A homogenization result in finite plasticity

We carry out a variational study for integral functionals that model the stored energy of a heterogeneous material governed by finite-strain elastoplasticity with hardening. Assuming that the composite has a periodic microscopic structure, we establish the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}Γ-convergence of the energies in the limiting of vanishing periodicity. The constraint that plastic deformations belong to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{SL}(3)$$\end{document}SL(3) poses the biggest hurdle to the analysis, and we address it by regarding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{SL}(3)$$\end{document}SL(3) as a Finsler manifold.


Introduction
This paper deals with the homogenization of some variational integrals related to the theory of finite plasticity.Precisely, we focus on those materials governed by finite-strain elastoplasticity with hardening.Starting from the microscopic level, we aim at retrieving effective macroscopic descriptions by means of a Γ-convergence approach.
Let Ω ⊂ R 3 be an open, bounded, connected set with Lipschitz boundary, and let us denote by SL(3) the group of 3 × 3 real matrices with determinant equal to 1.
Our main result describes the limiting behavior of the functionals F ε (y, P ) := ˆΩ W x ε , ∇y(x)P −1 (x) dx + ˆΩ H x ε , P (x) dx + ˆΩ |∇P (x)| q dx, (1.1) where y ∈ W 1,2 (Ω; R 3 ), P ∈ W 1,q (Ω; K) with q > 3 and K ⊂ SL(3) compact, and W and H encode, respectively, the periodic nonlinear elastic energy density and the hardening of the elastoplastic material sitting in Ω.The precise statement is contained in Theorem 2.2, and the exact mathematical setting of the analysis is detailed in Section 2.
As we touched upon before, we work within the classical mathematical theory of elastoplasticity at finite strains (see, e.g., [29]).We assume that the elastic behavior of our sample Ω is independent of preexistent plastic distortions.This can be rephrased as the assumption that the deformation gradient ∇y associated with any deformation y : Ω → R 3 of the body decomposes into an elastic strain and a plastic one.Within the framework of linearized elastoplasticity, the decomposition would have an additive nature; in the setting of finite plasticity, instead, a multiplicative structure is traditionally assumed [27,28].Following this approach, as in [30,31], we suppose that for any deformation y ∈ W 1,2 (Ω; R 3 ) there exist an elastic strain F el ∈ L 2 (Ω; R 3×3 ) and a plastic strain P ∈ L 2 (Ω; SL (3)) such that ∇y(x) = F el (x)P (x) for a. e. x ∈ Ω.Such multiplicative decomposition, recently justified in the setting of dislocation systems and crystal plasticity in [10,11], motivates the definitions of the energy functionals in (1.1).Other constitutive models in finite plasticity have been also taken into account in the mathematical literature (see, e.g., [15,21,22,33]).We point out that the stored energy in (1.1) has already been considered in [32] in a variational setting.We refer to [31] for more general models in finite plasticity including hardening variables.Concerning the regularization via ∇P in (1.1), we note that it is common in engineering models and prevents the formation of microstructures, see [4,7].We also refer to [18] for a discussion of its drawbacks and for alternative higher order regularizations.
In the small strain regime, the homogenization of the plasticity equations has been undertaken in the series of works [34,26,25] both in the periodic and in the aperiodic and stochastic settings.The framework of perfect plasticity in the case of a linear elastic response has been completely characterized in the seminal work [20].See also [24] for a homogenization result in the small strain regime with hardening.To the authors' knowledge, the only available results in the large strain setting are instead confined to the framework of crystal plasticity for stratified composites.In particular, we refer to [8,9] for the superlinear growth case, to [14] for the linear growth scenario, and to [17] for some first homogenization analysis in the evolutionary setting.Theorem 2.2 fills a gap in the study of elastoplastic microstructures by providing a static homogenization result in the large strain regime.As an application of our contribution, in [16] the analysis of high-contrast elastoplastic materials is carried out.
We conclude this introduction with a few words on the proofs.The equicoervicity of {F ε } and the semicontinuity of the elastic term require uniform bounds on the plastic strains (cf.(2.1), see also [32]).To enforce them, we prescribe the effective domain of the hardening to be a compact neighborhood of the identity in SL (3).The existence of an integral Γ-limit, instead, is based on the localization technique in the context of integral representation results.The biggest hurdle here is devising a version of the so-called fundamental estimate that complies with the constraint that plastic deformations must belong to the compact neighborhood of the identity.Note that this poses an additional difficulty with respect to, e.g., [2,12], which deal with manifold-valued Sobolev functions.Indeed, not only we constrain our maps to take values in a manifold, but we also require them to range in a specific set.To accommodate for this, an adequate notion of convexity is needed.Therefore, we will regard SL(3) as a manifold endowed with a Finsler metric, and impose the compact neighborhood of the identity to be also geodesically convex.We refer to Section 3 for a review of the localization method and of the Finsler structure on SL(3).The characterization of the limiting energy density is obtained by a perturbative argument that grounds on standard homogenization results, see Theorem 3.1.
Outline.The setup of our analysis and the main result, Theorem 2.2, is presented in Section 2. Section 3 contains some reminders about the technical tools to be used in the sequel.The proof of Theorem 2.2 is the subject of Section 4, which we conclude with the discussion on compactness of bounded energy sequences and convergence of (almost) minimizers.

Mathematical setting and results
Throughout the paper, Ω is an open, bounded, and connected set with Lipschitz boundary in R 3 (the analysis would not change significantly if we settled the problem in R d with d = 2 or d > 3).We use R 3×3 and R 3×3×3 to denote real-valued 3 × 3 and 3 × 3 × 3 tensors, respectively.We use the notation I for the identity matrix.The symbol | • | is indiscriminately adopted for the Euclidean norms in R 3 , R 3×3 and R 3×3×3 .To deal with plastic strains, we recall the classical notation SL(3) := {F ∈ R 3×3 : det F = 1}.In Subsection 3.2 we will endow SL(3) with a metric structure and regard it as a Finsler manifold.
If A ⊂ R 3 is a measurable set, we will denote by L 3 (A) its three-dimensional Lebesgue measure.
The building block of our study is the following variational notion of convergence: Definition 2.1.Let X be a set endowed with a notion of convergence.We say that the family and all infinitesimal sequences {ε k } k∈N the following holds: (1) for every sequence {x k } k∈N ⊂ X such that x k → x, we have When X is equipped with a topology τ , we sometimes use expressions such as Γ(τ )-convergence or Γ(τ )-limit to emphasize the underlying convergence for sequences in X.We elaborate on the connections between Γ-convergence and homogenization in Subsection 3.1.In what follows, for notational convenience, we indicate the dependence on ε k by means of the subscript k alone, e.g., We collect here the assumptions under which the Γ-convergence of the functionals in (1.1) will be proved.
Let Q := [0, 1) 3 be the periodicity cell.We assume that the elastic energy density W : R 3 × R 3×3 → [0, +∞] satisfies the following: . E2: It is 2-coercive and has at most quadratic growth, i.e., there exist 0 < c 1 ≤ c 2 such that for a. e. x ∈ R 3 and for all E3: It is 2-Lipschitz: there exists c 3 > 0 such that for a. e. x ∈ R 3 and for all In E2-E3 the exponent 2 may be well replaced by some generic p > 1.We do not pursue this direction just for notational convenience.
Let us compare E1-E3 with the common requirements for physically admissible elastic energy densities.Setting a couple of natural conditions are W ≡ +∞ on R 3×3 \ GL + (3), which rules out interpenetrations, and lim det F →0 W (F ) = +∞, which means that it should take infinite energy to squeeze a small block of material down to a point.Feasible elastic densities are also required to be frame indifferent and to attain their minimum, conventionally set to 0, on the identity matrix I.
In our analysis, it would be particularly challenging to preserve the constraint of positive determinant for deformation gradients.Therefore, we work in a simplified setting and consider the whole space of 3×3 matrices as domain for W .Similarly, the quadratic growth from above in E2 rules out the blow up of the energy densities, but it is a rather common assumption in variational studies.Frame indifference, that is, W (x, RF ) = W (x, F ) for a. e. x ∈ R 3 , for all F ∈ GL + (3) and all rotations R, does not play any role in our analysis, thus we ignore it.Finally, up to adding a constant to W , the growth condition E2 is compatible with the requirement min W (x, • ) = W (x, I) = 0 for a. e. x ∈ R 3 .
As for the hardening functional, we assume that H : R × R 3×3 → [0, +∞] meets the ensuing requirements: Let us spend some words on H2.This rather strong hypothesis prescribes that the effective domain of H(x, • ), namely the set {F ∈ SL(3) : H(x, F ) < +∞}, is an xindependent compact set containing I.This kind of constraint has already appeared in the variational literature, see [32,Formulas (2.6a)-(2.6b)].A fundamental consequence of technical advantage is that uniform L ∞ -bounds on the plastic strains become available, provided the latter give rise to finite hardening.Indeed, if K is as in H2, then there exists c K > 0 such that because SL(3) is by definition well separated from 0. From such bound we infer that for any These bounds will be employed several times in our analysis to obtain estimates on the deformation gradient ∇y starting from a control on the elastic strain ∇yP −1 .
The further requirement that the effective domain K is geodesically convex (in the Finsler sense we mentioned, see Subsection 3.2 for the details) roots in the localization argument that we adopt to establish the Γ-convergence (see Subsection 3.1).We recall that a subset of a Finsler manifold is said to be geodesically convex if, for any couple of points in the set, there is a unique shortest path contained in the set that joins those two points.The existence of a compact set K complying with H2 is ensured by Proposition 3.3 and Remark 3.5 below.
We are now ready to state the main result of this paper, namely the homogenization formula for the functionals in (1.1).Recalling that we chose q > 3, we work in the space W 1,2 (Ω; R 3 ) × W 1,q (Ω; SL(3)) endowed with the topology τ characterized by ( The topology τ is the "right one" from a variational perspective in view of Corollary 2.3. Theorem 2.2.Let F ε be the functionals in (1.1), which we extend by setting If W and H satisfy E1-E3 and H1-H3, respectively, then for all y ∈ L 2 (Ω; R 3 ), P ∈ L q (Ω; SL(3)) the Γ-limit exists.We also have that where By standard Γ-convergence arguments, we obtain the following.

Preliminaries
We gather in this section the technical tools to be employed in the sequel.

Localization and integral representation.
To the aim of laying the ground for the proof of Theorem 2.2, we briefly outline the localization technique in the context of integral representation results for Γ-limits.More detail and a thorough treatment of Γ-convergence, which we introduced in Definition 2.1, may instead be found in the monographs [6,13,5].For definiteness, we report on a well-known result (see, e.g., [6,Theorem 14.5]) that is underpinned by the localization method.Theorem 3.1.Let g : R 3 × R 3×3 → [0, +∞) be a Borel function that is Q-periodic in its first argument and that satisfies standard p-growth conditions for some p ∈ (1, +∞).For ε > 0 and y ∈ L p (Ω; R 3 ) we define Then, we have where the Γ-limit is taken with respect to the strong L p (Ω; R 3 )-topology, and g hom : R 3×3 → [0, +∞) is a quasiconvex function characterized by the asymptotic homogenization formula As starting point to establish this theorem, one resorts to a general property of Γ-convergence which ensures that if X is a separable metric space, then any family {G ε } with G ε : X → [−∞, +∞] has a Γ-convergent subsequence (see, e.g., [6,Proposition 7.9]).Such Γ-compactness principle yields that, up to subsequences, an abstract Γ(L p )-limit of {G ε } exists.One is then naturally led to wonder whether the limit is in turn a functional of integral type.In order to show that this is in fact the case, a localization argument is employed.This amounts to regard G ε as a function of the pair (y, A), where y ∈ L p (Ω; R 3 ) and A ∈ A(Ω) := {open subsets of Ω}; more precisely, setting and G ε (y, A) := +∞ otherwise, the idea is to focus on the properties of G ε (y, • ) as a set function.In this respect, we recall some terminology.
Definition 3.2.Let α : A(Ω) → [0, +∞] be a set function.We say that α is The De Giorgi-Letta criterion (see, e.g., [6,Theorem 10.2]) states that an increasing set function is a restriction to A(Ω) of a Borel measure if and only if subadditive, superadditive and inner regular.Therefore, since the Γ(L p )-limit of {G ε } must be an increasing set function and we work under p-growth conditions, the integral representation boils down to proving that Γ(L p )-lim G ε is subadditive, superadditive and inner regular.The subadditivity and inner regularity are the most delicate points, and they hinge in turn upon the so-called fundamental estimate (see, e.g., [6,Definition 11.2]).Roughly speaking, given A, B ∈ A(Ω), y ∈ L p (Ω; R 3 ) and sequences {y ′ ε } and ), the fundamental estimate allows to construct a third sequence {y ε } converging to y in L p and such that . The exact form of the fundamental estimate to be used in our analysis is contained in (4.3), while its application to establish subadditivity and inner regularity is discussed in Proposition 4.6.

Finsler structure on SL(3).
To the purpose of devising a form of the fundamental estimate that suits the functionals under consideration, it is convenient to endow SL(3) with a metric structure.In order to link the latter to the physics of the system we would like to model, we follow the approach in [30], which is grounded on the concept of plastic dissipation.
We recall some basic facts about the geometry of SL(3), which is a smooth manifold with respect to the topology induced by the inclusion in R 3×3 .For every F ∈ SL(3) the tangent space at F is characterized as In particular, T I SL(3) coincides with sl(3) := {M ∈ R 3×3 : trM = 0}.We equip SL(3) with a Finsler structure starting from a function ∆ I : sl(3) → [0, +∞) on which we make the following assumptions (cf.[ We point out that we work under stronger regularity assumptions than the ones in [30].This is mainly due to the fact that we borrow results developed within the context of differential geometry, where smoothness is customarily required.As a consequence, some models, such as single crystal plasticity, are not covered by our analysis; on the other hand, our assumptions encompass Von Mises plasticity, see the considerations in Example 3.6 or [23,30] for a wider discussion.
We also recall that Finsler structures are known to appear as homogenized limits of periodic Riemannian metrics [1].
Essentially, ∆ I is a Minkowski norm on sl(3), which we can "translate" to the other tangent spaces by setting where TSL(3) is the tangent bundle to SL(3).Then, it can be checked that (SL(3), ∆) is a C 2 Finsler manifold.We refer to the monograph [3] by Bao, Chern & Shen for an introduction to Finsler geometry.
Let now C(F 0 , F 1 ) be the family of piecewise C 2 curves Φ : [0, 1] → SL(3) such that Φ(0) = F 0 and Φ(1) = F 1 .We define a non-symmetric distance on SL(3) as follows: where Φ is the velocity of the curve.The function D is positive, attains 0 if and only if it is evaluated on the diagonal of SL(3) × SL(3), and fulfills the triangle inequality; in general, however, D(F 0 , F 1 ) = D(F 1 , F 0 ).From a physical viewpoint, if P 0 , P 1 : Ω → SL(3) are admissible plastic strains, the integral of D(P 0 , P 1 ) over Ω is interpreted as the minimum amount of energy that is dissipated when the system moves from a plastic configuration to another.
Note that under assumptions D1-D3 it follows that ∆ is subadditive (see [3, Theorem 1.2.2]), hence convex.Therefore, by an application of the direct method of the calculus of variations (cf.[30,Theorem 5.1]) it can be proved that for every We call such Φ a shortest path between F 0 and F 1 .The following result, which summarizes the content of [3, Exercise 6.3.3], is crucial for the proof of the fundamental estimate.
Proposition 3.3.Assume that D1-D3 hold.For any point F in the Finsler manifold SL(3) there exists a relatively compact neighborhood U of F such that for any F 0 , F 1 ∈ U there exists a unique shortest path Φ joining F 0 and F 1 , and such path depends smoothly on its endpoints F 0 and F 1 .
The smooth dependence of shortest paths on their endpoints follows from the fact that each of them is a geodesic, and thus a solution of an ODE.We recall that a path between F 0 and F 1 is called a geodesic if it is a critical point of the length functional under variations that do not change the endpoints, and that, given (F, M ) ∈ T SL(3) such that ∆(F, M ) is sufficiently small, there exists a unique geodesic γ F,M : (−2, 2) → SL(3) satisfying γ F,M (0) = F and γF,M (0) = M .Therefore, we have (see [3, p. 126]) that, for M in a neighborhood of 0 ∈ T F SL(3), we can define the so-called exponential map as For each [3,Section 5.3]).In particular, there exp(F, • ) is invertible.Denoting by exp −1 F its inverse, if Φ is the unique shortest path from F 0 to F 1 (Proposition 3.3), one sees that necessarily We conclude this section by noting that, while D1-D3 characterize the Finslerian structure in an intrinsic way, D4 instead relates the Minkowski norm to the Euclidean one in the ambient space.In general, when such an assumption is made, the Finslerian distance between two points of an embedded Finslerian manifold is locally comparable to the Euclidean one.A lower bound on the Finslerian distance in terms of the Euclidean one may be retrieved by relying on the first inequality in D4 and on the fact that shortest paths in R 3×3 are segments.Since such a bound is not needed in the sequel, in the next lemma we present only the upper one.The proof is based on the implicit function theorem.For notational convenience, we state it just in the specific case of (SL(3), ∆), which is an embedded 8-dimensional submanifold of R 3×3 .Lemma 3.4.Let D1-D4 hold, and let D be as in (3.1).For every F ∈ SL(3) there exist a bounded open neighborhood U of F and a constant c > 0 such that Proof.Observe that, by definition of D and ∆ and by employing the upper bound in D4, we have for any piecewise C 2 curve joining F 0 and F 1 .In order to prove the desired estimate, we need to choose Φ suitably.Since SL(3) is a submanifold of R 3×3 , we can locally represent it as a graph defined on the affine tangent space at a given point.To be precise, for any F ∈ SL(3), let π F be the affine tangent hyperplane to SL(3) at F , nF ∈ R 3×3 be a unit normal to SL(3) at F , and B r (F ) ⊂ R 3×3 be the Euclidean ball centered at F with radius r > 0.Then, there exist a number δ > 0, a bounded, open neighborhood U of F and a smooth function ψ : Now, given F ∈ SL(3) and the associated δ, U and ψ, consider F 0 , F 1 ∈ U and the matrices M 0 , M 1 ∈ π F ∩ B δ (F ) such that is a path in U joining F 0 and F 1 .We now choose Φ = Φ in (3.3).First, we observe that, since U is bounded and Φ(t) ∈ SL(3) for all t, there exist a U -dependent constant c > 0 such that |Φ −1 (t)| ≤ c for all t (cf.(2.1)).Therefore, from (3.3) we infer where we used that ∇ψ is continuous on π F ∩ B δ (F ).To conclude, it suffices to observe that, by (3.4), Remark 3.5.Grounding on Proposition 3.3 and Lemma 3.4, we can show that there exists a compact K meeting the requirements in H2.Let U be a relatively compact neighborhood of I ∈ SL(3) such that for any F 0 , F 1 ∈ U there is a unique shortest path Φ joining F 0 and F 1 .Up to restricting to a subset, we can suppose that U is such that Lemma 3.4 holds.By a Finsler variant of a theorem by Whitehead [3, Exercise 6.4.3], there exists an open neighborhood V of I that is compactly contained in U and geodesically convex.Since K := V ⊂ U , there is a unique shortest path Φ from F 0 to F 1 for any F 0 , F 1 ∈ K.By (3.2), the path depends smoothly on its endpoints.The fact that K is geodesically convex as well follows then by the same argument that proves that the closure of a convex set is still convex.

Example 3.6 (Von Mises plasticity). Von
Mises plasticity is a model for polycrystals, such as metals, where the presence of grains averages out the different crystallographic directions.As detailed in [30, Section 4.1], in such model the Minkowski norm ∆ I is given by a scalar product on the tangent space sl(3).This is equivalent to assuming that SL(3) is equipped with a left-invariant Riemannian metric.Exploiting the properties of sl(3), ∆ I can be put into the general form (see [23,Formula (5)] or [30, Formula (4.3)]) for some positive material parameters ), for any matrix in sl(3) there exists a number r > 0 such that the closure of the geodesic ball of radius r is geodesically convex.We recall that geodesic balls are defined as the images through the exponential map of balls centered at the origin on the tangent space, provided the exponential map is there a diffeomorphism (see [19, p. 70]).In conclusion, for the case of Von Mises plasticity, the set K in H2 can be chosen as the closure of a suitable geodesic ball centered at the identity.

Homogenization in finite plasticity
We devote this section to the proof of Theorem 2.2, that is, we exhibit the Γ(τ )limit of the functionals in (1.1).We recall that the topology τ was introduced in (2.3).
Remark 4.1.The energy functionals at stake depend on the plastic strain P through its inverse P −1 .In this respect, it is useful to notice that if P k → P uniformly, then P −1 k → P −1 uniformly as well.Indeed, recalling that for any k ∈ N we can write we deduce convergence for the sequence of inverses.
Since (W 1,2 (Ω; R 3 ) × W 1,q (Ω; SL(3)), τ ) is a separable metric space, we know from general properties of Γ-convergence that, up to extraction of subsequences, the functionals in (1.1) Γ(τ )-converge.Our main task is therefore to show that the limit is an integral functional, and we achieve this by a localization approach as the one streamlined in Subsection 3.1.Precisely, denoting by A(Ω) the family of open subset of Ω, with a slight abuse of notation, for any triple (y, ) where ε k is an infinitesimal sequence.
We first show that the limits of Γ-convergent subsequences are in turn integral functionals.Second, we characterize the limiting energy densities, proving as well that the whole family {F ε } Γ-converges.

Integral representation.
In this subsection we establish the following: Theorem 4.2.Let F k be as in (4.1),where W and H satisfy E1-E3 and H1-H3, respectively.Then, up to subsequences, {F k } Γ(τ )-converges and for all y ∈ L 2 (A; R 3 ) and P ∈ L q (A; SL(3)) 2) for some f ∈ L 1 loc (R 3 ) (which depends on the subsequence).As a first step, we introduce a version of the fundamental estimate fit for the functionals in (4.1).For our purposes (cf.Proposition 4.5), it suffices to formulate it for functions in W 1,2 (A; R 3 ) × W 1,q (A; K), which is the effective domain of the functionals We say that C satisfies uniformly the fundamental estimate if for every A, A ′ , B ∈ A(Ω) with A ′ ⋐ A and for every σ > 0 there exists a constant M σ > 0 with the following property: for all F ∈ C and for every (y 1 , P 1 ), (y 2 , P 2 ) ∈ W 1,2 (Ω; R 3 ) × W 1,q (Ω; K) there exist a cut-off function ϕ between A ′ and A and a path γ : 3) The use of the path γ in (4.3) is motivated by the simple observation that the "convex combination" ϕP 1 + (1 − ϕ)P 2 does not belong to SL(3) in general.For example, if we let we see that det P 1 = det P 2 = 1, but det(P 1 /2 + P 2 /2) = 0.However, not any path γ is appropriate for our purposes, because in general it may not be completely contained in K, that is, in the domain of H.Moreover, the regularization term ∇P calls for some regularity of γ with respect to its endpoints (see (4.5)) and for a specific bound on the velocity γ (see ( Proof.Fix A, A ′ , B ∈ A(Ω) with A ′ ⋐ A and σ > 0. Fix also (y 1 , P 1 ), (y 2 , P 2 ) ∈ W 1,2 (Ω; R 3 ) × W 1,q (Ω; K).We need to choose suitably the cut-off function ϕ.
Let δ := dist(A ′ , ∂A) > 0. For a fixed N ∈ N, N > 1, we define Observe that for all j we can construct a cut-off function ϕ j between C j−1 and C j such that |∇ϕ j | ≤ 2N/δ.Now, setting γ −1 (t, F, G) := (γ(t, F, G)) −1 for all t ∈ [0, 1], where Since P 1 , P 2 ∈ K, (2.1) holds.We now choose γ as the map such that γ(t, F, G) is the evaluation at t of the unique shortest path connecting F and G for all (t, F, G) ∈ [0, 1]×K ×K.By Proposition 3.3 we know that such γ exists, that it lies completely in K if F, G ∈ K (see H2), and that it depends smoothly on its arguments.Let us estimate the three summands above separately.For the elastic contribution we have where we used the fact that ( , the growth condition E2 and the uniform bound (2.1).Analogously, for the second summand we find where we exploited again the geodesic convexity of K and the fact that H is bounded on K, since it is Lipschitz.
We now estimate I reg .The chain rule yields whence, observing that by Proposition 3.3 the two differentials ∂ F γ and ∂ G γ are continuous functions restricted to compact sets, We now resort to the explicit expression of γ in terms of the exponential map, see (3.2).From (2.1) and recalling that by definition exp −1 , where ∆ I is the Minkowski norm on p. 8 and D is defined in (3.1).Therefore, recalling D4 and up to redefining the constant c, we obtain where the last inequality follows from Lemma 3.4.We hence conclude so that By gathering (4.4)-(4.7)we obtain thus there certainly exists ℓ ∈ 1, . . ., N such that where in the last inequality we exploited the fact that the set (A\A ′ )∩B is contained in both A and B, together with the growth condition from below in E2.Therefore, we obtain Finally, choosing N such that and letting we see (4.3) is satisfied, and the proof is complete.
Proof.We need to prove the inequalities only when F ′ (y, P, A), F ′′ (y, P, A) and F ′′ (y, P, B) are finite.By the definitions in (4.8), there exist sequences Recalling (3.2), we find k , P ′ k ) → P uniformly.Taking the lower limit in (4.9) we obtain and the first inequality in the statement follows by letting σ → 0. The second one is obtained by taking the upper limit in (4.9) and arguing in a similar way.
Proof.We prove the two statements separately.
(1) Let us focus on F ′ .According to Definition 3.2 we need to prove that F ′ (y, P, A) = sup F ′ (y, P, B) : Since F ′ (y, P, • ) is an increasing set function, it is trivial that F ′ (y, P, A) is larger than the supremum in the formula above.As for the reverse inequality, let us fix A ∈ A(Ω) and let C ⊂ A be compact.There exist We now apply Proposition 4.5: By the arbitrariness of C, we can let C invade A and we obtain The same argument applies to F ′′ . ( Now, by the monotonicity of F ′′ (y, P, • ) and Proposition 4.5, By taking the supremum over C, inner regularity yields F ′′ (y, P, A ∪ B) ≤ F ′′ (y, P, A) + F ′′ (y, P, B).
By gathering the previous results, we infer the existence of a Γ-limit admitting an integral representation.
Proof of Theorem 4.2.Let {ε k } k∈N be such that ε k → 0 + , and let F ′ and F ′′ be as in (4.8).It follows from the definitions that they are increasing set functions, and Proposition 4.6 yields inner regularity for both and subadditivity for F ′′ .By standard arguments (see, e.g., [6,Theorem 10.3]), we deduce that, upon extraction of a subsequences, F(y, P, A) := Γ(τ )-lim k→+∞ F k (y, P, A) exists for all triple (y, P, A).From the definition of Γ-limit and from the fact that F k (y, P, • ) is a measure for each k, it follows that F(y, P, • ) is superadditive.We are then in a position to apply the De Giorgi-Letta criterion (see Subsection 3.1), which ensures that F(y, P, • ) is the restriction of a Borel measure to A(Ω) for (y, P ) ∈ W 1,2 (A; R 3 ) × W 1,q (A; K).Thanks to (4.10) we infer also that it is absolutely continuous with respect to the Lebesgue measure.Thus, by Radon-Nikodym theorem there exists f ∈ L 1 loc (R 3 ) such that F(y, P, A) = ˆA f (x) dx.

Characterization of the limiting energy density.
So far, we have proved a Γ(τ )-compactness result for the functionals (4.1), and we have shown that the limit is actually an integral functional.We now provide an explicit formula for the limiting energy density.
Proof of Theorem 2.2.Let y ∈ W 1,2 (Ω; R 3 ), P ∈ W 1,q (Ω; K).Our aim is to identify f in (4.2) as f (x) = W hom (∇y(x), P (x)) + H hom (P (x)) + |∇P (x)| q for a. e. x ∈ Ω. (4.11)This also entails (see [6,Proposition 7.11]) that the whole family {F ε } Γ(τ )-converges to F as in the statement of the theorem For all From E1 and E2 we deduce that (1) W is a Borel function and G) is 2-coercive and has at most quadratic growth, uniformly in x and G: there exist 0 < c1 ≤ c2 such that for a. e. x ∈ R 3 , for all F ∈ R 3×3 and for all Note that the independence of c1 and c2 from G is a consequence of (2.2) and (2.1), respectively.
For A ∈ A(Ω), we now introduce the functional where G ∈ K is fixed, and we observe that From Theorem 4.2 we know that We next show that (4.11) holds by exploiting these two convergence results.From now on, let x 0 ∈ Ω be a Lebesgue point of the functions f , ∇y and ∇P .Notice that almost every x 0 ∈ Ω has such property.Let A ∈ A(Ω) contain x 0 , and let {(y k , P k )} ⊂ W 1,2 (A; R 3 ) × W 1,q (A; K) be a generic sequence.We use E3 and (2.1) to estimate the difference between the elastic contribution in F k (y k , P k , A) and  We now prove that f (x) ≥ W hom ∇y(x), P (x) + H hom (P (x)) + |∇P (x)| q for a. e. x ∈ Ω. (4.14)To this aim, we select a recovery sequence {(y k , P k )} ⊂ W 1,2 (A; R 3 ) × W 1,q (A; K) for F(y, P, A), namely Owing to the growth assumptions on the energy densities, we can without loss of generality suppose that y k ⇀ y weakly in W 1,2 (A; R 3 ), P k ⇀ P weakly in W 1,q (A; K).Observe that the lower bound on F(y, P, A) that we have just proved holds for any set A containing x 0 .In particular, we select A = B r (x 0 ) and divide both sides of the corresponding estimates by L 3 (B r (x 0 )).By letting r → 0, we apply Lebesgue differentiation theorem to deduce (4.14) for x = x 0 .Indeed, To establish the estimate, we let {y k } ⊂ W 1,2 (A; R 3 ) be a recovery sequence for G(y, P 0 , A), that is, lim k→+∞ G k (y k , P 0 , A) = G(y, P 0 , A), with y k ⇀ y weakly in W 1,2 (A; R 3 ).We consider estimate (4.13) for this particular {y k } and the constant sequence given by P k = P for all k.Exploiting this time the 2-coercivity of G k we obtain Arguing as above with A = B r (x 0 ) and exploiting the 2-growth conditions of W hom , we conclude that (4.16) holds for x = x 0 .
Since we can select almost every point in Ω as x 0 , the conclusion follows from (4.14) and (4.16).

4.3.
Convergence of minimum problems.We conclude this section with the proof of equicoercivity and convergence of (almost) minimizers, that is, Corollary 2.3.

Proof of Corollary 2.3. (i)
Let {(y k , P k )} be a bounded energy sequence such that y k L 2 ≤ c, uniformly in k.From the definition of F k , for all k ∈ N ∇P k L q ≤ c. (4.17) Besides, for all k, assumption H2 and the uniform bound (2.1) yield

.18)
As for the elastic part, assumption E2 entails
1 and β 2 .Here | • | denotes the matrix Frobenius norm, that is, the (element-wise) Euclidean norm | • | of R 3×3 .Note that ∆ I 4.6)).To tackle such issues, in Proposition 4.4 we select γ suitably.The next statement grants that the class C = {F k } meets the definition above.
Let us fix σ > 0. The fundamental estimate (4.3) gives a constant M σ , a sequence {ϕ k } k of cut-off functions between A ′ and A, and a sequence {γ k } k of shortest paths from P ′′ k to P ′ k such that , which, together with (2.2) and (4.18), gives∇y k L 2 ≤ c. (4.19)