T h e r e c i p r o c i t y t h e o r e m f o r p o r o u s a n i s o t r o p i c m e d i a 

E . B O S C H I ( * ) 

Received oil .July 17tli, 1972 

SUMMARY. •— In this p a p e r we give a reciprocity t h e o r e m for aniso-
tropic porous media in t h e q u a s i - s t a t i o n a r y case. T h e d i s t r i b u t i o n of t h e 
pores is assumed s t a t i s t i c a l l y h o m o g e n e o u s . 

RIASSUNTO. - Viene s t a b i l i t o un t e o r e m a di r e c i p r o c i t à per mezzi 
porosi a n i s o t r o p i nel caso quasi-stazionario. L a distribuzione dei pori è 
a s s u n t a s t a t i s t i c a m e n t e omogenea. 

1 . - I N T R O D U C T I O N . 

The theory of deformation of porous materials containing a fluid 
has been developed by Biot f1-2), for t h e case of an elastic solid. I n 
the last years various generalizations a n d particular applications have 
been considered. 

A porous solid is represented as an elastic skeleton, having com-
pressibility a n d shear rigidity, with a statistical distribution of inter-
connected pores containing a compressible fluid. I t is understood 
t h a t t h e term " p o r o s i t y " refers to the effective porosity, namely, t h a t 
encompassing only the intercommunicating void spaces as opposed 
to those pores which are sealed off. I n the following, t h e word " p o r e " 
will refer to t h e effective pores while t h e sealed pores will be consi-
dered as a p a r t of t h e solid. 

(*) I s t i t u t o di Fisica, U n i v e r s i t à di Bologna, I t a l i a . 



( ¡ 0 8 E . B O S C H I 

This s y s t e m of fluid and solid is a general elastic system with 
conservation properties. The deformation of a u n i t cube is assumed 
to be completely reversible. 

By d e f o r m a t i o n is m e a n t , here, t h a t determined by both strain 
tensors in t h e solid and in the fluid phases. 

I e f a n ("8) lias established a m e t h o d by means of which he was 
able to give reciprocity theorems in t h e d y n a m i c t h e o r y of continua, 
w i t h o u t using Laplace t r a n s f o r m s in the case of nonhomogeneous 
initial conditions. Moreover, by this m e t h o d , he obtained reciprocity 
relations which involve only t h e displacement vector a n d t h e known 
functions. 

Boschi (;i) has obtained a reciprocity theorem for isotropic porous 
homogeneous m e d i a in the quasi-stationary case. H e (4) has also 
been able to give a v a r i a t i o n a l theorem in t h e linear t h e o r y of porous 
media, a n d in t h e t h e o r y of viscoelastic porous m e d i a (5). 

This kind of speculations is of great interest because t h e y f o r m 
t h e basis for t h e solution of problems arising in m a n y diversified fields 
such as seepage in soil mechanics, ground w a t e r hydrology, petroleum 
engineering, w a t e r purifications, acoustic engineering a n d so on. 

2. - BASIC EQUATIONS. 

T h r o u g h o u t this p a p e r we employ a rectangular coordinate sys-
tem, Oxk (k = 1, 2, 3), and t h e usual indicial n o t a t i o n s . L e t V be 
a regular (in the sense of Kellog) region of space occupied by a n ani-
sotropic porous solid, whose b o u n d a r y is J7. Moreover V is t h e in-
terior of V, ni are t h e components of t h e u n i t o u t w a r d n o r m a l to Z. 

F o r convenience a n d clarity in presentation, all regularity hypo-
theses on considered f u n c t i o n s will be omitted. 

On these basis t h e field equations for anisotropic porous solids, 
in t h e qnasi-static case, are: 

— t h e constitutive equations: 

da = Gijki eici + eia e [ l . a ] 

[ l . b ] a = an en + fie 

— t h e equations of equilibrium: 

(<fu + aòi,),i +qFi = 0 [2.a] 



T H E R E C I P R O C I T Y T H E O R E M FOR P O R O U S A N I S O T R O P I C M E D I A till 

The D a i r y ' s law: 

a,i + oi F( = ht, ( U] — ùj). [2.b] 

The equation [2.b] is t h e D a r c y ' s law in a generalized form for 
an anisotropic medium. 

— t h e strain-displacement relations: 

In these equations we h a v e used the following notations: ui, 
t h e components of t h e displacement vector for t h e solid phase; Ut, 
t h e components of t h e displacement vector for the fluid inside the 
pores; oij, en, the components of the strain a n d of t h e stress tensors 
for t h e solid phase; a, the h y d r o s t a t i c s t a t e of stress of t h e liquid 
filling t h e pores, i.e., if we consider a cube of u n i t size of the bulk 
material, a represents the t o t a l normal tension force applied to t h e 
fluid p a r t of t h e faces of such a cube; e<j, t h e components of the strain 
tensor for t h e fluid phase; e — En, dilatation of t h e fluid; q, qi den-
sities of t h e solid and of t h e fluid, respectively; Cini, bn, characteristics 
of t h e solid phase; a n , (i, characteristics of t h e fluid phase (2). L e t us 
also remember t h a t a comma denotes p a r t i a l derivation with respect 
to the space variables, xk, a n d a dot denotes p a r t i a l derivation with 
respect to t h e t i m e /. 

F u r t h e r m o r e t h e following s y m m e t r y relations: 

are satisfied. 

To t h e system of t h e field equations we m u s t a d j o i n t h e initial 
conditions: 

13. a] 
[3.b] 

Ctjici = C)iki — Cxni 
an = an 
bu = bu 

[La] 
[ L b l 

Ui (x, 0) = at 
U, (x, 0) ='vli 

[5.a] 
[5.b] 

a n d t h e b o u n d a r y conditions: 

{an + <rôn) >'i = Vh on I [«J 

where at, Ai, pi are prescribed functions. 



(¡10 E . BOSCHI 

3 . - P R E L I M I N A R I E S . 

L e t /, g a n d h be f u n c t i o n on V x [0, oo), continuous on [0, oo) 
with respect t o t h e time t for each x e V. W e denote b y / * g t h e con-
volution of / a n d </: 

t 

[/ * 9] 0 = | / (®, t — s) g (.;•, s) (h. 
o 

W e will h a v e occasion to use t h e following well-known properties 
of the convolution (9): 

/ * 9 = 9 * / 
/ * (g *h) = (/ * g) *h — / * g * h 

f * (9 + M = f * 9 + f * h-

H e n c e f o r t h we will denote by I t h e f u n c t i o n defined on [0, oo) b y : 
l(t) = 1. 

I t is easy t o show t h a t : 

l*(Uj—11 j) = Uj — u j — (A} — a,) [7] 

I * {a,, + ei Ft) = I * + Qil* Fi. [8] 

Then from the equations [2.b], [7] and [8], we get: 

l*tt,i + q,=btJ(Ui — u,) [9] 

where: 
q, = QI I * Ft -f blf (A] — ai). [10] 

T h e equations [9] are equivalent to t h e equations [2.b] a n d to t h e 
initial conditions [5.a], [5.b], 

4 . - T H E R E C I P R O C I T Y T H E O R E M . 

L e t us now consider t h e body subjected to two different systems 
of elastic loadings: 

L^ = {F(,a\p({a),A,ia),a,ia),} a = 1 , 2 . [11] 



T H E R E C I P R O C I T Y T H E O R E M F O R P O R O U S A N I S O T R O P I C M E D I A t i l l 

T h e t w o corresponding configurations a r e : 

/-i(a) — / 77 (a) „, (a) „ (a) .(a) _ (a) _.(a) \ „ 10 n on O =\Ut , Ht ,eit ,e ,(Ju ,a , j- a — l, J. | L-!J 

From the equations [l.a], [l.b], we get: 

_ (a) „ _(a) rt „ (a) J TiJ —tti) £ = OiiklSki I 
> a = 1 2 

jo — at, e
(a) = p £

(a) j 

I t is t r u e t h a t 

(crij*1 > — cn, e<D) * e£i<2) = Gijki e*i<D * 
(o-fj«2» — au e(2>) * c<j (1) = Ciik, * 

A d d i n g these t w o relations, we h a v e : 

(at,'1' — a,j e'1») * eti121 = (,at,<2> — a « e<2>) * e t , »>. [13] 

I t is also t r u e t h a t : 

(ffi1) — an ey(D) * £'2> = e<i> * e<2> 
(ffi2) — an eti&>) * «(1) = /? e<2) * £(1) 

a n d , b y a d d i t i o n , we o b t a i n : 

(cr<n — an <?<j(1)) * «(2) = (fff2) — a»j e«(2>) * e«1'. [14] 

A d d i n g t h e relations [13] a n d [14] we g e t : 

a n a ) * e t / 2 > + o a > * e < 2 > = a n i S ) * e < i a ) + ff'21 * e ( 1 ) - [ 1 5 ] 

If we i n t r o d u c e t h e n o t a t i o n : 

= I I * ( f f « ( a ) * + ff(a) * e^) dV-, a, — 1, 2 [16] 
v 

f r o m e q u a t i o n [15] we h a v e : 

Ii2 = /21 [17] 



<>12 F.. BOSCH I 

lTsing t h e relations [3], [9], [10] a n d the divergence theorem, we g e t : 

Iafi = | i * v r ] * ut& dZ + | / * a'a) * I!,'?' n< dZ + 
V V 

+ I I * o F,(a) * m'P' dV — | / * aH1) * dV — 

v v 

- j b„(Ui,a) — u,,a)) * Ut'P' (IV + J q,M) * U,^ (IV, [181 
v v 

a,P = 1 , 2 

Finally f r o m equations [17] a n d [18] we can s t a t e the reciprocity 
theorem for anisotropic porous solids: 

If a n anisotropic porous solid is s u b j e c t e d to two different systems 
of elastic loadings [11], t h e n between t h e t w o corresponding 
configurations [12] there is tlie following reciprocity relation: 

j I * />,<>> * W(<2> dZ + | I * c' 1 ' fii * r/V2> dZ + 

+ | I g * F,ID * Mi<2) (IV — j Z * cr") * e<2> rfF — 

F F 

[6(/ (f.^'1» — M/«1») + d] * r/f(2) f / r = 

I l * p,<*> * u,<'I dZ + | I * ff'2> ill * {/,»> d r + 

+ J I * Q Ft<2> * Mi») dV — J Z * a<2> * ell) 

F F 

- J (?7/(2) — Wi'2») + R/I(21] * E > RFF. [19] 

W e should w a n t to p u t in evidence t h a t , t h r o u g h o u t this p a p e r , 
we did not use a n y restrictive assumption. The theorem [19] is valid 
under very general conditions. 



T H E R E C I P R O C I T Y T H E O R E M F O R P O R O U S A N I S O T R O P I C M E D I A till 

I n t h e special case of homogeneous b o u n d a r y conditions, of great 
interest in t h e applications, t h e relation [19] reduces t o : 

I * (Q JV1) *Mt<a' — <rll> *e<2>)— [bt)(Ui^ — uj'1') + * Ut,2> dV = 

J 11* (q — ff«) *e^) — [btl(U]^ — uii2)) + ?i, 2 )] * E7i(1) | dV. 

Tlie reciprocity theorem derived in this p a p e r can be used to 
obtain variational theorems in t h e t h e o r y of porous anisotropic mate-
rials. These theorems will be developed in a n o t h e r p a p e r . 

0 ) B I O T M. A., 1955. - " J . A p p i . P h y s . " , 26, 182. 
( 2 ) B I O T M. A., 1956. - " J . A p p i . P l i y s . " , 27, 210. 
( 3 ) B O S C H I E . , 1 9 7 2 . - " L e t t . N u o v o C i m e n t o " , 4 , 9 7 3 . 
( 4 ) B O S C H I E . , 1972. - To be p u b l i s h e d . 
( 6 ) B O S C H I E . , 1972. - To be p u b l i s h e d . 
(6) IEÇAN D., 1967. - " R e n d . A c a d . Sc.", P a r i s , 265, 271. 
(7) IESAN D., 1969. - " I n t . J . E n g n . Sci.", 7, 1213. 
( 8 ) I E § A N D., 1970. - " R e v . R o u m . M a t h . P u r e s et A p p i . " , 15, 1181. 
( " ) M I K U S I N S K I J . , 1959. - Operational Calculus, P e r g a m o n Press, New Y o r k . 

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