Elasticity of the Earth - Quantitative Plate Tectonics - page 249

( u y D A sin . ¨ t kx/
@ u y
@t D A¨ cos.¨t kx/

This equation says that all the derivatives of
W are linear functions of the strain tensor com-
ponents. Therefore, considering that this function
must be zero in the undeformed state of equilib-
rium, we have that W must be a homogeneous
bilinear function of the strain components:

(8.52)

where A is the wave amplitude, ¨ is the angular
frequency, and k D ¨/“. The density of kinetic
energy can be calculated using ( 8.43 ) :

W D D ijhk © ij © hk (8.49)

1
2 ¡A 2 ¨ 2 cos 2 .¨t kx/

K D

(8.53)

Taking the first derivative of this expression
gives:

Therefore, the average kinetic energy density
over a wavelength is:

@W
@© rs D D rshk © hk C D ijrs © ij D D rshk © hk C D hkrs © hk
D .D rshk C D hkrs /© hk

1
4 ¡A 2 ¨ 2

h K iD (8.54)

Let us consider now the potential energy den-
sity. In this example, the only non-zero compo-
nents of the strain tensor are:

A comparison with ( 8.48 ) allows to express
the elastic tensor in terms of components of D :

1
2

@ u y
@x D

1
2 Ak cos .¨t kx/
(8.55)

© xy D © yx D

C hkrs D D rshk C D hkrs

Therefore, using ( 8.47 ) we see that ( 8.49 ) can
be rewritten as follows:

Therefore, using Hooke's law ( 8.1 )wehave
that the non-zero components of the stress tensor
are:

2 D ijhk C D hkij © ij © hk

1

W D D ijhk © ij © hk D

1
2 C ijhk © ij © hk D

1
2 £ ij © ij

£ xy D £ yx D 2© xy D Ak cos.¨t kx/
(8.56)

(8.50)

D

In order to give a physical significance to the
strain-energy function W , we first calculate its
time derivative:

Substituting these expressions into ( 8.50 )
gives the strain-energy function associated with
this wave:

2 C ijhk P © ij © hk C © ij P © hk

1

W D

1
2 A 2 k 2 cos 2 .¨t kx/

W D (8.57)

1
2 £ ij P © ij C

1
2 £ hk P © hk D £ ij P © ij

D (8.51)

Also in this case it is useful to consider the
spatial average over a wavelength. It will be given
by:

where we have used an obvious symmetry prop-
erty of the elastic tensor: C ijhk D C hkij .Acom-
parison of this expression with ( 8.45 ) showsthat
W represents the potential energy density of the
material in the deformed state. Let us consider
now a monochromatic SH planewavethatis
propagating in the x direction. In this instance,
the displacements occur in the y direction at any
time, so that:

1
4 A 2 k 2 D

1
4 ¡A 2 ¨ 2

h W iD Dh K i (8.58)

Therefore, the average kinetic energy
coincides with the average potential energy. The
same result can be easily obtained in the case of
monochromatic P plane wave.