Diploma Thesis - Erich Schmid Institute

Polycrystalline and monocrystalline Model
Therefore the compliance components of the Voigt average defined by compliance
constants of a cubic material are:
s
V
12
s
V
11
=
= 2 s
11
1 ⎛
⎜
⎜−
s
2 ⎝
s
11
− s
12
+ 3s
2
5 ( s11
− s12
)
−
3s
− 3s
+ s
12
5 ( s
11
12
44
2
5 ( s11
− s12
)
+
3s
− 3s
+ s
− s
11
) s
12
44
⎟ ⎞
⎠
V
11 12 44
44 = (equ 7.3)
3s11
− 3s12
+ s44
The sample can be rotated around three different angles ω, ψ and ϕ, but the rotation
cannot be performed in any order of these angles. In figure 7.1 the sample and two
coordinate systems are shown. The first one is the laboratory coordinate system eL
which will be fixed and independent of the sample rotation. In this system the plane
spacing is measured, because G will always be parallel to eL3. The second one is the
sample coordinate system eS that is associated with the sample and will change
position during rotation. The problem is, that the strains are measured in the
laboratory system, but the stresses must be expressed in the sample system, so a
rotation order must be performed to describe the sample’s tilt during the
measurement. Let us start at a position where both systems coincide. The first
rotation performed by the diffractometer is around eS1 direction by an angle ω,
marked with the double arrow. In this position the new rotation is done around the
new eS2 axis and the last one around the new eS3 axis. This is similar to the Euler
angles procedure.
a.) starting position b.) ψ-rotation c.) ϕ-rotation d.) end position
and ω-rotation
Figure 7.1: Rotation of the sample with respect to the laboratory system
According to chapter 4.2 the laboratory system can be transformed by rotation into
the sample system.
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