Linear Models of a Stiffened Plate via $\Gamma$-convergence

We consider a family of three-dimensional stiffened plates whose dimensions are scaled through different powers of a small parameter $\varepsilon$. The plate and the stiffener are assumed to be linearly elastic, isotropic, and homogeneous. By means of $\Gamma$-convergence, we study the asymptotic behavior of the three-dimensional problems as the parameter $\varepsilon$ tends to zero. For different relative values of the powers of the parameter $\varepsilon$, we show how the interplay between the plate and the stiffener affects the limit energy. We derive twenty-three limit problems.


Introduction
Thin-walled structures are widely used in many engineering fields, such as aeronautic and aerospace structures, vessels, civil and mechanical constructions. Since the widespread use of such structures, many models, based on a priori kinematical assumptions, have been proposed in the history of Mechanics, in order to predict the behavior of loaded structures. Even though these models have been used successfully by generations of practitioners, they generally rely on heuristic assumptions. From a theoretical point of view, as reasonable as these assumptions may sound, they are hypotheses that jeopardize the validity of the mechanical models.
Over the last few years, attention has been paid to the rigorous justification of the classical mechanical theories and models: beams, shells, plates, etc. The underlying idea is to study the asymptotic behavior of actual three-dimensional variational formulations and to let some "smallness parameter" go to zero, so to fetch the essential features of the primitive problem in the resulting "simplified" asymptotic one.
A way to justify the mechanical models is via Γ-convergence, a variational convergence notion appeared for the first time in a the seminal work by De Giorgi and Franzoni [1]. The underlying idea is to replace the functional ruling the actual problem at hand by a new one, more handy, such that it may capture the major features of the primitive problem. More in detail, Γ-convergence derives the aimed asymptotic functional in such a way to achieve the convergence of minima and minimizers of the primitive problem and of the asymptotic one. The reader is addressed, for instance, to the monographs by Dal Maso and Braides [2,3] for an exhaustive exposition of the topic. Many interesting results have been obtained in Mechanics by Γ-convergence. One of the first works are by Acerbi et al. [4] and by Anzellotti et al. [5]. In the former, the asymptotic behavior of a string is derived in the framework of non-linear elasticity. In the latter, it is studied a plate and a beam in the context of linearized elasticity. These works represent a milestone, since they developed a fundamental modus operandi for the

Notation
In this paper, we work in the real Euclidean three-dimensional space R 3 . We use upper-case bold letters to indicate tensors and lower-case bold letters to indicate vectors. The Euclidean (Frobenius) product is indicated with · and the corresponding induced norm by | · |. We denote by Lin the set of linear transformations, and by Sym, Skw the subsets of linear symmetric and antisymmetric ones, respectively. tr(·) denotes the trace operator, whilst diag(a, b, c) is the diagonal matrix with elements a, b, c on the principal diagonal. R + denotes the set of all strictly positive real numbers, while N denotes the non-negative integer ones. Let S ⊂ R n (n ∈ {1, 2, 3}) be open. For any function v : S → R 3 , we shall denote its gradient by Hv := ∇v, and its unique decomposition in a symmetric and skew-symmetric part by Hv = Ev + Wv, where We denote by L 2 (S, R q ) := v : S → R q : v L 2 (S,R q ) < ∞ The corresponding Sobolev' spaces of functions on S with values in R q are the Banach spaces defined as follows (we will need only the cases for which l ∈ {1, 2}): W l,2 (S, R q ) := v : S → R q : v ∈ L 2 (S, R q ), ∇ α v ∈ L 2 (S, R n α ×q )∀α(∈ N) ≤ l .
P r e p r i n t They are endowed with the norm v 2 W l,2 (S,R q ) := v 2 L 2 (S,R q ) + l ∑ α=1 ∇ α v 2 L 2 (S,R n α ×q ) .
Note that the (high-order) gradients ∇ α (·) shall be understood in the sense of distributions. We shall furthermore consider the Sobolev' spaces W l,2 0 (S, R q ) := {v ∈ W l,2 (S, R q ) : v = 0 in ∂ D S} as the set of functions belonging to W l,2 (S, R q ) that assume value zero on a certain subset ∂ D S of the boundary of S. In this paper, we will make use of standard results concerning Sobolev' spaces. The reader is addressed to the classical monograph by Adams [27] or to the more recent book by Leoni [28]. If r ∈ N ∪ {∞}, then C r (S, R q ) denotes the space of r-times continuously differentiable functions on S with values in R q , and C ∞ 0 (S, R q ) denotes the space of functions belonging to C ∞ (S, R q ) that assume value zero in a neighborhood of the boundary of S. We will refrain to specify the codomain R in the notation of the functional spaces: for instance, we will simply write W 1,2 (S) instead of W 1,2 (S, R), and so forth.
We will denote by W −1,2 the dual space of W 1,2 0 . We recall that the following compact embeddings hold C ∞ 0 → W 1,2 0 → L 2 → W −1,2 , and that, if the operator T ∈ W −1,2 , where < T, x > denotes the dual pairing W −1,2 ×W 1,2 0 . If not specified, we adopt Einstein' summation convention for indices. Indices α, β , γ, δ take values in the set {1, 2}, indices a, b, c, d in the set {2, 3}, and indices i, j in the set {1, 2, 3}. With the notation A B we mean that there exists a constant C > 0 such that A ≤ C B. Such constant may vary line to line. As it is usual, we denote by´S f (·) dx the average value of the function f (·) over its integration domain, i.e. 1 meas(S)´S f (·) dx, meas(S) being the Lebesgue measure of the set S. We denote the strong convergence (convergence in norm) with the symbol →, whilst the weak convergence will be denoted by . Finally, with the symbol x ↓ y, we mean that x is approaching y from above (i.e., x → y + ).

General Setting
Let us introduce the real parameter ε that takes values in a sequence of positive numbers converging to zero. With reference to Fig. 1, we introduce in R 3 an orthonormal absolute reference system, denoted by We consider a plate-like body (hereafter, with a slightly abuse of language, just plate) occupying the regionΩ ε := (−L, L) × (−L, L) × ε(0, T ) and a blade-like stiffener body (hereafter, with a slightly abuse of language, just stiffener) occupying the regionΩ ε : be the overlapping region, that hereafter we refer to as junction region. We assume that h < 1 and that W < L. The first assumption implies that the height of the stiffener is larger than the thickness of the plate, while the second assumption is simply made to assure that the plate is larger than the width, ε w W , of the stiffener even in the case w = 0. The domain Ω ε :=Ω ε ∪Ω ε is depicted in Fig. 1. We shall consider the body clamped at x 1 = L, i.e., the displacement field is null in all points with coordinate x 1 equal to L (hence ∂ D S := Ω ε ∩ {x 1 = L}): this condition will be hereafter referred to as boundary condition.
The stored-energy functional W ε : Ω ε → R + is defined by P r e p r i n t x 1 x 2 where C is a fourth-order elasticity tensor, positive definite and having the usual major and minor symmetries.
In particular, this implies that there exists a positive constant µ such that In this paper, we consider a linear homogeneous isotropic material. It can be shown (see, for instance, [29, Article 68]) that for this kind of materials the stored energy density admits the unique representation where µ > 0 and λ > − 2 3 µ are called Lamé parameters. The energy density (3) is also known as Saint Venant-Kirchhoff's.
The functional in (1) can be decomposed into the sum of two contributions The function χ ε allows to associate half of the energy of the junction region to the stiffener and the other half to the plate, or, more simply, it avoids to consider the energy of the region Ω Jε twice.
Proof. The proof is similar to that of Lemma 4.1. Since C is positive-definite and from Korn inequality, it follows that Hence, up to a subsequence, ε kǔ ε ǔ for a certainǔ ∈ W 1,2 in L 2 (Ω) (not summed on a).

The Junction Conditions
The present section is devoted to establish the relationship existing between the limit fieldsû andǔ. This is carried out by studying the junction conditions on Ω J .

The Junction Conditions for Displacements
From (4) and (7), the following equality must be satisfied for the identity ofû andǔ in Ω Jε : which may be rewritten more explicitly for all (x 1 , It is noteworthy that, in this way, the junction region, which originally depends on ε, has been transformed into the fixed domain Ω J . The following two technical Lemmata express an approximation of the trace operator; similar results can be found in [15,16]. We recall that h < 1. P r e p r i n t Lemma 5.1. Let w ∈ W 1,2 (Ω J ) and w ε ∈ W 1,2 (Ω J ) be a sequence such that w ε w in W 1,2 (Ω J ). Then, the sequence of functions converges in the norm of L 2 ((−L, L) × (−W,W )) to the trace of the function w on (−L, L) × (−W,W ) × {0}. The trace will be denoted simply by w(x 1 , x 2 , 0).
where we have used a change of variable, the Fundamental Theorem of Calculus, and the Cauchy-Schwartz inequality (twice). The thesis follows from the continuity of the trace operator for functions in W 1,2 (see, for instance, [27,Theorem 5.36]) and from the general assumption h < 1.
With a similar argument, one can prove the following the counterpart for the stiffener of the previous Lemma.
The limit junction conditions for the displacements are derived in the next result.
Lemma 5.3. The following equalities hold for almost every x 1 ∈ (−L, L): 1} are defined as follows: P r e p r i n t Proof. We provide a detailed proof for the first component of the displacement, the others being similar. We consider the first of (11) averaged with respect to x 2 and x 3 : Since ε kǔ Similarly, sinceû 1ε By passing to the limit in (13), taking into account (14) and (15), we obtain SinceˆW −Wǔ (16) implies the first identity of (12).
For the other components of the displacement, from (11), we find: and arguing as above we complete the proof.

The Junction Conditions for the Torsion Angle
This Section is dedicated to establish a further relationship existing between the limit fields, involving the torsion angle of the stiffener. Hereafter, M := max{w, h}, m := min{w, h}. As a trivial consequence, M ≥ m, with equality holding if and only if h = w.
We start with a scaled Korn-type inequality.

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Proof. We use an argument from [6,Theorem 3.2]. DecomposeΩ ε in parallelepipeds having cross-section ε M × ε M . For each of them, apply the scaled Korn inequality of the type in [5, Theorem A.1.] (see also [31,Theorem 2]). Then, summing up, the thesis follows.
Lemma 5.5. With the assumptions and the notation of Lemma 4.2, we have In particular, up to a subsequence, with ϑ a function in L 2 (Ω).
Proof. The proof of (17) follows immediately by changing variables and scaling into the fixed domain the result of Lemma 5.4, and by recalling Lemma 4.2. From this bound we deduce that there exists a subsequence, not relabeled, of ε k+MȞ εǔε and aȞ ∈ L 2 (Ω, R 3×3 ), such that Accordingly,Ȟ is a skew-symmetric tensor field. Moreover, from the definition (8) and the convergence of ε kǔ The proof is completed by setting ϑ :=Ȟ 32 .
The function ϑ defined in Lemma 5.5 can be interpreted as the rotation angle, or torsion angle, of the stiffener cross-section around the longitudinal axis x 1 . It is noteworthy that if w = h the first two sequences of (18) have non-trivial limits simultaneously.
We shall now characterize the torsion angle ϑ . The idea is to show that the cross-sectional displacement field of the stiffener can be approximated by a rigid one. To do so, let us introduce the set of infinitesimal rigid displacements onω: . We indicate with P the projection of L 2 (ω, R 2 ) onto R(ω). It can be shown (see [32,Theorem 2.5]) that a two-dimensional Korn inequality holds for all functions w ∈ W 1,2 (ω, R 2 ): Denoting with (x 2 (G), x 3 (G)) = (0, H 2 ) the coordinates of the centre of mass ofω, we define (with summation convention and a, b, c, d ∈ {2, 3}): P r e p r i n t Lemma 5.6. Let ε kǔ ε be a sequence satisfying the assumptions of Lemma 5.5. Then, Proof. Taking into account (19) and (9), we havê where we have used the fact that one among w and h equals M (while the other, by definition, equals m), and the fact that ε 2M ≤ ε M+m ≤ ε 2m .
Lemma 5.7. With the assumptions and notation of Lemmata 5.5 and 5.6, we have where ϑ and ϑ ε are defined in (18) and (20), respectively.
Finally, we derive the last junction condition involving the torsion angle ϑ .

Different Regimes for the Limit Junction Conditions
Considering (12) and (23), ten different cases are possible for the joining conditions, depending on the values of w and h. The general scenario is graphically depicted in Fig. 2, whereby the four lines represents the conditionŝ i =ˇ i = 1 andˆ ϑ =ˇ ϑ = 1. The possible cases have been labeled with letters from "A" to "J".
Analyzing Fig. 2, it is noteworthy that for no combination of w and h (recall that w and h are strictly positive) the four joining conditions in (12) and (23) are non-trivial at the same time. The fictitious intersection, where all joining conditions would be non-trivial at once, is at point (w = 0, h = 1). This corresponds to a non scaling of the stiffener in direction x 2 and to scaling with the same velocity the plate thickness and the stiffener dimension along x 3 . This scaling leads the stiffener to degenerate into a prismatic portion of the plate: thus, the problem is equivalent to the asymptotic scaling of only a plate. 13 P r e p r i n t In this Section, we characterize the limit stored energy, and we will prove our main Γ-convergence result.
To begin, in the next two Lemmata we characterize some components of the limit strain. Proof. It is sufficient to notice that Ê εûε αβ = ∂ αûβ ε +∂ βûαε 2 and to apply Lemma 4.1.
Lemma 6.2. With the assumptions and notation of Lemma 5.5 we have, up to subsequences, in the weak L 2 (Ω) topology: where η 13 ∈ L 2 (Ω) is independent of x 2 , η 12 ∈ L 2 (Ω) is independent of x 3 , and Φ ∈ L 2 ((−L, L),W 1,2 (ω)) is the so-called torsion function, solution of the following boundary value problem: where n is the outer normal to ∂ω and ∆(·) is the Laplacian operator.
Proof. To prove (32), it is sufficient to notice that ε k Ě εǔε 11 = ε k ∂ 1ǔ1ε and to apply Lemma 4.2. We have already deduced that, up to subsequences, ε k Ě εǔε 1a Ě 1a in L 2 (Ω). To characterizeĚ 1a , note that in the sense of distributions. Hence, for ψ ∈ C ∞ 0 (Ω), we havêΩ We note that, in L 2 (Ω), Passing to the limit in (36), we findΩ Thus, To conclude this part of the proof, we set η 13 := γ 13 and η 12 := γ 12 − H 2 ∂ 1 ϑ . For the case M = h = w, in [33, Lemma 4.1] it is shown that there exists a function Φ ∈ L 2 ((−L, L),W 1,2 (ω)) satisfying the boundary value problem (35) and such that it can be writtenĚ 13 Since we characterized only some components of the limit strain, the others will be defined by minimization of the stored energy density (see (3)). In fact, the convergences stated in Lemmata 4.1 and 4.2 determine only some components of the limit strain energy, whilst the others remain undetermined. The minimization problem will select them in such a way to render the energy as small as possible. For this reason, we definê A direct computation shows that witĥ we have where E := µ(2µ+3λ ) µ+λ is the Young modulus and ν := λ 2(λ +µ) is the Poisson ratio. In particular, for the stiffener, we have the following characterization of the stored-energy density: Remark 6.1. The characterization (40) of the stored-energy density for the stiffener, combined with the ten regimes provided by all the possible junction conditions (see Sec. 5.3 and Fig. 2), results into twenty-three different limit problems. In particular, we have nine cases (i.e., A, B, C, D, E, F, G, I, J) for M = w = h, seven cases (i.e., A, B, C, E, F, G, H) for M = h = w and seven cases (i.e., A, B, C, E, F, G, H (or I)) for M = h = w.

Proof.
P r e p r i n t (a) Liminf inequality We start by proving the weak sequential lower semicontinuity of the family of stored energy functionals. Without loss of generality, we can suppose that lim inf n→∞ W ε n (û ε n , ε k nǔ ε n ) < ∞ otherwise there is nothing to prove. Hence, sup n W ε n (û ε n , ε k nǔ ε n ) < ∞, and Lemmata 4.1, 4.2, 5.5, 6.1, and 6.2 hold. Taking into account the decomposition given in (10), we need to show the weak sequential lower semicontinuity of the stored energy contributions due to the plate and the stiffener, i.e.
for every sequenceû ε n û in W 1,2 (Ω, R 3 ) and ε k nǔ ε n ǔ in W 1,2 (Ω, R 3 ), respectively. It is easy to prove, by an application of Fatou Lemma, thatŴ ε n (û ε n ) is sequential lower semicontinuous with respect to the strong W 1,2 (Ω, R 3 ) topology. However, the convexity of the integrand functionf (·) is sufficient (yet not necessary in the vector-valued case) to ensure the sequential lower semicontinuity also with respect to the weak W 1,2 (Ω, R 3 ) topology (see, for instance, [2,Proposition 1.18], [34,Theorem 2.6]). By using (37), we infer Similarly, for the stiffener, we infer and from Lemma 6.2, we havêΩf A direct computation shows that, for the latter two cases, the estimation is independent from η 12 and η 13 . We provide the computation for the case M = w = h only, the other one being conceptually similar. We havêΩf since the integral of x 2 ∂ 1 ϑ η 13 is zero because η 13 does not depend on x 2 by Lemma 6.2. Consequently, we have that (b) Existence of a recovery sequence case i) We start by proving case i). So, let k = 0, h + w = 1, 1/2 < M = w < 1, and 0 < m = h < 1/2. Let (û,ǔ, ϑ ) ∈ A . From the definition of A we have that P r e p r i n t for appropriate functionsξ ξ ξ andξ ξ ξ . With the values in consideration of k, h and w, Lemmata 5.3 and 5.8 imply thatξ To start, we assumeξ i ∈ C ∞ ((−L, L) × (−L, L)), andξ 3 , ϑ ∈ C ∞ ((−L, L)). Moreover, we assume that all these functions have value zero in a neighborhood of x 1 = L and, in view of (42), we may also assume thatξ 3 is equal to zero in a neighborhood of (−L, L) × {0}.
From the definition of A we have that for appropriate functionsξ ξ ξ andξ ξ ξ . In the case under study, Lemmata 5.3 and 5.8 state that We first build the recovery sequence assuming thatξ i ∈ C ∞ ((−L, L) × (−L, L)), andξ i , ϑ ∈ C ∞ (−L, L). Moreover, we assume that all these functions have value zero in a neighborhood of x 1 = L, and in view of (43), thatξ i are equal to zero in a neighborhood of (−L, L) × {0}.

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To define the recovery sequence for the stiffener,ǔ ε , we shall use the continuous piece-wise affine functioň ψ(x 3 ) : [0, H] → [0, 1] defined as in the previous case. We set The recovery sequence for the plateû ε is defined bŷ Sinceξ i are equal to zero in a neighborhood of (−L, L) × {0}, for ε small enough the junction conditions (11) are automatically satisfied. For x 3 ∈ [0, T ] we have thať and since −k, 1 − h − k, and −k − w are strictly greater than zero, it follows that .
A tedious calculation then shows thatÊ where (see Eq. (38))Ẑ The proof is concluded as in case i) case iii) We finally prove case iii). Let k > 0, k + w > 0, k + h − 1 > 0 and k + M − 1 > 0. In this case, from Lemmata 5.3 and 5.8 we deduce thatξ i = 0 and ϑ = 0. As a consequence, the limit displacementû as well as the limit strain tensorĚ (andŽ) of the beam are identically equal to zero.
To start, we assume (û, 0, 0) ∈ A ∩ C ∞ (Ω, R 3 ) ×C ∞ (Ω, R 3 ) ×C ∞ ((−L, L)) and to have value zero in a neighborhood of x 1 = L. From the definition of A , there exist smooth functionsξ ξ ξ , characterizing the KL 0 displacement type. We recall that, in this particular case,ξ i = ϑ = 0. Let us introduce the sequencesû ε and ε kǔ ε defined as follows: It can be easily verified that the pair (û ε , ε kǔ ε ) satisfies the boundary conditions at x 1 = L, the junction conditions (11), and verifies the following convergenceŝ withẐ Z Z defined as in (46). The proof is concluded as in case i) Remark 6.2. As it can be noticed, the construction of the recovery sequence is quite cumbersome, and general construction rules do not exist. For this reason, we decided to provide the recovery sequence for two extreme cases (ii and iii) and for an intermediate one (i), see also Fig. 2. As a consequence, in the statement of Theorem 6.1 we have explicitly considered, in the part concerning the existence of a recovery sequence, only three out of the twenty-three possible cases. We stress the fact that the given Liminf inequality proof is valid for every choice of the scaling parameters (in the admissible ranges), i.e. for all the twenty-three cases. We did not check all the twenty-three cases, but in all the presented cases we checked we were able to construct a recovery sequence. We are therefore confident that Theorem 6.1 holds for all the twenty-three cases, even if it is stated for only three.
We can explicitly write the expression of the limit stored-energy functional. Considering the case i) of Theorem 6.1 for the sake of simplicity, we have: P r e p r i n t where A :=´ω dx 2 dx 3 = 2W H is the cross-sectional area of the stiffener, S 2 :=´ω x 3 dx 2 dx 3 = W H 2 is the static moment with respect to the x 2 axis, J 2 :=´ω x 2 3 dx 2 dx 3 = 2 3 W H 3 is the moment of inertia with respect to the x 2 axis, J t := 4´ω x 2 2 dx 2 dx 3 = 8 3 HW 3 is the torsional moment of inertia. Similarly, we havê where we have posed

Strong Convergence of Minima and Minimizers
So far, we just considered the stored energy functional. However, the equilibrium problem is ruled by the total energy F ε , which is the sum of the stored energy W ε minus the work done by external loads L ε where the work of external loads is assumed to be and where b belongs to L 2 (Ω ε ). After the scaling of Section 4, we have where we have posedb In particular, we consider loads of the form whereb i ∈ L 2 (Ω),b i ∈ L 2 (Ω) and m(x 1 ) ∈ L 2 ((−L, L)).
It is easy to see that the external loads contributions continuously converge in the sense of the convergence used in Theorem 6.1 to the limit functionalŝ In the next Theorem, we show that the external loads do not impact on our Γ-limit result. This is the reason why we focused only on the stored energy in the previous part of the paper. Theorem 7.1. As ε ↓ 0, the sequence of functionals F ε (û ε , ε kǔ ε ) := W ε (û ε , ε kǔ ε ) − L ε (û ε , ε kǔ ε ) Γ-converges to the limit functional F (û,ǔ, ϑ ) := W (û,ǔ, ϑ ) − L (û,ǔ, ϑ ) in the sense specified in Theorem 6.1.

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Proof. The proof follows from the well-known stability of Γ-convergence with respect to continuous, realvalued perturbations (see [2,Proposition 6.20]). We conclude the paper by showing that the sequence of minima and minimizers from the sequence of three-dimensional total energies converges to the unique solution of the variational Γ-limit problem. From a mechanical point of view, it can be interpreted as follows: the equilibrium configurations of the sequence of three-dimensional problems converge towards the equilibrium configuration provided by the Γ-limit functional minimization. Moreover, we show that the convergence is actually strong.
Theorem 7.2. Suppose Theorems 6.1 and 7.1 hold. As ε ↓ 0, the sequence of three-dimensional minimization problems for the functional F ε (û ε , ε kǔ has a unique solution for each term in the sequence. The solution is denoted by (û # ε , ε kǔ# ε ). Similarly, the minimization problem for the Γ-limit functional F (û u u,ǔ u u, ϑ ) := W (û u u,ǔ u u, ϑ ) − L (û u u,ǔ u u, ϑ ) min (û,ǔ,ϑ )∈A F (û u u,ǔ u u, ϑ ) admits a unique solution denoted by (û u u # ,ǔ u u # , ϑ # ). Moreover, we have that Proof. The existence of a solution for problems (49), (50) can be proved through the Direct Method of the Calculus of Variations; the uniqueness follows from the strict convexity of the functionals F ε and F . From Theorem 7.1, Propositions 6.8 and 8.16 (lower-semicontinuity of sequential Γ-limits), Theorem 7.8 (coercivity of Γ-limits) and Corollary 7.24 (convergence of minima and minimizer) of [2], it follows that the weak convergence counterpart of points 1, 2, 3 is satisfied, and that point 4 is also proved. To show that the convergence is actually strong, we adapt some arguments proposed in [7,15].
Let us denote by a ε the approximate minimizer of problem (50), defined as the recovery sequence(s) appearing in Theorem 6.1, but with (ξ ξ ξ ,ξ ξ ξ , ϑ ) replaced by (ξ ξ ξ # ,ξ ξ ξ # , ϑ # ), related to the pair (û # ,ǔ # ). By part (b) of Theorem 6.1 and by Theorem 7.1 we have In particular, lim inf As a preliminary observation, quadratic forms (3) satisfy the identity for every A, U ∈ R 3×3 . By the coercivity condition (2), we obtain the following inequality: P r e p r i n t Then, we havê since,χ ε ,χ ε ≥ 1 2 . For brevity sake, we introduce the following notation (to be specialized for the plate and the stiffener with the usual symbolsˆandˇ): so that (52) rewriteŝ To start, we prove that the first integral appearing in the right hand side of both of (52) tends to zero in the limit of ε ↓ 0. The integrands of these two integrals rewrite as 2µ Â i jε∆i jε + λ Â iiε∆ j jε , ε 2k [2µ Ǎ i jε∆i jε + λ Ǎ iiε∆ j jε ].