A The equations of motion
Consider a small rectangular parallelepiped aligned in a cartesian coordinate
system, x, y and z, with sides of length d_{x}, d_{y} and
d_{z} respectively, as shown in Figure . Remembering that force equals
stress by area we begin by decomposing the
surfaces forces into stresses acting on each of the faces.
Figure 1: Coordinate system used to derive the equations of motion
Assume that at the point, P,
centred within the parallelpiped the normal stresses in the directions x, y and
z are given by s_{xx}, s_{yy} and s_{zz}. The shear stress in the
direction normal to x and parallel to y is given by t_{xy} and parallel to z
by t_{xz}. Similarly the shear stresses normal to y are given by
t_{yx} and t_{yz}, and normal to z by t_{zx} and
t_{zy}. Then the forces acting in the direction x on the faces BB'CC'
and AA'DD' are, respectively:

ì í
î

s_{xx} + 
1 2


¶s_{xx} ¶x

d_{x} 
ü ý
þ

d_{y} d_{z} 

and
 
ì í
î

s_{xx}  
1 2


¶s_{xx} ¶x

d_{x} 
ü ý
þ

d_{y} d_{z} 
 (1) 
where the negative sign is due the fact that stresses are treated as positive
in tension and negative in compression.
Across the faces A'B'C'D' and ABCD the forces are

ì í
î

t_{xy} + 
1 2


¶t_{yx} ¶y

d_{y} 
ü ý
þ

d_{x} d_{z} 

and
 
ì í
î

t_{xy}  
1 2


¶t_{yx} ¶y

d_{y} 
ü ý
þ

d_{x} d_{z} 
 (2) 
Across the faces DCC'D' and ABB'A' the forces are

ì í
î

t_{xz} + 
1 2


¶t_{zx} ¶z

d_{z} 
ü ý
þ

d_{x} d_{y} 

and
 
ì í
î

t_{yz}  
1 2


¶t_{zx} ¶z

d_{z} 
ü ý
þ

d_{x} d_{y} 
 (3) 
The body force acting on the volume with density, r, in the direction x is
given by
The total force in the x direction is then

ì í
î


¶s_{xx} ¶x

+ 
¶t_{yx} ¶y

+ 
¶t_{zx} ¶z

+ rX 
ü ý
þ

d_{x} d_{y}d_{z} 
 (5) 
If the component of displacement of point P in the x direction is u then
Newton's second law gives
r 
¶^{2}u ¶^{2}t

= 
¶s_{xx} ¶x

+ 
¶t_{yx} ¶y

+ 
¶t_{zx} ¶z

+ rX 
 (6) 
Similar results obtain for the y and z directions. In tectonic settings
accelerations can be regarded as neglible, and the only body force is gravity which
acts in the vertical direction, taken to be z, and using the convention for
summation over repeated indices, the equations of motion can be reduced to:
0 = 
¶t_{i,j} ¶x_{i}

+ a_{i} rg 
 (7) 
where a_{i} is the unit vector (0,0,1). Note that in the convention for indices adopted
in Eqn 7 the coordinates x, y and z are given by x_{1}, x_{2} and x_{3},
respectively, while s_{ii} = t_{ii}. Equation 7 is general and can be applied
to many problems related to tectonic phenomona. However, since it is couched in terms of the
components of the stress tensor it must be rendered useful through combination with constitutive equations
relating stresses to displacements.
B Calculation of ridgepush force
In
order to solve Eqn.
we need to formulate the density distribution
appropriate to Figure a. The appropriate density distribution is (Figure b):



 


r_{z} = r_{m}[1 + a (T_{m}  T_{z})], 


 (8) 
 

where r_{m} is the density of mantle at T_{m}, the temperature of the
asthenosphere, r_{w} is the density of water, a is the
volumetric coefficient of thermal expansion of peridotite. The density
distributions defined by Eqn give the following variation
(s_{zz})_{z}:

(s_{zz})_{z} = r_{m} g z, 



(s_{zz})_{z} = r_{w} g z , 



(s_{zz})_{z} = r_{w} g w + g 
ó õ

w+z_{l}
w

r_{z} dz , 


 (9) 
 

F_{1} and F_{2} are given by:
F_{1} = 
ó õ

w+z_{l}
0

(s_{zz})_{z} dz = 
r_{m} Êg Ê(wÊ+Êz_{l})^{2} 2


 (10) 
F_{2} = 
ó õ

w
0

(s_{zz})_{z} dz = 
r_{w} Êg Êw^{2} 2


 (11) 
Since in the lithosphere the density is itself a function of depth the third term, F_{3}, in
Eqn is given by:
F_{3} = 
ó õ

w+z_{l}
w

(s_{zz})_{z} dz 
 (12) 
Assuming that the lithospheric geotherm at t_{1} is linear in depth, the
temperature at the surface of the lithosphere T_{s} = 0^{o}C, and a is
independent of temperature then:
T_{z} = T_{m} 
z z_{l}

, r_{z} = r_{m} 
æ ç
è

1Ê+Êa ÊT_{m}Ê 
æ ç
è

1Ê 
z z_{l}

ö ÷
ø

ö ÷
ø



then
g 
ó õ

w+z_{l}
w

Êr_{z} dz = g z_{l} r_{m} + 
gÊ z_{l}Ê r_{m} ÊaÊ T_{m} 2

= g z_{l} r_{m} 
æ ç
è

1Ê+Ê 
a ÊT_{m} 2

ö ÷
ø



and
F_{3} = r_{w} g w z_{l} + 
g Êz_{l}^{2} Êr_{m} 2


æ ç
è

1Ê+ÊÊ 
aÊ T_{m} 2

ö ÷
ø


 (13) 
Thus Eqn is given by
F_{R} = 
r_{m} Êg Ê(wÊ+Êz_{l})^{2} 2

 
r_{w} ÊgÊ w^{2} 2

 
æ ç
è

r_{w} Êg Êw Êz_{l}ÊÊ+ÊÊ 
gÊ z_{l}^{2} Êr_{m} 2


æ ç
è

1Ê+ÊÊ 
a ÊT_{m} 2

ö ÷
ø

ö ÷
ø


 (14) 
The condition of isostatic compensation at depth, w + z_{l}, requires that
s_{zz (z = w + zl, t = 0)} = s_{zz (z = w +zl, t = 1)} 

Solving for z_{l} gives:
z_{l} = 
wÊ(r_{m}r_{w}) a Êr_{m} ÊT_{m}



File translated from T_{E}X by T_{T}H, version 2.25.
On 7 Oct 2000, 11:26.