Chemistry Reference
In-Depth Information
All other terms are cancelled out. Figure 9.5(c) is a plot of De
fc
versus sin 2c
þ
.
The slope of the plot can be calculated froma linear fitting of the data points. The shear
stress is then given by
qðDe
fc
Þ
qðsin 2c
þ
Þ
1
b
t
f
¼
¼
ð9
:
45Þ
2
S
fhklg
2
S
fhklg
1
1
2
2
where b is the slope of the De
fc
- sin 2c
þ
plot. The value of sin 2c
þ
reaches a
maximum at c ¼ 45
,soac-tilt of up to 45
is preferred to resolve shear stress.
Further increase in the c-tilt beyond 45
is not necessary for resolving shear
component. The shear stress components s
13
and s
23
can be obtained by measure-
ments at f ¼ 0
and f ¼ 90
, respectively. The e
fhklg
fc
-sin
2
c plot may deviate from a
straight line due to other reasons—for instance, texture, stress gradient, or composi-
tion gradient, as shown in Figure 9.5(d).
9.2.4 Stress Measurement Without
d
0
In the above e
fhklg
fc
-sin
2
c plots, the measured strain e
fhklg
fc
is given by Eq. (9.31), which
requires both the measured d-spacing d (or 2u) and the stress-free d-spacing d
0
(or 2u
0
). d
0
is not always available. For most metals or alloys, due to variations in
composition, processing, and laboratory temperature, it is very hard to obtain an
accurate d
0
(or 2u
0
). Assuming an inaccurate stress-free d-spacing, d
0
is used in the
calculation of e
fhklg
fc
. With the true strain definition, we have
¼ ln
d
fhklg
d
0
¼ ln
d
fhklg
d
0
¼ ln
d
fhklg
d
0
þ ln
d
0
e
fhklg
fc
fc
fc
d
0
fc
d
0
d
0
¼ e
0fhklg
þ e
ph
ð9
:
46Þ
fc
where e
ph
¼ ln d
0
=
d
0
is the pseudohydrostatic strain caused by the error in d-spacing
d
0
. The effect of pseudohydrostatic strain is as if the sample is first deformed by a
hydrostatic strain e
ph
so the d-spacing of {hkl} in all directions changes from d
0
to d
0
.
Introducing this effect into Eq. (9.36),
e
0
fhklg
fc
¼ S
fhklg
1
2
S
fhklg
s
w
sin
2
c e
ph
1
ðs
11
þs
22
Þþ
ð9
:
47Þ
2
By taking the first derivative with respect to sin
2
c, we obtain
qe
0
hkfg
fc
qðsin
2
cÞ
¼
2
S
fhklg
1
s
w
¼ m
0
ð9
:
48Þ
2
One can easily conclude that the error in stress-free d-spacing only shifts the
height of the e
fhklg
fc
-- sin
2
c plot, but does not change the slope (m ¼ m
0
). Therefore, in a
biaxial stress state or biaxial stress state with shear, it is not necessary to use an
accurate stress-free d-spacing when the sin
2
c method is used for stress measurement.
Typically the d-spacing measured at c ¼ 0, d
fhklg
f;c¼0
, is used in place of d
0
.
