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WHY ROTATION SEISMOLOGY:
CONFRONTATION BETWEEN
CLASSIC AND ASYMMETRIC
THEORIES
 Need of Modern Seismological
World Network
Roman Teisseyre
Institute of Geophysics, Polish Academy of Sciences,
ul. Ksiecia Janusza 64, 01-452 Warszawa, Poland
e-mail: rt@igf.edu.pl
1
• Classical elasticity has many insufficiencies.
• Angular motions are not incorporated in it; only
artificialy we can introduce it defining a point and arm
of the moment
Numeroeus attempts to improve this theory:
• Cosserat brothers’ theory with displacements and
rotations (Cosserat, E. and F. 1909)
• The micropolar and micromorphic theories (Eringen
and Suhubi, 1964; Mindlin, 1965; Nowacki, 1986;
Eringen 1999, 2001): a powerful tool for many
complicated material problems (Jones 1973); these
theories seem to be sometimes too complicated and
therefore are not in common use in the seismological
studies (some examples of seismological application,
see: Teisseyre, 1973; 1974).
Example of some unsolved problems faced by the
classical elasticity:
2
● Angular motions introduced artificially with
length and reference rotation point (angular
momentum balance ↔ symmetric stresses)
● Fault slip solutions rely on the additional
friction constitutive law along a fault
● Advanced deformations, like granulation and
fragmentation - not included
● Earthquake fracture geometry reveals an
asymmetric pattern with the main slip plane
(premonitory micro-fractures are probanly also
asymmetric)
● Edge dislocations present asymmetry in
relation to the strain components
● Direct differential relation for the density of
edge dislocations and stresses cannot be
3
derived in the classical theory
ASYMMETRIC THEORY:
Conventions used
•
NOTE: vector field could be replaced by antisymmetric tensor field :




ik 
 ikss ;





or in vector notation, this tensor is :     

 
•
•
when indexes are ordered (1,2,3; 2,3,1; 3,1,2),
missordered or repeating
 iks  1, 1, 0
•
NOTE: any symmetric tensor can be split into the axial and deviatoric
tensors.
•
D(ns)  ns D  D
A
kk
•
•
D
ns
:
D
A
(kk ) =
Dkk
and
Summation convention for repeating indexes:
D
1
Dns  ns
3
D
ns =
Dkk   Dkk
k
Dkk
4
Material destruction is a very complicated surely nonlinear
process; many fields are released, some are inter-related.
However, it is much more complicated to understand the earthquake
events as, in this case, our knowledge of the external conditions is
still very limited.
Therefore, it is very important to refer to the boundary or
initial conditions: these are related to strains: axial ,
deviatoric and moment related (rotational).
Seismometers record not directly displacements but deformations,
which may be related either to real displacenents or to strains:
u k
u k 
 xi  u k    u k
 xi
 xi
where
is a length of a seismometer platform much more rigid
than soil layer
Displacements, u, appear as integrating effect.
5
Fundamental Point Motions and Deformations
The problem of rotation waves becomes again actual due to
the recent observations based on very precise instruments
able to measure very small rotation motions and due to
development and new approach to the continuum theory.
We start our consideration recalling that between
serious defaults of the classical approach, like the known
problem how to include into continuum theory the angular
motions and related moments, there is a necessity to
include there the constitutive laws to account for response to
the applied moments and angular motions in continuum.
WE
SHALL
POSTULATE
THAT
EACH
DEFORMATION FIELD SHALL BE DESCRIBED
BY THE INDEPENDENT
BALANCE RELATION !!!
6
Axial point deformation (scalar), (in 3D, 2D, 1D):
Shear point deformation: string-string
(vector ┴ to the plane )
7
LEFT: ROTATION motion, (vector ┴ to the plane); its
velocity means spin;
RIGHT: SHEAR AXES OSCILLATIONS (vector ┴ to
the plane) this SHEAR_TWIST is equivalent to
STRING - STRING basic deformation
8
Shear-Twist means the rotational oscillation of the main
shear axes with the changes of shear magnitude as
caused by the internal fractures)
9
Three Reference Motions (3): displacements; (or rotations
3) are equivalents: we will refer to displacements u): remains 3
Seven STRAINS (7): Axial Strain – 1; Shears (strikestrike) - 3 anitisymmetric strains - Rotations – 3;
All deformations appear due to specific load conditions: rotation
strains (external moment or internal friction); axial strain – to
external pressure (or dilatancy); strike-strike deformation – to
external shears
All together we have 10 independent fields:
ALL these fields shall have the independent balance relations !!
Some of them may be inter-related as all deformations
could be refer to the unique diaplacement (or rotation) field or
related to a number of independently generated
displacements or directly appearing due to load stress 10
conditions (symmetric or, and antisymmetric – moment related)
Basic point motions(related to the Planck dimension) - given by vectors:

DISPLACEMENTS, u, and ROTATIONS,
ω
present the reference fields; we may refer to DIFFERENT u !!!
Strain defermations:
us
E( ss)  E 
xs
AXIAL STRAIN (refering to: u )
SHEAR STRAIN (refering to: uˆ ) :


ˆ
ˆ
ˆ

u

u

u
1
1
k
i
s
E  Eˆ(ik )  




ik
2

x

x
3

x
i
k
s


(
ROTATION STRAIN (refering u
to:
):
D
(ik )
(
1  uk ui 
E[ik ]  


2  xi xk 
Displacement rotation means the antisymmetic strain !
11
ELEMENTARY FRACTURE PROCESS
can join some different reference displacements:
(
u , uˆ , u
RELEASED CONSECUTIVELY WITH PHASE SHIFTS:
(
0
us   us , uˆs  e us , us   us
0
0


0


,  , e   0,  1,  i
0
0
We may refer these motions to a unique slip
displacements, u, released in a final stage of an
earthquake process:
The phase shift constants include both the advanced or
delayed motions in an elementary source process.
12
Strain deformations - unique reference field
E ( ss )  E ( ss )
u s
0 u s


x s
xs
                      
1  uˆk uˆi
 

2  xi xk
 1 uˆs

   ik
 3 xs
E
 Eˆ ( ik )
 
 1  uk ui  1 us 
e  


   ik
2  xi xk  3 xs 

                  
(
(
(
ui 
1  uk ui 
0 1  uk
E[ik ]  


 

2  xi xk 
2  xi xk 
D
( ik )
0
 

double
ii 
rebound process:
1
e , 
0
0
,
0
  0, 1,  i
13
ASYMMETRIC THEORY: STRESSES and STRAINS:
Skl  S( kl )  Sik 
D
(ik )
E( ss )  E ; E
Eik =E(ik ) +E[ik ] :
(
(
 Eˆ ; E  E
( ik )
ik 
ik 
Symmetric part of stresses: we take the classical constitutive law and
supplement with the Shimbo law for the antisymmetric part:
A
ss
S(kl)  kl Ess  2Ekl  S   3  2 E
(
D
ˆ
S(ik )  2 E(ik )  Sik   2 Eik 
In these constitutive relations we put – for simplicitythe same constants as for shears and rotations
14
FOR ANY REFERENCE DISPLACEMENTS
 2 us
 2 un
2
(   )

  2 un  F
xn xs
xs xs
t
After differentiating: for derivatives:
 Fn Fl 
3uk
2  ul un  2  ul un 

  
     2    2(  )
xkxk  xn xl  t  xn xl 
xnxlxk  xl xn 
Adding (or substracting) another with interchanged indexes n,l
2
2
2
n
n
s
n
2
k k
l
l
n l
s
l
                      
 u
 u
 u F


(  )

x x x
t x
x x x x
2
2


ul un

  ul un   Fn Fl 

 
    2    
xk xk  xn xl 
t  xn xl   xl xn 
15
C
l
a
s
AXIAL
 Fs
2
2
(  2  )
E ( ss )  
E ( ss ) 
2
STRAINS:
xs xs
t
xs
2
2



E( nl )   2 E( nl ) 
SYMMETRIC STRAINS:
xs xs
t
i
n
f
i
e
l
d
r
o
t
a
t
i
o
n
2
1  Fn Fl 
(    )
E( ss )  


xn xl
2  xl xn 
 2E(ss) nl 2E(ss) 
 2Eˆ(nl)  2Eˆ(nl)

     


2
DEVIATORIC x x

t
xnxl 3 xsxs 
s s

STRAINS:
1  Fn Fl  nl Fs
= 
 
2  xl xn  3 xs
ROTATION STRAINS:
2
2


1  Fn Fl 

E[ nl ]   2 E[ nl ]  


16
xs xs
t
2  xl xn 
F
o
r
•
,
C
l
a
s
 Fs
2
2
(  2  )
E ( ss )  
E ( ss ) 
2
xs xs
t
xs
i
n
V  VP

AXIAL STRAINS:
f
i
e
l
d
2
2
2
1  Fn Fl 

E(nl )   2 E(nl )  (  )
E( ss)  


xsxs
t
xnxl
2  xl xn 
r
o
t
a
t
i
o
n
V  VS

2
2


1  Fn Fl 

E[ nl ]   2 E[ nl ]  


xs xs
t
2  xl xn 
•
F
o
r
STRAINS:
,
ROTATION STRAINS:
V  VS
17
REPEAT
 Fs
2
2
(  2  )
E ( ss )  
E ( ss ) 
2
xs xs
t
xs
AXIAL STRAIN :
DEVIATORIC STRAINS:
2 Eˆ(nl )
:
 Fs
 2 u s
 2 u s
(  2  )


2
xs xs xs
t xs
xs
2 Eˆ(nl )
 2 E(ss) 1 2 E( ss)  1  Fn Fl 2 Fs 


      
 nl

 nl
  
:
2

xsxs
t
 xnxl 3 xsxs  2  xl xn 3 xs 
2  uˆn uˆl 2 uˆs 
2  uˆn uˆl 2 uˆs 


 nl

 nl

 2 

xsxs  xl xn 3 xs 
t  xl xn 3 xs 
 2
1
2  us  Fn Fl 2 Fs 
2(  ) 
 nl


 nl


 xnxl 3 xsxs  xs  xl xn 3 xs 
ANTISYMMETRIC (ROTATION) STRAINS :
2 (
2 (
1  Fn Fl 

E[nl ]   2 E[nl ]  
 :
xsxs
t
2  xl xn 
(
(
(
(
2  un ul 
2  un ul   Fn Fl 

  2 
  
 

xsxs  xl xn 
t  xl xn   xl xn 
18
Release Rebound Fracture Theory:
• e.g.,: first: break of molecular bonds
(rotation)
• then:
phase shifted delayed slip
• Theory shall help to discriminate such
inter-related motions
• Elementary process shall follow this
release rebound theory:
19
Three wave fields (at F=0) :
2
 E
E  2
 0
VP t
 Eˆ(nl )
2
2
2



1

Eˆ( nl )  2        
 nl
 E( ss)
VS t
 xnxl 3 xs xs 
(
E[nl ] 
(
 E[nl ]
2
2
S
V t
2
0
20
At :
 Eˆ( nl )
2
| Eˆ( nl ) 
2
S
V t
0 ,
E  0
(
E[ nl ] 
(
 E[ nl ]
2
2
S
V t
2
0
In 4D The homogeneus wave fields equivalent to
Maxwell – like relations
( Eˆ
rot E 
0 ;
VS t
(
E
ˆ
rot E 
0
VS t
Remark: These two relations are equivalent when the RELEASEREBOUND process in source runs with the phase shift
(
E  i Eˆ
±π/2 :
21
• shear-rotation interaction in waves
22
Experimental data
• Some experimental data confirm appearence of the
correlated motions
• (especialy between shears and rotations )
• with immediate correlation or with phase shift π/2
•
•
or
(
ˆ
E E
(
ˆ
E iE
or at the double rebound process:
(
E  iiEˆ   Eˆ
on POSTER by Krzysztof P. Teisseyre
23
Defect Induced Stresses
(rearrandgement of stresses)
• The important relation (M. Peach and J.S.
Koehler, 1950, The Forces Exerted on
Dislocations and the Stress Fields Produced by
Them, Phys. Rev. 80, 436–439) defines the
• forces exerted on dislocations (Peach and Koehler,
1950)
• Considering the continous defect fields we define
the induced stresses under the applied stress
system and internal defects

• Rearrendgement of the stress system
24
(
For the axial, S , shear, Sˆ , and rotation stresses, S
F
n


n
s
q
S
s
k
b
k

q
, applied we obtain the generalized forces action
on defects das defined by the Burgers vector and
versor of defect edge: bk q
1
Fn   nsq S  sk bk q =  nkq Sbk q ,
3
Fˆn   nsq Sˆ( sk ) bk q ,
(
(
Fn  M n   nsq S [ sk ]bk q
25
F
n


n
For the defect density
s
q
S
s
k
b
k

q
 qk   q bk
We define the INDUCED stresses
IND
np
S
LOAD
p sk
 Fn np   nsqqk n S
1
S   nkq S  qk n p ,
3
IND
ˆ
Snp   nsq S( sk ) qk n p ,
(
IND
Snp   nsq S[ sk ] qk n p
IND
np
26
Particular CASES
DEFECT PLANES PARRALLEL TO EARTH’ SURFACE
1
S   nkq S  qk
3
IND
ˆ
S n 3   nsq S( sk ) qk
(
IND
S n 3   nsq S[ sk ] qk
IND
n3
27
1
1
S

S  zy 
S  yz
Planes
3
3
parallel
1
1
IN D
S yz 
S  xz 
S  zx
3
3
to
1
1
IN D
S zz

S  yx 
S  xy
surface
3
3
S xzIND  Sˆ( yx) zx  Sˆ( zx ) yx  Sˆ( yy ) zy  Sˆ( zk ) yy  Sˆ( yz ) zz  Sˆ( zz ) yz
IN D
xz
S yzIND  Sˆ( zx) xx  Sˆ( xx ) zx  Sˆ( zy ) xy  Sˆ( xy ) zy  Sˆ( zz ) xz  Sˆ( xz ) zz
S zzIND  Sˆ( xx) yx  Sˆ( yx ) xx  Sˆ( xy ) yy  Sˆ( yy ) xy +Sˆ( xz ) yz  Sˆ( yz ) xz
(
(
(
(
(
(
S  S[ yx]zx  S[ zx] yx  S[ yy]zy  S[ zy] yy  S[ yz]zz  S[ zz] yz
(
(
(
(
(
(
IND
Syz  S[ zx]xx  S[ xx]zx  S[ zy]xy  S[ xy]zy  S[ zz ]xz  S[ xz]zz
(
(
(
(
(
(
IND
Szz  S[ xx] yx  S[ yx]xx  S[ xy] yy  S[ yy]xy  S[ xz ] yz  S[ yz ]xz
IND
xz
28
1
1
S
 S  zy  S  yz
3
3
1
1
IN D
S yz  S  xz  S  zx
3
3
1
1
IN D
S zz  S  xy  S  yx
3
3
IN D
xz
S
IND
rz
IND
z
S
S
IND
zz
S
IND
rz
 Sˆ( k ) zk  Sˆ( zk )  k
S
IND
z
 Sˆ( zk ) rk  Sˆ( rk ) zk
S
IND
zz
 Sˆ( rk )  k  Sˆ( k ) rk
(
(
 S[ k ] zk  S[ zk ] k
(
(
 S[ zk ] rk  S[ rk ] zk
(
(
 S[ rk ] k  S[ k ] rk
29
For rotational squeeze we put
n
1
1
S  S  z n  S  z n
3
3
IND
S
 Sˆ( zr ) rr n  Sˆ( rr ) zr n  Sˆ( z ) r n  Sˆ( r ) z n  Sˆ( zz ) rz n  Sˆ( rz ) zz n
(
(
(
(
(
(
IND
S z  S[ rr ] r n  S[ r ] rr n + S[ r ] n  S[ ] r n + S[ rz ] z n  S[ z ] rz n
IND
r
AND FOR LOAD:
1
1 ˆ
p   S   S( rr )  Sˆ( zz )
3
3

1
1
S

S  z n  S   z n
3
3
IN D
S   Sˆ ( zz ) rz n  Sˆ ( rr ) zr n
(
IN D
S z  S [ rr ]  r n

IN D
r
30
We have in
particular
S  S rr  S  S zz
We note an appearance of the significant component
E
in strain measurement related to earthquake events
(Gomberg and Agnew, 1996). The authors original
statement was that this squeeze strain angular squueze
component can be estimated from the solution for the

displacement potentials. However, the obtained
asymptotic solution of the wave equations, for the scalar
and vector potentials of displacements, indicated that
31
such field according to theory is negligable
E
IND
S
 S( zz )
n  S( rr )
rz 
However, according to the asymmetric theory this
component
E
shall be estimated from wave equation for the
independent wave equation for the axial strains.
For the case with a pressure gradient (z-direction)
we may have an angular symmetry and angular
squeeze in the plane parallel to the Earth surface.
We can obtain the following obtain the following
asymptotic solution of the wave equations for this
component in the cylindrical coordinates (r,φ,z)32
The required solution for the axial strain may be
expressed directly by an expansion of the Bessel
functions and the cylindrical harmonics; we obtain
significant result :
LOAD
E
S

3  2
2
exp i  kr r  k z z   t   / 4  
 kr r
assuming the same constants for angular squeeze.
33
Reasumming our theretical assumptions
related to the Asymmetric Continuum
Theory explains also the discovered
experimentaly
significant squeeze
cylindrical strains.
E
34
• Our final remarks concern a need
to create the modern worldwide
seismological network to record
the important and independent
strain fields;
• To this end we present sensitivity
and discrimination ability of the
different sensors.
35
Sensitivity of the different sensors
Source
Processes
and Fields:
Seismometer Seismic
Array
Rotation
Sensors
e.g., Sagnac
type
Rotation
Strain-meter
seismometer Array
Displacement
X
x
-
-
? Scale ?
Rotation Strain
X
x
x
x
-
Shear Strain
X
x
-
x
x
Axial Strain
X
x
-
-
x
•
Rotation seismometer: two pendulums used to record the rotational motions: two
perpendicularly oriented rotation seismometers detect the intensity of rotations).
36
Discrimination ability for motions or deformations
Source
Processes
and Fields:
Seismometer Seismic
Array
Rotation
Sensors
e.g., Sagnac
type
Rotation
Strain-meter
seismometer Array
Displacement
-
x
-
-
-
Rotation Strain -
x
x
x
-
Shear Strain
-
x
-
x
x
Axial Strain
-
x
-
-
x
37
• These tables show how poor equipment
we have in seismology !!!
• Seismometer records do not discriminate
and explain exactly to what deformations
the recorded displacement may belong:
some information come from recorded
time (P, or S or other seismic phases) but
still we do not know if these motions
belong to real displacements or to axial
stresses or shears or rotational strains !
38
Example: Element of
Simple Network
W
W
d
W
W
The 4 – squares mean the 3 – component
seismometer system; circle - control rotation sensor
39