An asymptotic strain gradient Reissner-Mindlin plate model

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An asymptotic strain gradient Reissner-Mindlin plate model

Michèle Serpilli, Françoise Krasucki, Giuseppe Geymonat

To cite this version:

Michèle Serpilli, Françoise Krasucki, Giuseppe Geymonat. An asymptotic strain gradient Reissner-

Mindlin plate model. Meccanica, Springer Verlag, 2013, pp.1-13. �10.1007/s11012-013-9719-6�. �hal-

00757357�

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An asymptotic strain gradient Reissner-Mindlin plate model

Michele Serpilli · Franc¸oise Krasucki · Giuseppe Geymonat

Received: date / Accepted: date

Abstract In this paper we derive a strain gradient plate model from the three-dimensional equations of strain gradient linearized elasticity. The deduction is based on the asymptotic analysis with respect of a small real parameter being the thickness of the elastic body we consider. The body is constituted by a second gradient isotropic linearly elastic material. The obtained model is recognized as a strain gradient Reissner-Mindlin plate model. We also provide a mathematical justification of the obtained plate model by means of a variational weak convergence result.

Keywords Asymptotic analysis·Strain gradient elasticity·Plate models·Micro-plates PACS 46.25.Cc·46.70.De

Mathematics Subject Classification (2000) 74K20·74A60

1 Introduction

Higher order gradient continuum theories in linear and nonlinear elasticity have recently raised the interest on many scientists, since modern technologies involving multi-scale ma- terials exhibit size effects and a strong dependence on internal (material) lengths. A possible generalization of Cauchy model has been proposed in the pioneering works by Toupin, [14], Mindlin, [12], and Germain, [7]. In these papers, the stored deformation energy is assumed

M. Serpilli

Department of Civil and Building Engineering, and Architecture,

Universit`a Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy Tel.: +39-071-2204554

E-mail: m.serpilli@univpm.it F. Krasucki

Institut de math´ematique et de mod´elisation de Montpellier, UMR-CNRS 5149,

Universit´e Montpellier II, CC 051, place Eug`ene-Bataillon, 34095 Montpellier cedex 5, France E-mail: krasucki@math.univ-montp2.fr

G.Geymonat

Laboratoire de M´ecanique des Solides, UMR- CNRS 7649, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau cedex, France´ E-mail: giuseppe.geymonat@lms.polytechnique.fr

to depend not only on the strain, but also on the strain gradient. These general continua are called second gradient continua by Germain, or strain gradient continua. In second gradient continua approaches it is necessary to generalize the concept of Cauchy contact actions, see [6], and the constitutive laws, see [5]. For a general overview on second gradient elasticity theories and their applications it is worth mentioning the work by Askes and Aifantis [1].

Thin plate theories have found recently several applications in the areas of micro-mechanics and nano-mechanics. Micro-mechanical systems and nano-mechanical systems show size effects and non local behavior, hence strain gradient elasticity theories find their natural and appropriate application. Furthermore, granular materials, porous materials and, generally, materials endowed with a microstructure, in which the stresses do not depend only on the local strain, can be described by strain gradient elasticity theories. For instance, Lazopoulos has derived a mechanical model for the bending behavior of strain gradient Kirchhoff-Love plates (see [9] and [10]) and shallow shells (see [11]).

In the present paper we derive a strain gradient plate model starting from the three- dimensional equations of strain gradient linearized elasticity through an asymptotic analysis.

We consider a plate-like domain filled by an isotropic second gradient linearly elastic ma- terial. By defining a small real parameterε, which represents the thickness of the plate-like domain, we apply the asymptotic expansion method, following the approach by Ciarlet in [4]. Then, we characterize the leading terms of the asymptotic expansion and the associated limit problems. In order to have a mathematical justification of the obtained model we study the weak convergence of the solution of the three-dimensional problem towards the solution of the limit problem in a precise functional framework.

The asymptotic analysis is a widely used technique for the formal derivation and justi- fication of classical theories of thin structures, starting from the classical three-dimensional elasticity, (see [4], in the case of plate models). For what concerns with the derivation of plate models, it is well-known that if we apply the asymptotic methods to the classical lin- ear or nonlinear elasticity equations, we are capable to derive only Kirchhoff-Love plate models. In order to obtain the Reissner-Mindlin plate model through an asymptotic anal- ysis or variational convergence, we need to generalize the stored elastic energy by adding some appropriate second gradient terms, see [13], or by using a different continuum model as starting point, like the micropolar continuum, see [2]. As we already mentioned, in the present approach, we use a second gradient continuum constituting the plate-like body and, by performing an asymptotic analysis, we derive a second gradient Reissner-Mindlin plate model.

The layout of this paper is as follows. In Section 2, we introduce the mathematical problem associated with the equilibrium of a strain gradient linearly elastic plate; we de- fine a small real parameterε which is related to the thickness of the plate; then we apply the asymptotic methods to obtain the simplified models. In Section 3, we present the main Ansatz (25) for the asymptotic expansions of the displacement field. Then we derive the limit displacement field, which corresponds to the Reissner-Mindlin kinematics, and its as- sociated limit problem. In Section 4, we give a mathematical justification of the obtained model by presenting a weak convergence result.

2 Statement of the problem

In the sequel, Latin indices range in the set {1,2,3}, while Greek indices range in the set{1,2}and the Einstein’s summation convention with respect to the repeated indices is adopted.

Letω∈²be a smooth domain in the plane spanned by vectorseα, letγ₀be a measurable subset of the boundaryγ of the set ω, such that lengthγ0 >0, and let 0<ε<1 be an dimensionlesssmallreal parameter which will tend to zero. For eachε, we define

Ω^ε:=ω×(−ε,ε),

Γ₀^ε:=γ₀×[−ε,ε], Γ_±^ε:=ω× {±ε}. (1) Hence the boundary of the setΩ^ε is partitioned into the lateral face γ×[−ε,ε] and the upper and lower facesΓ₊^ε andΓ₋^ε, and the lateral face is itself partitioned asγ×[−ε,ε] = (γ₀×[−ε,ε])∪(γ₁×[−ε,ε]), whereγ₁:=γ−γ₀. In order to avoid inessential complications in the sequel we suppose thatγ=γ₀, and thusγ₁=/0.

We assume that the setΩ^ε is the reference configuration of a strain gradient linearly elastic plate of thickness 2ε and middle surfaceω. We study the physical problem corre- sponding to the mechanical behaviour of a strain gradient plate. The plate is completely clamped onΓ₀^ε=Γ^ε, in the sense that the boundary conditions of place, imposed to the displacements, are

u^ε_i =0 and ∂_n^εu^ε_i =0 on Γ₀^ε, (2) where∂_n^εrepresents the derivative operator with respect to the unit outer normal vector(n^ε_i) along the boundaryΓ₀^ε. Moreover, we supposed that the plate is subjected to body forces (f_i^ε):Ω^ε→R³, and surface forces(g^ε_i):Γ₊^ε∪Γ₋^ε→R³.

We finally assume that the strain gradient linearly elastic material constituting the plate Ω^εis homogeneous and isotropic. The constitutive laws for this kind of material (see [5]) take the following form:

σ^ε(u^ε):=A^εe^ε(u^ε),

P^ε(u^ε):=B^ε∇^εe^ε(u^ε), (3) or, componentwise,

σ_{i j}^ε(u^ε) =A^ε_{i jkℓ}e^ε_kℓ(u^ε), with

A^ε_{i jkℓ}:=λ^εδi jδkℓ+2µ^ε(δikδjℓ+δiℓδjk), p^ε_{i jk}(u^ε) =B^ε_{i jkℓpq}e^ε_ℓpq(u^ε), with

B^ε_{i jkℓpq}:=c^ε₁(δi jδ_kℓδpq+δi jδk pδ_ℓq+δikδjqδ_ℓp+δiqδjkδ_ℓp) +c^ε₂δi jδkqδ_ℓp+ +c^ε₃(δikδjℓδpq+δikδj pδ_ℓq+δiℓδjkδpq+δipδjkδ_ℓq) +c^ε₄(δiℓδj pδkq+ +δipδ_jℓδkq) +c^ε₅(δ_iℓδjqδk p+δipδjqδ_kℓ+δiqδ_jℓδk p+δiqδj pδ_kℓ),

(4)

whereσ^ε= (σ_{i j}^ε)is the classical Cauchy stress tensor,P^ε= (p^ε_{i jk})is the hyperstress tensor, e^ε(u^ε) = (e^ε_{i j}(u^ε)):=

1

2(∂_i^εu^ε_j+∂^ε_ju^ε_i)

is the linearized strain tensor and∇^εe^ε(u^ε) = (e^ε_{i jk}(u^ε)):= (∂_k^εe^ε_{i j}(u^ε))is the gradient ofe^ε(u^ε).A^ε= (A^ε_{i jkℓ})andB^ε= (B^ε_{i jkℓpq})represent, respectively, the fourth order classical isotropic elasticity tensor and the sixth order isotropic strain gradient isotropic elasticity tensor. The components of the Cauchy stress tensor and the components of the hyperstress tensor can be also written as follows:

σ_{i j}^ε:=λ^εe^ε_ppδi j+2µ^εe^ε_{i j},

p^ε_{i jk}:=c^ε₁(e^ε_{pp j}δik+2e^ε_{k pp}δi j+e^ε_ppiδjk) +c^ε₂e^ε_ppkδi j+ +2c^ε₃(e^ε_{j pp}δik+e^ε_ippδjk) +2c^ε₄e^ε_{i jk}+2c^ε₅(e^ε_{ik j}+e^ε_jki),

(5) We assume thatA^εgives rise to a positive definite quadratic form on the vector space of symmetric matrices. As it is well known, this condition is satisfied if and only if 3λ^ε+2µ^ε>

0 and µ^ε>0. We also assume thatB^ε gives rise to a positive definite quadratic form on

the vector space of all third order symmetric matrices with respect to the first two indices.

In order to ensure this condition Dell’Isola et al. in [5] have proved the suffciency of the following inequalities:

c^ε₄>0, −c^ε₄

2 <c^ε₅<c^ε₄, 5c^ε₂+4c^ε₄>2c^ε₅,

c^ε₃>c^ε₂(3c^ε₄+c^ε₅) +2(c^ε2₄ −5c^ε2₁ −6c^ε₅c^ε₁−2c^ε2₅ +c^ε₄(2c^ε₁+c^ε₅)) 4c^ε₅−10c^ε₂−8c^ε₄ .

(6)

To begin with, we introduce some notations that will be used in the sequel. We let:

a·b:=aibi, A:B:=ai jbi j and C∴D:=ci jkdi jk, (7) for, respectively, all vectorsa= (ai)andb= (bi), for all symmetric second order matrices A= (ai j)andB= (bi j), and for all third order matricesC= (ci jk)andD= (di jk), symmetric with respect to the first two indices.

The displacement fieldu^ε= (u^ε_i)satisfies the following variational problem defined over the variable domainΩ^ε:

Z

Ω^ε{σ^ε(u^ε):e^ε(v^ε) +P^ε(u^ε)∴∇^εe^ε(v^ε)}dx^ε=l^ε(v^ε), (8) for allv^ε∈V(Ω^ε), where

V(Ω^ε):=

v^ε= (v^ε_i)∈H²(Ω^ε;R³); v^ε=0and∂_n^εv^ε=0onΓ₀^ε , (9) and

l^ε(v^ε):=

Z Ω^ε

f^ε·v^εdx^ε+ Z

Γ_±^ε

g^ε·v^εdΓ^ε. (10) Componentwise, we get:

Z Ω^ε

n

σ_{i j}^ε(u^ε)e^ε_{i j}(v^ε) +p^ε_{i jk}(u^ε)e^ε_{i jk}(v^ε)o dx^ε=

Z

Ω^εf_i^εv^ε_idx^ε+ Z

Γ_±^εg^ε_iv^ε_idΓ^ε, (11) for allv^ε∈V(Ω^ε). We suppose that f_i^ε∈L²(Ω^ε)andg^ε_i ∈L²(Γ_±^ε).

Proposition 1 From the assumption on the positive definiteness of the elasticity tensorsA^ε andB^ε, the variational problem (8) has a unique solutionu^εin V(Ω^ε).

For a more general formulation of the three-dimensional strain gradient non linear elas- ticity theory, the reader can refer to the work by dell’Isola et al. [5].

In order to perform an asymptotic analysis, we need to transform problem (11) posed on a variable domainΩ^εonto a problem posed on a fixed domain (independent ofε). Accord- ingly, we let

Ω:=ω×(−1,1),

Γ₀:=γ₀×[−1,1], Γ_±:=ω× {±1}. (12) Hence, we define the following change of variables (see [4]):

π^ε:x:= (˜x,x₃)∈Ω7→x^ε:= (x,εx˜ ₃)∈Ω^ε,with ˜x= (xα). (13) By using the bijectionπ^ε, one has∂_α^ε=∂αand∂₃^ε=¹_ε∂₃.

With the unknown displacement fieldu^ε= (u^ε_i)∈V(Ω^ε), we associated the scaled dis- placement fieldu(ε) = (ui(ε)):Ω→R³defined by:

u^ε_α(x^ε) =εuα(ε)(x) and u^ε₃(x^ε) =u3(ε)(x)for allx^ε=π^εx∈Ω^ε. (14) We likewise associate with any test functions vector fieldv^ε= (v^ε_i)∈V(Ω^ε), the scaled test functions vector fieldv= (vi):Ω→R³, defined by the scalings:

v^ε_α(x^ε) =εvα(x) and v^ε₃(x^ε) =v3(x)for allx^ε=π^εx∈Ω^ε. (15) We make the following assumptions on the data, and, thus, we require that the Lam´e con- stants and the second gradient elastic moduli satisfy the following relations:

λ^ε=λ, µ^ε=µ, c^ε_k=ck,k∈ {1,2,3,4,5}. (16) HenceA^ε_{i jkℓ}=Ai jkℓ andB^ε_{i jkℓpq}=Bi jkℓpqare both independent ofε. We also ask that the applied body and surface forces take the following forms:

f_α^ε(x^ε) =εfα(x) and f₃^ε(x^ε) =f₃(x) for allx^ε=π^εx∈Ω^ε,

g^ε_α(x^ε) =ε²gα(x) and g^ε₃(x^ε) =εg₃(x)for allx^ε=π^εx∈Ω^ε. (17) The elastic constantsλ,µandck, and functions fi∈L²(Ω)andgi∈L²(Γ₊∪Γ₋)are inde- pendent ofε.

Let us define, respectively, the rescaled components of the linearized strain tensorei j(u(ε)) and of its gradientei jk(u(ε)). According to the previous assumptions on the displacement field, one has:

e^ε_αβ(u^ε) =εeαβ(u(ε)) =^ε₂(∂αuβ(ε) +∂βuα(ε)), e^ε_α3(u^ε) =εeα3(u(ε)) =¹₂(∂3uα(ε) +∂αu3(ε)) e^ε₃₃(u^ε) =e₃₃(u(ε)) =¹_ε∂₃u₃(ε),

(18) and,

e^ε_{αβ γ}(u^ε) =εeαβ γ(u(ε)) =ε∂γeαβ(u(ε)), e^ε_αβ3(u^ε) =e_αβ3(u(ε)) =∂₃e_αβ(u(ε)), e^ε_α3β(u^ε) =eα3β(u(ε)) =∂βeα3(u(ε)), e^ε_α33(u^ε) =¹_εeα33(u(ε)) =¹_ε∂3eα3(u(ε)), e^ε_33α(u^ε) =¹_εe33α(u(ε)) =¹_ε∂αe₃₃(u(ε)), e^ε₃₃₃(u^ε) =_ε¹₂e₃₃₃(u(ε)) = _ε¹₂∂₃e₃₃(u(ε)).

(19)

By virtue of the relations above, we can compute the components of the rescaled hyperstress tensorpi jk(u(ε)):=B_{i jkℓ}pqe_ℓpq(u(ε))as follows

pαβ γ(u(ε)) =εp¹_{αβ γ}(u(ε)) +_ε¹p⁻¹_{αβ γ}(u(ε)), p_αβ3(u(ε)) =p⁰_αβ3(u(ε)) +_ε¹₂p⁻²_αβ3(u(ε)), p_α3β(u(ε)) =p⁰_α3β(u(ε)) +_ε¹₂p⁻²_α3β(u(ε)), pα33(u(ε)) =εp¹_α33(u(ε)) +¹_εp⁻¹_α33(u(ε)), p33α(u(ε)) =εp¹_33α(u(ε)) +¹_εp⁻¹_33α(u(ε)), p333(u(ε)) =p⁰₃₃₃(u(ε)) +_ε¹2p⁻²₃₃₃(u(ε)),

(20)

where

p¹_{αβ γ}(u(ε)):=c1[δαγeσ σ β(u(ε)) +2δαβeγσ σ(u(ε)) +δβ γeσ σ α(u(ε))]

+c₂δ_αβeσ σ γ(u(ε)) +2c₃[δαγe_{β σ σ}(u(ε)) +δ_{β γ}eασ σ(u(ε))]+

+2c₄e_{αβ γ}(u(ε)) +2c₅[e_αγβ(u(ε)) +e_{β γα}(u(ε))], p⁻¹_{αβ γ}(u(ε)):=c₁[δαγe_33β(u(ε)) +2δ_αβeγ33(u(ε)) +δ_{β γ}e33α(u(ε))]+

+c2δαβe33γ(u(ε)) +2c3[δαγeβ33(u(ε)) +δβ γeα33(u(ε))], p⁰_αβ3(u(ε)):=2c1δαβe3σ σ(u(ε)) +c2δαβeσ σ3(u(ε)) +2c4eαβ3(u(ε))+

+2c₅[e_α3β(u(ε)) +e_β3α(u(ε))], p⁻²_αβ3(u(ε)):= (2c1+c2)δαβe333(u(ε)),

p⁰_α3β(u(ε)):=c₁δ_αβeσ σ3(u(ε)) +2c₃δ_αβe3σ σ(u(ε)) +2c₄e_α3β(u(ε))+

+2c₅e_αβ3(u(ε)), p⁻²_α3β(u(ε)):= (c₁+2c₃)δ_αβe₃₃₃(u(ε)), p¹_α33(u(ε)):=c₁eσ σ α(u(ε)) +2c₃eασ σ(u(ε)),

p⁻¹_α33(u(ε)):= (c1+2c5)e33α(u(ε)) +2(c3+c4+c5)eα33(u(ε)), p¹_33α(u(ε)):=2c1eασ σ(u(ε)) +c2eσ σ α(u(ε)),

p⁻¹_33α(u(ε)):=2(c₁+2c₅)eα33(u(ε)) + (c₂+2c₄)e33α(u(ε)), p⁰₃₃₃(u(ε)):= (2c1+c2)eσ σ3(u(ε)) +2(c1+2c3)e3σ σ(u(ε)), p⁻²₃₃₃(u(ε)):= (4c₁+c₂+4c₃+2c₄+4c₅)e₃₃₃(u(ε)).

(21)

We can now reformulate the problem on the fixed domainΩ. From Proposition 1 it follows that for everyε >0 the rescaled displacement field u(ε)∈V(Ω)is the unique solution of the following rescaled problem:

1

ε⁴a₋₄(u(ε),v) + 1

ε²a₋₂(u(ε),v) +a₀(u(ε),v) +ε²a₂(u(ε),v) =l₀(v) +ε²l₂(v), (22)

for allv∈V(Ω), where

V(Ω):=

v= (vi)∈H²(Ω;R³); v=0and∂nv=0onΓ₀ . (23)

The bilinear forms a₋₄, a₋₂, a₀, a₂ :V(Ω)×V(Ω)→R and the linear forms l₀, l₂ : V(Ω)→Rare respectively defined as follows:

a₋₄(u(ε),v):=

Z

Ωp⁻²₃₃₃(u(ε))e333(v)dx, a₋₂(u(ε),v):=

Z

Ω[(λ+2µ)e₃₃(u(ε))e₃₃(v) +p⁻²_αβ3(u(ε))e_αβ3(v)+

+2p⁻²_α3β(u(ε))e_α3β(v) +2p⁻¹_α33(u(ε))eα33(v)+

+p⁻¹_33α(u(ε))e33α(v) +p⁰₃₃₃(u(ε))e333(v)]dx, a₀(u(ε),v):=

Z

Ω[λeσ σ(u(ε))e₃₃(v) +λe₃₃(u(ε))eσ σ(v) +4µeα3(u(ε))eα3(v)+

+p⁻¹_{αβ γ}(u(ε))e_{αβ γ}(v) +p⁰_αβ3(u(ε))e_αβ3(v) +2p⁰_α3β(u(ε))e_α3β(v)+

+2p¹_α33(u(ε))eα33(v) +p¹_33α(u(ε))e33α(v)]dx, a4(u(ε),v):=

Z

Ω[λeσ σ(u(ε))eττ(v) +2µe_αβ(u(ε))e_αβ(v)+

+p¹_{αβ γ}(u(ε))e_{αβ γ}(v)]dx, l0(v):=

Z

Ωf3v3dx+ Z

Γ_±g3v3dΓ, l₂(v):=

Z

Ωfαvαdx+ Z

Γ_±gαvαdΓ.

(24) 3 Asymptotic analysis

We can now perform an asymptotic analysis of the rescaled problem (22). Since it has a polynomial structure with respect to the small parameterε, we can look for the solution of the problem as a series of powers ofε:

u(ε) =u⁰+ε²u²+ε⁴u⁴+ε⁶u⁶+. . . . (25) By substituting (25) into the rescaled problem (22), and by identifying the terms with iden- tical power ofε, we obtain, as customary, the following set of problems, defined for all v∈V(Ω):

P₋₄:a₋₄(u⁰,v) =0,

P₋₂:a₋₄(u²,v) +a₋₂(u⁰,v) =0,

P₂: a₋₄(u⁴,v) +a₋₂(u²,v) +a0(u⁰,v) =l0(v),

P₄: a₋₄(u⁶,v) +a₋₂(u⁴,v) +a₀(u²,v) +a₂(u⁰,v) =l₂(v),

P_2j: a₋₄(u^2j+4,v) +a₋₂(u^2j+2,v) +a₀(u^2j,v) +a2(u^2j−2,v) =0, j≥2.

(26)

To proceed with the asymptotic analysis we need to solve each problem above and char- acterize the limit displacement fieldu⁰and the associated limit problem.

We start by solving problemP₋₄. Let us choose test functionsv=u⁰∈V(Ω):

Z

Ωp⁻²₃₃₃(u⁰)e333(u⁰)dx= Z

Ω(4c1+c2+4c3+2c4+4c₅)(e333(u⁰))²dx=0. (27) Since 4c1+c2+4c3+2c4+4c5>0, by virtue of the positive definiteness ofB, we get e333(u⁰) =0, which implies that

u⁰₃(x,˜x₃) =w⁰(x) +˜ x₃b⁰₃(˜x). (28)

Let us consider problemP₋₂. Sincee₃₃₃(u⁰) =0, we get thatp⁻²_α3β(u⁰) =p⁻²_αβ3(u⁰) = 0, thus one has:

Z

Ω[p⁻²₃₃₃(u²)e333(v) + (λ+2µ)e33(u⁰)e33(v) +2p⁻¹_α33(u⁰)eα33(v)+

+p⁻¹_33α(u⁰)e33α(v) +p⁰₃₃₃(u⁰)e333(v)]dx for allv∈V.

(29) If we choose test functionsv=u⁰∈V(Ω), problemP₋₂reads as follows:

Z Ω

"

(λ+2µ)(e₃₃(u⁰))²+4(c₃+c₄+c₅)

eα33(u⁰) + 2c₅+c₁ c3+c4+c5

e33α(u⁰) 2

+ +

c₂+2c₄−(2c₅+c1)² c₃+c₄+c₅

(e33α(u⁰))²

dx=0.

(30)

Since the assumptions onAandBimply that the coefficients multiplying the quadratic terms are positive, we obtain that

e₃₃(u⁰) =e33α(u⁰) =eα33(u⁰) =0. (31) By virtue of relations (31), the displacement fieldu⁰can be updated as follows:

u⁰_α(˜x,x3) =u¯⁰_α(˜x) +x3ϕ_α⁰(x),˜

u⁰₃(˜x,x₃) =w⁰(˜x). (32) The above displacement field corresponds to the well-known Reissner-Mindlin kinematics assumptions for a plate. Since we want to focus our attention on the flexural behavior of the plate, in the sequel we neglect the in-plane displacements ¯u⁰_α, which are associated with the membrane behavior of the plate. Hence,

u⁰_α(˜x,x₃) =x₃ϕ_α⁰(˜x),

u⁰₃(˜x,x₃) =w⁰(˜x). (33) Finally, by substitutingvα=0 andv₃=v₃(˜x,x₃)inP₋₂, we have

Z

Ω p⁻²₃₃₃(u²) +p⁰₃₃₃(u⁰)

∂₃₃v₃dx=0 for allv₃∈V(Ω), (34) which is verified whenp⁻²₃₃₃(u²) =−p⁰₃₃₃(u⁰)and so, we obtain the following characteriza- tion foru²₃:

u²₃(˜x,x₃) =a²₃(˜x) +x₃b²₃(x)˜ −x²₃ 2 ˜c

(c₁+2c₃)∂σ σa⁰₃+ (3c₁+c₂+2c₃)∂σb⁰_σ

(˜x), (35) with ˜c:=4c₁+c₂+4c₃+2c₄+4c₅.

ProblemP₀reads as follows:

Z Ω

h

p⁻²₃₃₃(u⁴)e₃₃₃(v) + (λ+2µ)e₃₃(u²)e₃₃(v) +p⁻²_αβ3(u²)e_αβ3(v)+

+2p⁻²_α3β(u²)eα3β(v) +2p⁻¹_α33(u²)eα33(v) +p⁻¹_33α(u²)e33α(v)+

+p⁰₃₃₃(u²)e333(v) +λeσ σ(u⁰)e33(v) +4µeα3(u⁰)eα3(v)+

+p⁰_αβ3(u⁰)e_αβ3(v) +2p⁰_α3β(u⁰)e_α3β(v)+

+2p¹_α33(u⁰)eα33(v) +p¹_33α(u⁰)e33α(v) dx=

Z

Ω f₃v₃dx+ Z

Γ_±g₃v₃dΓ,

(36)

for all v∈V(Ω). Let us choose test functions v∈V(Ω)such that vα(˜x,x₃) =bvα(x) +˜ x3ηα(˜x)andv3(˜x,x3) =η3(x), i.e.,˜ e333(v) =e33(v) =eα33(v) =e33α(v) =0. Hence, prob- lemP₂becomes:

Z Ω

h

p⁻²_αβ3(u²) +p⁰_αβ3(u⁰)

e_αβ3(v) +2

p⁻²_α3β(u²) +p⁰_α3β(u⁰)

e_α3β(v)+

+4µeα3(u⁰)eα3(v) dx=

Z

Ω f3η3dx+ Z

Γ_±g3η3dΓ for allv∈V(Ω). (37) Letvα=0, then we find the first limit problem verified byw⁰andϕ_α⁰:

h Z

ω

h(C₁∂_{β β}w⁰+C₂∂_βϕ_β⁰)∂ααη₃+

c₄(∂_αβw⁰+∂_βϕ_α⁰) +c₅(∂αϕ_β⁰+∂_βϕ_α⁰)

∂_αβη₃+ +µ(∂αw⁰+ϕ_α⁰)∂αη3

dx˜= Z

ωqη3dx,˜

(38) for allη₃∈V(ω):={η= (ηi)∈H¹(ω,R²)×H²(ω);η=0,∂νη₃=0 onγ₀}( by virtue of the assumptionγ₀=γ, one hasV(ω) =H₀¹(ω,R²)×H₀²(ω)), where

q(˜x):=

Z 1

−1f3(˜x,x3)dx3+g3(˜x,±1), (39) and

C1:=c3−^(c1−2c_c˜ ³⁾²,

C₂:=c₁+c₃−^(3c1+c2+2c_c˜^3)(c1+2c3). (40) If we choosev₃=0 in problem (37), we obtain the second limit problem satisfied byw⁰ andϕ_α⁰:

Z ω

h

C₂∂_{β β}w⁰+C₃∂_βϕ_β⁰

∂αηα+µ(∂αw⁰+ϕ_α⁰)ηα+ +

c4∂_βϕ_α⁰+ (c4+2c₅)(∂_αβw⁰+∂_βϕ_α⁰+∂αϕ_β⁰)

∂_βηα

i dx˜=0,

(41) for allηα∈V(ω), where

C₃:=2c₁+c₂+c₃−^(3c1+c2_c˜^+2c3)2. (42) By integrating by parts problem (38) and (41), we obtain the following differential sys-

tem: 





h(C₁∆−µ)∆w⁰+h(C₂∆−µ)divϕ⁰=q inω, (C₂∆−µ)∇w⁰+ (C₃∆−µ)ϕ⁰+C₄∇(divϕ⁰) =0inω, w⁰=0, ∂nw⁰=0, ϕ⁰=0, onγ0,

(43) where∆ φ:=∂ααφis the two-dimensional Laplacian operator applied toφ, divφ:=∂αφαis the divergence operator applied toφ= (φα),∇φ:= (∂αφ)is the two-dimensional gradient operator applied toφ, and

C₁:=c4+C1, C₂:=c4+2c5+C2,

C₃:=2(c4+c₅), C₄:=c4+2c₅+C3. (44) Remark 1.We notice that the partial differential operator associated with system (43) is self adjoint, because it comes from a symmetric bilinear form associated with the varia- tional problem (37).

Remark 2.In several works (see, for instance, [3]), Aifantis has proposed a simplified strain gradient isotropic linearly elastic constitutive law, in whichc1=c3=c5=0,c2=ℓ²λand c4=ℓ²µ, whereℓis an internal length connected to the micro-structure. The simplified strain gradient constitutive law gets the following expression:

σ_{i j}^ε(u^ε):=λ^εe^ε_pp(u^ε)δi j+2µ^εe^ε_{i j}(u^ε),

p^ε_{i jk}(u^ε):=ℓ²∂_k^εσ_{i j}^ε(u^ε) =ℓ²∂_k^ε(λ^εe^ε_pp(u^ε)δi j+2µ^εe^ε_{i j}(u^ε)). (45) In this particular case the limit problem takes the following form:





hµ(ℓ²∆−1)(∆w⁰+divϕ⁰) =q inω,

µ(ℓ²∆−1)(∇w⁰+ϕ⁰) +µℓ²∆ ϕ⁰+ℓ^2µ(2µ+3λ)_λ+2µ ∇divϕ⁰=0inω, (46) or, analogously,

(hµ(ℓ²∆−1)(∆w⁰+divϕ⁰) =qinω,

12ℓ²

h² D∆divϕ⁰=−q inω, (47)

whereD:=^µ(λ+µ)h_3(λ+2µ)³ =_12(1−ν^Eh3₂₎is the classical rigidity modulus of the plate.

The coefficient 12(ℓ/h)²usually appears in strain gradient plate theories and it repre- sents the ratio between the intrinsic length of the microstructure and the actual thickness of the plate. Its influence is high for small thicknesses, when the intrinsic lengthℓis compa- rable to the thickness of the plate. Besides, it has been shown in [10] that, by comparing the deflections of a classical Kirchhoff-Love plate and the deflections of a strain gradient Kirchhoff-Love plate, 12(ℓ/h)²has the effect of increasing the global stiffness of the plate.

4 A weak convergence results

In this section we establish a convergence result of the solution of the three-dimensional problem towards the solution of the simplified limit problem.

With the scaled displacement fieldu(ε)∈H²(Ω;R³), we associate the following tensors κ(ε) = (κi j(ε))and∇κ(ε) = (κi jk(ε)):= (∂kκi j(ε)), withκi j(ε)∈H¹(Ω)andκi jk(ε)∈ L²(Ω), defined by

κ_αβ(ε):=εe_αβ(ε), κα3(ε):=eα3(ε), κ₃₃(ε):=_ε¹e₃₃(ε), κ_{αβ γ}(ε):=εe_{αβ γ}(ε), κ_αβ3(ε):=e_αβ3(ε), κ_α3β(ε):=e_α3β(ε), κα33(ε):=¹_εeα33(ε), κ33α(ε):=¹_εe33α(ε), κ₃₃₃(ε):=_ε¹₂e₃₃₃(ε).

(48)

With an arbitrary vector fieldv∈H²(Ω;R³), we likewise associate the tensorsκ(ε;v) = (κi j(ε);v)and∇κ(ε;v) = (κi jk(ε;v)):= (∂kκi j(ε;v)). In particular, one hasκ(ε) =κ(ε;u(ε)) and∇κ(ε) =∇κ(ε;u(ε)).

Then the rescaled problem (22) takes the particularly condensed form:

Z

Ω(Aκ(ε):κ(ε;v) +B∇κ(ε)∴∇κ(ε;v))dx=l0(v3) +ε²l2(vα), (49) for allv∈V(Ω).

The main result of this section is claimed in the following theorem.

Theorem 1 For eachε>0, letu(ε)denote the (unique) solution of (49). Then u3(ε)⇀u¯3 in H²(ω),

∂₃uα(ε)⇀ϕ¯α in H¹(ω), (50) whereu¯₃andϕ¯αare the solutions of the limit problems (38)- (41)

Z ω

(C1∂β βu¯3+C2∂βϕ¯β)∂ααη3+ c4(∂αβu¯3+∂βϕ¯α) +c5(∂αϕ¯β+∂βϕ¯α)

∂αβη3+ +µ(∂αu¯₃+ϕ¯α)∂αη₃]dx˜=1

h Z

ωqη₃dx,˜ Z

ω

C₂∂_{β β}u¯₃+C₃∂_βϕ¯_β

∂αηα+µ(∂αu¯₃+ϕ¯α)ηα+ + c₄∂βϕ¯α+ (c4+2c₅)(∂αβu¯₃+∂βϕ¯α+∂αϕ¯β)

∂βηα dx˜=0,

(51) for allηi∈V(ω):={η= (ηi)∈H¹(ω,R²)×H²(ω);η=0,∂νη₃=0onγ₀}.

Proof For the sake of clarity the proof is divided into two parts. Let us define at first the followingL²-norms:

|κ(ε)|_0,Ω:=

(

∑

i,j

|κi j(ε)|²_0,Ω )1/2

, |∇κ(ε)|_0,Ω:=

(

∑

i,j,k

|κi jk(ε)|²_0,Ω )1/2

(52) (i) By lettingv=u(ε)in (49), the variational problem takes the following simple form:

Z

Ω(Aκ(ε):κ(ε) +B∇κ(ε)∴∇κ(ε))dx=l0(u3(ε)) +ε²l2(uα(ε)), (53) By virtue of the positive definiteness of the bilinear form and, by definition ofκ(ε)and

∇κ(ε), one has Z

Ω(Aκ(ε):κ(ε) +B∇κ(ε)∴∇κ(ε))dx≥C

|κ(ε)|²_0,Ω+|∇κ(ε)|²_0,Ω ≥

≥C n

ε²∑_α,β|e_αβ(ε)|²_0,Ω+∑_α|eα3(ε)|²_0,Ω+_ε¹₂|e₃₃(ε)|²_0,Ω+ +ε²∑_α,β,γ|eαβ γ(ε)|²_0,Ω+∑_α,β

|eαβ3(ε)|²_0,Ω+|eα3β(ε)|²_0,Ω + +_ε¹₂∑_α

|eα33(ε)|²_0,Ω+|e33α(ε)|²_0,Ω

+_ε¹₄|e₃₃₃(ε)|²_0,Ωo .

(54)

On the other side, by virtue of the continuity of the linear forms and sinceε≤1, we get:

l₀(u₃(ε)) +ε²l₂(uα(ε))≤C

ku₃(ε)k_1,Ω+ε²∑_αkuα(ε)k_1,Ω ≤

≤Cn

|κ(ε)|²_0,Ω+|∇κ(ε)|²_0,Ωo1/2

. (55)

In order to prove the inequality above, we notice that, sinceui(ε) =0 onΓ₀, the norm kui(ε)k_1,Ωis equivalent to{∑j|∂jui(ε)|²_0,Ω}^1/2. Since∂nui(ε) =0 onΓ0andn= (n1,n2,0), one has the same inequality for any∂αui(ε). In particular we get

|∂αu3(ε)|²_0,Ω ≤C{∑_β|∂_αβu3(ε)|²_0,Ω+|∂α3u3(ε)|²_0,Ω},

|∂αu_β(ε)|²_0,Ω≤C{∑γ|∂αγu_β(ε)|²_0,Ω+|∂α3u_β(ε)|²_0,Ω}. (56)

Since∂αjui(ε) =eαi j(ε) +ejiα(ε)−eαji(ε), one has

∑_α|∂αu₃(ε)|²_0,Ω≤C n

∑_α,β

|e_αβ3(ε)|²_0,Ω+|e_α3β(ε)|²_0,Ω+|eα33(ε)|²_0,Ω+|e33α(ε)|²_0,Ωo ,

∑_α|∂αuβ(ε)|²_0,Ω≤C n

∑_α,β,γ

|eαβ γ(ε)|²_0,Ω+|eα3β(ε)|²_0,Ω+|eαβ3(ε)|²_0,Ωo .

(57) Hence we have that

ku3(ε)k_1,Ω+ε²∑_αkuα(ε)k_1,Ω≤

≤C n1

ε²|e₃₃(ε)|²_0,Ω+∑_α,β

|e_αβ3(ε)|²_0,Ω+|e_α3β(ε)|²_0,Ω + +_ε¹2∑_α

|eα33(ε)|²_0,Ω+|e33α(ε)|²_0,Ω

+ε²∑_α,β|eαβ(ε)|²_0,Ω+ +ε²∑_α,β,γ|e_{αβ γ}(ε)|²_0,Ωo1/2

≤C n

|κ(ε)|²_0,Ω+|∇κ(ε)|²_0,Ωo1/2

.

(58)

The inequalities above imply that the norms|κ(ε)|_0,Ωand|∇κ(ε)|_0,Ω are bounded indepen- dently ofε. Since the sequences(κi j(ε))_ε>0and(κi jk(ε))_ε>0are bounded inL²(Ω), there exist a constantCsuch that

|e₃₃₃(ε)|_0,Ω ≤Cε²,

|e33α(ε)|_0,Ω≤Cε,

|e33(ε)|_0,Ω≤Cε,

|eα33(ε)|_0,Ω≤Cε,

|e_α3β(ε)|_0,Ω≤C,

|eα3(ε)|_0,Ω≤C,

|e_{αβ γ}(ε)|_0,Ω≤^C_ε,

|e_αβ3(ε)|_0,Ω≤C,

|e_αβ(ε)|_0,Ω≤^C_ε.

(59) Hence, from the first set of inequalities, we obtain that_ε¹₂e₃₃₃(ε)⇀e¯₃₃₃,e₃₃₃(ε):=∂₃₃u₃(ε)→ 0,e33α(ε):=∂α3u₃(ε)→0 ande₃₃(ε):=∂₃u₃(ε)→0 inL²(Ω), and thus∂₃u₃(ε)→0 in H¹(Ω). Moreover, one has∂αu₃(ε)⇀z¯α(˜x)inL²(Ω).

¿From the second set of inequalities, we get thateα3(ε)⇀e¯α3,eα33(ε):=∂₃eα3(ε)→0 andeα3β(ε):=∂βeα3(ε)⇀e¯α3βinL²(Ω). Thuseα3(ε)⇀e¯α3(˜x)inH¹(Ω)and so, ¯eα3β=

∂βe¯α3(˜x). By definition ofeα3(ε), we obtain that∂3uα(ε)⇀2 ¯eα3(˜x)−z¯α(x)˜ inL²(Ω).

Thanks to (59), we notice that∂_αβu3(ε) =e_α3β(ε) +e_β3α(ε)−e_αβ3(ε)is bounded in L²(Ω). Therefore,|∂i ju3(ε)|_0,Ω≤Cand by (57) one hasu3(ε)bounded inH₀²(ω)and so:

u₃(ε)⇀u¯₃(˜x)inH₀²(ω). (60) The limit ¯u3=u¯3(˜x)is independent ofx3. This implies that ¯zα=∂αu¯3and, thus

∂₃uα(ε)⇀ϕ¯α(˜x):=2 ¯eα3(x)˜ −∂αu¯₃(x)˜ inH₀¹(ω). (61) Finally, from the third set of inequalities, we deduce thatεe_{αβ γ}(ε)⇀e¯_{αβ γ},εe_αβ(ε)⇀

¯

e_αβande_αβ3(ε)⇀e¯_αβ3inL²(Ω). We notice that ¯e_αβ3=∂αe¯_β3+∂_βe¯α3−∂_αβu¯₃=¹₂(∂αϕ¯_β+

∂_βϕ¯α)inL²(Ω).

(ii) Now we characterize the limits ¯u3and ¯ϕα. Let us consider the rescaled variational problem (22). Let multiply it byε²and letεtend to zero. Then we find that

Z

Ω(˜ce¯333+ (2c1+c2)e¯σ σ3+2(c1+2c3)∂σe¯3σ)e333(v)dx=0, (62) for allv∈V(Ω). This relation is satisfied when ˜ce¯₃₃₃+(2c₁+c₂)e¯σ σ3+2(c₁+2c₃)∂σe¯3σ= 0, which implies that

¯

e₃₃₃=−2c₁+c₂

˜

c e¯σ σ3−2(c₁+2c₃)

˜

c ∂σe¯3σ =

=−3c₁+c₂+c₃

˜

c ∂σϕ¯σ−c₁+2c₃

˜

c ∂σ σu¯₃.

(63)