Elasticity of the Earth - Quantitative Plate Tectonics - page 248

Z

@

@x j P u i £ ij C f i P u i dV

( 8.41 ) represents a point source located at the ori-
gin and having time-varying magnitude D f ( t ).

D

R

Z @£ ij

@x j C f i

Z

£ ij @ P u i D P u i dV C

@x j dV

8.4 Seismic Energy

R R

Z ¡ @ P u i dV

@ P u i
@x j

D

@t P u i C £ ij

Seismic waves carry energy both in the form
of kinetic energy, associated with the motion of
volume elements, and potential energy related
to deformation. If K and U are respectively the
kinetic and the potential energy per unit volume,
then the total energy density E in a material
during the travel of a seismic wave is given by:

R

Z ¡ @ P u i dV

1

2 £ ij @ P u i

1

2 £ ji @ P u j

D

@t P u i C @x j C @x i

R

Z ¡ @ P u i

@t P u i C £ ij P © ij dV

D

R

Z Z

1
2

@
@t

E D K C U (8.42) D ¡ P u i P u i dV C £ ij P © ij dV

R R

The kinetic energy density is clearly deter-
mined by the local velocity of displacement, so
that:

Z

@K
@t C

D £ ij P © ij dV

(8.45)

R

2 ¡ " @ u 1

2 #

2 @ u 2

@t

2 @ u 3

@t

1
2 ¡ P u 2

1

where we have used the momentum Eq. ( 7.50 ) ,
the symmetry of the stress tensor, and the defini-
tion ( 8.44 ) of kinetic energy density. It is useful
at this point to introduce a function W of the
strain components, which allows to generate the
corresponding stress tensor by its derivatives:

K D

D C C

@t

(8.43)

Regarding the potential energy, let us assume
that an elastic body starts deforming under the
action of external forces. The rate of mechanical
work P ( t ) depends from both internal body forces
operating in the region R and surface forces
exerted along the boundary, ( R ), of R :

@W
@© ij

£ ij D (8.46)

The function W is called the strain - energy
function and has the dimensions of an energy den-
sity [J m 3 ]. The most general form of Hooke's
law, which expresses the linear dependence of
stress from strain, can be written as follows:

I Z

P.t/ D T i P u i dS C f i P u i dV (8.44)

. R /

R

where T D T ( n ) is the traction exerted along the
surface element d S D n dS and f D f ( r ) is the body
forces field. Using Cauchy's theorem ( 7.3 ) and
Gauss' divergence theorem (see Appendix 1 ) in
( 8.44 )gives:

£ ij D C ijhk © hk (8.47)

This constitutive equation is more general that
( 8.1 ), because it also holds in the case of non-
isotropic materials. The tensor C ijhk at the right-
hand side is referred to as the elastic tensor .
Substituting ( 8.47 )into( 8.46 )gives:

I Z

P.t/ D £ ij n j P u i dS C f i P u i dV

. R / R

I Z

P u i £ ij n j dS C

D f i P u i dV

@W
@© ij D C ijhk © hk

(8.48)

. R / R