Periodically wrinkled plate model of the Föppl-von Kármán type

Periodically wrinkled plate model of the F¨oppl-von K´arm´an type

IGORVELCIˇ C´

Abstract. In this paper we derive, by means of0-convergence, the periodically wrinkled plate model starting from three dimensional nonlinear elasticity. We assume that the thickness of the plate ish²and that the mid-surface of the plate is given by(x1,x2)!(x1,x2,h²✓(^x_h¹,^x_h²)), where✓is[0,1]²periodic function.

We also assume that the strain energy of the plate has the orderh⁸=(h²)⁴, which corresponds to the F¨oppl-von K´arm´an model in the case of the ordinary plate. The obtained model mixes the bending part of the energy with the stretching part.

Mathematics Subject Classification (2010): 74K20 (primary); 74B20 (sec- ondary).

1. Introduction

The study of thin structures is the subject of numerous works in the theory of elas- ticity. Many authors have proposed two-dimensional shell and plate models and we come to the problem of their justification. There is a vast literature on the subject of plates and shells (see [9, 10]).

The justification of the model of plates and shells, by using0-convergence is well established. The first works in that direction are [17, 18]. The thickness of the plate is assumed to beh, a small parameter, and the external loads are assumed to be of the order 0. The obtained model for plate and shells differs from the one obtained by the formal asymptotic expansion in the sense that additional relaxation of the energy functional is done.

From the pioneering work of Friesecke, James, M¨uller [13] higher order mod- els of plates and shells are justified from three dimensional nonlinear elasticity (see [13–16, 20, 21]. Here, higher order, relates that we assume that the magni- tude of the external loads (i.e. of the strain energy) behaves likeh^↵,↵ > 0 (i.e.

h , >0). Depending on different parameter↵different lower-dimensional mod- els are obtained (see [14]).

Different influence of the imperfections of the domain on the model is also discussed in the literature. In [5, 7] it is assumed that the stored energy is non- homogeneous function and oscillates with the orderh, as the thickness of the plate, Received April 21, 2011; accepted in revised form July 20, 2011.

but the strain energy (after divided by the order of volumeh) is assumed to be of the order 0. This model thus corresponds to the one given in [17] for the ordinary plate.

Also, the influence of non oscilating imperfections of the domain on the F¨oppl-von K´arm´an plate model is discussed in [22] (elastic pre-deformation is a special case of these imperfections). The special case of shallow shell and weakly curved rod is discussed in [28, 29]. All these models do not include periodic wrinkles which we discuss here. Here we assume that the thickness of the plate ish²and that the mid- surface of the plate is given by(x1,x2)!(x1,x2,h²✓(^x_h¹,^x_h²)), where✓is[0,1]² periodic function. We also assume that the strain energy of the plate (divided by the order of volumeh²) has the orderh⁸=(h²)⁴, which corresponds to the F¨oppl-von K´arm´an model in the case of the ordinary plate. The obtained model mixes the bending part of the energy with the stretching part. This model is obtained in the procedure of simultaneous homogenization and dimensional reduction. Recently such procedure is done to obtain bending model of rods (see [24]). Some partial re- sults are obtained in the bending case of plates (see [23]). However, in this case we consider the F¨oppl-von K´arm´an model for plates and suppose that we have oscillat- ing elastic pre-deformation, but not oscillating changes of the material itself. Let us just mention that applying the same type of wrinkles in the case of von K´arm´an rod model would not influence the modeli.e. we would obtain the usual von K´arm´an rod model.

We could try to generalize these periodic wrinkles to the general prestress im- perfections of domain (like it is done in [22] for non oscillating prestress or in [23]

for the oscillating prestress in the bending case for rod). But it is important to see that our model depends on the pre-deformation ✓₀ (see (4.60)), thus not only on the derivatives of ✓. To deal with periodic wrinkles we use the tool of two-scale convergence. The wrinkled plates model, derived from two dimensional linear Koi- ter shell model, are derived in [2, 3]. This model (and its linearization) is different from those ones, which is not unexpected, since we derive the model from three dimensional nonlinear theory and the thickness of the plate is of the same order as the amplitude of the mid-surface.

Throughout the paperA¯or{A} denotes the closure of the set. By a domain we call a bounded open set with Lipschitz boundary. Idenotes the identity matrix, by SO(3)we denote the rotations inR³and by so(3)the set of antisymmetric matrices 3⇥3. ByRⁿsym^⇥n we denote the set of symmetric matrices of the dimensionn⇥ n. eE1,Ee2,eE3 are the vectors of the canonical base inR^3. ! denotes the strong convergence and * the weak convergence. By A· Bwe denote tr(A^TB). We suppose that the Greek indices↵, take the values in the set{1,2}while the Latin indicesi,j take the values in the set{1,2,3}.

2. Setting up the problem

Let!be a two-dimensional domain with Lipschitz boundary in the plane spanned byeE1,eE2. The canonical cell inR² we denote byY = [0,1]²; the generic point in

Y isy = (y1,y2). By a periodically wrinkled plate we mean a shell defined in the following way. Let ✓ : R² ! R^{be aY}-periodic function of classC². We call✓ the shape function. We consider a three-dimensional elastic shell occupying in its reference configuration the set{ ˆ^h} , where

ˆ

^h=2E^h(^h), ^h =!⇥ h² 2 ,h²

2

!

;

the mapping2E^h : {^h} !R³is given by E

2^h(x^h)=

✓

x1,x2,h²✓

✓x1

h ,x2

h

◆◆

+x₃^hnE^h(x1,x2)

for allx^h = (x⁰,x₃^h)2¯^h, wherenE^h is a unit normal vector to the middle surface E

2^h(!)of the shell. By we denote ¹ and by x₃ we denote ^x_h^3h₂ and by x⁰ we denotex⁰=(x1,x2). At each point of the surface!¯ the vectornE^his given by

nE^h(x1,x2)=(n^h(x1,x2)) ^1/2

✓ h@1✓

✓x1

h ,x2

h

◆

, h@2✓

✓x1

h,x2

h

◆ ,1

◆ , where

n^h(x1,x2)=h²@₁✓

✓x1

h ,x2

h

◆2

+h²@₂✓

✓x1

h,x2

h

◆2

+1.

By inverse function theorem it can be easily seen that forhh0small enough2E^his aC¹diffeomorphism (the global injectivity can be proved by adapted compactness argument, see [10, Theorem 3.1-1] for the ordinary shell). Let us by✓₀ :R²! R denote the function:

✓₀=✓ h✓i, h✓i:=

Z

Y

✓dy. (2.1)

The following theorem is easy to prove.

Theorem 2.1. There existsh₀=h₀(✓) >0such that the Jacobian matrixr E2^h(x^h) is invertible for allx^h 2¯^h and allh h₀. Also there existsC >0such that for hh0we have

detr E2^h =1+h^{2 h}(x^h), (2.2) and

r E2^h(x^h)=I hC(x1,x2) h²D(x1,x2,x3)+h³R^h₁(x^h), (2.3) (r E2^h(x^h)) ¹=I+hC(x1,x2)+h²E(x1,x2,x3)+h³R^h₂(x^h), (2.4) k(r E2^h) IkL¹(^h;R^3⇥3),k(r E2^h) ¹ IkL¹(^h;R^3⇥3)<Ch, (2.5)

where

C(x1,x2) = 0

@

0 0 @₁✓(^x_h¹,^x_h²) 0 0 @₂✓(^x_h¹,^x_h²)

@₁✓(^x_h¹,^x_h²) @₂✓(^x_h¹,^x_h²) 0 1

A, (2.6)

D(x1,x2,x3)= (2.7)

0 B@

x₃@₁₁✓(^x_h¹,^x_h²) x₃@₁₂✓(^x_h¹,^x_h²) 0 x3@₁₂✓(^x_h¹,^x_h²) x3@₂₂✓(^x_h¹,^x_h²) 0

0 0 ¹₂⇣

@₁✓(^x_h¹,^x_h²)²+@₂✓(^x_h¹,^x_h²)²⌘ 1 CA,

E(x₁,x₂,x₃) = C²(x₁,x₂)+D(x₁,x₂,x₃)

= (E1(x1,x2,x3),E2(x1,x2,x3),E3(x1,x2,x3)), (2.8) E1(x1,x2,x3) =

0

@

@₁✓(^x_h¹,^x_h²)²+x3@₁₁✓(^x_h¹,^x_h²)

@₁✓(^x_h¹,^x_h²)@₂✓(^x_h¹,^x_h²)+x3@₁₂✓(^x_h¹,^x_h²) 0

1

A, (2.9)

E₂(x₁,x₂,x₃) = 0

@

@₁✓(^x_h¹,^x_h²)@₂✓(^x_h¹,^x_h²)+x₃@₁₂✓(^x_h¹,^x_h²)

@₂✓(^x_h¹,^x_h²)²+x3@₂₂✓(^x_h¹,^x_h²) 0

1

A, (2.10)

E3(x1,x2,x3) = 0 B@

0 0

1 2

⇣

@₁✓(^x_h¹,^x_h²)²+@₂✓(^x_h¹,^x_h²)²⌘ 1

CA, (2.11)

and ^h : ¯^h !R^{, Rh}k : ¯^h !R^{3⇥3, k}=1,2are functions which satisfy sup

0<hh0

max

x^h2¯^h| ^h(x^h)|C0, sup

0<hh0

maxi,j max

x^h2¯^h|R^h_{k,i j}(x^h)|C0, k=1,2, for some constantC0>0.

Proof. It is easy to see nE^h(x1,x2)= Ee3 h@1✓

✓x1

h,x2

h

◆

eE1 h@2✓

✓x1

h,x2

h

◆ Ee2

h² 2 @₁✓

✓x1

h ,x2

h

◆2

+@₂✓

✓x1

h,x2

h

◆2!

Ee3+h³o^h₁(x1,x2),(2.12) where

sup

0<hh0

max

x^h2¯^h|o^h₁(x1,x2)|C, sup

0<hh0

max

x^h2¯^h|@_↵o^h₁(x1,x2)| C h,

for someC>0 and↵=1,2. From the definition2E^h we conclude that r E2^h(x⁰,x₃^h) = I+h@1✓

✓x1

h ,x2

h

◆

eE3⌦ Ee1+h@2✓

✓x1

h ,x2

h

◆ Ee3⌦ Ee2

+

✓ x₃^h@₁nE^h

✓x₁ h ,x₂

h

◆

,x₃^h@₂nE^h

✓x₁ h,x₂

h

◆ ,nE^h

◆

. (2.13) The relation (2.3) is the direct consequence of the relations (2.12) and (2.13). The relations (2.2), (2.4), (2.5) are the direct consequences of the relation (2.3).

The starting point of our analysis is the minimization problem for the wrinkled plate. The strain energy of the wrinkled plate is given by

K^h(Ey)= Z

ˆ^h

W(r Ey(x))dx,

where W : M^3⇥3 ! [0,+1]is the stored energy density function. W is Borel measurable and, as in [13–15], is supposed to satisfy

i) W is of classC²in a neighborhood of SO(3);

ii) W is frame-indifferent, i.e., W(F) = W(RF)for everyF 2 R^{3⇥3 andR} 2 SO(3);

iii) W(F) CWdist²(F,SO(3)), for someCW >0 and allF2R^{3⇥3, W(F)}=0 ifF2SO(3).

By Q3:R^3⇥3!Rwe denote the quadratic formQ3(F)= D²W(I)(F,F)and by Q2:R^2⇥2!Rthe quadratic form,

Q₂(G)= min

aE2R³Q₃(G+aE⌦ Ee₃+eE₃⌦ Ea), (2.14) obtained by minimizing over the stretches in the x3 directions. Using ii) and iii) we conclude that both forms are positive semi-definite (and hence convex), equal to zero on antisymmetric matrices and depend only on the symmetric part of the variable matrix,i.e.we have

Q3(F)= Q3(symF), Q2(G)=Q2(symG). (2.15) Also, from ii) and iii), we can conclude that both forms are positive definite (and hence strictly convex) on symmetric matrices. For the special case of isotropic elasticity we have

Q3(F) = 2µ|F+F^T

2 |²+ (trF)², Q2(G) = 2µ|G+G^T

2 |²+ 2µ

2µ+ (trG)². (2.16)

Here µ, are Lam´e constants (see e.g. [8]). Since the strain energy is the most difficult part to deal with (see Theorem 4.15) we shall look for the 0-limit of the functional

I^h(Ey)= 1 h⁸

1

h²K^h(y).E

The reason why we divide by h⁸ is because we are interested in the F¨oppl-von K´arm´an type model of the wrinkled plate (the thickness of the plate is h²). The reason why we additionally divide K^h byh² is because the volume ofˆ^h is de- creasing with the order h². In the third section we prove some technical results about two scale convergence which we need later, in the forth section we prove the 0-convergence result. To prove it we firstly need the compactness result which tells us how the displacements of the energy orderh⁸look like and secondly we have to prove lower and upper bound which is standard in0-analysis.

3. Two-scale convergence

For the notion of two scale convergence see [4, 25]. Here ! ⇢ Rⁿ is a bounded Lipschitz domain andY = [0,1]ⁿ. Fork = (ki)_i=1,...,n 2Zⁿ we denote by|k| = (Pn

i=1k_i²)^1/2.

We denote by C_#^k(Y)the space of k-differentiable functions with continuous k-th derivative in Rⁿ which are periodic of periodY. Then L²_#(Y)(respectively H_#^m(Y)is the completion for the norm ofL²(Y)(respectively H^m(Y)ofC_#¹(Y)).

Remark that L²_#(Y) actually coincides with the space of functions in L²(Y) ex- tended by Y-periodicity to the whole ofR^{n. H}#¹(Y)coincides with the space of functions in H¹(Y)which areY-periodic at the boundary in the sense of traces and H_#²(Y)coincides with the space of functions in H²(Y) which are with their first derivativesY-periodic at the boundary in the sense of traces etc. Using the Fourier transform on torus it can be seen that (seee.g.[27])

H_#^m(Y)=

⇢

u, u(y)= X

k2Zⁿ

c_ke^2⇡ik·y,c_k =c _k,X

k2Zⁿ|k|^2m|c_k|²<1 , (3.1) and the normkukH_#^m(Y) is equivalent to the norm{P

k2Z²(1+ |k|^2m)|ck|²}^1/2. We also set

H˙_#^m(Y)=

⇢

uof type(3.1), c0=0} = {u,u2 H_#^m(Y), Z

Y

u=0 . The restricted normkukH˙_#^m(Y)is equivalent to the norm{P

k2Z²|k|^2m|ck|²}^1/2. The space L²(!;C#(Y))denotes the space of measurable and square integrable func- tions inx 2!with values in the Banach space of continuous functions,Y-periodic

iny. In the analogous way one can defineL²(!;H_#^m(Y)). Using the representation (3.1) it can be seen that

L²(!;H_#^m(Y)) =

⇢

u, u(x,y)= X

k2Zⁿ

c_k(x)e^2⇡ik·y,c_k 2L²(!;R^n),ck =c _k, X

k2Zⁿ|k|^2mkckk²_L²_(!)<1 , (3.2) where we have by c_k denoted the conjugate of c_k. The normkukL²(!;H_#^m(Y)) is equivalent to the norm

⇢ X

k2Z²

(1+ |k|^2m)kckk²_L²_(!)

1/2

.

The space D^(!;C_#¹(Y)) denotes the space of infinitely differentiable functions which take values inC_#¹(Y)with compact support in!. It is easily seen that this space is dense inL²(!;H_#^m(Y)). In fact it can be seen that the space of finite linear combinations

F L(!;C_#¹(Y)) =

⇢

u,9n2Nu(x,y)= X

k2Z^n,|k|n

c_k(x)e^2⇡ik·y,

ck 2C¹₀ (!;R^2),ck =c k

is dense inL²(!;H_#^m(Y)).

Definition 3.1. A sequence(uh)_h>0of functions inL²(!)converges two-scale to a functionu0belonging toL²(!⇥Y)if for every 2L²(!;C#(Y)),

Z

!

uh(x)

✓ x,x

h

◆

! Z

!

Z

Y

u0(x,y) (x,y) ash!0.

By**we denote the two-scale convergence. The following theorems are given in [4].

Theorem 3.2. Let f 2 L¹(!;C_#(Y)). Then f(x,_h^x)is a measurable function on

!for which it is valid:

kf

✓ x,x

h

◆

kL¹(!)  Z

!

sup

y2Y|f(x,y)|dy =:kfkL¹(!;C#(Y)), (3.3) and

hlim!0

Z

!

f

✓ x,x

h

◆ dx =

Z

!

Z

Y

f(x,y)dydx. (3.4)

Theorem 3.3. From each bounded sequence(uh)_h>0 in L²(!)one can extract a subsequence, and there exists a limitu0(x,y) 2 L²(!⇥Y)such that this subse- quence two-scale converges tou₀.

The following theorem tells us about the form of oscillations of order h of weakly convergent sequences inH¹(!).

Theorem 3.4. Let (uh)_h>0 be a bounded sequence in H¹(!) which converges weakly to a limit u 2 H¹(!). Then uh two-scale converges to u(x), and there exists a unique functionu1(x,y)2L²(!; ˙H_#¹(Y))such that, up to a subsequence, (ruh)_h>0two-scale converges torxu+ryu1(x,y).

Remark 3.5. In the definition of two-scale convergence we have taken the test functions to be in the spaceL²(!;C#(Y)). When we are dealing with the sequence of the functions which are bounded in L²(!)it is enough to take the test function to be in the spaceD^(!;C_#¹(Y)).

The following lemmas will be needed later.

Lemma 3.6. Let(u_h)_h>0 be a bounded sequence in H¹(!)and let there exists a constant C > 0 such that ku_hkL²(!)  Ch². Then we have that (u_h)_h>0 and (ruh)_h>0two-scale converge to 0.

Proof. That(u_h)_h>0 two-scale converges to zero is the direct consequence of the fact that strong convergence implies two-scale convergence to the same limit (not depending ony2Y). Let us now take 2D^(!;C_#¹(Y)). Then we have

Z

!

@_iuh(x)

✓ x,x

h

◆ dx =

Z

!

uh(x)@_x

✓ x,x

h

◆ dx 1

h Z

!

u_h(x)@_y

✓ x,x

h

◆ dx.

Since the both terms on the right hand side converge to 0, due to the fact that ku_hkL²(!) Ch²we have the claim.

The following characterization of the potentials is needed

Lemma 3.7. LetuE 2 L²(!⇥Y;Rⁿ⁾be such that for each E 2 D^(!;C_#¹(Y))ⁿ which satisfiesdivy E =0, 8x,ywe have that

Z

!

Z

Yu(x,E y)E(x,y)dydx=0.

Then there exists a unique functionv 2 L²(!; ˙H_#¹(Y))such that ryv =u. In the same way, letUE 2 L²(!⇥Y;R^n⇥n)be such that for each9E 2D^(!;C_#¹(Y))^n⇥n which satisfiesPn

i,j=1@_yiyj9E^{i j} =0, 8x,ywe have that Z

!

Z

Y

UE(x,y)·9(xE ,y)dydx=0. (3.5) Then there exists a unique functionv2L²(!; ˙H_#¹(Y))such thatr²yv=UE.

Proof. We shall only prove the second claim since the first goes in the analogous way. Let us define the operator r²y : L²(!; ˙H_#²(Y)) ! L²(!⇥ Y;R^{n⇥n) by} v ! r²yv. Let us identify the space L²(!; ˙H_#²(Y)) with the sequences of functions

L²(!; ˙H_#²(Y))=

⇢

(ck)_k2Zⁿ,ck 2L²(!;R^2),ck =c k, Z

!

X

k2Zⁿ|k|⁴|ck(x)|²dx <1 , with the norm

k(ck)_kk²= Z

!

X

k2Zⁿ|k|⁴|ck(x)|²dx and the spaceL²(!⇥Y;R^n⇥n)with

L²(!⇥Y;R^n⇥n) =

⇢

(c_k^{i j})_k2Zⁿ_,i,j=1,...,n,c_k^{i j}2L²(!;R^{2),ci j}k =c^{i j}_k, Z

!

X

k2Zⁿ

X

i,j=1,...n

|c^{i j}_k (x)|²dx <1 ,

with the norm

k(c^{i j}_k )_kk²= Z

!

X

k2Zⁿ

X

i,j=1,...,n

|c_k^{i j}(x)|²dx.

The operatorr²y operates in the following way

r²y(ck)_k =((kikjck)_i,j=1,...,n)_k2Zⁿ.

It is easily seen that r²y is continuous and one to one. We shall prove that it is enough to demand the condition (3.5) for 9E 2 F L(!;C_#¹(Y))^n⇥n. Using the properties of the Fourier transform the condition (3.5) can be interpreted in the following way: For given((UE_k^{i j})_i,j=1,...,n)_k2Zⁿ 2 L²(!⇥Y;(R^{2)n⇥n)and every} ((d_k^{i j})_i,j=1,...,n)_k2Zⁿ 2F L(!;C_#¹(Y))^n⇥nwhich satisfies the property

X

i,j=1,...,n

kikjd_k^{i j} =0, 8k 2Z^{n, (3.6)}

we have that

Z

!

X

k2Zⁿ

X

i,j=1,...n

UE_k^{i j}(x)d^{i j}_k(x)dx =0.

By fixingk⁰2Z^{nand takingd}_k^{i j0} 2C₀¹(!;R²), which satisfies X

i,j=1,...,n

k⁰_ik⁰_jd_k^{i j}₀=0

and definingd^{i j}_k0 =d^{i j}_k0,d_k^{i j} =0, 8k6=k⁰, k⁰,i,j =1, . . . ,nwe conclude that Z

!

Re(UE_k^{i j}0(x)d^{i j}_k0(x))dx =0.

From this it can be easily seen that there exists v_k0 2 L²(!;R^{2)such thatUE}_k^{i j0} = k⁰_ik⁰_jv_k0, for alli,j = 1, . . . ,n. This is valid for an arbitrary k0 2 Z^{n and we} can easily conclude from the fact ((UE_k^{i j})_i,j=1,...,n)_k2Zⁿ 2 L²(! ⇥ Y;(R^2)n⇥n) thatR

!

P

k2Zⁿ|k|⁴|v_k(x)|²dx < 1. Now we have the claim by takingv(x,y)= (v_k(x))k ⌘P

k2Zⁿv_k(x)e^2⇡ik·y.

Lemma 3.8. Let(uh)_h>0be a sequence which converges strongly touin H¹(!).

Let(vE_h)_h>0be a sequence which is bounded inH¹(!;Rⁿ⁾and for which is valid kruh vE_hkL²(!;Rⁿ⁾Ch², (3.7) for someC>0. Then there exists a uniquev2L²(!; ˙H_#²(Y))such thatr Ev_h **

r²u(x)+ry²v(x,y).

Proof. It is easily seen thatvE_h ! ru weakly in H¹(!;R^{n)and thusu}2H²(!).

By using Theorem 3.3 we conclude that there exists8E^{i j} 2L²(!⇥Y)such that r Evⁱ_h **(@_i1u(x)+8Eⁱ¹(x,y), . . . ,@_inu(x)+8Eⁱⁿ(x,y)).

To show the existence of v we shall use Lemma 3.7. Let us take 9E 2 D^(!; C_#¹(Y))^n⇥n which satisfies

Xn i,j=1

@_yiyj9E^{i j} =0, 8x,y (3.8)

and let us calculate Z

!

Z

Y

E

8·9dydxE = lim

h!0

Z

!

X

i,j=1,...,n

(@_jvE_hⁱ(x) @_{i j}u(x))·9E^{i j}

✓ x,x

h

◆!

dx

= lim

h!0

Z

!

X

i,j=1,...,n

(vEⁱ_h(x) @_iu(x))@_xj9E^{i j}

✓ x,x

h

◆ dx

hlim!0

1 h

Z

!

X

i,j=1,...,n

(vEⁱ_h(x) @_iu(x))@_yj9E^{i j}

✓ x,x

h

◆ dx using (3.7) = lim

h!0

1 h

Z

!

X

i,j=1,...,n

(@_iu^h(x) @_iu(x))@_yj9E^{i j}

✓ x,x

h

◆ dx

= lim

h!0

1 h

Z

!

X

i,j=1,...,n

(u^h(x) u(x))@yjxi9E^{i j}

✓ x,x

h

◆ dx

hlim!0

1 h²

Z

!

X

i,j=1,...,n

(u^h(x) u(x))@yiyj9E^{i j}

✓ x,x

h

◆ dx using (3.8) = lim

h!0

Z

!

X

i,j=1,...,n

(@_ju^h(x) @_ju(x))@xi9E^{i j}

✓ x,x

h

◆ dx +lim

h!0

Z

!

X

i,j=1,...,n

(u^h(x) u(x))@xixj9E^{i j}

✓ x,x

h

◆

dx =0.

Remark 3.9. In the special caseC = 0 Lemma 3.8 is just the generalization of Theorem 3.4. In fact what lemma tells us is that the claim is also valid if we are closer to the gradient than the order of the oscillations.

Lemma 3.10. LetQ :Rⁿ !Rbe a convex function which satisfies

|Q(x)|C(1+ |x|²), 8x 2R^{n, (3.9)} for someC > 0. Let(uE^h)_h>0 ⇢ L²(!;Rⁿ⁾be a sequence which two-scale con- verges touE₀2 L²(!⇥Y;Rⁿ). Then we have that

Z

!

Z

Y

Q(uE0(x,y))dydx lim inf

h!0

Z

!

Q(uE^h(x))dx. (3.10) Proof. Let us take an arbitrary E 2 (L²(!;C#(Y)))ⁿ. It is well known that if a convex function is finite on an open set than it is continuous. ThusQis continuous.

Also an arbitrary convex function is a pointwise limit of an increasing family of smooth convex Lipschitz functions Qn. To see this first we use the fact that there exists an increasing family Qen of piecewise affine functions (with finitely many

cuts) which pointwise converge to Q. Then we defineQee_n = Qen 1

n. Finally we smooth everyQee_n by an appropriate mollifier to preserve the fact that the sequence should be increasing. We obtain for eachn2N,h>0

Z

!

Qn(uE^h(x))dx Z

!

Qn

✓ E

✓ x,x

h

◆◆

dx +

Z

!

D Qn

✓ E

✓ x,x

h

◆◆ ✓

uE^h(x) E

✓ x,x

h

◆◆

dx.

(3.11)

By letting h ! 0 and using the definition of two-scale convergence and Theo- rem 3.2 and the fact that the convexity of Qn and|Qn(x)| C(1+ |x|²)implies

|D Qn(x)|C(1+ |x|)we obtain for eachn2N lim inf

h!0

Z

!

Qn(uE^h(x))dx Z

!

Z

Y

Qn( (x,E y))dydx +

Z

!

Z

Y

D Q_n( (x,E y))(uE₀(x,y) (x,E y))dx. (3.12)

By using an arbitrariness of E and the density of L²(!;C#(Y))inL²(!⇥Y)we conclude that for eachn2N

lim inf

h!0

Z

!

Qn(uE^h(x))dx Z

!

Z

Y

Qn(uE0(x,y))dydx. (3.13) SinceQn< Qwe conclude

lim inf

h!0

Z

!

Q(uE^h(x))dx Z

!

Z

Y

Qn(uE0(x,y))dydx. (3.14) Lettingn! 1and using (3.9) we conclude (3.10).

Lemma 3.11. Let(uh)_h>0be a bounded sequence inL²(!)which two-scale con- verges tou0(x,y) 2 L²(!⇥Y). Let (v_h)_h>0be a sequence bounded in L¹(!) which converges in measure tov₀2L¹(!). Thenv_huh **v₀(x)u0(x,y).

Proof. We know that(v_huh)_h>0is bounded inL²(!)and that there exists a subse- quence of(v_h)_h>0 such thatv_h !va.e. in!. Let us take 2D^(!;C#(Y))and write,

Z

!

v_h(x)uh(x) (x,x h)dx =

Z

!

(v_h(x) v(x))uh(x)

✓ x,x

h

◆ dx +

Z

!

v(x)uh(x)

✓ x,x

h

◆ dx.

(3.15)

The second converges to Z

!

Z

Y

v(x)u0(x,y) (x,y)dydx,

by the definition of two-scale convergence. We have to prove that the first term in (3.15) converges to 0. By Egoroff’s theorem for an arbitrary" > 0 there exists

E⇢!such that meas(E) <"andv_h !vuniformly on E^c. We write Z

!

(v_h(x) v(x))u_h(x)

✓ x,x

h

◆ dx=

Z

E

(v_h(x) v(x))u_h(x)

✓ x,x

h

◆ dx +

Z

E^c

(v_h(x) v(x))uh(x)

✓ x,x

h

◆ dx.

(3.16)

The second term in (3.16) converges to 0 and can be made arbitrary small. For the first term by the Cauchy inequality we have that there existsC>0 such that

Z

E

(v_h(x) v(x))u_h(x)

✓ x,x

h

◆

dx Cp

"sup

h>0ku_hkL²(!). (3.17) By the arbitrariness of"we have the claim.

4. 0-convergence

We shall need the following theorem which can be found in [13].

Theorem 4.1 (on geometric rigidity). Let U ⇢ R^m be a bounded Lipschitz do- main, m 2. Then there exists a constantC(U)with the following property: for everyvE2H¹(U;R^m)there is associated rotationR2SO(m)such that

kr Ev RkL²(U) C(U)kdist(r Ev,SO(m)kL²(U). (4.1) The constant C(U) can be chosen uniformly for a family of domains which are Bilipschitz equivalent with controlled Lipschitz constants. The constant C(U)is invariant under dilatations.

In the sequel we supposeh₀ ¹₂ (see Theorem 2.1). If this was not the case, what follows could be easily adapted. Let us by P^h :  ! ^h denote the map

P^h(x⁰,x3)=(x⁰,h²x3). Byrhwe denote rh=re_E1,Ee2+ 1

h²re_E3.

By rE^h : R² ! R² we denote the mappingrE^h(x1,x2) = (^x_h¹,^x_h²). In the same way as in [14, Theorem 10, Remark 11] (see also [20, Lemma 8.1]) we can prove the following theorem. For the adaption we only need Theorem 4.1 and the facts thatC(U)can be chosen uniformly for Bilipschitz equivalent domains and that the normskr E2^hk,k(r E2^h) ¹kare uniformly bounded on^hforh ¹₂.

Theorem 4.2. Let! ⇢ R²be a domain. Let2E^h be as above and leth  ¹₂. Let yE^h 2H¹(ˆ^h;R³⁾be such that

1 h²

Z

ˆ

^h

dist²(r Ey^h,SO(3))dx Ch⁸,

for someC>0. Then there exists mapR^h 2H¹(!,SO(3))such that

k(r Ey^h) 2E^h P^h R^hkL²() Ch⁴, (4.2) krR^hkL²(!) Ch². (4.3) Moreover there exist a constant rotationQ¯^h 2SO(3)such that

k(r Ey^h) 2E^h P^h Q¯^hkL²()  Ch², (4.4) kR^h Q¯^hk^Lp(!)  Cph², 8p<1. (4.5) Here all constants depend only on!(and on pwhere indicated).

Remark 4.3. Since2E^h is Bilipschitz map, it can easily be seen that the map Ey! yE 2E^h is an isomorphism between the spaces H¹(^h;R^m)andH1(ˆh;R^m)(see e.g.[1]).

To prove0-convergence result we need to prove the compactness result, the lower and the upper bound.

4.1. Compactness result

We need the following version of Korn’s inequality which is proved in a standard way by contradiction.

Lemma 4.4. Let! ⇢R²be a Lipschitz domain. Then there existsC(!) >0such that for an arbitraryuE2H¹(;R^{2)we have}

kEukH¹(!;R²⁾C(!)

✓

ksymr EukL²(!;R²⁾+ Z

!udxE + Z

!

(@₂uE₁ @₁uE₂)dx

◆ . (4.6) Lemma 4.5. LetyE^h 2H¹(ˆ^h;R³⁾be such that

1 h²

Z

ˆ

^h

dist²(r Ey^h,SO(3))dx  Ch⁸. (4.7) Then there exists mapsR^h 2 H¹(!,SO(3))and constantsR¯^h 2 SO(3),Ec^h 2 R³ such that

eEy^h:=(R¯^h)^TyE^h cE^h