Stress and Strain - Quantitative Plate Tectonics - page 228

hypothesis. In this instance, to the first order
the components of the differential displacements
field are given by:

@ u i
@x j dx j

d u i .r/ D (7.30)

In general, this field does not describe cor-
rectly the deformation, because it may include a
rotational component not associated with changes
of shape. To isolate the component of true defor-
mationin( 7.30 ), let us decompose the Jacobian
J ij D @ u i /@ x j into symmetric and antisymmetric
parts:

Fig. 7.6 Displacement field in a deformed continuum
body

d u D u ( r C d r )- u ( r ), as illustrated in Fig. 7.6 .
However, even a description of the deforma-
tion in terms of differential displacements is not
adequate. For example, if a metal bar having
length l D 1mandfixedatoneendisuniformly
stretched to a length l 0 D 1.2 m, the deformation
could be described saying that the magnitude
of the displacement vectors increases uniformly
from zero at the fixed end to 0.2 m at the opposite
side, thereby, we have differential displacements.
However, if the bar were simply rotated about
a hinge coinciding with the fixed end, the field
of differential displacements would be different
from zero anyway, despite this time the body has
not changed its shape.
Furthermore, in the example mentioned above
of a homogeneously stretched metal bar, a single
number would be sufficient to describe this de-
formation, because the magnitude of the displace-
ment vectors is always 0.2 x / l , x being the distance
from the fixed end. This example suggests that a
better description of the deformation should be
basedonthe relative variations of the displace-
ment field, rather than on absolute changes. Let us
consider a Taylor expansion of the displacement
field, stopped at the first order:

J ij D © ij C ¨ ij (7.31)

where © ij are the components of a symmetric
rank-two tensor, © ij D © ji , known as strain tensor .
The components of the strain tensor are given
by:

@ u i
@x j C

@ u j
@x i

1
2

© ij D (7.32)

while the tensor ¨, which is antisymmetric

(¨ ij D ¨ ji ), has components:

@ u i
@x j

1
2

@ u j
@x i

¨ ij D (7.33)

It is not difficult to prove that ¨ describes
a rigid rotation without deformation. In fact, by
its anti-symmetry, ¨ has a null diagonal, so that
there are only three independent components. Let
us consider the vector , having components:

1
2 © ijk ¨ ij

k D (7.34)

where © ijk is the Levi-Civita tensor (see

Appendix 1 ) . Using the identity:

@ u i
@x j

u i .r C dr/ D u i .r/ C dx j (7.29)

© ijk © stk D © kij © kst D • is • jt • it • js (7.35)

we find that:

With this approximation, we are limiting our
attention to geologic processes that involve only
infinitesimal deformations . The phenomena con-
sidered in seismology generally conform to this

2 ¨ ij ¨ ji D ¨ ij
(7.36)

1
2 © ijk © stk ¨ st D

1

© ijk k D