On the plane strain in a theory f o r self-gravitating 
elastic configuration with initial static stress field (*) 

E . BOSCHI ( * * ) 

R e c e i v e d on March 22nd, 1974 

SUMMARY. — This paper is concerned w i t h the plane strain in a theory 
for an arbitrary, uniformly rotating, self-gravitating, perfectly elastic Earth 
model with a hydrostatic initial stress field. Using the associated matrices 
method, a representation of Galerkin t y p e is given. This representation 
enables us to derive the solution of the vibration problem corresponding to 
concentrated b o d y forces. 

RIASSUNTO. — Tn questo lavoro si t r a t t a il problema della deformazione 
piana nell'ambito di un modello terrestre arbitrario, uniformemente ruo-
tante, autogravitante, p e r f e t t a m e n t e elastico e soggetto ad un campo idro-
statico di sforzo iniziale. Usando il metodo delle matrici associate, viene 
data una rappresentazione di tipo Galerkin. Questa rappresentazione per-
mette la soluzione del problema delle vibrazioni corrispondenti a forze di 
massa concentrate. 

INTRODUCTION. 

D a h l e n ( 3 ) has d e v e l o p e d t h e l i n e a r i z e d e q u a t i o n s a n d l i n e a r i z e d 

b o u n d a r y a n d c o n t i n u i t y c o n d i t i o n s g o v e r n i n g s m a l l e l a s t i c - g r a v i t a -

t i o n a l d i s t u r b a n c e s a w a y f r o m e q u i l i b r i u m of an a r b i t r a r y u n i f o r m l y 

r o t a t i n g , s e l f - g r a v i t a t i n g , p e r f e c t l y e l a s t i c E a r t h m o d e l w i t h an ar-

b i t r a r y i n i t i a l s t a t i c stress f i e l d [see, also, B o s c h i ( ' ) ] . 

( * ) This w o r k has been made during a tenure of a C . N . R . fellowship. 
Cavendish L a b o r a t o r y , University of Cambridge. 

( * * ) On leave f r o m i s t i t u t o di Geofisica, Università di Bologna, and 
Istituto di Scienze della Terra, Università di A n c o n a . 



2 1 4 E . B O S C H I 

In this paper we consider this theory in the case of a hydrostatic 
initial stress field and derive the equations for the plane strain. The 
medium is assumed homogeneous and isotropic. By making use of 
the associated matrices method (5), we give a representation of Ga-
lerkin type. This representation enables us to obtain the solution 
corresponding to concentrated loads in an infinite medium for the 
case of stationary vibrations. Analogous problems have been studied 
in other fields. («• 2) 

BASIC EQUATIONS. 

In the following we employ a rectangular coordinate system Oxk 
and the usual indicial notations. The Greek indices are supposed to 
take the values 1, 2 and the Latin indices the values 1, 2, 3. 

Let 2 be a plane region occupied by the considered medium. 
We denote by sa the components of the displacement vector and 

by 0 i the perturbation in the gravitational potential. 
In the case of plane strain, we have: 

sa = sa («1, t) , 01 = 01 («1, »2, t), S3 = 0. [1] 

From the equations established by Dahlen (3), we can derive the 
following basic equations for the plane strain problem in the case 
of a hydrostatic initial stress tensor T°j = — p 0 S u (pa = const.): 
— the equations of motion: 

aa = [2] 

Q0 Si — 2 Q0 Q3 S2 = — go Zltl + p + ^ 

g0 Sj — 2 g0 Q3 = — go + Tp».p + 

— the constitutive equations: 

T n = (2 + 2 ft — 2 p0) si.i + (A — po) S2.2 

T 2 2 = (A — po) si.i + (A + 2 /i — 2 p0) s2,2 [ 4 ] 

Tl2 = (1 «2,1 + (fl Po) Si,2 

T2I = ¡1 Si ,2 + (fl Po) 82,1 

In the above equations, we have used the following notations: 

Tap - the components of the incremental pseudostress tensor; Fa -

the components of body forces; Q = (0, 0, Q3) - the steady angular 



ON T H E P L A N E S T R A I N I N A T H E O R Y F O R S E L F - G R A V I T A T I N G E T C . 215 

velocity rotation; A, ¡i - the appropriate constants of the material; 
Qo - the mass density; G - the gravitational constant; a comma 
denotes partial differentiation with respect to space variables and a 
superposed dot denotes partial differentiation with respect to the 
time t. 

Using equations [4], the differential equations [2] and [3] may 
be written as: 

p A + (A + p — 2 p . ) 

+ 2 Qo Q 

V-

3 %i2 

7) 

<)2 
• Qo 

5 t2 
« i + (A + p - 2 p a ) + 

0X1 0 X2 

5 t 
S2 — Po — 0! = F1 

Dxi 

(X+fi—2 p0) 
t)X2 

• 2 g o Qz 
5 t 

Si 

32 

DX22 

a2 
- Qo 

at2 
S 2 — p0 

a X2 

p A + (A + p — 2 Po) 

0 i = — ^ 2 [ 5 ] 

3 a 
4 71 Qo G • Si + 4 n Qo G - — s2 + A 0i = 0 

diCi 1)X2 

G A L E R K I N REPRESENTATION. 

Using the associated matrices method (5), we obtain the following 
representation of Galerkin type: 

si = p 
L-'2 1 V22 DX22 

i 2 

+ p Qo 

(A + p — 2 po) 

2 a 2 Qz a 2 

V22 *X22 

a 
+ 2Q0Q3 

ciXi 3,»2 ai 
A + 4 71 Po2 G " | A 

c hXl DX2 ) 

• 2 hXi V21 t)X2 i)t r3. 

(X + p—2p0) • — ^ 2QoQZ^-
r t>x2 ai 

A + 4 71 po2 G 
y-

+ p 

+ p Qo 

N 

• 

vi 2 a2 

V22 '¿Xi2 
A + A 

2 Qz a2 

2 V22 i>i 

D22 a^i2 

n 

A 

[6] 



21(5 E . B O S C H I 

01 = Ì7I Qo G [X • 3 2 Q 3 Ì>2 

4 71 Qo G ¡1 • 
2 ÛiCl V22 Ì>X2 ìli 

S 2 Q 3 i 2 

A 

A 

where 

• 

2 òa;2 ®22 ûa;i ì)i 

4 Q 3 2 ì>2 

1 2 Di2 «22 ~ì)t2 
r. 

L a2 
V ' A = ô a ,a2 

®12 = (X + 2 n — 2 Po) I go [ 7 ] 

®22 = [X j Qo 

The functions A = A («i, ìc2, ì) satisfy to the following equations: 

1 
A , 

where 

where 

D F a ~ fx (X + 2 ¿t — 2 p „ ) 

n n = o 

1 ¡>2 

M2 
t, 2 = „ 2 M 

fi 2 \ - i 
71 QOG 

[8] 

[9] 

STATIONARY VIBRATIONS. 

I n what follows, we assume that: 

Fa = Be j j 1 * (an, x2) 
icot 

In this case we seek the solution in the form: 

icot ] 

io)t 

a = Be l S (Xi, x2) e 

cpl = Re 10* (x„ x2) e 

[10] 

[11] 



ON T I I E P L A N E S T R A I N I N A T H E O R Y F O R S E L F - G R A V I T A T I N G E T C . 2 1 7 

From the Galerkin representation [6], by putting 

icut 
Pi = Be ITi (xi, xi) e 

we obtain 

« :=/* 

Qo 

(A — 2 Po) 

Vl2 a2 A 
4 71 Qo G a2 1 

V22 0X22 
n 

V22 a#22 ) 

+ 2 i Qo to 

• i)Xi 
~òXi ÒX2 

2 i ü3 co a 
V22 ÒX2 

A 

A + 4 7t Qo2 G 
a2 

ì)Xi ÒX2 
r 2 

a 

s* = -
2 (X + /t—2p0) 

a2 

+ P 

+ fi Qo 

* 2 i ' l 2 Ò 2 

*2 a 2 

P . . T 5 T + -

¡tei ~òX2 

A + 

' i Qo Qs CO 

V2 

0 * = — 4 71 Qo G fl 
, *2 () 
n 2 + 

4 71 Qo G a2 ) 
V22 a»i2 j 

a 
r* 13 òxi 
r* 13 

2 i Q3 co a 
V2 2 <1X2 

A + 4 7t £>o2 G——-—! A 
0x2 

r2 

r; 

4 71 go G u • a 2 Ì Ü3 CO a a * 
V22 

a * 

Po) • *2 *2 
1 2 

1 ih 2m2 
Po) • *2 *2 

1 2 Vi2 V22 

[12] 

where 

• *2 = zl + - - -^ a Va2 
[13] 

Tlie functions A {xi, x2) satisfy the equations 

1 
d * ra* = — 

fi (X + 2 ¡1 — 2 p0) 

d * a * = 0. 

Fn 
[14] 



218 

where 

with 

E . B O S C H I 

D' 
, , *2 , *2 • • 

1 2 

4:71 Qo G 

® 2 2 3 ' 

• " = A + co21 v32. 

[15] 

The operator D can be written in the form: 

D* = A (A — A-I2) (A — A-22), 

where fti2, 7i22 are the roots of the equation 

k" + fc2 TO
2 ft)2 + 4 TI G 

I H 2 " ^ Ï 2 2 " 
+ 

CO* / ft)2 

î)22 
4 TI OO G ft)2 

Vi- X>3 

[16] 

0 [17] 

EFFECT OF CONCENTRATED FORCES. 

L e t us examine the effect of a body force acting along the axis 
Xi. In this case, we have: 

F t = 0, [18] 

and from equations [14], we can take: 

r* = r3* = 0 

The solution of the problem is given by: 

r " V22 V2 i>X22 

4 71 Qo G 52 ) * 
1 1 ) 

1)2® ~ÒX22 ' 

«2 = (A + / * — 2 V o ) 
ì)Xi ÒX2 

2 1 n0 D-3 CO 
0Xl ÒX2 

[ 1 9 ] 

„ 1 . 3 co2 3 , 2 i Q3 co û \ _ * 
01 = ± 71 Qo G [x\A —— — — 1\ , 

r X ÏXl V22 ì)®! V22 i)X2 ' 



ON T H E P L A N E S T R A I N I N A T H E O R Y F O R S E L F - G R A V I T A T I N G E T C . 2 1 9 

where 1\" is the solution of the equation: 

A (A — 7ci2) (A — A-22) n* = 
[X (X + 2 ¡x — 2 p0) 

W e can write the solution of equation [20] in the form: 

r-r * 
1 1 = Gi & + 

7ci2 (fci2 —• fe2 (7.-i2 —/C22) ' 7ci2 À-2 

where the functions Gi satisfy the equations: 

1 
(A — V ) ©a = 

Zl (?3 = -

IX (X + 2 ̂  — 2 p0) 

1 

Fi* , 

¡x (X + 2 /t — 2 p„) 
A 

[20] 

(?3 . [21] 

[22] 

Let us consider the concentrated body force 

Fi* = 6 (xi)d(x2). 

In this case the functions Gi are given by: 

1 
Ga = 

G3 = 

2 n [x (X + 2 [x — 2 p0) 

1 

2 j r / i (2 + 2 , « — 2 po) 

j f o (fc0 r) 

In r 

[23] 

[24] 

where K0 (z) is the modified Bessel function of third kind, and 
r3 - xi~ + ®22. Using equations [21] and [24], the solution of equation 
[20], for concentrated body force, is given by: 

r * 
2 71 fx (X + 2 (X — 2 p0) 

i o (7ci r ) i i o (7c2 r ) 
fcl2 (fcl2 — fc22) fc22 (fcl2 — 7i22) 

Ad2 fc22 
In r [25] 

Thus, the solution of the considered problem is given by [19], 
where f \ * has the expression [25]. I n a similar way, we can obtain 
the solution for a concentrated body force acting along the axis Ox2. 



220 E. BOSCHI 

ACKNOWLEDGEMENTS. 

I wish to thank Professor A . H . Cook F.R.S. for his kind hospita-
lity in Cambridge, aud Professor M. Caputo for valuable discussions 
and suggestions. 

R E F E R E N C E S 

(X) BOSCHI, E., 1973. - Reciprocity theorem and elastic dislocation theory for an 
earth model with an initial static stress field. " J . Geophys. R e s . " , 7 8 , 8584. 

( 2 ) BOSCIII, E., 1973. - On the plane strain in a Generalized Theory of Ther-
moelasticity. Int. " J . E n g . Sci.," 1 2 , 433. 

( 3 ) T) AH LEX, F . A . , 1972. - Elastic Dislocation Theory for a Self-Gravitating 
Elastic Configuration with an Initial Static Stress Field. " G e o p h y s . 
J. R . Astr. S o c . " , 28, 357-383. 

( 4 ) IESAN, D., 1908. - On the plane coupled micropolar thermoelasticity. I , 
" B u l l . A c a d . Polon. S c i . " , Scr. sci. Teclin., 1 6 , 379-484. 

( 5 ) MOISIL, Gr. C., 1952. - Teoria preliminara, a sistemelor de ecuatii cu de-
rivate partiale lineare cu coeficienti constanti, Boletinul Stiintific al 
A c a d . R. P . R., Seria A , 4, 319-326. 

( 6 ) NOWACKI, W . , 1904. - Green functions for the thermoelastic medium. I I , 
" B u l l . A c a d . Polon. S c i . " , Ser. sci. Teclin., 1 2 , 405-472.