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Noname manuscript No.(will be inserted by the editor)

A Rigorous Derivation of the Equations for the ClampedBiot-Kirchhoff-Love Poroelastic Plate

Anna Marciniak-Czochra · Andro Mikelic

to appear in the Archive for Rational Mechanics and Analysis, 2014, doi: 10.1007/s00205-014-0805-2

Abstract In this paper we investigate the limit behavior of the solution to quasi-staticBiot’s equations in thin poroelastic plates as the thickness tends to zero. We chooseTerzaghi’s time corresponding to the plate thickness and obtain the strong conver-gence of the three-dimensional solid displacement, fluid pressure and total poroelas-tic stress to the solution of the new class of plate equations. In the new equationsthe in-plane stretching is described by the 2D Navier’s linear elasticity equations,with elastic moduli depending on Gassmann’s and Biot’s coefficients. The bendingequation is coupled with the pressure equation and it contains the bending momentdue to the variation in pore pressure across the plate thickness. The pressure equationis parabolic only in the vertical direction. As additional terms it contains the timederivative of the in-plane Laplacian of the vertical deflection of the plate and of theelastic in-plane compression term.

Keywords Thin poroelastic plate · Biot’s quasi-static equations · bending-flowcoupling · higher order degenerate elliptic-parabolic systems · asymptotic methods

Mathematics Subject Classification (2000) MSC 35B25 · MSC · MSC 74F10 ·MSC 74K20 · MSC 74Q15 · MSC 76S

AM-C was supported by ERC Starting Grant ”Biostruct” 210680 and Emmy Noether Programme of Ger-man Research Council (DFG). The research of A.M. was partially supported by the Programme InterCarnot Fraunhofer from BMBF (Grant 01SF0804) and ANR.

Anna Marciniak-CzochraInstitute of Applied Mathematics, IWR and BIOQUANT, University of Heidelberg, Im Neuenheimer Feld267, 69120 Heidelberg , GermanyTel. : +496221544871E-mail: [email protected]

Andro MikelicUniversite de Lyon, CNRS UMR 5208, Universite Lyon 1, Institut Camille Jordan, 43, blvd. du 11 novem-bre 1918, 69622 Villeurbanne Cedex, FranceTel. : +33426234548Fax: +33956109885E-mail: [email protected]

2 Anna Marciniak-Czochra, Andro Mikelic

1 Introduction

A plate is a 3D body bounded by two surfaces of small curvature and placed at smalldistance. A plate is said to be thin if the distance between these surfaces, called thethickness, is much smaller than a characteristic size of the surrounding surfaces.

The construction of a linear theory for the extensional and flexural deformationof plates, starting from the 3D Navier equations of linear elasticity, goes back tothe 19th century and Kirchhoff’s work. Following a short period of controversy, thecomplete theory, nowadays known as the Kirchhoff-Love equations for bending ofthin elastic plates, was derived. Derivation was undertaken under assumptions, whichare referred to as Kirchhoff’s hypothesis. It reads as follows:

Kirchhoff’s hypothesis: Every straight line in the plate that was originally per-pendicular to the plate midsurface, remains straight after the strain and perpendicularto the deflected midsurface.

Since then a theory providing appropriate 2D equations applicable to shell-likebodies was developed. Later, due to the considerable difficulties with the derivationof plate equations from 3D linear elasticity equations, a direct approach in the sense ofTruesdell’s school of continuum mechanics, using a Cosserat surface, was proposed.We refer to the review paper of Naghdi [19] . In classical engineering textbooks onefinds a formal derivation, based on the Kirchhoff hypothesis. For details we refer toFung’s textbook [15].

A different approach is to consider the plate equations as an approximation to the3D elasticity equations in a plate domaine Ω ε = ω × (−ε,ε), where ω is the middlesurface and ε is the ration between the plate thickness and its longitudinal dimension.Here, for simplicity we suppose that plate is flat and of uniform thickness.

The comparison between 3D model and 2D equations was first performed for-mally in [14]. Then Ciarlet and collaborators developed systematically the approachwhere the vertical variable x3 ∈ (−ε,ε) was scaled by setting y3 = x3/ε . This changeof variables transforms the PDE to a singular perturbation problem on a fixed do-main. With such approach Ciarlet and Destuynder have established the rigorous errorestimate between the 3D solution and the Kirchhoff-Love solution to the 2D plateequations, in the limit as ε → 0. For details, we refer to the article [5], to the book[6] and to the subsequent work for details. The complete asymptotic expansion isdue to Dauge and Gruais (see [10], [11] and subsequent work by Dauge and collabo-rators). Handling general boundary conditions required the boundary layer analysis.For a review, discussing also the results by Russian school, we refer to [12]. Furthergeneralizations to nonlinear plates and shells exist and were obtained using Γ− con-vergence. We do not discuss it here and don’t undertake to give complete referencesto the plate theory.

Many living tissues are fluid-saturated thin bodies like bones, bladders, arteriesand diaphragms and they are interpreted as poroelastic plates or shells. For a reviewof modeling of bones as poroelastic plates we refer to [9]. Furthermore, industrialfilters are an example of poroelastic plates and shells.

Our goal is to extend the above mentioned theory to the poroelastic plates. Theseare the plates consisting of an elastic skeleton (the solid phase) and pores saturated

Derivation of the Equations for the Poroelastic Plate 3

by a viscous fluid (the fluid phase). Interaction between the two phases leads to anoverall or an effective behavior described by the poroelasticity equations instead ofthe Navier elasticity equations. In addition to the phase displacements, i.e. displace-ment of the solid and of the fluid phase, description of a poroelastic medium requiresthe pressure field and, consequently, an additional PDE. The equations were intro-duced by Biot in fifties based on phenomenological physical modeling. The originalderivation of the poroelasticity PDEs can be found in [2], [3] and in the selection ofBiot’s publications [27] .

The effective model corresponds to the homogenization of the complicated porelevel fluid-structure interaction problem formulated based on the continuum mechan-ics first principles, i.e. the Navier equations for the solid structure and by the Navier-Stokes equations for the flow. If the problem involves small deformations, a smallfluid compressibility and a slow viscous flow, the model can be simplified using lin-earization. If, in addition, we consider a periodic porous medium with connected fluidand solid phases, then the poroelasticity equations can be rigorously derived from thefirst principles using homogenization approach. The small parameter of the problemis the ratio between characteristic pore size and the domain size. Applying eitherLaplace’s transformation in time or the excitation by an external harmonic sourcewith a given frequency, we obtain a stationary system of PDEs containing the fre-quency as a parameter. In such setting, the two-scale poroelasticity equations can beobtained using formal two-scale expansions in the small parameter, applied to thepore level fluid-structure equations. The two-scale equations contain slow and fastvariables and the coefficients depend on frequency in a complicated way. For moredetails we refer to the book [21], the review [1] and the references therein.

First mathematically rigorous justification of the two-scale equations and one ofthe first major applications of the two-scale convergence technique was due to Nguet-seng [20]. Convergence in space and time variables was proved by Mikelic and col-leagues in [16], [7] and [13], where the effective coefficients were determined throughauxiliary problems. Furthermore, in case of inviscid fluid filling the pores, the scalesseparation was performed in [13] allowing to reduce the two-scale equations to theBiot original model. In case of the pores filled by a viscous fluid, Biot’s equations con-tain a viscodynamic operator. The corresponding scales separation in the two-scaleequations yields the exact expression of the viscodynamic operator. In case of poresfilled by a viscous fluid, the result depends on the contrast of property number, beingthe ratio between the viscosity, multiplied by the characteristic time, and the bulkmodulus of the solid structure. If it is of the order of the small parameter squared,then the homogenization leads to a diphasic macroscopic model of the fluid-solidmixture described by the diphasic Biot system. This setting can be always achievedby choosing the corresponding characteristic time, known as Terzaghi’s time.

Most of the quoted upscaling results concern the oscillations of the fluid saturatedstructures. The homogenization leads to nonstationary systems of PDEs containingmemory terms (see e.g. [17] for their study). In case of flows through deformableporous media, Terzaghi’s time is sufficiently large. Hence, such problems are quasi-static, i.e. the acceleration effects can be neglected. Derivation of the quasi-static Biotmodel from the first principles using homogenization techniques can be found in [18].


Following the above mentioned mathematical and engineering literature, it iswidely accepted that the deformable porous media, saturated by a fluid, are modeledusing Biot’s diphasic equations for the effective solid displacement and the effec-tive pressure. Biot’s equations are valid at every point of the plate and the averagedphases coexist at every point. For the direct continuum mechanics approach to Biot’sequations, we refer to the monograph of Coussy [8].

Until recently, few articles have addressed plate theory for poroelastic media.Nevertheless, in the early paper [4], Biot examined the buckling of a fluid-saturatedporous slab under axial compression. This can be considered as the first study of aporoelastic plate. The goal was to have a model for the buckling of porous media,which is simpler than general poroelasticity equations. The model was obtained inthe context of the thermodynamics of irreversible processes.

More recently, in [26] Theodorakopoulos and Beskos used Biot’s poroelastic the-ory and the Kirchhoff theory assuming thin plates and neglected any in-plane motionto obtain purely bending vibrations. A systematic approach to the linear poroelasticplate and shell theory was undertaken by Taber in [24] and [25], using Kirchhoff’sapproach (see e. g. [23]).

In this paper we follow the approach of Ciarlet and Destuynder, as presented inthe textbook [22] and rigorously develop equations for a poroelastic plate.

Our strategy of deriving the poroelastic plate model is similar to that for the dualporosity model of the Darcy law. In case of Darcy law, upscaling of the Navier-Stokesequations, valid in the pores of a rigid porous medium, yields a linear relationshipbetween the velocity and the pressure gradient, valid at every point, see e.g. [21] andreferences therein. The result is independent of the size of the porous medium. Thehomogenization requires that the size of the porous medium is much larger than thepore size. For a poroelastic plate, the plate width is much smaller than its length.Nevertheless, it is much larger than the pore size. Hence, derivation of the poroelasticplate model from the pore scale first principles is based on two separated scalingprocedures. First, it requires the homogenization of the microscopic model to themesoscopic level. Then, passing from the 3 dimensional poroelastic model to theporoelastic plate model corresponds to passing from mesoscopic to macroscopic levelof description and it is based on the singular perturbation technique introduced byCiarlet and colleagues. Our approach does not work if one considers a tissue withjust few layers of cells. Then the width of such structure is of the size of pores andasymptotic analysis should be performed directly on the pore scale.

Successful recent approaches to the derivation of linear and nonlinear plate mod-els use the elastic energy functional. In our situation, considering the steady statesetting is not of interest and the quasi-static problem is time-dependent and non-symmetric. The acceleration term is negligible and the equations for the effectivesolid skeleton displacement also contain the pressure gradient. They have the struc-ture of a generalized Stokes system, with the velocity field replaced by displacement.Mass conservation equation is parabolic in the pressure and contains the time deriva-tive of the volumetric strain. The structure of the system requires that we use theoriginal approach of Ciarlet and Destuynder. Strong convergence is obtained by ex-tracting the limit functions and at the end our result corresponds to the analogousΓ -convergence results in the purely elastic case.


The quasi-static Biot system is well-posed only if there is a relationship betweenBiot parameters multiplying the pressure gradient in the displacement system and thetime derivative of the divergence of the displacement in the pressure equation. Weare able to obtain the energy estimate. For the nonlinear Biot system, balancing theappropriate terms is an open mathematical problem.

In addition, there exists a major difference, with respect to the limit of the nor-malized e33 term, compared to a classical derivation of the Kirchhoff-Love plate. Inour poroelastic case, the limit also contains the pressure field. Consequence of themore complicated limit term is that the time derivative of the Laplacian of the de-flexion appears in the pressure equation and in the Laplacian of the bending pressuremoment in the bending equation. We note that the deflexion does not depend on thevertical variable, but the pressure oscillations persist. We prove that the limit prob-lem is well-posed. Nevertheless, it has a richer structure than the classical bendingequation. We expect more complex dynamics in this model.

For slender poroelastic bodies, more difficulties with boundary layers are to beexpected.

2 Setting of the problem

We study the deformation and the flow in a poroelastic plate Ω ℓ = (x1,x2,x3) ∈ωL × (−ℓ/2, ℓ/2), where the mid-surface ωL is a bounded domain in R2 with asmooth boundary ∂ωL ∈C1. For simplicity, we suppose that the poroelastic plate Ω ℓ

is an isotropic material. Σ ℓ (respectively Σ−ℓ ) is the upper face (respectively lowerface) of the plate Ω ℓ. Γ ℓ is the lateral boundary, Γ ℓ = ∂ωL × (−ℓ/2, ℓ/2). We recallthat the ratio between the plate thickness and the characteristic horizontal length isε = ℓ/(2L)<< 1.

Table 1 Parameter and unknowns description

SYMBOL QUANTITYG shear modulusν drained Poisson ratioγG inverse of Biot’s modulusα effective stress coefficientk permeabilityη viscosityL and ℓ midsurface length and plate width, respectivelyε = ℓ/(2L) small parameterT = ηL2/(kG) characteristic Terzaghi’s timed characteristic displacementP = dG/L characteristic fluid pressureu = (u1,u2,u3) solid phase displacementp pressure

We note that Biot’s diphasic equations describe behavior of the system at so calledTerzaghi’s time scale T = ηL2

c/(kG), where Lc is the characteristic domain size, η


is dynamic viscosity, k is permeability and G is the bulk modulus. For the list of allparameters see the Table 1.

In general, for the plate we have two possible choices of time scale:

1. Lc = L, leading to T = ηL2/(kG) and2. Lc = ℓ, leading to the Taber-Terzaghi transversal time Ttab = ηℓ2/(4kG).

If Terzaghi’s time is short, then it is necessary to study the vibrations of the poroe-lastic plate. For filters, Terzaghi’s time scale doesn’t correspond to short times, therescaled acceleration factors maxρ f ,ρsk2G/(η2Lc) are small and the accelerationis negligible. Only the time change of the variation of fluid volume per unit referencevolume ζ = γG p+α div u is not small. For more discussion of the scaling we referto [18].

Consequently, we study the simplest model of real applied importance: the quasi-static Biot system.

Following Biot’s classical work [2], the governing equations as written in [9] takethe following dimensional form:

σ = 2Ge(u)+(2νG

1−2νdiv u−α p)I in Ω ℓ, (1)

−Gu− G1−2ν

div u+α p = 0 in Ω ℓ, (2)

∂∂ t

(γG p+α div u)− kηp = 0 in Ω ℓ. (3)

We impose a given contact force σn = P±ℓ and a given normal flux − kη

∂ p∂x3

=U ℓ

at x3 =±ℓ/2.We recall that the effective displacement of the solid phase is denoted by u =

(u1,u2,u3), the strain tensor e is given by e(u) = symu, σ is the stress tensor andthe effective pore pressure is p. Note that Biot’s system (1)-(3) is formulated in theterms of customary effective parameters: the shear modulus G (which coincides withLame’s parameter µ), the Poisson ratio ν , the effective stress coefficient α , the inverseof Biot’s modulus γG, permeability k and fluid viscosity η . We refer to Table 1 for alist of the parameters. It is known that for Young’s modulus E it holds E = 2G(1+ν)

and Lame’s first parameter λ is given by λ =2νG

1−2ν.

Remark 1 It is of interest to comment on the relationship between equations (1)-(3) and Navier’s equations of linear elasticity. In the case of a medium containingonly an elastic solid, i.e. in the case of zero porosity, quasi-static Biot equations (1)-(3) should reduce to the Navier equations. Calculation of the effective poroelasticparameters is found in reference [18]. Using corresponding formulas from [18], wecan find the limit behavior when the porosity |Y f | tends to zero. Note that the porositycorresponds to the percentage of the volume filled by the fluid.

(i) Parameter α (called the effective stress coefficient, the pressure-storage couplingcoefficient or Biot’s coefficient) is found to be a symmetric matrix of the form

α = |Y f |I −BH .


Let n denote the unit normal vector to a surface. Matrix BH corresponds to theaverage over the solid structure part of the stress resulting from the contact force−n at the pore/structure interface.Based on (5.10), page 1250031-26 in reference [18], we conclude that BH

i j → 0as |Y f | → 0. Therefore, α → 0 as |Y f | → 0.

(ii) Since |ki j| ≤C|Y f |, permeability k tends to zero as |Y f | → 0.

Hence, in the zero porosity limit, displacement u = (u1,u2,u3), satisfies Navier equa-tions

−Gu− G1−2ν

div u = 0 (4)

and we receive the classical setting of derivation of the Kirchoff-Love plate.

Remark 2 The Biot’s modulus M = 1/γG is the sum of two terms combining poros-ity and compressibility of the fluid and the solid. The first term corresponds to theshear modulus times fluid compressibility times porosity. The second one is due tocompressibility arising from the fluid-solid interaction and it is nonzero even in thecase of an incompressible fluid phase. We refer to reference [18] page 1250031-11and page 1250031-26, formula (5.11) for more detail. Hence, we observe that theeffective compressibility effects remain in the quasi-static model and they are notexclusively linked with acoustics.

Following, after some appropriate modifications, engineering textbooks approachto Kirchhoff-Love’s plate model, we can derive formally the poroelastic plate equa-tions. For this, in addition to the Kirchhoff hypothesis and following Taber’s papers[24] and [25], we assume that

the fluid velocity derivatives in the longitudinal directionare small compared to the transverse one. (5)

Along the same lines as in Fung’s textbook [15], we perform classical Kirchhofftype formal calculations and obtain the following equations

Gℓ∆x1,x2 uω +Gℓ(1+ν)

1−ν∇x1,x2divx1,x2 uω +

α(1−2ν)1−ν

∇x1,x2N

+2

∑j=1

(Pℓj +P−ℓ

j )e j = 0, (6)

(γG +α2(1−2ν)2G(1−ν)

)N =α(1−2ν)ℓ

1−νdivx1,x2(u

ω1 ,u

ω2 ), (7)

(γG +α2(1−2ν)2G(1−ν)

)∂∂ t

(pe f f +

Nℓ

)− k

η∂ 2

∂x23

(pe f f +

Nℓ

)= αx3

1−2ν1−ν

∂∂ t

∆x1,x2w, (8)

Gℓ3

6(1−ν)∆ 2

x1,x2w+α

1−2ν1−ν

∆x1,x2

∫ ℓ/2

−ℓ/2x3 pe f f dx3 =

ℓ

2

2

∑i=1

∂∂xi

(Pℓi −P−ℓ

i )+Pℓ3 +P−ℓ

3 , (9)


where w(x1,x2, t) is the effective transverse displacement of the surface, uω =(uω1 ,u

ω2 ),

j = 1,2, are the effective in-plane solid displacements, uωj (x1,x2, t)−x3

∂w∂x j

, j = 1,2,

approximate the physical displacements u j, pe f f = pe f f (x1,x2,x3, t) is the effective

fluid pressure and N = −∫ ℓ/2

−ℓ/2pe f f dx3 is the effective stress resultant due to the

variation in pore pressure across the plate thickness.

We note that D =Gℓ3

6(1−ν)is the flexural rigidity of the plate solid skeleton.

We refer to Appendix for the detailed formal calculation.The disadvantage of such approach to derive the equations is in using some ad

hoc hypothesis. The assumptions (106) that the stresses σi3, i = 1,2,3 are negligible,can only be satisfied approximatively. They are not even consistent with the fact thatwe use the averages which are non zero. To this classical difficulty of the plate theory,we add a new ad hoc assumption (5) on the pressure field.

As in the case of the justification of the elastic plate equations by Ciarlet et al,here also the two-scale asymptotic expansion approach gives the correct answer. Notethat the presence of the effective stress resultant pω , due to the variation in porepressure across the plate thickness, causes stretching even if the boundary conditionsfor (uω

1 ,uω2 ) are homogeneous.

Our goal is to extend the Kirchhoff-Love plate justification by Ciarlet et al and byDauge et al to the poroelastic case. We start with a weak convergence result, and then,after adding additional correction terms show the strong convergence. The obtainedresult is then of the same type as in Γ -convergence approach i.e. they would coincidein the case of linear elasticity. In poroelastic case, we have to deal with more involvednorms related to the energy of the Biot system.

In subsection 3.1 we present the dimensionless form of the problem. Subsec-tion 3.2 contains a sketch of the proof of existence a unique smooth solution for thestarting problem. In subsection 3.3 we formulate our convergence results. Section4 is consecrated to the introduction of the rescaled problem, posed on the domainΩ = ω × (−1,1). Then in Section 5 we study convergence of the solutions to therescaled problem, as ε → 0. In short Sections 6 and 7 we prove the strong convergencefor the corrected displacement and pressures and stresses, respectively. In Appendix8 we give an ad hoc derivation of the model, which follows mechanical engineeringtextbooks and which is justified a posteriori by our rigorous results.

3 Main Results

3.1 Dimensionless equations

We introduce the dimensionless unknowns and variable by setting


γ = γGG; P =GUL

; λ =2ν

1−2ν; T =

ηℓ2

4kG;

Uuε = u; Ppε = p; xL = x; tT = t.

After dropping wiggles the system (1)-(3) becomes

−uε − 11−2ν

div uε +α pε = 0 in Ω ε × (0,T ), (10)

σ ε = 2e(uε)+(2ν

1−2νdiv uε −α pε)I in Ω ε × (0,T ), (11)

∂∂ t

(γ pε +α div uε)− ε2pε = 0 in Ω ε × (0,T ), (12)

where uε = (uε1,u

ε2,u

ε3) denotes the dimensionless displacement field and pε the di-

mensionless pressure. We study a plate Ω ε with thickness 2ε = ℓ/L and sectionω = ωL/L. It is described by

Ω ε = (x1,x2,x3) ∈ ω × (−ε,ε),

Σ+ε (respectively Σ−ε ) is the upper face (respectively the lower face) of the plateΩ ε . Γ ε is the lateral boundary, Γ ε = ∂ω × (−ε,ε).

We suppose that a given dimensionless traction force is applied on Σ+ε ∪Σ−ε

and impose the frictional boundary conditions on Γ ε :

σ ε nε = (2e(uε)−α pε I +2ν

1−2ν( div uε)I)(±e3) =

Pε = ε2(P1,P2,0)+ ε3P3e3 on Σ+ε ∪Σ−ε , (13)uε = 0, on Γ ε . (14)

For the pressure pε , at the lateral boundary Γ ε = ∂ω × (−ε,ε) we impose agiven inflow/outflow flux V :

− pε ·n =V (x1,x2, t) ; n = (n1,n2,0) (15)

and at Σ+ε ∪Σ−ε , we set

−∂ pε

∂x3=U1(x1,x2, t) on Σ±ε . (16)

Finally, we need an initial condition for pε at t = 0,

pε(x1,x2,x3,0) = pin(x1,x2) in Ω ε . (17)

We write problem (10)-(17) in the variational form:


Let V ε =

z ∈ H1(Ω ε)3| z|Γ ε = 0. Find uε ∈H1(0,T,V ε), pε ∈H1(0,T ;H1(Ω ε))

such that it holds∫Ω ε

2e(uε) : e(φ)dx+2ν

1−2ν

∫Ω ε

div uε div φdx−α∫

Ω εpε div φdx

=∫

Σ+ε∪Σ−εPε φds, for every φ ∈V ε , and t ∈ (0,T ), (18)

γ∫

Ω ε∂t pε ζ dx+

∫Ω ε

α div ∂tuε ζ dx+ ε2∫

Ω ε∇pε ∇ζ dx

= −ε2∫

∂Ω εNε ζ ds, for every ζ ∈ H1(Ω ε), (19)

where

N ε =

V, on Γ ε ,U1, on Σ±ε .

pε |t=0 = pin, in Ω ε . (20)

3.2 Existence and uniqueness for the ε-problem

In this section we prove existence and uniqueness of a solution uε , pε ∈H1(0,T ;V ε)×H1(0,T ;H1(Ω ε)), pε |t=0 = pin of the problem (18) -(20). We con-struct a Galerkin approximation uε ≈ vN, pε ≈ pN and show its convergence touε , pε when N →+∞.

Assumptions 1 pin ∈ H2(Ω ε), Pε ∈ H2(0,T ;H1(∂Ω ε\Γ ε)) and N ε ∈H1

0 (0,T ;L2(∂Ω ε)).

Proposition 1 Let us suppose assumptions 1 . Then problem (18)- (20) has a uniquesolution uε , pε ∈ H1(0,T ;V ε)×H1(0,T ;H1(Ω)).

Proof Let β j be an orthogonal basis of V ε with respect to the scalar product

(f,g)V =∫

Ω ε2e(f) : e(g)dx

and ξ j an orthogonal basis of H1(Ω ε), orthonormal for L2(Ω ε). Then, the Galerkinapproximation is

∫Ω ε

2e(vN) : e(φ)dx+2ν

1−2ν

∫Ω ε

div (vN) div φdx−

α∫

Ω εpN div φdx =

∫Σ+ε∪Σ−ε

Pε φds, for every φ ∈VN , and t ∈ (0,T ), (21)

γ∫

Ω ε∂t pNζ dx+

∫Ω ε

α div ∂t(vN)ζ dx+ ε2∫

Ω ε∇pN∇ζ dx

=−ε2∫

∂Ω εNε ζ ds, for every ζ ∈ spanξ1, ...,ξN, (22)

pN |t=0 = pNin, (23)


where pNin is a projection of pin to spanξ1, ...,ξN and VN = spanβ1, ...,βN.

Therefore, vN = Σ Nl=1blN(t)βl(x) and pN = Σ N

l=1slN(t)ξl(x). Using Korn’s in-equality we obtain that the bilinear form

A (f,g)V =∫

Ω ε2e(f) : e(g)dx+

2ν1−2ν

∫Ω ε

div f div gdx

is V ε -elliptic. Let

Ak j =∫

Ω ε(2e(β j) : e(βk)+

2ν1−2ν

div β j div βk) dx, 1 ≤ k, j ≤ 3,

Mk j =∫

Ω εξ j div βk dx, Kk j = ε2

∫Ω ε

∇ξ j∇ξk dx, 1 ≤ k, j ≤ 3.

Then, using the definition of vN and pN , we obtain

Ab(t)−αM s(t) = F N(t), (24)

γddt

s(t)+αM τ ddt

b(t)+K s(t) = G N , s(0) = s0N , (25)

where b(t) =

b1N(t)...bNN(t)

and s(t) =

s1N(t)...sNN(t)

.

After inserting (24) into (25), we obtain a linear system of ordinary differential equa-tions for s = (s1N(t), ...,sNN(t))T ,

(γI +α2M τ A−1M )ddt

s(t)+K s(t) = G N −αM τ A−1 ddt

F N(t), s(0) = s0N .

Since the(M τ A−1M ξ ,ξ

)≥ c||M ξ ||2, for all ξ ∈ RN , it has a unique solutions for

all times. Moreover, s ∈ H2(0,T ) implying the same time regularity for vN and pN .Next, we show a priori estimates, uniform in N. Testing (21) by ∂tvN , and (22) bypN , integrating in space and summing up, we find out that the cross terms cancel outand obtain

∂t

∫Ω ε

|e(vN)|2dx+ν

1−2ν

∫Ω ε

| div vN |2dx+γ2

∫Ω ε

|pN |2dx

+∫

Ω εε2|∇pN(t)|2dx = ∂t

∫Σ+ε∪Σ−ε

Pε vNds− ε2∫

∂Ω εNε pNds

−∫

Σ+ε∪Σ−ε∂tPε vNds+

∫∂Ω ε

ε2∂tNε pNds.

It yields

∥vN∥L∞(0,T ;V ε )+∥∇pN∥L2(0,T ;L2(Ω ε ))+∥pN∥L∞(0,T ;L2(Ω ε ))

≤ C(ε)∥Pε∥H1(0,T ;L2(∂Ω ε\Γ ε ))+∥N ε∥H1(0,T ;L2(∂Ω ε ))

.

Next we use (21)-(22) to calculate ∂tvN and ∂t pN at t = 0. Taking the time derivativeprovides H1-regularity in time. Finally, passing to the limit N → ∞ and uniquenessare obvious.


3.3 Convergence results

In the remainder of the paper we make the following assumptionsAssumptions 2 For simplicity, we assume that pin = 0, that V has a compact supportin ∂ω × (0,T ] and that U1 and Pε have compact support in ω × (0,T ].First we formulate the boundary value problem in Ω = ω × (−1,1) for the effectivedisplacements and the mean pressure:

Let the scaled in-plane displacements and the mean pressure w0,πm ∈H1(0,T ;H1

0 (ω))3 be given by

(γ +α2(1−2ν)

2(1−ν))πm +

α(1−2ν)1−ν

divy1,y2(w01,w

02) = 0, in ω × (0,T ); (26)

−∆y1,y2(w01,w

02)−

1+ν1−ν

∇y1,y2 divy1,y2(w01,w

02)+

α(1−2ν)1−ν

∇y1,y2πm

=12(P1|y3=1 +P1|y3=−1,P2|y3=1 +P2|y3=−1), in ω × (0,T ). (27)

Theory of the stationary 2D Navier system of linear elasticity yields

Lemma 1 Under assumptions (1)-(2), problem (26)-(27) has a unique solutionw0,πm∈H1(0,T ;H1

0 (ω))3. Furthermore, it belongs to H1(0,T ;(H3(ω)∩H10 (ω))2

×H2(ω)∩H10 (ω)).

The in-plane displacement w0 will serve us to obtain the approximation for uε1,u

ε2,

once the effective vertical displacement w03 is determined.

Next we write the system for the pressure fluctuation πw and w03:

Find πw,w03 ∈ H1(0,T ;H1(Ω)×H2

0 (ω)) satisfying the system

(γ +α2 1−2ν2(1−ν)

)∂tπw − ∂ 2πw

∂y23

−α1−2ν1−ν

y3∆y1,y2∂tw03 = 0 in Ω × (0,T ), (28)

∂y3 πw|y3=1 = ∂y3 πw|y3=−1 =−U1 in (0,T ), πw|t=0 = 0 in Ω , (29)

43

11−ν

∆ 2y1,y2

w03 +

α(1−2ν)1−ν

∆y1,y2

∫ 1

−1y3πw dy3 = P3|y3=1 +P3|y3=−1+

divy1,y2(P1|y3=1 −P1|y3=−1,P2|y3=1 −P2|y3=−1) = Π1,−1 in Ω × (0,T ). (30)

Proposition 2 Under assumptions (1)-(2), problem (28)- (30) has a unique solutionw0

3,πw ∈ H1(0,T ;H20 (ω)×L2

0(Ω)), ∂y3 πw ∈ H1(0,T ;L2(Ω)).Furthermore, w03 ∈

H1(0,T ;H3(ω)).

Proof The existence and the uniqueness are based on the energy estimate. If wechoose ∂tw0

3 as a test function for equation (30) and πw as a test function for sys-tem (28)-(29), then integration and summing up yield the equality

∂t

∫Ω

y23

1−ν|∆y1,y2w0

3|2 dy+∫

Ω

y3α(1−2ν)1−ν

πw∆y1,y2∂tw03 dy+

∫Ω|∂y3 πw|2 dy+

12

∂t

∫Ω(γ +α2 1−2ν

2(1−ν))(πw)

2 dy−∫

Ω

y3α(1−2ν)1−ν

πw∆y1,y2∂tw03 dy =

−∫

ΩU1∂y3 πw dy+

∫Ω

Π1,−1∂tw03 dy. (31)


After noting that the second and the fifth terms cancel each other, we obtain the apriori estimate for w0

3,πw in L∞(0,T ;H20 (ω)× L2

0(Ω)), ∂y3 πw ∈ L2((0,T )×Ω).After taking the time derivative of the equations, we obtain the same estimate for∂tw0

3,πw.Now the existence and the uniqueness follow. Additional space regularity of w0

3follows from equation (30).

Remark 3 Let γ = γ +α2 1−2ν2(1−ν)

. Using separation of variables, we obtain the for-

mulas

πw(y1,y2,y3, t) =α(1−2ν)γ(1−ν)

y3∆y1,y2w03(y1,y2, t)+

2γ

+∞

∑j=1

(−1) j(∫ t

0exp−π2(2 j−1)2(t − τ)

4γ(U1(y1,y2,τ)+

α(1−2ν)γ(1−ν)

∆y1,y2w03(y1,y2,τ)

)dτ

)sin((2 j−1)

πy3

2), (32)

∫ 1

−1y3πw dy3 =− 16

π2γ

+∞

∑j=1

(∫ t

0exp−π2(2 j−1)2(t − τ)

4γ(U1(·,τ)+

α(1−2ν)γ(1−ν)

∆y1,y2w03(·,τ)

)dτ

)+

2α(1−2ν)3γ(1−ν)

∆y1,y2w03(·, t). (33)

After plugging formula (33) into equation (30), we observe memory effects in thebending equation.

Theorem 1 (Convergence of the displacement and the pressure) Let us supposeassumptions (1)-(2). Letw0,πm be given by (26)-(27) and w0

3,πw by (28)-(30).Furthermore let

E0cor(x1,x2,x3, t) =

α(1−2ν)2(1−ν)

(πw(x1,x2,x3

ε, t)+πm(x1,x2, t))−

ν1−ν

( div x1,x2w0(x1,x2, t)−x3

ε∆x1,x2w0

3(x1,x2, t)). (34)

Then we have the following convergences

limε→0

max0≤t≤T

1|Ω ε |

∫Ω ε

ε−2|∂uε

3∂x3

− εE0cor|2 dx = 0, (35)

limε→0

max0≤t≤T

1|Ω ε |

∫Ω ε

|ei j

(uε

ε−w0 +

x3

ε∇x1,x2w0

3

)|2 dx = 0, 1 ≤ i, j ≤ 2, (36)

limε→0

max0≤t≤T

1|Ω ε |

∫Ω ε

|e j3

(uε)

ε|2 dx = 0, 1 ≤ j ≤ 2, (37)

limε→0

max0≤t≤T

1|Ω ε |

∫Ω ε

|uε

j

ε−w0

j +x3

ε∇x1,x2w0

3|2 dx = 0, 1 ≤ j ≤ 2, (38)


limε→0

max0≤t≤T

1|Ω ε |

∫Ω ε

|uε3 −w0

3(x1,x2, t)|2 dx = 0, (39)

limε→0

max0≤t≤T

1|Ω ε |

∫Ω ε

| pε

ε−πw(x1,x2,

x3

ε, t)−πm(x1,x2, t)|2 dx = 0, (40)

limε→0

max0≤t≤T

1|Ω ε |

∫Ω ε

|∇x1,x2 pε |2 + |∂x3 pε −∂y3 πw|y3=x3/ε |2 dx = 0. (41)

Proof It is a direct consequence of Proposition 11 . It is enough to rescale back thevariables and the unknowns.

Remark 4 Convergence (41) justifies Taber’s hypothesis (5).

Theorem 2 (Convergence of Biot’s stress) Let us suppose assumptions from The-orem 1. Then we have

limε→0

max0≤t≤T

1|Ω ε |

∫Ω ε

|σ ε

j j

ε−2e j j(w0)+2

x3

ε∂ 2w0

3

∂x2j+2E0

cor|2 dx = 0, j = 1,2, (42)

limε→0

max0≤t≤T

1|Ω ε |

∫Ω ε

|σ ε

12ε

−2e12(w0)+2x3

ε∂ 2w0

3∂x1∂x2

|2 dx = 0, (43)

limε→0

max0≤t≤T

1|Ω ε |

∫Ω ε

|σ ε

j3

ε|2 dx = 0, j = 1,2,3. (44)

Proof It is a direct consequence of Proposition 12 . It is enough to rescale back thevariables and the stresses.

Remark 5 We note that convergences (44) justify Kirchoff’s hypothesis for the caseof the poroelastic plate.

4 The scaled problem

Our objective is to study the behavior of the displacement uε and the pressure pε

in the limit ε → 0. Since the problem is defined on the domain Ω ε = ω × (−ε,ε),which is changing with ε , it is convenient to transform it to a problem defined on afixed Ω = ω × (−1,1).

To the unknowns uε and pε , defined on Ω ε , we associate the scaled displacementand pressure fields w(ε) and π(ε) by the scalings y j = x j for j = 1,2 and y3 = x3

εwith x ∈ Ω ε , y ∈ Ω such that

w j(ε)(y1,y2,y3, t) =uε

j (x1,x2,x3, t)

ε, for j = 1,2,

w3(ε)(y1,y2,y3, t) = uε3(x1,x2,x3, t),

π(ε)(y1,y2,y3, t) =pε(x1,x2,x3, t)

ε. (45)

By direct calculation we find out that the scaled displacement and pressure fieldssatisfy the following variational problem:


Let V (Ω) =

z ∈ H1(Ω)3| z|Γ = 0, where Γ = ∂ω ×(−1,1) and denote w =

(w1,w2) and φ = (φ1,φ2).

Find w(ε) ∈ H1(0,T,V (Ω)), π(ε) ∈ H1(0,T ;H1(Ω)), such that it holds

2∫

Ω

2

∑i, j=1

ei j(w(ε))ei j(φ)dy+2ν

1−2ν

∫Ω

divy1,y2w(ε) divy1,y2 φdy

−α∫

Ωπ(ε) divy1,y2 φdy+

1ε2

4∫

Ω

2

∑j=1

e3 j(w(ε))e3 j(φ)dy

−α∫

Ωπ(ε)∂y3 φ3dy+

2ν1−2ν

∫Ω

divy1,y2w(ε)∂y3 φ3dy

+2ν

1−2ν

∫Ω

∂y3 w3(ε) divy1,y2 φdy+

2(1−ν)(1−2ν)ε4

∫Ω

∂y3 w3(ε)∂y3 φ3dy

=2

∑j=1

∫Σ+∪Σ−

P jφ jds+∫

Σ+∪Σ−P3φ3ds, ∀φ ∈V, t ∈ (0,T ), (46)

γ∫

Ω∂tπ(ε)ζ dy+

∫Ω

α divy1,y2∂tw(ε)ζ dy+ ε2∫

Ω∇y1,y2 π(ε)∇y1,y2 ζ dy

+αε−2∫

Ω∂y3 ∂tw3(ε)ζ dy+

∫Ω

∂π(ε)∂y3

∂ζ∂y3

dy =−ε∫ 1

−1

∫∂ω

V ζ ds

−∫

Σ+∪Σ−±U1ζ ds, ∀ζ ∈ H1(Ω), (47)

π(ε)|t=0 = 0, in Ω . (48)

5 Convergence of the scaled displacement and the pressure as ε → 0

Proposition 3 The following a priori estimates hold

∥w(ε)∥H1(0,T ;H1(Ω)3)+∥π(ε)∥H1(0,T ;L2(Ω)) ≤ C, (49)

∥ei j(w(ε))∥L2(Ω×(0,T )) ≤ C, 1 ≤ i, j ≤ 2, (50)

∥ei3(w(ε))∥L2(Ω×(0,T )) ≤ Cε, 1 ≤ i ≤ 2, (51)

∥e33(w(ε))∥L2(Ω×(0,T )) ≤ Cε2, (52)

∥∂y3 π(ε)∥L2(Ω×(0,T )) ≤ C, (53)

∥∇y1,y2 π(ε)∥L2(Ω×(0,T )) ≤Cε. (54)


Proof Testing equations (46) and (47) by φ = ∂tw(ε) and ζ = π(ε), respectively,integrating them and summing up, we obtain

12

ddt

2∫

Ω

2

∑i, j=1

∣∣ei j(w(ε))∣∣2 dy+

2ν1−2ν

∫Ω

∣∣ divy1,y2w(ε)∣∣2 dy+ γ

∫Ω|π(ε)|2 dy

+4ε2

∫Ω

2

∑j=1

∣∣e3 j(w(ε))∣∣2 dy+

4νε2(1−2ν)

∫Ω

divy1,y2w(ε)∂y3 u3(ε)dy

+2(1−ν)(1−2ν)ε4

∫Ω

∣∣∂y3 w3(ε)∣∣2 dy

+ ε2

∫Ω

∣∣∇y1,y2π(ε)∣∣2 dy+

∫Ω

∣∣∣∣∂π(ε)∂y3

∣∣∣∣2 dy

=ddt

2

∑j=1

∫Σ+∪Σ−

P jw j(ε)ds+∫

Σ+∪Σ−P3w3(ε) ds

−

2

∑j=1

∫Σ+∪Σ−

∂tP jw j(ε)ds−∫

Σ+∪Σ−∂tP3w3(ε) ds

−∫

Σ+∪Σ−±U1π(ε) ds− ε

∫ 1

−1

∫∂ω

V π(ε)ds. (55)

We can estimate the terms on the right hand-side of (55):∣∣∣∣∫Σ+∪Σ−±U1π(ε)ds

∣∣∣∣= ∣∣∣∣∫Ω

∂∂y3

(U1π(ε))ds∣∣∣∣≤C

∫Ω

∣∣∣∣ ∂∂y3

π(ε)∣∣∣∣ds (56)∣∣∣∣∫Σ+∪Σ−

P3w3(ε)ds∣∣∣∣= ∣∣∣∣∫Ω

∂y3

(w3(ε)

(P1 −P2

2y3 +

P1 +P2

2

))dy∣∣∣∣≤

C∫

Ω(∣∣∂y3 w3(ε)

∣∣+ |w3(ε)|) dy ≤︸︷︷︸using Korn’s inequality

C(∫

Ω|e(w(ε))|2 dy)1/2,

(57)∣∣∣∣∫Σ+∪Σ−∂tP jw j(ε) ds

∣∣∣∣= |∫

Ω∂y3(

P j|y3=1 −P j|y3=−1

2y3+

P j|y3=1 +P j|y3=−1

2)w j(ε)dy| ≤C

∫Ω

∣∣∂y3 w j(ε)∣∣dy+C

∫Ω

∣∣w j(ε)∣∣dy ≤

C(∫

Ω

∣∣∇w j(ε)∣∣2 dy

)1/2

≤C∥e(w(ε))∥L2(Ω). (58)

The initial value of w(ε) is calculated using pin and we have w(ε)|t=0 = 0. Usingequation (47) we find that ∂tπ(ε)|t=0 = 0, as well. The integration in time of (55)gives estimates (49)-(54), but with L2-time norms.

Next we calculate the time derivative of equations (46)-(47), test by φ = ∂ttw(ε)and ζ = ∂tπ(ε). After repeating the above calculations, we obtain estimates (49)-(54).


Proposition 4 The weak limit w∗,π0∈H1(Ω ×(0,T ))3×H1(0,T ;L2(Ω)), ∂y3 π0 ∈H1(0,T ;L2(Ω)) of the weakly converging subsequence w(ε),π(ε)ε>0 belongs toH1(0,T ;Vkl(Ω))×H1(0,T ;L2(Ω)), where

Vkl(Ω) = v ∈ H1(Ω)3, e j3(v) = 0 in Ω and v = 0 on ∂ω × (−1,1).

Proof Using weak compactness, we find out that there exists a subsequence, denotedby the same subscript ε , and elements w∗ ∈H1(Ω ×(0,T ))3 and π0 ∈H1(Ω ×(0,T ))such that

w(ε) w∗ weakly in H1(Ω × (0,T ))3 as ε → 0, (59)π(ε) π0 weakly in H1(0,T ;L2(Ω)) as ε → 0, (60)∂y3 π(ε) ∂y3 π0 weakly in H1(0,T ;L2(Ω)) as ε → 0, (61)

e(w(ε)) e(w∗) weakly in L2s (Ω × (0,T )) as ε → 0. (62)

Using (51)-(52) we obtain

∥e j3(w(ε))∥L2(Ω×(0,T )) ≤ Cε, j = 1,2, (63)

∥e33(w(ε))∥L2(Ω×(0,T )) ≤ Cε2. (64)

Hence, by weak lower semicontinuity of norm

∥e j3(w∗)∥L2(Ω×(0,T )) ≤ liminfε→0

∥e j3(w(ε))∥L2(Ω×(0,T )) = 0. (65)

Consequently, w∗ ∈ H1(0,T ;Vkl(Ω)).

Lemma 2 The space Vkl defined in Proposition 4 is characterized by the followingproperties

Vkl(Ω) = v ∈ H1(Ω)3, v j = g j −y3∂g3

∂y j, g j ∈ H1

0 (ω), j = 1,2; v3 = g3 ∈ H20 (ω).

Proof The proof of this Lemma is given by Ciarlet [6][p.23].

Corollary 1 There is w0 ∈ H1(0,T ;H10 (ω)3), w0

3 ∈ H1(0,T ;H20 (ω)), such that in

Ω × (0,T )

w∗j = w0

j(y1,y2, t)− y3∂y j w03(y1,y2, t), j = 1,2; w∗

3 = w03(y1,y2, t). (66)

Proposition 5 Let < π0 >= 12∫ 1−1 π0 dy3. Let

Ecor = Ecor(y1,y2, t) =α(1−2ν)< π0 >−2νdiv y1,y2(w

01,w

02)

2(1−ν).

Then,

1ε2 e33(w(ε)) =

∂w3(ε)∂y3

1ε2

α(1−2ν)π0 −2ν div y1,y2 w∗

2(1−ν)= E∗

cor =

Ecor +νy3

1−ν∆y1,y2w0

3 +α(1−2ν)2(1−ν)

(π0−< π0 >), (67)

weakly in L2(Ω × (0,T )), as ε → 0.


Proof After setting φ j = 0 in equation (46) and multiplying by ε2, we get

2∫

Ω

2

∑j=1

e3 j(w(ε))∂y3 φ3 dy−α∫

Ωπ(ε)∂y3 φ3dy+

2ν1−2ν

∫Ω

divy1,y2w(ε)∂y3 φ3 dy+2(1−ν)(1−2ν)ε2

∫Ω

∂y3 w3(ε)∂y3 φ3dy

= ε2∫

Σ+∪Σ−P3φ3ds.

Passing to the limit ε → 0 yields

∫ T

0

∫Ω

(2(1−ν)1−2ν

E∗cor +

2ν1−2ν

divy1,y2(w∗1,w

∗2)−απ0

)∂φ3

∂y3dy = 0

for all φ3 ∈C(0,T ;H1(Ω)), φ3|ω = 0. Now it is straightforward to conclude (67).

Proposition 6 It holds

e j3(w(ε))ε

0, weakly in L2(Ω × (0,T )), as ε → 0, j = 1,2. (68)

Proof Setting φ3 = 0 in (46) and multiplying by ε yield

2ε

∫ T

0

∫Ω

2

∑j=1

e3 j(w(ε))∂y3 φ j +2ν

(1−2ν)ε

∫ T

0

∫Ω

∂y3 w3(ε)divy1,y2 φ = O(ε).

In the limit ε → 0 we obtain

e j3(w(ε))ε

χ j, weakly in L2(Ω × (0,T )) (69)

and∫

Ω χ j∂y3 φ j = 0, ∀φ j ∈ H1(Ω), φ j|∂ω×(−1,1) = 0. It finishes the proof of Propo-sition 6.

Proposition 7 (w01,w

02) satisfies system (26)-(27) with πm =< π0 >.

Proof We test equation (46) by φ ∈Vkl(Ω). Since e j3(φ) = 0, we have

2∫

Ω

2

∑i, j=1

ei j(w(ε))ei j(φ)dy+2ν

1−2ν

∫Ω

divy1,y2w(ε) divy1,y2 φ dy

−α∫

Ωπ(ε) divy1,y2 φ dy+

2ν(1−2ν)ε2

∫Ω

∂y3 w3(ε) divy1,y2 φ dy

=2

∑j=1

∫Σ+∪Σ−

P jφ j ds+∫

Σ+∪Σ−P3φ3 ds. (70)


We use Proposition 5 and pass to the limit ε → 0 in equation (70). It yields∫Ω

(2

2

∑i, j=1

ei j(w∗)ei j(φ)−1−2ν1−ν

(απ0−

2ν1−2ν

divy1,y2w∗) divy1,y2 φ)dy =

2

∑j=1

∫Σ+∪Σ−

P jφ j ds+∫

Σ+∪Σ−P3φ3 ds (71)

and choice φ = (g1,g2,0), g j ∈ H10 (ω), gives equation (27).

Finally, we take ζ = ζ (y1,y2, t) as a test function in (47). Using Proposition 5 andzero initial data, we obtain the equality (26).

Proposition 8 The pressure equation reads

∂t(γ +α2 1−2ν

2(1−ν))π0 +α

1−2ν1−ν

div y1,y2(w01,w

02)− ∂ 2π0

∂y23−

α1−2ν1−ν

y3∆y1,y2∂tw03 = 0 in Ω × (0,T ), (72)

∂y3 π0|y3=1 = ∂y3 π0|y3=−1 =−U1(y1,y2, t) in (0,T ), (73)

π0|t=0 = 0 in Ω . (74)

Proof Passing to the limit ε → 0 in equation (47) yields

γ∫

Ω∂tπ0ζ dy+

∫Ω

α( divy1,y2 ∂tw∗+∂tE∗cor)ζ dy

+∫

Ω

∂π0

∂y3

∂ζ∂y3

dy =−∫

Σ+∪Σ−±U1ζ ds, ∀ζ ∈ H1(Ω). (75)

Using Proposition 5 and zero initial data, we obtain from (75) system (72)-(74).

Corollary 2 The function πw = π0−< π0 > satisfies system (28)-(29).

Proposition 9 The limit plate deflection w03 satisfies equation (30).

Proof We take as test φ = (−y3∂g3

∂y1,−y3

∂g3

∂y2,g3), with g3 ∈ H2

0 (ω). It yields

e11(φ) =−y3∂ 2g3

∂y21, e22(φ) =−y3

∂ 2g3

∂y22, e12(φ) =−y3

∂ 2g3

∂y1∂y2,

div y1,y2(φ1,φ2) =−y3∆y1,y2g3

Passing to the limit ε → 0 in equation (46) yields∫Ω

(2y2

3

2

∑i, j=1

∂ 2w03

∂yi∂y j

∂ 2g3

∂yi∂y j+ y3

1−2ν1−ν

(απ0 +2νy3

1−2ν∆y1,y2w0

3)∆y1,y2g3)dy

=−2

∑j=1

∫Σ+∪Σ−

P jy3∂g3

∂y jds+

∫Σ+∪Σ−

P3g3ds. (76)

Equation (76) implies (30).


Proposition 10 The whole sequence w(ε), p(ε) satisfies

w j(ε) w0j − y3

∂w03

∂y j, j = 1,2, weakly in H1(Ω × (0,T )) as ε → 0, (77)

w3(ε) w03 weakly in H1(Ω × (0,T )) as ε → 0, (78)

π(ε) π0 = πm +πw weakly in H1(0,T ;L2(Ω)) as ε → 0, (79)

∂y3 π(ε) ∂y3 π0 weakly in H1(0,T ;L2(Ω)) as ε → 0, (80)

where w0 and πm =< π0 > are given by (26)-(27) and w03,πw by (28)- (30).

6 Strong convergence

In this section we establish that the weak convergences from previous section implythe strong convergences.

With such aim, we introduce the corrected unknowns, for which the weak con-vergence to zero was already established in Subsection 5.

Let Ψε ∈C∞0 (ω) be a regularized truncation of the indicator function 1ω , equal to

1ω if dist (y,∂ω)≥ ε and such that ||∇y1,y2Ψε ||Lq(ω) =Cε1/q−1 . We setξ j(ε) = w j(ε)−w0

j + y3∂w0

3∂y j

, j = 1,2;

ξ3(ε) = w3(ε)−w03 − ε2E ,

κ(ε) = π(ε)−π0,

(81)

with E = Ψε(y1,y2)∫ y3

0E∗

cor(y1,y2,a, t) da. The choice of correcting terms is ex-

plained by the following result

Lemma 3 We have1ε

e j3(ξ (ε)) 0 weakly in H1(0,T ;L2(Ω)), j = 1,2,3, as ε → 0, (82)

ξ (ε) 0 weakly in H1(0,T ;H10 (Ω))3, as ε → 0, (83)

1ε2 e33(ξ (ε)) 0 weakly in H1(0,T ;L2(Ω)), as ε → 0, (84)

Proof First, using definition (81) we find out that

1ε

e j3(ξ (ε)) =1ε

e j3(w(ε))− ε2

∂E

∂y j︸︷︷︸

by(68)

0,

implying (82).(83) follows from Proposition 4 and Corollary 1 and smallness of the correcting

terms.Finally,

1ε2 e33(ξ (ε)) =

1ε2 e33(w(ε))− ∂E

∂y3︸︷︷︸

by(67)

0.


Next, using equation (71) we obtain

2∫

Ω

2

∑i, j=1

ei j(w0 − y3∇y1,y2 w03)ei j(φ)dy+

2ν1−2ν

∫Ω

divy1,y2(w0

−y3∇y1,y2 w03) divy1,y2 φdy−α

∫Ω

π0 divy1,y2 φdy+2∫

Ω

2

∑j=1

∂E

∂y je3 j(φ)dy

+1ε2

−α

∫Ω

π0∂y3 φ3dy+2ν

1−2ν

∫Ω

divy1,y2(w0 − y3∇y1,y2 w0

3)∂y3 φ3dy

+2ν

1−2ν

∫Ω

∂ (w03 + ε2E )

∂y3divy1,y2 φdy

+

2(1−ν)(1−2ν)ε4

∫Ω

∂y3(w03 + ε2E )∂y3 φ3dy

=2

∑j=1

∫Σ+∪Σ−

P jφ jds+∫

Σ+∪Σ−P3φ3ds+

2ν1−2ν

∫Ω(Ψε −1)E∗

cor divy1,y2 φ dy

+2∫

Ω

2

∑j=1

∂E

∂y je j3(φ) dy− α

ε2

∫Ω

π0∂y3φ3 dy+1ε2

2ν1−2ν

∫Ω

divy1,y2(w0−

y3∇y1,y2w03)∂y3φ3dy+

2(1−ν)(1−2ν)ε2

∫Ω

Ψε E∗cor∂y3 φ3dy, ∀φ ∈V, t ∈ (0,T ). (85)

Consequently, we observe that ξ (ε),κ(ε) satisfy the following variational equation

2∫

Ω

2

∑i, j=1

ei j(ξ (ε))ei j(φ)dy+2ν

1−2ν

∫Ω

divy1,y2 ξ (ε) divy1,y2 φdy

−α∫

Ωκ(ε) divy1,y2 φdy+

1ε2

4∫

Ω

2

∑j=1

e3 j(ξ (ε))e3 j(φ)dy

−α∫

Ωκ(ε)∂y3 φ3dy+

2ν1−2ν

∫Ω

divy1,y2 ξ (ε)∂y3 φ3dy

+2ν

1−2ν

∫Ω

∂y3 ξ3(ε) divy1,y2 φdy+

2(1−ν)(1−2ν)ε4

∫Ω

∂y3 ξ3(ε)∂y3 φ3dy =

−2∫

Ω

2

∑j=1

∂E

∂y je j3(φ) dy+

αε2

∫Ω

π0∂y3 φ3 dy− 2ν1−2ν

∫Ω(Ψε −1)E∗

cor divy1,y2 φ dy

− 2ν1−2ν

∫Ω

divy1,y2(w0 − y3∇y1,y2 w0

3)∂y3 φ3

ε2 dy

− 2(1−ν)(1−2ν)ε2

∫Ω

Ψε E∗cor∂y3 φ3dy, ∀φ ∈V, t ∈ (0,T ). (86)


Effective pressure equation (75) implies

γ∫

Ω∂tπ0ζ dy+

∫Ω

α( divy1,y2 ∂tw∗+ ε−2∂t∂y3(w03 + ε2E ))ζ dy

+∫

Ω

∂π0

∂y3

∂ζ∂y3

dy+ ε2∫

Ω∇y1,y2π0∇y1,y2 ζ dy = ε2

∫Ω

∇y1,y2π0∇y1,y2 ζ dy

−∫

Σ+∪Σ−±U1ζ ds+α

∫Ω(Ψε −1)∂tE∗

corζ dy, ∀ζ ∈ H1(Ω). (87)

Consequently the pressure equation for ξ (ε),κ(ε) is

γ∫

Ω∂tκ(ε)ζ dy+

∫Ω

α divy1,y2∂t ξ (ε)ζ dy+ ε2∫

Ω∇y1,y2 κ(ε)∇y1,y2 ζ dy

+αε−2∫

Ω∂y3 ∂tξ3(ε)ζ dy+

∫Ω

∂κ(ε)∂y3

∂ζ∂y3

dy =−ε∫ 1

−1

∫∂ω

V ζ ds

−ε2∫

Ω∇y1,y2π0∇y1,y2 ζ dy−α

∫Ω(Ψε −1)∂tE∗

corζ dy, ∀ζ ∈ H1(Ω). (88)

By the choice of the correcting terms, ξ (ε) = 0 on ∂ω×(−1,1). Now we test (86) byφ = ∂tξ (ε), (88) by ζ = κ(ε) and add the obtained equalities, to obtain the followingenergy equality

12

ddt

2∫

Ω

2

∑i, j=1

∣∣ei j(ξ (ε))∣∣2 dy+

2ν1−2ν

∫Ω

∣∣∣ divy1,y2 ξ (ε)∣∣∣2 dy+ γ

∫Ω|κ(ε)|2 dy

+4ε2

∫Ω

2

∑j=1

∣∣e3 j(ξ (ε))∣∣2 dy+

4ν(1−2ν)ε2

∫Ω

divy1,y2 ξ (ε)∂y3 ξ3(ε) dy

+2(1−ν)(1−2ν)ε4

∫Ω

∣∣∂y3 ξ3(ε)∣∣2 dy

+ ε2

∫Ω

∣∣∇y1,y2κ(ε)∣∣2 dy+

∫Ω

∣∣∣∣∂κ(ε)∂y3

∣∣∣∣2 dy

=ddt

−2

∫Ω

2

∑j=1

ε∂E

∂y j

e j3(ξ (ε))ε

dy+αε2

∫Ω

π0∂y3 ξ3(ε) dy−

2ν1−2ν

∫Ω

(divy1,y2(w

0 − y3∇y1,y2 w03)

∂y3 ξ3(ε)ε2 +(Ψε −1)E∗

cor divy1,y2 ξ (ε))

dy

− 2(1−ν)(1−2ν)ε2

∫Ω

Ψε E∗cor∂y3 ξ3(ε)dy

+2ε

∫Ω

2

∑j=1

∂ 2E

∂y j∂ te j3(ξ (ε))

εdy

− αε2

∫Ω

∂tπ0∂y3 ξ3(ε) dy+2(1−ν)(1−2ν)ε2

∫Ω

Ψε ∂tE∗cor∂y3 ξ3(ε)dy+

2ν1−2ν

∫Ω

(∂t divy1,y2(w

0 − y3∇y1,y2 w03)

∂y3ξ3(ε)ε2 +(Ψε −1)∂tE∗

cor divy1,y2 ξ (ε))

dy

−ε2∫

Ω∇y1,y2π0∇y1,y2κ(ε) dy− ε

∫ 1

−1

∫∂ω

V κ(ε)ds−α∫

Ω(Ψε −1)∂tE∗

corκ(ε) dy. (89)


Proposition 11 For the whole sequence ξ (ε),κ(ε) we have

ei j(ξ (ε))→ 0 strongly in H1(0,T ;L2(Ω)), i, j = 1,2, as ε → 0, (90)

ξ (ε)→ 0 strongly in H1(0,T ;L2(Ω))3, as ε → 0, (91)

ei3(ξ (ε))ε

→ 0 strongly in H1(0,T ;L2(Ω)), as ε → 0, (92)

∂ξ3(ε)ε2∂y3

→ 0 strongly in H1(0,T ;L2(Ω)), as ε → 0, (93)

κ(ε)→ 0 strongly in H1(0,T ;L2(Ω)), as ε → 0, (94)

∂y3 κ(ε)→ 0 strongly in H1(0,T ;L2(Ω)), as ε → 0, (95)

ε∇y1,y2κ(ε)→ 0 strongly in H1(0,T ;L2(Ω)), as ε → 0. (96)

Proof We follow proof of Proposition 3 and use equality (89). In fact only new term

to be estimated is ε∫ 1

−1

∫∂ω

V κ(ε)ds. We recall the well-known interpolation inequal-

ity

||ζ ||L2(∂ω×(−1,1)) ≤C||ζ ||1/2L2(Ω)

||ζ ||1/2L2(−1,1;H1(ω))

.

It yields

|ε∫ 1

−1

∫∂ω

V κ(ε) ds| ≤Cε||V ||L2(∂ω×(−1,1))||κ(ε)||1/2L2(Ω)

(||κ(ε)||1/2

L2(Ω)+

||∇y1,y2 κ(ε)||1/2L2(Ω)2

)≤C1ε2/3||κ(ε)||L2(Ω)+

ε2

2||∇y1,y2 κ(ε)||2L2(Ω)2 . (97)

Now we integrate equality (89) in time, use Proposition 10 and Lemma 3 and con-clude the strong convergence in L2(Ω ×(0,T )). Iterating the argument after calculat-ing the time derivative of equations (86) and (88), yields the convergences (90)-(96).

7 Convergence of the poroelastic stress

The rescaled stress σ(w(ε)) is, in analogy with (11), given by

σ(w(ε)) = 2e(w(ε))+(2ν

1−2νdiv w(ε)−απ(ε))I in Ω × (0,T ). (98)

Nevertheless, this quantity does not correspond to the rescaled dimensionless physi-cal stress σ ε . We introduce the rescaled poroelastic stress σ(ε) by

σ(ε)ε

=

2

∂w1(ε)∂y1

+D(ε) 2e12(w(ε))1ε

e13(w(ε))

2e12(w(ε)) 2∂w2(ε)

∂y2+D(ε)

1ε

e23(w(ε))1ε

e13(w(ε))1ε

e23(w(ε))2ε2

∂w3(ε)∂y3

+D(ε)

, (99)


where

D(ε) =2ν

1−2νdivy1,y2w(ε)−απ(ε)+

2ν1−2ν

1ε2

∂w3(ε)∂y3

. (100)

As a direct consequence of Proposition 11, we obtain the following convergencesfor the stresses

Proposition 12 Let w0,πm be given by (26)-(27) and w03,πw by (28)- (30). Let

E∗cor be defined by (67). Then we have

D(ε)→−2E∗cor in H1(0,T ;L2(Ω)) as ε → 0, (101)

σ j j(ε)ε

−2e j j(w0)+2y3∂ 2w0

3

∂y2j+2E∗

cor → 0

in H1(0,T ;L2(Ω)) as ε → 0, j = 1,2, (102)

σ12(ε)ε

−2e12(w0)+2y3∂ 2w0

3∂y1∂y2

→ 0 in H1(0,T ;L2(Ω)) as ε → 0, (103)

σ j3(ε)ε

→ 0 in H1(0,T ;L2(Ω)) as ε → 0, j = 1,2, (104)

σ33(ε)ε

→ 0 in H1(0,T ;L2(Ω)) as ε → 0. (105)

8 Appendix: A classical Kirchhoff type poroelastic plate equations derivation

We follow [15] and, together with Kirchhoff’s hypothesis, suppose that

1. The vertical deflection of the plate takes form u3 = w = w(x1,x2, t).

2. For small deflections w and rotations (ϑ1,ϑ2) we have ϑ1 =∂w∂x2

and ϑ2 =− ∂w∂x1

.

Then, Kirchhoff’s hypothesis implies that the displacements u satisfy

u1 = uω1 (x1,x2, t)− x3

∂w∂x1

; u2 = uω2 (x1,x2, t)− x3

∂w∂x2

, (106)

where (uω1 ,u

ω2 ) are the tangential displacements of points lying on the midsur-

face. Now a direct calculation gives e j3(u) = 0, j = 1,2. Since σ = 2Ge(u)+

(2νG

1−2νdiv u−α p)I, we conclude that also σ j3 = 0, j = 1,2.

3. For the pressure field, we impose the normal velocities U ℓ at the top and bottomsurfaces and suppose (5). Finally, σ33 is also supposed small throughout the plate.The later assumption gives

e33(u) =− ν1+ν

12G

(σ11 +σ22)+α(1−2ν)2G(1+ν)

p. (107)


4. For the other components of the strain tensor we have

e11(u) =∂uω

1∂x1

− x3∂ 2w∂x2

1; e12(u) =

12(

∂uω1

∂x2+

∂uω2

∂x1)− x3

∂ 2w∂x1∂x2

;

e22(u) =∂uω

2∂x2

− x3∂ 2w∂x2

2; e33(u) =− ν

1−ν(e11(u)+ e22(u))+

α(1−2ν)2G(1−ν)

p;

2νG1−2ν

div u−α p =2Gν1−ν

(e11(u)+ e22(u))−α(1−2ν)

1−νp

5. For the other components of the stress tensor we have

σ11 =2G

1−ν(e11(u)+νe22(u))−

α(1−2ν)1−ν

p (108)

σ22 =2G

1−ν(e22(u)+νe11(u))−

α(1−2ν)1−ν

p (109)

σ12 = 2Ge12(u) = G(∂uω

1∂x2

+∂uω

2∂x1

)−2Gx3∂ 2w

∂x1∂x2. (110)

Next we define the stress resultants (forces per unit length) N1,N2,N12 by

N1 =∫ ℓ/2

−ℓ/2σ11 dx3, N2 =

∫ ℓ/2

−ℓ/2σ22 dx3, N12 =

∫ ℓ/2

−ℓ/2σ12 dx3, (111)

the effective stress resultant due to the variation in pore pressure across theplate thickness N by

N =−∫ ℓ/2

−ℓ/2p dx3 (112)

and the external loading tangential to the plate

fi = σi3|x3=ℓ/2 −σi3|x3=−ℓ/2 = Pℓi +P−ℓ

i , i = 1,2. (113)

Next we average the equation (2) over the thickness and obtain

∂N1

∂x1+

∂N12

∂x2+ f1 = 0, (114)

∂N12

∂x1+

∂N2

∂x2+ f2 = 0. (115)

Inserting (108)-(110) into the formulas (111) gives

N1 =2Gℓ

1−ν(

∂uω1

∂x1+ν

∂uω2

∂x2)+

α(1−2ν)1−ν

N; (116)

N2 =2Gℓ

1−ν(

∂uω2

∂x2+ν

∂uω1

∂x1)+

α(1−2ν)1−ν

N; (117)

N12 = Gℓ(∂uω

1∂x2

+∂uω

2∂x1

). (118)

A substitution of equations (116)-(118) into the (114)-(115) yields the equations forstretching of a plate of uniform thickness (6).


We need one more equation to complete the system (6). We have

div u =1−2ν1−ν

(divx1,x2(u

ω1 ,u

ω2 )− x3∆x1,x2w

)+

α(1−2ν)2G(1−ν)

p, (119)∫ ℓ/2

−ℓ/2div u dx3 =

ℓ(1−2ν)1−ν

divx1,x2(uω1 ,u

ω2 )−

α(1−2ν)2G(1−ν)

N. (120)

Next we average equation (3) over thickness, use the assumption (5) and expres-sion (120) and get (7). Inserting (119) in equation (3) and using hypothesis (5) andequation (7), yields equation (8).

It remains to find the equation for transverse deflection and for the bending mo-ment due to the variation in pore pressure across the plate thickness.

The bending moment M due to the variation in pore pressure across the platethickness is given by

M =−∫ ℓ/2

−ℓ/2x3 p dx3. (121)

For the stress moments of the plate, which have as physical dimension the momentper unit length, we have:

1. The twisting moment M12 :

M12 =∫ ℓ/2

−ℓ/2σ12 x3dx3 =−Gℓ3

6∂ 2w

∂x1∂x2. (122)

2. The bending moments M1 and M2:

M1 =∫ ℓ/2

−ℓ/2σ11 x3dx3 =− Gℓ3

6(1−ν)(

∂ 2w∂x2

1+ν

∂ 2w∂x2

2)+α

1−2ν1−ν

M (123)

M2 =∫ ℓ/2

−ℓ/2σ22 x3dx3 =− Gℓ3

6(1−ν)(

∂ 2w∂x2

2+ν

∂ 2w∂x2

1)+α

1−2ν1−ν

M. (124)

3. The transverse shear (Q1,Q2): Qi =∫ ℓ/2−ℓ/2 σi3 dx3, i = 1,2.

4. Resultant external moment (m1,m2) : mi =ℓ2 (P

ℓi −P−ℓ

i ).

Next we multiply the equation (2) by x3 and integrate over the thickness to obtain

∂M1

∂x1+

∂M12

∂x2−Q1 +m1 = 0, (125)

∂M12

∂x1+

∂M2

∂x2−Q2 +m2 = 0. (126)

Averaging the third equation in (2) over the thickness yields:

∂Q1

∂x1+

∂Q2

∂x2+Pℓ

3 +P−ℓ3 = 0 (127)


Following Fung’s textbook, we eliminate Qi from (125)-(127) and obtain the equationof equilibrium in moments:

∂ 2M1

∂x21

+2∂ 2M12

∂x1∂x2+

∂ 2M2

∂x22

+∂m1

∂x1+

∂m2

∂x2+Pℓ

3 +P−ℓ3 = 0. (128)

After inserting the formulas (122)-(124) into (128), we obtain the poroelastic platebending equation (9).

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