Stress and Strain - Quantitative Plate Tectonics - page 231

Fig. 7.8 Forces on a
volume element dV D

dx 1 dx 2 dx 3 in equilibrium
conditions

order to have a net force on dV , a non-zero
spatial gradient of the stress field must exist
at the location of dV . In this instance, to the first
order the net force on the faces normal to the x j
axis is given by:
•F i e j D dF i e j ; r C e j dx j C dF i e j ; r
D £ ij r C e j dx j £ ij .r/ dx r dx s

@£ ij
@x j C f i

f to i D (7.49)

To apply the second law of mechanics, we
must balance the total force by an inertial term
R u i dm,where dm D ¡ dV is the mass of the volume
element and the second time derivative of the
displacement field represents, in the context of
continuum mechanics, the analogue of the point
mass acceleration. Therefore, considering force
densities, the equations of motion can be written
as follows:

@£ ij
@x j

dx j dx r dx s D

@£ ij
@x j

D

dV I

r;s ¤ j I no summation on j
(7.47)

@£ ij
@x j C f i

¡ R u i D (7.50)

Therefore, considering the net force exerted
on all pairs of faces, we have that the total
surface force per unit volume acting on dV has
components:

This is the fundamental equation of dynamics
for continuous media. It is often referred to as the
Cauchy momentum equation . In seismology, it is
generally possible to neglect the contribution of
body forces in absence of seismic sources. Fur-
thermore, in this context the second time deriva-
tive of the displacement can be calculated as a
partial derivative, because the location of a vol-
ume element does not change significantly during
earthquakes (this is not true in fluid dynam-
ics). In this instance, the momentum equation re-
duces to the following homogeneous equation of
motion :

@£ ij
@x j

f s u rf
i

D (7.48)

Let us introduce now the body force den-
sity , f D f ( r ), exerted on the volume element
dV (see Sect. 2.1 ) , which also gives a contri-
bution to the force on dV : d F D f ( r ) dV . Then,
the total force per unit volume will be given
by: