DLC Sign Conventions and Coordinate Systems

Topics discussed here:

Units

Global Coordinate System

Orientation of Dislocations (strike and dip conventions)

Displacements (Burgers vectors)

Stress and Strain Tensor Sign Conventions

Principal Stresses

Local Coordinate System attached to dislocations and planes

Tractions in Local Coordinate Systems

Dilatations

Observation Plane Coordinates Sytem

Rakes

Units:
------------------------
Units used are km for specifying dislocation geometry, 
meters for specifying displacements (Burger's vectors).
Units of stress depend on the units used for shear modulus.
(The default value is 300000 bars, yielding stress units of
bars unless a different value of shear modulus is substituted.)
Strains, dilatations, tilts, etc. are in units of microstrain.
Rotations are in microradians.


Global Coordinate System:
------------------------
A right-handed 1,2,3 system is used with 1 and 2 horizontal and
3 pointing down (as in Erickson, 1986).  Often 1,2,3 are North,
East, Down in applications.  (1 can be rotated relative to North, but
sometimes the increase in confusion does not repay the gain in 
simplicity!)  (Note that Okada (1992) uses a right-handed system 
with the 3-axis pointing up.)


Orientation of Dislocation:
---------------------------
The strike of a rectangular dislocation patch is take to be
in the direction of the left hand as one looks down dip.  Thus,
strike = dipdirection - 90.0.  Dip angles are assumed to lie 
between 0.0 and 90.0.  (Dip-direction would serve as a less
ambiguous orientation quantity if dips are always between 0 and 90.)
(Note that special considerations often need to be taken in defining conventions for rectangular dislocation surfaces that are horizontal or vertical.)


Dislocation Displacements (Burger's vectors):
---------------------------------------------
The convention used is the same as in Erickson (1986) for dips 
between 0 and 90:
   (1)  strike-slip:     + is LL               - is RL
   (2)  dip-slip:        + is dwndip(normal)   - is updip(reverse)
   (3)  open-mode:       + is expand           - is shrink
(However, note that Erickson also allows dips between 90 and 180 which 
converts a + dip-slip Burger's vector from normal to reverse.)
(Okada has signs switched on dip-slip from this convention.)
(It would be desirable for the conventions here to be the same as those
for the stress components below.)
Vertical triangles should be viewed from a clockwise side in assigning dip-slip.

Stresses and Strains (as in Erickson):
-------------------------------
  normal stresses are + in tension, - in compression
  shear stresses are + if they act in a + direction on a + face of a
      small cubic element (see Erickson, page 9).
(Note that these conventions are consistent with common usage in
elasticity theory, but contrary to common usage in rock and soil
mechanics and structural geology where normal forces are taken
to be positive when compressive (e.g., Jaeger and Cook (1979)).

Principal Stresses:
-------------------
Principal stresses are + in tension, - in compression.
(Hence the maximum compressive stress direction is the principal axis
associated with the smallest eigenvalue, sigmin.)

Local Coordinate System Attached to Dislocations and to Planes:
---------------------------------------------------------------------------
(This does not agree with Erickson's convention.)
Local right-handed coordinate system is attached to the plane like this.
     Axes:      1 = Vector in direction of strike.
                2 = Vector in down-dip direction. (See convention below).
                3 = Normal vector, got by 1 cross 2 - usually points
                      down and to the left of the strike direction
                      when the rectangle is viewed from the Earth's
                      surface.
Convention assumed (even for dip=0):
If one stands and extends one's left hand sideways in the direction
of strike, one is looking in the down-dip direction.  Normal vector is
cross-product of strike direction with down-dip direction, so it points
downward for dips .ge.0 and .lt.90


Tractions (Stress Components) in local coordinate system attached to plane:
----------------------------------------------------------------------------
   (1)  horiz-shear:     + is RL shear                   - is LL shear
   (2)  updip-shear:     + is updip (thrust) shear       - is downdip shear
   (3)  normal:          + is tension (unclamping)       - is compression
Note that for 1 and 2 the signs are different from the Burger's vector signs above,
although these are stress components now rather than Burger's vectors.  
(It would probably be desirable for the signs to be the same, if only as an aid to memory.)


Dilatations
-----------
Positive is expansion, negative is compression.


Observation (Viewing) Coordinate System Attached to Observation Planes (ELFGRID output):
--------------------------------------------------------------------------
               obsaxis3 = perpendicular to the plane and pointing toward
                           the observer.
               obsaxis1 = horizontal in the plane and pointing to the
                           observer's right as he faces the plane.
               obsaxis2 = forms a right hand system with the others.
(This coordinate system does not agree with the local coordinate system attached
to planes above.  As long as observation planes are not used to calculate tractions,
there is no problem, but the potential for confusion is present.)


Rakes (as in Aki and Richards):  
-------------------------------
0=LL, 180=RL, -90=Normal, +90=Rev