Geology Reference
In-Depth Information
Fig. 7.7 Geometry of
deformation in two
dimensions. a Dilatation, b
Compression, c Simple
shear, d Pure shear
This expression says that the off-diagonal
component © 13 is the average angular variation
in the plane x 1 x 3 . A similar conclusion can be
drawn for the other off-diagonal components.
Therefore, ( 7.43 ) furnishes a simple intuitive
interpretation of the off-diagonal components of
the strain tensor.
A consequence of the strain tensor symmetry
is that a system of principal strain axes m i exists
such that the tensor is diagonal. In this frame, and
assuming no rotations, the variations of displace-
ment du i have the same direction of the variations
of position dx i :
relation between these variables in a geodynamic
context. This can be done adapting the classic
Newton's equations of motion (second law of me-
chanics) to the case of a continuum deformable
body. Let us consider the forces exerted on a
volume element dV D dx 1 dx 2 dx 3 of the region R
occupied by the body (Fig. 7.8 ).
We know that the surface force exerted on a
face of dV is given by the traction on that face,
times the area. Therefore, the force d F on a face
at position r D ( x 1 , x 2 , x 3 ), with normal - e j and
area dx r dx s ( r , s ¤ j ), has components:
dF i e j ; r D T i e j dx r dx s
D £ ij .r/dx r dx s I r;s ¤ j
(7.45)
d u i D © ij dx j D œdx i
(7.44)
The three eigenvectors of ( 7.44 ) are called
the principal strains © 1 , © 2 ,and© 3 . With the
exception of a situation of hydrostatic strain ,such
that © 1 D © 2 D © 3 , some amount of shear strain is
always present also in the principal strain axes
coordinate system.
Similarly, the force on a face at position r C
e j dx j , with normal e j and area dx r dx s ( r , s ¤ j ),
has components:
dF i e j ; r C e j dx j D ij r C e j dx j
dx r dx s I r;s ¤ j
(7.46)
7.3
Cauchy Momentum Equation
Clearly, when the stress field is homo-
geneous the net force exerted on dV is
zero, because the forces on opposite sides
of the volume element balance each other:
d F ( e j , r ) D d F ( e j , r C e j dx j ). Therefore, in
In the previous sections, we have introduced the
concepts of traction, stress, displacement, and
strain for a continuum body in conditions of static
equilibrium. Now we are going to describe the
Search WWH ::




Custom Search