Bending of thin periodic plates

We show that nonlinearly elastic plates of thickness $h\to 0$ with an $\varepsilon$-periodic structure such that $\varepsilon^{-2}h\to 0$ exhibit non-standard behaviour in the asymptotic two-dimensional reduction from three-dimensional elasticity: in general, their effective stored-energy density is"discontinuously anisotropic"in all directions. The proof relies on a new result concerning an additional isometric constraint that deformation fields must satisfy on the microscale.


Introduction
Understanding the behaviour of elastic plates (and, more generally, thin elastic structures) from the rigorous mathematical point of view has attracted much attention of applied analysts over the recent years. The related activity was initiated by the paper [8] by Friesecke, James and Müller, concerning plate deformations u with finite "bending energy". This work appeared alongside the thesis [20] by Pantz and was followed by a study of other energy scalings [9]. It puts forward the idea that homogeneous plates of thickness h, viewed as three-dimensional nonlinearly elastic bodies, afford a special compactness argument for sequences of deformation gradients ∇u with elastic energy of order O(h 3 ) as h → 0. This argument is based on a new "rigidity estimate" [8] concerning the distance of local values of ∇u from the group of rotations SO (3). The observation that this bounds the distance from a constant rotation field (which depends on u) allowed the authors of [8] to show that the "limit" elastic energy functional as h → 0 is given by E lim (u) := 1 12 ω Q 2 (II(u)), u ∈ H 2 iso (ω), (1.1) where ω ⊂ R 2 represents the mid-surface of the undeformed plate, II(u) = (∇ ′ u) ⊤ ∇ ′ (∂ 1 u ∧ ∂ 2 u), ∇ ′ = (∂ 1 , ∂ 2 ), is the matrix of the second fundamental form of an isometric surface u : ω → R 3 , and Q 2 is a quadratic form derived via dimension reduction from a quadratic form appearing in the process of linearisation of elastic properties of the material in the small-strain regime. From the point of view of applications to real-world materials, it seems reasonable to ask in what way the above result is affected by a possible inhomogeneity of material properties of the plate in the directions tangential to its mid-surface. One can imagine, for instance, that the plate has a periodic structure of period ε > 0 and try to replace the ε, h-dependent family of h −3 -scaled energy functionals by an "effective" functional in the sense of variational convergence [6]. Such analysis should reveal, in particular, whether the behaviour of the plate really depends on the relative orders of smallness of the parameters ε and h. Work on this programme was started in the thesis [16] by Neukamm, where a series of similar questions was addressed for a periodically inhomogeneous rod, i.e. a one-dimensional analogue of a plate. Further, a recent paper [14] has investigated the behaviour of periodic plates in the cases when h ≫ ε and h ∼ ε. This was followed by the work [22], where the case ε 2 ≪ h ≪ ε is addressed. In the present paper we develop an approach (see Sections 3,4) that in our view simplifies the derivation of the corresponding Γ-limit, via a "smoothing" approximation procedure that precedes the two-scale compactness argument, see Section 3 of the present work. Smoothing is known to be useful in the asymptotic analysis of sequences of solutions to parameter-dependent PDE (see e.g. [10], [24]). In the present work we exploit similar considerations in the asymptotic analysis of bounded-energy sequences, where the energy is represented by an integral functional. Our smoothing approach replaces the approximation result [14,Lemma 3.8], when h ≫ ε or h ∼ ε, and an additional technical statement [22,Lemma 3.7], when ε 2 ≪ h ≪ ε, by a two-scale compactness theorem for the second gradients of smooth approximations (Theorem 3.4 below). Remarkably, in the situation where the deformation u h ∈ H 1 (Ω) has no second derivatives, the smoothing procedure takes us to a setting where the second derivatives exist and are bounded in the L 2 − norm, see Lemma 3.2.
The added value of our approach, however, is revealed through its ability to deal with the more difficult case h ≪ ε 2 , which has remained open until now. An additional key observation in handling this case is that the determinant of the matrix of second derivatives of the smoothed approximation can be rewritten in a form amenable to the use of a compensated compactness argument (see proof of Theorem 5.1) in order to derive an additional, "fast-scale", isometry constraint on the limit finiteenergy deformations, see (1.2). It also perhaps worth mentioning that the smoothing approach yields a somewhat shorter route to the statement of compactness of finite-energy sequences in the case of homogeneous plates considered in [8].
Notably, a recent paper [17] contains an analysis of "zero-thickness" plates, where functionals of the "limit" form (1.1) with explicit ε-periodic x-dependence of Q 2 are studied in the limit ε → 0. Our results are consistent with those of [17], in the sense of offering an alternative route to the same "supercritically-thin plate" limit functional. However, while the authors of [17] work with a twodimensional formulation from the outset, our derivation is in the spirit of "dimension reduction" from the full three-dimensional problem. In addition, our approach offers a shorter route to the proofs of [17], which is checked directly.
All of the above earlier works follow a traditional methodology of Γ-convergence relying solely on convergence properties of sequences with bounded energy combined with "structure theorems" for combinations of their gradients, without analysing their asymptotic structure as a two-scale series in powers of the parameters ε, h. Our approach, on the contrary, is guided by an analysis of the original sequence of energy functionals via two-scale power series expansions. (For a detailed discussion of the viewpoint provided by the method of asymptotic expansions, see Appendix B.) For "supercritical" scalings h ≪ ε 2 it suggests in particular that, to the leading order, the approximating ("exact" or "recovery") sequences have to satisfy an additional constraint det(II + ∇ 2 y ψ) = 0, (1.2) where ψ = ψ(x ′ , y), x ′ ∈ ω, y ∈ Y := [0, 1) 2 , is a term that appears in the two-scale limit of the original sequence of deformations and is responsible for the behaviour of the plate on ε-scale. One key aspect that facilitates this approach is that the limit stored-energy function Q 2 is quadratic with respect to the second fundamental form of the surface, which makes the analysis of ε, h-dependent plate energies somewhat amenable to a perturbation technique. The observation that the space of deformations with finite bending energy acquires the constraint (1.2), combined with a characterisation of isometric embeddings by Pakzad [19] and the earlier ansatz of [8], see (7.76), allows us to carry out a rigorous proof of variational convergence. Thanks to the two-scale asymptotic structure, the presence of the second gradient of the corrector ψ in the twoscale limit of deformation sequences provides a natural construction for the recovery sequences, cf. (4.36) for the moderate regime. The supercritical regime h ≪ ε 2 is studied in Section 7, which contains our main result, Theorem 7.2: the limit energy functional is given by where Q sc hom = min Y Q 2 (y, II + ∇ 2 y ψ)dy, ψ ∈ H 2 (Y ) periodic, subject to (1.2).
The isometry constraint (1.2) implies, in particular, that Q sc (II) is "discontinuously anisotropic in all directions of bending" as a function of the macroscopic deformation gradient ∇u, see Theorem 7.1 and the subsequent discussion. To the best of our knowledge, this is a new phenomenon for nonlinearly elastic plates. Finally, we remark that the analysis of the "critical" scaling h ∼ ε 2 is currently open.
Throughout the text I d denotes the d × d identity matrix, d = 2, 3. Unless indicated otherwise, we denote by C a positive constant whose precise value is of no importance and may vary between different formulae. We use the notation ∂ i , i = 1, 2, 3, for the partial derivative with respect to x i , ∂ yi , i = 1, 2, for the partial derivative with respect to y i , and ∇ y for the gradient with respect to y.

Setting of the problem
Let ω be a bounded convex 1 domain in R 2 , and let Ω h := ω × (− h 2 , h 2 ) be a reference configuration of an undeformed elastic plate, where h is a small positive parameter, 0 < h ≪ 1. A deformation u of the plate Ω h is an H 1 -function from Ω h into R 3 . We assume that the material properties of the plate Ω h vary over ω with a period ε, which goes to zero simultaneously with h. For a plate made of a hyperelastic material (see e.g. [4]), the elastic energy of a deformation u is given by the functional We next rescale the transverse variable x 3 in order to work on a fixed domain. Namely, we multiply the transverse variable by h −1 so that the new variable belongs to the interval I := (− 1 2 , 1 2 ). We reassign the notation x 3 to the new variable (this should not cause any misunderstanding since we will always deal only with the rescaled domain in the future), so x ∈ Ω := ω × I. We also use the rescaled gradient so that the rescaled functional is given by where u ∈ H 1 (Ω). We consider sequences of deformations u h in the bending regime, that is such that In the non-rescaled setting this corresponds to the energies of order h 3 .) In this paper we focus on the case when h ≪ ε. We consider ε as a function of h, i.e. h is our "main" parameter and ε is the "dependent" parameter. The non-degeneracy property (2.3) and the assumption (2.5) imply that In what follows, we say that a sequence satisfying (2.6) has finite bending energy.

Two-scale compactness and second gradients
The two cases when ε is of order h and h ≫ ε were considered in [14]. In these regimes the behaviour of the mid-surface of the plate is essentially macroscopic and the microscale only plays a role away from the mid-surface. In our case the microscopic behaviour of the mid-surface of the plate is very important. This is due to the fact that for each ε-cell the corresponding piece of the plate itself behaves like a miniature plate. This "micro-plate" behaviour becomes even more dominant when h ≪ ε 2 . In order to study this property we employ the method of two-scale convergence. The structure of the limit functional in the case of a homogeneous plate, namely, the fact that it is basically defined on the second gradient of the limit surface, suggests that the information about the microscopic behaviour should also appear in the form of a second gradient. The problem is that the original functional is defined only on H 1 -functions, and how one would obtain the second derivatives with respect to the fast variable y in the limit is not clear at the outset. However, while this is not possible in general, in the bending regime deformations possess an additional property which in a certain sense is equivalent to the existence of the second gradient, which we explain in detail in what follows. Let us recall some results from [8]. We denote where u = u(x ′ ) belongs to the space H 2 iso (ω) of isometric immersions, i.e. such maps u : ω → R 3 that (∇ ′ u) ⊤ ∇ ′ u = I 2 , and n := u ,1 ∧ u ,2 ; 2. There exists a piecewise constant map R h : 3. The difference between the values of R h in each pair of neighbouring h-cells is small in the sense that Apart from the strong compactness of the gradients ∇ h u h , which is true in our setting as well as for homogeneous plates, this theorem implies that oscillations of the gradients on the h-scale are bounded. The third property in Theorem 3.1 implies that the difference between values of R h in neighbouring h-cells divided by h (which basically is the difference quotient) is bounded on average and a similar statement can be formulated for ∇ h u h due to the second part of the theorem. Hence one can try to mollify u h , expecting that the second gradient of the mollification will be bounded. In what follows we always assume that u h is a sequence with finite bending energy.
Let ω ′ be a domain whose closure is contained in ω. For each sufficiently small h > 0, consider the functionũ h = u h (x ′ ), x ′ ∈ ω ′ , defined as the result of the simultaneous mollification of u h with ϕ h and averaging with respect to the variable x 3 : (3.7) (The above integral is well defined in ω ′ for sufficiently small h since in this case the support of ϕ h (ξ − x ′ ) is contained in ω.) It is well known that the mollification u h is infinitely smooth. We will need to estimate in H 1 the difference between u h and the average of u h with respect to x 3 , as well as its derivatives up to the second-order. Such estimates are obtained in the next statement.
Let ω ′ be a domain whose closure is contained in ω. The mollification (3.7) satisfies the following inequalities with a constant C > 0 independent of ω ′ and h: Proof. In the subsequent estimates we use a generalised Minkowski inequality (also sometimes referred to as the Minkowski integral inequality): where X and Y are measure spaces with measures µ X (x) and µ Y (y) respectively, F : X × Y → R is measurable and 1 < p < ∞, see e.g. [11]. We obtain the L 2 -estimate in (3.8) as follows: Now we estimate the difference of the gradients ofũ h and u h . First notice that by the properties of mollification, ∇ ′ u h equals to the mollification of ∇ ′ u h . Second, one has where R h stands for the first two columns of the matrix R h and Ω ′ := ω ′ × I. The bound (3.10) is a simple consequence of application of the generalised Minkowski inequality and the Hölder inequality. Further, the following estimate holds: where we use the generalised Minkowski inequality and Theorem 3.1.
Next we estimate the second gradient of u h , given by Since the mollifier ϕ is radially symmetric, ∂ ξ1 ϕ h (ξ) is an odd function of ξ 1 and an even function of ξ 2 , and analogously for ∂ ξ2 ϕ h (ξ), hence we write Using the generalised Minkowski inequality once again we have, for i = 1, 2 (3.12) The last inequality is obtained analogously to (3.11). Finally, we argue that ∇ ′ u h is bounded pointwise independently of h: for all points x ′ ∈ ω and sufficiently small values of h.
As one can see from the above, ∇ ′ u h is a "reasonably good" approximation of ∇ ′ u h . It remains to approximate h −1 ∂ 3 u h by a smooth vector function. Let n h be the unit vector orthogonal to the surface u h : We show that the L 2 -norm of n h − R h,3 is at most of order h, where R h,i stands for the i-th column of R h . To this end we first transform the above expression for n h , as follows: Since R h and ∇ ′ u h are L ∞ -bounded, the L 2 -norms of the last three terms are of order h, hence Additionally, it is clear from (3.12) and (3.9) that (3.14) Putting together the above estimates for n h , Theorem 3.2 and Theorem 3.1 we obtain the following result. Proposition 3.3. Let ω ′ be a domain whose closure is contained in ω. Under the assumptions of Theorem 3.1 the following estimates hold: As is customary in periodic homogenisation, and due to the fact that our problem is of twoscale nature, we require a suitable notion of convergence which takes into account the microscopic behaviour, e.g. the so-called two-scale convergence, introduced by Nguetseng in [18]. Out of several equivalent definitions of two-scale convergence, we will use the one based on periodic unfolding, see e.g. [5]. For convenience of the reader we give the definition of two-scale convergence that we use and some of its properties in Appendix A. Next we prove a general statement about the two-scale convergence of a second gradient, without any reference to a specific problem. This assertion may be used in various contexts where two-scale convergence of a second gradient is required. In the present paper we make use of this statement in Theorems 4.2 and 7.2. In the remainder of this section we consider an arbitrary d-dimensional space, returning to the case d = 2 in Section 4 onwards.
Theorem 3.4 (Second gradient and two-scale convergence). Let ω ⊂ R d be a bounded domain and Y := [0, 1) d . Assume that a sequence v ε ∈ H 2 (ω) is bounded in H 2 (ω). Then up to a subsequence where v ∈ H 2 (ω) is the weak limit in H 2 (ω) (hence, strong limit in H 1 (ω)) of the sequence v ε and ϕ ∈ L 2 ω; H 2 per (Y ) with zero mean with respect to y ∈ Y . Note that here v ε is a scalar function. The theorem directly applies to the vector case componentwise.
Proof. Following the idea of the proof of the two-scale convergence for the first gradients in [5] we introduce the following function of x and y: where T ε is the unfolding operator (see Appendix A), M Y (f ) := Y f dy is the operator of averaging with respect to the variable y, and y c := y − (1/2, . . . , 1/2) ⊤ . We calculate the first and second gradients of v ε with respect to y: Notice that the function v ε and its gradient ∇ y v ε have zero mean value in y. Hence, we can apply the Poincaré inequality: where the last bound follows from the assumptions of the theorem. Hence, the sequence v ε is bounded in L 2 ω; H 2 (Y ) and it converges weakly in this space, up to a subsequence: v ε ⇀ ϕ weakly in L 2 ω; H 2 (Y ) .
Let a function ϕ ∈ L 2 ω; H 2 (Y ) be such that Clearly, under this notation one has which is equivalent to (3.15). Now it only remains to prove the periodicity of ϕ with respect to y.
To this end, let ψ(x, y ′ ) be an arbitrary function from C ∞ 0 (ω × Y ′ ), where y ′ := (y 1 , . . . , y d−1 ) ⊤ and Y ′ := [0, 1) d−1 . We write the difference between values of v ε on the opposite faces of the cube Y corresponding to y d = 1 and y d = 0 (the proof for other components of y being exactly the same), , where e d is a unit vector in the direction of x d axis, hence we derive that Using a slight variation of the argument in Proposition 3.4 in [5] one can see that Z ε converges strongly in L 2 (ω; H 1 (Y )) to (y c ) ′ · ∇ ′ v as ε → 0. Hence, its trace We carry out the the same calculations for z(x, y) : Multiplying the last expression by ψ and integrating by parts we get Comparing the latter with (3.19) we conclude that ϕ(x, y) is periodic with respect to y.
Analogously we prove the periodicity of ∇ y ϕ. Indeed, we have Note that by the assumptions of the theorem and properties of two-scale convergence (see Appendix A) the sequence T ε (∇v ε ) converges to ∇v strongly in L 2 (ω; H 1 (Y )). Hence, by Proposition 3.5 below the trace T ε (∇v ε )(x, y ′ , 0) converges to ∇v weakly in L 2 (ω × Y ′ ), and therefore which implies the periodicity of ∇ y ϕ. Summarising all of the above we conclude that ϕ ∈ L 2 (ω; H 2 per (Y )).

Proposition 3.5.
Suppose that a sequence f n (x, y) converges to f (x, y) weakly in L 2 (ω; H 1 (Y )) and strongly in L 2 (ω × Y ). Then its trace on ω × ∂Y converges to the trace of f weakly in L 2 (ω × ∂Y ).
Proof. Without loss of generality we will assume that f = 0 and prove the assertion for the part of the boundary corresponding to y n = 0. We need to show that . Let us fix ψ and consider it as a function defined on ω × Y via ψ(x, y) := ψ(x, y ′ ). Let us define function h n (y) as ω f n ψ dx. It is easy to see that and strongly in L 2 (Y ). By the trace theorem h n (y ′ , 0) → 0 strongly in L 2 (Y ′ ). In particular, Now, we have all necessary tools to proceed to the Γ-convergence statement.
4 Limit functional and Γ-convergence for the "moderate" thickness-to-period ratio ε ≫ h ≫ ε 2 Our main result is establishing Γ-convergence of the rescaled functional E h to the limit homogenised functional defined on isometric surfaces. Γ-convergence is a standard tool to describe the asymptotic behaviour of nonlinear functionals, introduced by De Giorgi [6]. The general definition of Γ-convergence is the following: . . and F ∞ be some functionals defined on functional spaces H i and H ∞ . Then we say that the sequence of functionals F i Γ-converges to the functional F ∞ , if the following two properties hold: 1. Lower bound. For any sequence u i ∈ H i converging in a certain sense to u ∞ ∈ H ∞ one has

Recovery of the lower bound. For any
In the above definition the notion of convergence depends on the underlying properties of the functionals and may be quite non-standard, in particular, because the spaces H i and H ∞ often do not coincide, as in the present paper, for example. Moreover, the notion of convergence in the second part of the definition may be different from the convergence in the first part. In this case the former must normally be "stronger" that the latter in order to facilitate the main purpose of the definition (nevertheless, this requirement is not always critical, e.g. see [2]). The lower bound inequality ensures that the limit functional gives in the limit a lower bound for the values of the functionals F i . The recovery property implies that the lower bound F ∞ is the greatest lower bound. In particular, the definition implies that and, if a sequence u i of "almost minimisers" of F i converges to u ∞ , then the latter is a minimiser of F ∞ .
In the present work we obtain different Γ-limits for the regimes ε ≫ h ≫ ε 2 and ε 2 ≫ h. In this section we present our results for the "moderate" regime ε ≫ h ≫ ε 2 , for which the limit homogenised functional is given by where II is the matrix of the second fundamental form of u, and the homogenised stored-energy function is given by where Q 2 is the quadratic form defined on 2 × 2 matrices and is obtained from the form Q 3 by minimising when reducing to two dimensions (notice that Q 3 vanishes on skew-symmetric matrices): where we implicitly assume that the space R 2×2 matrices is naturally embedded in the space R 3×3 by adding zero third column and zero third row, cf. [8].
Then the rescaled sequence of functionals h −2 E h Γ-converges to the limit functional E m hom in the following sense.
1. Lower bound. For every bounded bending energy sequence u h ∈ H 1 (Ω) such that ∇ ′ u h converges to ∇ ′ u weakly in L 2 (Ω), u ∈ H 2 iso (ω), the lower semicontinuity type inequality holds: 2. Recovery of the lower bound. For every u ∈ H 2 iso (ω) there exists a sequence u rec h ∈ H 1 (Ω) such that ∇ ′ u rec h converges to ∇ ′ u strongly in L 2 (Ω) and Proof. Lower bound. Since ∇ h u is close to a rotation-valued function R h we can use the frame indifference property of the stored energy function W and its Taylor expansion near identity to linearise the functional. Roughly speaking, we will use the observation that However, the last asymptotic equality in (4.23) has to be made precise, since the smallness of the term hG h is not pointwise. We also need to understand the structure of the (weak) two-scale limit of G h . To this end we will use a special representation of u h . Let ω ′ be a domain whose closure is contained in ω and Ω ′ = ω ′ × I. For small enough h we can write u h in Ω ′ in the form where u h is the mollified mid-surface of u h defined in the previous section and n h is the corresponding normal. Then one has Notice that G h is bounded in L 2 (Ω ′ ) by Theorem 3.1. The first term on the right-hand side is bounded due to Proposition 3.3. The second term on the right-hand side is bounded by (3.14). Hence, the third term is also bounded. Since we perform the dimension reduction when passing to the limit, we are interested only in the behaviour of the top left 2 × 2 block of the matrix G h . Let us use the following notation: if A ∈ R 3×3 , then A denotes the first two columns of A, and A is obtained from A by omitting the third column and the third row. We have The limit of the first and the third terms on the right-hand side of (4.25) is of no importance for us apart from the fact that they do not depend on x 3 . This is obvious in case of the first term. As for the third term, let On the other hand we know that ∂ 3 w h vanishes in the limit, hence w is independent of x 3 : for some H ∈ L 2 (ω ′ × Y ), up to a subsequence. Now we examine the expression ( R h ) ⊤ ∇ ′ n h . The element (1, 1) of this matrix can be written as where the vector product operations take priority over the inner products. Consider the second term in the last expression (the third term can be dealt in the same way). By Theorem 3.1 and Theorem 3.2 one has On the one hand, the second term is bounded in L 2 (ω ′ ), and so converges weakly two-scale to some function from L 2 (ω ′ × Y ) up to a subsequence; on the other hand, for any C ∞ 0 (ω ′ × Y ) test-function we get zero in the limit. We conclude that the weak two-scale limits of the second and the third terms are zero. We can apply analogous argument to the other elements of ( 3 have the same weak two-scale limit (the "dot" product gives a 2 × 2 matrix whose elements are −∂ ij u h · R h,3 ). Applying Theorem 3.4 to u h one has up to a subsequence, for some ϕ ∈ L 2 (Ω ′ ; H 2 per (Y )), whereas R h,3 converges strongly to n due to Theorem 3.1. Hence, Putting together (4.27) and (4.28) we conclude that up to a subsequence We can proceed to the last step of the proof. In order to use the Taylor expansion of W we need to control G h pointwise. Let χ h be a characteristic function of the subset of Ω ′ on which hG h is relatively small, namely, Since W is non-negative we can restrict the integral to Ω ′ and discard the set of points on which |hG h | > h 1/2 by using χ h : (4.31) By the assumptions on W we have An elementary lemma below implies that Here it is implied, with some abuse of notation, that (f t (x)) −1 g(f t (x)) = 0 whenever f t (x) = 0.
We apply Lemma 4.3 to the expression (4.32) by setting t := h 2 , f t := χ h |hG h | 2 . Notice that the particular choice of the cut-off threshold in (4.30) is not important: instead of h 1/2 one can take any threshold that goes to zero slower than h. Now we estimate the quadratic term in (4.31). By definition of the form Q 2 ant the properties of the unfolding operator we have It is well known that convex integral functionals are lower semi-continuous with respect to the weak convergence in functional spaces, see e.g. [7]. Assume that we deal with a weakly converging (as in (4.29)) subsequence G h . Then passing to the limit and recalling that χ h converges strongly to 1 we obtain the following inequality: Note that in the second step in the above we integrate with respect x 3 ∈ I, which causes the cross term of x 3 (II + ∇ 2 y ψ) and H in the quadratic form Q 2 to disappear. At the last step we just used the definition of Q m hom . A simple argument by contradiction shows that the inequality holds for the whole sequence G h . Putting together (4.31), (4.32), (4.33) and (4.35) and noting that ω ′ is an arbitrary subset of ω concludes the proof of the lower bound.
Remark 4.4. The above proof applies to the general case of h ≪ ε, up to the last inequality in (4.34). We will use this observation in our analysis of the lower bound in the case h ≪ ε 2 in Section 7.
Recovery sequence. In order to build a recovery sequence one normally needs to ensure that the argument of the stored-energy density W is an L ∞ function, so that one can pass to the limit using the Taylor expansion (2.4). If this condition is not in place one would have to estimate W on the "bad" set where the argument might be uncontrollably large, which would require additional restrictions on W . For example in ( [8]) the authors used a truncation result for Sobolev maps, which consists of changing a function on some subset to make it together with its gradient essentially bounded and estimating the measure of this set. In this paper we will use a result of Pakzad from [19] (which appeared after [8]). It states that the space of infinitely smooth isometric immersions from a convex set ω ⊂ R 2 into R 3 is strongly dense in the Sobolev space of isometric immersion H 2 iso (ω). Let u ∈ H 2 iso (ω) ∩ C ∞ (ω) and II be the matrix of the corresponding second fundamental form. Let also ψ(x ′ , y) and d(x ′ , y) be correspondingly a scalar and a vector functions from C ∞ (ω ×Y ) periodic with respect to y.
Denote ψ ε := ψ(x ′ , ε −1 x ′ ), d ε := d(x ′ , ε −1 x ′ ) and consider the sequence Rd ε , (4.36) where n := ∂ 1 u ∧ ∂ 2 u, n ε := n + ε∂ y1 ψ ε ∂ 1 u + ε∂ y2 ψ ε ∂ 2 u and R := (∇ ′ u|n). Calculating the rescaled gradient of u h we obtain where U h denotes the rest of the terms: Due to our assumptions we see that (4.37) This implies that there exists an infinitely smooth function R h with values in SO(3) such that . (It can be constructed explicitly via Gram-Schmidt orthonormalisation for the columns of the matrix (∇ ′ u − ε∇ y (ψ ε n)|n ε ).) Further, consider the expression . Notice that R h converges to R uniformly on ω, ∇ 2 y ψ ε converges strongly two-scale to ∇ 2 y ψ, and d ε converges strongly two-scale to d. Hence, the convergence holds. Passing to the limit in we see that the remainder term vanishes since G h is uniformly bounded in L ∞ (ω), and for the quadratic term due to (4.38) we have We conclude the proof with a standard argument showing that the above assumption of smoothness of u, ψ, d is not restrictive. Let u be an arbitrary element of H 2 iso (ω) and take ψ(x ′ , y) ∈ L 2 (ω; H 2 per (Y )) to be a solution to the minimisation problem in (4.21) corresponding to the matrix of the second fundamental form of u, that is Let also d(x ′ , y) be such that There exist sequences of C ∞ functions u j , ψ j and d j converging to u, ψ and d in the corresponding function spaces such that For each j define u hj is the required recovery sequence.
5 A microscale isometry constraint on the limits of finite bending energy sequences in the "supercritical" regime h ≪ ε 2 During the process of constructing a recovery sequence we have encountered a problem in the case when h ≪ ε 2 . Namely, introducing the corrector terms −ε 2 ψ ε n and ε∂ y1 ψ ε ∂ 1 u + ε∂ y2 ψ ε ∂ 2 u in (4.36) incurs an error of order ε 2 (c.f. (4.37)) which becomes of order h −1 ε 2 when scaled in the subsequent argument. This error does not vanish if h ≪ ε 2 or h ∼ ε 2 . In order to understand the nature of the problem we write a formal asymptotic expansion for a recovery sequence, see Appendix B. In the process of eliminating the errors of order ε 2 and higher, it transpires that ψ ε needs to satisfy some solvability condition which is equivalent to the zero-determinant condition (5.40) below. As we show in the next theorem, this condition is intrinsic for the limits of finite bending energy sequences in the case h ≪ ε 2 . This is due to the fact that II + ∇ 2 y ψ ε approximates the matrix of the second fundamental form of the mollified mid-surface u h of the plate (defined in Section 3) on ε-scale. It appears that u h has to be almost isometric in the case h ≪ ε 2 for the expression (∇ ′ u h ) ⊤ ∇ ′ u h − I 2 to be of order o(h). Noting that the matrix of the second fundamental form of an isometric surface has zero determinant suggests the condition det(II + ∇ 2 y ψ) = 0, where ψ is the two-scale limit of the sequence ψ ε .
Theorem 5.1 (Zero determinant condition for the two-scale limit isometric surface.). Let u h be a finite bending energy sequence, u h be the mollified mid-surface of the plate defined by (3.7) and ω ′ ⋐ ω. Assume that h ≪ ε 2 . Then the weak two-scale limit where ψ := −ϕ · n.
Proof. The proof is inspired by the general idea of the proof of the identity det II = 0 for the H 2 isometric surfaces (see [19]). In our setting, however, the mollified mid-surface u h of the plate is not exactly isometric. This, together with the two-scale nature of our problem, requires in-depth analysis on the microscale. The following identity is easily checked by a straightforward calculation (recall that u h is a C ∞ function): Note that we can consider the third order tensor ∇ 2 u h as a 2 × 2 matrix whose elements ∂ ij u h are vector-valued functions. Thus when we take its determinant we understand the multiplication of entries as the scalar product of vectors. We will use this convention henceforth. We can rewrite the above identity applying the unfolding operator: We consider the left-hand side of the last identity as a scalar product of two "vectors" Since curl y f ε = 0 and div y g ε = 0, we can apply the compensated compactness theorem, see e.g. [23], which implies that the left-hand side of (5.41) *-weak converges to the scalar product of their weak limits, that is Note that x ′ plays the role of a parameter, and does not affect our argument. Let us now consider the right-hand side of (5.41). Due to Proposition 3.3 we have Multiplying the last expression in (5.42) by an arbitrary function η ∈ C ∞ 0 (ω ′ × Y ) and integrating by parts twice we see that it converges *-weakly to zero given that h/ε 2 At the last step of our proof we need one additional property of the two-scale limit ϕ(x ′ , y). We know that the second-order derivatives of u(x ′ ) are parallel to the vector n(x ′ ) : one has since at least two of the indices i, j, k coincide. Notice further that hence ∂ ij u = −II ij n. In particular, one has det ∇ ′2 u = − det II. In order to derive (5.40) from (5.43) we need a similar property for ϕ to be held. In fact, we require a much weaker property of ∂ yi ϕ being orthogonal to R i for i = 1, 2. We establish it with the help of the following statement.
In the proof of this proposition we will need the following simple inequality, which is a sort of Poincaré inequality.
Proof. Since the set of C ∞ (Y ) functions is dense in H 1 (Y ) we only need to prove the lemma for f ∈ C ∞ (Y ). First we write down a pointwise estimate for |f (y) − f (ζ)|.
We finish the proof with Proposition 5.2. Taking the gradient of the function z ε and using the properties of the unfolding operator we have Let us write the L 2 -norm of the right-hand side (dropping the coefficient ε −1 ) in a special form: where in the last inequality we used the property of boundedness of the second gradient of u h , see Theorem 3.2. Due to Proposition 3.3 we can replace T ε (∇ ′ u h ) in the last estimate by T ε ( R h ): Similarly to (5.46) we can write the expression in (5.45) as As follows from Proposition 3.3, in order to prove the statement it is enough to show that Notice that the integrand in the latter is non-negative everywhere. The expression under the sign of integral has the form (a − b) · a at every point, where a and b are unit vectors. We use the following elementary identity, 2(a − b) · a = |a − b| 2 , together with (5.48) to conclude the proof of the proposition: Theorem 3.4 implies that the sequence z ε converges weakly in L 2 ω ′ ; H 2 (Y ) to the function (cf. *-weakly, i.e. with test functions taken from C ∞ 0 (ω ′ × Y ). (In fact, with some effort it is possible to show weak convergence in L 2 (ω ′ × Y ), however, *-weak convergence is sufficient for our purposes.) Further, since the second-order derivatives of u are parallel to the vector b, we have Lastly, it follows from Proposition 5.2 that the *-weak limit of ∂ yi z ε · T ε (R h,i ) equals zero, hence ∂ yi ϕ · R i = 0, i = 1, 2.

The structure of isometric immersions
The idea of the construction of a recovery sequence for the supercritical regime h ≪ ε 2 , which is performed in the next section, relies on the properties of isometric surfaces from H 2 iso (ω). There has recently been a number of works devoted to this topic, see [12], [13], [15], [19]. The results of Section 5 (as discussed at the beginning of Section 7) imply a discontinuous dependence of the homogenised energy density function on the direction of the bending. This observation drastically reduces the number of manipulations one can perform with a surface in order to construct a recovery sequence. In particular, we can not use the result on approximation of H 2 iso (ω) surfaces by infinitely smooth ones because this approximation does not preserve the orientation of the bending directions. In our paper we will use mostly the results of [19], which we review below. We then give an example of an isometric surface for which the set of rational directions is irregular (cf. Theorem 7.1), and prove a statement on regularisation of isometric surfaces required for the construction of a recovery sequence.

Auxiliary observations
For any u ∈ H 2 iso (ω) domain ω can be partitioned into "bodies" and "arms", according to the following definition. Definition 6.1. A body is a maximal subdomain of ω on which u is affine and whose boundary contains at least three segments inside ω.
A differentiable curve Γ : [0, l] → ω is referred to as a leading curve if u is affine when restricted to the lines normal to Γ [0, l] , which we refer to as leading segments. In the approximation procedure below we also use the term "leading curve" for the curve γ := (u • Γ) on the surface.
For a given leading curve Γ, generatrices are the lines tangent to the surface u at the points of γ and orthogonal to the latter.
An arm ω[Γ] is a subdomain of ω covered by leading segments corresponding to a leading curve Γ. (See Remark 6.3 below.) Remark 6.2. Note that a maximal subdomain of ω on which u is affine and hose boundary contains at most two segments inside ω is always an arm or part of an arm. For each t ∈ [0, l], the set Γ(t) + sN (t) : s ∈ R ∩ ω is the corresponding leading segment.
Since leading curves are not uniquely defined, it is more reasonable to define the vector field N on the whole ω[Γ] as a continuous vector field of unit vectors parallel to leading segments, and the vector field T as the continuous unit vector field orthogonal to N . In particular, we have T = Γ ′ for any leading curve Γ. In what follows we consider N and T either as vector fields N (x ′ ), T (x ′ ) or the normal N (t) and the tangent T (t) of the curve Γ(t), depending on the context. We will say that the value of T (equivalently, N ) is rational if the ratio of its components is rational. We will also say in this situation that the corresponding leading segment is rational. Remark 6.4. It can be shown that: a) For any given body or arm the part of its boundary that is situated in the interior of ω is a union of straight segments whose vertices belong to ∂ω; b) As follows from [19], all leading curves are in the class C 1,1 loc . A combination of this observation with the result of [15] establishing that H 2 iso ⊂ C 1 (ω) implies the fact that leading curves γ on the surface u are in the class C 1 .
Since the leading segments do not intersect in ω, the vector field N (and, hence, T ) is locally Lipschitz in ω. In is not difficult to prove, using a geometric argument, the following lemma.
Lemma 6.5. The local Lipschitz constant This lemma implies, in particular, local Lipschitz continuous differentiability of leading curves. Note that for any leading curve Γ, unit tangent vectors of γ are also tangent to the surface u : R 2 → R 3 . Therefore, the surface u is reconstructed on each arm from the knowledge of the corresponding leading curve and the non-vanishing principal curvature. Indeed, denote by κ Γ the curvature of Γ, so that Γ ′′ = T ′ = κ Γ N . For a given γ, we define a Darboux frame (τ, ν, n) by the tangent τ = γ ′ to the curve γ, the unit vector ν = (∇ ′ u)N tangent to u and orthogonal to τ , and n = τ ∧ ν. This Darboux frame satisfies the system of equations (see e.g. [21, p. 277], [19] which has a unique solution subject to appropriate initial values. Given γ(t) and ν(t) we can write where s parametrises the generatrix corresponding to the value t on the curve Γ.
Remark 6.6. There is freedom in the choice of the sign of κ Γ and the direction of the vector field N . We make this choice in such a way that the normal n in the above Darboux frame coincides with the normal n defined as ∂ 1 u ∧ ∂ 2 u in the discussion preceding the current section.
Let us write down the formulae for for the gradients ∇ ′ u, ∇ ′2 u and the matrix II in terms of the leading directions and curvatures. Since x ′ is related to (t, s) via x ′ (t, s) = Γ(t) + sN (t). We have and therefore The above expression is well defined since 1 − sκ Γ is positive everywhere on ω[Γ], cf. Lemma 6.5. Then we can write the gradient and the second gradient of u in terms of the geometry of the curves γ and Γ: Notice that (cf. (5.44)) (6.53) Let us introduce the following notation (6.54) As will be shown in the next section, the homogenised stored energy density Q sc hom in the supercritical regime is in general a discontinuous function of its argument II. More precisely, it is continuous with respect to the coefficient −κ(1 − sκ Γ ) −1 , but discontinuous with respect to II (the direction T ). This suggests that one can modify the curvature κ in a continuous way (i.e. without changing much the homogenised energy), however, a perturbation of the vector field T may lead to uncontrollable changes in the homogenised energy.

Regularisation of irregular surfaces.
The results of [19] and [15] assert some regularity of isometric immersions u ∈ H 2 iso (ω), namely, u ∈ C 1 (ω) and its associated leading curves in ω have locally Lipschitz derivative. Nevertheless, isometric immersions can be quite irregular for the purpose of this paper. To illustrate this, we give an example of a surface for which the set of "rational" directions for the corresponding leading curve is the union of a fat Cantor set and a set of zero measure.
Example. Let us recall the construction of a fat Cantor set. At the first step we remove from the middle of [0, 1] an open interval B 1 of length 1/4 and denote the remaining set A 1 . At the next step from the middle of each the two subintervals of A 1 we remove open intervals of length 1/16 each, the remaining set is denoted by A 2 and its complement by B 2 and so on: at the n-th step we remove 2 n−1 disjoint intervals of length 1/2 2n , whose union is a set of measure 1/2 n+1 . Thus we have λ 1 (A) = λ 1 (B) = 1 2 , where A := ∩ n A n , B := ∪ n B n . By λ d we denote the Lebesgue measure in R d . The Fat Cantor Set A is closed and is nowhere dense (i.e. contains no interior points). For each n we define a leading curve via its unit tangent vector T n (t) given as follows. Let ϕ be a function that is continuously differentiable on the interval [0, 1], has zero mean and is such that where ϕ n,k = β n ϕ(2 2n t − t n,k ) is the scaled (by a coefficient β n , |β n | < 1, and the length of the interval 1/2 2n ) and appropriately translated (by the left end t n,k of the interval I n,k ) version of ϕ, which "fits" into this interval. We then integrate T n to obtain the curves Γ n : Note that at each step of this procedure we "redefine" the vector T n only on the set B n \ B n−1 .
The above example illustrates that for a leading curve Γ(t) the set on which T (t) has rational directions can not in general be represented as a union of intervals (up to a null set). In what follows we argue that one can replace an arbitrary isometric surface u by a "sufficiently regular" isometric surface, in order to obtain an explicit construction of a recovery sequence. Note that one can not simply approximate u by an infinitely smooth isometric surface, since Q sc hom , in general, is an everywhere discontinuous function of its argument. Thus the matrix II, see (6.54), of the replacement surface should be different from the one of u only on a relatively small set. First we need to recall some properties of isometric surfaces. We reformulate and complement some results of [19] (Lemmas 3.8 and 3.9) in the following statement. Theorem 6.7. Let u ∈ H 2 iso (ω). For any δ > 0 there exists v ∈ H 2 iso (ω) such that the surface v has a finite number of bodies and arms, each maximal connected region with nonzero mean curvature lies between two regions where v is affine, the corresponding leading curves are Lipschitz differentiable and do not have common points with ∂ω.
The above properties of v are partly derived from the proofs of Lemmas 3.8 and 3.9 in [19]. Next, we briefly elucidate them. Every maximal connected region (we denote it by Σ) where u has non-zero mean curvature is attached to either one or two regions where u is affine. If Σ is situated between two regions where u is affine, then the lengths of the leading segments that cover Σ are bounded below by a positive constant. Lemma 6.5 implies that on any closed subset of ω the vector field T is Lipschitz continuous. In particular, we can cover Σ by a finite number of leading curves so that they do not have common points with the boundary of ω, and hence have Lipschitz continuous derivative. If Σ has just one "affine" region attached to it or Σ = ω, then it may happen that the infimum of the length of leading segments covering Σ is zero. Then one can not control the Lipschitz constant on the sequence of segments which lengths tend to zero (such sequence will converge to a point on the boundary of ω). In this case we can replace u in a small region near the boundary point 2 A direct calculation shows that |Γ ′′ n | ≤ |ϕ ′ n,k |(1 − ϕ 2 n,k ) −1/2 = 2 2n βn|ϕ ′ |(1 − β 2 n ϕ 2 ) −1/2 , which e.g. for the choice ϕ(t) = sin(2πt) 3 , t ∈ [0, 1], and β ≤ 2 √ 2 gives the value C = 4π 2/3.
where such degeneration of leading segments occurs by an affine function making sure that the new surface is still an element of H 2 iso (ω). Then we get to the situation discussed previously. As was mentioned at the beginning of this section we aim at the construction of recovery sequences for the homogenised functional E sc hom in the supercritical case h ≪ ε 2 , given by (7.66). Theorem 6.7 implies, in particular, that for δ > 0 there exists v ∈ H 2 iso (ω) with a finite number of bodies and arms that depends on δ so that E sc hom (u) − E sc hom (v) ≤ δ. The next theorem ensures that we can replace u ∈ H 2 iso (ω) with an isometric surface such that the set on which its leading segments are rational can be represented as a union of finite number of arms and a set of arbitrary small measure. Theorem 6.8. Let u ∈ H 2 iso (ω) be such that the subset ω r (u) of ω covered by rational leading segments has positive measure. Then for each δ > 0 there exists a surface u δ ∈ H 2 iso (ω) such that: a) The surface u δ has a finite number of bodies and arms, which we generically denote by ω[Γ]; b) The subset ω r (u δ ) of ω covered by rational leading segments can be represented as where each arm ω[Γ r ] consists of parallel rational segments (in particular, each leading curveΓ r is a straight segment and u(ω[Γ r ]) is cylindrical); c) The set S δ is measurable; d) The estimate holds. Moreover, u δ can be chosen in such a way that on each arm the corresponding curvature κ (cf. (6.50) and (6.53)) is infinitely smooth and II δ ∈ L ∞ (ω), where the latter in the matrix of the second fundamental form of u δ . Remark 6.9. The conditions that the curvature κ is smooth and II δ ∈ L ∞ (ω) is required for the construction of a recovery sequence. In particular, we take advantage of these facts when using the Taylor expansion formula for W , see (2.4).
Proof. In view of Theorem 6.7 we may assume without loss of generality that u has a finite number of bodies and arms from the outset. Let us consider an arm ω[Γ] with the corresponding leading curve Γ(t), t ∈ [0, l], and assume that the subset K = K(Γ) of [0, l] on which the directions of T = T (t) are rational has positive measure λ 1 (K) > 0. Since the set of rational directions is countable, we can order its elements in such a way that λ 1 (K 1 ) ≥ λ 1 (K 2 ) ≥ . . ., where K i is the preimage of a particular rational direction T i , i = 1, 2, 3, ..., so that K = ∪ i K i . We take an arbitrary η > 0 and let m = m(η) be such that for Note that all K i are closed, in particular, the distance between any two of these sets is positive. This implies that we can remove a finite number of open intervalsL i ⊂ [0, l] \ K η , i = 1 . . . , k, so that the remaining set L η := [0, l] \ (∪ k i=1L i ) (we may also remove the points 0 and l from L η if they are isolated) possesses the following properties: 1. The estimate λ 1 (L η \ K η ) < η holds.
2. The set L η is the union of a finite number of mutually disjoint closed intervals L j , j = 1, . . . , n.
3. For each j one has L j ∩ K η = L j ∩ K i for some i ∈ {1, 2, ..., k} and both ends of the interval L j belong to K i .
Let us consider a particular interval L j = [a j , b j ]. The part of ω[Γ] covered by the restriction of Γ to L j , which we denote by ω[Γ| Lj ], is contained between two parallel segments Γ(a j ) + sN i and Γ(b j ) + sN i corresponding to the rational direction T i . In fact, loosely speaking, ω[Γ| Lj ] is mostly covered by the leading segments which are parallel to N i . On the rest of the set, corresponding to the restriction Γ| Lj \Ki , where the leading segments are not parallel to N i , we replace them with new leading segments that are parallel to N i . Notice that the difference between N (t) and N i on the set L j \ K i is not greater than Cη, where C is the Lipschitz constant of N (t). We can write We carry out this procedure for each interval L j . In this way we change the leading segments covering ω[Γ] only on a small set ω η [Γ]. We estimate the measure of this set as follows: (6.56) Let us denote by N η and T η the vector fields corresponding to the new set of leading segments. We have We denote by S Γ η the union of all rational leading segments that do not belong to the set n j=1 ω[Γ| Lj ] and estimate its measure as follows: For each arm we build a family of leading curves Γ η and isometric surfaces U η , which will be shown to converge to the surface u uniformly as η → 0. We assemble the approximating parts U η on all arms and bodies in a continuous fashion. We then choose a sufficiently small value of η so that the resulting surface denoted by u δ satisfies the inequality (6.55).
We define Γ η as the maximal solution in ω[Γ] to the following ODE: where T η is the field of unit vectors orthogonal to the modified set of leading segments of ω [Γ]. Notice that Γ η uniformly tends to Γ as η → 0. In particular, for small enough values of η, the new curve will cover the whole ω[Γ], i.e. one has ω[Γ] = ω[Γ η ]. We denote the domain of Γ η by [0, l η ], noting that in general l η = l. Both T an T η are Lipschitz continuous on any compact subset of ω, the measure of the set ω η [Γ] on which T = T η is proportional to η. This implies that on ω[Γ η ] one has a point-wise estimate |T (t) − T η (t)| ≤ Cη, where C is proportional to the Lipschitz constant. It follows that the solution of (6.59) converges to Γ uniformly as η → 0. To each leading segment of ω[Γ] correspond unique t ∈ [0, l] andt ∈ [0, l η ] such that Γ(t) and Γ η (t) belong to this segment. Let us denote by ζ η the corresponding map ζ η : t →t. It is not difficult to see that ζ η is differentiable with ζ ′ η converging to 1 as η → 0. A direct calculation yields the following expression for the directional derivative of T (x ′ ) in the direction of T (x ′ ): where (t, s) is such that x ′ = Γ(t) + sN (t). In particular, on the set ω Now we can construct a new surface on ω[Γ η ]. To this end we will use the equations analogous to (6.50) and (6.51). First, we need to adjust the non-zero principal curvature of the surface κ to compensate the shift of the leading curve Γ η along the leading segments, so let .
We define the leading curve on the surface by with a unique Darboux frame satisfying the following system of equations: with the initial conditions τ η (0), ν η (0), n η (0) = τ (0), ν(0), n(0) , where ζ −1 η is the inverse of ζ η . Then the new surface U η is defined by for suitable values oft and s.
with the initial conditions τ η • ζ η (0), ν η • ζ η (0), n η • ζ η (0) = τ (0), ν(0), n(0) . Due to (6.60) and the properties of ζ η the coefficients on the right hand side of (6.63) converge to the coefficients of system (6.50) in L ∞ norm. This implies Analogously we have where the terms on the right hand sides are functions oft ∈ [0, l ε ], cf. (6.62). Let us substitutê . We obtain, in particular, via (6.60) and (6.61) that Then from (6.56), (6.57) and (6.64) we derive the following convergence properties: where II η is the matrix of the second fundamental form of U η . The estimates (6.58), (6.65) imply the validity of (6.55). Indeed, since the number of bodies and arms of u is finite, we can carry out the above construction for every arm (on bodies the surface remains unchanged), and then assemble an H 2 (ω) isometric surface starting from an arbitrary arm or body and attaching sequentially to a current piece the next one by applying a rigid motion, so that the resulting surface satisfies the properties analogous to (6.65). Then choosing a sufficiently small η and denoting the corresponding surface by u δ and the union of all S Γ η by S δ yields the estimate (6.55). Finally, we split each arm of u δ into smaller arms ω[Γ r ] corresponding to the cylindrical parts with rational directions according to the construction and their complement.
We conclude the proof with an argument for additional regularity of u δ , which is claimed in the statement. On each arm we can approximate the corresponding curvature κ in the L 2 -norm by an infinitely smooth function. We then reconstruct the pieces of the new surface using the equations (6.50), (6.51) and γ ′ = τ with appropriate initial conditions, and attach them together in a fashion similar to the above. Taking sufficiently good approximations of κ we ensure the validity of the estimate (6.55) for the new surface.
The following scaling argument allows us to obtain the boundedness of II δ in ω. Due to the smoothness of κ and (6.53) the matrix of the second fundamental form II δ is bounded on any compact subset of ω. Without loss of generality we can assume that ω contains the origin. Since ω is convex by the assumption, it is a star-shaped domain, i.e. ω η := (1 − η)ω ⊂ ω, where we use η again to denote a small positive parameter. Let us define the scaled surface: u δ,η (x ′ ) := (1−η) −1 u δ (1−η)x ′ , x ′ ∈ ω. It is easy to see that u δ,η is an isometric surface (this is why we need to use the coefficient (1 − η) −1 in front of u δ ). In addition, one has II δ,η (x ′ ) = (1 − η)II δ (1 − η)x ′ and II δ,η ∈ L ∞ (ω). Therefore, for any quadratic form Q, in particular for Q sc hom , we obtain Hence, choosing small enough η and re-denoting u δ,η by u δ we arrive at (6.55).
7 Limit functional and Γ-convergence in the "supercritical" regime h ≪ ε 2 As follows from the compactness result of Section 3 and the isometry constraint of Section 5, in the case when h ≪ ε 2 we should define the homogenised energy functional as Zero determinant condition is rather restrictive in the case of periodic functions. Next we state a theorem describing the structure of periodic functions ψ subject to this condition, first recalling that II can be represented in the form (6.53). The coefficient −κ(1 − sκ Γ ) −1 is one of the two principal curvatures of u, the other principal curvature is always zero.
Proof. The proof is based on a result by Pakzad in [19]. Let us denote v(y) := 1 2 y 2 1 II 11 + y 1 y 2 II 12 + 1 2 y 2 2 II 22 + ψ, so that det(∇ 2 y v) = 0 is equivalent to equation (7.68). Then ∇v is continuous and for any regular convex domain D in R 2 and any y ∈ D there exist a segment of line passing through y and connecting two points on the boundary of D such that ∇ y v = IIy + ∇ y ψ is constant along this segment. Since the periodicity and continuity (by the Sobolev Embedding Theorem) of ψ imply its boundedness, it follows that the segment along which ∇ y v is constant has to be parallel to the vector (−T 2 , T 1 ) ⊤ : along this direction IIy is constant. Hence ∇ y ψ is constant in the direction (−T 2 , T 1 ) ⊤ , and the derivative of ψ in this direction has to be zero due to the boundedness of ψ. The rest of the statement of the theorem easily follows from the periodicity of ψ.
The above theorem implies that at the points x ′ where the principal directions of the isometric surface u are irrational we have Q sc hom (II) = Y Q 2 (y, II)dy, while on the rational directions the corrector ψ is non-trivial and in general one has Q sc hom (II) < Y Q 2 (y, II)dy. This means that in general Q sc hom (II) is an everywhere discontinuous function of the direction T . We can write ψ in the form ψ(y 1 , y 2 ) = − κ 1 − sκ Γ ψ T (T 1 y 1 + T 2 y 2 ), (7.69) where ψ T (t) is the solution of the minimisation problem min Y Q 2 y, II(1 + ψ ′′ T (T 1 y 1 + T 2 y 2 )) dy, ψ T (t) ∈ H 2 per (0, P ), (7.70) with II defined by (6.54). Notice that the Sobolev embedding theorem implies that ψ T (t) is continuously differentiable.

2.
Recovery of the lower bound. For every u ∈ H 2 iso (ω) there exists a sequence u rec h ∈ H 1 (Ω) such that ∇ ′ u rec h converges to ∇ ′ u strongly in L 2 (Ω) and Proof. Lower bound. The proof follows the derivation of the lower bound in the case when ε ≫ h ≫ ε 2 , up to the last inequality in (4.34), whose right-hand side can have Q m hom replaced by Q sc hom given by (7.67), thanks to Theorem 5.1. Recovery sequence. A natural method of approximating the deformation of an elastic plate is to first write an expression u = u(x ′ ) for the mid-surface of the deformed plate, which takes into account the microscopic oscillations in the definition of the homogenised functional (represented by the function ψ), and then "add thickness" to this surface. One possible approach to deriving the mid-surface expression is via an asymptotic expansion, with u(x ′ ) as the leading order term followed by a series of correctors in sequential powers of ε. In the case h ≪ ε 2 , the first corrector involving the function ψ is shown to satisfy equation (7.68), see Section B, which is the solvability condition for the equation at the next order in ε. However, higher-order equations all involve two unknown functions and their solution is not straightforward. As was already mentioned in the previous section, in our construction we adopt a different approach based on the properties of isometric immersion, which we describe next. The details of the formal asymptotic argument are given in Section B.
As was shown in Theorem 7.1, the corrector ψ is constant whenever T is irrational. This implies that if the measure of the set on which T is rational is zero, then the limit elastic energy is simply Thus, the only case that remains to be considered is when the above mentioned set has non-zero measure.
It follows immediately that ∇ ′ u ε − ∇ ′ u δj L ∞ (ω[Γr]) → 0. (7.74) We carry out this procedure for all rational arms ω[Γ r ] leaving the surface unchanged on the rest of ω. We then assemble a new surface denoting it by u ε in the same way it was done in the proof of Theorem 6.8 using rigid motions. It is clear from the construction that u ε → u δj in W 1,∞ (ω) and in H 1 (ω).
We conclude that whered j := (∇ ′ u, n) ⊤ d j , and ω = ω r ∪ ω ir ∪ S δ is the natural decomposition of ω into the union of rational arms ω r = Γr ω[Γ r ], the residual set of rational segments S δ and the rest of the set ω ir . We understand the notation ψ T,j as a family of functions whose members are related to rational arms ω[Γ r ] via the corresponding directions T . Now we are ready to define a recovery sequence and prove the convergence of the elastic energy. We use the frame indifference of W , its Taylor expansion near I 3 , the unfolding operator and (7.78) to get ωir ∩S δ Y Q 3 y, T ε (II|d j ) dx ′ dy + 1 12 ωr Y Q 3 y, T ε (−(κ + κψ ′′ T,j ) II|d j ) dx ′ dy + Ω O(ε)dx Notice that we can use the Taylor expansion of W due to the fact that II ∈ L ∞ (ω) by Theorem 6.8. Passing to the limit in the above as h → 0 and taking into account the properties of two-scale convergence we arrive at lim h→0 h −2 E h (u δj ,h ) = 1 12 ωir∩S δ Y Q 3 y, (II|d j ) dx ′ dy + 1 12 ωr Y Q 3 y, (−(κ + κψ ′′ T,j (T · y)) II|d j ) dx ′ dy.
For the term corresponding to the residual set S δj we have, in general, however thanks to the estimate (7.71) it can be controlled. Thus, for each j we can choose ψ T,j and d j so that The latter, together with (7.71), implies that we can choose a subsequence h j → 0 so that Passing to the limit as j → ∞ we get the convergence of energies. Thus u rec h := u δj ,hj is a recovery sequence. The convergence of the gradients follows from (6.55) (7.75) and the construction procedure.
A Appendix: Two-scale convergence and its properties Let ω ⊂ R d be a bounded domain and Y := [0, 1) d (in the present paper d = 2). We denote by [x] the integer part of x, and by {x} its fractional part, i.e. [x] ∈ Z d , {x} := x − [x] ∈ Y . The definition uses the unfolding operator which maps functions of one variable x to functions of two variables x and y, where the "fast" variable y is responsible for the function behaviour on ε-scale: otherwise, for a measurable on ω function v(x). Notice that T ε (v)(x, y) is set to be zero on the cells intersecting with the boundary of ω. This implies, in particular, that if v ∈ H 1 (ω) then T ε (v) ∈ L 2 (ω; H 1 (Y )). One has where Λ ε := {x ∈ ε(ξ + Y ) ∩ ω : ε(ξ + Y ) ∩ ∂ω = ∅, ξ ∈ Z d }.
Definition A.1. We say that a bounded in L p (ω) sequence v ε converges weakly (strongly) two-scale to v ∈ L p (ω × Y ) as ε → 0 and denote this by v ε converges to v weakly (strongly) in L p (ω × Y ).
We refer the reader to [5] for a comprehensive analysis of the properties of the unfolding operator T ε , which, in particular, links the convergence of sequences in L 2 (ω) to convergence of their unfoldings in L 2 (ω × Y ).
We also need a formal definition of two-scale convergence when not all variables have their fast scale counterparts, which is just a variation of the standard definition.

B Appendix: Formal asymptotic expansion for a recovery sequence
Eliminating the terms that contain derivatives of ϕ and recalling that det II = 0 yields det(II + ∇ 2 y ψ) = 0.