A.1 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 A.1. Remembering
that force equals stress by area we begin by decomposing the surfaces forces
into stresses acting on each of the faces.
Figure A.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} 

(A.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} 

(A.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} 

(A.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} 

(A.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 

(A.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 

(A.7) 
where a_{i} is the unit vector (0,0,1). Note that in the convention for
indices adopted in Eqn A.7 the coordinates x, y and
z are given by x_{1}, x_{2} and x_{3}, respectively, while
s_{ii} = t_{ii}. Equation A.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.
A.2 Calculation of ridgepush force
In order to solve Eqn. 7.7 we need to formulate the
density distribution appropriate to Figure 7.2a. The appropriate
density distribution is (Figure 7.2b):







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



(A.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 7.7 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 , 



(A.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



(A.10) 
F_{2} = 
ó
õ 
w
0

(s_{zz})_{z} dz = 
r_{w} Êg Êw^{2}
2



(A.11) 
Since in the lithosphere the density is itself a function of depth the third term,
F_{3}, in Eqn 7.7 is given by:
F_{3} = 
ó
õ 
w+z_{l}
w

(s_{zz})_{z} dz 

(A.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 


(A.13) 
Thus Eqn 7.7 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 


(A.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}


