Brittle membranes in finite elasticity

This work is devoted to the variational derivation of a reduced model for brittle membranes in finite elasticity. The main mathematical tools we develop for our analysis are: (i) a new density result in $GSBV^{p}$ of functions satisfying a maximal-rank constraint on the subgradients, which can be approximated by $C^{1}$-local immersions on regular subdomains of the cracked set, and (ii) the construction of recovery sequences by means of suitable $W^{1,\infty}$ diffeomorphisms mapping the regular subdomains onto the fractured configuration.

In the framework of variational fracture mechanics [14], few dimension reduction results have been obtained in linearized [2,3,10,33,38,46] and nonlinear elasticity [9,51].In the mentioned papers an explicit non-interpenetration constraint in the form of positive sign of the Jacobian of the deformation was not considered in the membranes setting.In order to fill this gap, we propose here to study the limit as ρ → 0 of the 3-dimensional functional (1) F ρ (v) := defined on the thin reference configuration Σ ρ := Σ×(− ρ 2 , ρ 2 ) of thickness ρ > 0 and middle surface Σ, an open bounded subset of R 2 with Lipschitz boundary ∂Σ.In (1), L 0 (Σ ρ , R 3 ) is the space of measurable functions on Σ ρ with values in R 3 and GSBV p (Σ ρ , R 3 ) for p ∈ (1, +∞) denotes the space of generalized special functions of bounded variations [4,Section4.5]whose approximate gradient ∇v belongs to L p (Σ ρ , M 3×3 ) and whose jump set J v has finite 2-dimensional Hausdorff measure H 2 (J v ).The stored elastic energy density W : M 3×3 → [0, +∞] is assumed to satisfy the following conditions: (H 4 ) there exists C 1 > 0 such that for every (H 5 ) for every δ > 0 there exists c δ > 0 such that for every F ∈ M 3×3 with det F ≥ δ As usual in finite elasticity, (H 3 ) forces the non-interpenetration constraint det ∇v > 0 a.e. in Σ ρ for every v ∈ GSBV p (Σ ρ , R 3 ) with F ρ (v) < +∞.Moreover, we notice that (H 3 ) is incompatible with a p-growth condition.Nevertheless, condition (H 5 ) ensures that W satisfies a weak p-growth condition which degenerates as det F approaches 0 (see also [5,7]).
The main difficulty in the identification of the Γ-limit of the sequence ρ −1 G ρ is indeed the lack of an uniform p-growth of the bulk energy density W , which prevents us from directly applying classical representation theorems for free discontinuity functionals [13,15].
When fracture is not present, the Γ-limit of the elastic energy (3) has been identified in [5,7] and is given by where the reduced elastic energy density W 0 : M 3×2 → [0, +∞] is defined as (4) W 0 (E) := inf and QW 0 denotes the quasi-convex envelope of W 0 .The proof of such Γ-convergence is based on the density in Sobolev spaces of local immersions, i.e., of C 1 -functions satisfying a maximum-rank condition on the Jacobian.Such density result builds upon a classical theorem by Gromov and Eliashberg (see [40,41]), which holds on regular domains.A first step towards the Γ-limsup inequality in [5,7] is indeed the characterization of the Γ-limit of I ρ as the relaxation of the functional Σ W 0 (∇u) dx for u ∈ W 1,p (Σ, R 3 ) local immersion.
Then, a local construction of an optimizer (see [7,Lemma 5.4] and [11]) allows one to estimate the Γ-limsup of I ρ with the functional where RW 0 indicates the rank-one convex envelope of W 0 .In particular, it turns out that RW 0 has p-growth (see [7,Lemma 5.2] and Lemma 4.11 below).Thus, the aforementioned density of local immersions and standard relaxation results in Sobolev spaces entail the desired Γ-convergence.
Coming to brittle fracture, the presence of the jump set makes the identification of the Γ-limit of ρ −1 G ρ more involved.Our aim is to show that the sequence ρ −1 G ρ Γ-converges to (5) G 0 (u) := where we have defined the set of limit deformations A as In particular, notice that G 0 is purely 2-dimensional, as each deformation u ∈ A can be identified with its trace on Σ, which we still denote by u ∈ GSBV p (Σ, R 3 ) throughout the paper.
Although such convergence result may be rather expected, the strategy necessary to prove it requires a nontrivial adaption of the arguments of [5,6,7], as well as the introduction of new tools.As a first step in our proof we show the density in GSBV p (Σ, R 3 ) of piecewise affine functions with polyhedral jump set and approximate gradient with maximal rank (see Theorem 3.1).Such result, which can be extended to higher dimensions if 1 < p ≤ 2 (see Remarks 3.3 and 3.4), generalizes the approximation theorems of [24,25,32].The crucial point is to perform our approximation for GSBV p -functions with polyhedral jump.In such scenario, the key idea, already used in [32], is that a domain with smooth cuts can be deformed into a Lipschitz domain through a bi-W 1,∞ -diffeomorphism arbitrarily close to the identity.This is clearly the best one can hope for, as bi-Lipschitz diffeomorphisms can not exist due to the low regularity of domains with cuts.We further remark that such diffeomorphisms can be made piecewise affine.On the Lipschitz domains we can apply density results in Sobolev spaces to obtain a piecewise affine approximation satisfying a maximum-rank constraint.In particular, since our limit deformations take values in a higher-dimensional space, the results in [41] via discretization allow us to ensure that the constraint is also satisfied by the subgradients at vertices and interfaces.We refer to Theorem 3.1 and Lemma 3.2 for more details.
With the above density result at hand, the crucial step to prove Γ-convergence is to estimate the Γ-limsup of ρ −1 G ρ with the functional defined for functions u ∈ A whose trace on Σ is piecewise affine, has polyhedral and essentially closed jump set, and has maximum-rank subgradients.We refer to (19) for the precise formulation of G w 0 and to Theorem 4.4 for the Γ-limsup estimate.To this aim, at first we approximate each deformation in the domain of G w 0 with C 1 -local immersions, but only on regular subdomains of the non-smooth set Σ \ J u .This is shown in Lemma 4.5.With this and a delicate adaption of the arguments of [7,Proposition 5.3], the desired estimate of the Γ-limsup by means of G w 0 can be obtained.We remark that in order to define recovery sequences on the whole domain, we again use the diffeomorphisms introduced in Lemma 3.2.In turn, this requires some refinements of the constructions in [5,7] (see Lemmas 4.9 and 4.10).
The final part of the Γ-limit process follows quite closely the arguments of [7, Theorem 2.6], provided one replaces the density of local immersions with Lemma 3.1 (see Theorem 4.1 and Proposition 4.14).
We stress the self-containedness of our paper, although some arguments are inspired by [7] and adapted to a fractured domain.Indeed, many adaptions and careful controls of the involved constants are necessary in the limit procedure, which made a complete proofs presentation preferable.Therefore, no previous knowledge of the results contained in [5,7,11,52,53] is needed.
To the best of our knowledge, this is the first result in the framework of reduced models for brittle membranes in finite elasticity.We believe our result may pave the way to further reduced theories with different geometries, e.g.shells [2,18], or constraints, such as incompressibility [23,53] and weaker growth assumptions than (H 5 ).

Notations and Preliminaries
2.1.Notations.For n, k ∈ N, we denote by L n the Lebesgue measure in R n and by H k the k-dimensional Hausdorff measure in R n .The symbol M n×m stands for the space of matrices with n rows and m columns with real coefficients and I is the identity.For every r > 0 and every x ∈ R n , we denote by B r (x) the open ball in R n of radius r and center x.
where A1 and A 2 are the columns of A. We also say that two matrices A, B ∈ M 3×2 are rank one connected or have a rank one connection if rank(A − B) = 1.We call span(A) the spanning of the columns of A as vectors of R 3 .Finally, given two vectors u, v ∈ R 3 , we denote by u ∧ v the usual vector product of u and v. Let ∂f i ∂x j (x) for i = 1, . . ., m and j = 1, . . .n.
In the dimension reduction problem, we will need the definition of quasi-convex and rank one envelops.We recall here their definitions.Definition 2.1.Let f : M m×n → [0, +∞] be a Borel measurable function.We define the quasiconvex envelope of f , denoted as Qf , the function defined as where B 1 ⊂ R n is the unit ball.
Remark 2.2.If f : M m×n → [0, +∞) is a Borel measurable function and locally bounded then, by [26, Theorem 6.9], for every ξ ∈ M m×n it holds We say that f is rank one convex if for every λ ∈ (0, 1) and every ξ, ξ ′ ∈ M m×n with rank(ξ − ξ ′ ) = 1, (ii) By the rank one convex envelope of f , that we denote by Rf , we mean the greatest rank one convex function which is less than or equal to f .2.2.Density Results in Sobolev Spaces.We begin by recalling some density results in Sobolev spaces (see e.g.[1], Chapter 3).
Definition 2.4 (Piecewise affine functions).We say that a function ψ on a domain Ω ⊆ R n is piecewise affine if and only if ψ is continuous and there exists a finite family {V i } i∈I of open disjoint subsets of Ω such that L n (Ω \ ∪ i∈I V i ) = 0 and for every i ∈ I, the restriction of ψ to V i is affine (notice that we can assume each V i to be the intersection of Ω with an n-dimensional simplex).Let us denote We call triangulation of u, and we denote it by T u , the collection of T i ∩ Ω where T i is an ndimensional simplex such that u| T i ∩Ω is affine.We call diameter of the triangulation T the quantity max{diam(T i ) : We call vertex of the triangulation each point in Ω which is a vertex of some T i .
In particular, if n = 2, for every element V i of the triangulation, we call each closed segment of ∂T i ∩ Ω a side of the triangulation.Finally, in this case, we call vertices of the triangulation of u the endpoints of the sides of the triangulation.
In order to obtain an approximation with locally injective functions we need the following definitions (see [21,Section 2.6]).
) and m > n, then v is locally injective in Ω.Indeed, let x 0 ∈ Ω and take δ > 0 small such that B δ (x 0 ) ⊂ Ω and ∂v(x) ⊆ ∂v(x 0 ) for every x ∈ B δ (x 0 ).Given y, z ∈ B δ (x 0 ) the segment connecting y and z intersects at most K triangles of T v denoted by T 1 , . . ., T K .Let us fix x k ∈ T k for every k = 1, . . ., K. We have where We present a density result due to Gromov and Eliashberg (see [40, Section 2.2.1] and [41]).A similar (actually stronger) result is proven, for the case n = 2 and m = 3, by Conti and Dolzmann in [22,Proposition 4.1].
Theorem 2.11.Let 1 ≤ p < ∞ and let 1 ≤ n < m be two integers and let M be a compact n-dimensional manifold which can be immersed in R m .Then, for each ψ ∈ C 1 (M ; R m ) there exists a sequence ψ j ∈ C 1 (M, R m ) such that ψ j is an immersion for every j ≥ 1 and C 1 -immersions can be approximated by locally injective functions in Aff * by discretization, as we state below.Proposition 2.12.Let n ≤ m, R ⊂ R n be an open n-dimensional rectangle and u ∈ C 1 (R, R m ).Then, the following facts hold: Proof.Item (i) follows by a standard discretization argument, using the fact that u ∈ C 1 (R, R m ).
As for item (ii), let us fix η > 0 such that det((∇u(x)) T (∇u(x))) ≥ η.It follows by (i) and continuity of determinant that for σ small we have Using Theorem 2.11 (if m > n) and Proposition 2.12 we can actually show a stronger density result than the one stated in Remark 2.7 which involves locally injective functions.
We can use a classical topological argument applied to the restriction of v in R (see e.g.[52,Proposition 3.1.6])in order to obtain w ∈ Aff * (R, R m ).Thus allowing us to conclude in the case m = n.
To conclude it is enough to apply Proposition 2.12 to each φ j .The general case follows by density using Theorem 2.6.

Density in GSBV.
We now briefly recall some basic definitions and results in the space GSBV.We refer also to [4,Section 4.5] for more details on this topic. For we say that a ∈ R m is the approximate limit of v at x if In that case we write aplim y→x v(y) = a.
We say that x ∈ Ω is an approximate jump point of v, and we write In particular, for every x ∈ J v the triple (a, b, ν) is uniquely determined up to a change of sign of ν and a permutation of a and b.We indicate such triple by The space BV(Ω, R m ) of functions of bounded variation is the set of u ∈ L 1 (Ω; R m ) whose distributional gradient Du is a bounded Radon measure on Ω with values in M m×n .Given u ∈ BV(Ω, R m ) we can write Du = D a u + D s u, where D a u is absolutely continuous and D s u is singular w.r.t.L n .The set J u is countably rectifiable and has approximate unit normal vector ν u , while the density ∇u ∈ L 1 (Ω, M m×n ) of D a u w.r.t.L n coincides a.e. in Ω with the approximate gradient of u.That is, for a.e.x ∈ Ω it holds ap- The space SBV(Ω, R m ) of special functions of bounded variation is defined as the set of all u ∈ BV(Ω, R m ) such that |D s u|(Ω \ J u ) = 0.Moreover, we denote by SBV loc (Ω, R m ) the space of functions belonging to SBV(U, R m ) for every U ⋐ Ω.For p ∈ [1, +∞), SBV p (Ω, R m ) stands for the set of functions u ∈ SBV(Ω, R m ), with approximate gradient ∇u ∈ L p (Ω, M m×n ) and H n−1 (J u ) < +∞.
We say that u ) whose gradient has compact support.Also for u ∈ GSBV(Ω, R m ) the approximate gradient ∇u exists L n -a.e. in Ω and the jump set J u is countably H N −1 -rectifiable with approximate unit normal vector ν u .Finally, for p ∈ [1, +∞), we define GSBV p (Ω, R m ) as the set of functions u ∈ GSBV(Ω, R m ), with approximate gradient ∇u ∈ L p (Ω, M m×n ) and H n−1 (J u ) < +∞.Definition 2.14.We denote by W(Ω; R m ) the space of all functions u ∈ SBV(Ω; R m ) with the following properties: Let us state a classic approximation result for GSBV p functions, obtained in [24] and subsequentially refined in [25].
Then, there exists a sequence of functions for every A ⊂⊂ Ω open and every upper semicontinuous function g Remark 2. 16.By [25, Remark 3.2], if g is locally bounded near ∂Ω we can choose the sequence {u j } j in such a way that (7) holds for every A ⊆ Ω, in this case A must be replaced by the relative closure of A in Ω.
Remark 2.17.Using [32, Lemma 5.2] we can always assume that J u j ⋐ Ω for each j ≥ 1, where {u j } j is the sequence of approximants given in Theorem 2.15.Moreover, if 1 < p ≤ 2, the capacitary argument of [24,Corollary 3.11] holds.Thus, we can additionally assume that the jump set of the approximants is composed by the finite union of disjoint (n − 1)-dimensional simplexes.
We finally present a result concerning the integral representation of functionals defined on GSBV p .More details can be found in [ Then the lower semicontinuous envelope of the functional with respect to the convergence in measure is given by where Qf is the quasi convex envelope of f .
From now on we will consider mainly the case n = 2.
Definition 2.19.Let Ω ⊂ R 2 be an open bounded set with Lipschitz boundary.We denote by W(Ω; R m ) the space of all functions u ∈ W(Ω, R m ) with the following property: each connected component of the jump set of u is either a segment or the union of two segments intersecting only at one endpoint and whose convex hull is contained in Ω.
In Theorem 2.15, even if p > 2, we can take the sequence of approximants such that it is contained in W(Ω, R m ).Indeed, the following Lemma holds.
The above property is essentially proved in [32, Lemma 5.2] which however has a different statement in terms of C 1 manifolds.Actually, the proof is essentially based on an argument for polyhedral sets which also allows one to deduce the statement in Lemma 2.20.The slight adaptions which are required to this aim are sketched in the Appendix for completeness.

Approximation Results with a Maximal Rank Condition
The goal of this section is proving an approximation result for GSBV p -functions in the spirit of Theorem 2.15 with a maximal rank constraint on the Jacobian of the approximating sequence.Theorem 3.1.Let Ω ⊂ R 2 be an open bounded set with Lipschitz boundary and 1 < p < ∞.Let u ∈ GSBV p (Ω, R m ) with m ≥ 2.Then, there exists a sequence of functions To prove Theorem 3.1, we need the following technical Lemma which deals with the construction of an homeomorphism.Lemma 3.2.Let Π ⊂ R 2 be a segment or the union of two segments intersecting only at one endpoint.Then, for δ > 0 small enough, there exist ∆ ⊂ R 2 depending on δ and a piecewise affine homeomorphism Φ δ : R 2 \ Π → R 2 \ ∆, such that the following properties hold: Proof.We divide the proof into two cases.Case 1: We first assume that Π is a segment.Without loss of generality, we can suppose that Π ⊂ {x 2 = 0}.
We claim that there exists a continuous piecewise affine function f : R → R ≥0 such that for every δ > 0: To prove the claim, let p 1 and p 2 be the endpoints of Π and for z = (z 1 , 0) ∈ Int(Π), let p 0 = (z 1 , 1).On Π we define f as the piecewise affine function whose graph is given by the two segments connecting p 1 with p 0 and p 0 with p 2 .We further set f = 0 on R \ Π.By construction, f satisfies the requirements.Observe that the set ∆ is a triangle for every Notice that Φ δ is a continuous bijection from R 2 \ Π and R 2 \ ∆ and Φ δ (x) = x outside (Π) δ .It either holds Since f δ is piecewise affine and continuous, we have that Φ δ is piecewise affine on R 2 \ Π.Moreover, by a direct computation we get As for the differential of Φ −1 δ it either holds Finally, since f δ and f ′ δ are uniformly bounded with respect to δ, we have that Case 2: Now we assume that Π is the union of two segments intersecting only at one endpoint.We call γ the segment connecting the endpoints of Π.As in the previous case, it is not restrictive to assume γ ⊂ {x 2 = 0}.We further define T := conv(Π), and observe that ∂T = Π ∪ γ.
Step 1: We claim that there exists a piecewise affine homeomorphism Ψ : R 2 → R 2 such that Ψ(x) = x out of an arbitrarily small neighborhood U of T , Ψ(Π) = γ and Ψ is bi-Lipschitz of constant M = M (U ).
To construct Ψ, let T − be a triangle contained in the half-plane opposite to T , having a side coinciding with γ and such that the height of T − is arbitrarily small.Consider two triangles T and T − with basis γ, with the property T ⊂ T , T − ⊂ T − in such a way that |( T \ T ) ∪ ( T − \ T − )| is arbitrarily small.Finally suppose that all their free vertices are aligned on a line perpendicular to γ.We can construct Ψ piecewise affine, as depicted in Figure 1, such that Ψ( Notice that the triangulation of the set {Ψ(x) = x} is composed by the eight triangles depicted in Figure 1.Furthermore, by construction Ψ(x) = x for every x / ∈ T ∪ T − , which is contained in a small neighborhood U of T , and Finally, it follows by construction that (10) Step 2: Let Ψ, M and γ be given by the previous step.Fix δ ∈ (0, 1/2), as in Case 1 we construct Θ δ : R 2 \ γ → Ω such that the image set Ω is a Lipschitz domain (depending on δ) and Θ δ (x) = x for every x ∈ R 2 \ (γ) δ/M .Observe that by the construction given in Case 1, see (8), we may assume that Up to taking δ small enough, {Θ δ = Id} ⊆ T , where T is given in the previous step.
Combining these properties with (10), we infer that We are now in a position to define Φ δ .Let δ > 0 small enough such that (11) holds.
x and this proves property (ii).Observe that Θ δ (R 2 \ γ) is the complementary set of a triangle.Hence ∆, which is the image of such a triangle through the piecewise affine bi-Lipschitz homeomorphism Ψ, is a polygon without self intersections.It follows that the complementary set R 2 \ ∆ is a Lipschitz open set.This is property (i).
Remark 3.3.We notice en passant that the same construction can be generalized for n ≥ 3 when Π is an (n−1)-dimensional simplex.In this case, the set ∆ is an n-dimensional simplex with base Π and the homeomorphism can be constructed analogously to Case 1. Finally, since ∆ is a convex polyhedron we have that R n \ ∆ is a Lipschitz domain by [39, Theorem 1.2.2.3].
We are now in a position to prove Theorem 3.1 Proof of Theorem 3.1.By a double sequence argument and Theorem 2.15 it suffices to prove that given ), for all ε > 0, there exists a function ũ with the desired properties and such that For i = 1, . . ., K denote the connected components of J u with Π i .Correspondingly, we fix A i ⋐ Ω an open smooth neighborhood of Π i .We can assume that the sets A i are pairwise disjoint.
Using Lemma 3.2 we construct W 1,∞ -piecewise affine homeomorphisms Φ i : x outside a neighborhood of Π i compactly contained in A i .Moreover, using (iii) of Lemma 3.2 we can assume that for for every M ∈ ∂Φ i (x) and for every x Moreover, by (12) and using the fact that the A i 's are pairwise disjoint, we have that for every M ∈ ∂Φ(x) and for every In the following, we restrict ourselves to the case m > 2. Let us fix an open rectangle R containing V .Using Theorem 2.6 and Theorem 2.11 we find η = η(ε) > 0 and φ ∈ Let σ > 0 small which will be determined later.By Proposition 2.12 there exists w ∈ Choosing σ suitably small ( 14) and ( 15) also implies (16) det(M T M ) ≥ η 2 for every M ∈ ∂w(x) and every x ∈ V .
For every x ∈ Ω \ J u , set ũ(x) := w(Φ(x)).We now check that the function ũ has all the desired properties.By ( 13), ( 14) and ( 15) we deduce that . By construction we have J ũ ⊆ J u .Moreover, since ũ is piecewise affine and J u is polyhedral, we have that J ũ has a finite number of connected components, which implies ũ ∈ W(Ω, R m ).
To this aim we denote by T the triangulation of ũ on Ω \ J ũ and we fix x ∈ Ω \ J ũ.If x ∈ T \ ∂T for some T ∈ T then property (17) follows immediately from ( 13) and ( 16).
We are only left to estimate the L p distance between ∇ũ and ∇u.Using (13) ( 14) and ( 15), we have To conclude, the argument to prove the case m = 2 is analogous to the one used for m > 2. Actually it is simpler due to the definition of Aff * when m = n.Remark 3.4.Proposition 3.1 can be immediately generalized to the setting n > 2 and 1 < p ≤ 2 a sequence of functions u j ∈ Aff * (Ω \ J u j , R m ) ∩ W(Ω, R m ) satisfying properties (i)-(iii) for m ≥ n .Indeed in this case by Remark 2.17 we can assume that each approximant obtained using Theorem 2.15 is such that the connected components of its jump set are (n − 1)-dimensional simplexes compactly contained in Ω.Hence, using Remark 3.3, we can repeat the construction in the proof of Proposition 3.1.5) and ( 4) for the definitions of ρ −1 G ρ , G 0 and the reduced density W 0 .Our main result is the following.

Dimension Reduction under Noncompenetration
) with respect to the topology induced by the convergence in measure.
For every ρ > 0 it holds From the arbitrariness of ρ > 0 we infer that (ν u ) 3 = 0 H 2 -a.e. on J u , hence u does not depend on x 3 and u ∈ A.
Finally, by definition of W 0 in (4) we have where in the last inequality we have used Theorem 2.18.
We conclude this section by recalling some properties of W 0 (see [7,Lemma 2.4]) which will be used later on in the paper.(i) W 0 is continuous as an extended valued function and satisfies (H 4 ) with the same C 1 and p of W ; (ii) for A ∈ M 3×2 , W 0 (A) = +∞ if and only if A 1 ∧ A 2 = 0, (iii) for every δ > 0, there exists c δ > 0 such that for every

An intermediate step:
Γ-limsup inequality in terms of an auxiliary functional.We start by estimating the Γ-limsup of the functionals ρ −1 G ρ in terms of an auxiliary functional defined on a subset of the functions in GSBV p (Σ 1 , R 3 ) not dependent on the third variable.To this aim, we start by fixing some notations.Let Σ ⊂ R 2 be an open set and let u be a function defined on Σ 1 = Σ × (−1/2, 1/2) not depending on the third variable.In the following, to short the notation, we will denote with u also the corresponding trace of u on Σ. Set Y ⊂ GSBV p (Σ 1 , R 3 ) as We introduce the functional where G w 0 is the relaxation of G w 0 .The proof of Theorem 4.4 needs some preliminary work.We now state a result that provides a suitable smooth approximation for functions in Aff * in the spirit of [11,Lemma 7].∈ Aff * (U, R 3 ).Then, there exists a sequence v j ∈ C 1 (U , R 3 ) with the following properties: (ii) there exists θ > 0 such that for every j ≥ 1 Proof.For ε > 0 let us denote Let ε j → 0 as j → +∞ such that for every x ∈ (U ) −2ε j there exists z x ∈ B ε j (x) satisfying ∇v(y) ∈ ∂v(z x ) for every y ∈ B ε j (x).
Let {ρ τ } τ >0 be a family of standard mollifiers.We define , for every x ∈ U .By the properties of convolution, we have that for every x ∈ (U ) −2ε j , ∇u j (x) ∈ ∂u(z x ).This in turn implies that |∂ 1 u j (x) ∧ ∂ 2 u j (x)| ≥ η for every x ∈ (U ) −2ε j .By uniform continuity of ∇u j , we can extend this property to (U ) −2ε j .
Since U has Lipschitz boundary, there exists a sequence of diffeomorphisms Furthermore, for j large enough it holds Therefore, ( 21) is satisfied with θ := η/2.
In the following three lemmas we give some results about the representation of the integral of W 0 .Let us start giving a general result about W and its reduced counterpart W 0 .Lemma 4.6.Let α, K > 0 and set Then, there exists β = β(α, K) > 0 such that for every A ∈ Λ(α, K) we have Proof.By definition of W 0 , to prove (22) we show that there exists β = β(α, K) > 0 such that for every A ∈ Λ(α, K), the following holds: Let A ∈ Λ(α, K) and ζ ∈ R 3 be such that W (A|ζ) ≤ W 0 (A) + 1.By the assumptions (H 4 ) and (H 5 ) on W , we have that This implies that |ζ| ≤ As for the lower bound on the determinant, let us assume by contradiction that for every n ∈ N there exist By continuity of W 0 (see Lemma 4.3) we have W 0 (A n ) → W 0 (A).Since A ∈ Λ(α, K), we have that W 0 (A) < +∞.On the other hand, in view of (24) and of hypotheses (H 1 ) and (H 3 ) on W we get which is a contradiction.Hence, there exists β 2 > 0 such that for every A ∈ Λ(α, K) and we infer (23) and hence (22).
Lemma 4.7.Let Ω ⊂ R 2 be an open bounded set, η > 0, and G : Ω → M 3×2 be an uniformly continuous function such that |G 1 (x) ∧ G 2 (x)| ≥ η for every x ∈ Ω.Let Λ j : Ω ⇒ R 3 be the multifunction defined for j ≥ 1 by Then, there exists j(η, G L ∞ ) such that for every j ≥ j(η, G L ∞ ) we have Proof.By Tietze extension theorem and uniform continuity of G, we may assume that G satisfies the assumptions of the Lemma in a neighborhood Ω ′ of Ω, up to taking a smaller η.
Let h > 0 and let T h be a regular triangulation of R 2 such that diam(T ) ≤ h for every T ∈ T h .We fix G h any piecewise constant interpolation of G on T h .Using Lemma 4.6 with j ≥ β(η, G L ∞ ) from ( 22), for every T ∈ T h we pick ϕ T ∈ R 3 such that W (G h (x)|ϕ T ) = W 0 (G h (x)) for every x ∈ T .We define φ h : Ω ′ → R 3 as φ h (x) = ϕ T for x ∈ T .For δ > 0, we notice that by uniform continuity of G and by continuity of W , for every h small enough it holds (26) W (G(x)|φ h (x)) ≤ W 0 (G(x)) + δ for every x ∈ Ω.
Observe that by uniform continuity of G we have for every x ∈ Ω ′ hence, we can find an uniform ε > 0 not depending on x 0 and h, such that Let {ρ ǫ } ǫ be a family of standard mollifiers and set φ ε,h = ρ ε * φ h .By linearity of the determinant as a function of the columns and by the properties of convolutions we deduce that det(G(x)|φ ε,h (x)) ≥ 1 3j for every x ∈ Ω.For every h > 0 we have that φ ε,h → φ h in L p (Ω, R 3 ) for every p < ∞.By the uniform control of the determinant, by (H 3 ) and by (26) we have Since δ is arbitrary we conclude.
Proof.It is enough to combine Lemma 4.7 with Lemma 4.6 with the choice α = η and K = G L ∞ .
We now prove an analogous result to Corollary 4.8, which gives that for any subset U ⊂ Ω there exists an almost optimal function φ for W 0 on U which satisfies the determinant constraint in the whole Ω.
We are now in a position to give the proof of Theorem 4.4.
By construction u j,ρ ∈ SBV p (Σ 1 , R 3 ).Observe that the approximate normal ν u j,ρ to J u j,ρ has third component (ν u j,ρ ) 3 = 0, so that ψ ρ (ν u j,ρ ) = 1 on J u j,ρ .Moreover, by definition of Φ δ it holds J u j,ρ ⊆ J u .Thus, we have By the change of variables induced by Φ −1 δ , we obtain lim sup 1 ) = 0. We now estimate ρ −1 G ρ (u j,ρ ).Since we have already computed the limit of the surface integral in (38), we restrict ourselves to the bulk part.Let us keep in mind that ∇ϕ j ∈ L ∞ (Ω δ 1 , R 3 ) for j fixed, although possibly not uniformly.However this is no problem since we first take the limit in ρ.By (30), we deduce We now perform a change of variables with Φ −1 δ .After that, using ( 34) and ( 37), we get We can now pass to the limit on j by Vitali dominate convergence theorem using properties (i)-(iv) of Lemma 4.5 and Lemma 4.3.Finally, by a change of variables with Φ δ , in view of (34) we have Thus, keeping in mind (38), (33) is proved.By the arbitrariness of ε, we infer Γ − lim sup ρ→0 ρ −1 G ρ (u) ≤ G w 0 (u) for every u ∈ Y .For the remaining u ∈ GSBV p (Σ 1 , R 3 ) \ Y , the above inequality is obvious.By lower semicontinuity of the Γ-limsup (see [27,Proposition 6.8]) we then conclude.4.3.Proof of the Main Theorem.In this last subsection we give the proof of Theorem 4.1.Let us summarize our strategy.So far, using Proposition 4.2 and Theorem 4.4, for every u ∈ GSBV p (Σ, R 3 ) we have proven that Observe now that we cannot use Theorem 2.18 to represent G w 0 as an integral functional, as no p-growth from the above is satisfied by the bulk energy.Following the lines of [7] and [11], our aim is to prove that G w 0 = G R 0 , where Above, RW 0 being the rank one convexified of W 0 .Indeed, in Lemma 4.11 we show that RW 0 is finite valued, continuous and satisfies an uniform p-growth condition.This allows us to apply the integral representation of Theorem 2.18 to G R 0 .The proof of the equality G w 0 = G R 0 relies on Proposition 4.14, where we make use of the density result proved in Theorem 3.1.
We start with some properties of the rank-one and quasi-convex envelope of W 0 (see Definitions 2. 3 and 2 We will also need the following result which follows from Kohn and Strang [43, Section 5C].Lemma 4.12.Define the sequence {R i W 0 } i≥0 by R 0 W 0 = W 0 and for every i ≥ 0 and every We have that R i W 0 is upper semi-continuous for every i ≥ 0 and finite valued for every i ≥ 2.Moreover, R i+1 W 0 ≤ R i W 0 for every i ≥ 0 and RW 0 = inf i≥0 R i W 0 . Proof.Notice that since W 0 satisfies (i) and (iii) of Lemma 4.3, we have that R 1 W 0 (A) is finite for every A ∈ M 3×2 such that A = 0, thus R 2 W 0 is finite valued and R 2 W 0 ≤ RW 0 .This allows us to apply the iterative scheme of Khon and Strang to W 0 even if it is not finite valued.For the upper semicontinuity, it is enough to notice that the infimum of upper semi-continuous functions is upper semi-continuous as well and reason by induction.
As in [7] the following technical lemma will be crucial to the asymptotic analysis.We give the proof of this result in the Appendix.Lemma 4.13 (Belgacem).Let V ⊂ Σ an open set with |∂V | = 0 and A ∈ M 3×2 with rank(A) = 2.There exists {φ n,l,q } n,l,q≥1 ⊂ Aff c (V, R 3 ) such that: (i) for every l, q ≥ 1 we have lim n→+∞ φ n,l,q = 0 in L p (V, R 3 ), (ii) for every n, l, q ≥ 1 the function x → φ n,l,q (x) We now show the Γ-limsup inequality in terms of the auxiliary functional G R 0 .Proposition 4.14.Let G R 0 be given by (39).For every u ∈ GSBV p (Σ, R 3 ) we have (40) Γ − lim sup Proof.We divide the proof in two steps.
Step 1: We first prove that for every According to Lemma 4.12, it is enough to show that for every i ≥ 0 and for every We show (42) by induction on i.Since R 0 W 0 = W 0 the inequality is satisfied by definition of G w 0 if i = 0. Assume now that (42) holds for a certain i ∈ N. By definition of v, there exists a finite family ∂V j and for every j = 1, . . ., M we have v(x) = A j x + c in V j , where A j ∈ M 3×2 , rank(A j ) = 2 and c ∈ R 3 .For every j = 1, . . ., M consider φ j k,l,q k,l,q≥1 ⊂ Aff c (V j , R 3 ) as in Lemma 4.13 applied with with V = V j and A = A j .

Now set
) for every k, l, q ≥ 1 by (ii) of Lemma 4.13.Hence, the inductive hypothesis implies that ) for every k, l, q ≥ 1.
By Lemma 4.13 (i) for every l, q ≥ 1 we see that Ψ k,l,q → v in L p (Σ, R 3 ) for k → ∞.By lower semicontinuity of G w 0 , it follows that ) dx + H 1 (J v ) for every k, l, q ≥ 1.
Proof of Theorem 4.1.By (i) and (ii) in Lemma 4.11, we are allowed to apply Theorem 2.18 to the functional G R 0 , yielding that, for every u ∈ GSBV p (Σ, R 3 ), Combining with Proposition 4.2, we deduce Therefore, for every u ∈ GSBV p (Σ, R 3 ), it holds It remains to show that QW 0 = Q(RW 0 ).The "≥" inequality is obvious.By (iii) in Lemma 4.11 and the definition of rank-one convex envelope, we have that QW 0 ≤ RW 0 .With this and Remark 2.2 applied to RW 0 , we deduce QW 0 ≤ Q(RW 0 ).This concludes the proof of the theorem.

Appendix
This section is devoted to give a sketch of the proofs of Lemma 2.20 and Lemma 4.13.
Proof of Lemma 2.20.Since we reason component-wise, we only consider the case m = 1.Suppose that J u is composed by the union of K closed segments Π and {Π i } K−1 i=1 with Π ⊂ {x 2 = 0} and that Π ∩ (∪ K−1 i=1 Π i ) = {P j } M j=1 .We denote by Γ j the M + 1 connected components of Π \ {P j } M j=1 .Fix 0 < λ << η.For each j = 1, . . ., M + 1, we denote by Z j each set composed by the union of a segment Γ λ j := x ∈ Γ j | dist(x, ∪ M j=1 P j ) ≥ λ and two segments S 1 j and S 2 j of length η, with an endpoint coinciding with one endpoint of Γ λ j and the other contained in {x 2 > 0} (see Figure 2).The same construction as in [32,Lemma 5.2] shows that, provided λ and η small enough we can construct a function v ∈ W(Ω) ∩ C 1 (Ω \ J v ) such that Notice that, for some constant C > 0 independent of η and λ, one can also ensure Now we want to exploit the same construction in order to transform each portion of the jump Z 1 into three disjoint components, the first being given by Γ λ 1 and the remaining two being arbitrarily small "L-shaped" sets entirely contained in {x 2 > 0}.This can be done again by fixing 0 < µ << δ << η (see again Figure 2) and repeating the construction of [32, Lemma 5.2, proof of (A.8)].This results in a function v 1 with Without entering in details (for which we refer again to [32, Lemma 5.2]), we mention here that a proper choice of µ and δ in dependence of ( 45) is needed to bound the L p distance of the gradients.Finally, by iterating this procedure for j = 1, . . ., M + 1 and then i = 1, . . ., K −2, and summing on the inequalities in (46), one can obtain the required approximation u ε ∈ W(Ω).
Since Lemma 4.13 is essentially proved in [11] and [7,Lemma B.5], we only give a sketch of the proof highlighting the main differences.
Before proving Lemma 4.13, we need the following result.

Definition 2 . 8 (
Clarke subdifferential).Let f : Ω ⊆ R n → R m be a locally Lipschitz function defined on an open set Ω of R n .For every x ∈ Ω we define the Clarke subdifferential ∂f (x) of f in x as ∂f (x) := conv lim k→+∞ ∇f (x k ) : x k → x, ∇f (x k ) is well defined , where conv denotes the convex hull.Definition 2.9.Let Ω ⊂ R n be a bounded open set and let n ≤ m.If m > n, we define Aff

Corollary 2 . 13 .
Let 1 ≤ p < ∞, n ≤ m and Ω ⊂ R n be an open bounded set with Lipschitz boundary.For every 16, Theorem 3.5 and Theorem 3.8].Theorem 2.18.Let Ω ⊂ R n open, let f : M m×n → R + be a Borel function, and assume that there exist c, C > 0 and p

Theorem 4 . 1 .
Let Σ be an open bounded subset of R 2 with Lipschitz boundary and 1

Lemma 4 . 5 .
Let U be an open bounded subset of R 2 with Lipschitz boundary and let

Lemma 4 . 9 .
Let Ω and G be as in Corollary 4.8.Then, for every open set U ⊂ Ω, there exists
By (i) and (ii) of Lemma 4.11 and by the L p -convergence of ∇u k to ∇u we have thatlim k→+∞ Σ RW 0 (∇u k ) dx = Σ RW 0 (∇u).