a finite element solution procedure for porous medium ...

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng, 14, 381-392 (1998)

A FINITE ELEMENT SOLUTION PROCEDURE FOR

POROUS MEDIUM WITH FLUID FLOW AND

ELECTROMECHANICAL COUPLING

MILOS KOJIC,I* NENAD FILIPOVIC,1 SNEZANA VULOVICI AND SRBOLJUB MIJAILOVIC2 IFaculty of Mechanical Engineering, University of Kragujevac, 34000 Kragujevac, Serbia, Yugoslavia 2Harvard School of Public Health, Cambridge, MA 02139, U.S.A.

SUMMARY We consider a coupled problem of the deformation of a porous solid, flow of a compressible fluid and the electrical field in the mixture. The governing equations consist of balance of the linear momentum of solid and of fluid, continuity equations of the fluid and current density, and a generalized form of Darcy's law which includes electrokinetic coupling. The compressibility of the solid and the fluid are taken into account. We transform these equations to the corresponding finite element relations by employing the principle of virtual work and the Galerkin procedure. The nodal point variables in our general formulation are dis­ placements of solid, fluid pore pressure, relative velocity of the fluid and electrical potential. Derivation of the FE equations is presented for small displacements and elastic solid, which can further be generalized to large displacements and inelastic behaviour of the solid skeleton. According to this formulation we can include general boundary conditions for the solid, relative velocity of the fluid, fluid pressure, current density and electrical potential. The dynamic-type non-symmetric system of equations is solved through the Newmark procedure, while in the case of neglect of inertial terms we use the Euler method. Numerical examples, solved by our general-purpose FE package PAK, are taken from biomechanics. The results are compared with those available in the literature, demonstrating the correctness and generality of the procedure presented. © 1998 John Wiley & Sons, Ltd. KEY WORDS

porous medium; fluid flow; electromechanical coupling; FEM

1. INTRODUCTION Finite element solution procedures of coupled problems, which include deformation of solid and fluid flow through a porous medium, have been the subject of investigation of many authors. 1­ 5 Implementation of these methods has been directed to geomechanics and to biomechanics. Many important practical problems such as settlement of foundation under loading and with water flow through porous solid skeletons have been solved numerically, confirming the correctness of the theoretical basis and numerical (FE) procedures. Application of the FE analysis to these coupled problems in biomechanics is still in the development phase. Various approaches have been adopted, starting basically from the same fundamental equations, in formulation of the governing relations for flow through porous deformable media. * Correspondence to: Milos Kojic, Faculty of Mechanical Engineering, University of Kragujevac, 34000 Kragujevac, Serbia, Yugoslavia.

CCC 1069-8299/98/040381-12$17·50 © 1998 John Wiley & Sons, Ltd.

Received 17 January 1997 Accepted 10 November 1997

I ~

382

M. KOJIC ET AL.

There are differences in some details related to compressibility of solid material. In our formula­ tion we follow approach of Lewis. 1 In transformation of the basic relations to the FE formula­ tion, some authors use displacements of solid and fluid pressure as the nodal point variables (u-p formulation), like Lewis l and Siriwardane,2 leading to a symmetrical system of equations; Simon 3 proposes two formulations: (i) displacements of solid and relative displacements of fluid - with a symmetric system; and (ii) u-p formulation - with a non-symmetric system; Gaj 0 4 uses displacements of solid and fluid and pressure with a symmetric system of equations. Small displacements are considered in References 1-4, with material non-linearity of the solid,I,2 while in Reference 5 a large displacement formulation is given. In our presentation we give in some detail the derivation of dynamic FE equations for small displacements and linear material behaviour, following Simon et al. 3 and Lewis et al: for the fluid continuity equation. This approach relies on the balance of the linear momentum of solid and fluid, and the continuity of fluid, which takes into account the compressibility of solid and of fluid. The paper represents a generalization of the approach given in Kojic et al. 6 We include electrokinetic coupling by using linearized electrokinetics and a continuity equation for the current density according to References 7 and 8. From the electrokinetic coupling follows a correction of the equation of balance of linear momentum for fluid. The above formulation is suitable for general use, since we can implement all boundary conditions appearing in the applications: boundary conditions for solid, relative fluid velocity, given current density and electrical potential. In the next Section we summarize the fundamental equations of the coupled problem described above, and then, in Section 3 we derive the basic FE equations for linear dynamics. In Section 4 some typical numerical examples from biomechanics are presented, and finally we give some concluding remarks in Section 5.

2. FUNDAMENTAL EQUATIONS

We present here the fundamental equations, supposing first that displacements of the solid are small, which represent the basis for the FE formulation in Section 3. The assumptions are that the solid is linear elastic, with compressible material of the skeleton, and that the fluid is com­ pressible. Also, we consider the dynamic problem and take into account inertial forces of the solid and the fluid. We first present all governing equations in the case when there is no electro­ kinetic coupling and then give corrections due to this effect. The balance equation of the solid in the case of no electric coupling can be written in the form (1 - n)L T (Js

+ (1

- n)psb

+ k-1nq -

(1 - n)psii == 0

(1)

where (Js is stress in the solid phase, n is porosity, k is the permeability matrix, Ps is the density of the solid, b is the body force per unit mass, q is the relative velocity of the fluid, ii is the acceleration of the solid material and L T is the differential operator. The balance equation of the fluid phase (no electrokinetic coupling) is

-nVp Commun. Numer. Meth. Engng, 14, 381-392 (1998)

+ nprb -

k-1nq - nprvr == 0

(2) © 1998 John Wiley & Sons, Ltd.

A FINITE ELEMENT SOLUTION PROCEDURE

383

where p is the pore fluid pressure, Pr is the density of the fluid and vr is the acceleration of the fluid. This equation is also known as the generalized Darcy law. Both equilibrium equations are written per unit volume of the mixture. Combining equations (1) and (2) we obtain T

+ pb -

L (J

(3)

pii - PrCt == 0

where (J is the total stress, which can be expressed in terms of (J sand p as

==

(J

(1 - n)(Js - nmp

(4)

. and p == (1 - n)ps + nPr is the density of mixture. Here m is a constant vector defined as m T == {Ill 0 0 O} to indicate that the pressure contributes to normal stresses only. Also, we have taken into account that the pressure has a compressive character, while tensional stresses and strains are considered positive. In further analysis we use the effective stress (J', defined as (J' == (J

+ mp

(5)

which is relevant for the constitutive relations of the solid. Using the definition of relative velocity q as the volume of fluid passing in unit time through unit area of the mixture (also called the Darcy velocity), i.e.

q == n(vr - 0)

(6)

Pr· 0 - Vp + Pr b - k -I q - Pro.0 - -q == n

(7)

we transform (3) to a form

The next fundamental equation is the constitutive relation of the solid, (J , == C E(e - ep )

(8)

where CE is the elastic constitutive matrix of the solid skeleton, e is total strain· and ep is the deformation of the solid material due to pressure,1

m ep == - 3K P

(9)

s

Here Ks is the bulk modulus of the solid grains. Now we write the continuity equation for the fluid, which takes into account the compress­ ibilities of the solid and fluid, as well as the elastic constitutive law for the solid skeleton, I

E

mTC n - 'mTcEm) p== . 0 VTq+ (T m -_ .).e+ (1---n+ 3K K K 9~ 2

s

s

(10)

r

Now we c0nsider electrokinetic coupling. The basic relations, which encompasses both Ohm's and Darcy's laws as special cases, are

~. }.

{. J © 1998 John Wiley & Sons, Ltd.

== [

-kll k 21

k I2 ]{. Vp } -k22 V4J

(11)

Commun. Numer. Meth. Engng, 14, 381-392 (1998)

iC:

~

384

M. KOJIC ET AL.

where ¢ is the electrical potential, k ll is the (short-circuit) Darcy hydraulic permeability, k 22 is the electrical conductivity, and k l2 and k 21 are electrokinetic coupling coefficients that are equal by Onsager reciprocity, according to References 7 and 8. Next, we generalize the equation of balance of linear momentum for the fluid, (7), to include electrokinetic coupling. The resistance force F w for the fluid follows from the first equation of system (11),

Fw == -k~/q +k~/kI2V¢

(12)

and the equation of balance of linear momentum, (8), now changes to

I t7 b - k-llq+ k-Ik t7~ •• Pr· 0 -vP+Pr II 12 v lfJ-Pr u --q==

(13)

n

Note that this equation corresponds to isotropic conditions of the solid skeleton, and represents a form of the generalized Darcy law. Finally, we can use the continuity equation for the current density in the form

VTj == 0

(14)

Substituting the current density j from the second equation of system (11) into (14), we obtain T

T

k 21 V Vp - k 22 V V¢

==

(15)

0

3. DERIVATION OF FINITE ELEMENT EQUATIONS In this Section we transform the fundalnental relations of Section 2 into the finite element equations. We present derivations assuming small displacements of solid and elastic material. These equations can be generalized to large displacements of the solid and inelastic deformation of the material by a standard procedure. 9­ 12 First, by employing the principle of virtual work and supposing that the material is elastic, from equation (4) we obtain the following equation:

[beTCEe dV+ [beT(~:~ - m)p dV + [bUT pii dV + [bUT Prq dV = [bUT pb dV + buTt dA

i

(16)

Next, in the case of electrokinetic coupling we multiply equation (13) by the interpolation matrix H~ for the relative velocity of fluid q, and integrate over the finite element volume according to

the Galerkin method. The resulting equation is

- lv{ H qT Vp dV -

- Jvr

T

-I

{T

k ll k 12 lv H q V4J dV

JvrH q Prnit dV = 0

{

T

+ lv HqPr b dV -

k ll

T lv{ Hqq dV

T

(17)

Commun. Numer. Meth. Engng, 14,381-392 (1998)

© 1998 John Wiley & Sons, Ltd.

H Pr ii dV q

385

A FINITE ELEMENT SOLUTION PROCEDURE

Further, we multiply the continuity equation (10) by the interpolation matrix U~ for pressure (which is vector-column) and obtain

1

UTVT dV

V

p

q

+

1

HT

V

p

(

m

E T - mTC ). dV 3K

e

s

+

1

U T (1 -

K

p

V

n s

+ !!..K f

mTcEm). dV == 0 (18) 9K2 s

P

Finally, we multiply the continuity equation (10) by the interpolation matrix U~ for electrical potential and integrate over the volume V,

k 21

Jv{ H",TVTVp dV -

k 22

Jv{ H",TVTV4J dV = 0

(19)

Note that (usually) in practical applications interpolation functions for displacements H u and for relative velocities H q are quadratic, while Up for pressure and H for electrical potential are linear. We employ the standard procedure of integration over the element volume in equations (16)­ (19) and use the Gauss theorem. The resulting FE system of equations is 0

0

0ll !!

0

0

0

o I I I! I

m qu

0

0

0

0

0

0

0

r0

I

muu

~l

0

Cuq

cpu

Cpp

0

0

0

Cqq

0

0

0

+

k uu

k up

0

0

!!

0

0

k pq

0

I!

+1

:l

!!

i!

:J I:

fu fp

}

-

0

k qp

k qq

kq

~

fq

0-

k4>p

0

k

~

f

(20)

The matrices and vectors in this equation are:

muu

Cuq

= [H~PHu dV = mquT =

Cpp = -

(

1

T v HuPrHq dV

= [ H~PrHu Cpu

=-

dV

1

r

E

v HpT( m T - mTC 3K ) B d V

T(1 - n n mTcEm) ~ + K - 9K; H p dV

Jv H p

© 1998 John Wiley & Sons, Ltd.

mqu

s

C qq

== {HTPfU dV

lv

q

n

q

Commun. Numer. Meth. Engng, 14, 381-392 (1998)

~~

386

M. KOJIC ET AL.

k uu

=

Jv{ BT CEB dV

kUp

= Cpu =

kpq

= [H~XHq dV

kqp

= [H~Hp.x dV

k qq

= [H~k-IHq

T

k q",

dV

(CErn ) Jv{ T B 3K - m H p dV s

= -ki/kl2 [ H~H",.x dV (21)

k",q

=

k 21

[

H~.xHq.x d V

fu

= [H~Pb dV +

fq

= [H~Pfb dV

k",,,,

i H~t f",

=

dA

= -k22 [ H~.xH",.x d V fp

i H~DTj

(

T T

= JA HpD

q dA

dA

In these expressions n is the normal vector to the boundary and B is the strain-displacement transformation matrix. As we can see from (20) the nodal point variables are displacements of solid !!, relative velocities q, pressures p and electrical potential l/J. Boundary conditions include general boundary conditions for the solid, relative velocities, SUrface pressures, current densities and electric potential. The system of equations (20) is non-symmetrical in general. In the case when inertial forces are neglected the system becomes symmetric. A standard Newmark method can be employed for time integration of the system (20), as it is done in Reference 8. We note that in the case of no electrokinetic coupling we drop-out terms corresponding to the electric potential ~.

4. NUMERICAL EXAMPLES We give two typical examples from biomechanics and demonstrate that our results agree with those available in the cited references. 13 - IS 4.1. Example 1. One-dimensional electrokinetic transduction

We consider electrokinetic transduction in cartilage in one dimension. Two transductions are analysed: (i) mechanical-to-electrical; and (ii) electrical-to-mechanical. We impose on the top: 1. displacement with amplitude Uo' which elicits stress and electrokinetic response, and 2. current density with amplitude Jo which generates a mechanical response. Both the prescribed displacement and the current density have an oscillatory character, as given in Figure l(a). The cartilage is constrained at the bottom surface z == 0 by no..;flow, no-displacement and impermeability with respect to currency; also, pressure p == 0 at z == h. Commun.Numer. Meth. Engng, 14,381-392 (1998)

© 1998 John Wiley & Sons, Ltd.

387

A FINITE ELEMENT SOLUTION PROCEDURE

Displacements and fluid velocities in the cartilage have an oscillatory character. In Figures I(b), (c) and (d) we show distributions along the cartilage depth for various frequencies of excitation. They are practically the same as those reported in Reference 15, obtained analytically and experimentally. 4.2. Example 2. Cartilage subjected to current density

In this example we analyse the response of cartilage poroelastic material subjected to current density on the top surface, according to the function 8 Jy

= J o cos(2nxjA -

u(t)=U ocos rot

rot) E=l.l MPa v=O.l k=3xI0· 15 rn4/Ns h=680 J.l.rn 10= I Nm 2 Uo=IO Jlm

bJ-

b ll

b l2

(b"

h

_

10 15 Ns

10

m4

[ 107 l!­ Am

7

VS]

m2

10m

Figure l(a) Caption on next page

---f=0.001 ........•....... f=0.01 ......,,'*' f=0.1 1.0xlO·S

8.0xlO·'

~

6u

~ A

6.0xIO·'

,i

, ti' ..

..!

i'

~

4.0xlO·'

lOxIO·'

I

t

I

4.0xlO· 4

6.0xlO·4

8. Oxl 0.4

Depth Figure 1(b) Caption on next page

© 1998 John Wiley & Sons, Ltd.

Commun. Numer. Meth. Engng, 14, 381-392 (1998)

388

M. KOJIC ET AL.

---f=O.001 -·-f=0.01 ..... >A f=O.1 5xlO·1

•• • • •

4xlO·1

5 §

... I· •

~..

3xlO·1

I

~

••••••••

•••

••

••

... .. • '­

••••••••••

•

•

• • •I I • •• J•• •

t)

.!! ..,..,.~

•••

2xlO· 7

/' •

•• I!I.t4'

lxlO·1

j,

I'

~A~

•• \

• ~

•

• • • • \• • \ ,\ ••~\ \

~.

r

. AAi.:£,4,.·4A... ·.££~~A~A~.· Ac.. . . .A£.AA."t...,~&.k.~.~

.•..a.:"

.

:~.

...

·~·k~:.,.

-,

~

'&'i::,..

A 44

o•

I

I

2.0xlO·4

4.0xlO· 4

i

0.0

I'"

i

8.0xlO·4

6.0xlO· 4

Depth Figure l(c)

---f=0.03 ---f=0.1 .." ,A f=O.3

/y.. A.-

1.0xlO·S

...,.··XII.II .tt.~:

..,'Y

1

8.0xlO·9

.~ t) 0

6.0xlO·

lr!.i. • .. ~

;' . ,I

4.0xlO·

g

~

~

/

:f l

••••••••••••

••••

•

•••• •

•

'.,

T

~

•

-~--f=1

\

~

.~,..

•

:~

I

•• -.. "'" .. \ \

~t.';.

~.

\

.. • .•

.\~ .,~\ ~' ','.

~ ~

• •

--.

".

\'

•

•

~I

\~

\

•

~~ If.!' I

\ \

.. ~\

~~

\_ ·I.~

••

••~

~I

0.0 •

0.0

\

'=:

•

j.1 .'

2.0xlO·9

AA

•

J,.

>

:a

Ji.'

r .~

J

A

~

•

~

~

..••

~..

~*.

r .A,. • • I .;./

g

UI.xXXX·...", ~:... ....

.,

2.0xlO·4

4. Oxl 0.4

6.0xlO·4

• 8.0xlO·4

Depth Figure led) Figure l(a-d). One-dimensional electrokinetic transduction in cartilage: (a) geometrical and material data; (b) distribution of displacement amplitudes generated by displacement on top surface; (c) distribution of displacement amplitudes generated by current density on top surface; (d) distribution of fluid velocity amplitudes generated by current density Commun. Numer. Meth. Engng, 14,381-392 (1998)

© 1998 John Wiley & Sons, Ltd.

389

A FINITE ELEMENT SOLUTION PROCEDURE

where J o is the amplitude, A is the wavelength and w is the angular frequency (m == 2nf, The medium is infinite in the x-direction and is subjected to plane strain conditions in the z-direction. We use the length 1==3 mm in the analysis, which, for given A == 2 mm, provides repetition of the solution in the x-direction. Boundary conditions are shown in Figure 2(a): top and bottom surfaces have zero displacements and no normal fluid flow; current density flux is zero through the bottom surface. We have solved this example by using appropriate tilne steps, depending on frequency I Solutions repeat after time period T == 1/1 Results are shown in Figures 2(b)-(e). They agree with those in Reference 8, obtained analytically and by experimental investigation.

f == frequency).

5. CONCLUSIONS

We have presented a general numerical procedure for FE analysis of coupled problems which include deformation of a porous medium with flow of compressible fluid, and with electrokinetic coupling. The generalized Darcy law is extended to include electrokinetic coupling. The formulation also takes into account inertial forces of solid and fluid.

v

Jy= Jocos(21r.xlA-rot)

~Om-----------------

Y=-l l l l l l l l l l l l! ~! !~!~ !I!1 1 1 1 1 1 1 ~ ),»)/////,

))///"//

~

3

E= 0.5xl06 Pa p=0.306xI03 kg/m 3 3 v= 0.4 PFO.2977x10 kg/m b ll

1015

b l2

_

(b bJ -[ 10 ll

7 :

Ns m4

10 7

VS]

m2

10m

Jo=IAlm2

1=3mm h= 1 mm Figure 2(a) Caption on page 391

,,

....-

"­

\

"

"

~ I

."

~ \

"

Tl~ /

1

'-..

11, .....

-"

~ /

~

l

P'

\

.....

Figure 2(b) Caption on page 391

© 1998 John Wiley & Sons, Ltd.

Commun. Numer. Meth. Engng, 14,381-392 (1998)

390

M. KOJIC ET AL.

---f=0.001 ---f=0.01 ·······..·······--··1=0.1

1.4x1O-7 1.2x10·7 I.OxIO·7

~

6 iu c..

I

0

/.\

.,.....

8.0xI0-S

~_. /

6.0xIO·a

~

00

. I·

~

/I~ ...........~.I//

./

\

. . .,.,..•'/

I

.\\-..~.~_.

;.~"

\'

I

4.0xl0·S

/

I

lOxIO-S

I

>.",.~"'-"" ,'I

0.0 '" -1.0xI0·3

'.,\

'

\'\

JIo' /./JIo .•..•• "j,"

"' , . / / '

••...•.•. ,........

.

i

-8.0xI0··

-6.0xIO- 4

,\\

i

".

-2.0xIO· 4

-4.0xIO- 4

0.0

Depth Figure 2(c) Caption opposite

---f=0.001 ----f=0.01 ................ f=0.1

I.OxI0·7

/---.-.

.~//..~-_.. ------~

8.0xI0·S

~

6 iu •

6.0xIO·&

~

4.0xIO·&

~ ~

2.OxIO·

0.0

,/

.~:._,._ V

S

-1.Oxlo-3

/; :./'~/­

'

//

,.

~.

\~.

~ ~

......._.._."' .. . .. r..,.._.--...

i -8.0xI0··

\~~

\

...... "'~-- ....'--. •

,./~

•• J

.•

k

'.

"......:... \

,

,

-6.0xlo-4

-4. Oxl 0··

I

-lOxIO·4

Depth Figure 2{d) Caption opposite

Commun. Numer. Meth. Engng, 14,381-392 (1998)

© 1998 John Wiley & Sons, Ltd.

391

A FINITE ELEMENT SOLUTION PROCEDURE

+_

-S.8..+888 -4.8..+888 -3."1 ..

-2."1.. +888.

-J..6.. +888 -6 ..... -eeJ.

4.S .. -eeJ. J..S..+888 2.S..+888 3.6 .. 4.6..

+_ +_

Uie.. P\lint

SanSott CAD v2.8

St...... -

kOIOPOn..nta VV

[9

Figure 2(e) Figure 2(a-e). Cartilage subjected to current density (two-dimensional infinite medium): (a) geometrical and material data; (b) relative velocity field, f = 0·001, A. = 2 mm, t = I s. Maximum value is 9·3 nm; (c) magnitude of horizontal displacement I U y I, f = 0·1 to f = 0·001, x = 1·75 mm, A. = 2 mm, t = 1000 s; (d) magnitude of vertical displacement I U y l,f= 0·1 tof = 0·001, x= 1·75 mm, A. = 2 mm, t = 1000 s; (e) normal stress (IT)'y) field,f=O·ool, A. = 2 mm, t = I s

Generalized nodal point variables are: displacements of solid, fluid pressure, relative fluid velocity and electric potential. With these nodal point variables all boundary conditions appearing in practice can be employed, including given current density. The linear formulation which has a general form can be extended to geometrically and materially non-linear problems. Therefore, we have presented a methodology applicable to general problems in geomechanics and biomechanics. We have employed the implicit time integration, with equilibrium iteration in time step, and hence selection of time~step size is dictated by the excitation characteristics of the system. Although we have a non-symmetric system of equations with more variables per node than in other formulations in the cited refer­ ences (for example, the u-p formulation), the presented approach offers generality and suitability for applications. We have implemented the presented algorithm into our general-purpose FE package PAK as a special module PAK_p I6 , so that 2D and 3D practical problems in geomechanics and bio­ mechanics can be analysed. Numerical examples show the accuracy of our solutions, since the results agree very well with those obtained with other FE formulations and other methods. © 1998 John Wiley & Sons, Ltd.

Commun. Numer. Me/h. Engng, 14, 381-392 (1998)

\

392

, M. KOJIC ET AL. REFERENCES

1. R. W. Lewis and B. A. Schrefler, The Finite Element Method in the Deformation and Consolidation of Porous Media, Wiley, 1987. 2. H. J. Siriwardane and C. S. Desai, 'Two numerical schemes for nonlinear consolidation', Int.}. numer. methods eng., 17,405-426 (1981). 3. B. R. Simon, J. S.-S. Su, O. C. Zienkiewicz and D. K. Paul, 'Evaluation ofu-w and u-p finite element methods for the dynalnic response of saturated porous media using one-dimensional models', Int.}. numer. anal. methods geomech., 10, 461-482 (1986). 4. A. Gajo, A. Saetta and R. Vitaliani, 'Evaluation of three- and two-field finite element methods for dynamic response of saturated soil', Int.}.;, numer. methods eng., 37, 1231-1247 (1994). 5. M. E. Levenston, E. H. Frank and A. J. Grodzinsky, 'Variationally derived 3-field finite element formulations for quasistatic poroelastic analysis of hydrated biological tissues', private communication. 6. M. Kojic, N. Filipovic and S. Mijailovic, 'A general formulation for finite element analysis of flow through a porous deformable mediuln', Theor. Appl. Mech. (Yugoslavian), 23, 67-82 (1997). 7. E. H. Frank and A. J. Grodzinsky, 'Cartilage electromechanics - I. Electrokinetic transduction and the effects of electrolyte pH and ionic strength', J. Biomech., 20,615-627 (1987). 8. J. R. Sachs and A. J. Grodzinsky, 'An electromechanically coupled poroelastic medium driven by an applied electric current: surface detection of bulk material properties', PCH, 11,585-614 (1989). 9. M. Kojic, R. Slavkovic, M. Zivkovic and N. Grujovic, PAK - Program Package for Linear and Nonlinear Structural Analysis, Heat Conduction, Mass and Heat Transfer (Fluid Mechanics), Laborat­ ory for Engineering Software, Faculty of Mechanical Engineering, Kragujevac, 1996. 10. M. Kojic and K. J. Bathe, Inelastic Analysis of Solids and Structures, Wiley, in preparation. 11. K. J. Bathe, Finite Element Procedure in Engineering Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1982. 12. M. Kojic, R. Slavkovic, N. Grujovic and M. Vukicevic, 'Implicit stress integration algorithm for the modified Cam-Clay material', Theor. Appl. Mech. (Yugoslavian), 20, 95-118 (1994). 13. B. R. Simon, J. S. S. Wu, M. W. Cartlo!),J. H. Evans and L. E. Kazarian, 'Structural models for human spinal motion segments based on a poro~lastic view of the intervertebral disk', J. Biomech. Eng., 107, 327-335 (1985). . 14. Y. Kim, L. J. Bonassar and A. J. Grodzinsky, 'The role of cartilage streaming potential, fluid flow and pressure in the stimulation of chondrocyte biosynthesis duritigdynamic compression', J. Biomech., 28, 1055-1066 (1995). 15. E. H. Frank and A. J. Grodzinsky, 'Cartilage electromechanics - II. A continuum model of cartilage electrokinetics and correlation with experiments', J. Biomech., 20, 629-639 (1987). 16. M. Kojic, R. Slavkovic, M. Zivkovic, N. Grujovic, N.Filipovic and S. Vulovic, PAK-P Program Module for FE Analysis of Flow Through a Porous Deformable Medium, Laboratory for Engineering Software, Faculty of Mechanical Engineering, Kragujevac, 1996.

Commun. Numer. Meth. Engng, 14, 381-392 (1998)

© 1998 John

Wiley & Sons, Ltd.

:.,