Phase-field approximation of functionals defined on piecewise-rigid maps

We provide a variational approximation of Ambrosio-Tortorelli type for brittle fracture energies of piecewise-rigid solids. Our result covers both the case of geometrically nonlinear elasticity and that of linearised elasticity.


Introduction
According to the Griffith theory of crack-propagation in brittle materials [41], the equilibrium configuration of a fractured body is determined by balancing the reduction in bulk elastic energy E e stored in the material with the increment in fracture energy E f due to the formation of a new free surface. For those materials for which crack-growth can be seen as a quasi-static process, the equilibrium configurations are obtained, at each time, by solving a minimisation problem involving the total free energy of the system; i.e., E := E e + E f .
For hyperelastic brittle materials a prototypical elastic energy E e is of the form E e (u, K) = µ Ω\K W (∇u) dx, (1.1) where Ω ⊂ R 3 is open, bounded and represents the reference configuration of a body which is fractured along a sufficiently regular closed surface K ⊂ Ω, and u : Ω\K → R 3 is the deformation map, which is smooth outside K. In (1.1) the constant µ > 0 represents the shear modulus of the material and W : M 3×3 → [0, +∞) the stored elastic energy density. In the setting of nonlinear elasticity W is assumed to be frame indifferent and to vanish only on SO(3), the set of 3 × 3 rotation matrices; moreover, close to SO(3) the function W (·) behaves like dist 2 (·, SO (3)).
In a simplified isotropic setting, the fracture energy of a brittle material obeys the Griffith criterion and is proportional to the area of the crack-surface K; i.e., E f (K) = γ H 2 (K), (1.2) to be interpreted as the deformation gradient outside the crack, and a surface contribution concentrated along the crack-set J u . In the SBV (Ω)-setting the energy E then becomes (1.3) and its minimisation can be carried out by applying the direct methods to F, or to its relaxed functional (cf. [4]). Functionals as in (1.3) are commonly referred to as free-discontinuity functionals and play a central role both in fracture mechanics [36,11,12] and in computer vision [51] and have been extensively studied in the last decades [4,14].
In this paper we are interested in the case when the material parameters in (1.3) satisfy the relation µ/γ ≫ 1, or, up to a renormalisation, when µ ≫ 1 and γ = O(1). This parameterregime is typical of rigid solids; i.e., of solids which deform without storing any elastic energy. In fact, being µ large (and γ = O(1)), a deformation u shall satisfy W (∇u) = 0 which is equivalent to asking ∇u ∈ SO(3) almost everywhere in Ω. However, since u ∈ SBV (Ω), the differential constraint ∇u ∈ SO(3) does not prevent u to jump and thus fracture to occur. Hence, for µ ≫ 1 the energy-functional F models those brittle solids which exhibit a rigid behaviour in a number of subregions of Ω which are separated from one another by a discontinuity surface. Mathematically, these configurations are described by the so-called piecewise-rigid maps on Ω and are denoted by P R(Ω). Namely, u ∈ P R(Ω) if (1.4) where, for every i ∈ N, A i ∈ SO(3), b i ∈ R 3 , and (E i ) is a Caccioppoli partition of Ω. Then, up to a lower-oder bulk-contribution, in the regime µ ≫ 1 the total energy E of a brittle rigid solid can be identified with its fracture energy; the latter coincides with the surface term in (1.3) where now the deformation-variable u belongs to the space P R(Ω) (see [38]). Despite their simple analytical expression, energy functionals of type γ H n−1 (J u ) are notoriously difficult to be treated numerically, due to their explicit dependence on the discontinuity surface J u . To develop efficient methods to compute their energy minimisers and to analyse phenomena like crack-initiation, crack-branching or arrest in nonlinearly elastic brittle materials, in the engineering community suitable "regularisations" have been recently proposed, where the surface J u is replaced by an additional phase-field variable v ∈ [0, 1] (see e.g., [22,59] and references therein). In these models the phase-filed variable v interpolates between the sound state (corresponding to v = 1) and the fractured state of the material (corresponding to v = 0) and it is to be interpreted as a damage variable in the spirit of [52,53,54].
The purpose of the present paper is to establish a rigorous mathematical connection between damage models of Ambrosio-Tortorelli type and variational fracture models for brittle piecewiserigid solids. In other words, in this work we provide an elliptic approximation of functionals of type F (u) = γ H n−1 (J u ), u ∈ P R(Ω), (1.5) where the constraint u ∈ P R(Ω) is reminiscent of the nonlinear elastic energy density W , which satisfy W −1 ({0}) = SO(n).
Namely, we show that for (u, v) ∈ W 1,2 (Ω; R n ) × W 1,2 (Ω), 0 ≤ v ≤ 1 the family of functionals converges, in the sense of De Giorgi's Γ-convergence [27,13], to the functional (1.5), under the assumption that k ε → +∞, as ε → 0. As in the case of the Modica-Mortola functional [47,48] and of the Ambrosio-Tortorelli approximation [5,6], in (1.6) the singular-perturbation parameter ε > 0 determines the thickness of the diffuse interface around the limit discontinuity surface J u , while the diverging parameter k ε is proportional to the stiffness of the material, hence to the constant µ appearing in (1.3). More precisely, in Theorem 3.3 we prove that if the zero-set of the bulk energy density W coincides with SO(n) and for every A ∈ M n×n it holds for some α > 0, then the family (F ε ) Γ-converges to F , in the L 1 (Ω; R n ) × L 1 (Ω) topology. The proof of Theorem 3.3 takes advantage of a number of analytical tools. First, to determine the set of the limit deformations we use a piecewise-rigidity result in SBV (Ω) by Chambolle, Giacomini and Ponsiglione [20] (cf. Theorem 2.4). The latter is the counterpart of the Liouville rigidity Theorem for deformations of brittle elastic materials and provides a characterisation of discontinuous deformations with zero elastic energy as a collection of an at most countable family of rigid motions defined on an underlying Caccioppoli partition of Ω. To match the assumptions of Chambolle, Giacomini, and Ponsiglione's result we use a global argument of Ambrosio [3] which is based on the co-area formula and is tailor-made to gain compactness in SBV . Namely, starting from a pair (u ε , v ε ) ⊂ W 1,2 (Ω, R n ) × W 1,2 (Ω) with equi-bounded energy F ε we use the co-area formula to find a suitable sublevel set of v ε in which u ε can be modified to obtain a new sequence (ũ ε ) ⊂ SBV (Ω) which differs from u ε on a set of vanishing measure and moreover satisfies sup ε>0 k ε Ω dist 2 (∇ũ ε , SO(n)) dx + H n−1 (Jũ ε ) < +∞.
The estimate above, combined with a result of Zhang which guarantees that the zero-set of the quasiconvexification of dist(·, SO(n)) coincides with SO(n), proves that any L 1 limit u ofũ ε satisfies ∇u ∈ SO(n) a.e. in Ω. Eventually, the Chambolle-Giacomini-Ponsiglione piecewise-rigidity Theorem yields that u is a piecewise-rigid map. The construction of Ambrosio additionally provides us with the sharp lower bound. Indeed, the perimeter of the sublevel sets of v ε chosen as above prove to be asymptotically larger than the interfacial energy-contribution of the piecewise rigid limit deformation. As in the case of the Ambrosio-Tortorelli functional, the sharp interfacial energy is defined in terms of a one-dimensional optimal profile problem. The upper bound is then proven first by resorting to a density argument and then by an explicit construction. Namely, we use the density in P R(Ω) of finite partitions subordinated to Caccioppoli sets which are polyhedral [15]. Then, for these partitions, a recovery sequence matching asymptotically the sharp lower bound can be contructed by creating a layer of order ε around the jump set of the target function u, in which the transition is one-dimensional and is obtained by a suitable scaling of the optimal profile. As for the Ambrosio-Tortorelli approximation of the Mumford-Shah functionals (see also, e.g., [33,7,8,9,58]) also in our case the regularised bulk and surface energy in (1.6) separately converge to their sharp counterparts. Namely, in this case the bulk term in (1.6) vanishes in the limit due to the presence of the diverging parameter k ε , that is, equivalently, limit deformations have (approximate) gradients in SO(n) a.e. in Ω. Similarly, the Modica-Mortola term in (1.6) approximates the limit surface energy, which in our model carries the whole energy contribution.
It is worth mentioning that the arguments in Theorem 3.3 can be extended (resorting to by-now standard modifications) to cover the case of anisotropic surface-integrals which model the presence of preferred cleavage planes in single crystals (cf. Remark 3.4).
In Theorem 4.4 we generalise the approximation result Theorem 3.3 to the case of energy densities W vanishing on a compact set K ⊂ M n×n for which a piecewise-rigidity result analogous to the one for SO(n)-valued discontinuous deformations holds true. In fact in [20] piecewise rigidity is proven, more in general, for those K for which a quantitative L p -rigidity estimate holds (see Section 4 for more details). In this way, multiple incompatible wells can be also taken into account. From a mechanical point of view, the incompatibility describes those solids for which no fine-scale phase-mixtures are allowed in solid-solid transformations (see [50]). A list of non trivial examples of possible compact sets K fulfilling the assumptions of Theorem 4.4 is also included.
Finally, in Theorem 4.10 a further approximation result is provided, which covers the case of linearised elasticity.

Notation.
In what follows Ω ⊂ R n denotes a bounded domain (i.e., an open and connected set) with Lipschitz boundary. We use a standard notation for Lebesgue and Sobolev spaces, and for the Hausdorff measure. The Euclidean scalar product in R n is denoted by ·, · .
We refer the reader to the book [4] for a comprehensive introduction to the theory of functions of bounded variation and of (generalised) special functions of bounded variation (G)SBV (Ω) and to [28] for the definition and main properties of generalised special functions of bounded deformation GSBD(Ω). In any of these cases we shall deal with the proper subspaces of these functional spaces in L 1 (Ω, R n ).
Below we briefly recall the notation and the main results for one-dimensional sections of GSBV functions, in a form that we need, since we will make an extensive use of these.
Moreover, if w ∈ GSBV (Ω) n and ξ ∈ R n \ {0} the following properties hold: (a) (w ξ y ) ′ (t) = ∇w (y + tξ) ξ for L 1 -a.e. t ∈ Ω ξ y and for H n−1 -a.e. y ∈ Π ξ ; We observe that (c) simply follows from a Fubini type argument noting that for every x ∈ J w , where [w](x) is the difference of the one-sided traces of w at x ∈ J u . We also note that, if w k , w ∈ L 1 (Ω, R n ) and w k → w in L 1 (Ω, R n ), then for every ξ ∈ S n−1 there exists a subsequence (w k j ) of (w k ) such that We recall here the definition of Caccioppoli-affine and piecewise-rigid function. Moreover we also recall the piecewise-rigidity result [20, Theorem 1.1] in a variant which is useful for our purposes.
The set of piecewise-rigid functions on Ω will be denoted by P R(Ω).
The measure theoretic properties of Caccioppoli-affine functions are collected in the result below (cf. [23, Theorem 2.2]).
Below we recall a slight generalisation of the piecewise-rigidity result by Chambolle, Giacomini, and Ponsiglione [20, Theorem 1.1] originally stated in the SBV -setting.

Setting of the problem and main result
In this section we introduce a family of functionals of Ambrosio-Tortorelli type (cf. [5,6]) and we prove that this family converges to a surface functional of perimeter type which is finite only on piecewise-rigid maps.
Let W : Ω×M n×n → [0, +∞) be a Borel function such that W (x, A) = 0 for every A ∈ SO(n). Assume moreover that for every x ∈ Ω and every A ∈ M n×n it holds on Ω, +∞ otherwise.
(3.2) In the following proposition we show that the Γ-limit of (F ε ) (if it exists) is finite only on the set of piecewise-rigid maps.
We first note that up to subsequences (not relabelled) we have from which the convergence v ε → 1 in L 1 (Ω) easily follows. In fact, for η > 0 we have Since V −1 ({0}) = 1, the minimum in the left hand side of (3.4) is strictly positive. Therefore, gathering (3.4) and (3.3) implies that v ε → 1 in measure. The latter, together with the uniform bound satisfied by (v ε ) immediately gives v ε → 1 in L 1 (Ω).
The next theorem establishes a Γ-convergence result for the functionals F ε .
Proof. We divide the proof into two steps.
Without loss of generality, up to the extraction of a subsequence, we may assume that the liminf in (3.13) is a limit; therefore we have Then Proposition 3.1 readily implies that u ∈ P R(Ω) and v = 1 a.e. in Ω. To prove (3.13) we start noticing that by (3.6) and the Fatou Lemma we have then, to conclude it suffices to show that for L 1 -a.e. s ∈ (δ, 1 − δ), and then let δ → 0 + . The estimate in (3.15) can be obtained via slicing similarly as in [33,34]. Specifically, fix s ∈ (δ, 1 − δ) for which the left-hand side of (3.15) is finite, and set Ω ε := {v ε < s}; we notice that L n (Ω ε ) → 0, as ε → 0. We now claim that for every open subset U ⊂ Ω and every ξ ∈ S n−1 we have Thanks to (2.2) for H n−1 -a.e. ξ ∈ S n−1 we have H n−1 (J u \ J ξ u ) = 0, therefore from (3.17) we also infer lim inf Then, (3.15) follows from (3.18) passing to the supremum on a dense sequence (ξ j ) in S n−1 and invoking [4,Lemma 2.35], also noticing that the function is superadditive on pairwise disjoint open subsets of Ω.
Hence we are now left to prove (3.16). To this end we start observing that for every A ∈ M n×n it holds Therefore, for every A ∈ M n×n and every ξ ∈ S n−1 we get where [t] + denotes the positive part of t ∈ R. In view of (3.1), (3.14), (3.19) and by the Fatou Lemma we can find a subsequence (u ε j , v ε j ) of (u ε , v ε ) such that and for H n−1 -a.e. y ∈ π ξ (Ω) and lim inf for some constant c > 0 (which may depend on y). Let y ∈ π ξ (Ω) be fixed and such that both Indeed, if s h > 0 for some h ∈ {1, ..., l}, then for j sufficiently large we would get Hence, Rellich-Kondrakov's Theorem and (3.21) would imply that the slice u ξ y belongs to W 1,1 (I h , R n ), thus contradicting the assumption H 0 J u ξ y ∩ I h > 0. Then, thanks to (3.21) and (3.23) we can find (r i j ) ⊂ I i such that Eventually, the subadditivity of the liminf and the arbitrariness of l yield so that (3.16) follows by integrating the previous inequality on π ξ (U ) and using the Fatou Lemma.
Step 2: Existence of a recovery sequence. Let (u, v) ∈ L 1 (Ω, R n ) × L 1 (Ω) be arbitrary, in this step we will construct a sequence ( We start by noticing that the inequality in (3.24) is trivial unless we additionally assume that u ∈ P R(Ω) and v = 1 a.e. in Ω. Therefore, in particular we can write u as where A i ∈ SO(n), b i ∈ R n for every i ∈ N, and (E i ) is Caccioppoli partition of Ω. By standard density and continuity arguments (cf. [13, Remark 1.29]) we notice that it is enough to prove (3.24) in a subset X of P R(Ω), which is dense in P R(Ω) in the following sense: for every u ∈ P R(Ω) there exists (u j ) ⊂ X such that for j → +∞. We now claim that X is given by those u ∈ P R(Ω) of the form where A i ∈ SO(n),b i ∈ R n , and E i is a polyhedral set, for every i = 1, . . . , N . Indeed given u as in (3.25) the sequence (u N ) defined as clearly satisfies u N → u in L 1 (Ω, R n ), as N → +∞. Moreover, by lower semicontinuity we have that Further, given the finite partition of Ω into sets of finite perimeter , as j → +∞. Eventually, the desired sequence (u j ) satisfying (3.26)-(3.27) can be obtained by a standard diagonal argument.
We now construct a recovery sequence (u ε , v ε ) ⊂ W 1,2 (Ω, R n ) × W 1,2 (Ω) for F ε when u is as in (3.27). Therefore we have that, in particular, up to a set of zero H n−1 -measure where S 1 , . . . , S M ⊂ R n are a finite number of (n − 1)-dimensional simplexes. For every i ∈ {1, . . . , M } we denote with Π i the (n − 1)-dimensional hyperplane containing the simplex S i ; we have that Π i = Π ℓ , for i = ℓ.
Moreover, for every δ > 0 we define Now let γ i ε be a cut-off function between S ε i and S 2ε From the very definition of v i ε we have that 0 ≤ v i ε ≤ 1 and v i ε ∈ W 1,∞ (R n ). Moreover, by using the following facts: |∇γ i ε | ≤ c/ε, π i is Lipschitz with constant 1, and |∇d i | = 1, we also get where to establish the last equality we have used that ξ ε ≪ ε and H n−1 (S ε i ) → H n−1 (S i ), as ε → 0. We now estimate the second term in the right-hand side of (3.35). To do so it is convenient to write and Therefore, since |∇d i | = 1 a.e., we have as ε → 0, where to establish the last inequality we have used (3.30). Furthermore, from (3.33) it is immediate to show that as ε → 0. Now the idea is to combine together the sequences (v i ε ) in order to define a new sequence (v ε ) which belongs to W 1,2 (Ω) and in every B ε i coincides with (v i ε ), up to a set where the surface energy is negligible. Moreover the sequence (v ε ) shall satisfy: v ε → 1 in L 1 (Ω) and where u is as in (3.27).

by (3.34) and (3.42) we readily deduce that
as ε → 0, the second term in the right hand side of (3.44) is negligible. Hence, to get (3.41) we are left to estimate the surface energy in Ω ∩ (B ε \ A ε ). We claim that lim sup We notice that where the sets H ε i and I ε i are defined as in (3.37) and (3.38), respectively. Since moreover We now estimate the terms S 1 ε and S 2 ε separately. To this end, we start observing that hence, invoking (3.39) and (3.40), we readily get as ε → 0. Moreover, appealing to (3.33) easily gives We now claim that lim for every i, j ∈ {1, . . . , M }. Indeed, since Π i = Π j then the set S i ∩ S j is contained in an (n − 2)-dimensional affine subspace of R n , so that by (3.37) and (3.38) we can deduce that (3.50) as ε → 0, where the constant c > 0 depends only on the angle between Π i and Π j and on H n−2 (S i ∩ S j ). Hence, (3.50) immediately yields (3.49). Finally, gathering (3.46) and (3.47) entails (3.45), as desired. Therefore, to conclude the proof of the upper bound we now have to exhibit a sequence (u ε ) ⊂ W 1,2 (Ω, R n ) such that u ε → u in L 1 (Ω, R n ) and To this end, set let ϕ ε ∈ C ∞ 0 (A ε ) be a cut-off function between (A ε ) ′ and A ε , and define u ε := (1 − ϕ ε )u.
hence using that u ∈ P R(Ω) together with the fact that for every x ∈ Ω the function W (x, ·) vanishes in SO(n) we immediately get in Ω \ A ε and hence the claim.
on Ω, +∞ otherwise, (3.52) where the euclidean norm in F ε is now replaced by a Finsler norm φ. That is, φ : Ω × R n → [0, +∞) is a continuous function which is convex in its second variable and satisfies the two following properties: i. for every (x, z) ∈ Ω × R n and for every t ∈ R φ(x, tz) = |t|φ(x, z); ii. for every (x, z) ∈ Ω × R n there exist 0 < m ≤ M < +∞ such that In this case it can be proven that the family of functionals (F φ ε ) Γ(L 1 (Ω, R n ) × L 1 (Ω))-converges to the following inhomogeneous and anisotropic functional F φ : L 1 (Ω, R n ) × L 1 (Ω) −→ [0, +∞] defined on piecewise rigid maps as: where ν u denotes the exterior unit normal to J u .

Incompatible wells and linearised elasticity
In this section we are going to address two possible extensions of Theorem 3.3. We first discuss a generalisation of Theorem 3.3 to the case where the zeros of the potential W lie in a suitable nonempty compact set K. Then, we show that our proof-strategy also applies to the case of linearised elasticity. Similarly as in Section 3, also in these cases the key tools for the analysis are two suitable variants of the piecewise-rigidity property stated in Theorem 2.4 (cf. Theorem 4.2 and Theorem 4.6).
4.1. The case of K piecewise-rigid maps. Let U ⊂ R n be a bounded domain with Lipschitz boundary, and let K ⊂ M n×n be a nonempty compact set satisfying the following L p -quantitative rigidity estimate for some p ∈ (1, n/(n − 1)): there exists a constant C > 0 (depending only on p and n) such that for every u ∈ W 1,p (U, R n ) (4.1) We notice that (4.1) implies the rigidity of the differential inclusion v ∈ W 1,∞ (U, R n ) and ∇v(x) ∈ K a.e. U , (4.2) in the sense explained in Lemma 4.1 below. In the statement of Lemma 4.1 we use the same terminology adopted in [55,Chapter 8] (see also [50,Section 1.4] and [45]).
Lemma 4.1. Let U ⊂ R n be a bounded domain with Lipschitz boundary. Let K ⊂ M n×n be a nonempty compact set satisfying (4.1). Then, the following statements hold true: (1) the differential inclusion (4.2) is rigid for exact solutions; i.e., the only solutions to (4.2) are affine functions; (2) the differential inclusion (4.2) is rigid for approximate solutions; i.e., if dist(∇u j , K) → 0 in measure in U , (u j ) converges to u weakly* in W 1,∞ (U, R n ), u j = Ax on ∂U for some A ∈ M n×n , then (∇u j ) converges in measure to ∇u in U and u is affine; (3) the differential inclusion (4.2) is strongly rigid; i.e., if dist(∇u j , K) → 0 in measure in U and (u j ) converges to u weakly* in W 1,∞ (U, R n ), then (∇u j ) converges in measure to ∇u in U and u is affine; (4) we have K = K qc , (4.3) where K qc denotes the quasiconvex envelope of K; i.e., For the readers' convenience the proof of Lemma 4.1 is included in the Appendix A.
Below we give a list of nonempty compact sets K ⊂ M n×n for which (4.1) holds true. The most prominent examples are due to Ball and James [10, Proposition 2] and to Friesecke, James, and Müller [39, Theorem 3.1] and correspond, respectively, to the case of two non rank-1 connected matrices and to that of SO(n).
We notice that in the examples (1) and (3) below, property (4.1) directly follows from an incompatibility condition for the approximate solutions of (4.2), as shown in [21] (see also [31,Theorem 1.2]). This condition reduces rigidity for multiple-wells to a single-well rigidity statement. We recall here that two disjoint compact sets K 1 , K 2 ∈ M n×n are incompatible for the differential inclusion (4.2), with K = K 1 ∪ K 2 , if for any sequence (u j ) ⊂ W 1,∞ (U, R n ) such that dist(∇u j , K 1 ∪ K 2 ) → 0 in measure, then either dist(∇u j , K 1 ) → 0 or dist(∇u j , K 2 ) → 0 in measure. In this case K 1 and K 2 are also called incompatible energy-wells.
In the examples (4) and (5) listed below, property (4.1) is instead a consequence of the Friesecke, James, and Müller rigidity estimate [39,Theorem 3.1] for (2), and of the above mentioned incompatibility for approximate solutions of (4.2). Although equality (4.3) is a consequence of (4.1) as established by Lemma 4.1, for each example in the list below we also give a precise reference to a direct proof of (4.3). We refer the reader to [50], [45] and [55,Chapter 8] for more details on these topics.
n×n are not rank-1 connected, see [10,Proposition 2], (see also [60,Example 4.3]); (2) K = SO(n) [39, Theorem 3.1] (for (4.3) see [44] and also [60,Example 4.4]); R n×n are such that K has no rank-1 connections, see [56,Section 4]; (2), where A i ∈ R 2×2 are such that detA i > 0 for all i ∈ {1, 2, 3} and K has no rank-1 connections, see [57, Theorem 2 and Remark 1]; 0 and h 2 = 1 (the latter condition is equivalent to K having no rank-1 connections). Additionally, one of the following two conditions must hold true: We now recall an extension of the piecewise-rigidity result contained in Theorem 2.4 to the case of a compact set K for which (4.1) holds true. We state this result for GSBV -functions and we refer the reader to [20,Theorem 2.1] for the original statement in the SBV -setting.
Theorem 4.2. Let K ⊂ M n×n be a nonempty compact set for which (4.1) holds true and let u ∈ GSBV (Ω, R n ) be such that H n−1 (J u ) < +∞ and ∇u ∈ K a.e. in Ω. Then, u ∈ P R K (Ω); i.e., (4.4) with A i ∈ K for every i ∈ N, b i ∈ R n , and (E i ) Caccioppoli partition of Ω.
By combining Proposition 4.3 and Theorem 4.4 below we can identify the Γ-limit of F K ε . Now using Theorem 4.2 in place of Theorem 2.4, these results can be proven by following exactly the same arguments employed in the proofs of Proposition 3.1 and Theorem 3.3, respectively. For the readers' convenience we notice that in the proof of Proposition 4.3 the following characterisation of K qc is needed: for every q ∈ [1, +∞) (cf. [60,Proposition 2.14], and also [50,Theorem 4.10]) together with Lemma 4.1, statement (4).
Instead, the proof of Theorem 4.4 is identical to that of Theorem 3.3 up to replacing the coercivity estimate (3.19) with which holds true for every A ∈ M n×n and every ξ ∈ S n−1 .
The following proposition shows that the Γ-limit of (F K ε ) (if it exists) is finite only on P R K (Ω).
In addition, the following Γ-convergence result holds true. Definition 4.5. A map u : Ω → R n is called a piecewise-rigid displacement if there exist skewsymmetric matrices A i ∈ M n×n skew and vectors b i ∈ R n such that with (E i ) Caccioppoli partition of Ω. The set of piecewise-rigid displacements on Ω will be denoted by P RD(Ω). In what follows e(u) denotes the symmetrized approximate gradient of u ∈ GSBD(Ω) (cf. [28]).
To approximate interfacial energies defined on P RD(Ω), for ε > 0, we consider the functionals in Ω +∞ otherwise, (4.8) where k ε → +∞, as ε → 0, W : Ω × M n×n → [0, +∞) is a Borel function such that for every x ∈ Ω and for every A ∈ M n×n it holds α|A sym | 2 ≤ W (x, A) for some α > 0. Here we denote by A sym the symmetric part of A, namely A sym := A+A T 2 . We use standard notation for the strain e(u) = (∇u) sym of u ∈ W 1,2 (Ω, R n ).
Remark 4.7. We refer the reader to [25,Remark 4.14] for an explicit example of a nonconvex, polyconvex function which depends non-trivially on the skew-symmetric part of A and satisfies the bounds α|A sym | 2 ≤ W (x, A) ≤ β(|A sym | 2 + 1) for some α, β > 0, for every x ∈ Ω and for every A ∈ M n×n .
Another example can be obtained by taking W (A) = h 2 (A), A ∈ M n×n , where h is a onehomogeneous quasiconvex function such that for every A ∈ M n×n and for some α, β > 0 with h depending non-trivially on A skew := A−A T 2 . We notice that, in particular, h (and therefore W ) is not convex. A function as above can be obtained by slightly modifying Müller's celebrated example [49] similarly as in [17,Section 7].
In the following proposition we show that the Γ-limit of (E ε ) (if it exists) is finite only on piecewise-rigid displacements.
Remark 4.9. We notice that from the weak convergence of e(ũ ε ) to e(u) in L 2 (Ω, M n×n sym ) and the convergence of e(ũ ε ) L 2 (Ω,M n×n sym ) to e(u) L 2 (Ω,M n×n sym ) , we conclude that e(ũ ε ) → e(u) in L 2 (Ω, M n×n sym ). (4.10) Arguing as in the proof of Theorem 3.3, on account of Proposition 4.8 we can now prove the following Γ-convergence result for the family (E ε ). Proof. The proof is very similar to that of Theorem 3.3, therefore we highlight only the necessary changes, referring for the notation and the details to that proof. In order to establish the lower bound inequality in Step 1 there, we have to show (3.15). Assuming that (u ε , v ε ) converges to (u, 1) in L 1 (Ω, R n+1 ) and sup ε E ε (u ε , v ε ) < +∞, we can find a subsequence (u ε j , v ε j ) of (u ε , v ε ) such that (3.20) is satisfied, and in place of (3.21) and (3.22) we have for H n−1 -a.e. y ∈ π ξ (Ω) (u ε j ) ξ y , ξ , (v ε j ) ξ y → ( u ξ y , ξ , 1) in L 1 (Ω ξ y , R n ) × L 1 (Ω ξ y ), and for some constant c > 0 (which may depend on y). To deduce (4.12) we have used that | Aξ, ξ | 2 ≤ |A| 2 for every ξ ∈ S n−1 and for every A ∈ M n×n sym . Noting that by linearity ((u ε j ) ξ y ) ′ , ξ = (u ε j ) ξ y , ξ ′ , the rest of the argument is exactly the same as the corresponding one in Theorem 3.3 now replacing (u ε j ) ξ y and u ξ y with (u ε j ) ξ y , ξ and u ξ y , ξ , respectively. The proof of the upper bound inequality in Step 2 of Theorem 3.3 remains unchanged up to replacing rotation matrices with antisymmetric ones.  (1) and (2) immediately follow (we notice that actually the validity of (1) and (2) is equivalent to (3), as shown in [55,Corollary 8.9]). To this end let (u j ) and u be as in (3); then (dist(∇u j , K)) converges to 0 in L p (U ), indeed it converges to 0 in measure and ∇u j is bounded in L ∞ (U, M n×n ). Thanks to (4.1) we can find A j ∈ K such that ∇u j − A j L p (U,M n×n ) ≤ C dist(∇u j , K) L p (U ) .
For an arbitrary subsequence (j k ), we extract a further subsequence (j k h ) such that A j k h converges to some A ∈ K. Therefore, (∇u j k h ) converges to A in L p (U, M n×n ). This convergence combined with the weak* convergence of (u j ) to u in W 1,∞ (U, R n ) immediately gives ∇u = A a.e. on U . Being the limit independent of the subsequence, the Urysohn property implies that the whole sequence (∇u j ) converges to A in L p (U, M n×n ) and hence the claim.