Archive of Applied Mechanics 63 (1993) 413-423

Archiveof

Applied Mechanics ~, Springer-Verlag 1993

Flexural vibrations of fissured poroelastic plates D. D. Theodorakopoulos and D. E. Beskos, Patras Summary: The governing equations of flexural vibrations of thin, fluid-saturated, poroelastic plates are derived in detail. The plate material obeys the Aifantis-Beskos theory of poroelasticity with two degrees of porosity, while the plate theory employed is the one due to Kirchhoff. The dynamic response of a rectangular, simply supported, poroelastic plate with two degrees of porosity to harmonic load is obtained analytically-numericallyand the effects of porosities and permeabilities on the response are studied. A comparison between the single and double porosity material models is also made. The quasi-static problem is analysed as as special case of the dynamic one.

Biegeschwingungen gekliifteter poroelastischer Platten Ubersicht: Es werden die Gleichungen,die die Biegeschwingungen von diinnen,mit Fliissigkeit gefiillten, poroelastischen Platten beschreiben, im einzelnenhergeleitet. Das Materialverhalten der Platte wird durch die Poroelastizit/itstheorie yon Aifandis und Beskos ffir Materialien mit doppelter Porositfit beschrieben, w/ihrend fiir die Platten die Kirchhoffsche Plattentheorie angewandt wird. Man erh/ilt auf analytisch-numerischem Weg das dynamische Verhalten einer rechteckigen,einfach gestiitzten, poroelastischen Platte mit doppelter Porosit/it bezfiglich einer harmonischen Belastung. Der EinfluB der Porosit/it und Permeabilit/it auf das dynamische Verhalten wird untersucht. Weiterhin werden die Modelle fiir Materialien mit einfacher und doppelter Porosit/it miteinander verglichen. Das quasi-statische Problem wird als Spezialfall des dynamischen Problems analysiert.

1 Introduction

The problem of determining the response of fluid saturated, poroelastic, flexural plates to dynamic lateral forces is an important problem in structural and geotechnical engineering, bioengineering and geodynamics. This problem was recently studied by Theodorakopoulos and Beskos [1] on the assumption that the plate material is characterized by one degree or kind of porosity and obeys Blot's [2] classical theory of poroelasticity. The present paper studies the same problem on the assumption that the plate material is characterized by two degrees or kinds of porosity and obeys the Aifantis-Beskos [3, 4] theory of poroelasticity. It has been observed that in many applications involving fully saturated, fissured, poroelastic rock media there is a need for using more refined models than the one developed by Biot [2]. Indeed fissured, poroelastic rocks have a permeability of fissures much higher than the permeability of pores and a porosity of porous blocks, separated by fissures, much larger than the porosity of the fissures. This clearly indicates that there are two degrees or kinds of porosity, one due to the fissures and the other due to the pores. Other examples characterized by two degrees of porosity involve the flow of water in the stems of plants and the diffusion of gases in metals (Aifantis [5]). Beskos [4] has discussed the problem of modelling the dynamic behavior of fissured, poroelastic rocks in a recent paper, which also contains an extensive literature review on the subject. The same paper also describes the construction of the governing equations of motion for fully saturated, poroelastic rocks on the basis of Aifantis' [3] model of double porosity and the basic principles of the continuum theory of mixtures. In three subsequent papers Beskos et al. [6, 7] and Vgenopoulou and Beskos [8] studied analytically-numericallythe propagation of plane harmonic body and Rayleigh waves and the dynamic column and borehole problems of rock mechanics on the basis of the rock model of Aifantis-Beskos [3, 4].

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This paper starts with the derivation of the governing equations of flexural vibrations of thin, fluid saturated, poroelastic rock plates by combining the governing equations of motion for the double porosity rock model in Beskos [4] with the classical flexural theory of thin plates due to Kirchhoff (Szilard [9]). The dynamic response of a rectangular, simply supported, poroelastic plate with two degrees of porosity to harmonic lateral load is obtained analytically-numerically. The effects of porosities and permeabilities on the response are investigated and a comparison between the single and double porosity rock models is also made. The quasi-static problem is analysed as as special case of the dynamic one.

2

Aifantis-Beskos' governing equations of motion

A fissured poroelastic rock medium consists of porous elastic blocks separated from each other by a system of fissures, i.e., it is a poroelastic medium with two porosities, one corresponding to the fissures and the other to the pores of the blocks. By utilizing the rock modelling of Aifantis [3] and employing well-known principles of continuum mechanics, Beskos [4] succeeded in developing the governing equations of motion for such a fully saturated rock medium. These are 11 partial differential equations with 11 unknowns which, for reasons of completeness, are given here explicitly as (,~ + #) ttk,ki + #Ui,kk = --vlQi 1 -- v2Qi 2 + -fl~P.i~t

v

~s~ii,

(2)

~Qiot + qfQi" ~ + O~fii,

(#~+ n~) fi,,, + (?~ + 6n,3 f

+ O~,, = - ( -

(1)

1)~ x(p z - pl)

(3)

in which indices i and k take the values 1, 2, and 3 and imply summation when they are repeated, while the index a takes the values 1 (denoting fissures) and 2 (denoting pores) and does not imply summation when it is repeated; commas and dots indicate differentiation with respect to space and time, respectively; ui are the displacement components of the solid phase of the medium, n~ are the porosities due to the fissures (a = 1) and the pores (~ = 2); f are the fluid pressures in the fissures (~ = 1) and the pores (~ = 2); 2 and # are the Lam6 elastic constants; 0~ and 0~ are the relative solid and fluid mass densities, respectively, while Of is the actual fluid mass density; the coefficients fl, express the deformability of the solid as this affects the flow in the fissures (a = 1) and the pores (a = 2); 6 is the fluid compressibility; the coefficients 7~ measure the compressibility of the fissures (e = 1) and the pores (0~ = 2); the coefficient z measures the transfer of fluid from the pores to the fissures; and the seepage force factors v~and the relative specific discharges Q? in the fissures (~t = 1) and the pores (a = 2) are given by v, = fl,v/K,,

(4)

(2? = n~(a? - u3,

(5)

in which v is the dynamic viscosity of the fluid; K~ are the Muskat permeabilities of the fissures (a = 1) and the pores (ct = 2) and u~~ are the displacement components of the fluid in the fissures (~ = 1) and the pores (a = 2). The relative mass densities of the composite medium are given in terms of the actual ones by the relations Os = Os[1 - (nl + n2)],

(6)

O, = n,es.

(7)

In a fissured poroelastic medium the total stress tensor a~j (i,j = 1, 2, 3) is given by =

+

+

(8)

where a~ and a~ are the stress tensors acting on the solid, fluid in fissures (~ = l) and fluid in pores (a = 2), respectively. Stress tensors ai~j and tr~ are expressible in terms of deformation as ai~ ~-- 2l~eij + 2eli j,

tr~j = Z~3ij ---- --fl~p~6ij,

(9) (10)

D. D. Theodorakopoulos and D. E. Beskos: Flexural vibrations of fissured poroelastic plates

415

where 6 u is the Kronecker's delta and the strain tensor e u and dilatation e of the solid are expressible as

(11) (12)

'~ij = ( l / 2 ) " (ui,j + uj,i), 8 = 8kk = Uk, k .

In view of (9) and (10), (8) can take the form (13)

17ij = 21Zgij + ~G(~ij -~- TI(~/j -31- "~2 t~ij '

while using (2) for c~ = 1, 2 in turn and adding the resulting equations together with Eq. (1) with uis, being replaced by eu's one obtains 2[18ij,j + 2g,i -- fllP,li -- fl2P, 2 : Qy(Oi I +

Qi 2) + (0s + 01 + 02)/~/i.

(14)

Spatial differentiation of (13) and use of (14) yields (Tij,j = e f ( Q i I Jr Qi 2) "~ (Os "~ 01 + 02) /~i"

(15)

Substituting (5) into (2) and (15) one obtaines r ~,1 = v~n,(f~i ~ - ui) + 0,i/i';

c~ = 1, 2

(16)

ai~,j + z 1",,+ z 2, = 01iii 1 + 02fii 2 + Osfii.

(17)

3

Flexural motion of a poroelastic plate

Consider a thin plate, whose middle surface coincides with the (xl, x2) plane, experiencing lateral flexural motion along the x3 direction, as shown in Fig. 1. The thickness of the plate is h and its upper (x3 = - h/2) surface is subjected to transverse loading (or pressure) of intensity q = q ( x l , x2, t). The material of this plate is a fissured poroelastic - viscous fluid system of the Aifantis-Beskos [3, 4] type, while Kirchhoffs linear flexural theory of plates is assumed to govern the motion of the plate (Szilard [9]). It is further assumed that the in-plane fluid flow relative to the solid is negligible c o m p a r e d to the lateral one, i.e., Ul ~ R1 1 ~ Ul 2 ,

/'/2 ~ I/21 ~---U22 ,

/13 = W ::~ /,/3~,

(18)

where u3 = Ua(Xl, x2, t) = w = w ( x l , x2, t) is the lateral solid displacement of the middle surface and u3 ~ = Ua~(Xl, x2, t) are the lateral fluid displacements. On the basis of the above assumptions one has (19)

0"33 : O, where a33 is the normal total stress c o m p o n e n t along the x3 direction, and H i -~ tli ~t =

--X3W,i

eu = --x3wu

(20)

i = 1, 2

,

= Xa(1/R),

(21)

i , j = 1, 2

where (l/R) is the middle surface curvature. /4

a

,,/ X3=-h/2

TTT i ~

xl,3 = -q~

~---q~ Xl, Ul

Xa=+h/2 -t,13 :ql 2 ; "1;'23: q22

3 w

\ 7 ~ middle surface

Fig. 1. Laterally loaded rectangular plate

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At this point it is assumed that the dynamic disturbance is harmonic in time and hence the plate response is also harmonic, meaning that any function f(xl, x2, t) of the problem is expressible as

f(xx, X2, t)

(22)

= f ( X l , X2) e i~

where f i s the amplitude of f, i = ~ and 03 is the circular frequency of vibration. In the following the factor e i~'' is omitted for simplicity. Considering (16) in the frequency domain and solving them for ad one obtains U- + (i03v~n~)ui. i03v~n~- 0320~ '

ui~

c~ = 1, 2.

(23)

Substituting (23) and the frequency domain version of (9) into the frequency domain version of (17) one has

(. !03V! ~1.+ 2#gq,j + 2(i + \i03v, - 0320y/ '~

(i03v2 2i03v2 ~-0320j/e~

= --032 Os "-[- 01 At- 02 -]- 9 03 Of e l 2 032Qf02 + 103Vl -- 03 e f i03v-2a~ - : e y J

ui

(24)

i.e., an equation involving only five variables (tT~,i = 1, 2, 3 and P, ~ = 1, 2) instead of eleven (g~, ti~', i = 1, 2, 3 and f', e = 1, 2) it involved before. (24) in view of the frequency domain version of (13), suggests that ff~j for a fissured poroelastic material can be written in the form

(i Ol

6iJ = 2#giJ + 2g(~iJ "~ \i03vl - 0320f//fl(~ij -~ \i03v2 -- coZof,] ~26ij"

(25)

Multiplying the first two (for i = 1, 2) of (24) by x3 taking into account the boundary conditions cr~3 = 0 on the two plate surfaces and integrating over the thickness one obtaines the equations mll,1

-[- J~12,2 -- Q1 = 0 ,

(26)

1~r

+ ]~r

(27)

-- Q2 = 0,

where the moment M~ and shear 0~ resultants are defined by h/2

]~/lij =

~ ~ijX3 dx3;

h/2 ~ ei3 dx3;

Qi =

i = l, 2

(28)

-h/2

i = 1, 2.

(29)

-hi2

Integrating the third (for i = 3) of (24) over the plate thickness and satisfying the boundary conditions t733 = --t~

for

x3 = -h/2

~33 = 0

for

X3 =

(30)

h/2

one obtains the equation _

(~1,1 + (~z,2 + q = _032 Os + 01 + 02 +

---~ Q--ye--!~

i03vl - 03M

+

i03

OQd

(31)

~h.

Elimination of (~x and (~z between (26), (27) and (31) yields the equation

/~lt 11 + 2~'t12 9

2

12 "-[- ) ~ 2 2 22 71- I~ = --(D 2 0 s , ,

-F O1 "-[- 02 -[-

-

(D 0.r01 + i(D/.) 1 -- (,02Qf

03M&

i03v2 --

O)2~f/

kh.

(32)

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417

It should be emphasized here that indices i,j = 1, 2, indicate a state of stress defined by (19) and therefore 6ij that appear in (28) should not be taken from (25), which is valid for i,j = 1, 2, 3, but from the equation resulting from (25) after the elimination of g33 with the aid of (19). This equation reads

2tz 2# 21~ A2~Z(~ij ' ~ij = 2[Jgij q- ")~2flfl ~AA ~kk(~ij ~- 2~t +-~-2hl'~lt~iJ q- 2t, + ~

k = 1, 2,

(33)

where, icoG 2 ; l(.O/)ct -- O) Of

A~-.

~=1,2.

(34)

Thus, use of ~ej as given by (33) into (28) in conjunction with (21), result in Mzj = -D[(1 - v) u?,,i + (V~,kk + B~P + B z I 2) 6q],

k = 1, 2

(35)

where

D = Eh3/12(1 - v2) B ~ = - 12vA~/2h 3 ; h/2

~ = 1, 2

]'~ =

~ = 1, 2

I UX 3 dx3; -h/2

(36)

and use was made of the fact that # = E/2(1 + v) and 2 = Ev/(1 + v) (1 - 2 0. Finally substitution of Mij in (32) by its expression in (35) yields the first governing equation of flexural motion of the poroelastic plate, which reads (D2OfO 1

(-02Of02

"~

--D[V4r + B' V 2 I 1 + B2 V2F] + q = --0.) 2 0s q- 01 -]- 02 q- IO)UI' _ (.020f -1- i~o727-jojj #h,

(37)

where V e = O2/axa 2 + 9 and V4 = V2(V:). Solving now Eq. (33) for g under the condition g33 = 0 of (19) and also using (21), one can obtain g=

1 (AI'~ 1 q- A2"~2) 2kt + 2

2# x 3 V21~. 2# + 2

(38)

Elimination of g between the frequency domain version of (3) for e = l, 2 and (38) after using the frequency domain versions of(5), (10), (12) and (23), finally yields the other two governing equations of flexural motion of the fissured porelastic plate in the form ~lifll-

2 fl,Aa]i~vl--f~

fllico(7, + 6 n l ) + z +ico(fl,+nlA1)21t +

~2[

fll

- - r ~--Xflz2 +

ico(fil + nlA1) filA2"~ ie)vl -- co20~ 2p+2 ) io)

2/t

21~ + 2 (ill + nlA1) fl~(ioovl -- co20f) x3 V2u~ = 0,

(39)

f12 i~(fl2 + n2A2) / i~v2 -- (D2~f ~2ifl 2 __ ~i __ ~ f l l -~ 2~ -~- /. fl2A1/ ifo --'~2Ii(D(~2-~- (~n2' -~- )~ -'~ i0)([~2 -~~ l'12A2' ~ fl2A21 i(d)v2 io) -

2g (f12 + 2/~+2

n2A2)/~2(i~v2-

~ 2 e s) x3 V2~ = O.

(40)

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The system of (37), (39) and (40) for the flexural motion of a fissured poroelastic plate in the frequency domain is easily reducible to the equation

- D V4ff~ + gl = -o920~ ~h,

(41)

when the plate material becomes simply elastic with mass density ~ = ~. Equations (37), (39) and (40) can also be written in the much simpler form C1 V%~ + C2 V2I a + C3 V2I 2 + C4~ + ~ = 0,

(42)

C5~li i q- C6 fl -F C7 ~2 -F Csx 3 V2w ~-- O,

(43)

C9f2i + Clo~ ~ + Clxf z + C12x3 V2~ = O,

(44)

where

Ci = - D = -Eh3/12(1

-

1 - 2v

-DB 1 -

C2 =

V2),

-

1 -

C3 = - D B 2

1-2v

v

ic0vl ie)vt

-- e)20z

'

i~v2

-

1 -

v

i~v

2 -

~2~f,

C4 = ~2h(~s + ~l + ~2 "J- B1 -~- B2) = (-02h ~s + ~1 + ~2 + icov-~Z co-~ql + ico 2 --

C5=fll,

C6=-

[ C7---- --

fla --Eft2 +

[

i~~

+ nlA1) -2;q- ~

f12

C12 =

' (45)

i(.o

'

fll(io)vl - coz~i),

io)(fl2 + n2A2)n

C9 "~- ~2,

" ] i~Ov2 - ~ o 2O:

.:lj

I

ic~176 ioJ

J ic~ -o~ fllA2

Cs = -(2#/(2/~ + 2)) (1~1 -[- nlA1)


ir176 21*+2

io@1+6nl)+~+

~f/'

io,

'

io(fl2 + ~2A2)R a ~ i c o v 2 - co2O:

-(2~/(2~ + 2)) (f12 + n2A2) f12 (kov2 - co2~fl.

A more convenient form of (43) and (44) useful in applications can be obtained by following a method used by Theodorakopoulos and Beskos [1] in the case of plates with one degree of porosity. Thus by first multiplying (43) and (44) by x3, then integrating them over the thickness and finally employing integration by parts, one can obtain h3 h (C5 V 2 + C6) r 1 + CTr2 + ~ Cs V2k + ~ C5(q21 - c~11)- Cs(?21 - "711) = 0,

(46)

h3 h (C9 V2 + Cl1) ~2 + ClflU + ] 2 C12 V2~ + ~ C9(q22 - ql 2) - C9(?22 - fl 2) = 0.

(47)

Equations (42), (46) and (47) constitute a system involving only/-1, r2 and k as well as the boundary values ~, s s ~21, s ~2z, "c73(h/2) and/~.3(- h/2). In general one has the following possible boundary conditions in the frequency domain ~ = ~1 ~

and

~2 = ~12

or

~:3 = - 0 ~ ~

= 1721

and

?2 = f22

or

-, = q21 "/7,3

~1

and and

~23 = - 0 ~ 2 f23 = ~2z

on

on x3 =

x3 =

h/2 9

-h/2

(48) (49)

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In the above, ( 4 8 ) 1 , 2 o r (49)1,z denote permeable surfaces, while (48)3,4 or (49)3,4 with ql" = q 2 ~ = 0 (~ = 1, 2) impermeable surfaces. In a specific boundary value problem four of those boundary conditions, two on each plate surface will be known.

4

Speciailization to one degree of porosity and comparison with Biot's theory

On the assumption that the volume of the fissures shrinks to zero, i.e., 7~ + 6n~ = 0, fi~ = 0, na = 0 and ~ = 0, the medium becomes one of single porosity and (42) to (44) with the simultaneous dropping of the remaining index 2 reduce to Cl V41~ .vff C3 V2lqt_ C4 ~ + 0 = 0,

(50)

C 9 f i i q- Cll'~ -J- C12x 3 V2w = 0,

(51)

while the dynamic governing equations for plates with one degree of porosity based on Biot's theory [2] are given in Theodorakopoulos and Beskos [1] by

(52) (53)

Fa v a # + F2 V2r~ + F3~ + c~ = 0, F4"~,ii -~

Fsf +

F6X3 V2w = 0.

In the above two sets of equations C1=

F2-

F1 = - D = -Eh3/12(1 1 -2v(Q

1 -- v

+

--

v2),

1 - 2v ico(vfl/K) 1 - v ico(vfl/K) - ~oz~f ' I C4 = r 05(1 -- n) + noy +

C3 -

ico(vn2/K)+m2012~ - - ~ , im(vn2/K) eo 0z2J

F 3 -~- ~02h 0s(1 - n) + n0f + .

o,2n2/

--

--- 2

lco(vn/K) -

o)

-]

022A

,

C9 = fl,

O)2rtOf2

]

ko(v3-~oo2oiJ'

F4 = R,

Ca 2 = -(2#/(2p + 2))[kofl(vfl/K)(fl + n) + o)2(-fiz0fl],

(54)

F6 = --(2~/(2/~ + 2))[ko(vn2/K)(Q + R) + 0)2(R~21-Qoz2)],

C,~=-

ko(vfi/K)(y+an)--oo2oy(y+6n)+

Fs = -[iog(vne/K) - ~oeoze + ( ~ 2 v

~C3

\

2#)_]

F2) ( - - ~ ) 1.

A comparison of (50)-(51) with (52)-(53) easily reveals that, apart from the extra term [(1 - 2v)/(1 - v)] (Q/R) in coefficient F 2 of (52), the form of these two sets of equations is exactly the same. The differences that exist in some of the coefficients as well as the existence of the above extra term, are due to different definitions of partial stresses, seepage forces, material coefficients and mass densities between the Beskos theory [4] and that of Biot [2]. If the realistic assumption that Q12 = 0 is made, then ~)22 = n~f and consequently a closer agreement between the coefficients is established.

5

Dynamic analysis of a fissured poroelastic rectangular plate

Consider a fluid saturated fissured poroelastic, simply supported, rectangular plate with sides a and b experiencing harmonic flexural motion due to the presence of lateral load of intensity q acting on its upper surface, and varying harmonically in time. The motion of this plate is governed by the system of (42), (46), and (47) subjected to appropriate boundary conditions, as explained at the end of section 3. The solution procedure of this problem closely follows the one used by Theodorakopoulos and Beskos [1] in solving the corresponding poroelastic plate problem of single degree of porosity.

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Since the plate sides are simply supported, the solution is assumed to be expressible in double Fourier series i.e., W(X1, X2) = /I(x1, X2) =

~

~ ~m. sinmn - - xl sin nn x2, G T

(55)

~.] ~ _-,,i . FF/'/'C /'/~ Iron Sm - - Xl sin x2,

(56)

m=l n=l

m=l n=l

a

=

T

Ira, sin - - xx sin m=X n=l

a

x2.

(57)

T

Similar series expansions with amplitudes f~',.., 8.,. are also assumed for f f and c~, where ~ --- 1, 2 and i = 1, 2. Thus substitution of(55)-(57) into (42), (46), and (47) yields the system of linear algebraic equations

Glff~mn+ G2lln q- G312m.= G4, Hlff)mn+ Hz[lmn+ H312mn= H4,

(58)

SiWmn "at- S2ffLn --[-S3~2mn= $4,

(60)

(59)

where G1 = C1 A2 -]- C4,

G2 = - C2 A,

H2 - - C 5 A -1- C6,

H 3 = C7,

S1 = -(h3/12) C12A, $4 =

-(h/2)

G3 = - C 3 A,

G4 = --t~,..,

H1 = -(h3/12) C8 A,

H 4 = -(h/2) C5(q1.,. - ql,..) + C5(f2~,.. - flm.), $3 = - C 9 A + Cll,

S 2 = Clo ,

(61)

C9(~12mn- q2mn) -'1-C9('~2mn-- "~2mn),

A = am2 + bn2,

a m = mn/a,

b. = nn/b.

Solving (58)-(60) for fire., rim.and r2. and making use of (55)-(57) one can finally obtain the frequency domain response of the fissured poroelastic plate in the form G4 G2 G3 [ H4 H2 H3 W(X1, X2, CO) = ~

~

S4

$2

$3

sin a.,xl sin b.x2,

(62)

m=ln=llG1slHi 82H2G2Hzs3G3 I G1 H1

G4 G3[ H4 H3 $4 $3

GI H1

G2 G3[ Ha H3

S1

S2

$3

H1

H2

H4

=

ra=l . = i

[2(xl, x2, co) = ~

~

m=l n=l

G1 H1 $1

G2 H2 $2

G31 H3 $3

sin a.,xl sin b,,x2,

(63)

sin amXl sin b,x2.

(64)

D. D .

6

Theodorakopoulos and

D. E.

Beskos: Flexural vibrations of fissured poroelastic plates

421

Numerical results and discussion

The numerical results of this section have been obtained on the basis of the numerical values of the coefficients given in Table 1 which correspond to a water-fully-saturated, poroelastic, fissured, sandstone medium. The values of these coefficients have been obtained on the basis of information provided in Beskos et al. [6]. The physical quantity of interest in this investigation is the frequency-dependent lateral deflection amplitude of the plate k. The case of a square plate 4.00 x 4.00 x 0.20 m, with permeable surfaces subjected to a harmonic lateral load of amplitude 1 000 N / m 2 w a s examined. The response of the plate as a function of frequency is shown in Figs. 2 - 4 for various combinations of porosities and permeabilities n, and K~ (c~ = t, 2), respectively. One can argue on the basis of physical reasons that n~ and K~ are the two most important material parameters of the problem [4]. Responses near resonance were calculated at frequency intervals of 0.0001 Hz in order to accuratelly capture their sharp variation there. Figure 2 shows the variation of the plate response amplitude [~1 versus frequencyf = co/2n for various values of total porosity n = nl + n2 (with n~ = n 2 ) and for a ratio of permeabilities Kt/K2 = 1 000 (with K 2 = 1 x 10 -s m 2) under dynamic and quasi-static conditions. The value of n = 0 corresponds to an elastic solid medium without fissures and pores. The dynamic behavior of this special double porosity plate was found to be the same with that of a single porosity plate [1], as expected. Thus, Fig. 2 indicates that for the kind of dynamic load used, inertial effects are important only around the resonance area. It also shows that the first natural frequency of the elastic plate, found to be 18.85 Hz (Szilard [9]), increases with increasing total porosity and this shifting can be attributed to the fact that increasing values of the total porosity decrease the mass more than the stiffness of the Table 1. Numerical values of coefficients for water-fully saturated poroelastic, fissured sandstone constant constant Poisson's ratio Solid density Water density Water dynamic viscosity Water compressibility Compressibility of fissures and pores Porosity change due to dilatation Lam6 Lain6

)o = 2.76 x 109 N / m 2 p = 2.76 x 109 N / m z v = 0.25 G = 2660 QI = 1 0 0 0 v = 1 x 10 6 = 4.67 x

71 = 4.34 x 10

9 I/N 9m 2

72 = 4.24 x 10 - 9 I / N -Irl 2 fll = 0.20 fi2 = 0.70 z = 2 x 10

Water transfer from fissures to pores

I~1

Kg/m 3 Kg/m 3 .3 Ns/m 2 1 0 - lO I / N . m 2

1~ N / m :

-s

displ. [m] lOO

K1/K2=1000 K2= 1 x'iO-8m 2 N1= R2= n/2

10

n:0 n = 0.10 0.1

n : 0.26 , /t~/

0.01

/n=0.50

1.000E-O:

Fig. 2. Plate amplitude response versus frequency for various values of total porosity

1.000E-0,

9

11

13

15

17

19 21 freq [Hz]

23

25

27

29

31

422

Archive of Applied Mechanics 63 (1993)

plate. It is also observed from Fig. 2 that the viscous flow dissipation reduces the amplitude response with increasing total porosity, as expected, because of the fact that dissipation forces in both fissures and pores increase with their porosities. Figure 3 shows the variation of [wl versus frequency for various values of the ratio nl/n where n = nl + n2 = 0.26 and for K1/K 2 = 1000 (K2 = 1 x 10-8 m2). The ratio nl/n = 0 corresponds to the single-porosity model, i.e., to a poroelastic medium without fissures. It is apparent from Fig. 3 that the effect of porosities on the response is negligible for all frequencies before and after the resonance region, whereas in the resonance region, the amplitude response increases with increasing values of the n~/n ratio. This is due to the fact that an increase in porosity nl with the value of total porosity being kept constant, means that part of the pores with a given permeability, K, and of a given coefficient fl, is substituted by fissures of greater permeability, K, and smaller coefficient, fl, resulting in smaller dissipative forces. It is also observed that the first natural frequency of the elastic plate slightly increases with increasing nl/n ratios for analogous reasons as in the previous case of Fig. 2. Figure 4 shows the variation of the plate response amplitude [v~[ versus frequency for nl = n2 = n/2 = 0.13 and various values of K1/K2 (with K2 = 1 x 10 -8 m2). It is evident that the effect of the permeabilities on the response of the plate is negligible even in the resonance area, especially for values of K1/K2 greater than 10. I~,1

0.1

displ. Ira] KI/K 2 = 1000 K2 = 1 xlO-Sm 2 n= nl. nz=0,26

j ]t

~

nl/n=0"90

Jb~ ~n~/n=0.50 l/ ~ n ,

0.01

In- 0.10

1.000E-03

1.000E-04

9

0.01

I

k

J

L

I

11

13

15

17

19

L

I

I

21 23 25 freq [Hz]

Fig. 3. Plate amplitude response versus frequency for various combinations of porosities

I

I

i

I

27

29

31

33

35

displ. [rnl -8 2

~

K1/K2 =100,1000

1.000E-03

1.000E-04

i

~

I

I

11

13

15

17

t

I

19 21 freq [Hz]

i

I

I

i

23

25

27

29

Fig. 4. Plate amplitude response versus frequency for various combinations of permeabilities

D. D. Theodorakopoulos and D. E. Beskos: Flexural vibrations of fissured poroelastic plates

7

423

Conclusions

The governing equations of flexural vibrations of thin, fluid-saturated fissured-poroelastic plates were derived in detail. These equations were speciallized to the one degree of porosity model and compared with the corresponding ones based on Biot's theory. The dynamic response of a square, simply supported, poroelastic plate with two degrees of porosity to harmonic load showed that the inertial effects are important only around the resonance area. It was found that the resonance frequency of the plate increases with total porosity, whereas the viscous flow dissipation reduces the amplitude response with increasing total porosity. It was also evident that the presence of fissures of relatively higher permeability and lower deformability than in the pores has a negligible effect on the response outside the resonance area, whereas the resulting decreased viscous flow dissipation increases the amplitude response in the resonance area. This increase was only marginal when only the parameter of permeability was increased.

Acknowledgements The authors would like to express their appreciation to Mr. A. A. Stamos for his help with the numerical computations.

References 1. Theodorakopoulos, D.; Beskos, D. E.: Flexural vibrations of poroelastic plates. (accepted for publication in Acta Mechanica) 2. Biot, M.A.:Theoryofpropagationofelasticwavesinafluid - saturated porous solid I : Low-frequency range. TheJournal of the Acoustical Society of America: JASA 28 (1956) 168 178 3. Aifantis, E. C.: On the problem of diffusion in solids. Acta Mechanica 37 (1980) 265-296 4, Beskos, D. E.: Dynamics of saturated rocks l: Equations of motion. J. Engng. Mech. 115 (1989) 982-995 5. Aifantis, E. C.: A new interpretation of diffusion in high-diffusivity paths - a continuum approach. Acta Metallurgica 27 (1979) 683-691 6. Beskos, D. E.; Vgenopoulou, I.; Providakis, C. E: Dynamics of saturated rocks II: Body waves. J. Engng. Mech. 115 (1989) 996-1016 7. Beskos, D. E.; Papadakis, C. N.; Woo, H. S.: Dynamics of saturated rocks III: Rayleigh waves. J. Engng. Mech. 115 (1989) 1017 1034 8. Vgenopoulou, I.; Beskos, D. E.: Dynamics of saturated rocks IV: Column and borehole problems. (accepted for publication in J, Engng. Mech.) 9. Szilard, R.: Theory and analysis of plates: classical and numerical methods. Englewood Cliffs, N. J.: Prentice-Hall 1974 Received October 12, 1992

Asst. Prof. Dr. D. D. Theodorakopoulos Prof. Dr. D. E. Beskos Department of Civil Engineering University of Patras GR-26500 Patras Greece