T h e Mechanism of the Earthquakes 

in T e r m s of the Dislocation T h e o r y 

S. D R O S T E - R . TEISSEYRE 

Da ring the last years there originated quite a number of model 
theories of seismic foci. In such models the dynamic proces in the focus 
is substituted by the action of a classic system of equivalent mass forces. 
Among works on this subject the publications of the school of W . I. 
Keylis-Borok, as well as of H . Honda should be mentioned in the first 
order. 

Let us consider the simplest model of a focus, that is the case of 
a simple point force. The expression for displacement ii in such a case 
is given by the classic Love's formulas. The dependence of the force 
from time is here expressed by an arbitrary function x (t). If this func-
tion will be discontinuous, as for example in case of a step function 
(x = 0 for t < 0, x = 1 for t > 0), than Love's formula for u determines 
the creation of a infìnitesimal dislocation element at time t = 0. This 
problem is discussed in detail by Nabarro (10). 

In this manner model theories represent the creation of dislocation 
elements for such cases in whicli the time dependence of active forces 
is discontinuous. 

To this category there belong also the publication of A. Wwie-
denskaja (I 2) on the mechanism of earthquakes, represented as a crea-
tion of a finite dislocation of the disc type. Formally her work differs 
from the preceding group, because of direct use of the theory of physical 
dislocation. 

The localisation of the dislocation created — should be determined 
by the stresses occurring in the medium however no such relation is 
clearly given by the theories discussed above. 

W e should like to present a new theory of the mechanism of earth-
quakes. The starting point of this theory is the assumption, that there 
exists a nonhydrostatic part of stress in the medium, as well as a field 
of inhomogenities in the largest sense. Such inhomogenities, even very 



180 S. D R O S T E - l l . T E I S S E T R E 

small in size, may serve as points of attachment. They institute the 
nucleus of dischargement of the strain energy occurring in the medium 
and together with stresses they determine the earthquake's area. 

These obvious assumptions may be considered further in the light 
of the theory of physical dislocations, which may be interpreted as the 
theory of inhomogenities in a field of forces. The theory of physical 
dislocation was developped in the solid state physics. 

Before going further on we will stop for a short while on basic de-
flnitions and consequences of the physical dislocation theory. The sim-
plest example is represented by linear dislocation. Classically is will 
be defined, as foUows: a cut is executed along a chosen halfplane and 
afterwards the material on both sides of the cut is displaced relatively 
by a Constant weetor b (what is called Burgers' vector). The edge of the 
dislocated halfplane is called dislocation Une. 

The main part of the selfenergy of dislocation, determined as the 
energy of the deformation created, is acumulated in the neighbourhood 
of the dislocation line. The displacement of the dislocation line gives 
an increase in the dislocation area. 

I t may be said that the potential energy of physical dislocation 
expresses the tendency of the whole medium's deformation. This poten-
tial energy is liberated by the movement in this moment when the dis-
location reaches the medium's boundary. 

In order to defìne fully dislocation Burgers' sing convention which 
serves to determine the relation between the circulation along the dis-
location line and the direction of relative displacement of the medium 
on both sides of dislocation should be used. 

The dislocation vector b which is situated in the cut piane (which 
is called also the slip piane) may be parallel to the dislocation Une — 
in this case the dislocation is of the screw type, or it may be vertical to 
the dislocation line in which case we caU the dislocation edge dislocation. 

The external stress field exerts it's influence on the dislocation 
through interaction with the field of dislocation. This influence may be 
expressed as the action of force on an element of the dislocation Une da 

F = ( P -b) X da [1] 

where P being the stress tensor. 
Similary tliere exist an interaction between dislocation. Dislocations 

of opposite signs are attracted to each other, of the same signs are being 
repulsed. These forces are inversely proportional to distance between 
them. 



T I I E M E C H A N I S M OF T H E E A R T Q U A K E S , E T C . 1 8 1 

Dislocation which have curved dislocation lines, especially, closed 
ones, will serve as a generalisation of dislocations discussed above. Dis-
locations having curved closed dislocation lines are called loop dislo-
cations. W e will consider now very small loop dislocations. These liave 
interesting properties, which deserve more detailed study. 

W e are considering now dislocations situated in a common slip 
piane. Loop dislocations of the same signs are now attracted to eacli 
other. This is caused by the fact that the elements of dislocations si-
tuated nearest one to each other have opposite signs. A s the result of 
such atraction two loop dislocations may approach each other and after-
wards join into one. A s a consequence of such a synthesis a dislocation 
comprising a greater area is created. Different kinds of dislocations 
m a y arise by means of a chain of such syntheses. 

N o w to the basic assumptions of the present work we will add that 
the value of the nonhydrostatic part of the stress field and density of 
inhomogenities of the medium are extrenally great in the neighbourliood 
of a suppossed piane (y = 0). A s we will see this piane and it's neigh-
boui'hood constitute the stage upon which dislocative processes and 
accompaning earthquakes take place. This piane, together with it's 
neighbourhood institute the area of seismic activity. I t may be called 
also the hypocentric piane. 

L e t us assume that the nonhydrostatic stresses are represented 
by the shear stress pyz. N o w we can proove tliat the Constant fìeld pyz 
on the piane may be represented by an equal distribution of loop micro-
dislocations characterized by Burgers vector b, and by normal vector 
of the direction — y. If n will represent the number of microdislocations 
for an unit of surface and o0 will represent their radius, than we will 
obtain: 

Here we can go to the limit n = oo, bz 0, q„ -> 0, but under 

the condition lim = const. 

Assuming the existence of inhomogenities on the considered piane, 
bigger loop dislocations will arise around them under the influence of 
the field pyz. A s an example let us consider the extreme case of inho-
mogenities in the forni of small incompressible intrusions. Inside the 
area occupied by such an intrusion no microdislocation will appear 
because of the incompressibility of the medium constituting this intru-

do 



182 S. D R O S T E - l l . T E I S S E T R E 

sion. Instead there appears a bigger loop dislocation around the intrusion. 
In this manner inhomogenities quantize, as it were, the field pyz. The 
loop dislocations created around inhomogenities we will consider as 
dementar loop dislocations. I t may be said that dementar loop dis-
locations represent inhomogenities in the field pyz. Later the action 
of a sufficiently great field pyz and the interaction between dislocations 
causes their deformation by an increase in surface which dislocation 
comprises. 

Now we must assume that a fluctuation in density of inhomogenities 
exist on the considered piane. I t could be now said also in other words 
that there exist a fluctuation in the density of loop dislocations. In the 
region of the maximal concentration of inhomogenities, there arise a 
greater dislocation by it's own growth and by syntheses with neigh-

Z 1 

U A 
®òz 4 

U A 
®òz 4 t t , ^ t X 

Fig. 1. 

bouring dislocation. The creation of greater dislocation elements may 
be of violent nature giving source to seismic shocks. I t should be men-
tioned, however, that the greater part of the work of the field pyz is 
consumed by an incorease in internai energy of the dislocation. Therefore 
it is to note, the earthquakes caused by violent creation of dislocations 
or of their movements will belong to the weaker category. The dislo-
cation created in the manner described above may be represented by 
a pair of edge dislocations limited by sides of screw dislocations (fig. 1). 
In an extreme case we obtain a pure pair of edge dislocations infinitely 
long, composed with two dislocation of opposite signs. 

Further under the influence of the field pyz the dislocation 1+ and 1~ 
go asunder. This may occur under the condition that, the forces of the 



T I I E M E C H A N I S M OF T H E E A R T Q U A K E S , E T C . 1 8 3 

field pyz must be greater, than the force of attraction, between the pair 
of dislocations (forces are usually expresses for an unit of length). 

W e will now pay attention to particular cases of dislocative proces-
ses on the discussed piane. Firstly we will examine the case when there 
exists one center of generation of dislocations. L e t us assume that it 
is concentrated along the line " 0 " (fig. 2). 

A 

Fig. 2. 

Many pairs of dislocations are created around the line " 0 " as the 
result of Constant action of the field pyz. Each of these pairs go asunder, 
as it was described above. I n consequence a situation as on drawing is 
created. I n such case the neighbouring dislocation 1+ and 2+ are of the 
sanie sign and are repulsing each other. This leads, as it were, to " jost-
ling " of the extremely situated dislocation by dislocations situated near 
centre " 0 " towards external directions. 



1 8 4 S. D R O S T E - R . T F I S S E T R E 

When there exists also the second center " 0' " of generation of 
the dislocation ( 3 + , 3~), then the dislocation 1 + , 3~ will attract and 
approach to each other. 

In the area ( " b " ) the dislocations 3~ and 1+ will join and annihilate 
themselves, with accompanying dischargement of their energy. W e 
assume that initially the approach is slow, when they reach the distance 
L a violent movement will occur and the dislocations will join and di-
sapear. Their energy at distance L is (Koehler (3)) 

where a being Poisson's coefficient, r„ being a radius of the dislocation 
Une (a small value), la the length of the dislocation. 

The expression [3] presents the selfenergy and interaction energy 
of the dislocation 3~ and 1 + . In the result of the movement the 
energy [3] is partly converted into kinetic energy and by the confluance 
of dislocations is discharged in the whole. The value of energy 
discharged will be of the same order of magnitude as in formule [3]. 
The work of the field pyz and of resistance forces was not taken into ac-
count. The displacement ù" originating from the annihilation of the 
dislocations may be expressed, as follows: 

where us reoresents the static field of a pair of edge dislocations 
(tabarro (10)): 

[3] 

- a - s - c 
U = U U [4] 

ut = 0 
X 

z 

u' 

(k - 1 - ~ , + j 



T I I E M E C H A N I S M OF T H E E A R T Q U A K E S , ETC. 1 8 5 

and uc represents the field of creation of a pair of edge dislocations (13): 

u° — 0 , 
x ' 

1 c2 , _/ y2at , Aat 4y2Aat\(, r 
K ~ — — H for — < < 

Y TI a2 \ R*A rl r" ) \ a 

1 , _ / y2 et , Get iy2Cct , 1 et \ /, r \ 

- • £ - ^ o + -F - - V + 1 - ( 1 E o r < '1' c 

l e 2 , ( z2at , Aat 4z2Aat 
K = - 7.1T -I- - U ^ f o r - < t -n a2 \ r1 A r4 t 

r 

a 

1 , r ( z2 et Gct 4 z2 G et , 1 et \ L r \ 

~ ~ - ^ + ^ + J WG ( f o r T ^ V 

( A = j/a212 — r2 , G = |/c2<2 — r2 ) . 

The creation (annihilation) takes place at the moment t = 0. 
The annihilation of the dislocation 1 + and 3~ is accompanied by 

the confluence of the dislocation areas limited by them into one defor-
mation area. N o w we must pay the attenuation to the region " a " near 
the Earth's surface. 

The influence of the free surface may be expressed by means of the 
well known image method. The field of the dislocation in the presence 
of free earth surface may be expressed as the total of the dislocation 
field and it's image, the image being a dislocation of opposite sign. Such 
resultant field leads to disparition of stresses on the earth surface. I n 
such a manner the action of the free earth surface may be represented 
as the interaction between the dislocation and it's image. This action 
manifests itself always as an attraction of the dislocation to the earth 
surface. Naturally this will concern in first order the dislocation 3 + and 
after 2 + . 

I n consequence of movements of dislocation 3+ the dislocation 
area approach to the Earth's surface. When the dislocation reaches it, 
then it's stresses will be released. A s in this case the field of stresses 
equals the total of the dislocation and it's image, the energy of dischar-
gement will be expressed as in the former case but with a coefficient 
equals to 1/2 

* 4 r ^ - - i c r b r ) z ° v ( 5 ) 



1 8 6 S. D R O S T E - l l . T E I S S E T R E 

where L represents now the distance between the Earth's surface and 
the dislocation at this moment, when the rapid movement of the dis-
location in the direction of the surface begins. One would say, at this 
moment when the dislocation is about to leap to the earth surface. 

I n the area " c " the dislocation 2~ moves deep into the Earth ac-
companied by the deformation of the medium. I n such a manner the 
slip increases also in this direction. I t may be postulated that in deep 
strata of the Earth there will also occur a dischargement of dislocation 
stresses. The character of this dischargement, however, may be more 
graduai and it is difficult to say something precise about it at this moment. 

I will amplify the above, presenting a f e w data concerning the 
earthquakes of type " a " that is to say of this type in which the dischar-
gement of energy take place on the Earth's surface and the deformations 
represented by dislocations reach this surface. 

For example let us consider several great earthquakes for which 
the length and magnitude of slip of dislocations on the earth surface 
are known (Table 1). 

The mechanism of earthquakes presented above constitutes at the 
same time a theory of origin of dislocations. From the above conside-
rations follows that they originate as the result of displacement of phy-
sical dislocations along a given piane. The physical dislocations are 
connected closely with the existance of stresses and inhomogenities 
of the media. 

Table 1 

Earlhquake 
Length 
of slip 

lo 

Magnitudo 
ni slip 

K 
Energy (lormula |5|) 

Magnitudo 
of 

earthquake 
M 

Ennrgy Irnm eq. : 

logE = U+l,6M 

California 450 km 7 m 2 1524 d erg 8 1/4 1,6 IO24 erg 
1906 

8 1/4 IO24 erg 

Nevada 30 km 5 m 1- IO23 d erg 7 3/4 2,1 IO23 erg 
1915 

7 3/4 IO23 erg 

Mino-Ovari 65-120km 7 m 3-5 IO23 d erg 7 1/2 1,0 IO23 erg 
1891 

d erg 7 1/2 IO23 erg 

Assam 20 km 12 m 2,5- IO23 d erg 8 1/2 4,0 IO24 erg 
1897 

8 1/2 IO24 erg 

d = lo 
T, 1 

d = lo cr — d = lo 
° r 0 4(1 - o r ) 

" — • — 

d being of order 2-4 



T I I E M E C H A N I S M OF T H E E A R T Q U A K E S , E T C . 1 8 7 

To the end of this communication we shoud like to add that the 
field pyz acting in the neighbourhood of the piane y = 0 characterized 
by great inhomogenities, acts also on neighbouring elements of the 
surfaces z = const. The processes occurring in these planes will be of 
course less important and may be considered as only accompaning the 
basic plienomena along the piane y = 0. 

RIASSUNTO 

Viene esposto un nuovo meccanismo per terremoti, alla luce della 

teoria delle dislocazioni. I presupposti base sono resistenza di una parte non 

idrostatica delle tensioni nel mezzo (py.) e una certa distribuzione delle 
inomogeneità. Questa distribuzione determina un piano e il suo intorno, 

come il piano del processo di dislocazione attiva (piano di slittamento, 

y = 0). 
Si dimostra che le inomogeneità possono essere rappresentate da dislo-

cazioni a cappio, in presenza del campo pyz. Le dislocazioni a cappio 

possono aumentare sotto l'influsso del campo pyz e sotto l'interazione fra 

le medesime (sintesi dei cappi). Come risultato, conseguono coppie di dislo-

cazioni torcenti e di confine. Allora le dislocazioni della stessa coppia sono 

costrette, dall'intenso campo pyz, a procedere separatamente. L'area di scor-

rimento, in tal modo, aumenta. La fusione delle due dislocazioni associate 

alle coppie vicine, deternami il loro annientamento e la liberazione della loro 

energia, il che provoca un ulteriore .aumento dell'area di scorrimento omo-

geneo. Similmente, quando la dislocazione raggiunge la superficie terrestre, 

la sua energia verrà liberata (nel qual caso, va preso in considerazione il 

fattore % nci confronti con il caso precedente). 

Il confronto dell'energia liberata, così determinata, è in buon accordo 

con i dati di osservazione. 

ABSTRACT 

A new mechanism of earthqualces is evolved in the light of the physical 

dislocation theory. The basic assumptions are the existence of a nonhydros-

tatic part of stresses in the medium (pyz) and a supposed distribution of 

inhomogenities. This distribution determines a given piane and it's neigh-

bourhood, as the piane of active dislocation processes (slip piane, y = 0). 



188 S. D R O S T E - R . T E I S S E Y R E 

It is shown that the inhomogenities could be represented by the loop 

dislocations in the presence of the field pyz. The loop dislocations may 

increase under the influence of the field pyi and under the interaction be-

tween themselves (synthesis of loops). As the result, there arise pairs of the 

screw as well as edge dislocations. Then the dislocations from the same 

pair are constrained by the strong field pyz to go asunder. In such manner 

the slip area increases. The blending of the tivo dislocations from the neigh-

bouring pairs give their anniliilation and dischargement of their energy, 

this is accompanied by further increase in tlie homogeneus slip area. Si-

milarly when the dislocation reaches the Earth's surface it's energy will 

be discharged (noiv the factor 1/2 in the comparison with the preceding case 

must be taken into account). 

The comparison of the dischargement's energy is in good agreement 

with the observation data. 

R E F E R E N C E S 

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( " ) F R A N K , F . C . , Proc. Phys. Soc. A.* 6 2 , 1 3 1 ( 1 9 4 9 ) . 

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( « ) K E Y L I S - B O R O K , W . I . , D. A. N. USSB., 7 0 , 6 ( 1 9 5 0 ) . 

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v