameter	0.1 mm
	Number of strands	3870
	Winding wire packing factor	0.56
	Winding equivalent resistance at 60°C (normal/degraded) Rac	16.5 / 13.6 mΩ
	Calculated winding power loss at 60°C (normal/degraded)	206 / 347 W
	Leakage inductance Lac	16 µH
	Magnetizing inductance (approx. value)	1 mH
	Primary winding capacitance (approx. value)	65 pF
	Secondary winding capacitance (approx. value)	60 pF
	Primary-secondary capacitance	80 pF
	Core material	ferrite 3C90 I-core
	Core cross-section	12.5 cm 2
	Maximum flux density	0.27 T
	Calculated core power loss at 120°C	255 W
	Insulation voltage	5 kV
	Insulation class	H
	Cooling	forced air
	Weight	36 kg