A Rigorous Derivation of the Equations for the ... - Andro.Mikelic

Institute of Applied Mathematics, IWR and BIOQUANT, University of .... which is simpler than general poroelasticity equations. ..... 8 we give an ad hoc derivation of the model, which follows mechanical engineering ...... a viscous fluid and an elastic body, M3AS: Mathematical Models and Methods in Applied Sciences,. Vol.
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A Rigorous Derivation of the Equations for the Clamped Biot-Kirchhoff-Love Poroelastic Plate Anna Marciniak-Czochra · Andro Mikeli´c

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-static Biot’s equations in thin poroelastic plates as the thickness tends to zero. We choose Terzaghi’s time corresponding to the plate thickness and obtain the strong convergence of the three-dimensional solid displacement, fluid pressure and total poroelastic stress to the solution of the new class of plate equations. In the new equations the 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 bending equation is coupled with the pressure equation and it contains the bending moment due to the variation in pore pressure across the plate thickness. The pressure equation is parabolic only in the vertical direction. As additional terms it contains the time derivative of the in-plane Laplacian of the vertical deflection of the plate and of the elastic in-plane compression term. Keywords Thin poroelastic plate · Biot’s quasi-static equations · bending-flow coupling · 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 German Research Council (DFG). The research of A.M. was partially supported by the Programme Inter Carnot Fraunhofer from BMBF (Grant 01SF0804) and ANR. Anna Marciniak-Czochra Institute of Applied Mathematics, IWR and BIOQUANT, University of Heidelberg, Im Neuenheimer Feld 267, 69120 Heidelberg , Germany Tel. : +496221544871 E-mail: [email protected] Andro Mikeli´c Universit´e de Lyon, CNRS UMR 5208, Universit´e Lyon 1, Institut Camille Jordan, 43, blvd. du 11 novembre 1918, 69622 Villeurbanne Cedex, France Tel. : +33426234548 Fax: +33956109885 E-mail: [email protected]

2

Anna Marciniak-Czochra, Andro Mikeli´c

1 Introduction A plate is a 3D body bounded by two surfaces of small curvature and placed at small distance. A plate is said to be thin if the distance between these surfaces, called the thickness, is much smaller than a characteristic size of the surrounding surfaces. The construction of a linear theory for the extensional and flexural deformation of plates, starting from the 3D Navier equations of linear elasticity, goes back to the 19th century and Kirchhoff’s work. Following a short period of controversy, the complete theory, nowadays known as the Kirchhoff-Love equations for bending of thin elastic plates, was derived. Derivation was undertaken under assumptions, which are referred to as Kirchhoff’s hypothesis. It reads as follows: Kirchhoff’s hypothesis: Every straight line in the plate that was originally perpendicular to the plate midsurface, remains straight after the strain and perpendicular to the deflected midsurface. Since then a theory providing appropriate 2D equations applicable to shell-like bodies was developed. Later, due to the considerable difficulties with the derivation of plate equations from 3D linear elasticity equations, a direct approach in the sense of Truesdell’s school of continuum mechanics, using a Cosserat surface, was proposed. We refer to the review paper of Naghdi [19] . In classical engineering textbooks one finds a formal derivation, based on the Kirchhoff hypothesis. For details we refer to Fung’s textbook [15]. A different approach is to consider the plate equations as an approximation to the 3D elasticity equations in a plate domaine Ω ε = ω × (−ε , ε ), where ω is the middle surface 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 formally in [14]. Then Ciarlet and collaborators developed systematically the approach where the vertical variable x3 ∈ (−ε , ε ) was scaled by setting y3 = x3 /ε . This change of variables transforms the PDE to a singular perturbation problem on a fixed domain. With such approach Ciarlet and Destuynder have established the rigorous error estimate between the 3D solution and the Kirchhoff-Love solution to the 2D plate equations, 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 is due to Dauge and Gruais (see [10], [11] and subsequent work by Dauge and collaborators). Handling general boundary conditions required the boundary layer analysis. For a review, discussing also the results by Russian school, we refer to [12]. Further generalizations to nonlinear plates and shells exist and were obtained using Γ − convergence. We do not discuss it here and don’t undertake to give complete references to the plate theory. Many living tissues are fluid-saturated thin bodies like bones, bladders, arteries and diaphragms and they are interpreted as poroelastic plates or shells. For a review of modeling of bones as poroelastic plates we refer to [9]. Furthermore, industrial filters are an example of poroelastic plates and shells. Our goal is to extend the above mentioned theory to the poroelastic plates. These are 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 an overall or an effective behavior described by the poroelasticity equations instead of the Navier elasticity equations. In addition to the phase displacements, i.e. displacement of the solid and of the fluid phase, description of a poroelastic medium requires the pressure field and, consequently, an additional PDE. The equations were introduced by Biot in fifties based on phenomenological physical modeling. The original derivation of the poroelasticity PDEs can be found in [2], [3] and in the selection of Biot’s publications [27] . The effective model corresponds to the homogenization of the complicated pore level fluid-structure interaction problem formulated based on the continuum mechanics first principles, i.e. the Navier equations for the solid structure and by the NavierStokes equations for the flow. If the problem involves small deformations, a small fluid compressibility and a slow viscous flow, the model can be simplified using linearization. If, in addition, we consider a periodic porous medium with connected fluid and solid phases, then the poroelasticity equations can be rigorously derived from the first principles using homogenization approach. The small parameter of the problem is the ratio between characteristic pore size and the domain size. Applying either Laplace’s transformation in time or the excitation by an external harmonic source with a given frequency, we obtain a stationary system of PDEs containing the frequency as a parameter. In such setting, the two-scale poroelasticity equations can be obtained using formal two-scale expansions in the small parameter, applied to the pore level fluid-structure equations. The two-scale equations contain slow and fast variables and the coefficients depend on frequency in a complicated way. For more details we refer to the book [21], the review [1] and the references therein. First mathematically rigorous justification of the two-scale equations and one of the first major applications of the two-scale convergence technique was due to Nguetseng [20]. Convergence in space and time variables was proved by Mikeli´c and colleagues in [16], [7] and [13], where the effective coefficients were determined through auxiliary problems. Furthermore, in case of inviscid fluid filling the pores, the scales separation was performed in [13] allowing to reduce the two-scale equations to the Biot original model. In case of the pores filled by a viscous fluid, Biot’s equations contain a viscodynamic operator. The corresponding scales separation in the two-scale equations yields the exact expression of the viscodynamic operator. In case of pores filled by a viscous fluid, the result depends on the contrast of property number, being the ratio between the viscosity, multiplied by the characteristic time, and the bulk modulus 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-solid mixture described by the diphasic Biot system. This setting can be always achieved by choosing the corresponding characteristic time, known as Terzaghi’s time. Most of the quoted upscaling results concern the oscillations of the fluid saturated structures. The homogenization leads to nonstationary systems of PDEs containing memory terms (see e.g. [17] for their study). In case of flows through deformable porous media, Terzaghi’s time is sufficiently large. Hence, such problems are quasistatic, i.e. the acceleration effects can be neglected. Derivation of the quasi-static Biot model from the first principles using homogenization techniques can be found in [18].

4

Anna Marciniak-Czochra, Andro Mikeli´c

Following the above mentioned mathematical and engineering literature, it is widely accepted that the deformable porous media, saturated by a fluid, are modeled using Biot’s diphasic equations for the effective solid displacement and the effective pressure. Biot’s equations are valid at every point of the plate and the averaged phases coexist at every point. For the direct continuum mechanics approach to Biot’s equations, 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-saturated porous slab under axial compression. This can be considered as the first study of a poroelastic 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 in the context of the thermodynamics of irreversible processes. More recently, in [26] Theodorakopoulos and Beskos used Biot’s poroelastic theory and the Kirchhoff theory assuming thin plates and neglected any in-plane motion to obtain purely bending vibrations. A systematic approach to the linear poroelastic plate and shell theory was undertaken by Taber in [24] and [25], using Kirchhoff’s approach (see e. g. [23]). In this paper we follow the approach of Ciarlet and Destuynder, as presented in the textbook [22] and rigorously develop equations for a poroelastic plate. Our strategy of deriving the poroelastic plate model is similar to that for the dual porosity model of the Darcy law. In case of Darcy law, upscaling of the Navier-Stokes equations, valid in the pores of a rigid porous medium, yields a linear relationship between the velocity and the pressure gradient, valid at every point, see e.g. [21] and references therein. The result is independent of the size of the porous medium. The homogenization requires that the size of the porous medium is much larger than the pore 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 poroelastic plate model from the pore scale first principles is based on two separated scaling procedures. First, it requires the homogenization of the microscopic model to the mesoscopic level. Then, passing from the 3 dimensional poroelastic model to the poroelastic plate model corresponds to passing from mesoscopic to macroscopic level of description and it is based on the singular perturbation technique introduced by Ciarlet and colleagues. Our approach does not work if one considers a tissue with just few layers of cells. Then the width of such structure is of the size of pores and asymptotic analysis should be performed directly on the pore scale. Successful recent approaches to the derivation of linear and nonlinear plate models use the elastic energy functional. In our situation, considering the steady state setting is not of interest and the quasi-static problem is time-dependent and nonsymmetric. The acceleration term is negligible and the equations for the effective solid skeleton displacement also contain the pressure gradient. They have the structure of a generalized Stokes system, with the velocity field replaced by displacement. Mass conservation equation is parabolic in the pressure and contains the time derivative of the volumetric strain. The structure of the system requires that we use the original approach of Ciarlet and Destuynder. Strong convergence is obtained by extracting the limit functions and at the end our result corresponds to the analogous Γ -convergence results in the purely elastic case.

Derivation of the Equations for the Poroelastic Plate

5

The quasi-static Biot system is well-posed only if there is a relationship between Biot parameters multiplying the pressure gradient in the displacement system and the time derivative of the divergence of the displacement in the pressure equation. We are able to obtain the energy estimate. For the nonlinear Biot system, balancing the appropriate terms is an open mathematical problem. In addition, there exists a major difference, with respect to the limit of the normalized e33 term, compared to a classical derivation of the Kirchhoff-Love plate. In our poroelastic case, the limit also contains the pressure field. Consequence of the more complicated limit term is that the time derivative of the Laplacian of the deflexion appears in the pressure equation and in the Laplacian of the bending pressure moment in the bending equation. We note that the deflexion does not depend on the vertical variable, but the pressure oscillations persist. We prove that the limit problem is well-posed. Nevertheless, it has a richer structure than the classical bending equation. We expect more complex dynamics in this model. For slender poroelastic bodies, more difficulties with boundary layers are to be expected.

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 a smooth 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 lower face) of the plate Ω ℓ . Γ ℓ is the lateral boundary, Γ ℓ = ∂ ωL × (−ℓ/2, ℓ/2). We recall that the ratio between the plate thickness and the characteristic horizontal length is ε = ℓ/(2L) 0 belongs to H 1 (0, T ;Vkl (Ω )) × H 1 (0, T ; L2 (Ω )), where Vkl (Ω ) = {v ∈ H 1 (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, denoted by the same subscript ε , and elements w∗ ∈ H 1 (Ω ×(0, T ))3 and π 0 ∈ H 1 (Ω ×(0, T )) such that w(ε ) ⇀ w∗ weakly in H 1 (Ω × (0, T ))3 as ε → 0, π (ε ) ⇀ π 0 weakly in H 1 (0, T ; L2 (Ω )) as ε → 0, ∂y3 π (ε ) ⇀ ∂y3 π 0 weakly in H 1 (0, T ; L2 (Ω )) as ε → 0, ∗

e(w(ε )) ⇀ e(w ) weakly in

Ls2 (Ω

× (0, T )) as ε → 0.

(59) (60) (61) (62)

Using (51)-(52) we obtain ∥e j3 (w(ε ))∥L2 (Ω ×(0,T )) ≤ Cε ,

j = 1, 2,

∥e33 (w(ε ))∥L2 (Ω ×(0,T )) ≤ Cε . 2

(63) (64)

Hence, by weak lower semicontinuity of norm ∥e j3 (w∗ )∥L2 (Ω ×(0,T )) ≤ lim inf ∥e j3 (w(ε ))∥L2 (Ω ×(0,T )) = 0. ε →0

(65)

Consequently, w∗ ∈ H 1 (0, T ;Vkl (Ω )). Lemma 2 The space Vkl defined in Proposition 4 is characterized by the following properties Vkl (Ω ) = {v ∈ H 1 (Ω )3 , v j = g j − y3

∂ g3 , g j ∈ H01 (ω ), j = 1, 2; v3 = g3 ∈ H02 (ω )}. ∂yj

Proof The proof of this Lemma is given by Ciarlet [6][p.23]. Corollary 1 There is w0 ∈ H 1 (0, T ; H01 (ω )3 ), w03 ∈ H 1 (0, T ; H02 (ω )), such that in Ω × (0, T ) w∗j = w0j (y1 , y2 ,t) − y3 ∂y j w03 (y1 , y2 ,t), j = 1, 2; w∗3 = w03 (y1 , y2 ,t). Proposition 5 Let < π 0 >=

1 ∫1 0 2 −1 π

Ecor = Ecor (y1 , y2 ,t) =

(66)

dy3 . Let

α (1 − 2ν ) < π 0 > −2ν div y1 ,y2 (w01 , w02 ) . 2(1 − ν )

Then, ˜∗ α (1 − 2ν )π 0 − 2ν div y1 ,y2 w 1 ∂ w3 (ε ) 1 ∗ e (w( ⇀ ε )) = = Ecor = 33 ε2 ∂ y3 ε 2 2(1 − ν ) ν y3 α (1 − 2ν ) 0 Ecor + ∆y1 ,y2 w03 + (π − < π 0 >), 1−ν 2(1 − ν ) weakly in L2 (Ω × (0, T )), as ε → 0.

(67)

18

Anna Marciniak-Czochra, Andro Mikeli´c

Proof After setting φ j = 0 in equation (46) and multiplying by ε 2 , we get ∫

2

2

∑ e3 j (w(ε ))∂y3 φ3 dy − α

Ω j=1



2ν 1 − 2ν = ε2



π (ε )∂y3 φ3 dy +

˜ ε )∂y3 φ3 dy + divy1 ,y2 w(







Σ + ∪Σ −

2(1 − ν ) (1 − 2ν )ε 2

∫ Ω

∂y3 w3 (ε )∂y3 φ3 dy

P3 φ3 ds.

Passing to the limit ε → 0 yields ∫ T∫ ( 2(1 − ν ) ∗ Ecor +

1 − 2ν



0

) 2ν 0 ∂ φ3 ∗ ∗ divy1 ,y2 (w1 , w2 ) − απ dy = 0 1 − 2ν ∂ y3

for all φ3 ∈ C(0, T ; H 1 (Ω )), φ3 |ω = 0. Now it is straightforward to conclude (67). Proposition 6 It holds e j3 (w(ε )) ⇀ 0, ε

weakly in L2 (Ω × (0, T )),

as ε → 0, j = 1, 2.

(68)

Proof Setting φ3 = 0 in (46) and multiplying by ε yield 2 ε

∫ T∫

2

∑ e3 j (w(ε ))∂y3 φ j +

Ω j=1

0

2ν (1 − 2ν )ε

∫ T∫ Ω

0

∂y3 w3 (ε )divy1 ,y2 φ˜ = O(ε ).

In the limit ε → 0 we obtain e j3 (w(ε )) ⇀ χ j, ε

weakly in L2 (Ω × (0, T ))

(69)



and Ω χ j ∂y3 φ j = 0, ∀φ j ∈ H 1 (Ω ), φ j |∂ ω ×(−1,1) = 0. It finishes the proof of Proposition 6. Proposition 7 (w01 , w02 ) satisfies system (26)-(27) with πm =< π 0 >. Proof We test equation (46) by φ ∈ Vkl (Ω ). Since e j3 (φ ) = 0, we have ∫

2

2



Ω i, j=1

−α





ei j (w(ε ))ei j (φ )dy +

π (ε ) divy1 ,y2 φ˜ dy + 2

=





+ − j=1 Σ ∪Σ

2ν 1 − 2ν

∫ Ω

2ν (1 − 2ν )ε 2

P j φ j ds+



˜ ε ) divy1 ,y2 φ˜ dy divy1 ,y2 w(

∫ Ω

∂y3 w3 (ε ) divy1 ,y2 φ˜ dy

Σ + ∪Σ −

P3 φ3 ds.

(70)

Derivation of the Equations for the Poroelastic Plate

19

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

2

2



ei j (w∗ )ei j (φ ) −

i, j=1

) 2ν ˜ ∗ ) divy1 ,y2 φ˜ dy = divy1 ,y2 w 1 − 2ν

2



1 − 2ν (απ 0 − 1−ν



+ − j=1 Σ ∪Σ

P j φ j ds+

∫ Σ + ∪Σ −

P3 φ3 ds

(71)

and choice φ = (g1 , g2 , 0), g j ∈ H01 (ω ), gives equation (27). Finally, we take ζ = ζ (y1 , y2 ,t) as a test function in (47). Using Proposition 5 and zero initial data, we obtain the equality (26). Proposition 8 The pressure equation reads { } ∂ 2π 0 1 − 2ν 1 − 2ν ∂t (γ + α 2 )π 0 + α div y1 ,y2 (w01 , w02 ) − − 2(1 − ν ) 1−ν ∂ y23 1 − 2ν α y3 ∆y1 ,y2 ∂t w03 = 0 in Ω × (0, T ), 1−ν ∂y3 π 0 |y3 =1 = ∂y3 π 0 |y3 =−1 = −U 1 (y1 , y2 ,t) in (0, T ),

π |t=0 = 0 0

in

Ω.

(72) (73) (74)

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

γ



∂t π ζ dy +





+



0





∗ ˜ ∗ + ∂t Ecor α ( divy1 ,y2 ∂t w )ζ dy

∂ π0 ∂ ζ dy = − ∂ y3 ∂ y3

∫ Σ + ∪Σ −

±U 1 ζ ds,

∀ζ ∈ H 1 (Ω ).

(75)

Using Proposition 5 and zero initial data, we obtain from (75) system (72)-(74). Corollary 2 The function πw = π 0 − < π 0 > satisfies system (28)-(29). Proposition 9 The limit plate deflection w03 satisfies equation (30). Proof We take as test φ = (−y3 e11 (φ ) = −y3

∂ 2 g3 , ∂ y21

∂ g3 ∂ g3 , −y3 , g3 ), with g3 ∈ H02 (ω ). It yields ∂ y1 ∂ y2

e22 (φ ) = −y3

∂ 2 g3 , ∂ y22

e12 (φ ) = −y3

∂ 2 g3 , ∂ y1 ∂ y2

div y1 ,y2 (φ1 , φ2 ) = −y3 ∆y1 ,y2 g3 Passing to the limit ε → 0 in equation (46) yields ∫ ( Ω

2y23

) ∂ 2 w03 ∂ 2 g3 1 − 2ν 2ν y3 ∑ ∂ yi ∂ y j ∂ yi ∂ y j + y3 1 − ν (απ 0 + 1 − 2ν ∆y1 ,y2 w03 )∆y1 ,y2 g3 dy i, j=1 2

2

=−∑



+ − j=1 Σ ∪Σ

Equation (76) implies (30).

P j y3

∂ g3 ds + ∂yj

∫ Σ + ∪Σ −

P3 g3 ds.

(76)

20

Anna Marciniak-Czochra, Andro Mikeli´c

Proposition 10 The whole sequence {w(ε ), p(ε )} satisfies w j (ε ) ⇀ w0j − y3

∂ w30 , j = 1, 2, weakly in H 1 (Ω × (0, T )) as ε → 0, ∂yj

w3 (ε ) ⇀ w03 weakly in H 1 (Ω × (0, T )) as ε → 0,

π (ε ) ⇀ π = πm + πw weakly in 0

∂y3 π (ε ) ⇀ ∂y3 π 0 weakly in where

w0

and πm

=< π 0

H (0, T ; L (Ω )) 1

H 1 (0, T ; L2 (Ω ))

> are given by (26)-(27) and

as ε → 0,

2

as ε → 0,

{w03 , πw }

(77) (78) (79) (80)

by (28)- (30).

6 Strong convergence In this section we establish that the weak convergences from previous section imply the strong convergences. With such aim, we introduce the corrected unknowns, for which the weak convergence to zero was already established in Subsection 5. Let Ψε ∈ C0∞ (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 ) = Cε 1/q−1 . We set  ∂ w0    ξ j (ε ) = w j (ε ) − w0j + y3 3 , j = 1, 2; ∂yj (81) 2 0  E , ξ ( ε ) = w ( ε ) − w − ε 3 3   3 κ (ε ) = π (ε ) − π 0 , with E = Ψε (y1 , y2 )

∫ y3 0

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

plained by the following result Lemma 3 We have 1 e j3 (ξ (ε )) ⇀ 0 weakly in H 1 (0, T ; L2 (Ω )), j = 1, 2, 3, as ε → 0, ε ξ (ε ) ⇀ 0 weakly in H 1 (0, T ; H01 (Ω ))3 , as ε → 0, 1 e33 (ξ (ε )) ⇀ 0 weakly in H 1 (0, T ; L2 (Ω )), as ε → 0, ε2 Proof First, using definition (81) we find out that

(82) (83) (84)

1 ε ∂E 1 e j3 (ξ (ε )) = e j3 (w(ε )) − ⇀ 0, ε ε 2 ∂ y j |{z} by(68)

implying (82). (83) follows from Proposition 4 and Corollary 1 and smallness of the correcting terms. Finally, 1 ∂E 1 e33 (ξ (ε )) = 2 e33 (w(ε )) − ⇀ 0. 2 ε ε ∂ y3 |{z} by(67)

Derivation of the Equations for the Poroelastic Plate

21

Next, using equation (71) we obtain ∫

2

2



Ω i, j=1

˜ 0 − y3 ∇y1 ,y2 w03 )ei j (φ )dy + ei j (w ∫

−y3 ∇y1 ,y2 w03 ) divy1 ,y2 φ˜ dy − α



2ν 1 − 2ν

π 0 divy1 ,y2 φ˜ dy + 2

∫ Ω



˜0 divy1 ,y2 (w

∂E

2

∑ ∂ y j e3 j (φ )dy Ω j=1

∫ ∫ 1{ 2ν 0 ˜ 0 − y3 ∇y1 ,y2 w03 )∂y3 φ3 dy π ∂y3 φ3 dy + divy1 ,y2 (w + 2 −α ε 1 − 2ν Ω Ω ∫ ∫ } ∂ (w03 + ε 2 E ) 2ν 2(1 − ν ) ∂y (w0 + ε 2 E )∂y3 φ3 dy + divy1 ,y2 φ˜ dy + 1 − 2ν Ω ∂ y3 (1 − 2ν )ε 4 Ω 3 3 2

=





+ − j=1 Σ ∪Σ



+2

P j φ j ds +



Σ + ∪Σ −

∂E α ∑ ∂ y j e j3 (φ ) dy − ε 2 j=1 2



y3 ∇y1 ,y2 w03 )∂y3 φ3 dy +

P3 φ3 ds +



π 0 ∂y3 φ3 dy +



2(1 − ν ) (1 − 2ν )ε 2

2ν 1 − 2ν







∗ (Ψε − 1)Ecor divy1 ,y2 φ˜ dy

1 2ν ε 2 1 − 2ν

∗ Ψε Ecor ∂y3 φ3 dy,



∫ Ω

˜ 0− divy1 ,y2 (w

∀φ ∈ V,t ∈ (0, T ).

(85)

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



2

2



Ω i, j=1

−α

∫ Ω

ei j (ξ (ε ))ei j (φ )dy +

κ (ε ) divy1 ,y2 φ˜ dy +



2ν 1 − 2ν

1{ 4 ε2 ∫



∫ Ω

divy1 ,y2 ξ˜ (ε ) divy1 ,y2 φ˜ dy

2

∑ e3 j (ξ (ε ))e3 j (φ )dy Ω j=1

2ν κ (ε )∂y3 φ3 dy + −α divy1 ,y2 ξ˜ (ε )∂y3 φ3 dy 1 − 2ν Ω Ω ∫ ∫ } 2ν 2(1 − ν ) + ∂y3 ξ3 (ε ) divy1 ,y2 φ˜ dy + ∂y ξ3 (ε )∂y3 φ3 dy = 1 − 2ν Ω (1 − 2ν )ε 4 Ω 3

−2



2

∂E

α

∑ ∂ y j e j3 (φ ) dy + ε 2 Ω j=1







π 0 ∂y3 φ3 dy −

2ν 1 − 2ν





∗ (Ψε − 1)Ecor divy1 ,y2 φ˜ dy

∂y φ3 2ν ˜ 0 − y3 ∇y1 ,y2 w03 ) 3 2 dy divy1 ,y2 (w 1 − 2ν Ω ε ∫ 2(1 − ν ) ∗ Ψε Ecor − ∂y3 φ3 dy, ∀φ ∈ V,t ∈ (0, T ). (1 − 2ν )ε 2 Ω −

(86)

22

Anna Marciniak-Czochra, Andro Mikeli´c

Effective pressure equation (75) implies

γ

∫ Ω

∂t π 0 ζ dy +



+ −

∫ Ω

˜ ∗ + ε −2 ∂t ∂y3 (w03 + ε 2 E ))ζ dy α ( divy1 ,y2 ∂t w

∂ π0 ∂ ζ dy + ε 2 ∂ y3 ∂ y3





∫ Ω

±U 1 ζ ds + α

Σ + ∪Σ −

∇y1 ,y2 π 0 ∇y1 ,y2 ζ dy = ε 2





∗ (Ψε − 1)∂t Ecor ζ dy,

∫ Ω

∇y1 ,y2 π 0 ∇y1 ,y2 ζ dy

∀ζ ∈ H 1 (Ω ).

(87)

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

γ



∂t κ (ε )ζ dy +



+αε −2 −ε 2

∫ Ω

∫ Ω

∫ Ω

α divy1 ,y2 ∂t ξ˜ (ε )ζ dy + ε 2

∂y3 ∂t ξ3 (ε )ζ dy +

∫ Ω

∇y1 ,y2 π 0 ∇y1 ,y2 ζ dy − α

∫ Ω

∇y1 ,y2 κ (ε )∇y1 ,y2 ζ dy

∂ κ (ε ) ∂ ζ dy = −ε ∂ y3 ∂ y3





∫ 1∫

∗ (Ψε − 1)∂t Ecor ζ dy,

−1 ∂ ω

V ζ ds

∀ζ ∈ H 1 (Ω ). (88)

By the choice of the correcting terms, ξ (ε ) = 0 on ∂ ω × (−1, 1). Now we test (86) by φ = ∂t ξ (ε ), (88) by ζ = κ (ε ) and add the obtained equalities, to obtain the following energy equality 1 d{ 2 2 dt +

4 ε2





2



Ω i, j=1 2

ei j (ξ (ε )) 2 dy + 2



∑ e3 j (ξ (ε )) Ω j=1

2(1 − ν ) + (1 − 2ν )ε 4

dy +

2ν 1 − 2ν



∫ 2 |κ (ε )|2 dy divy1 ,y2 ξ˜ (ε ) dy + γ



4ν (1 − 2ν )ε 2



∫ Ω

divy1 ,y2 ξ˜ (ε )∂y3 ξ3 (ε ) dy

2 ∫ ∫ } ∂y ξ3 (ε ) 2 dy + ε 2 ∇y ,y κ (ε ) 2 dy + ∂ κ (ε ) dy 3 1 2 ∂ y3 Ω Ω Ω







2 d{ α ∂ E e j3 (ξ (ε )) π 0 ∂y3 ξ3 (ε ) dy − ε dy + 2 −2 ∑ dt ε ε Ω Ω j=1 ∂ y j ) ∫ ( ∂y ξ3 (ε ) 2ν ∗ ˜ 0 − y3 ∇y1 ,y2 w03 ) 3 2 divy1 ,y2 (w divy1 ,y2 ξ˜ (ε ) dy + (Ψε − 1)Ecor 1 − 2ν Ω ε ∫ ∫ 2 } 2(1 − ν ) ∂ 2 E e j3 (ξ (ε )) ∗ − Ψ E ∂ ξ ( ε )dy + 2 ε dy ε y 3 ∑ cor 3 (1 − 2ν )ε 2 Ω ε Ω j=1 ∂ y j ∂ t

=





α 2(1 − ν ) ∗ ∂t π 0 ∂y3 ξ3 (ε ) dy + Ψε ∂t Ecor ∂y3 ξ3 (ε )dy + ε2 Ω (1 − 2ν )ε 2 Ω ) ∫ ( ∂y ξ3 (ε ) 2ν ∗ ˜ 0 − y3 ∇y1 ,y2 w03 ) 3 2 + (Ψε − 1)∂t Ecor ∂t divy1 ,y2 (w divy1 ,y2 ξ˜ (ε ) dy 1 − 2ν Ω ε



−ε 2





∇y1 ,y2 π 0 ∇y1 ,y2 κ (ε ) dy − ε

∫ 1∫

−1 ∂ ω

V κ (ε )ds − α





∗ (Ψε − 1)∂t Ecor κ (ε ) dy. (89)

Derivation of the Equations for the Poroelastic Plate

23

Proposition 11 For the whole sequence {ξ (ε ), κ (ε )} we have ei j (ξ (ε )) → 0

strongly in

H 1 (0, T ; L2 (Ω )), i, j = 1, 2, as ε → 0,

(90)

ξ (ε ) → 0 strongly in H (0, T ; L (Ω )) , as ε → 0, ei3 (ξ (ε )) → 0 strongly in H 1 (0, T ; L2 (Ω )), as ε → 0, ε ∂ ξ3 (ε ) → 0 strongly in H 1 (0, T ; L2 (Ω )), as ε → 0, ε 2 ∂ y3 κ (ε ) → 0 strongly in H 1 (0, T ; L2 (Ω )), as ε → 0,

(91)

∂y3 κ (ε ) → 0 strongly in H (0, T ; L (Ω )), as ε → 0,

(95)

ε ∇y1 ,y2 κ (ε ) → 0 strongly in H 1 (0, T ; L2 (Ω )), as ε → 0.

(96)

1

2

1

3

2

(92) (93) (94)

Proof We follow proof of Proposition 3 and use equality (89). In fact only new term to be estimated is ε ity

∫ 1∫

−1 ∂ ω

V κ (ε )ds. We recall the well-known interpolation inequal1/2

1/2

||ζ ||L2 (∂ ω ×(−1,1)) ≤ C||ζ ||L2 (Ω ) ||ζ ||L2 (−1,1;H 1 (ω )) . It yields |ε

∫ 1∫ −1 ∂ ω

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

) ε2 1/2 ||∇y1 ,y2 κ (ε )||L2 (Ω )2 ≤ C1 ε 2/3 ||κ (ε )||L2 (Ω ) + ||∇y1 ,y2 κ (ε )||2L2 (Ω )2 . 2

(97)

Now we integrate equality (89) in time, use Proposition 10 and Lemma 3 and conclude the strong convergence in L2 (Ω × (0, T )). Iterating the argument after calculating 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 div w(ε ) − απ (ε ))I in Ω × (0, T ). 1 − 2ν

(98)

Nevertheless, this quantity does not correspond to the rescaled dimensionless physical stress σ ε . We introduce the rescaled poroelastic stress σ (ε ) by   ∂ w1 (ε ) 1 + D(ε ) 2e12 (w(ε )) e13 (w(ε )) 2   ∂ y1 ε  σ (ε )  ∂ w2 (ε ) 1   =  2e12 (w(ε )) 2 e23 (w(ε ))  , + D(ε )   ε ∂ y2 ε  1  1 2 ∂ w3 (ε ) + D(ε ) e13 (w(ε )) e23 (w(ε )) 2 ε ε ε ∂ y3

(99)

24

Anna Marciniak-Czochra, Andro Mikeli´c

where D(ε ) =

2ν 2ν 1 ∂ w3 (ε ) ˜ ε ) − απ (ε ) + divy1 ,y2 w( . 1 − 2ν 1 − 2ν ε 2 ∂ y3

(100)

As a direct consequence of Proposition 11, we obtain the following convergences for the stresses ˜ 0 , πm } be given by (26)-(27) and {w03 , πw } by (28)- (30). Let Proposition 12 Let {w ∗ Ecor be defined by (67). Then we have ∗ D(ε ) → −2Ecor in

σ j j (ε ) ε in

H 1 (0, T ; L2 (Ω )) as ε → 0,

∂ 2 w03 − 2e j j (w0 ) + 2y3 ∂ y2j 1 2

(101)

∗ + 2Ecor →0

H (0, T ; L (Ω )) as ε → 0, j = 1, 2,

∂ 2 w03 σ12 (ε ) − 2e12 (w0 ) + 2y3 → 0 in H 1 (0, T ; L2 (Ω )) as ε → 0, ε ∂ y1 ∂ y2 σ j3 (ε ) → 0 in H 1 (0, T ; L2 (Ω )) as ε → 0, j = 1, 2, ε σ33 (ε ) → 0 in H 1 (0, T ; L2 (Ω )) as ε → 0. ε

(102)

(103) (104) (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). ∂w ∂w 2. For small deflections w and rotations (ϑ1 , ϑ2 ) we have ϑ1 = and ϑ2 = − . ∂ x2 ∂ x1 Then, Kirchhoff’s hypothesis implies that the displacements u satisfy u1 = uω 1 (x1 , x2 ,t) − x3

∂w ∂w ; u2 = uω , 2 (x1 , x2 ,t) − x3 ∂ x1 ∂ x2

(106)

ω where (uω 1 , u2 ) are the tangential displacements of points lying on the midsurface. Now a direct calculation gives e j3 (u) = 0, j = 1, 2. Since σ = 2Ge(u) + 2ν G ( div u − α p)I, we conclude that also σ j3 = 0, j = 1, 2. 1 − 2ν 3. For the pressure field, we impose the normal velocities U ℓ at the top and bottom surfaces and suppose (5). Finally, σ33 is also supposed small throughout the plate. The later assumption gives

e33 (u) = −

ν 1 α (1 − 2ν ) (σ11 + σ22 ) + p. 1 + ν 2G 2G(1 + ν )

(107)

Derivation of the Equations for the Poroelastic Plate

25

4. For the other components of the strain tensor we have e11 (u) =

∂ uω ∂ 2w 1 ∂ uω ∂ uω ∂ 2w 1 − x3 2 ; e12 (u) = ( 1 + 2 ) − x3 ; ∂ x1 2 ∂ x2 ∂ x1 ∂ x1 ∂ x2 ∂ x1

∂ uω ∂ 2w ν α (1 − 2ν ) 2 − x3 2 ; e33 (u) = − (e11 (u) + e22 (u)) + p; ∂ x2 1−ν 2G(1 − ν ) ∂ x2 2ν G α (1 − 2ν ) 2Gν div u − α p = (e11 (u) + e22 (u)) − p 1 − 2ν 1−ν 1−ν

e22 (u) =

5. For the other components of the stress tensor we have 2G α (1 − 2ν ) (e11 (u) + ν e22 (u)) − p 1−ν 1−ν 2G α (1 − 2ν ) σ22 = (e22 (u) + ν e11 (u)) − p 1−ν 1−ν ∂ uω ∂ uω ∂ 2w . σ12 = 2Ge12 (u) = G( 1 + 2 ) − 2Gx3 ∂ x2 ∂ x1 ∂ x1 ∂ x2

σ11 =

(108) (109) (110)

Next we define the stress resultants (forces per unit length) N1 , N2 , N12 by ∫ ℓ/2

N1 =

−ℓ/2

σ11 dx3 , N2 =

∫ ℓ/2 −ℓ/2

σ22 dx3 , N12 =

∫ ℓ/2 −ℓ/2

σ12 dx3 ,

(111)

the effective stress resultant due to the variation in pore pressure across the plate thickness N by N=−

∫ ℓ/2

−ℓ/2

p dx3

(112)

and the external loading tangential to the plate fi = σi3 |x3 =ℓ/2 − σi3 |x3 =−ℓ/2 = Piℓ + Pi−ℓ , i = 1, 2.

(113)

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

∂ N1 ∂ N12 + + f1 = 0, ∂ x1 ∂ x2 ∂ N12 ∂ N2 + + f2 = 0. ∂ x1 ∂ x2

(114) (115)

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

∂ uω 2Gℓ ∂ uω α (1 − 2ν ) ( 1 +ν 2 )+ N; 1 − ν ∂ x1 ∂ x2 1−ν ∂ uω 2Gℓ ∂ uω α (1 − 2ν ) N2 = ( 2 +ν 1 )+ N; 1 − ν ∂ x2 ∂ x1 1−ν ∂ uω ∂ uω N12 = Gℓ( 1 + 2 ). ∂ x2 ∂ x1 N1 =

(116) (117) (118)

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

26

Anna Marciniak-Czochra, Andro Mikeli´c

We need one more equation to complete the system (6). We have ) α (1 − 2ν ) 1 − 2ν ( ω divx1 ,x2 (uω p, 1 , u2 ) − x3 ∆ x1 ,x2 w + 1−ν 2G(1 − ν ) ℓ(1 − 2ν ) α (1 − 2ν ) ω div u dx3 = divx1 ,x2 (uω N. 1 , u2 ) − 1−ν 2G(1 − ν )

div u = ∫ ℓ/2 −ℓ/2

(119) (120)

Next we average equation (3) over thickness, use the assumption (5) and expression (120) and get (7). Inserting (119) in equation (3) and using hypothesis (5) and equation (7), yields equation (8). It remains to find the equation for transverse deflection and for the bending moment due to the variation in pore pressure across the plate thickness. The bending moment M due to the variation in pore pressure across the plate thickness is given by M=−

∫ ℓ/2

−ℓ/2

x3 p dx3 .

(121)

For the stress moments of the plate, which have as physical dimension the moment per unit length, we have: 1. The twisting moment M12 : ∫ ℓ/2

M12 =

−ℓ/2

σ12 x3 dx3 = −

Gℓ3 ∂ 2 w . 6 ∂ x1 ∂ x2

(122)

2. The bending moments M1 and M2 : ∫ ℓ/2

M1 =

−ℓ/2

∫ ℓ/2

M2 =

−ℓ/2

Gℓ3 ∂ 2w ∂ 2w 1 − 2ν M ( 2 +ν 2 )+α 6(1 − ν ) ∂ x1 1−ν ∂ x2

(123)

Gℓ3 ∂ 2w 1 − 2ν ∂ 2w M. ( 2 +ν 2 )+α 6(1 − ν ) ∂ x2 1−ν ∂ x1

(124)

σ11 x3 dx3 = − σ22 x3 dx3 = −

4. Resultant external moment

∫ ℓ/2

−ℓ/2 σi3 dx3 , i = 1, 2. (m1 , m2 ) : mi = 2ℓ (Piℓ − Pi−ℓ ).

3. The transverse shear (Q1 , Q2 ): Qi =

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

∂ M1 ∂ M12 + − Q1 + m1 = 0, ∂ x1 ∂ x2 ∂ M12 ∂ M2 + − Q2 + m2 = 0. ∂ x1 ∂ x2

(125) (126)

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

∂ Q1 ∂ Q2 + + P3ℓ + P3−ℓ = 0 ∂ x1 ∂ x2

(127)

Derivation of the Equations for the Poroelastic Plate

27

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

∂ 2 M1 ∂ 2 M12 ∂ 2 M2 ∂ m1 ∂ m2 +2 + + + + P3ℓ + P3−ℓ = 0. 2 ∂ x1 ∂ x2 ∂ x1 ∂ x2 ∂ x1 ∂ x22

(128)

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

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