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S0301-679X(16)30117-7 http://dx.doi.org/10.1016/j.triboint.2016.05.025 JTRI4206

To appear in: Tribiology International Received date: 30 January 2016 Revised date: 13 May 2016 Accepted date: 20 May 2016 Cite this article as: Ory Serge Tanguy Gbehe, Mohamed El Khlifi, Mohamed Nabhani and Benyebka Bou-Saïd, Numerical modelling of poro-elasticity effects on squeeze film of parallel plates using homogenization method, Tribiology International, http://dx.doi.org/10.1016/j.triboint.2016.05.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Numerical modelling of poro-elasticity effects on squeeze film of parallel plates using homogenization method

Ory Serge Tanguy Gbehea*, Mohamed El Khlifia, Mohamed Nabhania, Benyebka Bou-Saïdb

a

Hassan II University of Casablanca, Faculty of Sciences and Techniques, Laboratory of Mathematics, Cryptography, Mechanics and Numerical Analysis, PO Box 146, 20650 Mohammedia, Morocco b University of Lyon, CNRS INSA-Lyon, LaMCoS, UMR5259, 18–20 Avenue Albert Einstein,69621 Villeurbanne Cedex, France *

Corresponding author. [email protected]

Abstract This paper deals with the combined effects of elastic deformation and permeability on hydrodynamic performances of squeeze film between infinitely long parallel plates. The lower plate is an elastic porous matrix saturated by a Newtonian fluid. The poro-elasticity is taken into account by the means of homogenization method for periodic structures. The squeeze film analysis considers the Beavers-Joseph slip condition at the film – poro-elastic plate interface. The governing equations are discretized by finite differences method and solved iteratively using Gauss-Seidel method. The fluid film and poro-elastic plate coupling is managed using iterative fixed point. A Fortran program is developed for this purpose. The numerical results show that the poro-elasticity effects are significant. Keywords: Squeeze film; Poro-elasticity; Homogenization method; Beavers-Joseph slip velocity.

Nomenclature

c Cf

Elastic stiffness tensor Friction coefficient

Cijkh

Index notation of elastic stiffness tensor, Cijkh

eff C ijmn

Effective elastic stiffness tensor

D(v f ) Rate of strain tensor, D(v f ) E

1 T v f 2

Young modulus

Emn ( ) Macroscopic strain tensor, Emn e

E E ij kh 1 1- 2 1 ik jh

1 2 xm xn

Width of the elementary cell 1 e(u s ) Strain tensor, e(u s ) T u s 2

1

e y ,ij

2 yi y j

Microscopic strain tensor, e y ,ij 1

F G H h h0 k L l m Ni p pf pf0 Q t Ub

Viscous friction force Upper plate position Poro-elastic plate thickness Total fluid film thickness Initial fluid film thickness Permeability of the poro-elastic plate Plate length Elementary cell length Mass of the upper plate Component of the unit normal vector of the pore boundary Γ Pressure in the fluid film Pressure in the poro-elastic plate Homogenized pressure in the poro-elastic plate Leakage rate Time Beavers-Joseph slip velocity

us

Displacement vector

u vi

0 s

Macroscopic displacement vector Component of velocity vector of fluid into the fluid film

vf

Velocity vector of fluid into the poro-elastic plate 0

vf

Vsqueeze

Darcy velocity vector in the poro-elastic plate Upper plate velocity, Vsqueeze

dg dt

W0 W xi yi Γ Ω Ωf Ωs ∂Ωs α αij δ

Constant load capacity Load capacity per unit width of the squeeze Variables of macroscopic space Variables of microscopic space Common interface between the pore domain and solid matrix domain Area of the elementary cell Pore domain saturated by the fluid Solid matrix External boundaries of the elementary cell solid matrix Dimensionless slip constant Biot effective stress coefficient tensor Film – Poro-elastic plate interface deflection

ij

Identity tensor Index notation of the identity tensor

ε

Small parameter Microscopic solid displacement vector within the elementary cell due to a unit Value of the liquid pressure 2

2 μ ν kh

Truncation of the first order asymptotic expansion Dynamic viscosity of the fluid Poisson coefficient Microscopic solid displacement vector within the elementary cell due to a unit Value of the macroscopic strain tensor

Stress tensor of the fluid into the poro-elastic plate

f

s ijT

Stress tensor of the elastic solid matrix into the poro-elastic plate Macroscopic total stress tensor of the poro-elastic plate saturated

ϕ x

Poro-elastic plate porosity Differential operator following xi

y

Differential operator following yi

<>

Averaging symbol on the elementary cell *

:

Doubly contracted tensor product

1 * d

1. Introduction The squeeze film phenomenon plays an important role in engineering practice and has many applications in industry and in biomechanics. This phenomenon occurs when the two lubricating surfaces move towards each other in the normal direction. Such phenomenon has extensively been studied. Commonly, as the lubricant is considered to be viscous, it takes a long time for the impermeable surfaces to come into contact. During this process high friction can be generated which may rapidly damage the surfaces. Many papers [1-5] deal with the use of porous materials in lubrication and show beneficial effects on the surface lifetime. In all these studies the conventional no-slip condition at the film-porous material interface is used. Nevertheless, experiments related to laminar flow of water or oil in flat rectangular ducts with porous wall have shown the existence of a slip velocity [6, 7]. The study of slip velocity effects on porous wall was performed by Sparrow et al. [8] for porous annular disk, Wu [9] for porous rectangular plates, Prakash and Vij [10] for porous disk and Patel [11] for porous circular disk. They showed that a substantial gain in friction can be obtained thanks to a relevant selection of porous materials leading to an increase of slip velocity. In most of squeeze film situations with porous material, structure deformation must be taken into account during the squeeze film process. Indeed, the high pressure generated in the fluid film leads to elastic deformation of the film-porous interface. Several investigators considered this effect to analyze hydrodynamic performances of various bearings’configuration in squeeze situation such as Hou et al. [12], Bujurke et al. [13], Bujurke and Kudinatti [14, 15] and Naduvinamani and Savitramma [16] for knee joint, Naduvinamani and Savitramma [17] for hip joint, Higginson and Norman [18] for sphere-plane contact, Savitramma and Naduvinamani [19] for rectangular plates, Nabhani et al. [20] for parallel circular discs and Shimpi and Dehiri [21] for annular plates. 3

The present paper deals with a numerical investigation of the combined effects of elastic deformation and permeability upon the hydrodynamic performances of a squeeze film between two infinitely long parallel plates. The lower plate is poro-elastic and saturated by a Newtonian fluid. The poro-elasticity is taken into account by the means of homogenization method [22]. This method is a very efficient tool in the modeling of complex phenomena in heterogeneous media. The mathematical modelling of this physical phenomenon is rather complex because of the presence of heterogeneities. The objective of homogenization, introduced here for the first time to study poro-elastic contacts, is to obtain an equivalent macroscopic continuous medium which behaves “on average” like the heterogeneous material. This method is based on the periodic property of elementary cells which constitute the porous material. Thus one can deduce the macroscopic effective poro-elastic parameters from an analysis of a microstructural geometry and mechanical properties of an elementary cell. The modified Reynolds equation, including the poro-elasticity effects and the BeaversJoseph slip velocity at the film – poro-elastic plate interface, is derived for Newtonian fluid film. Discretized by finite difference method, the governing equations in the fluid film and in the poro-elastic plate are coupled using a sequential algorithm and solved iteratively using Gauss-Seidel over-relaxation method. The numerical results obtained from the computational FORTRAN program developed in this study are presented for three different values of Young modulus and permeability. 2. Problem definition and governing equations The squeeze film geometry is shown in Fig. 1. Consider two parallel infinitely long flat plates of length L immersed in a lubricant. The lower plate of thickness H, in Cartesian reference frame (O, Ox1, Ox2), is fixed and poro-elastic. δ represents the deflection of the film – poroelastic plate interface. The upper plate position is located by g(t) and is subjected to a constant load W0. It is considered rigid and has a squeezing movement of instantaneous velocity – dg/dt along the Ox2 axis.

2.1. Governing equations in the fluid film The fluid film is considered Newtonian and incompressible. The flow is laminar and axisymmetric. Using the thin film assumption [23] in absence of body forces, the continuity and motion equations in Cartesian coordinates read: v1 v2 0 x1 x2

(2.1)

2 v1 p 2 x1 x2

(2.2)

p 0 x2

(2.3)

where p, μ, v1 and v2 are respectively the pressure, the dynamic viscosity and the components of fluid velocity vector in x1 and x2 directions. 4

Using the no slip condition at the upper plate, the velocity component v1 is obtained from integration of (2.2): v1 x1 , x2 , t

U 1 p 2 x2 g t x2 g t b g t x2 2 x1 h

(2.4)

where Ub is the slip velocity at the film – poro-elastic plate interface and h is the total fluid film thickness. Introducing equation (2.4) into the continuity equation (2.1) and integrate across the film. To obtain the modified Reynolds equation one uses the conditions on v2 of no slip at the upper plate and of continuity at the film – poro-elastic plate interface. It reads as: x1

3 p U b h dg h 6 12 12 v 2 x1 , dt x1 x1

(2.5)

where v2 x1, is the fluid velocity in x2 direction at the film – poro-elastic plate interface. To analyze the squeezing process it is necessary to know at every moment the upper plate position. The equation of motion of the upper plate can be written as follows: m

d 2 g t W t W0 dt 2

(2.6)

where m is the mass of the upper plate and W(t) is the load capacity per unit width generated at the squeeze time t. It is obtained from integration of fluid film pressure on the contact surface: L

W t pdx1

(2.7)

0

The squeeze velocity of the upper plate, for each time t + Δt, is deduced from the motion equation (2.6) using Euler first order explicit scheme:

dg t t dg t t W t W0 dt dt m

(2.8)

Finally, the upper plate position is obtained from a second order Taylor expansion: g t t g t t

2 dg t t W t W0 dt 2 m

(2.9)

The load capacity, the squeeze velocity and the upper plate position are assumed to be known at the squeeze time t. The total fluid film thickness h including the film - poro-elastic plate interface deflection is given by: h g t

(2.10)

5

2.2. Governing equations in the poro-elastic plate In this work, the homogenization method for periodic structures is used. It’s based on the assumption of scales separation. Thus, the poro-elastic plate is considered homogenous, isotropic and composed of periodically reproduced elementary cell such as shown in Fig. 2. The elementary cell Ω, of length l and width e, is composed of solid matrix domain Ωs and pore domain Ωf saturated by a Newtonian viscous fluid. The solid matrix is supposed to be elastic and thus the magnitude of displacement vector is considered to be sufficiently small compared to the poro-elastic plate thickness H to guarantee this hypothesis (the authors verify that the deformation is lower than 20%). The fluid saturating the pore domain has the same dynamic viscosity as that of the fluid film and its movement is slow. Neglecting body forces, the dynamic equations in solid and fluid parts are given by: s 0

within Ωs

(2.11)

with s c : e(u s ) is the stress tensor of the solid matrix, e(u s ) , u s and c are respectively the strain tensor, the displacement vector and the elastic stiffness tensor.

vf 0

within Ωf

(2.12)

f 0

within Ωf

(2.13)

with f p f 2 D(v f ) is the fluid stress tensor, D(v f ) , , p f and v f are the rate of strain tensor, the identity tensor, the pressure and the velocity vector. Due to the assumption of scales separation, the unknown fields u s , v f and p f are functions of both microscopic space variables (y1, y2) and macroscopic space variables (x1, x2) and thus can be written in the form of asymptotic expansion in powers of :

(2.14)

(2.15)

u s x, y, t u s x, y, t u s x, y, t 2 0

1

v f x, y, t v f x, y, t v f x, y, t 2 0

1

p f x, y, t p 0f x, y, t p1f x, y, t 2

(2.16)

where usi, vf i and pf i are Ω-periodic functions depending on variables (y1, y2), as the elementary cells are periodically reproduced inside the poro-elastic plate, and ε is a small scale parameter. The subscripts 0 and 1 define respectively the zero and the first order. Incorporating the asymptotic expansions (2.14), (2.15) and (2.16) into equations (2.11), (2.12) and (2.13), using the expression of the gradient operator [22]: x 1 y

(2.17) 6

and averaging over the elementary cell Ω, the macroscopic homogenized equations are given as [24, 25]: In the fluid part: 0

x v f 0

vf

0

(2.18)

x p 0f

(2.19)

k

Equation (2.19) is the Darcy’s law describing the fluid flow into the poro-elastic plate.

In the solid part:

ijT x j

0

(2.20)

The averaging symbol < > on the elementary cell means *

1 * d , where Ω is the cell

area. k is a constant permeability of the poro-elastic plate and ijT is the macroscopic total stress tensor given by: eff ijT Cijmn Emn u s ij p 0f 0

(2.21)

0 where Emn u s is the macroscopic strain tensor associated with the macroscopic 0

eff displacement vector u s . C ijmn and ij represent respectively the effective elastic stiffness

tensor and the Biot effective stress coefficient tensor given by:

kh eff Cijmn Cijkh Cijmn e y,mn ij ij Cijkh e y ,kh

with is the poro-elastic plate porosity, e y ,ij represents the local strain tensor field depending on the microscopic variables (y1, y2).

7

The vectors kh and are particular solutions of the equilibrium equation (2.22) within the

elementary cell respectively for p 0f 0; E mn u s 0

1 km hn kn hm , with k and h fixed and 2

0 0 p f 1; E mn u s 0 : 1 0 Cijmn ey , mn u s Cijmn Emn u s 0 y j

in Ωs

(2.22)

subject to the boundary conditions: 1

u s is Ω-periodic depending on variables (y1, y2)

C e u1 C E u 0 N p 0 N ijmn mn s f ij j ijmn y , mn s j

on Γ

(2.23)

where Nj is a component of the unit normal vector of the pore boundary Γ. The elastic stiffness tensor Cijkh of the solid matrix is given by: Cijkh

E E ij kh 1 1 2 1 ik jh

where E is a Young modulus and ν is the Poisson coefficient. The resolution of the macroscopic homogenized equations gives the deflection δ as the normal macroscopic displacement of the film – poro-elastic plate interface: x1 u s0, 2 x1 ,0

(2.24)

2.3. Boundary conditions Solving these governing equations requires realistic boundary conditions on pressure, displacement and velocity.

Velocity boundary conditions

On the upper plate, x2 = g(t), the no-slip condition is applied:

v1 0 , v2

dg dt

(2.25)

On the film – poro-elastic plate interface, x2 = - δ, the continuity of normal velocity and the Beaver-Joseph slip condition of the tangential velocity are assumed: v2 v 0f , 2

(2.26) 8

v1 v1 v 0f ,1 x 2 k

(2.27)

where α is a dimensionless slip constant which depends on the poro-elastic plate characteristics [6].

Pressure boundary conditions

On the symmetry axis, x1 = 0: 0 p p f 0 x1 x1

(2.28)

At the output of the lubricated contact, into the fluid film and the poro-elastic plate, x1 = L/2: p p 0f 0

(2.29)

On the film – poro-elastic plate interface, x2 = ˗ δ, the continuity of the pressure is applied: p p 0f

(2.30)

On the lower wall of the poro-elastic plate, x2 = - H, the impermeability condition is used: p 0f x2

0

(2.31)

x2 H

Displacement boundary conditions

On the symmetry axis x1 = 0 and at the output of the poro-elastic plate x1 = L/2: u s0 0 x1

(2.32)

On the film – poro-elastic plate interface, x2 =- δ, the continuity of normal and tangential stresses is applied:

v1 x 2

p

12T

(2.33)

x 2

x 2

T 22

(2.34)

On the lower wall of the poro-elastic plate, x2 = - H, the nullity of displacement is applied: (2.35) u s0 0

3. Numerical procedure 9

The modified Reynolds equation (2.5), the equilibrium equation (2.22), the macroscopic homogenized equations (2.18 – 2.20) and the boundary conditions (2.25– 2.35) are discretized using the finite difference method. The resulting algebraic equations are iteratively solved using the successive over-relaxation scheme. The coupled problem, film – poro-elastic plate, is solved using a sequential coupling algorithm. The fluid film and the saturated poro-elastic plate problems are solved independently. At t = 0 s, the fluid film thickness is h0. The squeeze velocity and the fluid film pressure are initialized from analytical solution for rigid impermeable case given by Frêne et al. [26]. At each time step, the squeeze velocity and position of the upper plate are calculated from the previous time step using the equations (2.8) and (2.9). The global iterative process starts by taking an initial value to pressure from the analytical solution. Then this fluid film pressure is used for solving the problem of macroscopic homogenized equations (2.18 – 2.20) for the pressure pf 0 calculation, the 0

displacement vector u s and the deflection δ at the film – poro-elastic interface. As the homogenized equilibrium equation (2.20) depends on the solutions of equilibrium equation within the elementary cell (2.22), this equation is solved before the global iteration process. Once the total film thickness (2.10) is reached, the new pressure within the fluid film is obtained solving the modified Reynolds equation (2.5). The convergence for the global process is obtained when the relative root mean square change for pressure into the fluid film between two successive iterations is less than 10-3. The load capacity per unit width is then L

expressed in difference form by using rectangular method W t p dx1 x1 0

p

i

, where

i

x1 is the grid size used in the finite difference scheme. This process is repeated for each squeeze time step until a given minimum fluid film thickness limit is reached. A FORTRAN computational program summarized in Fig. 3 was developed. 4. Results and discussion 4.1. Validation The validation of the present numerical model was limited to the comparison with the analytical solution [26] in a rigid impermeable case. It’s found that the numerical results of load capacity for k = 0 m² and Young modulus E of 200 GPa magnitude are almost coincident with the analytical solution (Fig.4). In the following, the effects of permeability and elastic deformation of poro-elastic plate on lubricated contact performances are examined. The numerical results are presented in dimensional form to make their physical meaning more comprehensible, for different values of Young modulus and permeability. The data used in this work are listed in table I. The mechanical property values for example can be those of cellular foam [27] or arthritic joint cartilage [28]. 4.2. Load capacity Fig. 5 shows the elasticity effects of the film – poro-elastic plate interface layer on the load capacity. The load capacity increases at the initial stage of the squeeze, due to the upper plate acceleration (equation 2.6). Then it decreases sharply and becomes almost constant at the end of the squeeze. The deformation of the film – poro-elastic plate interface reduces the load 10

capacity. This trend was also observed by Ramanaiah and Sundarammal [29] for circular and rectangular plates and by Shimpi and Deheri [30] for rough annular plates. This is due to film – poro-elastic plate interface, which tends to damp the pressure in the lubricant during the deformation. This decrease in pressure results in decrease of the load capacity. Moreover, it is noted that the deformation of the poro-elastic plate decreases the load capacity at the initial stage of the squeeze. The effects of permeability of poro-elastic plate on the load capacity are presented in Fig. 6. It is observed that the load capacity decreases with permeability. Increasing permeability makes easier the fluid infiltration into the poro-elastic plate and thus reduces the fluid film pressure. Thus, the upper plate moves faster and consequently has greater acceleration. The fluid film, therefore, provides greater load to overcome the plate inertial force (equation 2.6). 4.3. Squeeze velocity The variation of squeeze velocity as function of time is depicted on Fig. 7 for three different Young modulus values. The squeeze velocity decreases first sharply at the squeeze beginning and then progressively. The squeeze velocity is higher at the initial stage of the squeeze when the film – poro-elastic plate interface is deformable. The effects of permeability of the poro-elastic plate on squeeze velocity are observed in Fig. 8. The upper plate goes down more rapidly when the poro-elastic plate permeability is high. This is due to fluid motion which more quickly flows into the poro-elastic plate and makes easier the upper plate motion. 4.4. Upper plate position Fig. 9 and 10 depict the motion of the upper plate with respect to time. The position of the upper plate decreases sharply at the beginning stage of the squeeze. Its variation becomes less significant at the end of the squeeze process. It is shown in Fig. 9 that the decreasing of the poro-elastic plate Young modulus allows the upper plate to reach a lower position. These results are consistent with those presented in figure 7. The permeability effects of the poro-elastic plate depicted in Fig. 10 show that, compared to the impermeable case (k = 0 m²), the upper plate reaches a lower position. The movement of the upper plate faces less resistance due to the fluid flow in the poro-elastic plate. 4.5. Friction coefficient The friction coefficient is calculated as: C f t

F

(4.1) W t where W(t) is the load capacity given by equation (2.7) and F is the viscous friction force by unit width of the fluid film applied on the poro-elastic plate interface: v F 1 x 2 0 L

h p U b dx1 dx1 2 x1 h x2 0 L

(4.2)

11

Fig. 11 and 12 present the variation of the friction coefficient with time. The friction coefficient decreases with respect to time. The deformation effects (see Fig. 11) increase slightly the friction coefficient during the beginning stage of squeeze and decrease it later. This is consistent with the fact that the friction coefficient is inversely proportional to the load-carrying capacity. The permeability effect of the poro-elastic plate (see fig. 12) reduces the friction coefficient. The presence of the pore leads to the infiltration of the fluid film into the poro-elastic plate, and then induces a less shearing of the fluid film between the two plates. 4.6. Leakage rate The leakage rate at the outlet of the contact is given by: g

L Qt v1 , x2 , t dx2 2

(4.3)

The variation of the leakage rate as a function of time is presented in Fig. 13 and 14. The leakage rate decreases sharply during the initial stage of the squeeze and then decreases slowly after that. The effects of the deformation presented in Fig. 13 show that the leakage rate increases during initial stage of the squeeze and then becomes quite the same whatever the value of the Young modulus. This is due to the gradient pressure increase at the outlet of the contact during deformation and which leads to fluid film velocity increase. This result is consistent with the variation of the squeeze velocity presented in figure 7. The influence of permeability (see Fig. 14) shows that the leakage rate increases during the initial stage of squeeze. The permeability increases the slip velocity, which increases the velocity of the fluid in the fluid film. 5. Conclusion The combined effects of elastic deformation and permeability on the hydrodynamic performances of a poro-elastic squeeze film are examined in this paper. The modified Reynolds equation is derived for the squeeze film using the slip velocity conditions at the film – poro-elastic plate interface. The poro-elastic plate is saturated by an incompressible Newtonian fluid. The poro-elasticity is taken into account by the means of homogenization method. The governing equations in the fluid film and the poro-elastic plate are discretized by finite differences method and solved iteratively using the Gauss-Seidel method. The numerical results show that the elastic deformation and the permeability have noticeable effects especially on load capacity and friction coefficient. The deformation of the poro-elastic plate reduces the load capacity and increases the friction coefficient. However, for high value of the permeability, the load capacity is enhanced and the friction coefficient is reduced. This is highly desirable for the increase of contact lifetime. References 12

[1] Wu H. Squeeze-film behavior for porous annular disks. Journal of lubrication technology, ASME 1970; 92(4):593-6. [2] Wu H. An analysis of squeeze film between porous rectangular plates. Journal of lubrication technology, ASME 1972; 94(1):64-8. [3] Murti PRK. Squeeze films in porous circular disks. Wear 1973;23(3):283-9. [4] Murti PRK. Squeeze film behavior in a spherical porous bearing. Journal of lubrication technology, ASME 1975; 97(4):638-41. [5] Elsharkawy AA, Nassar MN. Hydrodynamic lubrication of squeeze-film porous bearings. Acta Mechanica 1996; 118(1):121-34. [6] Beavers GS, Joseph DD. Boundary conditions at a naturally permeable wall. Journal of Fluid Mechanics 1967; 30(1):197-207. [7] Beavers GS, Sparrow EM, Magnuson RA. Experiments on coupled parallel flows in a channel and a bounding porous medium. Journal of Fluid Engineering, ASME 1970;92(4):843-8. [8] Sparrow EM, Beavers GS, Hwang IT. Effect of velocity slip on porous-walled squeeze films. Journal of lubrication technology, ASME 1972; 94(3):260–4. [9] Wu H. Effect of velocity-slip on the squeeze film between porous reactangular plates. Wear 1972; 20(1):67–71. [10] Prakash J, Vij SK. Effect of velocity slip on porous-walled squeeze films. Wear 1974; 29(3):363–72. [11] Patel KC. The hydromagnetic squeeze film between porous circular disks with velocity slip. Wear 1980; 58(2):275–81. [12] Hou JS, Mow VC, Lai WM, Holmes MH. An analysis of the squeeze-film lubrication mechanism for articular cartilage. Journal of Biomechanics 1992; 25(3):247–59. [13] Bujurke NM, Salimath CS, Kudenatti RB, Shiralashetti SC. Wavelet-multigrid analysis of squeeze film characteristics of poroelastic bearings. Journal of Computational and Applied Mathematics 2007; 203: 237–48. [14] Bujurke NM, Kudenatti RB. Surface roughness effects on squeeze film poroelastic bearings. Applied Mathematics and Computation 2006; 174:1181–95 [15] Bujurke NM, Kudenatti RB. An analysis of rough poroelastic bearings with reference to lubrication mechanism of synovial joints. Applied Mathematics and Computation 2006; 178:309–20 [16] Naduvinamani NB, Savitramma GK. Squeeze film lubrication between rough poroelastic rectangular plates with micropolar fluid: A special reference to the study of synovial joint lubrication. ISRN Tribology 2013;2013:9 pages [17] Naduvinamani NB, Savitramma GK. Effect of surface roughness on the squeeze film lubrication of finite poroelastic partial journal bearings with couple stress fluids: A special reference to hip joint lubrication. ISRN Tribology 2014;2014:13 pages [18] Higginson GR, Norman R. The lubrication of porous elastic solids with reference to the functioning of human joints. Journal Mechanical Engineering Science 1974; 16(4):250-7.

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Savitramma GK, Naduvinamani NB. Micropolar fluid squeeze film lubrication between rough anisotropic poroelastic rectangular plates: special reference to synovial joint lubrication. Tribology - Materials, Surfaces & Interfaces 2012; 6(4):174-82 [20] Nabhani M, El Khlifi M, Bou-Saïd B. Non-Newtonian couple stress poroelastic squeeze film. Tribology International 2013;64:116-27. [21] Shimpi ME, Deheri GM. Magnetic fluid-based squeeze film behaviour in curved porous-rotating rough annular plates and elastic deformation effect. Advances in Tribology 2012;2012:12 pages [22] Sanchez-Palencia E. Non-Homogeneous Media and Vibration Theory. Heidelberg: Springer–Verlag; 1980. [23] Cameron A. Basic Lubrication Theory. New delhi: Wiley; 1981. [24] Auriault JL, Boutin C, Geindreau C. Homogenization of coupled phenomena in heterogenous media. Hoboken: Wiley; 2009. [25] Boutin C, Auriault JL. Dynamic behaviour of porous media saturated by a viscoelastic fluid. Application to bituminous concretes. International Journal of Engineering Science 1990;28(11):1157-81. [26] Frêne J, Nicolas D, Degueurce B, Berthe D, Godet M. Hydrodynamic lubrication: bearings and thrust bearings. Amsterdam: Elsevier; 1997. [27] Etchessahar M, Sahraoui S, Brouard B. Vibrations of poroelastic Plates: mixed displacement-pressure modelisation and experiments. Acta Acustica United with Acustica 2009;95:857-65. [28] Richard F, Villars M, Thibaud S. Viscoelastic modeling and quantitative experimental characterization of normal and osteoarthritic human articular cartilage using indentation. Journal of the Mechanical Behavior of Biomedical Materials 2013;24:4152. [29] Ramanaiah G, Sundarammal Kesavan. Effect of bearing deformation on the characteristics of squeeze films between circular and rectangular plates. Wear 1982;82(1):49-55. [30] Shimpi ME, Deheri GM. A study on the performance of a magnetic fluid based squeeze film incurved porous rotating rough annular plates and deformation effect. Tribology International 2012;47:90-9

Fig. 1. Squeeze film geometrical configuration Fig. 2. Porous medium: (a) macroscopic view, (b) elementary cell Fig. 3. Computer program flow chart Fig. 4. Comparison between analytical and numerical result of load capacity W variationwith time t Fig. 5. Variation of load capacity W as function of time t for different Young modulus E Fig. 6. Variation of load capacity W as function of time t for different permeability values k 14

Fig. 7. Variation of squeeze velocity Vsqueeze as function of time t for different Youngmodulus E Fig. 8. Variation of squeeze velocity Vsqueeze as function of time t for different permeabilityvalues k Fig. 9. Variation of upper plate position g as function of time t for different Young modulus E Fig. 10. Variation of upper plate position g as function of time t for different permeabilityvalues k Fig. 11. Variation of friction coefficient Cf as function of time t for different Youngmodulus E Fig. 12. Variation of friction coefficient Cf as function of time t for different permeabilityvalues k Fig. 13. Variation of leakage rate Q at the contact edge as function of time t for differentYoung modulus E Fig. 14. Variation of leakage rate Q at the contact edge as function of time t for differentpermeability values k

Table 1. Geometrical and physical data

15

Plate length (mm) Porous medium thickness (mm) Initial fluid film thickness (mm) Elementary cell length (μm) Elementary cell width (nm) Dynamic viscosity (Pa.s) Porous plate porosity Dimensionless slip constant Upper plate mass (kg) Constant load (N) Young modulus (MPa)

Poisson coefficient Porous plate permeability (m²)

L H h0 l e µ ϕ α m W0 E

ν k

40 1.6 1.2 23 161 0. 1 0.4 0.1 30 300 0.1 1 2.105 0.12 0 1.10-14 5.10-12 5.10-11

16

17

18

19

20

21

22

23

24

25

26

27

Highlights

Numerical analysis of poro-elastic squeeze films between two parallel plates.

The homogenization method for periodic structures is used to take into account the poroelasticity of the porous plate.

The squeeze film analysis considers the Beavers-Joseph slip velocity condition at the porous medium interface.

28