Chemistry Reference
In-Depth Information
S
3
e
fy
y
S
2
S
1
L
f
FIGURE 9.4 Schematic showing the strain measured by X-ray diffraction in the sample
coordinates.
the measured normal strain and the strain tensor expressed in the sample coordinates
is as given in Ref. [2].
e
fc
¼ e
11
cos
2
f sin
2
cþ e
12
sin 2f sin
2
cþ e
22
sin
2
f sin
2
c
þ e
13
cos f sin 2cþ e
23
sin f sin 2cþ e
33
cos
2
c
ð9
:
30Þ
where e
fc
is the measured strain in the orientation defined by f and c and
e
11
; e
12
; e
22
; e
13
; e
23
;
and e
33
are strain tensor components in the sample coordinates
S
1
S
2
S
3
. The above equation can be easily obtained from the unit vector in the e
fc
direction expressed in the sample coordinates. The unit vector is given by
2
4
3
5
¼
2
4
3
5
h
fc
1
h
fc
2
h
fc
3
cos c
sin f sin c
cos f sin c
h
fc
¼
ð9
:
31Þ
Then the relationship between the measured strain e
fc
and the strain tensor can be
given by
e
fc
¼ e
ij
h
wc
h
wc
j
ð9
:
32Þ
i
The scalar product of the strain tensor with the unit vector in the above equation
is the sum of all components in the tensor multiplied by the components in the
unit vector corresponding to the first and the second indices. Equation (9.30) can
then be obtained by the expansion of this equation for i and j values of 1, 2, and 3.
