D E F O R M A T I O N I N A C ON T I N U OU S M E D I U M
Content
Introduction
One dimensional strain
Two-dimensional strain
Three-dimensional strain
Conditions for homogeneity in two-dimensions
An example of deformation of a line
Infinitesimal deformations
INTRODUCTION
Deformation is the response of a body of rock to stress. Because we are considering the elastic
wave field we will consider deformations that too small to reach the yield strength of the rock and
change shape or size permanently. We describe a general deformation field as a function of position
in three-dimensional space:
u  u( x1 ,x2 ,x3 )
We choose to examine the deformational behavior of a very small portion of rock say about a few
mineral grains, at a point

P  P( x1, x2 , x3 ) .
 
We refer to deformation at this point by u (P ) . By reducing the size of our problem we are less
likely to err in approximating a general function of deformation by one that is only linear. This is
analogous to expanding a function using a Taylor series expansion but only using the first two terms
of the expansion.
One-dimensional strain
In the one-dimensional case, deformation, u( x1 ) , can be calculated in the vicinity of the origin
O (where , x1  0 , i.e. Taylor Series) as a function of the distance away from the origin:
u ( x1 ) O  u O   x1
2
du x1  O  1
2 d u  x1  O 
 x1  O 
 .....
2
dx1
2!
dx1
Substituting for a specific value of x1  a , we have
u (a )  u 0   x1
du a  0 
dx1
if we only consider the first term of the expansion.
Because the higher-order terms are assumed insignificant, the deformation is described as a
simple linear function linearly. The deformation at any point along the x1 axis is a function of a small
deformation at the origin plus an additional value proportional to distance from the origin.
Curiously, there is a connection between homogeneous, linear strain (above) and the expansion
of the universe. If we assume that the universe is expanding at the same throughout (homogeneous)
and that it is proceeding at a constant rate, this is equivalent to saying that strain is homogeneous.
In a 1-D sense while the rate of deformation is homogeneous, the accumulated effect of the
strain as seen from the origin is greater at greater distances from the observer where the cumulative
velocity of the expansion is linear with offset between the observer and the target (sensor, in our
case). The 2011 Nobel Prize in physics was awarded to those (Saul Perlmutter, Brian P. Schmidt,
Adam G. Riess) who discovered that the rate of expansion of the universe is actually increasing and
not constant.
(http://ircamera.as.arizona.edu/astr_250/Lectures/Lec_23sml.htm)
Two-dimensional case (i=1,2; j=1,2)
Three-dimensional case (i=1,2,3; j=1,2,3)
There is a Taylor series expansion for functions with three variables. In three-dimensions, by
analogy to the one-dimensional conditions, the deformation function can be approximated by the
first two terms of the expansion in Taylor series:
   
u P  u O  u  P  ,

where u is the discrete differential of the function, valid for the very small volumes of rock we
consider but whose limit for a continuous medium, is the differential du . Remember that u
describes the behavior of deformation in all 3-diemnsions at once. The total deformation will have
contributions from all three directions. We can examine contributions to deformation from each of
these three directions
u  P  
 
u P 
For example term
x1 x
u  P 
x1
x1 ˆx1 
x2 ,x3
u  P 
x2
x2 ˆx2 
x1 ,x3
u  P 
x3
x3 ˆx3
x1 ,x2

is read as “the partial differential of the function u with respect to
2 , x3
the x̂1 basis vector direction”.
The vertical bar with the sub-indices signifies that during

differentiation any of the other two components of u in either the x̂2 or x̂3 directions are constant

and differentiate to zero. Note that because u is itself a function of a three-variable function of

position, u1 in the x̂1 direction can have terms that relate to the x2 and x3 value of P.
In figure 3 we see that there are two types of three-dimensional vectors used to describe the

deformation. One vector relates position of any point in the undeformed state P  ( x1, x2 , x3 ) with

respect to the origin in the undeformed state ( O ). Another relates the new position of the point

P  ( x1 ,x2 ,x3 ) after the deformation, with respect to the new origin ( O ). The difference in the
deformation
between
the
position
u  P   ( u1  P  ,u2  P  ,u3  P  ) where
at
the

origin

(O)
u  P   u1  P  ,u2  P  ,u3  P 

and
is
at
the

P is
general
deformation function that we approximate in a linear fashion.
After deformation,
O  O and P  P .
P  is the vector that describes the deformation with respect to the new origin O , that is we have
experienced a coordinate transformation.
O is related to O through the displacement vector
 
u O  ,
and P is related to P  through the displacement vector:

 
   
u P   u O   u P , (1)
(1.1)
similar to the Taylor expansion for the one-dimensional case.
u is the spatial change of the displacement vector. In the most general case, each component
of the displacement vector is a function of three other components:
u  f ( x1 ,x2 ,x3 )
We have, using (1)
P  u  P   P  u  O 
P  u  O   u  P   P  u  O  ,
P  P  u  P 
If we express the displacements as a function of three components:
 u1  P1 
u1  P   


x1
 u2  P2 
u2  P   


x1
x1 
 u3  P3 
u3  P   


x1
u1  P1 
x1 
x2
u2  P2 
x2
x1 
x2 
x2 
u3  P3 
x2
  x  xˆ
u1 P1
x3
u2  P2 
x3
x2 
3


1

x3  ˆx2


u3  P3 
x3
(2)

x3  ˆx3


By assuming that P is a scalar function (!) and in indicial notation, ui 
ui
xj
x j
From (1) and (2) we have that
P1  x1 '  x1  u1
 x1 
u1
u
u
x1  1 x2  1 x3
x1
x2
x3
P2  x2 '  x2  u2
 x2 
u2
u
u
x1  2 x2  2 x3
x1
x2
x3
etc.,
Using indicial notation we can rewrite the above expressions as:
Pi  xi  xi 
ui
xj ,
x j
Using matrix notation we can also write this as:
 u1
u1
1 
x2
 x1   x1
u
 x    u2
1 2
 2   x
x2
1
 x3  
u3
u3

x2
 x1
u1 

x3   x 
1
u2   
x2
x3   
 
u   x3 
1 3 
x3 
(3)
Note that I intentionally use the symbol “  ” in order to remark on its difference, if only
temporarily, from the gradient of a scalar field that we have already seen. This symbol  does not
represent the gradient of a scalar field, but the gradient tensor of the displacement vector field.
Compare this to the simple gradient of a scalar field:
( u ) 
u
u
u
ˆx1 
ˆx2 
ˆx3
x1
x2
x3
Conditions for homogeneity in two-dimensions
We can now revert to two-dimensions to convey the same concept but with less notation than
the three-dimensional case;
u1 
 u1
1


 x1 
x1
x2   x1 



 x 
u  
 2   u2
1  2   x2 
 x1
x2 
(4)
Deformation is homogeneous throughout a space when the following two conditions are met:
(1) Parallel material lines in the undeformed space remain parallel following the deformation and
(2) if straight lines in the undeformed space remain as straight lines following the deformation.
These two conditions are equivalent mathematically to stating that
u1 u1 u2 u2
,
,
,
x1 x2 x1 x2
(or
ui
in indicial notation),
x j
does not change through space and can be considered to be of constant value: say, a , b , c , d .
That is, we can write (4) as:
 x1  a b   x1 
 x    c d   x  (5)
 2 
 2 
where a  1 
c
u1
,
x1
b
u1
,
x2
u2
u
, and d  1  2
x1
x2
Actually, the assumption of homogeneous strain is scale-dependent. For a sufficiently small scale
we can always consider the deformation to be homogeneous. Beyond the scale of homogeneity, we
must start to deal with inhomogeneous deformation, also known as non-homogeneous or
heterogeneous deformation. In order to deal with the inhomogeneity we would expand the equation
(2) to include more terms. However, when dealing with elasticity, such as the propagation of waves,
normally we have that the deformations are very small, a few percent at most and
u1 u1 u2 u2
,
,
,
x1 x2 x1 x2
(or
ui
in indicial notation) << 1.
x j
In geological applications these quantities can be quite large and the values for a , b , c , d are
not infinitesimal.
An example of deformation of a line
We can now demonstrate the condition of homogeneity by showing that a straight line remains
straight throughout the deformation. Consider a line prior to deformation:
x2  mx1  k
(6)
From (5) and (6) we have that
x1  ax1  bx2  ax1  b(mx1  k )
x2  cx1  dx2  cx1  d (mx1  k )
By replacing the value of x1 in the second equation with the value of x1 taken from the first
equation x1 
x1  bk
we have:
a  bm
 x  bk 
 x  bk 
x2  c 1
  dm 1
  dk
 a  bm 
 a  bm 
x1
x1
  bc    bdm 
 a  bm 
c
 dm
 k
  k
  kd

a  bm
a  bm
 a  bm   a  bm 
 a  bm 
c  dm
  bc  bd  ad  bd 

x1  k 
(7)

a  bm
a  bm


 c  dm    ad  bc 

 x1  
k
 a  bm 
 a  bm 
 m  k 
We have shown that the deformed straight line is also another line! Because m does not depend
on k , two parallel lines (same m but a different k ) will remain parallel after the deformation. They
will have the same slope m . We can also demonstrate that a circle will transform into an ellipse and
that one ellipse will transform into another ellipse. A series of incremental homogeneous
transformations can be described by a single finite-deformation ellipse.
(TBD: Conditions for principal strain axes: The case of general transformation:
extension and contraction along two perpendicular axes plus a rotation)
Let us consider the homogeneous deformation:
 x1   A B   x1 
 x   C D  x 
 2 
 2 
(8) or,
A  TA , where
 A B   cos
T

C D   sin 
sin   a b 
cos  b d 
(9)
T is matrix for deformation via a rotation transformation, A is the matrix of another
deformation and A is the resultant general deformation matrix. Note that matrix A is symmetric so
that it can be diagonalized. In this deformation, two perpendicular lines exist that remain
perpendicular after the deformation in the same direction as before the deformation (eigenvectors).
Along these directions there is either only extension or only shortening. These directions are the
principal directions of deformation along which the deformation is either maximum or minimum.
We have that :
A  a cos  b sin 
B  b cos  d sin 
C  a sin   b cos
D  b sin   d cos
As a result, the rotation angle  is related to the general deformation components by:
tan  
BC
A D
(9’)
Once we arrive at a value for  , we can multiply the left-hand side of equations for A and C by
sin  and cos to obtain
a  A cos  C sin 
b  A sin   C cos
We also multiply the left hand side of equations for B and D by sin  and cos , respectively
to get:
d  B sin   D cos
(9’’)
In decomposing A from equation (9) the four components are shown to be trigonometric
functions.
We could now just find the eigenvectors and show they are at right angles to each other (See
exercises)! However, an alternative approach is to show that if two perpendicular lines exist in the
deformed system they are also the principal deformation axes.
Let’s start by considering two perpendicular lines:
x2  mx1 , x2  
1
x1
m
(10)
After application of the deformation matrix as given by:
 x1   a
 x    b
 2 
b   x1 
d   x2 
(11)
we have after substitution by (10)
bx1 

ax

1


x
x
ax

bmx

 1
 1  1
1
m  which solve as:

and





 x  bx  dmx 
 x2  bx1  dx1 
1
 2  1
m

x2 
md  b
mb  d
x1 and x2 
x1 , respectively. We want to know whether
mb  a 1
ma  b 1
m
md  b
1 mb  d
and  
?
mb  a 1
m ma  b 1
These conditions can be shown as true by substituting into the identity so that:
1
 1
m     0.
 m
From the equation above we obtain that the deformed lines are also at right angles if the
following combination is met:
m2b  ma  d   b  0
(13)
Graphically, this can be expressed as:
Question: Show that the homogeneous deformation of a circle:
x2  y 2  4
is an ellipse and draw the result.
Infinitesimal deformations
We now want to demonstrate that the strain theory we have looked at so far for finite geological
deformations can also be applied to the very small deformations that are used in wave propagation
theory and seismology.
Form equation (9) we have;
 A B   cos
T

C D   sin 
A 1
C
u1
,
x1
sin   a b 
cos  b d  where
B
u1
,
x2
u2
u
, and D  1  2
x1
x2
Because deformations are very small, recall that
be negligible. From equation (9’) we have now:
ui
<< 1 and that higher order derivatives will also
x j
1  u1 u2 
u1 u2




2  x2 x1 
1  u u 
x2 x1
tan  

  1  2 
u u
2  1  2 1  1  u1  u2  2  x2 x1 
x1 x2
2  x1 x2 
As well,
tan   sin    and cos  1
(14a)
and to a first order equation (9’’) becomes
a  A  C
b  A  C
(14b)
d  B  D
 
 
For u P   P  P , in the two-dimensional case we have:
 u1   cos
u    sin 
 2  
sin   a b  1 0  x1 


cos  b d  0 1  x2 
(15)
and in the infinitesimal case, using approximations (13) and (14):
 u1   1    A  C
u     1   A  C

 2  
 A  1  C
 
 A  C
A  C  1 0  x1 


B  D  0 1  x2 
A  C 
 A  C
 

D  1  B 
 c  A
If we disregard those second-order terms we get:
A  1  C 
u1
x1
D  1  B 
u2
x2
B  D   x1 

 A  C   x2 
(16)
 u 
u
u
u
u
A  C  1  1   2    1   2    2
x1
x1
x
x1
 x1 
Similarly:
 A 2  C   0

 B 2  D   B 2  D  D  1 

u2 
  
x2 
C  A  
Then:

u1

 u1  
x1
u    
 2   1  u1  u2 
 2  x2 x1 


1  u1 u2 




2  x2 x1 
 0 1  x 
 
   or,

u2
  1 0   y 


x2


if we use equation 13 and 14(a):

u1

 u1  
x1
u    
 2   1  u1  u2 
 2  x2 x1 

(17)

1  u1 u2 



0

2  x2 x1 



 1  u1 u2 
u2



 
x2
2

x

x
2
1




1  u1 u2  

 

2  x2 x1   x1 

  x2 
0


The first term on the right hand of the equation corresponds to the infinitestimal
internal deformation of the rock body whereas the second term corresponds to an
infinitesimal clockwise rotation by angle  .
 x1 
 0 1  x1 
 x      1 0   x 

 2 
 2
 x 
 2 
 x1 
Finally we arrive at the equation for the infinitesimal strain tensor e :


u1
1  u1 u2 



0


 u1  
x1
2  x2 x1 


u    

 1  u1 u2 
u2
 2   1  u1  u2 




 
 2  x2 x1 

x
2

x

x
2
2
1






u1
1  u1 u2 





x
2

x

x
1
1 
 2

 1  u1 u2 

u2


 

x2
 2  x2 x1 

1  u1 u2  

 

2  x2 x1   x1 

  x2 
0


In indicial notation we can write the infinitesimal strain tensor as:
1  u u 
1
eij   i  j  , or for the general case: eij  ui , j  u j ,i 


2
2  x j xi 
Rotational Seismology
“This shows that it is necessary to have three
components of translation, six components of strain, and three components of
rotation to fully characterize the change in the medium around point x”
{Suriyanto, #4249}