Linear Models for Composite Thin-Walled Beams by $Gamma$-Convergence. Part I: Open Cross Sections
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(2) LINEAR MODELS FOR COMPOSITE THIN-WALLED BEAMS. 3297. (i) the component parallel to the short side of the rectangle is bounded in H 1 ; (ii) the component parallel to the long side of the rectangle divided by ε is bounded in H 1 ; (iii) the component parallel to the axis of the beam divided by ε2 is bounded in H 1 . Thin-walled beams with a multirectangular cross section were studied in [7]. Each rectangle composing the cross section had sides that scaled with ε and ε2 , respectively. The analysis was carried out by “splitting” the beam in different rectangular thinwalled beams, in order to use the compactness theorem of [6], and then “recomposing” the beam by means of appropriate junction conditions. This procedure circumvented the necessity to prove a compactness theorem specific for the type of beams considered. The assumptions of homogeneity and isotropy were completely removed in [8], where, still for a rectangular thin-walled beam, very detailed convergence results for the displacements were obtained. In [4, 5] a hierarchy of models for homogeneous anisotropic thin-walled beams with rectangular cross sections have been deduced starting from the three-dimensional theory of nonlinear elasticity. In the nonlinear setting the scaling of the energy determines the limit model: for “very small” energy a linear model is obtained, while for “large” energies different nonlinear limit models are deduced. Some of the compactness results used in the present paper were inspired by the nonlinear counterpart studied in [4, 5]. In this paper we consider a fully anisotropic and inhomogeneous thin-walled beam with arbitrary geometry of the cross section clamped at one of its bases. More precisely, the cross section we take into consideration is a tubular neighborhood, whose thickness scales with a parameter δε > 0, of a simple planar curve γ whose length scales with ε. The curve γ can be either open or closed but not completely straight. As in [4, 5], the thinness of the “wall” is characterized by the assumption that lim. ε→0. δε = 0. ε. By allowing the thickness of the wall to scale with δε , instead of simply ε2 , we can study beams with cross section having different degrees of slenderness. We measure this quantity by means of a parameter s defined by ε2 . ε→0 δε. s := lim. Without loss of generality, it suffices to consider three cases s ∈ {0, 1, +∞}. In this paper we consider only s ∈ {0, 1}. As in the case of rectangular cross sections, our analysis is based on a compactness theorem. Roughly, it establishes that a sequence of displacements with equi-bounded energy is such that (i) the projection on the plane of the cross section divided by δε /ε is bounded in H 1 ; (ii) the component parallel to the axis of the beam divided by δε is bounded in H 1 . In the case of rectangular thin-walled beams the compactness theorem follows directly from a “rescaled” Korn’s inequality. For beams with a curved cross section, treated in this paper, it follows from a rescaled Korn’s inequality and a detailed study of the sequence of the strain components in the direction of the axis of the beam and in the direction tangent to the midline curve γ of the cross section (see Theorem 5.8). The.
(3) 3298. C. DAVINI, L. FREDDI, AND R. PARONI. argument used in the proof works only if the curve γ is not completely straight; thus our result is complementary to that derived in [6] for rectangular thin-walled beams. Recently Davoli [3] generalized the works [4, 5] by studying thin-walled beams with both open and curved cross section, starting from the three-dimensional theory of nonlinear elasticity. Her impressive work does not contain a compactness theorem as detailed as ours and thus the Γ-convergence analysis is carried out in terms of strains and not, as customary, in terms of displacements. A very important kinematical parameter in any beam model is the twist of the cross section, hereafter denoted by ϑ. This parameter is “generated by” a sequence, whose terms involve derivatives of the displacement, bounded in L2 . (See (i) of Corollary 5.4 for a precise definition.) Thus, a priori ϑ is only an L2 -function. We show that for s = 0 the twist ϑ is an H 1 -function and in the case s = 1 it is even an H 2 -function. This augmented regularity of the twist is essentially a consequence of the “structure” of the limit displacements. All the compactness results presented up to section 6 hold for beams with open and closed cross sections, but clearly, a closed cross section imposes more constraints on the limiting displacements than an open one. Although these constraints will be explored in the second part of this paper, it is worth mentioning a few results. For a closed cross section it will be proved that ϑ is identically equal to zero. This result essentially states that the sequence that generates the twist in the case of open cross sections is too “crude” and needs to be refined and further rescaled in the case of closed cross sections. As will be shown in Part II, such a refined sequence strongly depends on the geometry of the cross section. Moreover, still in the case of closed cross sections, further constraints on the limit strains will emerge and these will drastically affect the Γ-limit. As already stated, in this paper we consider a fully anisotropic and inhomogeneous material. This generality makes part of the analysis quite involved, particuarly the proof of the so-called recovery sequence condition. Contrary to what is customary, we do not construct a recovery sequence but instead prove that there exists one. In order to do this, we set up a sequence of “auxiliary” minimization problems and show that the sequence of minimizers is indeed a recovery sequence. This procedure allows us to circumvent awkward constructions of the full recovery sequence and to limit ourselves to produce only few partial and simple “recoveries” that are needed in order to show that the sequence of minimizers is indeed a recovery sequence. The paper is organized as follows. In section 2 we define the geometry of the cross section, with the relative scaling parameters, and we set up the problem. The curved cross section is, by a change of variables, “straightened” in section 3, while in section 4 it is “rescaled” to a fixed domain, i.e., independent of ε. In the same section we define a system of curvilinear components that will be used throughout the paper. Several compactness results are proved in section 5. In particular, in subsection 5.1 we obtain compactness results for appropriately rescaled components of the displacement, while in subsection 5.2 correspondent results for strains are proved. Section 6 anticipates the energy density of the limit problem and studies some of its properties, while section 7 is devoted to the characterization of the Γ-limit. The proof of the two compactness theorems is given in the appendix. 1.1. Notation. Throughout this article, and unless otherwise stated, we index vector and tensor components as follows. Greek indices α, β, and γ take values in the set {1, 2} and Latin indices i, j, k, l in the set {1, 2, 3}. With (e1 , e2 , e3 ) we shall denote the canonical basis of R3 . Lp (A; B) and H s (A; B) are the standard Lebesgue and.
(4) LINEAR MODELS FOR COMPOSITE THIN-WALLED BEAMS. 3299. Sobolev spaces of functions defined on the domain A and taking values in B. When B = R, or when the target set B is clear from the context, we will simply write Lp (A) or H s (A); also, in the norms we shall systematically drop the target set. Convergence in the norm, that is, the so-called strong convergence, will be denoted by →, while weak convergence is denoted with . With a little abuse of language, and because this is a common practice and does not give rise to any confusion, we use “sequences” even for those families indicized by a continuous parameter ε which, throughout the whole paper, will be assumed to belong to the interval (0, 1]. Throughout the paper, the constant C may change from expression to expression (and even in the same line). The scalar product between vectors or tensors is denoted by ·. R3×3 skw denotes the vector space of skew-symmetric 3 × 3 real matrices. For A = (aij ) ∈√R3×3 we denote the Euclidean norm (with the summation convention) by |A| = A · A = √ tr(AAT ) = aij aij . Whenever we write a matrix by means of its columns we separate the columns with vertical bars (·| · |·) ∈ R3×3 . ∂i stands for the distributional ∂ . For every a, b ∈ R3 we denote by a b := 12 (a ⊗ b + b ⊗ a) the derivative ∂x i symmetrized diadic product, where (a ⊗ b)ij = ai bj . From section 5 on, when a function of three variables is independent of one or two of them we consider it as a function of the remaining variables only. This means, for instance, that a function u ∈ H 1 ((0, ) × (0, L); Rm ) will be identified with a corresponding u ∈ H 1 ((0, ) × (−h/2, h/2) × (0, L); Rm ) such that ∂2 u = 0 and a function v ∈ H 1 ((0, L); Rm ) will be identified with a corresponding v ∈ H 1 ((0, ) × ∂1 v = ∂2 v = 0. The notation da stands for the (−h/2, h/2) × (0, L); Rm ) such that area element dx1 dx2 . As usual, − denotes the integral mean value. 2. Setting of the problem. We consider a sequence of thin-walled beams whose cross section is a tubular neighborhood, with thickness scaling with a parameter δε , of a simple curve γ whose length scales with ε, as detailed below. Let (e1 , e2 , e3 ) be an orthonormal basis of R3 and ε, δε two positive parameters converging to zero. We consider a simple curve γ ∈ W 3,∞ (I; R2 × {0}), where I is an interval of length > 0, for two distinct instances: • I = (0, ) with lims→ γ (s) = lims→0 γ (s) if lims→ γ(s) = lims→0 γ(s); • I = [0, ] with γ( ) = γ(0) and γ ( ) = γ (0). In the former case we will say that the curve is open, in the latter that it is closed. We assume γ to be parameterized by the arclength parameter s ∈ [0, length(γ)], so that t := γ is a unit tangent vector contained in the plane spanned by e1 and e2 . We denote by n := e3 ∧ t the unit normal to γ and by κ := t · n its curvature, so that t = κn and n = −κt. We assume that κ is not identically equal to zero. Let ε ∈ (0, 1), I˜ε := εI, and γ˜ ε : I˜ε → R2 × {0} be the map defined by γ˜ ε (s) := ˜ ε = e3 ∧ t˜ε , and κ ˜ ε = t˜ε · n ˜ ε so that t˜ε (s) = t(s/ε), εγ(s/ε). We set t˜ε = γ˜ ε , n ε ε n ˜ (s) = n(s/ε), and κ ˜ (s) = κ(s/ε)/ε. Let h > 0. We consider a beam with cross section of diameter that scales with ε and constant thickness δε h. To deal with thin-walled beams, we assume that (2.1). lim. ε→0. δε = 0. ε. To define the region occupied by the beam in the reference configuration we set δε h δε h 2 ˜ ε := ω ω ˜ ε := (˜ , x1 , x ˜2 ) ∈ R : x ˜1 ∈ Iε and x˜2 ∈ − ˜ ε × (0, L), , Ω 2 2.
(5) 3300. C. DAVINI, L. FREDDI, AND R. PARONI. x3 ce. We x2 x3. x1 re. We. W x1 x2 ε, Ω ε , and Ω. Fig. 1. The domains Ω. Fig. 2. Two cross sections with lims→0 γ(s) = lims→ γ(s) but one closed and the other open.. ε → R3 where L > 0 denotes the length of the beam, and we consider the map χ ˜ε : Ω defined by (2.2). x) := γ˜ ε (˜ x1 ) + x ˜2 n ˜ ε (˜ x1 ) + x ˜3 e3 . χ ˜ε (˜. The region occupied by the beam in the reference configuration is (2.3). ε := χ ε ), Ω ˜ε (Ω. ε is an open set independently of the fact that the curve see Figure 1. We note that Ω γ is open or closed. If the curve γ is open (closed) we say that the thin-walled beam has an open (closed) cross section (see Figure 2). Henceforth we denote by (2.4). Eu ˆ(ˆ x) := sym(∇ˆ u(ˆ x)) :=. ∇ˆ u(ˆ x) + ∇ˆ u(ˆ x)T 2. ε → R3 . the strain corresponding to the displacement u ˆ:Ω In what follows we consider an inhomogeneous linear hyperelastic material with ε ) satisfy the major and minor elasticity tensor Cε , whose components Cεijkl ∈ L∞ (Ω ε ε ε symmetries, i.e., Cijkl = Cijlk = Cklij . We further suppose Cε to be uniformly positive definite, that is, there exists c > 0 such that (2.5). x)M · M ≥ c|M |2 Cε (ˆ. for almost every x ˆ and for all symmetric matrices M ..
(6) LINEAR MODELS FOR COMPOSITE THIN-WALLED BEAMS. 3301. We assume the beam to be clamped at x3 = 0, and we denote by. 1 ε ; R3 ) : uˆ = 0 on ∂ Ω ˆ ε ∩ {x3 = 0} . Hdn (Ωε ; R3 ) := u ˆ ∈ H 1 (Ω 1 The energy functional of the beam Fˆε : Hdn (Ωε ; R3 ) → R is given by. (2.6). 1 Fˆε (ˆ u) := 2.
(7) ε Ω. Cε E u ˆ · Eu ˆ dˆ x − Lˆε (ˆ u),. where Lˆε (ˆ u) denotes the work done by the loads on the displacements u ˆ. Our analysis will focus on the asymptotic behavior of the elastic energy; the work done by the loads will be considered only in Remark 7.3. 3. Representation of the deformation gradient and the strains as fields ε . We shall use the curvilinear coordinates {˜ on Ω x1 , x˜2 , x ˜3 } and the natural basis (˜ g1ε , g˜2ε , g˜3ε ), where the basis vectors are defined by g˜iε :=. ∂χ ˜ε . ∂x ˜i. A simple computation yields g˜1ε = (1 − x ˜2 κ ˜ε )t˜ε ,. g˜2ε = n ˜ε,. g˜3ε = e3 .. The dual basis (˜ gε1 , g˜ε2 , g˜ε3 ), defined by the set of equations g˜εi · g˜jε = δji with δji the Kronecker’s symbols, turns out to be g˜ε1 =. t˜ε , 1−x ˜2 κ ˜ε. g˜ε2 = n ˜ε,. g˜ε3 = e3 .. Since x) = 1 − x ˜2 κ ˜ ε (˜ x1 ) > 1 − det ∇χ ˜ε (˜. δε h|κ(˜ x1 /ε)| >0 ε 2. for any ε small enough, due to assumption (2.1), then χ ˜ε is locally invertible and, in fact, it is a diffeomorphism, up to a set of measure zero in the case of closed ε. ε and Ω cross sections, between Ω 3 ε → R3 be defined by. For every uˆ : Ωε → R , let u˜ : Ω u ˜ := uˆ ◦ χ ˜ε so that u ˆ(ˆ x) = u ˜(˜ x), where xˆ = χ ˜ε (˜ x). By the chain rule we find (3.1). εu H ˜ := ∇˜ u(∇χ ˜ε )−1 = (∇ˆ u) ◦ χ ˜ε .. We set (3.2). ε u˜ := symH εu E ˜ = (E u ˆ) ◦ χ ˜ε .. ε since it will have no use in our analysis. We refrain from writing the energy on Ω.
(8) 3302. C. DAVINI, L. FREDDI, AND R. PARONI. 4. Problem on a fixed domain. In order to rewrite the problem on a domain 1 and which does not depend on ε, let us introduce the notation ω := ω ˜ 1 and Ω := Ω ε consider the rescaling map r : Ω → Ωε defined by rε (x) := (εx1 , δε x2 , x3 ). Let giε := g˜iε ◦ rε , gεi := g˜εi ◦ rε , i.e., −1 δε δε g1ε = 1 − x2 κ t, gε1 = 1 − x2 κ t, ε ε. g2ε = gε2 = n,. g3ε = gε3 = e3 .. ε → R3 the function u : Ωε → R3 defined by Accordingly, we associate to u˜ : Ω u := u ˜ ◦ rε so that u(x) = u ˜(˜ x), where x˜ = rε (x). By the chain rule we have. 1. 1 ∂1 u ∂2 u ∂3 u = (∇˜ u) ◦ r ε . ε δε Observing that (4.1). g2ε |˜ g3ε , ∇χ ˜ε = g˜1ε |˜. then we have (4.2). H ε u :=. ε −1 1 2 3 T ∇χ ˜ = g˜ε |˜ gε |˜ gε ,. 1. T ε ε 1 u ∂1 u ∂2 u ∂3 u gε1 |gε2 |gε3 = H ˜ ◦r ε δε. and (4.3). ε u ˜) ◦ rε . E ε u := sym H ε u = (E. We note that (4.2) implies. 1. 1. ∂1 u ∂2 u ∂3 u = H ε u g1ε |g2ε |g3ε , (4.4) ε δε that is, (4.5). 1 ∂1 u = H ε u g1ε , ε. 1 ∂2 u = H ε u g2ε , δε. ∂3 u = H ε u g3ε .. Let χε := χ ˜ε ◦ rε . By means of (3.2) we may rewrite (4.3) as ε u E ε u = (E ˜) ◦ rε = (E u ˆ ) ◦ χε . √ Then, by setting Lε (u) := Lˆε (u ◦ χε−1 )/(εδε ) and g ε := 1 − (δε /ε)x2 κ, we get
(9) √ Fˆε (ˆ u) 1 (4.6) = CE ε u · E ε u g ε dx − Lε (u) =: Fε (u), εδε 2 Ω where we have assumed that C := Cε ◦ χε is, in fact, independent of ε. The functional Lε will be discussed in Remark 7.3. The domain of the energy functional Fε becomes. 1 Hdn (Ω; R3 ) := u ∈ H 1 (Ω; R3 ) : u = 0 on ω ˜ × {0}.
(10) LINEAR MODELS FOR COMPOSITE THIN-WALLED BEAMS. 3303. for an open cross section and. 1 1 H#dn (Ω; R3 ) := u ∈ Hdn (Ω; R3 ) : u(0, ·, ·) = u( , ·, ·) for a closed cross section. From the assumptions made on the elasticity tensor Cε (see (2.5)) it follows that Cijkl ∈ L∞ (Ω), that Cijkl = Cijlk = Cklij , and that there exists c > 0 such that C(x)M · M ≥ c|M |2. (4.7). for almost every x and for all symmetric matrices M . For later use we write (H ε u)ij := giε · H ε u gjε , which give the components of H ε u in the local basis {g1ε , g2ε , g3ε }. From (4.5), they are. (4.8). 1 ε g · ∂1 u, ε 1 1 = g2ε · ∂1 u, ε 1 = g3ε · ∂1 u, ε. 1 ε g · ∂2 u, δε 1 1 = g2ε · ∂2 u, δε 1 = g3ε · ∂2 u, δε. (H ε u)11 =. (H ε u)12 =. (H ε u)13 = g1ε · ∂3 u,. (H ε u)21. (H ε u)22. (H ε u)23 = g2ε · ∂3 u,. (H ε u)31. (H ε u)32. (H ε u)33 = g3ε · ∂3 u.. We also note that (4.9). (E ε u)ij := giε · E ε u gjε =. (H ε u)ij + (H ε u)ji . 2. 5. Kinematic results. Throughout the section we consider a sequence of func1 tions uε ∈ Hdn (Ω; R3 ) such that (5.1). sup ε. that. 1 E ε uε L2 (Ω) < +∞. δε. Theorem 5.1. There exists a sequence {W ε } ⊆ H 1 ((0, ) × (0, L); R3×3 skw ) such. (i) H ε uε − W ε L2 (Ω) ≤ Cδε , (ii) W ε L2 (Ω) + ∂3 W ε L2 (Ω) ≤ C, (iii) ∂1 W ε L2 (Ω) ≤ Cε for a suitable C > 0 and every ε small enough. 1 Hdn ((0, L); R3×3 skw ) such that, up to a subsequence, (5.2). Moreover, there exists W ∈. Wε W. in H 1 ((0, ) × (0, L); R3×3 skw ). Proof. The proof is given in the appendix (section 8). 1 Theorem 5.2. If {uε } ⊆ H#dn (Ω; R3 ) is a sequence for which (5.1) holds, then the conclusions of Theorem 5.1 hold for a sequence {W ε } such that W ε (0, ·) = W ε ( , ·) for every ε. Proof. The proof is given given in the appendix (section 8). For the rest of this section W ε and W will be as in Theorem 5.1..
(11) 3304. C. DAVINI, L. FREDDI, AND R. PARONI. Corollary 5.3. The following propositions hold up to subsequences: (i) H ε uε → W in L2 (Ω; R3×3 ), 1 1 (Ω; R3 ) such that uε u in Hdn (Ω; R3 ). (ii) there exists u ∈ Hdn Proof. Item (i) follows from (5.2), Rellich’s compactness theorem, and part (i) of Theorem 5.1. From (4.5) and the definition of giε it follows that. . ε 1 ε ε ε ε ε ε 1 ε ε ∇u 2 H u g1 |g2 |g3 2 ∂ ∂ ≤ u ∂ u u = 1 2 3 . L (Ω) L (Ω) ε δε L2 (Ω) ε ε ≤ C H u 2 , L (Ω). and hence (ii) follows from (i). 1 Hereafter we set ϑ := n · W t. By the regularity of γ we have ϑ ∈ Hdn (0, L). 3×3 2 Corollary 5.4. There exists B ∈ L ((0, ) × (0, L); Rskw ) such that, up to subsequences, ∂1 W ε B ε. (5.3). in L2 ((0, ) × (0, L); R3×3 skw ). Then ε 1 (i) W21 := g2ε · W ε g1ε ϑ in Hdn ((0, ) × (0, L)), (ii) W = ϑ(n ⊗ t − t ⊗ n), (iii) Be3 = ∂3 ϑ n, (iv) u = 0. Proof. The existence of B is a consequence of (iii) of Theorem 5.1. Item (i) follows from (5.2) and the uniform convergence of g1ε and g2ε to t and n, respectively. Let W13 := t · W e3 and W23 := n · W e3 . From (4.8) and (i)–(ii) of Corollary 5.3 we deduce that W13 = t · ∂3 u = ∂3 (u · t),. (5.4). W23 = n · ∂3 u = ∂3 (u · n).. We now claim that Be3 = ∂3 W t.. (5.5). Indeed, let ψ ∈ C0∞ ((0, ) × (0, L)). By (i) of Theorem 5.1, by (i) of Corollary 5.3, and taking into account (4.4) we have that
(12) ε ε
(13) H u e3 ∂3 uε W ε e3 W ε e3 − − 0 = lim ∂1 ψ dx = lim ∂1 ψ dx ε→0 Ω ε→0 Ω ε ε ε ε
(14)
(15) ∂1 uε ∂1 W ε e3 ∂1 W ε e3 ∂3 ψ + ψ dx = lim ψ dx H ε uε g1ε ∂3 ψ + = lim ε→0 Ω ε→0 Ω ε ε ε
(16)
(17) = W t∂3 ψ + Be3 ψ dx = (−∂3 W t + Be3 )ψ dx, Ω. Ω. from which we get (5.5). Since B is a skew-symmetric matrix field we have that e3 · Be3 = 0, and hence by (5.5) it follows that 0 = −e3 · ∂3 W t = t · ∂3 W e3 . Since t is not constant and also e3 · ∂3 W e3 = 0, it follows that ∂3 W e3 = 0..
(18) LINEAR MODELS FOR COMPOSITE THIN-WALLED BEAMS. 3305. 1 Thus W e3 = 0, because W e3 = 0 at {x3 = 0} as W ∈ Hdn ((0, ) × (0, L); R3×3 skw ). Hence, W13 = W23 = 0 and this implies (ii). From (5.5) it follows that t · Be3 = 0, since W is a skew-symmetric matrix field, and still from (5.5) we have that n · Be3 = ∂3 (n · W t) = ∂3 ϑ. Hence also (iii) has been proved. 1 (Ω; R3 ), from (5.4) we We finally prove (iv). Since W13 = W23 = 0 and u ∈ Hdn deduce that u · t = u · n = 0. Therefore, to conclude the proof it suffices to show that ε = 0, it follows that u3 = 0. Indeed, from (4.8) and (i) of Theorem 5.1, and since W33 ε 2 ∂3 u3 → 0 in L (Ω). Hence, by (ii) of Corollary 5.3, we have ∂3 u3 = 0, which implies that u3 = 0, since u3 = 0 on {x3 = 0}. Remark 5.5. The regularity W 3,∞ of the curve γ is fully exploited in (i) of ε Corollary 5.4. Indeed, ∂1 W21 involves ∂1 g1ε , which, in turn, involves the derivative of the curvature κ of γ. Thus, under the W 3,∞ regularity of the curve γ we have that ε is an L2 function. ∂1 g1ε is in L∞ and ∂1 W21 If the regularity of the curve γ were assumed to be W 2,∞ , we could only claim ε that W21 weakly converges in L2 ((0, ); H 1 (0, L)). Thus, a weakening of the regularity ε . of γ also weakens the convergence of W21. 5.1. Limit of rescaled displacements. Since the limit of uε is equal to zero, we look at the following rescaled components of uε : (5.6). v¯ε :=. uε − uε3 e3 , δε /ε. v3ε :=. uε3 . δε. To measure the slenderness of the cross section we introduce the parameter s defined by ε2 , ε→0 δε. s := lim. where we have assumed that the above limit exists. Without loss of generality, it suffices to consider three cases s ∈ {0, 1, +∞}. In this paper we consider only the cases s ∈ {0, 1}. Let
(19) γG := − γ(x1 ) dx1 . 0. Lemma 5.6. Up to a subsequence we have
(20) ε W ε e3 W e3 −− dx1 ∂3 ϑe3 ∧ (γ − γG ) ε ε 0 in H 1 ((0, ); L2 ((0, L); R3 )). Proof. Let
(21) ε W e3 W ε e3 −− dx1 . b := ε ε 0 ε.
(22) 3306. C. DAVINI, L. FREDDI, AND R. PARONI. By Poincare’s inequality and (iii) of Theorem 5.1 it follows that bε is bounded in H 1 ((0, ); L2 (0, L)). Thus, there exists a b ∈ H 1 ((0, ); L2 ((0, L); R3 )) such that bε b in H 1 ((0, ); L2 ((0, L); R3 )), and
(23) − b dx1 = 0.. (5.7). 0. Since ∂1 b = ∂1 W e3 /ε, from (5.3) we deduce that ∂1 b = Be3 . Thus, by (iii) of Corollary 5.4 we have that ε. ε. ∂1 b = ∂3 ϑn = ∂3 ϑe3 ∧ t = ∂1 (∂3 ϑ e3 ∧ γ), and from this identity and (5.7) we find b = ∂3 ϑe3 ∧ (γ − γG ). Lemma 5.7. For s ∈ {0, 1}, we have that
(24) v¯ε − − v¯ε da s ϑe3 ∧ (γ − γG ) ω. in H 1 (Ω; R3 ). Proof. By (i) of Theorem 5.1 we have that ε ε H u e3 W ε e3 (5.8) ≤ Cε, δε /ε − δε /ε 2 L (Ω) and by Jensen’s inequality we then have that
(25) H ε uε e3 W ε e3 (5.9) ≤ Cε. − δε /ε − δε /ε da 2 ω L (0,L) Since.
(26)
(27) H ε uε e3 ∂3 uε ∂3 uε H ε uε e3 −− da = −− da δε /ε δε /ε δε /ε ω δε /ε ω
(28) H ε uε e3 W ε e3 W ε e3 H ε uε e3 = − −− − da δε /ε δε /ε δε /ε δε /ε ω 2 ε
(29) ε W e3 W ε e3 −− da + , ε ε δε ω. (5.10). by taking into account (5.1), (5.6), (5.8), (5.9), and Lemma 5.6 we deduce that
(30) ε ε ∂3 v¯ − − v¯ da s∂3 ϑe3 ∧ (γ − γG ) ω. in L (Ω; R ). In particular, it follows that the sequence vˇε := v¯ε − −ω v¯ε da is bounded in H 1 ((0, L); L2 (ω; R3 )), hence vˇε vˇ in H 1 ((0, L); L2 (ω; R3 )), up to a subsequence. Since ∂3 (ˇ v − sϑe3 ∧ (γ − γG )) = 0 and vˇ − sϑe3 ∧ (γ − γG ) = 0 on {x3 = 0}, it follows that vˇ = s ϑe3 ∧ (γ − γG ). To conclude the proof it suffices to show that the sequences {∂1 vˇε } and {∂2 vˇε } are bounded in L2 (Ω; R3 ). By means of (4.5) we find
(31) ε ε 1 1 ε ε ε ε ε ε ε H u − W g1 = ∂1 u − W g1 = ∂1 u − − u da − W ε g1ε ; ε ε ω 2. 3.
(32) LINEAR MODELS FOR COMPOSITE THIN-WALLED BEAMS. 3307. thus, by the definition of vˇε , (5.6), (4.4), and from (i)–(ii) of Theorem 5.1, we find ∂1 vˇε L2 (Ω) ≤ C. ε2 ε2 ( (H ε uε − W ε )g1ε L2 (Ω) + W ε g1ε L2 (Ω) ) ≤ C , δε δε. which is bounded for s ∈ {0, 1}. Similarly we find ∂2 vˇε L2 (Ω) ≤ ε( (H ε uε − W ε )g2ε L2 (Ω) + W ε g2ε L2 (Ω) ) ≤ Cε. 1 Theorem 5.8. Let s ∈ {0, 1}. There exists m ¯ ∈ Hdn ((0, L); R3 ) such that
(33) − v¯ε da m ¯ ω 1 ((0, L); R3 ), up to a subsequence. Moreover, there exists m3 ∈ L2 (0, L) such in Hdn that, setting. (5.11) (5.12). v¯ := m ¯ + sϑe3 ∧ (γ − γG ),.
(34). v3 := m3 − ∂3 m ¯ · (γ − γG ) + s ∂3 ϑ. 0. x1. (γ − γG ) · n ds,. up to a subsequence we have 1 (i) v¯ε v¯ in Hdn (Ω; R3 ), ε 1 (ii) v3 v3 in Hdn (Ω). Proof. Since ε ε (E u )33 ∂3 uε3 = sup = sup ∂3 v3ε L2 (Ω) , +∞ > sup δε δε L2 (Ω) ε ε ε L2 (Ω) by the Poincar´e inequality we have that sup v3ε H 1 ((0,L);L2 (ω)) < +∞,. (5.13). ε. and there exists v3 ∈ H 1 ((0, L); L2 (ω)) such that, up to a subsequence, v3ε v3 in H 1 ((0, L); L2 (ω)). Also, by (4.8) and (4.9) we have ∂2 v3ε = g3ε ·. ∂2 uε = (H ε uε )32 = 2(E ε uε )32 − (H ε uε )23 . δε. By (5.1), (i) of Corollary 5.3, and (ii) of Corollary 5.4, this implies that ∂2 v3ε → W23 = 0 in L2 (Ω). Thus v3ε v3. (5.14). in H 1 ((−h/2, h/2) × (0, L); L2 (0, )) and (5.15) Let. ∂2 v3 = 0.
(35) m ¯ := − v¯ε da. ε. ω.
(36) 3308. C. DAVINI, L. FREDDI, AND R. PARONI. From (4.8) and (4.9) we have that 2(E ε uε )13 1 1 ε = g3ε · ∂1 uε + g · ∂3 uε = ∂1 v3ε + g1ε · ∂3 v¯ε δε /ε δε δε /ε 1 = ∂1 v3ε + g1ε · ∂3 m ¯ ε + g1ε · ∂3 (¯ vε − m ¯ ε ).. (5.16). Let 2(E ε uε )13 − g1ε · ∂3 (¯ vε − m ¯ ε ) = ∂1 v3ε + g1ε · ∂3 m ¯ε δε /ε. hε :=. and note that, by (5.1) and Lemma 5.7, we have that sup hε L2 (Ω) < +∞.. (5.17). ε. Let ψ ∈. C0∞ (0, ),. 2. let ϕ ∈ L (0, L) with ϕ L2 (0,L) ≤ 1, and denote by
(37) Mϕε :=. We then have.
(38). L. 0. ∂3 m ¯ ε ϕ dx3 ..
(39) ∂1 v3ε ψϕ + g1ε · ∂3 m ¯ ε ψϕ dx Ω
(40)
(41) =− v3ε ∂1 ψϕ dx + Mϕε · g1ε ψ da. hε ψϕ dx = Ω. Ω. and therefore. ω.
(42). Mϕε · g1ε ψ da ≤ ψ W 1,∞ (0,) ( hε L2 (Ω) + v3ε L2 (Ω) ).. ω. Since.
(43).
(44) ω. we find that. Mϕε. ·. g1ε ψ da. =. Mϕε. · ω.
(45) g1ε ψ da. =. Mϕε. ·. tψ da, ω.
(46) ε. Mϕ · tψ da ≤ ψ W 1,∞ (0,) ( hε L2 (Ω) + v3ε L2 (Ω) ).. ω. Since t is not constant, by choosing two appropriate functions ψ one can show that |Mϕε | ≤ C( hε L2 (Ω) + v3ε L2 (Ω) ), which implies that ∂3 m ¯ ε L2 (0,L) =. sup ϕL2 (0,L) ≤1. |Mϕε | ≤ C( hε L2 (Ω) + v3ε L2 (Ω) ).. 1 Thus, taking into account that v¯ε ∈ Hdn (Ω; R3 ) and using (5.13) and (5.17), we ε deduce that supε m ¯ H 1 (0,L) < +∞, and hence, up to a subsequence,. m ¯ε m ¯.
(47) LINEAR MODELS FOR COMPOSITE THIN-WALLED BEAMS. 3309. 1 in Hdn (0, L), which is the first part of the statement. By Lemma 5.7 then we have. ¯ + sϑe3 ∧ (γ − γG ) v¯ε m. (5.18). 1 in Hdn (Ω; R3 ), which proves (i). By (5.16) it follows that. 2(E ε uε )13 − g1ε · ∂3 v¯ε . δε /ε. ∂1 v3ε =. (5.19). Hence, also ∂1 v3ε is bounded in L2 (Ω) and the convergence in (5.14) is, in fact, weak in H 1 (Ω), as stated in (ii). By using (5.1) and (5.18) to take the limit in (5.19) we deduce that ∂1 v3 = −t · ∂3 (m ¯ + s ϑe3 ∧ (γ − γG )) = −t · ∂3 m ¯ + s∂3 ϑ(γ − γG ) · n
(48) x1 ¯ · (γ − γG ) + s ∂3 ϑ (γ − γG ) · n ds . = ∂1 −∂3 m 0. Taking into account (5.15), we conclude that
(49) ¯ · (γ − γG ) + s ∂3 ϑ v3 = m3 − ∂3 m. (5.20). 0. x1. (γ − γG ) · n ds. with m3 ∈ L2 (0, L). In fact, (5.11) and (5.12) imply further regularity on m3 , m, ¯ and ϑ. Theorem 5.9. Let s, m3 , m, ¯ and ϑ be as in Theorem 5.8. Then, 1 (0, L), (i) m3 ∈ Hdn 2 (ii) m ¯ ∈ Hdn (0, L; R3 ) := {z ∈ H 2 (0, L; R3 ) : z(0) = ∂3 z(0) = 0}, 2 (iii) if s = 1, then ϑ ∈ Hdn (0, L). In particular, the displacement v¯ defined in (5.11) belongs to the space H 2 ((0, ) × (0, L); R3 ). 1 (0, L) it is Proof. Let us consider the case s = 0 first. To prove that m3 ∈ Hdn enough to take the integral over (0, ) with respect to the variable x1 in (5.12). In 1 1 this way we get m3 = −0 v3 dx1 , which implies m3 ∈ Hdn (0, L) because v3 ∈ Hdn (Ω). 3,∞ Statement (ii) can be proved similarly. Indeed, since the curve γ is W and not x a straight line, then there exist x1 , x1 ∈ (0, ) such that the vectors 0 1 (γ − γG ) ds x and 0 1 (γ − γG ) ds are linearly independent and can be used as a basis in the plane x3 = 0. Then (ii) follows by integrating (5.12) with respect to x1 over the intervals (0, x1 ) and (0, x1 ) and by taking a linear combination. Consider now the case s = 1. Without loss of generality we may take the origin of the axes coincident with γG , that is, γG = 0, and rotate axes e1 , e2 , if necessary, so that they coincide with the principal axes of inertia of the curve γ:
(50) 0. . γ1 γ2 dx1 = 0.. We also introduce a point γSC in the plane of the cross section and a scalar c to be chosen in what follows. Then, by means of the identity
(51) x1
(52) x1 (5.21) γ · n ds = (γ − γSC ) · n ds + γSC · e3 ∧ (γ − γ(0)), 0. 0.
(53) 3310. C. DAVINI, L. FREDDI, AND R. PARONI. the displacement v can be written as (5.22). v¯ = ξ¯ + ϑe3 ∧ (γ − γSC ),. where (5.23).
(54). x1. ψ := c + 0. (5.24) (5.25). v3 = ξ3 − ∂3 ξ¯ · γ + ψ∂3 ϑ,. (γ − γSC ) · n ds,. 1 ((0, L); R3 ), ξ¯ := m ¯ + ϑe3 ∧ γSC ∈ Hdn ξ3 := m3 − ∂3 ϑ c + e3 ∧ γ(0) · γSC ∈ L2 (0, L).. By means of (5.21) it can be shown that γSC and c are uniquely determined by the requirements
(55)
(56)
(57) (5.26) ψ da = ψγ1 da = ψγ2 da = 0. ω. ω. ω. For such a choice of γSC and c, from (5.22) and (5.26) we deduce that
(58)
(59) ψv3 da = ψ 2 da ∂3 ϑ, ω. hence ϑ ∈. 2 Hdn (0, L),. and that. ω.
(60) − v3 da = ξ3 . ω. 1 Hdn (0, L),. 1 Then ξ3 ∈ since the left-hand side of the above equality is in Hdn (0, L). 1 By (5.25), this implies in particular that m3 ∈ Hdn (0, L). Finally, we deduce from 2 (5.22) that ξ¯ ∈ Hdn ((0, L); R3 ). The claimed regularity of m ¯ follows from (5.24). In the technical literature the point γSC defined by (5.26) is called the shear center and plays a special role in the uncoupling of the torsional and flexural effects. Also, ψ describes the warping of the cross section, and the scalar c introduced in (5.23) defines the warping at point x1 = 0 on the curve γ.. 5.2. Limit of rescaled strains. From (5.1) it follows that there exists E ∈ L2 (Ω; R3×3 ) such that (5.27). E ε uε E in L2 (Ω; R3×3 ) δε. up to a subsequence. The following lemmas give a characterization of some components of E. Lemma 5.10. Let E33 := e3 · Ee3 . Then E33 = ∂3 v3 . Proof. Indeed, from (4.8), (4.9), and (5.6), we have (E ε uε )33 g ε · ∂3 u ε = 3 = ∂3 v3ε δε δε and the thesis follows by applying Theorem 5.8. Lemma 5.11. Let E13 := t · Ee3 . Then E13 = −x2 ∂3 ϑ + η2 , where η2 ∈ L2 (Ω) and ∂2 η2 = 0..
(61) LINEAR MODELS FOR COMPOSITE THIN-WALLED BEAMS. 3311. Proof. Let ϑε :=. (H ε uε )21 − (H ε uε )12 . 2. We claim that ∂3 ϑε =. (5.28). 1 1 ∂1 (E ε uε )23 − ∂2 (E ε uε )13 , ε δε. in the sense of distributions. Assuming this, note that ϑε = (H ε uε )21 − (E ε uε )21 → ϑ in L2 (Ω) by (5.1), (i) of Corollary 5.3, and the definition of ϑ (see Corollary 5.4). Thus, using (5.27) to take the limit in H −1 in (5.28), we find ∂3 ϑ = −∂2 E13 . The statement of the lemma follows from the equality above. Let us now prove (5.28). Since 2ϑε =. 1 ε 1 g · ∂1 uε − g1ε · ∂2 uε , ε 2 δε. we have that 1 ε 1 g · ∂1 ∂3 uε − g1ε · ∂2 ∂3 uε ε 2 δε 1 ε 1 1 ε 1 g2 · ∂3 uε − ∂1 g2ε · ∂3 uε − ∂2 = ∂1 g1 · ∂3 uε + ∂2 g1ε · ∂3 uε . ε ε δε δε. 2∂3 ϑε =. Since ∂2 g1ε = (δε /ε)∂1 g2ε , then 1 ε 1 ε 1 1 g2 · ∂3 uε − ∂2 2∂3 ϑε = ∂1 g1 · ∂3 uε = ∂1 (H ε uε )23 − ∂2 (H ε uε )13 ε δε ε δε 1 1 1 1 = ∂1 (2E ε uε )23 − ∂1 (H ε uε )32 − ∂2 (2E ε uε )13 + ∂2 (H ε uε )31 ε ε δε δε 1 1 ε 1 1 1 1 ε ε ε ε ε ε ε g · ∂1 u − ∂1 = ∂1 (2E u )23 − ∂2 (2E u )13 + ∂2 g · ∂2 u ε δε δε ε 3 ε δε 3 1 1 = ∂1 (2E ε uε )23 − ∂2 (2E ε uε )13 , ε δε and hence (5.28) has been proved. Lemma 5.12. Let E11 := t · Et. Then, (5.29). E11 = x2 η3 + η1. with η1 ∈ L2 (Ω), η3 = t · Bn, and ∂2 η1 = ∂2 η3 = 0. Proof. Note that 1 1 ∂2 uε 1 ∂1 uε 1 = ∂1 ∂2 (H ε uε g1ε ) = ∂2 = ∂1 (H ε uε g2ε ) δε δε ε ε δε ε ε ε H u 1 = ∂1 g2ε + H ε uε ∂1 g2ε , ε ε.
(62) 3312. C. DAVINI, L. FREDDI, AND R. PARONI. and by means of this identity we have that ∂2. (E ε uε )11 1 1 1 = ∂2 (g1ε · H ε uε g1ε ) = ∂2 g1ε · H ε uε g1ε + g1ε · ∂2 (H ε uε g1ε ) δε δε δε δε ε ε H u 1 1 = ∂2 g1ε · H ε uε g1ε + g1ε · ∂1 g2ε + g1ε · H ε uε ∂1 g2ε . δε ε ε. Since ∂1 g2ε =. ε κ gε, ∂2 g1ε = − δε 1 − (δε /ε)x2 κ 1. we deduce that (5.30). ∂2. 2κ (E ε uε )11 (E ε uε )11 H ε u ε ε δε g2 − = g1ε · ∂1 . δε ε ε 1 − (δε /ε)x2 κ δε. From (i) of Theorem 5.1 and (5.3) it follows that ∂1. H ε uε → B in H −1 (Ω; R3×3 ), ε. and from (5.27) we have ∂2. (E ε uε )11 ∂2 E11 in H −1 (Ω). δε. Hence from (5.30) we deduce that ∂2 E11 = t · Bn. Since B, t, and n do not depend on x2 we have the claim. 6. Reduced energy densities for open cross sections. In this section we introduce and study some properties of two reduced energy densities that will appear in the Γ-convergence result presented in the next section. Lemmas 5.10, 5.11, and 5.12, give a partial characterization of the components E33 = e3 · Ee3 , E13 = t · Ee3 , and E11 = t · Et of the limit strain E defined by (5.27). No information is instead given on the components E12 = t · En, E22 = n · En, and E23 = n · Ee3 . Motivated by this fact, with f (x, M ) :=. 1 C(x)M · M, 2. M ∈ R3×3 sym ,. we define (6.1). f0 (x, M11 , M13 , M33 ) := min f x, M11 t(x1 ) t(x1 ) + 2A12 t(x1 ) n(x1 ) Aij. + 2M13 t(x1 ) e3 + A22 n(x1 ) n(x1 ) + 2A23 n(x1 ) e3 + M33 e3 e3 . That is, f0 is obtained from f by keeping fixed the components that have been partially characterized in section 5.2 and by minimizing over the remaining components. For open cross sections by Lemmas 5.10, 5.11, and 5.12, we have that E11 = η1 + x2 η3 ,. E13 = −x2 ∂3 ϑ + η2 ,. E33 = ∂3 v3 ,.
(63) LINEAR MODELS FOR COMPOSITE THIN-WALLED BEAMS. 3313. where ηi ∈ L2 (Ω), i = 1, 2, 3, are functions of (x1 , x3 ) that have not been characterized in terms of v3 and ϑ. This leads us to a second minimization and hence to the definition of a second reduced energy density. In contrast to the minimization performed in (6.1), the minimization over ηi is not completely local because of the presence of the variable x2 in E11 and E13 . For x1 and x3 fixed and a, b ∈ R, let ηiopt = ηiopt (x1 , x3 , a, b) be the minimizers of
(64) h/2 inf (6.2) f0 (x, η1 + x2 η3 , −x2 a + η2 , b) dx2 . ηi ∈R. −h/2. In section 6.2 it is proved that ηiopt , for i = 1, 2, 3, are unique and that they can be written as a linear combination of a and b with L∞ -coefficients. In particular, the maps ηiopt : (0, ) × (0, L) × R × R → R defined by (x1 , x3 , a, b) → ηiopt (x1 , x3 , a, b) are measurable. Then we can define the function f00 : Ω × R × R → R as (6.3). f00 (x, a, b) := f0 (x, η1opt + x2 η3opt , −x2 a + η2opt , b),. where the ηiopt are evaluated in (x1 , x3 , a, b). From the definition of f0 it follows that there exists a constant c > 0 such that (6.4). 2 2 2 f0 (x, M11 , M13 , M33 ) ≥ c (M11 + 2M13 + M33 ),. and from this inequality and the definition of f00 we deduce that there exists a constant c > 0 such that
(65) h/2 (6.5) f00 (x, a, b) dx2 ≥ c (a2 + b2 ). −h/2. The given definitions are sufficient to prove the so-called liminf inequality (see Theorem 7.1). On the other hand, to provide the so-called recovery sequence (see Theorem 7.2) several properties of f0 , f00 and their minimizers are needed. These properties will be determined in the next lemmas. In order to keep the notation compact we shall not use the components of E, as done in (6.1) and (6.3), but work with tensors. For fixed x1 , let S(x1 ) := span{t(x1 ) n(x1 ), n(x1 ) n(x1 ), n(x1 ) e3 } and S ⊥ (x1 ) := span{t(x1 ) t(x1 ), t(x1 ) e3 , e3 e3 } so that ⊥ R3×3 sym = S(x1 ) ⊕ S (x1 ).. Thus any tensor M ∈ R3×3 sym can be uniquely written as M = MS + M⊥.
(66) 3314. C. DAVINI, L. FREDDI, AND R. PARONI. with M S := 2(t · M n) t n + (n · M n) n n + 2(n · M e3 ) n e3 ∈ S(x1 ) and M ⊥ := (t · M t) t t + 2(t · M e3 ) t e3 + (e3 · M e3 ) e3 e3 ∈ S ⊥ (x1 ). The decomposition of E, as given by (5.27), is such that E ⊥ contains the components of the strain E that have been partially characterized by Lemmas 5.10, 5.11, and 5.12, and E S contains the remaining components. With a slight abuse of notation we may write (6.6). f0 (x, M ⊥ ) =. min. M S ∈S(x1 ). f (x, M S + M ⊥ ).. We shall denote by E0 M ⊥ the “full” tensor achieving the (unique) minimum in (6.6), i.e., (6.7). E0 M ⊥ := M0S + M ⊥. for f (x, M0S + M ⊥ ) =. min. M S ∈S(x1 ). f (x, M S + M ⊥ ).. Hence E0 maps the fixed part of the strain M ⊥ to the “full” minimizer of (6.6) as stated by (6.7). Since f is a quadratic function we have that E0 is a linear operator from S ⊥ to R3×3 sym . We also have (6.8). f0 (x, M ⊥ ) = f (x, E0 M ⊥ ). and (6.9). (I − P)E0 M ⊥ = M ⊥. for every M ⊥ ∈ S ⊥ , since (I − P)M S = 0 for every M S ∈ S. To write the mapping E0 explicitly, and in compact form, we introduce the projection operator (6.10). P(x1 ) : R3×3 sym → S(x1 ), M → PM := M S .. Then M ⊥ = (I − P)M , where I denotes the identity. Lemma 6.1. The mapping E0 defined by (6.7) is given by (6.11). E0 = I − (PCP)−1 PC.. Moreover, for every M ∈ R3×3 Sym we have that (6.12). PCM = 0 if and only if M = E0 M ⊥. and (6.13). CE0 M ⊥ · B ⊥ = CE0 M ⊥ · E0 B ⊥ for every B ⊥ ∈ S ⊥ .. ˜ 0 in place ˜ 0 := I − (PCP)−1 PC. We first prove that (6.12) holds with E Proof. Set E of E0 . Note that PCP : S → S is an invertible operator, since PCPU S · U S = C(PU S ) · (PU S ) = CU S · U S ≥ c|U S |2 ,.
(67) LINEAR MODELS FOR COMPOSITE THIN-WALLED BEAMS. 3315. where we used the facts PU S = U S for every U S ∈ S and PU · V = PV · U for every ⊥ U , V ∈ R3×3 sym . Since M = PM + M , then PCM = 0 holds if and only if PC(PM + M ⊥ ) = 0, that is, PCP(PM ) = −PCM ⊥ or PM = −(PCP)−1 PCM ⊥ , which is equivalent to ˜ 0 M ⊥. PM + M ⊥ = E ˜ 0 in place of E0 . Thus, (6.12) holds for E S Let M0 ∈ S(x1 ) be the minimizer of (6.6), i.e., f (x, M0S + M ⊥ ) =. min. M S ∈S(x1 ). f (x, M S + M ⊥ ).. The minimality conditions for (6.6) are ⎧ S ⊥ ⎪ ⎪C(M0 + M ) · t n = 0, ⎨ C(M0S + M ⊥ ) · n n = 0, ⎪ ⎪ ⎩ C(M0S + M ⊥ ) · n e3 = 0, ˜ 0 in place of E0 , as proved above, that is, PC(M0S + M ⊥ ) = 0. Thus, by (6.12) with E we deduce that ˜ 0 M ⊥, M0S + M ⊥ = E ˜ 0 = E0 . and hence, from (6.7), we conclude that E To prove (6.13) note that CE0 M ⊥ · B ⊥ = CE0 M ⊥ · (I − P)E0 B ⊥ = (I − P)CE0 M ⊥ · E0 B ⊥ = CE0 M ⊥ · E0 B ⊥ , where the first identity follows from (6.9) while the last one has been obtained by applying (6.12). The previous lemma characterizes the minimizer of f0 . We now study the minimizers of f00 . 6.1. Properties of f00 for beams with an open cross section. Let E be the limit strain defined by (5.27). Then E ⊥ = (η1 + x2 η3 )t t + 2η2 t e3 + E K (∂3 ϑ, ∂3 v3 ), where E K (∂3 ϑ, ∂3 v3 ) is the part of E ⊥ that is known in terms of v3 and ϑ, that is, (6.14). E K (a, b) := −2x2 a t e3 + b e3 e3 ,.
(68) 3316. C. DAVINI, L. FREDDI, AND R. PARONI. for every a, b ∈ R. By means of the minimizers ηiopt of (6.2) we define the mapping E00 by (6.15). E00 E K (a, b) := (η1opt + x2 η3opt )t t + 2η2opt t e3 + E K (a, b),. so we rewrite (6.3), with the slight abuse of notation introduced in (6.6), as f00 (x, a, b) = f0 (x, E00 E K (a, b)).. (6.16). Thus E00 essentially associates to the known part of the strain the “full” minimizer of (6.2) as stated by (6.16). Thus the mappings E0 and E00 allow us to rewrite the densities f0 and f00 in terms of f f00 (x, a, b) = f0 (x, E00 E K (a, b)) = f (x, E0 E00 E K (a, b)),. (6.17). as follows from (6.8) and (6.16). Hereafter, to keep the notation compact, we denote by
(69) h/2 · := − ·dx2 −h/2. the average over the x2 variable. Lemma 6.2. For a, b ∈ R let M ∈ R3×3 sym be a tensor field such that M ⊥ = (η1 + x2 η3 )t t + 2η2 t e3 + E K (a, b),. (6.18). where ηi = ηi (x1 , x3 , a, b) and E K is defined by (6.14). Then,. (6.19). M⊥. ⎧ CE0 M ⊥ · t t = 0, ⎪ ⎪ ⎨ = E00 E K (a, b) (i.e., ηi = ηiopt ) ⇐⇒ CE0 M ⊥ · t e3 = 0, ⎪ ⎪ ⎩ x2 CE0 M ⊥ · t t = 0.. Moreover, if M ⊥ = E00 E K (a, b), then (6.20). CE0 E00 E K (a, b) · E K (c, d) = CE0 E00 E K (a, b) · E0 E00 E K (c, d). for every a, b, c, d ∈ R. Proof. Let us denote by E U (ηi ) := (η1 + x2 η3 )t t + 2η2 t e3 so that E ⊥ = U E (ηi ) + E K (a, b). By (6.8) we may rewrite the minimization problem (6.2) as f (x, E0 (E U (ηiopt ) + E K (a, b))) = inf f (x, E0 (E U (ηi ) + E K (a, b))) ηi ∈R. whose minimality condition is CE0 (E U (ηiopt ) + E K (a, b)) · E0 E U (ξi ) = 0 ∀ξi ∈ R. Thus, since f is convex and E0 is a linear operator, we have that ηi = ηiopt ⇐⇒ CE0 (E U (ηi ) + E K (a, b)) · E0 E U (ξi ) = 0. ∀ξi ∈ R,.
(70) 3317. LINEAR MODELS FOR COMPOSITE THIN-WALLED BEAMS. which, by (6.13), is equivalent to ηi = ηiopt ⇐⇒ CE0 (E U (ηi ) + E K (a, b)) · E U (ξi ) = 0. ∀ξi ∈ R,. and this, in turn, is equivalent to (6.19). To prove (6.20) note that CE0 E00 E K (a, b) · E K (c, d) = CE0 E00 E K (a, b) · E00 E K (c, d) = CE0 E00 E K (a, b) · E0 E00 E K (c, d), where the first equality follows from (6.19) and the last from (6.13). Remark 6.3. By taking a = c and b = d in (6.20) and using that C is positive definite, we deduce that there exists a constant C ≥ 0 such that |E0 E00 E K (a, b)| ≤ C|E K (a, b)| for every a, b ∈ R. 6.2. Computation of the reduced energy densities. In this subsection we outline the computation of the energy densities. Let E11 := t · Et,. E13 := t · Ee3 ,. E33 := e3 · Ee3 .. From (6.8) we find f0 (x, E11 , E13 , E33 ) = f (x, E0 E ⊥ ) =. 1 CE0 E ⊥ · E0 E ⊥ 2. 1 CE0 (E11 t t + 2E13 t e3 + E33 e3 e3 ) 2 · E0 (E11 t t + 2E13 t e3 + E33 e3 e3 ) ⎞ ⎛ ⎞⎛ ⎞ ⎛ E11 E11 11 (x) 12 (x) 13 (x) ⎟ ⎜ 1⎜ ⎟⎜ ⎟ ⎟ ⎜ = ⎝ 12 (x) 22 (x) 23 (x)⎠ ⎝E13 ⎠ · ⎝E13 ⎠ , 2 E33 E33 13 (x) 23 (x) 33 (x). =. where. (6.21). ij. are given by 11. = CE0 (t t) · E0 (t t),. 22. = 4CE0 (t e3 ) · E0 (t e3 ),. 12. = 4CE0 (t t) · E0 (t e3 ),. 23. = 4CE0 (t e3 ) · E0 (e3 e3 ),. 13. = 2CE0 (t t) · E0 (e3 e3 ),. 33. = CE0 (e3 e3 ) · E0 (e3 e3 ).. The ηiopt may be computed by solving the system ⎛ ⎞ ⎛ opt ⎞ ⎛ η1 12 x2 11 11 x2 ⎜ ⎟ ⎜ opt ⎟ ⎜ 22 x2 12 ⎠ ⎝η2 ⎠ = ⎝ x2 (6.22) ⎝ 12 x2. 11 . x2. 12 . x22. 11 . η3opt. x22. 12 a. −. 13 b. 22 a. −. 23 b. 12 a. − x2. ⎞ ⎟ ⎠,. 13 b. since (6.22) is equivalent to (6.19). Indeed, let E ⊥ be as in (6.18) with ηi = ηiopt ; then by means of (6.13) and (6.21) we have 0 = CE0 E ⊥ · t t = CE0 E ⊥ · t t = CE0 E ⊥ · E0 (t t) =. opt 11 η1. + x2. opt 11 η3. +. opt 12 η2. − x2. 12 a. +. 13 b,.
(71) 3318. C. DAVINI, L. FREDDI, AND R. PARONI. and hence the first equation of (6.22) is equivalent to the first equation of (6.19). The equivalence of the other equations is proved similarly. We note that (6.22), and hence (6.19), has a unique solution since ⎛. . ⎜ ⎝ . 11 . . 12 . 12 . . 22 . ⎞⎛ ⎞ ⎛ ⎞ a1 a1 ⎟⎜ ⎟ ⎜ ⎟ x2 12 ⎠ ⎝a2 ⎠ · ⎝a2 ⎠ x2. 11 . x22 11 a3 a3 a1 + x2 a3 a1 + x2 a3 12 · dx2 a2 a2 −h/2 12 22 2
(72) h/2 . a1 + x2 a3 ≥c−. dx2 ≥ c (a21 + a22 + a23 ). a2 −h/2 x2 11 x2
(73) h/2 11 =−. 12 . for every a1 , a2 , a3 ∈ R. The first inequality above is a consequence of (6.4). Remark 6.4. From (6.22) we also deduce that the maps ηiopt are measurable. Indeed, from the measurability of C and the Lipschitz continuity of the projection P we deduce the measurability of E0 thanks to (6.1). Then, from (6.21) it follows immediately that the coefficients ij are measurable. 7. Γ-limit for beams with an open cross section. Let Jε : H 1 (Ω; R3 ) → R ∪ {+∞} be defined by ⎧ ε ε √ ε 1 ⎨1 if u ∈ Hdn (Ω; R3 ), 2 Ω CE u · E u g dx Jε (u) = (7.1) ⎩+∞ 1 if u ∈ H 1 (Ω; R3 )\Hdn (Ω; R3 ). In this section we shall prove the Γ-convergence of the functional Jε /δε2 under an appropriate topology. In order to define the Γ-limit we set 1+s 2 1 (0, L) : ∃ m ¯ ∈ Hdn (0, L; R3 ), ∃ m3 ∈ Hdn (0, L) As := {(v, ϑ) ∈ H 1 (Ω; R3 ) × Hdn. ¯ + sϑe3 ∧ (γ − γG ), such that v = v¯ + v3 e3 , where v¯ = m x ¯ · (γ − γG ) + s∂3 ϑ 0 1 (γ − γG ) · n ds}. and v3 = m3 − ∂3 m The Γ-limit will be the functional J0 : H 1 (Ω; R3 ) × H 1 (0, L) → R ∪ {+∞} defined by J0 (v, ϑ) =. Ω. f00 (x, ∂3 ϑ, ∂3 v3 ) dx. +∞. if (v, ϑ) ∈ As , otherwise.. We split the Γ-convergence analysis into two parts. In the next theorem we study the liminf inequality. Theorem 7.1 (liminf inequality). For every sequence {uε } ⊆ H 1 (Ω; R3 ) and every (v, ϑ) ∈ H 1 (Ω; R3 ) × H 1 (0, L) such that v¯ε + v3ε e3 v in H 1 (Ω; R3 ) and g2ε · H ε uε g1ε ϑ in H 1 (Ω),.
(74) LINEAR MODELS FOR COMPOSITE THIN-WALLED BEAMS. 3319. where v¯ε =. uε − uε3 e3 , δε /ε. v3ε =. uε3 , δε. g2ε · H ε uε g1ε =. 1 ε g · ∂1 u ε , ε 2. we have lim inf ε→0. Jε (uε ) ≥ J0 (v, ϑ). δε2. Proof. Without loss of generality we may assume that lim inf ε→0. Jε (uε ) Jε (uε ) = lim < +∞, ε→0 δε2 δε2. since otherwise the claim is trivially satisfied. By (2.5), it follows that the sequence {uε } satisfies (5.1) and hence all the theorems contained in section 5 hold. In particular, by Theorems 5.8 and 5.9 we have that (v, ϑ) ∈ As . From (5.27), the convexity of f , and (6.1) we find
(75)
(76) Jε (uε ) E ε uε √ ε = lim inf f x, g dx ≥ f (x, E) dx lim inf ε→0 ε→0 δε2 δε Ω Ω
(77) f0 (x, E11 , E13 , E33 ) dx ≥ Ω
(78) = f0 (x, η1 + x2 η3 , −x2 ∂3 ϑ + η2 , ∂3 v3 ) dx, Ω. where the last equality has been obtained by means of Lemmas 5.10, 5.11, and 5.12. Here ηi do not depend on x2 and have the regularity prescribed in the lemmas just quoted. Thus from (6.2) and (6.3) we deduce that
(79) L
(80)
(81) h/2 Jε (uε ) lim inf ≥ inf f0 (x, η1 (x1 , x3 ) + x2 η3 (x1 , x3 ) ε→0 δε2 0 0 ηi (x1 ,x3 ) −h/2 − x2 ∂3 ϑ + η2 (x1 , x3 ), ∂3 v3 ) dx2 dx1 dx3
(82) = f00 (x, ∂3 ϑ, ∂3 v3 ) dx = J0 (v, ϑ), Ω. and hence the theorem is proved. We now prove the existence of a recovery sequence. Theorem 7.2 (recovery sequence). For every (v, ϑ) ∈ H 1 (Ω; R3 ) × H 1 (0, L) there exists a sequence {uε } ⊆ H 1 (Ω; R3 ) such that v¯ε + v3ε e3 v in H 1 (Ω; R3 ), g2ε · H ε uε g1ε ϑ in H 1 (Ω), where v¯ε =. uε − uε3 e3 , δε /ε. v3ε =. uε3 , δε. g2ε · H ε uε g1ε =. and lim sup ε→0. Jε (uε ) ≤ J0 (v, ϑ). δε2. 1 ε g · ∂1 u ε , ε 2.
(83) 3320. C. DAVINI, L. FREDDI, AND R. PARONI. Proof. Let (v, ϑ) ∈ H 1 (Ω; R3 ) × H 1 (0, L) be given. To avoid trivial cases we assume that J0 (v, ϑ) < +∞. Thence (v, ϑ) ∈ As . Let (7.2). E K (∂3 ϑ, ∂3 v3 ) := −2x2 ∂3 ϑt e3 + ∂3 v3 e3 e3. and E opt := E0 E00 E K (∂3 ϑ, ∂3 v3 ).. (7.3). At the end of section 6 we remarked that the maps ηiopt are measurable. From (6.15) it follows that E0 and hence E opt are measurable. Moreover, from Remark 6.3 it follows immediately that E opt ∈ L2 (Ω; R3×3 ). 1 Define Rε : Hdn (Ω; R3 ) → R by ε ε
(84) 1 E u E u C − E opt · − E opt dx. Rε (u) := 2 Ω δε δε It follows that for each ε the functional Rε has a minimizer. Let uε be the minimizer, i.e., Rε (uε ) = inf Rε (u). u. By (4.7), we trivially find that E ε uε L2 (Ω) ≤ Cδε , and hence from Corollary 5.4 and Theorem 5.8 we deduce that ¯ + vˇ3 e3 =: vˇ in H 1 (Ω; R3 ), v¯ε + v3ε e3 vˇ g ε · H ε uε g ε ϑˇ in H 1 (Ω). 2. 1. Moreover, we also have E ε uε ˇ in L2 (Ω; R3×3 ). E δε We split the proof into several claims. ˇ ⊥ , thus f0 (x, Eˇ ⊥ ) = f (x, E). ˇ = E0 E ˇ Claim 1. E ⊥ ˇ ∂3 vˇ3 ), thus f00 (x, ∂3 ϑ, ˇ ∂3 vˇ3 ) = f0 (x, Eˇ ⊥ ). ˇ K (∂3 ϑ, ˇ = E00 E Claim 2. E Claim 3. ϑˇ = ϑ and vˇ = v. Assuming that the claims hold we easily conclude the proof. Indeed, from Claim 3 we deduce that v¯ε + v3ε e3 v in H 1 (Ω; R3 ), g2ε · H ε uε g1ε ϑ in H 1 (Ω), and from the three claims it follows that ˇ = E0 E00 E K (∂3 ϑ, ∂3 v3 ) = E opt , E where the last identity is a consequence of (7.3). The minimizer uε of Rε satisfies the following problem: ε ε
(85) E u Eεψ opt 1 (7.4) C −E dx = 0 for every ψ ∈ Hdn (Ω; R3 ). · δε δε Ω.
(86) LINEAR MODELS FOR COMPOSITE THIN-WALLED BEAMS. 3321. For later use we note that this problem is equivalent to ε ε
(87) E u H εψ opt 1 (7.5) C −E dx = 0 for every ψ ∈ Hdn (Ω; R3 ). · δε δε Ω By taking ψ = uε in (7.4) we find
(88)
(89) E ε uε E ε uε E ε uε C · dx = C · E opt dx δ δ δ ε ε ε Ω Ω and hence, by (6.17), lim ε.
(90)
(91) E ε uε E ε uε √ ε E ε uε E ε uε Jε (uε ) 1 1 = lim C · g dx = lim C · dx ε 2 Ω ε 2 Ω δε2 δε δε δε δε
(92)
(93) 1 E ε uε 1 ˇ · E opt dx = lim C · E opt dx = CE ε 2 Ω δε 2 Ω
(94) 1 = CE0 E00 E K (∂3 ϑ, ∂3 v3 ) · E0 E00 E K (∂3 ϑ, ∂3 v3 ) dx 2 Ω
(95)
(96) = f (x, E0 E00 E K (∂3 ϑ, ∂3 v3 )) dx = f00 (x, ∂3 ϑ, ∂3 v3 ) dx, Ω. Ω. = J0 (v, ϑ), which is the thesis of the theorem. We now prove the claims. Proof of Claim 1. With ϕi ∈ C0∞ (Ω), for i = 1, 2, 3, let
(97)
(98)
(99) ψ=. δε2. x2. −h/2. x2. ϕ1 (x1 , ζ, x3 ) dζ t +. −h/2. ϕ2 (x1 , ζ, x3 ) dζ n +. . x2. −h/2. ϕ3 (x1 , ζ, x3 ) dζ e3 .. Then H εψ → ϕ1 t ⊗ n + ϕ2 n ⊗ n + ϕ3 e3 ⊗ n δε uniformly. From (7.5) we deduce that
(100) C(Eˇ − E opt ) · (ϕ1 t ⊗ n + ϕ2 n ⊗ n + ϕ3 e3 ⊗ n) dx = 0, Ω. which implies that PC(Eˇ − E opt ) = 0,. (7.6). where P is defined in (6.10). Since, by (6.12), PCE opt = PCE0 E00 E K (∂3 ϑ, ∂3 v3 ) = 0 it follows that ˇ = 0. PCE. (7.7). Hence, Claim 1 follows from (6.12). Proof of Claim 2. Let
(101) x1
(102) ψ = δε ε ϕ1 (ζ, x3 )t(ζ) dζ + 0. 0. . x1. ϕ2 (ζ, x3 ) dζ e3.
(103) 3322. C. DAVINI, L. FREDDI, AND R. PARONI. with ϕi ∈ C0∞ ((0, ) × (0, L)) for i = 1, 2. Then H εψ → ϕ1 t ⊗ t + ϕ2 e3 ⊗ t δε uniformly. By passing to the limit in (7.5) we deduce that
(104) C(Eˇ − E opt ) · (ϕ1 t ⊗ t + ϕ2 e3 ⊗ t) dx = 0, Ω. and hence (7.8). C(Eˇ − E opt ) · t t = 0, C(Eˇ − E opt ) · t e3 = 0.. We now take ψ = δε εx2 ϕt − ε. 2.
(105) 0. x1. ϕn dx1. with ϕ ∈ C ∞ ((0, ) × (0, L)) and ϕ = 0 nearby x3 = 0. After some calculations we can check that Eεψ → x2 ∂1 ϕt ⊗ t δε uniformly. By passing to the limit in (7.4) we get
(106) ˇ − E opt ) · x2 ∂1 ϕt ⊗ t dx = 0. C(E x1. Ω. Let ϕ := 0 φ(ζ, x3 ) dζ with φ ∈ C0∞ ((0, ) × (0, L)). Then, the above equation implies that (7.9). ˇ − E opt ) · t t = 0. x2 C(E. ˇ ⊥ , from ˇ = E0 E Since E opt = E0 E00 E K (∂3 ϑ, ∂3 v3 ), and by Claim 1 we have that E (7.8), (7.9) we deduce that ⎧ CE0 Eˇ ⊥ · t t = 0, ⎪ ⎪ ⎨ CE0 Eˇ ⊥ · t e3 = 0, ⎪ ⎪ ⎩ ˇ ⊥ · t t = 0, x2 CE0 E since the part involving E opt in (7.8) and (7.9) disappears by applying Lemma 6.2. Thus, by Lemma 6.2, it follows that ˇ ∂3 vˇ3 ). Eˇ ⊥ = E00 E K (∂3 ϑ, 2 1 2 Proof of Claim 3. Let ζ¯ ∈ Hdn (0, L; R2 ), ζ3 ∈ Hdn (0, L), and φ ∈ Hdn (0, L). Set. δε ψ¯ := ζ¯ + εφe3 ∧ (γ − γG ) − δε φx2 t ε.
(107) LINEAR MODELS FOR COMPOSITE THIN-WALLED BEAMS. and. 3323. δε ψ3 := δε ζ3 − δε ∂3 ζ¯ · γ − γG + x2 n ε
(108) x1 + ε2 ∂3 φ (γ − γG ) · n ds − δε ε∂3 φx2 (γ − γG ) · t. 0. With ψ := ψ¯ + ψ3 e3 we have. δε ∂3 ψ3 Eεψ = −2x2 ∂3 φ 1 − κx2 g1ε e3 + e3 e3 , δε 2ε δε. and hence Eεψ → −2x2 ∂3 φt e3 δε
(109) + ∂3 ζ3 − ∂3 ∂3 ζ¯ · (γ − γG ) + s∂3 ∂3 φ. x1 0. (γ − γG ) · n ds e3 e3 ,. uniformly. By passing to the limit in (7.4) we deduce that
(110) (7.10) C(Eˇ − E opt ) · (−2x2 ∂3 φt ⊗ e3 + ∂3 w3 e3 e3 ) dx = 0, Ω. where.
(111) w3 := ζ3 − ∂3 ζ¯ · (γ − γG ) + s∂3 φ. x1 0. (γ − γG ) · n ds.. 2 (0, L). By density it Equality (7.10) holds for every w3 as above and every φ ∈ Hdn 1+s also holds for every φ ∈ Hdn (0, L). In view of (7.3) and Claims 1 and 2, we have
(112) CE0 E00 E K (∂3 (ϑˇ − ϑ), ∂3 (ˇ v3 − v3 )) · E K (∂3 φ, ∂3 w3 ) dx = 0, Ω. and by (6.20) it follows that also
(113) CE0 E00 E K (∂3 (ϑˇ − ϑ), ∂3 (ˇ v3 − v3 )) · E0 E00 E K (∂3 φ, ∂3 w3 ) dx = 0. Ω. By taking φ = ϑˇ − ϑ and w3 = vˇ3 − v3 and by using (6.17) we deduce
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