1. Internal stress measurement using XRD
Elasticity, for an isotropic elastic solid:
the elastic constant E and v
: Kroenecker’s delta
Written explicitly:
Shear modulus

2. Stress normal to a free surface ( ) must be zero at the
surface, i.e.,
Equation of equilibrium (satisfied at each point of the
material):
Transformation of the strain tensor (from one coordination system to another:
where defines the cosine of the angle between in
the old coordinate system and in the new coordinate
system.

3. Supplement
Vector transformation from one (X) to another (X’) coordination
system:
X system:
X’ system:

4. Consider the transformation of the sample coordinate system
to the laboratory coordinate
system .
Find out the transformation matrix for the above case:
1. Rotate along the axis by an angle ;
2. rotating an angle along the  y y x x

5.  transformation matrix for the coordinate system z z x x
 transformation matrix for the coordinate system

7. Change strain to stress
Interested in
13 and 31
Change strain to stress

8. Look at the 11 term, there are
Add and subtract one term
We get
Similar for 22 term 
For 33 term 

9. Let’s group the sin2 into one term, and the rest …
The quantity measured at angles  and . 
: d-spacing in the stresses sample (measured for
the plane whose normal is at angles ,  from the
sample coordinate system);
: d-spacing for the unstressed state is related

10. Three stress states of interests are: uniaxial, biaxial, and
hydrostatic states.
* uniaxial stress state:

11. * biaxial stress state:

13. * Hydrostatic stress state:
volumetric strain :
volumetric strain (hydrostatic stress): H
K: bulk modulus

14. Linear relation when the sample is in the biaxial stress state.
Slope ~
When the sample is in the triaxial
state  -splitting
asymmetric
The shear stress can lead to
compression of some plane spacing
and expansion of others
Presence of stress gradient, texture
and/or elastic and plastic anisotropic