A derivation of Griffith functionals from discrete finite-difference models

We analyze a finite-difference approximation of a functional of Ambrosio-Tortorelli type in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step $\delta$ is smaller than the ellipticity parameter $\varepsilon$, we show the $\Gamma$-convergence of the model to the Griffith functional, containing only a term enforcing Dirichlet boundary conditions and no $L^p$ fidelity term. Restricting to two dimensions, we also address the case in which a (linearized) constraint of non-interpenetration of matter is added in the limit functional, in the spirit of a recent work by Chambolle, Conti and Francfort.

antiplane shear (see, e.g., [12]) where the energy (1.1) reduces to the Mumford-Shah-type functional Ω |∇u| 2 dx + H d−1 (J u ), (1.2) for a scalar-valued displacement u ∈ SBV (Ω), the space of special functions of bounded variation. In view of the aforementioned numerical issues, a particular attention has been devoted over the last three decades to provide suitable discrete approximations, by means of both finite-difference and finite-elements, of the functional (1.2). A first approach, based on earlier models in Image Segmentation, has been proposed by Chambolle [16] in dimension d = 1, 2; there, the discrete model depends on finite differences through a truncated quadratic potential. In the case d = 2, the surface term of the variational limit is described by an anisotropic function ϕ(ν u ) of the normal ν u to J u depending on the geometry of the underlying lattice. As a matter of fact, this anisotropy can be avoided by considering alternate finite-elements of different local approximations of the Mumford-Shah functional, as showed, still in dimension two, by Chambolle and Dal Maso [22]. We refer to [8] (cf. also [11]) and to [27] for some other approximations using finite-elements and continuous finite-difference approximations of (1.2), respectively.
A different strategy consists in replacing the Mumford-Shah functional by an elliptic approximation (with parameter ε > 0) in the spirit of Ambrosio-Tortorelli [4,5], and then by discretizing these elliptic functionals by means of either finite-difference or finite-elements with mesh-size δ, independent of ε. For a suitable fine mesh, with size δ = δ(ε) small enough, these numerical approximations Γ-converge, as ε → 0, to the Mumford-Shah functional.
This suggests that a remarkable problem to be addressed is the so called "quantitative analysis": i.e., the study of the limit behavior of these approximations as δ and ε simultaneously tend to 0. Following on the footsteps of the approximation of the Modica-Mortola functional proposed by Braides and Yip [14], this analysis has been recently developed by Bach, Braides and Zeppieri in [6] for (1.2). They characterize the limit behavior of the energies α,β∈Ω∩δZ d |α−β|=δ showing the variational convergence to the functional (1.2) in the regime δ << ε. Other scalings of the parameters are also studied: in the regime δ ∼ ε, the surface energy is described by a function ϕ(ν u ) solution to a discrete optimal-profile problem, while if δ >> ε, the limit energy is the Dirichlet functional. Recently, approximations of (1.2) (thus without anisotropy in the limit) have been obtained even when δ ∼ ε, by employing discretizations on random lattices. In particular, [7] analyzes the random version of the discrete energies in [6], basing on [30] (cf. also [15]). Coming back to the problem of providing discrete approximations of the Griffith functional, we mention the finite-elements approximation in [29] and focus on the discrete-to-continuum analysis performed by Alicando, Focardi and Gelli [1]. They considered, in the spirit of [16] and in the planar setting d = 2, discrete energies of the form defined on a portion R ξ δ of Ω ∩ δZ d , where ρ is a positive kernel, θ is a positive constant, f (t) := min{t, 1}, D ξ δ u(x) denotes the difference quotient 1 δ (u(x + δξ) − u(x)) and div ξ δ u is a suitable discretization of the divergence which takes into account three-point-interactions in the directions ξ and ξ ⊥ (the vector orthogonal to ξ). In order to obtain compactness of sequences of competitors with equibounded energy, they require that ρ(ξ) > 0 for ξ ∈ {±e 1 , ±e 2 , ±(e 1 ± e 2 )}, which amounts to consider nearest-neighbors (NN) and next-to-nearest neighbors (NNN) interactions in the energies. Furthermore, an L ∞ bound has to be imposed, which is quite unnatural in Fracture Mechanics. Differently from [6], the characterization of the limit energy cannot be achieved with the reduction to a 1-dimensional case by means of slicing techniques (see, e.g., [13,17,27]), due to the presence of the divergence term. Hence, a different strategy has to be used, involving the construction of suitable interpolants (see [1,Proposition 4.1]). As it happened in [16], the surface term in the limit energy is still reminiscent of the underlying lattice, and only a continuous version of (1.3) allows to obtain H d−1 (J u ) as surface energy. Furthermore, a possible extension of the model to dimension d = 3, still involving NN and NNN interactions is proposed, but no compactness result is provided.
Our results: This leads us to the motivation of our paper, which complements the results of both [6] and [1]. On the one hand, we provide a discrete Ambrosio-Tortorelli approximation to the Griffith functional both in dimension d = 2 and d = 3, of the form where S d is a set of lattice directions (depending on the dimension d), D δ,ξ u and Div δ u are suitable discretizations of the symmetrized gradient and of the divergence of the vector-valued u, and the latter term is a discrete Modica-Mortola functional. Notice that Div δ u takes into account (d + 1)-point-interactions on a complete set of orthogonal directions (see (3.4)). Then we prove, as main result (Theorem 3.1), that (1.4) Γ-converges as ε → 0 to the Griffith's functional under the assumption that δ << ε.
On the other hand, we conclude the analysis started in [1] for the finite-difference approximation of (1.1) in dimension d = 3, although with a different approach, by both rigorously proving a compactness result under more general assumptions, and recovering an isotropic surface energy in the limit. We also stress the fact that the extension of the two-dimensional model to the case d = 3 is not just a minor modification but requires the introduction of additional interactions in the elastic term of the energies by specifying the set of directions S 3 (see (2.24)); namely, we need to take into account also next-to-next-nearest neighbors (NNNN) interactions, corresponding to lattice vectors ξ ∈ {±(e 1 ± e 2 ± e 3 )}.
The aforementioned compactness result, which is the content of Proposition 4.1, determines the functional space domain of the limit: we benefit from the recent results [21,25] and prove that sequences (u ε , v ε ) with equibounded energies (1.4) converge (up to subsequences) to a limit pair (u, v) ∈ GSBD 2 ∞ (Ω) × {1}. We refer the reader to Section 2 for a precise definition of this function space, where also the value ∞ is allowed. We underline that our compactness result, valid under the weaker assumption that δ ε be bounded, cannot be obtained through any slicing procedure (as it happened, on the contrary, in [6]). Indeed, while in the scalar-valued case controlling the total variation along d independent slices of u ε is enough to provide BV -compactness, no analogue procedure is at the moment known in GSBD (whose definition [26,Definition 4.1] in principle requires a uniform control of the symmetrized slices on a dense set of directions in the unit sphere). In fact, we are able to prove that a continuous Ambrosio-Tortorelli functional, defined on the standard piecewise affine interpolationsū ε of the u ε and on suitable piecewise constant interpolationsṽ min,ε of the v ε (different than the standard ones), bounds from below the discrete energies (1.4). To this aim, taking the additional (NNNN) interactions is crucial in dimension d = 3 . In addition, we do not need to add any L p fidelity term to the discrete energies, since compactness in GSBD 2 ∞ does not require such limitations and is also able to handle the fact that u may take value ∞.
The proof of the Γ-liminf inequality is subdivided into two steps. The lower semicontinuity of the elastic part of the limit energies (see Lemma 5.1 and Proposition 5.3) can be obtained by combining slicing arguments on suitable interpolations of u ε and v ε with a splitting into sublattices of δZ d , which are frequently used techniques to work with discrete energies with both short and long-range interactions (see, e.g., [1,13]). The lower bound for the surface term, instead, requires a more refined blow-up procedure (Proposition 5.4) and this is the very first technical point where we need to assume that δ ε → 0, in order to recover the optimal constant. Indeed a slicing argument under the weaker assumption that δ/ε be bounded would provide a lower bound with a wrong constant. We remark that, also in this proof, similar arguments as in Proposition 4.1 have to be used, in order to get compactness of a rescaled version of the u ε . Moreover, additional care is needed in order to deal with the fact that our limit displacements may assume the value infinity (see e.g. Step 2 in Proposition 5.3).
The construction of a recovery sequence (Proposition 6.1) relies on the density result for GSBD 2 functions [19, Theorem 1.1], recalled here with Theorem 2.3. The upper bound for the elastic term is obtained by first reducing the discrete energies to continuous ones by means of a classical translation argument (see, e.g. [1,Proposition 4.4]) and then by exploiting the upper estimates coming from the approximations of | (Eu)ξ, ξ | 2 dx and (div u) 2 dx outside an infinitesimal neighborhood of the jump set of the target function u. The limsup inequality for the surface term is developed as in [6,Proposition 4.2], by also employing the one-dimensional solution to the Ambrosio-Tortorelli optimal profile problem.
We conclude our analysis by investigating the compatibility of our two-dimensional model with the constraint of non-interpenetration. The answer is positive under the assumptions of [18] but, in order to obtain the desired upper bound, we need to require the stronger scaling δ ε 2 → 0 between the parameters.
As a final remark, we mention that our results also give a partial insight on the case δ ∼ ε. Indeed, the constructions in Sections 5 and 6 can also be used to show that, whenever the ratio δ/ε stays bounded, the Γ-limit of the energy (1.4) can be controlled from above and from below by functionals of the kind (1.1), with different constants appearing in the surface term. However, a precise characterization of the limit energy in this case has to face additional issues. If we compare with the analysis performed in [6] for the scalar-valued case, indeed, some major ingredients are still missing. First of all, in order to apply the global method for relaxation introduced in [10], an integral representation result for energies on spaces of functions of bounded deformation is needed, which is at the moment only known in the planar setting [23]. Secondly, and most importantly, a crucial step in this procedure consists in proving that a separation of bulk and surface contributions takes place in the limit. In an SBV -setting, this is for instance done in [7,Proposition 4.11] with the help of a weighted coarea formula, a tool which is not available when dealing with (G)SBD functions. The investigation of these issues has therefore to be deferred to further contributions.
Outline of the paper: The paper is organized as follows. In Section 2 we fix the basic notation and collect some definitions and results on the function spaces we will deal with. In Section 3 we introduce our discrete model and state the main results of the paper. Section 4 contains the compactness result of Proposition 4.1. Section 5 is devoted to the liminf inequality, proved with Proposition 5.4, while Section 6 deals with the upper inequality (Proposition 6.1). Eventually, in Section 7 we analyze the compatibility of the two-dimensional model with a non-interpenetration constraint.

Preliminaries
2.1. Notation. The symbol ·, · denotes the scalar product in R d , while | · | stands for the Euclidean norm in any dimension. For any x, y ∈ R d , [x, y] is the segment with endpoints x and y. The symbol Ω will always denote an open, bounded subset of R d . The Lebesgue measure in R d and the s-dimensional Hausdorff measure are written as L d and H s , respectively. We will often use the notation |A| for the Lebesgue measure of a Borel set A. The symbols and denote the boundedness modulo a constant. . For every µ ∈ M b (B; R m ), its total variation is denoted by |µ|(B). We write {e 1 , . . . , e d } for the canonical basis of R d .
2.2. GBD, GSBD, and GSBD 2 ∞ functions. We recall here some basic definitions and results on generalized functions with bounded deformation, as introduced in [26]. Throughout the paper we will use standard notations for the spaces SBV and SBD, referring the reader to [3] and [2,9,31], respectively, for a detailed treatment on the topics.

Theorem 2.3.
Let Ω ⊂ R d be a bounded open Lipschitz set, and u ∈ GSBD 2 (Ω; R d ). Then there exists a sequence u n such that (i) u n ∈ SBV 2 (Ω; R d ) ∩ L ∞ (Ω; R d ); (ii) each J un is closed and included in a finite union of closed connected pieces of C 1 -hypersurfaces; (iii) u n ∈ W 1,∞ (Ω\J un ; R d ), and u n → u in measure on Ω, (2.5) Eu n → Eu in L 2 (Ω; R d×d sym ), (2.6) Moreover, if ∂ D Ω ⊂ ∂Ω satisfies (2.4) and u 0 ∈ H 1 (R d ; R d ), then one can ensure that each u n satisfies u n = u 0 in a neighborhood U n ⊂ Ω of ∂ D Ω, provided that (2.7) is replaced by (2.8) A further approximation result, by Cortesani and Toader [24,Theorem 3.9], allows us to approximate GSBD 2 (Ω) functions with the so-called "piecewise smooth" SBV -functions, denoted W(Ω; R d ), characterized by the three properties J u is the intersection of Ω with a finite union of (d−1)-dimensional simplexes . (2.9) As observed in [20,Remark 4.3], we may even approximate through functions u such that, besides (2.9), also J u ⊂ Ω holds and the (d−1)-dimensional simplexes in the decomposition of J u may be taken pairwise disjoint with J u ∩Π i ∩Π j = ∅ for any two different hyperplanes Π i , Π j . Furthermore, in the assumption under which (2.8) holds true, we may also ensure that u = u 0 in a neighborhood of ∂Ω. We will employ these properties in Section 6. We recall the following general GSBD 2 compactness result from [21]. In the following, when we deal with sets of finite perimeter, such as A ∞ u , we identify the set with its subset of points with density 1, with respect to d-dimensional Lebesgue measure (cf. [3,Definition 3.60]), while we denote explicitly their essential boundary with the symbol ∂ * .
Theorem 2.4 (GSBD 2 compactness). Let Ω ⊂ R be an open, bounded set, and let (u n ) n ⊂ GSBD 2 (Ω) be a sequence satisfying Then there exists a subsequence, still denoted by u n , such that the set A ∞ u := {x ∈ Ω : |u n (x)| → +∞} has finite perimeter, and there exists u ∈ GSBD 2 (Ω) such that (2.10) GSBD 2 ∞ functions. Inspired by the previous compactness result, in [25] a space of GSBD 2 functions which may also attain a limit value ∞ has been introduced, as we recall. The spacē R d := R d ∪ {∞} (with its sum given by a + ∞ = ∞ for any a ∈R d ) is in a natural bijection with S d = {ξ ∈ R d+1 : |ξ| = 1} through the stereographic projection of S d toR d : induces a bounded metric onR d . Then Symbolically, we will also write u = uχ Ω\A ∞ u + ∞χ A ∞ u . Moreover, for any u ∈ GSBD 2 ∞ (Ω) In particular, on Ω and J u = J ut H d−1 -a.e. for almost all t ∈ R , (2.14) where u t is the function from (2.12). Hereby, we also get a natural definition of a normal ν u to the jump set J u , and the slicing properties described for GSBD 2 still hold in Ω \ A ∞ u . Finally, we point out that all definitions are consistent with the usual ones if u ∈ GSBD 2 (Ω); i.e., if A ∞ u = ∅. Since GSBD 2 (Ω) is a vector space, we observe that the sum of two functions in GSBD 2 ∞ (Ω) lies again in this space. A metric on GSBD 2 ∞ (Ω) is given by where dRd is the distance in (2.11). In Sections 4 and 5, when we work in an extended domain Ω, we will still write d(u, v) for Ω dRd (u(x), v(x)) dx. We say that a sequence (u n ) n ⊂ GSBD 2 ∞ (Ω) converges weakly to u ∈ GSBD 2 ∞ (Ω) if sup n∈N Eu n L 2 (Ω) + H d−1 (J un ) < +∞ and d(u n , u) → 0 for n → ∞ .
(2.16) 2.3. Some lemmas. For a < b, we introduce the space P C δ (a, b) of piecewise-constant functions on partitions of (a, b) ⊂ R with size at most δ; namely, For every v ∈ P C δ (a, b), we denote byv the corresponding piecewise-affine interpolation on the nodes of the same partition, defined aŝ , v ε ≥ 0, and let (v ε ) ε be the sequence of the corresponding piecewise-affine interpolations defined as in (2.17). Assume that there exists Then, setting we have: (a) for every fixed constant N C > 0 depending only on C, it holds that Proof. The assertion (b) immediately follows from (a). As for the proof of (a), let us fix N C := 4C and, arguing by contradiction, we assume that #I = N C + 1 and I = {s 1 , s 2 , . . . , s N C +1 }. For every such index i, we denote by (s i ε ) ε the sequence defined by (2.19) such that s i ε → s i and lim inf Moreover, we may assume that the subsequences of s i ε and t i ε realizing the liminf in (2.20) and (iii), respectively, have infinite terms of the sequences of the indices in common. Now, letŝ i ε and t i ε be the greatest nodes of the partition that are less or equal than s i ε and t i ε , respectively. Since Now, for every i and ε, lett i ε be the first node of the partition such thatt i ε ≥ŝ i ε and v ε (t i ε ) ≥ 1 2 , and let τ i ε be the first point in (ŝ i ε ,t i ε ) such thatv ε (τ i ε ) = 1 2 , whose existence is ensured by the Mean Value Theorem. We then have Now, by Young's inequality and (2.18), which gives a contradiction.
, for some measurable f and g. Then, In particular, For δ > 0, and for any measurable function u : z, e i e i and, for every t ∈ R, t denotes the integer part of t. We have that Moreover, the following result holds (see, e.g., [1, Lemma 2.11]).
and consider a kernel function σ : and σ(ξ) = 0 for every ξ ∈ S d ; we will often use the shortcut Then, defining S d and σ as before, it holds that Proof. We can rewrite the sum on left hand side of (2.25c) as (recall that {e 1 , . . . , e d } denote the canonical basis of R d ) which coincides with the right hand side of (2.25c).

Discrete models and approximation results
Let d ∈ {2, 3}, Ω ⊂ R d an open, bounded, Lipschitz set, with ∂Ω satisfying (2.4) and the related assumptions, and let u 0 ∈ H 1 (R d ; R d ). For any δ > 0, we consider the scaled lattice δZ d and set Ω δ := Ω ∩ δZ d . We introduce suitable discretizations for both the symmetrized gradient and the divergence. For ξ ∈ R d \{0}, δ > 0, and u : Ω → R d measurable we define For a scalar function v : Ω → R, we will often adopt the notation Then we define In order to impose a non-interpenetration constraint in the limit fracture energy, we treat differently in the approximation the positive and negative part of the discrete divergence. We set, for u : Ω → R d measurable, and Notice that F div − ε does not include any contribution in v. Moreover, we introduce the discrete Modica-Mortola-type functional It will be useful to introduce also a localized version of the functionals defined above. For every A ⊂ Ω open bounded set, the symbols F ξ ε (u, v, A), F div ε (u, v, A) and G ε (v, A) denote the energies as in (3.6a), (3.6b) and (3.9), respectively, where the sums are restricted to α ∈ R ξ δ (A) defined as in (3.7) with A in place of Ω. For Let us define the class of vector-valued piecewise constant functions on Ω and, analogously, the class of real-valued piecewise constant functions A δ (Ω; R); in order to deal with the Dirichlet boundary value problem, we set and A Dir δ (Ω; R) for real-valued functions, with u 0 replaced by the constant function 1.
Notice that G Dir λ,θ (u) = G Dir λ,θ ( u t , 1) for L d -a.e. t ∈ R d , by (2.14). Moreover, G NI,M λ,θ displays a noninterpenetration constraint, not present in G Dir λ,θ . We define it directly accounting for an L ∞ bound for |u| at level M , for technical reasons. Finally, we do not take into account the role of boundary conditions for the functional with non-interpenetration constraint, since we employ results from [18] (cf. Lemma 7.1), where the boundary value problem was not explicitly addressed.
We are now ready to state the main results of the paper. In the following we assume that u 0 , λ, θ are fixed and that lim ε→0 δ ε = 0. Theorem 3.1. Under the assumptions above, it holds that: (i) as ε → 0, (E Dir λ,θ ) ε Γ-converges with respect to the topology of the convergence in measure to G Dir λ,θ ; (3.14) We remark that any sequence of minimizers (u ε , v ε ) ε for (E Dir λ,θ ) ε satisfies, up to a subsequence, where the Γ-lim inf and Γ-lim sup above are with respect to the strong L 1 (Ω; R d ) × L 1 (Ω) topology.
In Sections 4 and 5 we actually work in the enlarged configuration Ω ⊂ R d satisfying (3.11) and Let us also fix once and for all λ, θ > 0.

Compactness
In this section we prove a compactness result (Proposition 4.1) for the discrete approximations of the Griffith energy, that holds under the assumption that δ ε be bounded. We show that sequences (u ε , v ε ) ε with equibounded energy E λ,θ ε are approximated, in the sense of the convergence in measure, by sequences with bounded continuous Griffith energy (for which compactness is known from Theorem 2.4).
For every simplex T ∈ Σ d , we denote by D T the set of the edges directions for T , which contains d(d + 1)/2 linearly independent vectors of S d . For any vector ξ ∈ R d , we denote by ξ,T j the coordinates of ξ ⊗ ξ in the basis {ν j ⊗ν j :ξ j ∈ D T } of R d×d sym , whereν j :=ξ j /|ξ j |. Finally, we define the triangulation of Ω induced by the partition Σ d as We then denote byû ε = (û 1 ε , . . . ,û d ε ) andv ε the piecewise-affine interpolations of u ε and v ε on T d ε , respectively. We also consider the piecewise constant functions The result will be an immediate consequence of the following crucial claim and of [1, Proposition A.1, Remark A.2], which hold true for any distance inducing the convergence in measure on bounded sets (in particular, for the metric d(u, v) defined in (2.15)).
Step 1: The preliminary remark is that from the equi-boundedness of the energies (4.1) we can get Let η > 0 be fixed, and consider Ω η := {x ∈ Ω : dist(x, R d \ Ω) > η}. Sinceû ε is the affine interpolation of u ε on each simplex of partition Σ d , we have that In order to prove (4.6), a simple computation based on (2.25d), (4.4) and (4.8) shows that where s j , s j +ξ j represent the only two vertices of T whose difference isξ j . Thus, by simple inequalities we infer that whence the assertion easily follows from (4.1) and by the arbitrariness of η. For what concerns (4.7), we notice thatv ε (x) can be rewritten on each simplex α + δT , with vertices α + δξ i , i = 0, 1, . . . , d (we use here the convention α We first prove that for δ small. Indeed, on the one hand, sincev ε is the piecewise affine interpolation of v ε on each simplex of the decomposition, we deduce that for every x ∈ α + δT , so that, by means of elementary inequalities, for δ sufficiently small we have that On the other hand, rewritingv ε (x) as in (4.9) on each symplex α + δT for every α ∈ δZ d ∩ Ω, with the convexity of z → (z − 1) 2 we obtain Hence, summing up on all simplices α + δT ∈ T d ε we finally get, for δ small enough, Now, as a consequence of (4.10), (4.1) and the Cauchy-Schwarz inequality we deduce that whence (4.7) follows by the arbitrariness of η.

Semicontinuity properties for the Griffith energy
This section is devoted to prove the semicontinuity inequality (3.14) in Theorem 3.1, assuming the convergence of u ε to u guaranteed in Section 4 on sequences with bounded approximating energies. In particular, we deduce the lower limit inequality for the Γ-convergence approximation of the classic Griffith energy, with Dirichlet boundary conditions.
As in Section 4, we work with the extended set Ω ⊂ R d , d ∈ {2, 3}, and functions in A Dir δ ( Ω; R d ), A Dir δ ( Ω; R). As observed in Section 3, if u ε ∈ A Dir δ ( Ω; R d ) are such that u ε →ū a.e. in Ω, then u = u 0 in Ω \ Ω. Then (recall the definition of u (3.12) and (3.13)), prove the lower limit inequality for (E Dir λ,θ ) ε is equivalent to prove the lower inequality for the energies ( E Dir λ,θ ) ε defined in the very same way of (E Dir λ,θ ) ε , but with all the integrals and corresponding notation considered in Ω in place of Ω. To ease the reading, in the following we keep the same notation of Section 3 for the functionals, just referring to the set Ω in place of Ω in integrals, in sets of nodes, and in A Dir δ ( Ω; R d ), A Dir δ ( Ω; R). We estimate separately from below the terms F ε and F div ε (Lemma 5.1, Lemma 5.2, and Proposition 5.3), and then address in Proposition 5.4 the lower bound for the Modica-Mortola part G ε , by a blow-up argument. We remark that the results concerning F ε and F div ε hold under the only assumption that δ = δ(ε) vanishes as ε → 0. In contrast, we use the assumption lim ε→0 δ ε = 0 to estimate the Modica-Mortola terms from below in Step 3 of Proposition 5.4.
w ei,y ε ∈ H 1 ( Ω ei,y ) for a.e. y ∈ Π ei , i = 1, . . . , d, (5.14) Proof. Notice that, under the assumption (5.13), from the identity Now, under assumptions (5.13)-(5.15), an analogous slicing argument as for the proof of Lemma 5.1 applied to w ei,y ε shows that for every g ∈ L 2 ( Ω \ A ∞ u ) and every i = 1, . . . , d. The proof of (5.19) can be developed in the case g = 0, the general case following by approximation of g ∈ L 2 ( Ω \ A ∞ u ) with piecewise constant functions on a Lipschitz partition of Ω.
As a consequence of Lemma 5.2, we deduce now the optimal lower bound for the functionals F div ε (u, v) as defined in Section 3.
Now, from the equi-boundedness of the energies (5.22), we infer that where H ζ is defined as in (5.10). Thus, the conclusion (5.16) of Lemma 5.2 holds with z ε andṽ ε in place of w ε and v ε , respectively. Therefore, with (5.31), it follows that which proves the claim (5.25).
We now observe that we have also, for every l and η small, In fact, (5.28)-(5.31) continue to hold, since the lattices Z l are just suitable translations of Z 1 ≡ Z, while the compact subset Ω η of Ω appears on the right-hand side. We deduce that (5.25) follows also for general F div,l ε in place of F div,1 ε . By (5.24) we eventually obtain that whence (5.23) follows by the arbitrariness of η > 0.
With the results proven before in this section, we are in position to prove the liminf inequality for (E Div λ,θ ) ε .
Proof. Let us fix a small ζ ∈ (0, 1). For every ε > 0, we define the discrete measures where 1 α denotes the Dirac delta in α. We observe that In view of Lemma 5.1 (recall Remark 2.9) and Proposition 5.3, the general proof will be a consequence of lim inf by the arbitrariness of ζ ∈ (0, 1). Therefore we prove (5.35) in the following. We divide the proof into three steps: in Step 1 we see that (5.35) is guaranteed from (5.37); in Step 2 we show that, after a blow up procedure around a fixed x 0 in a set of full H d−1 -measure of J u , (5.37) would follow from (5.46); in Step 3 we prove (5.46).
Step 2. Since u ∈ GSBD 2 ∞ ( Ω), we may subdivide In fact, the latter identity may be seen by considering the GSBD 2 function u t for a t for which J u = J ut , so that x 0 ∈ J ut . Thus the approximate limit of u t as x → x 0 in (Q ν ρ (x 0 )) + is t; on the other hand, we have that u t (x) = t if and only if |u(x)| = +∞, so we deduce the latter identity in (5.39).
Let us fix x 0 ∈ J u such that (5.36) and either (5.38 . For this, we first note that for every j and for every m we can find x j 0 ∈ δ j Z d and ρ m,j > 0 such that we introduce the functions u j,m ∈ A τm,j (Q ν ; R d ), v j,m ∈ A τm,j (Q ν ; R) characterized by the following "change of variables in the nodes" Let G σm,j and F σm,j be defined by replacing, in (3.9) for G σm,j , both δ m,j with τ m,j and ε m,j with σ m,j , and, in (3.6a) for F σm,j , δ m,j with τ m,j . We find that In particular we have that (5.42) Notice that we used above that ζ > 0 is fixed, and it holds indeed that lim m,j F σm,j (u j,m , v j,m ) = 0.
By (5.41), (5.42), Proposition 4.1, and Theorem 2.4, we obtain that (u j,m , v j,m ) j,m converges, up to a subsequence, towards a suitable couple in GSBD 2 ∞ ( Ω) × L 2 ( Ω). Moreover, setting u m (y) := u(x 0 +ρ m y) for y ∈ Q ν , it holds that (u m ) m converges in L 0 (Q ν ; R d ) to in Q ν,− and that tanh(|u m |)| Q ν,+ converges in L 1 (Q ν,+ ; R d ) to the constant function 1. Since, for fixed m, u j,m , v j,m converge in measure to u m , v m as j → +∞, by a diagonal argument we may find a sequence m j → +∞ such that the above properties hold for u j := u j,mj as j → +∞ in place of u m as m → +∞ and v j := v j,mj → 1 in L 2 (Q ν ), σ j := σ mj ,j → 0, τ j := τ mj ,j → 0, and We now collect these informations and the fact that (u j , v j ) j converges L d -a.e., up to a subsequence (see discussion below (5.42)). Therefore and v j → 1 in L 2 (Q ν ), where u 0 is given by (5.43) if x 0 ∈ J u ∩ ( Ω \ A ∞ u ) and by Thus, (5.37) (and then the result) would follow from that we show in the remaining part of the present proof.
For this, for every α ∈ τ j Z d ∩ Q ν we set From the equiboundedness of the energies (5.33) and an analogous argument as for the proof of (4.18), we deduce that there exists a constant C > 0 such that for every j, whence , for every j .

The upper limit for the Griffith energy
In this section we prove the Γ-limsup inequality for the convergence stated in Theorem 3.1. Differently to what done in the previous sections, here we argue for the reference configuration Ω. The constraint u ε ∈ A Dir δ (Ω; R d ), v ε ∈ A Dir δ (Ω; R) for the recovery sequence will follow from the part of the density result Theorem 2.3 concerning the treatment of Dirichlet boundary conditions. Proposition 6.1. Assume that lim ε→0 δ ε = 0, and let u ∈ GSBD 2 (Ω). Then there exists a sequence (u ε , v ε ) ∈ A Dir δ (Ω; R d ) × A Dir δ (Ω; R) such that (u ε , v ε ) → (u, 1) in measure on Ω × Ω and lim sup Proof. In view of Theorem 2.3 and remarks below, by a diagonal argument it is not restrictive to assume that u ∈ W(Ω; R d ) and that J u is a closed subset of the hyperplane Π e d = {x d = 0}, that we denote by K. To fix the notation we argue for d = 3, the case d = 2 being analogous. We recall from [6, (4.23)-(4.24)] the following fact about the optimal profile problem for the Ambrosio-Tortorelli functional: for fixed η > 0, there exist T η > 0 and f η ∈ C 2 ([0, +∞)) such that Let T > T η and γ ε > 0 be a sequence such that γ ε /ε → 0 as ε → 0. We set and A ε,δ := A ε ∩ δZ 3 , A ε,δ := A ε ∩ δZ 3 . Notice that, for ε small, recalling that K ⊂ Ω. Let φ ε be a smooth cut-off function between B ε and B ε , and set Since u ∈ W 1,∞ (Ω\J u ; R 3 ) we have u ε ∈ W 1,∞ (Ω; R 3 ). Moreover, since A ε is a compact set in Ω and u = u 0 in a neighborhood of ∂ D Ω, also u ε = u 0 in a neighborhood of ∂ D Ω. Note also that, by the Lebesgue Dominated Convergence Theorem, u ε → u in L 1 (Ω; R 3 ). If ψ ε is a cut-off function between K ε+ where the function h ε : [0, +∞) → R is given by By construction, v ε ∈ W 1,∞ (Ω) ∩ C 0 (Ω) ∩ C 2 (Ω\A ε ) and v ε → 1 in L 1 (Ω). We start proving that there exists a sequence (ū ε ,v ε ) ∈ A Dir δ (Ω; R 3 ) × A Dir δ (Ω; R) converging in measure to (u, 1) on Ω × Ω such that and lim sup since F ε (·,v ε ) ≤ F ε (·, 1) and F div ε (·,v ε ) ≤ F div ε (·, 1), it will be sufficient to prove both (6.3) and (6.4) for the pair of admissible functions (ū ε , 1). Notice thatv ε ∈ A Dir δ (Ω; R) by (6.2) and since A ε is a compact subset of Ω.
With the estimates (6.6), by summing over ξ ∈ S 3 and taking into account Remark 2.9 we infer that lim sup From (6.7), we deduce that lim sup Now, we adapt to our case the argument of the proof of [1,Proposition 4.4], which combined with (6.8)-(6.9) will give (6.3)-(6.4).
We provide now an estimate for G ε (v ε ). Setting, for α ∈ Ω δ such that α + δe k ∈ Ω, G α ε (v) := In particular, this permits to control from above the Γ-lim sup of E Dir λ,θ through a Griffith-type functional.

The non-interpenetration constraint
This section contains the proof of the Γ-convergence approximation in Theorem 3.2. The lower inequality relies on the results proven in Section 4. For the upper inequality we employ a density result for couples (u, v), here recalled in Lemma 7.1, which has been shown in dimension 2 in [18] to prove the upper bound in a continuum approximation for the Griffith energy with a linearized non-interpenetration constraint. We give first the proof of Theorem 3.2, keeping in the last part of the section the auxiliary results.
Summing over α and recalling the control on F div,NI ε (u ε , v ε ), (3.8b), and (3.7), we infer that if ε > 0 is small enough then div − z ε L 2 ( Ωη) ≤ C , for C > 0 depending on M and θ. In view of the L 1 convergence of z ε to u, we have that div z ε converges in the sense of distributions on Ω to Div u = Tr(Eu). Then, arguing as in e.g. [3, Proposition 1.62], we can see that div − z ε converges weakly in L 2 (Ω η ) to a suitable non negative function f , with f ≥ Div − u. Then the positive measure Div − u = Tr − (Eu) is indeed in L 2 (Ω). Now, computing the negative part of the trace of the identity (7.1), we obtain the non-interpenetration condition [u] · ν ≥ 0 H d−1 -a.e. on J u , since Div − u has no singular part. We deduce that G NI,M λ,θ (u, 1) = G λ,θ (u, 1; Ω) < +∞. Then, by Theorem 3.1 and E λ,θ ε ≤ (E NI λ,θ ) ε we conclude (i).
Proof of (iii). Let u β,l ε , v β,l ε the functions provided by Lemma 7.1, in correspondence to families Proof. Properties (i), (ii), (iii) are clear from the construction for the lim sup inequality for [18,Theorem 1], in [18, Subsections 3.1 and 3.2]. In particular, for (i) see (with the numeration in [18]) the definition of v l ε at the beginning of Subsection 3.1 and (17), for (ii) the very last sentence of Section 3, and for (iii) the definition of u ε below (24), where u has to be replaced by u I , as explained below (34).
As for (iv), this is a consequence of (18) for the Modica-Mortola part in v (with a minor modification since the Modica-Mortola term here is slightly different), of (27), that states that Eu β,l ε is a good approximation of Eu where v β,l ε = 0 (then one can treat separately Eu β,l ε and div + u β,l ε , as we did), and of (36)-(37) for the treatment of div − u β,l ε .