From SBD to SBH: the elastic-plastic plate

Interfaces and Free Boundaries 4, (2002) 137–165 From SBD to SBH: the elastic-plastic plate DANILO P ERCIVALE† Dipartimento di Metodi e Modelli Matematici, Università di Genova, Piazzale Kennedy, Fiera del Mare, Padiglione D, Genova, Italia AND F RANCO T OMARELLI‡ Dipartimento di Matematica Francesco Brioschi, Politecnico, Piazza Leonardo da Vinci 32, 20133, Milano, Italia [Received 18 October 2000 and in revised form 17 September 2001] A description of a flat elastic perfectly plastic plate is obtained as a variational limit of a thin elastic 3D body with damage at small scale. A model of a rigid plastic slab is described in a similar way. Keywords: Free discontinuity problems; Γ -convergence; bounded deformation; elasticity and damage. 0. Introduction The asymptotic behaviour of variational integrals depending on vector-valued functions with free discontinuities may describe damaged thin structures by mean of dimension reduction in the framework of Γ -convergence. In this paper we derive a variational model of an elastic-plastic plate with flat unstressed configuration by an asymptotic analysis of a thick elastic body with small mesoscopic cracks. We also study the case of a rigid plastic slab. The elastic-perfectly plastic behaviour of a plate undergoing small deformations has been studied by coupling bulk energy of elastic type and plastic energy concentrated on a priori unknown yield lines: existence and regularity of minimizers that are Special Bounded Hessian functions (SBH) have been proved in several papers [14–17, 43]. A simplified model of such stored energy is given by a functional of the following kind:   |[Dw]| dH 1 , (0.1) |∇ 2 w|2 dL2 + H 1 (S Dw ) + Σ S Dw where Σ ⊂ R2 is the reference configuration of the plate, w ∈ SBH(Σ ) is the scalar-valued transverse displacement, ∇ 2 w is the absolutely continuous part of D 2 w and S Dw denotes the singular set of Dw, L and H denote, respectively, the Lebesgue and Hausdorff measure. The stored energy of elastic bodies with small cracks undergoing small deformations can be described in the frame of Special Bounded Deformation functions (SBD). We introduce here the † ‡ Email: [email protected] Email: [email protected] c Oxford University Press 2002  138 D . PERCIVALE & F. TOMARELLI following model of the related stored energy:   2 3 2 |[v] ⊙ νv | dH 2 |E(v)| dL + H (Jv ) + (0.2) Jv U where the open set U ⊂ R3 is the reference configuration of the body, v : U ⊂ R3 → R3 is a vector field with special bounded deformation (that is to say, v belongs to SBD(U )), E(v) is the absolutely continuous part of the linear strain tensor e(v) = sym Dv, Jv is the jump set of v (the points x where v has two different one-sided Lebesgue limits v+ (x), v− (x) with respect to a suitable direction νv (x)), while [v] = v+ − v− and ⊙ denotes the symmetric tensor product. The first term in (0.2) represents the elastic energy in undamaged regions, the second one is a surface energy (area of material surfaces where damage occurs [32]), and the third one describes a weak resistance of the material to compression or crack opening and is related to the Barenblatt model of damage [7]. The last term allows us to deal with nontrivial loads, even without artificial confinement of the body. A stored energy similar to (0.2), but without the third term, has been introduced and studied in [3, 9]. For a different but related approach to the coupling of damage and strain in one dimension see [10]. On this subject we refer also to [27–29, 45]. Minimization of (0.2) is the linearized version of minimization of the nonlinear elastic energy coupled with a surface energy according to the Barenblatt model of quasi-static formation of cracks [7, 12]: note that the linear growth in the third term of (0.2) is essential in the analysis, since sublinearity would prevent coerciveness even in BD frame, while subadditivity is necessary in order to have semi-continuity [11: Proposition 7.1]. In this paper we study a variational relationship between the functionals (0.1) and (0.2) by showing that the elastic perfectly plastic energy (0.1) of the plate is the variational limit (as ε → 0+ ) of the fracture-elastic energy of a three-dimensional body with thickness of order ε described as follows, where appropriate weights are given to the various terms in (0.2): we set Σ ε = Σ × (−ε, ε) and   ε2 2 3 |[v] ⊙ νv | dH 2 . (0.3) |E(v)|2 dL3 + H (Jv ) + ε 2 Σε 2 Jv We prove that minimizers of energy (0.1) are limit of minimizers of (0.3) as the thickness 2ε goes to zero, and that ε−3 rescaled energy (0.3) converges to (0.1), without a priori assuming any formal asymptotic expansion of (0.3) minimizers. Our proofs are valid for general bulk loads. Here, for simplicity, the analysis is detailed assuming homogeneous Dirichlet boundary conditions. Additional information about functional (0.3) with small but strictly positive ε can be obtained by the recovery sequence (5.23) defined in Section 5: the total energy of such recovery sequence energy exhibits an excess from minimality of order ε 3 . Concrete slabs undergoing small deformations are sometimes described as thin plates whose elastic deformation turns out to be irrelevant if compared with plastic flow occurring along a priori unknown plastic-yield lines [41]. Such situation can be modelled [17] by coupling rigid transverse deformations with plastic hinges along an unknown pattern of lines, and assuming that on these lines the deformation is continuous but the gradient may have jump discontinuity of rank 1: the stored energy of the slab is then  H 1 (S Dw ) + |[Dw]| dH 1 , (0.4) S Dw FROM SBD TO SBH : THE ELASTIC - PLASTIC PLATE 139 where Σ ⊂ R2 is the reference configuration of the slab, and w ∈ SBH(Σ ) is the scalar-valued transverse displacement and satisfies the constraint ∇ 2 w ≡ 0 in Σ , that is to say, w is a piecewise affine function in Σ . Here we show that energy (0.4) is the variational limit of an energy of an elastic 2 ε-thick elastic board with small cracks, analogous to (0.3) but with a differently weighted first term: ε−1  Σε |E(v)|2 dL3 + ε2 2 H (Jv ) + ε 2  Jv |[v] ⊙ νv | dH 2 . (0.5) The main results are Theorem 2.1 (asymptotic behaviour of the constant in Korn–Poincaré inequality as the open set shrinks in one single direction), Theorem 3.1 (existence of minima of the approximating problems provided a safe load condition is fulfilled), Theorem 3.4 (variational approximation of the plate energy) and Theorem 3.6 (convergence of minimizers to equilibria of the elastic plastic plate), Theorem 3.8 (safe load condition for equilibrium of an elastic plastic plate) and Theorem 4.2 (variational approximation of the rigid plastic slab). The techniques of this paper rely on Γ convergence methods [21, 23]. The coupling of bulk energy and surface energy causes the main difficulty in the analysis since none of energies (0.1)–(0.5) is convex. We cannot use a straightforward scaling of the Korn–Poincaré inequality, since a dilation of the plate thickness does not map the space of rigid displacements into itself: to overcome this difficulty, we show explicitly the asymptotic behaviour of the constant appearing in the inequality, as the plate thickness goes to 0. Moreover, we never consider thickness averages (which could jeopardize the functional frame of allowed damage, since the projection of vertical cuts with finite total area may have infinite length), but we study the approximating functionals in a fixed re-scaled domain (blow-up of the thin plate and slab). We emphasize that all the results are proved without assuming the Kirchhoff cinematic restriction e(v) · n = 0 (see [33, 39]; obviously such restriction can be skipped in the purely elastic case too, see [1]). In the last section we show some analogies and differences in the analysis if Kirchhoff assumption is made: Theorem 6.2 and Remark 6.3 clarify the gap between the two frameworks. Actually, by assuming Kirchhoff cinematic restriction, we are able to deduce a quantitative estimate on the amount of damages (see Theorem 6.4). Such estimate may be interpreted as an analytical deduction of the fact that cracks compatible with equilibrium have a smaller size than the Griffith critical length [32]. The one-dimensional problem (approximation of an elastic plastic beam via micro-cracked thin rod) has been studied in [37] (flat beam with Kirchhoff hypothesis) and [38] (curved beam with Kirchhoff hypothesis). Boundary value and obstacle problems with energy (0.2) are studied in [20]. In the elastic context (both for plates and beams) and the Sobolev spaces framework (without allowing fractures in the approximating problems, and hence without plasticity in the limit problem) a statement analogous to Theorems 3.4, 3.6 was proved by [18] (see also [19]). The purely plastic frame for plates was studied in [36]. The approximation of membrane and thin films (obtained by different weights in (0.3)) is dealt with in [12, 13, 31]. Regularity for essential minimizers of energy (0.1) with prescribed transverse load and Neumann boundary condition were proven in [16] under smallness condition of the load in L q (Ω ) with q > 2: 140 D . PERCIVALE & F. TOMARELLI in such case energy (0.1) for a minimizer can be written as   |D 2 w|2 dL2 + H1 (S Dw ) + Σ \S Dw S Dw |[Dw]| dH1 (0.6) with w ∈ C 0 (Σ ) ∩ C 2 (Σ \ S Dw ) and H1 ((S Dw ∩ Σ ) \ S Dw )) = 0: that is, closing S Dw in Σ does not increase length. Regularity for minimizers of energy (0.4) with prescribed transverse load and Neumann boundary condition were proven in [17] under smallness condition of the load in L q (Ω ) with q > 2: in such case energy (0.4) for a minimizer can be written as  1 H (S Dw ) + |[Dw]| dH1 (0.7) S Dw where w ∈ C 0 (Σ ) is affine in each connected component of Σ \ S Dw and H1 ((S Dw ∩ Σ ) \ S Dw )) = 0. The paper has the following sections: (1) Functional framework (2) Asymptotic behaviour of the best constant in the Korn–Poincaré inequality for an ndimensional cylinder whose height tends to null (3) Approximation of the linear elastic plastic plate (LP) (4) Approximation of the rigid plastic slab (RS) (5) Proofs of convergence results (6) Some remarks about the Kirchhoff cinematic restriction 1. Functional framework We denote by U an open bounded subset of Rn with Lipschitz boundary; by Ln the n-dimensional Lebesgue measure and by {ei } the canonical basis of Rn . For a given set Q ⊂ Rn we denote by ∂ Q its topological boundary, by H m (Q) its m-dimensional Hausdorff measure and by |Q| its Lebesgue outer measure; p ′ = p/( p − 1) denotes the conjugate exponent of any p ∈ [1, +∞]. We denote by Bρ (x) the open ball {y ∈ Rn ; |y − x| < ρ}, and we set Bρ = Bρ (0). Moreover, s ∧ t = min{s, t}, s ∨ t = max{s, t} for every s, t ∈ R. spt denotes the support of a distribution.   −1 v dx ∀L—measurable set Q, and L—integrable function v in Q. v dx = |Q| Q Q Mk,n denotes the k ×n matrices and Ik the identity matrix in Mk,k ; given vectors a = {ai }, b = {bi }, and matrices A = {Ai j }, B ={Bi j }, we set a · b = i ai bi , (a ⊗ b)i j =  ai b j , (a ⊙ b)i j = 1/2 (ai b j + a j bi ), (A · b)i = k Aik Bk j , A : B = i Ai j bi , (AB)i j = j Ai j b j , (b · A) j =    2 = 1}. 2 = a · a, |A|2 = A : A = sup{ B A B : A B , |a| i j i j i j i j ij ij ij ij We say that a subset E of Rn is countably (Hn−1 , n − 1) rectifiable if (up to a set of vanishing Hn−1 measure) it is the countable union of C 1 images of bounded subsets of Rn−1 ; if in addition Hn−1 (E) < +∞ then we say that E is (Hn−1 , n − 1) rectifiable. For p ∈ [1, +∞], and Y a finite-dimensional space, we denote by L p (U, Y ) and by W 1, p (U, Y ) the Lebesgue and Sobolev spaces of functions with values in Y , endowed with the usual norms · L p FROM SBD TO SBH : THE ELASTIC - PLASTIC PLATE 141 and  · W 1, p respectively. M(U, Y ) denotes the space of the bounded measures on Ω with values in Y . For brevity, we write L p (U ), W 1, p (U ), M(U ) when Y = R. | · |M denotes the total variation of a measure in M(U, Y ), i.e.      0 2 |µ|M = ϕi j dµi j : ϕi j ∈ C0 (U ), d|µ| = sup ϕi j
1 in U . U ij U ij m s a µa = ddµ Lm is the absolutely continuous part of µ with respect to L and µ = µ−µ is the singular part. If O ⊂ U is any open set, then |µ|M(O) is defined in the same way, with ϕi j ∈ C00 (O), and we define a Borel measure |µ| by setting for every Borel set B ⊂ U   |µ|; B ⊂ O, O open . |µ|(B) = inf O Let v : U → Rk be a Borel function, (we write v in the scalar case, k = 1); for x ∈ U and z ∈ Rk = Rk ∪ {∞} (the one-point compactification of Rk ), we say that z is the approximate limit of v at x, and we write if, for every g ∈ C 0 (Rk ), z = ap lim v(y), y→x g(z) = lim Bρ (x) g(v(y)) dy |Bρ | ρ→0 . Sv := {x ∈ U : ap lim v(y) does not exist} The singular set y→x is a Borel set; for brevity we denote by v : Ω \ Sv → Rk the function v(x) = ap lim v(y). y→x Let x ∈ U \ Sv s.t. v(x) ∈ Rk : we say that v is approximately differentiable at x iff there is a k × n matrix ∇v(x) s.t. |v(y) − v(x) − ∇v(x)(y − x)| ap lim = 0. y→x |y − x| If v is a smooth function then ∇v coincides with the classical gradient. We recall the definition of the space of functions with bounded variation in U with values in Rk : BV(U, Rk ) = {v ∈ L 1 (Ω , Rk ) : Dv ∈ M(U, Mk,n )}  v BV U = v L 1 (U ) + |Dv| U where Dv = D j vi i=1,..k j=1,..m denotes the distributional derivatives of v. In the one-dimensional case (n = 1) we shall use the notation v̇ in place of ∇v and v′ instead of Dv. To simplify notation we set, for any n  1, v,i := ∂v = Di v = Dv · ei ∂ xi ∇i v := (ei · ∇) v For every v ∈ BV(U Rk ) the following properties hold: i = 1, . . . , n. 142 D . PERCIVALE & F. TOMARELLI (BV1) v(x) ∈ Rk for H n−1 almost all x ∈ U \ Sv (see [46: 5.9.6]); (BV2) Sv has null Lebesgue measure and is countably (H n−1 ; n − 1) rectifiable (see [46: 5.9.6]); (BV3) ∇v exists a.e. in U and coincides with the Radon–Nikodym derivative of Dv with respect to the Lebesgue measure (see [30: 4.5.9(26)]); (BV4) for H n−1 almost all x ∈ Sv there exist ν = νv (x) ∈ ∂ B1 , v+ (x), v− (x) ∈ Rk (outer and inner trace, respectively, of v at x in the direction ν) such that (see [46: 5.14.3] and [30: 4.5.9(15)])  |v(y) − v+ (x)| dy = 0, (1.1) lim ̺−n ̺→0+ lim ̺−n ̺→0+ |Dv|M   U  {y∈B̺ (x);(y−x)·ν>0} |v(y) − v− (x)| dy = 0, (1.2) |v+ (x) − v− (x)| dH n−1 (x); (1.3) {y∈B̺ (x);(y−x)·ν<0} ∇v(x) dx + (BV5) by setting jv = (v+ −v− )⊗νv dH n−1  Sv Sv , Cv = (Dv)s − jv , we have the decomposition Dv = ∇v dx + jv + Cv . The space of vector fields with bounded deformation has been introduced to deal with variational problems in perfect plasticity (see [42, 44]): BD(U ) = {v ∈ L 1 (Ω , Rn ) : e(v) := 12 (Dv + (Dv)T ) ∈ M(U, Mn,n )}  v B D(U ) = v L 1 (U ) + |e(v)|. U BD(U ) is the dual of a separable Banach space. For any v ∈ BD(U ) we define Jv := {x ∈ Sv : ∃v+ (x), v− (x) ∈ Rn , νv (x) ∈ ∂ B1 (0), s.t. (1.1), (1.2) hold with k = n} which is the subset of the singular set where v has one-sided approximate limits with respect to a suitable direction νv ‘normal’ to Jv . Jv is called the jump set of v and plays a role analogous to the singular set Sv in the theory of BV functions (see [3]). We notice that for v ∈ BV(U, Rk ), the set Sv \ Jv is H n−1 negligible, while it is not known whether the same property holds in BD(U ). Moreover, for every v ∈ BD (BD1) the linear strain tensor e(v) has the following decomposition: e(v) = ea (v) + es (v) = E(v) dx + e j (v) + ec (v) where ea (v) = E(v) dx and es (v) are respectively the absolutely continuous and the singular part of e(v) with respect to Ln ; e j (v) , ec (v) are respectively the restriction of es (v) to Jv and the restriction of es to its complement (say the jump and Cantor part of e(v)). Throughout the paper we denote by div v = Tr E(v) the absolutely continuous part of the distributional divergence of v. (BD2) E(v) can be interpreted as an approximate symmetric differential ([3: Theorem 4.3]):  |(v(y) − v(x) − E(v) · (y − x)) · (y − x)| 1 dy = 0 Ln a.e. x ∈ U. lim n + |y − x|2 ̺→0 ̺ B̺ (x) FROM SBD TO SBH : THE ELASTIC - PLASTIC PLATE 143 (BD3) Jv is a Borel set with null Lebesgue measure and is countably (H n−1 , n − 1) rectifiable (see [3: Proposition 3.5]), and there are νv = νv (x) ∈ ∂ B1 , v+ (x), v− (x) (respectively geometric measure theory normal, outer and inner trace in the ν direction) Hn−1 a.e. in Jv , s.t. e j (v) = (v+ − v− ) ⊙ νv Hn−1 Jv , and the jump part e j (v) can be represented on every Borel set B by the formula  [v] ⊙ νv dH n−1 , where [v] := v+ − v− . e j (v)(B) = B∩Jv (BD4) If R denotes the set of rigid displacements (the affine maps of type A · x + b where A ∈ Mn,n is skew-symmetric and b ∈ Rn ), then ([42: Proposition 2.2, 2.3 p. 155] and [3: Theorem 3.1]) for every bounded connected open set U with Lipschitz boundary, and every continuous linear map R : BD(U ) → R which leaves fixed the elements of R, there is a constant c1 = c1 (U, R) such that v − R(v) L n/(n−1) (U )
c1 (U, R)|e(v)|(U ) ∀v ∈ BD(U ). (BD5) If ψ is a continuous semi-norm on BD(U ) and a norm on R, then ψ(v) + U |e(v)| is a norm on BD(U ) equivalent to  · BD(U ) . In particular (BD6) (Korn-Poincaré inequality) If U is the unit cube (−1, 1)n and ∂l U denotes its lateral boundary {x ∈ ∂U : xn = ±1}, then there is a constant c2 = c2 (U ) such that    |v| dHn−1 ∀v ∈ BD(U ). v L n/(n−1) (U )
c2 (U ) |e(v)|(U ) + ∂l U The constants in BD4, BD6 are invariant by dilations ( εU , ε > 0 ) of the cube. (BD7) BD(U ) ⊂ L s (U ) ∀s ∈ [1, n/(n − 1)] with compact embedding if s < n/(n − 1). In the study of elastic–perfectly plastic plates the spaces BH(Ω ) (functions with bounded Hessian) and SBH(Ω ) have been introduced and studied in [14, 26, 40]; BH(U ) = {v ∈ W 1,1 (U ) : D 2 v ∈ M(U, Mn,n )} = {v ∈ L 1 (Ω ) : Dv ∈ BV(U, Rm )}, vBH(U ) = v L 1 (U ) + Dv L 1 (U ) + |D 2 v|M . BH(U ) endowed with this norm is the dual of a Banach space. Now we recall the definition and main properties of the following spaces: functions with special bounded variation (see [25]), vector fields with special bounded deformation (see [3]) and functions with special bounded hessian (see [14]), and point out some of their properties. These spaces are characterized by the property that some combinations of distributional derivatives are special measures in the sense of De Giorgi ([24]). We set SBV(U, Rk ) = {v ∈ BV (U, Rk ) : Cv ≡ 0}, SBD(U ) = {v ∈ BD(U ) : ec (v) ≡ 0}, SBH(U ) = {w ∈ W 1,1 (U ) : Dw ∈ SBV(U, Rn )}, SBV 2 (U ) = {w ∈ SBV(U ) : ∇w ∈ SBV(U, Rn )}. 144 D . PERCIVALE & F. TOMARELLI We note that v ∈ SBV(U, Rk ) if and only if v ∈ BV(U, Rk ) where H n−1 and   Dv = ∇v dx + v+ − v− ⊗ νv dH n−1 Sv , Sv (B) = H n−1 (B ∩ Sv ) for any Borel set B (see [2]). Moreover (by [9: Appendix]) SBD(U ) ∩ BV(U, Rn ) = SBV(U, Rn ), SBV(U, Rn ) ⊂ SBD(U ) ⊂ BD(U ). = = We remark that Dw = ∇w in SBH(U ) and in BH(U ), but Dw = ∇w in SBV 2 (U ). We set S Dw = n  i=1 S Di w ∀w ∈ SBH(U ), S∇w = n  i=1 S∇i w ∀w ∈ SBV 2 (U ), hence S Dw = S∇w , ∆a w = ∇ · Dw, ∀w ∈ SBH(U ). By definition SBH(U ) is a closed subspace of BH(U ) with respect to the strong norm, while it is not closed with respect to the w∗ –BH(U ) topology. In addition we have (see [14, 15]): (SBH1) U |D 2 w| = U |∇ 2 w| dx + S Dw |[Dw]| dH n−1 where [Dv] = (Dv)+ − (Dv)− . (SBH2) (embeddings) Let U ⊂ Rn (n > 1) be a bounded open set with the exterior cone property. Then BH(U ) ⊂ W 1,q (U ) with continuous embedding if q
n n−1 ; compact embedding if q < (1.4) n n−1 , hence BH(Ω ) ⊂ L s (Ω ) n (compactly when the inequality is strict) if n > 2; for s
n−2 for any s  1 (compactly for finite s) if n = 2. (SBH3) Let U ⊂ R2 be a an open set. If w ∈ BH(U ) has compact support in U , then  1 w L ∞ (U )
|D 2 w|. 4 U (1.5) (1.6) (SBH4) If a bounded connected Lipschitz open set U ⊂ R2 has boundary ∂U which is a union of finitely many C 2 regular arcs, then BH(U ) ⊂ C 0 (U ). (1.7) (SBH5) (Traces) Let U ⊂ Rn have the same property as in SBH4. Then, two bounded linear maps exist: γ0 : BH(Ω ) → W 1,1 (∂Ω ), γ1 : BH(Ω ) → L 1 (∂Ω ) such that   γ0 (v) = v  , ∂Ω  ∂v  γ1 (v) = ∂ N ∂ Ω for every v ∈ C 2 (Ω ), where N is the outward normal to ∂Ω . Moreover γ1 is onto. FROM SBD TO SBH : THE ELASTIC - PLASTIC PLATE 145 (SBH6) Let U ⊂ Rn be an open set and w ∈ SBH(U ). Then   ∂w 2 s n−1 S Dw = (a) (D w) = [Dw] ⊗ ν dH ν ⊗ ν dH n−1 S Dw , ∂ν         ∂w   2 s n−1   |[Dw]| dH n−1 = (b) , (D w)  =  ∂ν  dH S S U Dw     Dw  s   2 s ∆ w  , (c) (D w)  = U U 2 s s where ∂w ∂ν = ν · Dw, and (D w) and ∆ w denote respectively the singular part of the distributional Hessian and Laplacian of v with respect to Ln . We notice explicitly that C ∞ (Ω ) is neither dense in BH(Ω ) nor in SBH(Ω ) with respect to the strong topology. Nevertheless, if Ω is strongly Lipschitz, the density holds true with respect to the intermediate topology associated to the distance      2  2  |D v|  . d2 (u, v) = u − v L 1 (Ω ) +  |D u| − Ω Ω 2. Asymptotic behaviour of the best constant in the Korn–Poincaré inequality for an ndimensional cylinder whose height tends to null We describe the asymptotic behaviour, when ε goes to zero, of the constant in a Korn–Poincaré inequality (of type (BD6)) for a cylinder Ω ε (approximating the plate Ω ), and in general for an ε-fattened open subset of Rn−1 , by first estimating from above the blow-up rate of such constant and then by exhibiting a simple example which shows exactly such rate. The body is supposed to be fixed at the lateral boundary: this fact is analytically imposed by a support restriction of admissible deformations, which are defined in a set Σ ε bigger than the reference configuration Ω ε . The following analysis includes Theorem 4.1 in [37] about flat beams. T HEOREM 2.1 For n  2 let Ω ⊂ Rn−1 be a non-empty bounded connected Lipschitz open set, def Ω ⊂ (−L , L)n−1 = Σ (2.1) (2.2) and set Γ = ∂Ω , Ω ε = Ω × (−ε, ε), Σ ε = Σ × (−ε, ε), Γ ε = ∂Ω × (−ε, ε) ∀ε ∈ (0, 1]. (2.3) Then there exists a constant CΩ = C(Ω , L , n) > 0 independent of ε such that   v L n/(n−1) (Ω ε )
CΩ ε−1−1/n |e(v)|(Σ ε )
CΩ ε−1−1/n |e(v)|(Ω ε ) + Γε ε ∀ε ∈ (0, 1] and ∀v ∈ BD(Σ ) s.t. spt v ⊂ Ω ε .    1 1 1 |v| dH n−1 v L s (Ω ε )
(2|Ω |) s + n −1 CΩ ε s −2 |e(v)|(Ω ε ) + Γε   n ∀ε ∈ (0, 1], ∀s ∈ 1, and ∀v ∈ BD(Σ ε ) s.t. spt v ⊂ Ω ε . n−1 |v| dH n−1  (2.4) (2.5) 146 D . PERCIVALE & F. TOMARELLI Before proving Theorem 2.1, we recall some known results and state some preliminary lemmas. Let Q = (−1, 1)n be the open cube of Rn , ε > 0. By [42: Theorem 2.1 p. 148, Proposition 2.3, and Remark 2.5 p. 155–156], it is easy to see that here is a positive constant c, depending only on n, such that, for every v ∈ BD(ε Q),     n n−1 |v| dL |v| dH
c |e(v)|(Q) + Q ∂Q  n Q |v| n−1 dLn  ∃w ∈ R(ε Q) :  n−1  n
c |e(v)|(Q) + s Q n |v − w| dL 1/s  ∂l Q |v| dHn−1
ce(v)(ε Q)   n . ∀s ∈ 1, n−1  By scaling (xε = εx, Hn−1 (x) = Hn−1 (xε )/εn−1 , Ln (x) = Ln (xε )/εn ), we get for all v ∈ BD(ε Q)     1 n−1 n (2.6) |v| dH
c |e(v)|(ε Q) + |v| dL ε εQ ∂(ε Q)   n−1    n n n n−1 (2.7) |v| dH |v| n−1 dL
c |e(v)|(ε Q) + εQ  ∃w ∈ R(ε Q) : εQ s n |v − w| dL 1/s ∂l (ε Q)
cε 1− sn′ |e(v)|(ε Q)  n . ∀s ∈ 1, n−1  L EMMA 2.2 For every a, b ∈ Rn the following inequality holds:  ε |a + bz| dz  τ |b|ε 2 + 2(1 − τ )|a|ε ∀ε  0, ∀τ ∈ [0, 1]. (2.8) (2.9) −ε Moreover, if b = 0, then equality in (2.9) holds iff τ = 0; if b = 0, then equality in (2.9) holds iff a = 0 and τ = 1.  Proof. The case when a = 0 or b = 0 is trivial. If a = 0 = b then, by convexity of the Euclidean norm we have |a + bz|  |bz| + a · sign(bz)∀z = 0, |a + bz|  |a| + b · sign(a)z where sign denotes the vector-valued sign function: sign(a) = a/|a|. By integration  ε  ε  ε bz · a dz = ε 2 |b| |bz| dz + |a + bz| dz  −ε −ε −ε |b||z| and  ε −ε |a + bz| dz   ε −ε |a| dz +  ε −ε a · bz dz = 2ε|a| |a| and (2.9) is a convex combination of the above inequalities. If b = 0 the statement about equality is trivial. (2.10) FROM SBD TO SBH : THE ELASTIC - PLASTIC PLATE 147 A necessary condition for both equalities in (2.10) is that a, b are linearly dependent, say if b = 0, that a = λb for some λ ∈ R, but in this case  ε+λ  ε |s| ds = |b|ψ(λ) |a + bz| dz = |b| −ε+λ −ε ψ ′ (λ) ε ψ ′′ (λ) hence = 0, iff λ = 0 and = 2 for |λ|
ε. Therefore −ε |a + bz| dz achieves its strict 2 absolute minimum value |b|ε at λ = 0 and this proves the last part of the thesis.  L EMMA 2.3 For every a, b ∈ Rn , we have √ |a||b|/ 2
|a ⊙ b|
|a||b| and the first inequality is strict iff a · b = 0, the second is strict iff |a · b| = |a||b|. Proof. For every c, d ∈ Rn : |c| = |d| = 1, we have |c ⊙ d|2 = 1 2 1 (c · d)2 1  2 2 (ci d j + c2j di2 + 2ci d j c j di ) = (ci + di2 + 2ci di c · d) = + 4 i j 4 i 2 2 hence 0 < 2−1/2 = min{|c ⊙ d| : |c| = |d| = 1}
|a ⊙ b|
1 |a||b| ∀a, b ∈ Rn \ {0}.  In the following we denote respectively by x = (x1 , . . . , xn−1 ), z and X = (x, z) the variables in Ω , (−ε, ε) and Ω ε , and we show an estimate for piecewise rigid displacements on Ω ε , related to a finite partition of Σ ε in small cubes: we consider for every ε > 0 a finite family Q εk of pair wise disjoint open cubes of size 2ε with edges parallel to coordinate axes and centred in Xk = (xk , 0) so that  Σ ε = int Qε k . k We fix ε and we drop the index ε whenever there is no risk of confusion. We denote by ∂l Q k the lateral boundary of Q k , that is the portion of ∂ Q k outside the planes {z = ±ε}, by v+ , v− the interior and exterior traces of v on ∂l Q k , by R(Q k ) the space of rigid displacements in Q k and by 1 E (y) = 1 if y ∈ E, 1 E (y) = 0 else, the usual characteristic function of a set E.  L EMMA 2.4 There is cΩ = c(Ω , L , n) > 0 such that for every w = k wk 1 Q k , with spt w disjoint from ∂Σ × (−ε, ε) and wk in R(Q k ) ∀k, the following estimate holds: w L n/(n−1) (Ω ε )
cΩ ε−1−1/n |e(w)|(Σ ε ) ∀ε s.t. 0 < ε
1. (2.11) Proof. We denote by x, z, (x, z) respectively the variables on Σ , (−ε, ε) and Σ ε . In the following we denote by the same letter c suitable constants which may change in different inequalities, but are independent of ε. Since w is a rigid displacement in each small cube Q k , there are Ak ∈ Mn,n skew-symmetric matrices and mk ∈ Rn such that wk (x, z) = Ak · (x − xk , z) + mk ∀(x, z) ∈ Q k , ∀k. 148 D . PERCIVALE & F. TOMARELLI We denote by Ak ∈ Mn−1,n−1 the skew-symmetric matrix obtained from Ak by dropping the nth row and the nth column (Ak is equal to Ak trimmed by bk , −bkT and 0), then we get wk = (Ak · (x − xk ) + zbk , −bk · (x − xk )) + mk in Q k . In each Q k we set w(x) = (a(x) + zb(x), −b(x) · (x −xk )) + m(x) where a(x) = Ak · (x − xk ), m(x) = mk , b(x) = bk ∈ Rn−1 . We set also B(x) = − k bk (x − xk )1 Q k . Hence a, b ∈ SBV(Σ , Rn−1 ), m ∈ SBV(Σ , Rn ), b is constant on each Q k , ∇x (b) ≡ 0 in Σ ε , ∇x (m) ≡ 0 in Σ ε , Ex (a) ≡ 0 in Σ ε since wk ∈ R(Q k ), where Ex denotes the 2 × 2 absolutely continuous symmetric gradient with respect to the n − 1 variables x only. In addition Sa , Sb and Sm def are subsets of S = ∂l Q k ∩ Σ ε ∩ {z = 0} and Sw ⊂ S ⊗ (−ε, ε) = S ε . We may apply property BD6 in the (n − 1)-dimensional cube Σ taking into account Lemma 2.3 and the relationship between Sobolev exponents 1∗ (n) < 1∗ (n − 1): since Ex (a) ≡ 0 we obtain a L n/n−1 (Σ )
c b L n/n−1 (Σ )
c m L n/n−1 (Σ )
c  |[a]| dHn−2 (x) (2.12) S |[b]| dHn−2 (x) (2.13) S (2.14) S |[m]| dHn−2 (x).   By (2.12)–(2.14) and Minkowski inequality we get w n L n−1 (Σ ε )
c  ε  −ε Σ (n−1)/n (|a| + |ε||b| + |b · (x − xk )| + |m|)n/(n−1) dx dz  n−1 n {a L n/(n−1) (Σ ) + b L n/(n−1) (Σ ) + m L n/(n−1) (Σ ) } 
cε1−1/n (|[a]| + |[b]| + |[m]|) dHn−2 (x).
cε (2.15) S On the other hand, from (2.9) with τ = 1/2 Lemma 2.3 and (2.6) we get ε2  S (|[a]| + |[b]|+|[m]|) dHn−2 (x)    ε
c (|[B(x)]| + |[a + zb]| + |[m]|) dz dHn−2 (x) ε S  (|[B(x)]| + |[a + zb]| + |[m]|) dH n−1 (X)
c |e(w)|(Σ ε )
c (2.16) Sε and inequality (2.11) follows now by recalling Lemma 2.3 and joining (2.15) with (2.16) times ε−1−1/n . Proof of Theorem 2.1. Let now v ∈ BD(Σ ε ) with spt v ⊂ Ω ε . We choose rigid displacements wk as in Lemma 2.4 such that (2.8) holds in each Q k , with the choice s = n/(n − 1). In particular, we FROM SBD TO SBH : THE ELASTIC - PLASTIC PLATE 149 choose wk ≡ 0 in each Qk s.t. Qk ∩ Ω ε = ∅. By Minkowski inequality and (2.6), (2.7) we get    v L n/(n−1) (Ω ε )
v − w L n/(n−1) (Q k ) + w L n/(n−1) (Ω ε ) k
c  k c |e(v)|(Q k ) + ε(n+1)/n  c
c|e(v)|(Σ ε ) + By (2.6), (2.8), taking into account that 1 ε(n+1)/n  Sε Sε |[w]|  |w+ − v+ | + |w− − v− | + |[v]| . |[v]|
|e(v)|(Σ ε ), we get   + + − − |w − v | + |w − v | + |[v]| Sε |e(v)|(Σ ε ) + cε −1− n Sε   1 1
cε−2− n |w − v| + cε −1− n |e(v)|(Σ ε ) k
cε −1− n1 Qk |e(v)|(Σ ε ). We obtain (2.4) by gathering the previous inequalities together. Inequality (2.5) follows from Hölder inequality. 1 E XAMPLE 2.5 The rates ε −1−1/n and ε s −2 in (2.5) are optimal since, if n  2, and for x ∈ Σ , z ∈ (−ε, ε), we can choose ϕ ∈ C 2 (Σ ) with ϕ ≡ 0, spt ϕ ⊂ Ω and set v(x, z) = (−z Dϕ(x), ϕ(x))T .   −z D 2 ϕ 0 Clearly, spt v ⊂ Ω ε , v ∈ SBD(Σ ε ), Jvε = ∅ and E(v) = . Therefore, 0T 0   |E(v)| dx = ε 2 D 2 ϕ L 1 (Σ ) |e(v)| = Σε Σε while, as ε → 0+ ,  Σε |v|s 1 s 1 1 ∼ ε s 2 s ϕ L s (Σ ) which shows that the estimates (2.4), (2.5) are optimal. 3. Approximation of the linear elastic plastic plate (LP) From now on we deal with the physical situation, say n = 3, x = (x, y), and we assume Ω ⊂ R2 is a bounded connected Lipschitz open set (3.1) Ω ⊂ Σ = (−L , L)x × (−L , L) y (3.2) and we set, ∀ε ∈ (0, 1], Ω ε = Ω × (−ε, ε), Σ ε = Σ × (−ε, ε), Γ = ∂Ω , Γ ε = Γ × (−ε, ε). (3.3) 150 D . PERCIVALE & F. TOMARELLI We denote with x = (x, y) the variables on Ω , with z the variable on (−ε, ε) and with t the re-scaled vertical variable variable on (−1, 1). For every vector field W : Ω 1 ( respectively Ω ε ) → R3 we denote with w = (w1 , w2 ) its two horizontal components, with w3 its vertical one and for every X ∈ Ω 1 ( resp. Ω ε ) we set X = (x, t) and X = (x, z) with x = (x, y) ∈ Σ , t ∈ (−1, 1) and z ∈ (−ε, ε) respectively. Ω ε will be the reference configuration of a thick elastic plastic body which we will assume loaded by a dead force field Fε , not necessarily perpendicular to the middle surface Ω , such that ε2 G(x, y) 2 Fε (x, y) = G = (g1 , g2 , g) ∈ L p (Ω , R3 ), p ∈ [3, +∞]. (3.4) Without relabelling we denote by Fε and G also their trivial extensions on Σ ε (say spt F ε , spt G ⊂ Ω ε ). We assume that the stored energy due to a displacement V ∈ SBD(Σ ε ) is given by G ε (V) =    λ µ|E(V)|2 + | Tr E(V)|2 dX + θ ε ([V], νV ) dH 2 (X) 2 JV Σε  (3.5) where λ, µ are the Lamé constants, with µ > 0, 2µ + 3λ > 0, δ > 0, γ >0 θ ε (η, ξ ) = ε 2 δ|ξ | + εγ |η ⊙ ξ |. (3.6) (3.7) More explicitly JV θ ε ([V], νV ) dH 2 (X) = ε2 δH2 (JV ) + εγ JV |[V] ⊙ νV | dH2 (X). Now we introduce the load energy and the total energy associated to the displacement field V:   ε2 L (V) = G · V dX F · V dX = 2 Σε Σε (3.8) F ε (V) = G ε (V) − Lε (V) (3.9) ε ε and we state the weak formulation for the Dirichlet problem (see also [9]) min{F ε (V) : V ∈ SBD(Σ ε ), spt V ⊂ Ω ε }. (LPε ) T HEOREM 3.1 Assume (3.1)–(3.9) and 3− p √ 3 4 γ |Σ | 3 p G L p (Ω ) < CΩ where CΩ is the constant of Korn–Poincaré inequality (2.4) of Lemma 2.1. Then the problem LPε achieves a finite minimum. (3.10)  We notice that a smallness condition on loads, like the safe load condition (3.10), is not only sufficient but also necessary in order to find solutions of problem LPε , as usual in variational problems with linear growth (see [6]). FROM SBD TO SBH : THE ELASTIC - PLASTIC PLATE 151 Proof of Theorem 3.1. F ε is seq. w ∗ BD l.s.c. (see [9]). We apply the same method of Theorem 3.3 in [37] with n = 3 (see [6]) in the set Σ ε , a = ε2 δ, b = εγ . By Theorem 2.1, we get K Σ ε = CΩ ε−4/3 , say v L n′ (Ω ε )
K Σ ε |e(v)|(Σ ε ) ∀ε ∈ (0, 1), ∀v ∈ SBD(Σ ε ) : spt v ⊂ Σ ε . Hence  2  ε  1 1  p −1 ε 2+ p G p Fε  L p (Ω ε ) =  L (Ω )  2 G p ε = 2 L (Ω ) (3.10) γε b b < = = p−3 1 1 1 1 −1− 1p − − −4/3 ε ε CΩ (2ε|Σ |) 3 p CΩ (2|Σ |) 3 p K Σ ε |Σ ε | 3 p that is (3.6’) of [37] holds in this case.  From now on we will assume that Greek indices vary in the set {1, 2}, and Roman indices in {1, 2, 3}, and we will denote with ex and Ex the 2 × 2 tensor field whose components coincide with the horizontal components of e and E respectively. For every V ∈ SBD(Σ ε ) we define a vector field U in Σ 1 by a suitable re-scaling, and a family of functionals E ε to be evaluated on U (so that we can refer to a single fixed domain Σ 1 instead of varying domains Σ ε ):   U(x, t) = ε−1 v(x, εt , v3 (x, εt)), x ∈ Σ , z ∈ (−ε, ε), t ∈ (−1, 1), (3.11) E ε (U) = ε −3 F ε (V). We will see in Section 5 that actually such U belongs to SBD(Σ 1 ). We fix Vε ∈ argmin F ε and Uε (x, t) = (ε−1 vε (x, εt), v3ε (x, εt)). Obviously Uε ∈ argmin E ε and spt Uε ⊂ Ω ε . We study the asymptotic behaviour of the family Uε as ε → 0+ with respect to the strong convergence in L 1 (Σ 1 ) (denoted by σ ). In order to describe the variational limit of functionals (3.11) it is useful to recall the notion of Γ convergence (see [21, 22]). D EFINITION 3.2 Let ε > 0, (S, σ ) a complete metric space and I, I ε : S → R ∪ {+∞} a family of functionals. We say that Γ (σ − ) lim I ε (s) = I (s) ε→0+ if and only if the two following conditions are satisfied: (i) (ii) ∀s ∈ S ∀s ∈ S ∀εn → 0+ ∀εn → 0+ σ ∀ sn −→ s σ ∃ sn −→ s lim inf I εn (sn )  I (s) , n→∞ lim sup I εn (sn ) = I (s) . n→∞ The most important consequence of Γ -convergence is convergence of minimizers. T HEOREM 3.3 ([22: Corollary 2.4 ]) Assume I = Γ (σ − ) limε→0+ I ε . If s ε ∈ argmin I ε and σ sε −→ s, then I ε (s ε ) → I (s) = min{I (s) : s ∈ S}. 152 D . PERCIVALE & F. TOMARELLI Now we can state the two main results of this section. T HEOREM 3.4 Assume (3.1)–(3.9) and the safe load condition (3.10). Set A = {U ∈ SBD(Σ 1 ) : U = (u, u 3 ), u 3 = u 3 (x, y), spt u 3 ⊂ Ω , u(x, y, t) = ζ (x, y) − t Du 3 (x, y)}       2 2 + λ | div ζ |2 2 u |2 + λ |∆a u |2 + 2µ |E (ζ  ζ )| dxdy µ |∇  x 3 3  Σ 3 λ+2µ λ+2µ 0 2 E (U) = + JU (δ + γ |[U] ⊙ νU | dH (x, y, t) − Σ gu 3 dxdy if U ∈ A   +∞ otherwise. Then Γ (L 1 (Σ 1 , R3 )− ) lim E ε (U) = E 0 (U). ε→0+ R EMARK 3.5 It is worth noticing that if U = (u, u 3 ) ∈ A then u(., t) ∈ SBD(Σ ) for a.e. t ∈ (−1, 1) and hence ζ ∈ SBD(Σ ), spt ζ ⊂ Ω , spt u 3 ⊂ Ω , νU is a horizontal vector, say (νU )3 ≡ 0 H2 a.e. on JU , and u 3 ∈ SBH(Σ ), since e(U) = e(u) = e(ζζ ) − te(Du 3 ) = e(ζζ ) − t D 2 u 3 ∈ M(Σ ) a.e. t whence e(ζζ ) and D 2 u 3 belong to M and, since ex (U) has no Cantor part, u 3 ∈ SBH, ζ ∈ SBD. In particular, if U = (u, u 3 ) ∈ A and E 0 (U) < +∞, then ∇ 2 u 3 = ∇ Du 3 ∈ L 2 and E(ζζ ) ∈ L 2 . In addition to Theorem 3.4 it is possible to prove the following result concerning convergence of minimizers of (LPε ). T HEOREM 3.6 Assume (3.1)–(3.11). Then, ∀ε ∈ (0, 1], and for every Vε ∈ argmin(LPε ), by setting Uε (x, t) = (uε , u ε3 ) = (ε−1 vε (x, εt), v3ε (x, εt)) = (ε −1 vε (x, z), v3ε (x, z)), for x ∈ Σ , z ∈ (−ε, ε), t ∈ (−1, 1), we have, up to subsequences, w∗ −SBD(Σ ) (uε , u ε3 ) −−−−−−−−→ (−t Du 3 , u 3 ) and E ε (Uε ) −→ G 0 (u 3 ) for a suitable u 3 minimizer of min{G 0 (w) : w ∈ SBH(Σ ), spt w ⊂ Ω } where (LP)    λ 2 2 2 a 2 |∇ w| + G (w) = µ |∆ w| dx + 2δH1 (S Dw ) 3 λ + 2µ Σ   1 gw dx.  |[Dw]| dH (x) − +γ 0 S Dw Σ λ ∈ (−1, 1/2) and the stiffness We notice that (by denoting the Poisson ratio ν = 2(λ+µ)  4 λ+µ coefficient T = 3 µ λ+2µ the first integral in G 0 recovers exactly the classic linear elastic plate energy in the undamaged region Σ \ S Dw (see [34]):       λ 1 2 2 2 a 2 |∇ w| + (1 − ν)|∇ 2 w|2 + ν|∆a w|2 dx µ |∆ w| dx = T 3 λ + 2µ 2 Σ Σ   1 = 2T |∆a w|2 − 2(1 − ν) det ∇ 2 w dx. Σ FROM SBD TO SBH : THE ELASTIC - PLASTIC PLATE 153 R EMARK 3.7 The previous result makes precise in what sense the variational limit of the energies of 3D thick bodies with damage at mesoscopic scale describes the energy of an elastic plastic plate: indeed, the limit functional takes into account other terms depending on ζ but, since ζ is un-coupled with u 3 in E 0 , then the choice ζ ≡ 0 is optimal. So that any minimizer w of G 0 leads to a minimizer U = (−t∇w, w) of E 0 . The functional E 0 may have higher energy than G 0 on competing vector fields, nevertheless E 0 has the same minimizers of G 0 in (LP) (which describes the elastic plastic plate [14–16]), and the energy of minimizers are the same. This fact is well known for a linearly elastic plate: here we show that, even in presence of free gradient discontinuities there is no shear in the limit problem as for the Kirchhoff–Love plate. We conclude with a statement for equilibrium of an elastic plastic flat plate, clamped at the boundary and subject to transverse loads: actually a similar statement was already given ( [15: Theorem 4.1]); we observe that, with slightly stronger assumptions (e.g. g ∈ L p , p  3) in order to provide meaningful approximating problems in BD, the following Theorem 3.8 would be an easy consequence of Theorems 3.4, 3.6, since E 0 is the Γ limit of equi-coercive functionals. Anyway, the direct proof is a straightforward application of the direct method in calculus of variation. T HEOREM 3.8 If µ > 0, 2µ + 3λ > 0, δ > 0, γ > 0, and gM(Σ ) < 4 γ (3.12) then LP achieves a finite minimum. Proof. G 0 is sequentially w∗ l.s.c. in SBH. The quadratic growth with respect to the absolutely continuous part of D 2 w (due to µ > 0, 2µ + 3λ > 0) and the linear growth with coefficient γ with respect to the singular part, together with inequalities (1.6) and (3.12), provide enough coercivity to compensate the load g. The support constraint is sequentially w ∗ closed.  4. Approximation of the rigid plastic slab (RS) Now we plug a divergent weight (which blows up as ε → 0+ ) in the first integral of (3.5): as a consequence we find a stiffer structure in the limit (see [17, 41]). Throughout this section we assume (3.1)–(3.4) and (3.6)–(3.8) and the safe load (3.10). We assume that the stored energy due to a displacement V is given by     λ 2 2 ε −1 θ ε ([V], νV ) dH 2 (X). (4.1) T (V) = ε µ|E(V)| + | Tr E(V)| dX + 2 JV Σε and, referring to (3.8), that the total mechanical energy is given by Λε (V) = T ε (V) − Lε (V). (4.2) Now we can state the weak formulation of the Dirichlet problem min Λε (V) : V ∈ SBD(Σ ε ), spt V ⊂ Ω ε . T HEOREM 4.1 Assume (3.1)–(3.4), (3.6), (3.8), (3.10), (4.1), (4.2) and ε ∈ (0, 1). Then the problem RSε achieves a finite minimum. (RSε ) 154 D . PERCIVALE & F. TOMARELLI Proof. We argue as in the proof of Theorem 3.1.  For every V ∈ SBD(Σ ε ) we define a vector field U in Σ 1 by a suitable re-scaling, and a family of functionals W ε to be evaluated on U (so that we can refer to a single fixed domain Σ 1 ):   U(x, t) = ε−1 v(x, εt), v3 (x, εt) , x ∈ Σ , z ∈ (−ε, ε), t ∈ (−1, 1), (4.3) W ε (U) = ε−3 Λε (V). T HEOREM 4.2 Assume (3.1)–(3.4), (3.6), (3.8), (3.10), (4.1)–(4.3). Set U = {U ∈ SBD(Σ 1 ) : U = (u, u 3 ), u 3 = u 3 (x, y), spt u 3 ⊂ Ω , u(x, y, t) = ζ (x, y) − t Du 3 (x, y), Ex (ζζ ) ≡ 0 ≡ ∇ Du 3 },  (δ + γ |[U] ⊙ νU | dH 2 (x, y, t) − Ω gu 3 dx dy if U ∈ U, 0 W (U) = JU +∞ otherwise. Then Γ (L 1 (Σ 1 , R3 )− ) lim W ε = W 0 . ε→0+ Moreover, if Vε ∈ argmin Λε , then, up to subsequences, w∗ −SBD(Σ ) Uε = (uε , u ε3 )−−−−−−−−→(−t Du 3 , u 3 ) and W ε (Uε ) −→ T 0 (u 3 ) for a suitable minimizer u 3 of min{T 0 (w) : w ∈ SBH(Σ ), spt w ⊂ Ω , ∇ Dw ≡ 0} where T 0 (w) = 2δ H1 (S Dw ) + γ  S Dw |[Dw]| dH1 (x) −  gw dx. (RS)  Σ By arguing as in Theorem 3.8 one can prove the following statement. T HEOREM 4.3 If δ > 0, γ > 0, and (3.12) holds true, then RS achieves a finite minimum. 5. Proof of convergence results In this section we prove Theorems 3.4, 3.6 and 4.2. The proof of Theorem 3.5 (approximation of the elastic plastic plate) will proceed in two steps: in the first one we give a lower bound for the Γ lim inf and in the second one we obtain an upper bound for the Γ lim sup. We remark explicitly that both estimates are obtained without using any abstract result of representation for the Γ limit. Then we deal with convergence of minimizers by proving Theorem 3.6, and eventually the approximation of the rigid plastic plate (Theorem 4.2) is discussed. Throughout this section Γ denotes Γ (σ − ) where σ is the strong topology of L 1 (Σ 1 , R3 ). FROM SBD TO SBH : THE ELASTIC - PLASTIC PLATE 155 Proof of Theorem 3.4 Step 1: lower bound for the Γ limit. For every V = (v, v3 ) ∈ SBD(Σ ε ), we define a vector field in Σ 1 by U(x, t) = (ε−1 v(x, εt), v3 (x, εt)) and we set    def ∂u β α  eαβ (U) = eαβ (V) = 2ε ∂u xα , xβ = x, y +   ∂ x ∂ x α    β def ∂u 3 α (5.1) eαt (U) = eαz (V) = 21 ∂u xα = x, y ∂t + ∂ xα     def 3 ett (U) = ezz (V) = ε−1 ∂u ∂t . Since V ∈ SBD(Σ ε ) we deduce that e(U) is a measure without Cantor part and therefore e(U) is a measure without Cantor part too. Then U ∈ SBD(Σ 1 ) but U does not necessarily belong to A. So we can define     Eαβ (U) def xα , xβ = x, y = Eαβ (V) = 2ε ∇β u α + ∇α u β   def 1 Eα t (U) = Eαz (V) = 2 (∇t u α + ∇α u 3 ) xα , = x, y    E (U) def = Ezz (V) = ε −1 ∇t u 3 . tt Moreover, the change of variables (x, y, z) → (x, y, t) = (x, y, z/ε) (5.2) carries JV onto JU , hence, by setting here and in the following νU = ((νU )1 , (νU )2 ), we get !   where S = ε−2 |νV |2 + |(νV )3 |2 νU = S −1 ε −1 νV , (νV )3 e.g. (νV )3 = 0 entails νU = νV , while if (νV )3 = 0 then νU is squeezed on the x, y plane as ε → 0. By the area formula (see [30: Theorem 3.2.22(3)]) we get dH2 (x, y, z) JV = S −1 dH2 (x, y, t) JU and since θ ε (η, .) is 1-homogeneous we get    θ ε [V], νV dH2 (x, y, z) JV    = θ ε ε[u 1 ], ε[u 2 ], [u 3 ], εS(νU )1 , εS(νU )2 , S(νU )3 S −1 dH2 (x, y, t) JU    =ε θ ε ε[u 1 ], ε[u 2 ], [u 3 ], (νU )1 , (νU )2 , ε −1 (νU )3 dH2 (x, y, t). (5.3) JU By formulae (5.1)–(5.3), (3.4), (3.8) the functional F ε (V) may be rewritten in terms of U as follows:    λ µ|E(U)|2 + | Tr E(U)|2 dx dt F ε (V) =ε 2 Σ1    +ε θ ε ε[u], [u 3 ], νU , ε −1 (νU )3 dH 2 (x, t) (5.4) JU − ε3 2  Σ1 g · u 3 dx dt −  ε4 g · u dx dt. 2 Σ1 156 D . PERCIVALE & F. TOMARELLI From now on we will use the notation ν(W) instead of νW whenever the label W cannot be read easily when written as an index. We have to prove that given U, Uε ∈ SBD(Σ ε ), with ε ∈ (0, 1), and Uε → U in L 1 (Σ 1 ), then lim infε→0+ E ε (Uε )  E 0 (U). We may assume that E ε (Uε )
c < +∞ otherwise the inequality is trivial. So we fix, ∀ε ∈ (0, 1), Vε = (vε , v3 ε ) ∈ SBD(Σ ε ), with Uε = (uε , u ε3 ) = (ε−1 vε (x, εt), v3ε (x, εt) such that Uε → U in L 1 (Σ 1 ), E ε (Uε )
c. (5.5) By E ε (Uε ) = ε−3 F ε (Vε ) we get F ε (Vε )
cε 3 . Then by identity U(x, t) = (ε−1 vε (x, εt), v3ε (x, εt)), (2.5), BD5–BD7, Lemma 2.3, Hölder and Young inequalities, we get, for small ε,  ε2 G (V )
cε + G · Vε dx dy
cε 3 + 2(1− p)/ p ε2+1/ p G L p (Ω ) Vε  L p′ (Σ ε ) 2 Σε    p−3 3 −2/3 ε ε 2 ε 3 p p
cε + CΩ 2 ε|Ω | G L (Ω ) |[V ⊙ ν(V ]| dH (X) |E(V )| dX + ε 3 ε Σε
cε3 + CΩ 2−1/6 |Σ | 5 p−6 6p  3 JVε 1/2 |E(Vε )|2 dX ε 2 G L p (Ω ) Σε  p−3 + CΩ 2−2/3 ε|Σ | 3 p G L p (Ω ) |[Vε ⊙ ν(Vε )]| dH 2 (X)
cε3 +  µ 6 Σε J (5.6) ε   V p−3 ε 2 |E(V )| dX + ε CΩ 2−2/3 |Σ | 3 p G L p (Ω ) JVε |[Vε ⊙ ν(Vε )]| dH 2 (X). Taking into account that, by | Tr A|2
3|A|2 and λ− < 23 µ, G ε (Vε )    µ |[Vε ⊙ ν(Vε )]| dH 2 (X) + ε2 δH 2 (JVε ) |E(Vε )|2 dX + εγ 3 Σε JVε the safe load condition (3.10) and (3.7), (5.3), (5.6) yield  Σε |E(Vε )|2 dx dz + ε  JUε   θ ε ε[u ε1 ], ε[u ε2 ], [u ε3 ], (νUε )1 , (νUε )2 , ε−1 (νUε )3 dH2 (x, y, t)
Cε 3 and, by (5.4),  ε 2 Σ1 |E(U )| dx dt +  JUε   θ ε ε[u ε1 ], ε[u ε2 ], [u ε3 ], (νUε )1 , (νUε )2 , ε −1 (νUε )3 dH2 (x, y, t)
Cε 2 . (5.7) FROM SBD TO SBH : THE ELASTIC - PLASTIC PLATE 157 Then, by using Lemma 2.3 we obtain now  JUε  |[u ε3 ](νUε )3 | dH 2
cε 2 ; (5.8) |E(Uε )|2 dx dt
cε2 (5.9) |∇t u ε3 |2 dx dt
cε4 (5.10) Σ1   Σ1 |[Uεα ](νUε )3 + [u ε3 ](νUε )α | dH 2
cε;  (1 + |[Uε ]|)
C. JUε (α = 1, 2) (5.11) (5.12) JUε Therefore Uε is bounded in SBD(Σ 1 ) and H2 (JUε ) is bounded too. Hence Uε ⇀ U = (u, u 3 ) in w∗ SBD; moreover (5.10) implies ∇t u 3 = 0 a.e. in Σ 1 while (5.8) and Corollary 1.3 of [9] entail  2 JU |[u 3 ](νU )3 | dH
lim inf ε→0  JUε |[u ε3 ](νUε )3 | dH 2 = 0. (5.13) Then spt u 3 ⊂ Ω and D3 u 3 = E33 (U) dx dt + [u 3 ](νU )3 dH 2 (x, t) JU = 0 hence u 3 does not depend on z, and by taking into account (5.11), [u 3 ](νU )3 = [u 3 ] = (νUε )3 = 0. (5.14) By using (5.9) and (5.11) it is easy to see that Ex (Uε ), is bounded in L 2 (Σ 1 ), for α, β = 1, 2, Eα,β (Uε ) ⇀ Eα,β (U) weakly in L 2 , and eα3 (Uε ) → 0 in the strong topology of measures and then for every ϕ ∈ C00 (Ω ) we have &Uεα,3 + Uε3,α , ϕ' =−  Σ1 (Uεα ϕ,3 + u ε3 ϕ,α ) dx dt →−  Σ1 (Uα ϕ,3 + u 3 ϕ,α ) dx dt = 0 (5.15) say eα3 (U) = 0. Then, by U3 = u 3 (x), we get u(x, t) = ζ (x) − t Du 3 (x). (5.16) So far we have proved that E 0 (U) < +∞ if and only if U ∈ A. Now, by (3.7), (5.3), we have ε−2  JUε   θ ε ε[uε ], [u ε3 ], νUε , ε−1 (νUε )3 dH 2  δH 2 (JUε ) + γ  JUε |[Uε ] ⊙ ν(Uε )| dH 2 λ λµ |Tr Ex (Uε )|2 = ε−2 min(µ|E(Uε )|2 + |Tr (E(Uε ))|2 ) µ|Ex (U )| + λ + 2µ 2 ε 2 158 D . PERCIVALE & F. TOMARELLI where the minimum is taken over the third row and column of the matrix E(Uε ). Then, by (3.11), (5.4),  λ E ε (Uε ) = ε−2 (µ|E(Uε )|2 + |Tr (E(Uε ))|2 ) dx dt 2 Σ1    ε  1 gu 3 + εg · uε dx dt + ε−2 θ ε (ε[uε ], [u ε3 ], νUε , ε −1 (νUε )3 ) dH 2 − 2 Σ1 JUε    λµ |Tr Ex (Uε )|2 dx dt µ|Ex (Uε )|2 +  λ + 2µ Σ1    ε  1 |[Uε ] ⊙ νUε | dH 2 − + δH 2 (JUε ) + γ gu 3 + εg · uε dx dt 2 Σ1 JUε     2 λµ 2λµ dx  2µ|Ex (ζζ )|2 + | div ζ |2 + |∆a u 3 |2 µ|∇ 2 u 3 |2 + λ + 2µ 3 λ + 2µ Σ    2λµ (div ζ − t∆a u 3 )I2 : Ex (Uε − ζ + t Du 3 ) dx dt 2µ(Ex (ζζ ) − t∇ 2 u 3 ) + + λ + 2µ Σ1    ε  1 |[Uε ] ⊙ νUε | dH 2 − + δH 2 (JUε ) + γ gu 3 + εg · uε dx dt. (5.17) 2 Σ1 JUε In the second inequality we exploited cancellation of odd terms in t and the following inequality: ϕ(A)  ϕ(B) + Dϕ(B) : (A − B)) for C 1 , convex ϕ : M2,2 → R, with ϕ(A) = µ|A|2 + λµ 2 ε 2 ζ ζ λ+2µ | Tr A| , A = Ex (U ) and B = Ex (ζ − t Du 3 ) = Ex (ζ ) − t∇ u 3 . Taking into account (5.15) and (5.16) we get  2λµ (div ζ − t∆u 3 )I2 } : Ex (Uε − ζ + t Du 3 ) dx dt → 0 {2µ(Ex (ζζ ) − t∇ 2 u 3 ) + λ + 2µ Σ1 and by lower semi-continuity we get     ε ε 2 2 2 ε |[U ] ⊙ ν(U )| dH  δH (JU ) + γ lim inf δH (JU ) + γ ε→0 JUε JU |[U] ⊙ ν(U)| dH 2 . (5.18) Moreover    ε  1 gu 3 dx dy gu 3 dx dt + εg · uε dx dy dt → 2 Σ1 Ω and gathering together (5.17), (5.18) and (5.19) we obtain that lim inf E ε (Uε )  E 0 (U). ε→0 (5.19) (5.20) Proof of Theorem 3.4—Step 2: upper bound for the Γ limit. Since ∆a u 3 , div ζ belong to L 2 (Σ ) we can choose U ∈ A, that is U(x, t) = (ζζ (x) − t Du 3 (x), u 3 (x)), and, for every k > 0, two smooth scalar functions ϕk , ψk defined on Σ such that     λ 1 a  ϕk + ∆ u (5.21)
3  λ + 2µ k 2 L (Σ )     λ 1 ψk + div ζ  (5.22)
.   λ + 2µ k 2 L (Σ ) FROM SBD TO SBH : THE ELASTIC - PLASTIC PLATE 159 We define now the recovery sequences Uεk (x, t) = (ζζ (x) − t∇u 3 (x), u 3 (x) + ε2 tψk (x) − 21 (εt)2 ϕk (x)) ε V (x, z) = (v ε (x, z), v3ε (x, z)) where ε v (x, z) = εu ε (x, z/ε), v3ε (x, z) (5.23) = u ε3 (x, z/ε), then L 1 (Σ 1 , R 3 ) Uεk −−−−−−−−→U = (u, u 3 ) as ε → 0+ , for every k > 0. Moreover (νUεk )3 = 0, [Uεk,3 ] = 0 and, referring to (5.4), (3.11),     εEx (ζζ − t∇u 3 )  E Uεk =   ε2 t  t ∇ψk − ∇ϕk 2 2 E ε (Uεk )   t ∇ψk − ∇ϕk  2    −1 2 2 ε ε ψk − tε ϕk ε2 t 2   λ 2 2 2 2 µ|Ex (ζζ )| + µ|ψk | + | div ζ + ψk | = 2 Σ    2 2 ε ε 2 λ + µ|∇ 2 u 3 |2 + µ|ϕk |2 + |∆a u 3 + ϕk |2 + |∇ψk |2 + |∇ϕk |2 dx dy 3 2 3 5 (5.24)       g(u εk )3 + εg · uεk dx dt. δ + γ |[Uεk ] ⊙ ν(Uεk )| dH 2 (x, t) − +  Σ JUε k By (5.21) and (5.22) it is easy to see that there exists a constant C dependent of λ, µ but independent of ε such that, for every k > 0, lim sup E ε (Uεk ) ε→0       2 λ λ 2 2 a 2 2 2
µ |∇ u 3 | + |∆ u 3 | + 2µ |Ex (ζζ )| + | div ζ | dx dy (5.25) λ + 2µ λ + 2µ Σ 3   gu 3 dx dy = E 0 (U) + ck −2 . + ck −2 + (δ + γ |[U] ⊙ ν(U)|) dH 2 (x, y, t) − Σ JU Then by a standard diagonal procedure there exists Uεk ε such that Uεk ε → U in L 1 (Σ 1 ) and lim sup E ε (Uεk ε )
E 0 (U). (5.26) ε→0 Theorem 3.4 is proved by summarizing (5.20), (5.26) and Definition 3.2.  Proof of Theorem 3.6—Convergence of minimizers.  Let Vε = (vε , v3ε ) ∈ argmin(LPε ) and Uε (x, t) = ε−1 vε (x, εt), v3ε (x, εt) , then Uε minimizes E ε and E ε (Uε )
E ε (0) = 0. 160 D . PERCIVALE & F. TOMARELLI By arguing as in the proof of Theorem 3.4, we find that (5.6)–(5.15) hold in the present case too, that there exist ξ ∈ SBD(Σ ), ω ∈ SBH(Σ ) with support contained in Ω such that, up to subsequences, w∗ SBD(Σ 1 ) (uε , u ε3 )−−−−−−−−→(ξξ − t Dω, ω) = W and that, by Theorems 3.3, 3.4, W minimizes E 0 (U). We claim that ξ = 0: the first step is proving that Ex (ξξ ) = 0 ⊗ 0. Indeed if Ex (ξξ ) ≡ 0 ⊗ 0 then    λ 2 2 ξ ξ | Tr Ex (ξ )| dx > 0 µ |Ex (ξ )| + λ + 2µ Ω hence by Lemma 2.2 (with ε = 1, τ = 1) E 0 (W) > E 0 (−t Dω, ω) which contradicts the minimality of W. We show now that Jξ = ∅: if this identity were false, by (5.14), ν3 (W) = 0, hence ([30: Theorem 3.2.19]) ν(W) = (ν(Dω), 0) ν(W) = (ν(ξξ ), 0) = (ν(Dω), 0) 1 on J Dω \ Jξ if t = 0 H a.e. Jξ ∩ J Dω except at most one single value of t, then Lemma 2.3 and Lemma 2.2 with τ = ε = 1 yield,  0 gω dx E (W) +     Σ 2 λ = µ (δ + γ |[W] ⊙ ν(W)|) dH 2 (x, t) |∆a ω|2 + |∇ 2 ω|2 dx + 3 λ + 2µ Σ JW    2 λ |∆a w|2 + |∇ 2 w|2 dx  µ 3 Σ λ + 2µ    1 + + −1  1 −1  Jξ ∩J Dω (δ + γ |[ξξ − t Dω] ⊙ ν(ξξ − t Dω)|) dH 1 (x) dt 1 J Dω \Jξ (δ + γ |[ξξ − t Dω] ⊙ ν(Dω)|) dH (x) dt    λ 2 2 2 a 2 |∇ ω| + |∆ ω| dx > µ 3 λ + 2µ Σ  (2δ + γ |[∇ω] ⊙ ν(Dω)|) dH 1 (x) + + Jξ ∩J Dω 0  =E (−t Dω, ω) +  J Dω \Jξ (2δ + γ |[Dω] ⊙ ν(Dω)|) dH 1 (x) gω dx Σ which gives a contradiction too. Therefore ξ is a rigid plane displacement field in Σ . Recalling now that spt ξ ⊂ Ω we have that ξ = 0 in Σ and the proof of Theorem 3.6 is achieved.  Proof of 4.2—Approximation of the rigid plastic plate. FROM SBD TO SBH : THE ELASTIC - PLASTIC PLATE 161 The proofs of Theorems 3.4 and 3.6 may be easily adapted in order to obtain Theorem 4.2. Indeed by repeating the same arguments of estimates (5.6) and (5.7) we have that inequalities (5.8)– (5.12) continue to hold whenever Vε ∈ argmin(Λε ). Moreover, having in mind formulae (4.3), we have  |E(Uε )|2 dx dt
cε3 (5.27) Σ1 and therefore the technique of the proof of Theorem 3.4 permits to show that the Γ -limit W 0 is finite if and only if U ∈ A. More precisely, (5.27) yields that Ex (U) = Ex (ζζ ) − t∇ 2 u 3 = 0 ⊗ 0 for a.e. t ∈ (−1, 1), then Ex (ζζ ) = 0 = ∇ 2 u 3 a.e. in Σ , T 0 (U) < +∞ iff U ∈ U and W 0 has the explicit representation claimed in Theorem 4.2. In order to conclude the proof it is enough to take into account (5.27) and to repeat the arguments of the proofs of Theorem 3.6.  6. Some remarks about the Kirchhoff cinematic restriction In this section we assume the usual Kirchhoff cinematic restriction on the deformations of thin bodies, namely ‘the material fibres orthogonal to the middle surface before loading remain approximately orthogonal to it after loading and suffer negligible stretching’ [33]. Kirchhoff restriction can be written in the following way for linearly elastic bodies (see [39]): e(V) · n = 0 where n denotes the prescribed normal to the middle line (or surface). Explicitly, by assuming (3.1)–(3.3), the cinematic restriction is e(V) · e3 ≡ 0 in D′ (Σ ε , R3 ). (6.1) This hypothesis entails a precise (and simpler) geometric structure of admissible deformations and fractures, as shown by the following lemma, whose statement is well known and easily achievable for smooth or Sobolev displacements—but it is proved here for SBD functions. L EMMA 6.1 Let m = 3, V ∈ SBD(Σ ε ), V = (v, v3 ), v = (v1 , v2 ) such that e(V) · e3 = 0 in the sense of measures. Then v3 = v3 (x, y) ∈ SBH(Σ ) and there exists ζ ∈ SBD(Σ ) such that   v = ζ − z Dv3 . JV = Jζ ∪ S Dv3 × (−ε, ε) up to a set of null (6.2) H2 measure. (6.3) Moreover ex (V) = ex (v) − z D 2 v3 say, denoting by commas the distributional partial derivatives, v1 = ζ 1 − zv3,x , v2 = ζ 2 − zv3,y , and   1 ζ ζ 1,x − zv3,x x 2 (ζ 1,y + ζ 2,x ) − zv3,x y 0 e(V) =  21 (ζζ 1,y + ζ 2,x ) − zv3,x y (6.4) ζ 2,y − zv3,yy 0 0 0 0 Ex (V) = E(v) = E(ζζ ) − z∇ 2 v3 div V = Tr ea (V) = Tr Ex (V) = div ζ − z∆a v3 . (6.5) (6.6) Proof. Since V ∈ SBD(Σ ε ) and e(V) · e3 = 0 in the sense of measures we get that Dz v3 = 0, v1,z = −v3,x , v2,z = v3,y and  v = ζ − z Dv3 v3 = v3 (x, y) 162 D . PERCIVALE & F. TOMARELLI with ζ , v3 ∈ D′ (Σ ). Since V ∈ SBD(Σ ε ), Theorem 4.5 and Proposition 4.7 of [3] entail v(·, ·, z) ∈ SBD(Σ ) for a.e. z ∈ (−ε, ε) therefore ζ , Dv3 , ∈ SBD(Σ ) by a standard elimination procedure and a.e. in Σ ε . By e(Dv3 ) = D 2 v3 we get v3 ∈ SBH. Then (νV )3 = 0 and JV ⊂  (6.2) holds  Jζ ∪ J Dv3 × (−ε, ε): the two sets may differ only where there is cancellation (allowed, for H1 a.e. x, y, for at most one z) then (6.3) is proven. (6.4)–(6.6) follow by differentiating (6.2) and this concludes the proof.  Let now Fε , G, G ε , F ε , Lε as in (3.4)–(3.9); we consider here the following Dirichlet problem subject to the Kirchhoff cinematic constraint: min{F ε (V) : V ∈ SBD(Σ ε ), s.t. spt V ⊂ Ω ε , e(V) · e3 ≡ 0}. (KLPε ) The same techniques of Section 5 together with (6.1)–(6.6) show    λ µ|E(V)|2 + | Tr E(V)|2 dX 2 Σε       2 λ λ = ε2 µ |∇ 2 v3 |2 + |∆a v3 |2 dx + ε2µ |E(ζζ )|2 + (div ζ )2 dx 3 2µ 2µ Σ Σ and hence the following statement. T HEOREM 6.2 Assume that m = 3, (3.1)–(3.10) and (6.1) are satisfied. Then, for every ε > 0, ε (KLPε ) admits a solution Vε ∈ SBD(Σ ε ), Vε (x, z) = (ζζ ε − z∇v3ε , v3ε ) such that spt Vε ⊂ Ω ε−1ζ ε w∗ −SBD(Σ ) −→ 0, v3ε w∗ −SBH(Σ ) −→ v3 where v3 is a suitable minimizer of      λ 2 |∇ 2 w|2 + |[Dw]| dH1 − gw dx dy |∆a w|2 dx dy+2δ H1 (S Dw )+γ K(w) = µ 3 2µ Σ Σ S Dw among all w ∈ SBH(Σ ) such that w = 0 on Σ \ Ω . R EMARK 6.3 It is worth noticing that for every competing function w G 0 (w)
K(w) and the equality occurs if and only if either λ = 0 or ∆a w ≡ 0. A direct calculation allows us to show that in the elastic framework a strict inequality holds also for minimizers of G 0 and K respectively. Therefore we can say that when λ > 0, even if free gradient discontinuities develop, one may have min G 0 < min K for all loads and shapes such that at least one minimizer u 0 of K fulfills ∆a u 0 ≡ 0. Actually, such a case may happen: explicit examples can be shown for the elastic plastic beam. Finally we show an analytical deduction of a quantitative estimate for Griffith critical length from safe load condition and Kirchhoff assumption: the estimate depends explicitly of the geometry of the plate and the Lamé coefficients of the material and is dimensionally consistent. FROM SBD TO SBH : THE ELASTIC - PLASTIC PLATE 163 T HEOREM 6.4 If n = 3 and (3.1)–(3.9) hold, then the safe load (3.10) entails an estimate on the area of crack for minimizers Vε of problem (LPε ) H2 (JVε ) < 6 γ2 |Σ |ε. µδ (6.7) If Kirchhoff cinematic restriction (6.1) holds too, then the following estimate of the Griffith critical length follows: H1 (Jζ ε ∪ S Dv3ε ) < 3 γ2 |Σ |. µδ (6.8) Proof. From p  n = 3, 2µ + 3λ > 0, |Ω | < |Σ |, F ε (Vε )
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2(1− p)/ p ε 2+1/ p G L p (Ω ) Vε  L p′ (Σ ε )
G ε (Vε )
Lε (Vε )
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