Variational linearization of pure traction problems in incompressible elasticity

We consider pure traction problems, and we show that incompressible linearized elasticity can be obtained as variational limit of incompressible finite elasticity under suitable conditions on external loads.


Introduction
Let us consider a hyperelastic body occupying a bounded open set Ω ⊂ R 3 in its reference configuration. Equilibrium states under a body force field f : Ω → R 3 and a surface force field g : ∂Ω → R 3 are obtained by minimizing the total energy Here, y : Ω → R 3 denotes the deformation field, H 2 denotes the surface measure, and W I : Ω × R 3×3 → [0, +∞] is the incompressible strain energy density. We require incompressibility by letting W I (x, F) = +∞ whenever det F = 1. Moreover, we assume that W I (x, ·) is a frame indifferent function that is minimized at F = I with value 0.
If h > 0 is an adimensional small parameter, we rescale the displacement field and the external forces by letting f = hf , g = hg and y(x) − x = hv(x). We get We aim at obtaining the behavior of rescaled energies as h → 0 and at showing that the linearized elasticity functional arises in the limit. More precisely, we aim at proving that if v h is a sequence of almost minimizers for E I h ) then Here, E(v) := 1 2 (∇v + ∇v T ) is the infinitesimal strain tensor field and Q I (x, ·) is defined for every F ∈ R 3×3 by where W(x, F) := W I (x, (det F) −1/3 F) is the isochoric part of W I . Such a quadratic form is obtained by a formal Taylor expansion around the identity matrix (D 2 denoting the Hessian in the second variable). Since Q I (x, F) = +∞ if Tr F = 0, we see that E I (v) is finite only if div v = 0 a.e. in Ω. Therefore, E I is the linearized elastic energy with elasticity tensor D 2 W(x, I) and div v = 0 is the linearized incompressibility constraint.
Under Dirichlet boundary conditions, (1.1)-(1.2) have been obtained in [25], by means of a Γ-convergence analysis with respect to the weak topology of W 1,p (Ω, R 3 ), where the exponent p is suitably related to the coercivity properties of W I (see Section 2). On the other hand, in this paper we shall consider natural Neumann boundary conditions, i.e., the pure traction problem. In this case, it is crucial to impose suitable restrictions on the external forces. In particular, as done in [23,24], here we shall assume they have null resultant and null momentum with respect to the origin, namely and that they satisfy the following strict compatibility condition where R 3×3 skew denotes the set of real 3 × 3 skew-symmetric matrices. For suitable classes of external forces, the latter condition can be interpreted as an overall dilation effect on the body, see Remark 2.7 later on. Even under such restrictions, in this case it is not possible to obtain a sequential Γ-convergence result with respect to the weak convergence in W 1,p (Ω, R 3 ) or to the weak L p (Ω, R 3×3 ) convergence of infinitesimal strain tensors (we stress that the elastic part in E I h is not invariant by infinitesimal rigid displacements, see also Remark 2.6). However, in this context we will show that E I provides indeed an upper bound for the sequence h −2 E I h (v h ) in the limit h → 0, as soon as E(v h ) ⇀ E(v) weakly in L p (Ω, R 3×3 ) and det(I + h∇v h ) = 1. On the other hand, due to the lack of control on skewsymmetric parts, we may only obtain a lower bound in terms of the functional Clearly, we have F I ≤ E I . Interestingly, this still allows to obtain (1.1)-(1.2), since it is possible to show (see Lemma 7.1 later on) that if v ∈ argmin F I , then F I (v) = E I (v). Indeed, if v ∈ argmin F I , then the minimization problem inside the definition of F I is solved by W = 0.
A key step for obtaining to proof of (1.1)-(1.2) will be to approximate divergence-free vector fields in terms of vector fields v t having the property W I (x, I + t∇v t ) < +∞, i.e., det(I + t∇v t ) = 1 for any t > 0. Following the approach of [25], we define v t (x) : where y t is the flow associated to the divergence-free field v, starting from y 0 (x) = x ∈ Ω, and then by Reynolds transport formula we see that y t is a volume preserving deformation field. Indeed, for any A ⊆ Ω we have so that det ∇y t = det(I + t∇v t ) = 1. In this paper, we will further develop this approach in order to obtain the desired recovery sequence and the upper bound under the above conditions on external loads. Moreover, since we have natural boundary conditions we avoid the technical difficulties due to the need of keeping track of the boundary data through the construction of the recovery sequence. Therefore, we shall not need the strong regularity assumptions on ∂Ω that were imposed in [25].
Hyperlastic incompressible models are typically used to describe rubber-like materials such as artificial elastomers as well as biological soft tissues [9,16,17,19,28,30,36,40]. We refer to [5,6,7,26,39,41] for many examples of strain energy densities that are used for the nonlinear descprition of the stress-strain behavior of these materials. On the other hand, linear modeling is usually considered a good approximation in the small strain regime. Indeed, the classical theory of linearized elasticity is based on the smallness assumption on deformation gradients, see for instance [14,22,38]. Nevertheless, for a variational derivation, i.e., for the proof of (1.1)-(1.2), no a-priori smallness assumption is needed, leading to a rigorous justification of linearized elasticity (we also stress that small loads need not give rise to small strains in rubber-like materials, due to their high compliance in shear). The first rigorous variational derivation of linearized elasticity from finite elasticity is given in [11], where Γ-convergence and convergence of minimizers of the associated Dirichlet boundary value problems are proven in the compressible case. We refer to [1,2,3,10,18,23,24,25,37] for many other results of this kind, some of which including theories for incompressible materials [10,18,25]. The study of asymptotic properties of minimal energies, similar to (1.1)-(1.2), is also typical of dimension reduction problems, see for instance [4,21,31,32,33,34,35].
In the next section we rigorously state the main theorem, providing the proof of (1.1)-(1.2) for the pure traction problem. A related result has been recently obtained by Jesenko and Schmidt in [18] under different assumptions on the external loads, but in the more general framework of multiwell potentials that leads to a suitable quasiconvex envelope of the strain energy density in the limiting functional.
Plan of the paper. In Section 2 we collect the the assumptions of the theory and we state the main result about convergence of minimizers. The latter is based on suitable compactness properties of (almost) minimizing sequences that are established in section 4, after some preliminaries in Section 3. In section 5, we provide the lower bound. Section 6 delivers the upper bound. Section 7 completes the proof of the main result.

Main result
In this section we introduce all the assumptions and we state the main result. Let Ω ⊂ R 3 be the reference configuration of the body. We assume that for some m ∈ N Ω is a bounded open connected Lipschitz set, ∂Ω has m connected components.
Assumptions on the elastic energy density. We let W I : Ω × R 3×3 → [0, +∞] be L 3 ×B 9measurable. The assumptions on W I are similar to the ones in [2,25], i.e., for a.e. x ∈ Ω (W0) and we assume that W(x, F) is C 2 in a neighbor of rotations, i.e., (W3) there exists a neighborhood U of SO(3) s.t., for a.e. x ∈ Ω, W(x, ·) ∈ C 2 (U ), with a modulus of continuity of D 2 W(x, ·) that does not depend on x. Moreover, there exists K > 0 such that |D 2 W(x, I)| ≤ K for a.e. x ∈ Ω.
Assumptions on the external forces. We introduce a body force field f ∈ L 3p 4p−3 (Ω, R 3 ) and a surface force field g ∈ L 2p 3p−3 (∂Ω, R 3 ), where p is such that (W4) holds. The corresponding contribution to the energy is given by the linear functional We note that since Ω is a bounded Lipschitz domain, the Sobolev embedding W 1,p (Ω, We assume that external loads have null resultant and null momentum, i.e., and that they satisfy the following strict compatibility condition Some examples of external loads satisfying the above assumptions are provided in the remarks at the end of this section. Statement of the main result. The functional representing the scaled total energy is denoted by F I h : W 1,p (Ω, R 3 ) → R ∪ {+∞} and defined as follows We further introduce the functional of linearized incompressible elasticity E I : where if Tr B = 0 +∞ otherwise, and the limit energy functional F I : is finite only if v has constant nonpositive divergence (since W 2 is negative semi-definite for any W ∈ R 3×3 skew ). We are ready for the statement of the main result then there is a (not relabeled) subsequence such that where v * ∈ H 1 (Ω, R 3 ) is a minimizer of F I over W 1,p (Ω, R 3 ), and We close this section with several remarks about the main theorem.
Remark 2.2. It is worth noticing that the infimum in the right hand side of (2.4) is actually a minimum. Indeed if v ∈ H 1 (Ω, R 3 ) then either F I (v) = +∞ or div v is a non positive constant. In the latter case let (W n ) n∈N ⊂ R 3×3 skew be a minimizing sequence: then |W n | 2 = Tr(W T n W n ) = −TrW 2 n = −2 div v hence there exists W v such that, up to subsequences, Remark 2.3. If the function W is replaced by any other W, still satisfying (W3), such that W(x, F) = W(x, F) if det F = 1, then this does not affect functionals F I and E I . Indeed, if Tr B = 0 then det(exp(hB)) = exp(hTrB) = 1 and by Taylor's expansion we have for a.e.
Remark 2.4. A typical form of W I is the Ogden incompressible strain energy density, see [8,16,29], given by where N, µ k , α k are material constants, and extended to +∞ if det F = 1. If the material constants vary in a suitable range, the Ogden model satisfies the assumptions (W1), (W2), (W3), (W4). In particular, we refer to [2] and [25] for a discussion about the growth properties and the validity of (W4) for the Ogden strain energy density and other standard models.
Remark 2.5. It is worth to stress that Theorem 2.1 does not hold without suitable compatibility assumptions on external forces. Not even relaxing (L2) by requiring non strict inequality therein would work. Indeed, choose f = g ≡ 0, However, it has no subsequence that is weakly converging in W 1,p (Ω, R 3 ). Moreover, E(v j ) = c h −1 j W 2 for some c ∈ (0, 1] and some W ∈ R 3×3 skew such that |W| 2 = 2, as a consequence of the Euler-Rodrigues formula 3.2 for the representation of rotations that we shall recall in Section 3. Therefore, the sequence (E(v j )) j∈N has no subsequence that is bounded in L p (Ω, R 3×3 ).
Remark 2.6. Under the assumptions of Theorem 2.1, in general it is not possible to get weak W 1,p (Ω, R 3 ) compactness of (almost) minimizing sequences. Indeed, let us consider the following example. Let Ω = B 1 be the unit ball of R 3 , centered at the origin. Let W I be given by (2.7). Let f (x) = x and g ≡ 0. It is readily seen that (L1) and (L2) are satisfied. On the other hand, let the divergence-free vector field v * ∈ H 1 (Ω, R 3 ) be a minimizer of E I over W 1,p (Ω, R 3 ). Since Ω = B 1 and v * is divergence-free, there exists w * ∈ H 2 (Ω, R 3 ) such that v * = curl w * and by divergence theorem and since E I (0) = 0, we get E I (v * ) = 0. By Theorem 2.1, we deduce that 0 is the minimal value of both E I and F I over W 1,p (Ω, R 3 ). Let now W ∈ R 3×3 skew be such that |W| 2 = 2. Let α ∈ ( 1 2 , 1) and let us consider a vanishing sequence (h j ) j∈N ⊂ (0, 1) and the sequence The Euler-Rodrigues formula (3.2) implies that for any j ∈ N there holds y j ( . This implies det ∇y j = 1 for any j ∈ N and then by (2.7) we get The above right hand side goes to 0 as j → +∞, as shown after a computation making use of (2.8). Therefore the sequence (v j ) j∈N satisfies (2.6). On the other hand, (∇v j ) j∈N has no subsequence that is bounded in and by taking a vanishing sequence (h j ) j∈N ⊂ (0, 1) and v j : For general Ω, we notice that if the body is subject to a uniform boundary compressive force field then the above situation occurs. Indeed, if n denotes the outer unit normal vector to ∂Ω, and we choose g = λn with λ < 0 and f ≡ 0, then as j → +∞, as before. On the other hand, if λ > 0 we have a dilation effect on the body and (L2) is satisfied.
Remark 2.8. Let us consider external forces of the following form. Given p such that (W4) holds, let f = ∇φ, where φ ∈ W 1,r 0 (Ω), r = 3p 4p−3 , and let g = λn, where λ ∈ R and n is the unit exterior normal vector to ∂Ω, with Ω φ(x) dx < λ|Ω|. It is readily seen that in this case (L1) and (L2) are satisfied. Moreover, by the divergence theorem, L(v) = 0 for every divergencefree vector field v ∈ H 1 (Ω, R 3 ). Therefore, under the assumptions of Theorem 2.1, from the definition of E I and from the estimate (3.5) below we deduce that argmin W 1,p (Ω,R 3 ) E I coincides with the set of rigid displacements of Ω (i.e., displacements fields with vanishing infinitesimal strain tensor). From Theorem 2.1 we deduce in this case that the minimal value of both E I and F I is 0.

Notation and preliminary results
Through the paper, R 3×3 will denote the set of 3 × 3 real matrices. R 3×3 sym and R 3×3 skew denote respectively the sets of symmetric and skew-symmetric matrices and for every B ∈ R 3×3 we define sym B := 1 2 (B + B T ) and skew B : The codomain of functions of L r (Ω) or W 1,r (Ω) shall be R, R 3 or R 3×3 and we shall often omit it from the notation. Bold letters will be used for vector fields.
Euler-Rodrigues formula. For every R ∈ SO(3) there exist ϑ ∈ R and W ∈ R 3×3 skew , such that |W| 2 = 2 and such that exp(ϑ W) = R. By taking into account that W 3 = −W, the exponential matrix series exp(ϑ W) = ∞ k=0 ϑ k W k /k! yields the Euler-Rodrigues formula: We also recall that if W ∈ R 3×3 skew and |W| 2 = 2 then |W 2 | 2 = 2. Properties of W. Let assumptions (W0), (W1), (W2), (W3) and (W4) hold. We recall that W is defined by W(x, F) := W I (x, (det F) −1/3 F), thus W I ≥ W. It is clear that W itself satifies (W1) and (W2), so that by (W3), since W ≥ 0, we deduce Due to frame indifference there exists a function V such that hence (W4) and (2.3) imply that for a.e. x ∈ Ω, as soon as Tr B = 0, Moreover, by expressing the remainder of Taylor's expansion in terms of the x-independent modulus of continuity ω of D 2 W(x, ·) on the set U from (W3), we have Similarly, V(x, ·) is C 2 in a neighbor of the origin in R 3×3 , with an x-independent modulus of continuity η : R + → R, which is increasing and such that lim t→0 + η(t) = 0, and we have for any small enough h.
Sobolev-Poincaré inequality. Here and for the rest of this section, Ω is a bounded connected set with Lipschitz boundary. Let p ∈ (1, 2]. By Sobolev embedding, Sobolev trace embedding and by the Poincaré inequality for null-mean functions we have for any where K F is a constant only depending on Ω, p andv : Projection on rigid motions. Let p ∈ (1, 2] and let denote the space spanned by the set of the infinitesimal rigid displacements. We denote by Pv the unique projection of v ∈ W 1,p (Ω, R 3 ) onto R.
and Korn inequality, see for instance [27], entails the existence of a constant Q K = Q K (Ω, p) such that Moreover, by combining the latter with Sobolev and trace inequalities, we obtain the existence of a further constant Basic estimate on external forces. As a consequence of (3.10), if (L1) holds true we obtain the following estimate for functional L: for any v ∈ W 1,p (Ω, R 3 ), there holds and C K is the constant in (3.10).
Rigidity inequality. We recall the rigidity inequality by Friesecke, James and Müller [12], in its version from [13], [2]. Let g p the function defined in (2.2). There exists a constant C p = C p (Ω) > 0 such that for every y ∈ W 1,p (Ω, R 3 ) there exists a constant R ∈ SO(3) such that we have We close this section with a result about convergence of infinitesimal strain tensors.
If in addition we assume that ∇w n ⇀ G weakly in L p (Ω; R 3×3 ), then there exists a constant matrix W ∈ R 3×3 skew such that ∇w = G − W. Proof. The proof is given in [24,Lemma 3.2] for the case p = 2. Its extension to p ∈ (1, 2) is straightforward.

Compactness
We prove uniform L p (Ω, We start by showing that functionals F I h are uniformly bounded from below. There exists a constant C > 0 (only depending on Ω, p, f , g) such that F I h (v) ≥ −C for any h ∈ (0, 1) and any v ∈ W 1,p (Ω, R 3 ). Proof. Let v ∈ W 1,p (Ω, R 3 ) and let h ∈ (0, 1). Let y := i + hv and let R ∈ SO(3) be a constant matrix such that (3.12) holds. Let S := {x ∈ Ω : |∇y(x) − R| ≤ 1}. By taking advantage of assumption W4 and of the linearity of L, since g p (t) = t 2 for 0 ≤ t ≤ 1 and g p (t) ≥ t p for t ≥ 1, we get where c > 0 is a constant only depending on p and Ω. By the Sobolev-Poincaré inequality (3.8), letting u(x) := Rx and letting m denote the mean value of y − u on Ω, we have By the Euler-Rodrigues formula (3.2) we represent R as R = I + sin θW + (1 − cos θ)W 2 for a suitable skew-symmetric matrix W and some θ ∈ (−π, π]. Then, we notice that (L1) and (L2) entail L(u − i) ≤ 0 so that L(y − i) ≤ L(y − u − m). As a consequence, by means of the Hölder inequality and of (4.2) we get , and then by Young inequality we obtain where c is the constant appearing in (4.1). By joining together (4.1) and (4.3) we get Proof. The argument extends the one of [24, Lemma 3.6] to the weaker coercivity condition (W4). For any j ∈ N, by Lemma 4.1 there holds therefore by considering (4.4) it is not restrictive to assume that F I h j (v j ) ≤ 1 for any j ∈ N. We assume by contradiction that the sequence (t j ) j∈N , defined as t j := E(v j ) L p (Ω) , is unbounded, so that up to extraction of a not relabeled subsequence we have t j → +∞ as j → +∞ and moreover t j h j converge to a limit as j → +∞. We let w j = v j /t j , so that E(w j ) L p (Ω) = 1 for any j ∈ N, and along a not relabeled subsequence we have E(w j ) ⇀ E(w) weakly in L p (Ω) for some w ∈ W 1,p (Ω), thanks to Lemma 3.1. By defining y j := i + h j v j , we let R j be the corresponding constant rotation matrix such that (3.12) holds, so that by By inserting (3.11) that is, We claim that ∇w is the sum of a skew-symmetric matrix and an element of K, where K is defined by (3.1), and that E(w j ) → E(w) in L p (Ω), up to extraction of a further not relabeled subsequence. We shall prove the claim by separately treating the following three possible cases: h j t j → λ ∈ (0, +∞), h j t j → 0 and h j t j → +∞ as j → +∞.
Case 1: h j t j → λ as j → +∞ for some λ > 0. It is easy to check that for any x ≥ 0 and any a ≥ 0, so that (4.6) implies We define Since g p (t) = t 2 for 0 ≤ t ≤ 1 and g p (t) ≥ t p for t ≥ 1, taking advantage of (4.8) we get go to zero in L p (Ω) as j → +∞, since p ∈ (1, 2]. As a consequence, we obtain the convergence to zero of h −1 j t −1 j (I − R j ) + ∇w j in L p (Ω) as j → +∞. Therefore, ∇w j converge in L p (Ω), up to subsequences, to λ −1 (R − I) for some suitable R ∈ SO(3) (thus E(w j ) converge in L p (Ω) to E(w)) and Lemma 3.1 implies that ∇w is the sum of a skew-symmetric matrix and an element of K.
Case 2: h j t j → 0 as j → +∞. We assume wlog that t j h j ≤ 1 for any j ∈ N. Writing R j by means of the Euler-Rodrigues formula (3.2), from (4.6) we get where, for any j ∈ N, θ j ∈ (−π, π] and W j ∈ R 3×3 is skew-symmetric. Since |symF| ≤ |F| and g p is increasing, we deduce Therefore, (4.7) implies (since h j t j ≤ 1) By taking advantage of the latter estimate, since g p is increasing and satisfies g p (x) ≤ 2x p for any x ≥ 0, we get Since t j → +∞ as j → +∞, we obtain the existence of a positive constant C * (not depending on j) such that In particular, up to subsequences, we have for some suitable constant skew-symmetric matrix G, so that from (4.9) we deduce, since g p is continuous and increasing, By the same argument of Case 1, we conclude that E(w j ) → G 2 in L p (Ω) as j → +∞, hence E(w) = G 2 . We deduce that the skew-symmetric part of ∇w is a gradient field, hence a constant skew-symmetric matrix Λ, and that ∇w = Λ + 1 2 G 2 . By applying the Euler-Rodrigues formula (3.2), we deduce the existence of µ > 0, of R ∈ SO(3) and of Q ∈ R 3×3 skew such that ∇w = µ(R − I) + Q, so that indeed ∇w is the sum of an element of K and a skew-symmetric matrix.
Case 3: h j t j → +∞ as j → +∞. We may assume in this case that h j t j ≥ 1 for any j ∈ N. By applying (4.6) and (4.7) we get where the right hand side vanishes as j → +∞, and where I−R j h j t j vanishes as well, since R j − I is bounded. By the same argument of Case 1, we conclude that ∇w j → 0 in L p (Ω) as j → +∞, thus E(w j ) → E(w) = 0 in L p (Ω). By Lemma 3.1 we deduce that ∇w is a constant skewsymmetric matrix. This ends the last of the three cases and proves the claim.

Lower bound
In this section we prove the lower bound lim inf j→+∞ F I h j (v j ) ≥ F I (v) as E(v j ) ⇀ E(v) weakly in L p (Ω, R 3×3 ) and h j → 0. We start with two preliminary lemmas.
Then there exists a constant matrix W ∈ R 3×3 skew such that, up to subsequences, Proof. Through the proof, C will always denote a generic positive constant only depending on p, Ω, f and g. By taking into account (L1), (W4), and by applying (3.12) to y j := x + h j v j , we see that for any j ∈ N there exists a constant matrix R j ∈ SO(3) such that By (3.11) and by the boundedness of E(v j ) in L p (Ω, R 3×3 ) we get Due to the representation (3.2) of rotations, for every j ∈ N there exist ϑ j ∈ (−π, π] and W j ∈ R 3×3 skew , with |W j | 2 = 2, such that R j = exp(ϑ j W j ) = I + sin ϑ j W j + (1 − cos ϑ j ) W 2 j .

By setting
and again by (2.2) we also obtain
and where W ∈ R 3×3 skew is given by Lemma 5.1.
Proof. Again, through the proof we denote by C the various positive constants, possibly depending only on Ω, p, f , g. Let θ j and W j be as in the proof of Lemma 5.1, so that (5.5) holds. We claim that 1 B j E(v j ) is bounded in L 2 (Ω, R 3×3 ). To this aim it is useful to notice that by (2.2) for every δ > 0 there exists c = c(p, δ) > 0 such that g p (t) ≥ ct 2 for every t ∈ [0, δ]. Therefore by taking into account that for j large enough we may fix δ = 2|W| + 1 and obtain for any j large enough and by arguing as in the proof of Lemma 5.1, see (5.2) and (5.5), we get as claimed. On the other hand for every q ∈ (1, p) we have since |B c j | → 0 by Chebyshev inequality and by Lemma 5.1. By taking into account that E(v j ) = 1 B c j E(v j ) + 1 B j E(v j ) and by assuming wlog that 1 B j E(v j ) ⇀ u weakly in L 2 (Ω, R 3×3 ) we get E(v j ) ⇀ u weakly in L q (Ω, R 3×3 ) and recalling that E(v j ) ⇀ E(v) weakly in L p (Ω, R 3×3 ) we get u = E(v) ∈ L 2 (Ω, R 3×3 ) thus proving that v ∈ H 1 (Ω, R 3 ) by Korn inequality.
Proof. We may assume wlog that F I h j (v j ) ≤ C for any j ∈ N so that by setting D j := E(v j ) + 1 2 h j ∇v T j ∇v j we get in Ω, that is, By taking into account Lemma (5.1) we get √ h j ∇v j → W in L p hence, up to subsequences, h j ∇v T j ∇v j → −W 2 a.e. in Ω and 2h j (T r(D 2 j ) − (T rD j ) 2 ) − 8h 2 j det D j → 0 a.e. in Ω. Therefore 2 div v j → Tr W 2 a.e. in Ω and since the weak convergence of E(v j ) implies div v j ⇀ div v weakly in L p (Ω) we get 2 div v = Tr W 2 a.e. in Ω. On the other hand by Lemma 5.2, with B j defined by (5.7), we have Hence, by (3.7), (3.4) and (L1) we get for large enough j for large enough j, as η is increasing. Since h j ∇v T j ∇v j → −W 2 a.e. in Ω and |B c j | → 0 as j → +∞, and since Hence, by (W3), (5.9) and by the weak L 2 (Ω, R 3×3 ) lower semicontinuity of the map F → Ω F T D 2 W(x, I) F dx, we deduce In order to obtain (5.10), we have also used the fact that L(v j − Pv j ) → L(v) as j → +∞. Indeed, by (3.9) and (3.10) we obtain boundedness of v j − Pv j in W 1,p (Ω, R 3 ) and in L 3p 3−p (Ω, R 3 ). Therefore up to subsequences we get v j − Pv j ⇀ w weakly in L 3p 3−p (Ω, R 3 ) for some w ∈ W 1,p (Ω, R 3 ), and by trace embedding v j − Pv j ⇀ w weakly in L Eventually, since 2 div v = Tr W 2 a.e. in Ω, the result follows from (5.10).

Upper bound
The following result is an extension of [25,Lemma 4.1].
is an open set such that Ω ⊂ Ω ′ . Let (h j ) j∈N ⊂ (0, 1) be a vanishing sequence. There exists a sequence of vector fields (v j ) j∈N ⊂ W 2,∞ (Ω, R 3 ) and j * ∈ N such that for any j > j * (6.1) det(I + h j ∇v j ) = 1, Proof. We choose T ∈ (0, 1) small enough, such that y(t, x) ∈ Ω ′ for any x ∈ Ω and any t ∈ [0, T ], where y(·, x) is the unique solution to so that y is the flow associated to the vector field v. We have y ∈ C 1 ([0, T ]; W 2,∞ (Ω)), see [15,Corollary 5.2.8,Remark 5.2.9]. From (6.5), we have for any x ∈ Ω. We get therefore the basic estimate for any x ∈ Ω, and Gronwall lemma entails so that we have We for every t ∈ (0, T ] and for every x ∈ Ω. . From the definition of v t , from (6.6) and (6.7) we get for any x ∈ Ω and any t ∈ (0, T ]. From the latter we get in particular the convergence of v t to v in L 1 ∩ L ∞ (Ω) as t → 0.
Since the map Ω ∋ x → v(y(t, x)) is Lipschitz continuous, uniformly with respect to t ∈ (0, T ), we may take the gradient under integral sign in (6.6) and obtain (6.12) y(s, x))∇y(s, x) − ∇v(x)∇y(s, x)) ds for every x ∈ Ω and every t ∈ (0, T ]. Form the first equality of (6.12) and from (6.9) we get for any t ∈ (0, T ]. Moreover, by (6.12), (6.8), (6.13) and (6.9) we have for any x ∈ Ω and any t ∈ (0, T ]. Eventually, let us consider a vanishing sequence (h j ) j∈N ⊂ (0, 1). By defining j * as the smallest positive integer such that h j < T for any j > j * and by defining v j := v h j , the result follows from (6.10), (6.11), (6.13) and from the latter estimate.
Given a vanishing sequence (h j ) j∈N of positive numbers and given n ∈ N we may define (v εn j ) j∈N to be the sequence from Lemma 6.1, constructed from the W 2,∞ (Ω ′ , R 3 ) divergencefree vector field v εn : indeed, by applying Lemma 6.1 to v εn , we construct a sequence of W 2,∞ (Ω) vector fields (v εn j ) j∈N and a strictly increasing diverging function g : N → N such that (6.1), (6.2), (6.3), (6.4) are satisfied for any positive integers j, n with j > g(n), i.e., (6.14) det(I + h j ∇v εn j ) = 1, Therefore, by defining the diverging sequence (n(j)) j∈N ⊂ [0, +∞) as we obtain that (6.14), (6.15), (6.16), (6.17) are satisfied, with ε n(j) in place of ε n , for any j ∈ N, since n(j) < g −1 (j). We let therefore v j := v ε n(j) j and conclude by checking that the sequence (v j ) j∈N satisfies the desired properties. Property i) is already given by (6.14). Moreover, by the elementary estimates so that h j v ε n(j) W 1,+∞ (Ω ′ ) vanishes as j → +∞ and then (6.16) implies property ii). On the other hand, (6.18) similarly implies and . Thanks to (6.19), (6.20) and (6.21), from (6.15) and (6.17) we obtain This entails, since v ε → v in H 1 (Ω, R 3 ) as ε → 0 and since ε n(j) → 0 as j → +∞, Proof. It is enough to prove the result in case v ∈ H 1 div (Ω). Let us define E : We take the sequence (v j ) j∈N from Lemma 6.2. Property ii) of Lemma 6.2 yields I+h j ∇v j ∈ U for a.e. x in Ω if j is large enough, where U is the neighbor of SO(3) that appears in (W3). In particular, D 2 W(x, ·) ∈ C 2 (U ) for a.e. x ∈ Ω and we make use of (3.6) together with det(I + h j ∇v j ) = 1 to obtain The limit in the last line is zero since h j ∇v j → 0 in L ∞ (Ω), since ω is increasing with lim t→0 + ω(t) → 0 and since (v j ) j∈N is converging in H 1 (Ω) as j → +∞ by Lemma 6.2 . But the H 1 (Ω) convergence also entails E(v j ) → E(v) as j → +∞. Hence, Therefore, along the sequence (v j ) j∈N provided by Lemma 6.2, we get F I h j (v j ) → E I (v) as j → +∞. The result is proven.

Convergence of minimisers
We show that E I and F I have the same minimizers thus concluding the proof of the main result. Proof. Existence of minimizers of E I on W 1,p (Ω, R 3 ) follows by standard arguments. Indeed, (3.5) and (3.11) imply that a minimizing sequence (u n ) n∈N ⊂ H 1 div (Ω, R 3 ) of E I satisfies sup n∈N E(u n ) L 2 (Ω) < +∞, and Lemma 3.1 entails the existence of u ∈ H 1 (Ω, R 3 ) such that up to subsequences E(u n ) → E(u) weakly in L 2 (Ω, R 3×3 ). By (3.10) and (L1) we deduce L(u n ) → L(u), up to subsequences, as n → +∞. By the weak L 2 (Ω) lower semicontinuity of F → Ω F T D 2 W(x, I) F dx we deduce that u is a minimizer of E I over W 1,p (Ω, R 3 ). Moreover, first order minimality conditions show that all the minimizers of E I have the same infinitesimal strain tensor. In particular, minimizers of E I are unique up to rigid displacements.
By taking into account that F I (v) ≤ E I (v) for every v ∈ H 1 (Ω), and setting z W (x) := where last inequality follows by L(z W ) ≤ 0. Therefore also min F I exists on W 1,p (Ω, R 3 ) and (7.1) is proved so we are left to show (7.2). First assume v ∈ argmin F I and let If W v = 0 then, by setting z Wv (x) = 1 2 W 2 v x we get E(z Wv ) = ∇z Wv = 1 2 W 2 v and, by compatibility (L2) we obtain a contradiction. Therefore W v = 0, z Wv = 0, and all the inequalities in (7.4) turn out to be equalities, hence we get F I (v) = E I (v) = min E I = min F I , therefore v ∈ argmin E and argmin F I ⊆ argmin E I . In order to show the opposite inclusion, we assume v ∈ argmin E I and still referring to the choice (7.3) we get 2 div v = 0 = TrW 2 v = −|W v | 2 . Therefore E I (v) = F I (v) and v ∈ argmin F I .
Remark 7.2. The proof of lemma 7.1 shows that, although Theorem 2.1 is not true if (L2) is replaced by the weaker condition still E I and F I have the same minimal values under such weaker condition. As an example, we may consider f and g as in Remark 2.8, but with Ω φ(x) dx = λ|Ω| instead of Ω φ(x) dx < λ|Ω|. In this case L(W 2 x) = 0 for any W ∈ R 3×3 skew and then Theorem 2.1 does not apply (see Remark 2.5). However, still E I (v) is minimal (with minimal value 0) if and only if v is a rigid displacement. Moreover, F I is minimal on rigid displacements as well.
Proof of Theorem 2.1. We obtain (2.5) from Lemma 4.1. If (v j ) j∈N ⊂ W 1,p (Ω, R 3 ) is a sequence such that On the other hand by Lemma 6.3 for every v ∈ H 1 (Ω) there exists a sequence (v j ) j∈N ⊂ W 1,p (Ω) such that E(v j ) ⇀ E(v * ) weakly in L p (Ω) and (7.6) lim sup that is, F I (v * ) ≤ E I (v) for every v ∈ H 1 (Ω). Hence, F I (v * ) ≤ min W 1,p (Ω) E I and by Lemma 7.1 we obtain F I (v * ) ≤ min W 1,p (Ω) E I = min W 1,p (Ω) F I so that by (7.5), (7.6) lim j→+∞ F I h j (v j ) = F I (v * ) = min