Chemistry Reference
In-Depth Information
For the biaxial stress state, s
33
¼ s
13
¼ s
23
¼ 0, we have
sin u
0
sin u
p
11
s
11
þp
12
s
12
þp
22
s
22
þp
ph
s
ph
¼ ln
ð9
:
83Þ
Here, four or more independent measured strains are required. The biaxial stress
state corresponds to the straight line of the d-sin
2
c plot. The biaxial stress with shear
is the case when there is a split between the data points in þc side and c side. For
the equibiaxial stress state when s
11
¼ s
22
¼ s, there are no in-plane (S
1
-S
2
) shear
components, s
12
¼ 0. Therefore, we have
sin u
0
sin u
ðp
11
þp
22
Þs þp
ph
s
ph
¼ ln
ð9
:
84Þ
In this case, two or more independent measured strains are required. This stress
state exists typically over the surface of metal parts after surface treatment, such as
shot peening, or thin films with no texture or with fiber texture. In this stress state, there
are only normal stress components having the same value in any direction within the
plane.
9.3.4 True Stress-Free Lattice
d
-Spacing
In the 2D stress equations for the biaxial and biaxial with shear states, we have
assumed that s
33
is zero so that we can calculate stress with an approximation of d
0
(or 2u
0
). Any error in d
0
(or 2u
0
) will contribute only to a pseudo-hydrostatic terms
ph
.
The measured stresses are independent of the input d
0
(or 2u
0
) values. This can be
approved by an experiment on an Almen strip. The Almen strip is a thin strip of metal
sample used to quantify the intensity of a shot peening process. The Almen strip is
named after John O. Almen for his invention of the method [34], which has been
further modified by many [35]. The strip is placed in the shot peening chamber where
the machine parts are shot peened. The compressive stress introduced by the peening
operation can then be measured by the strip deformation. In this experiment, the
residual stress is measured by X-ray diffraction using a GADDS systemwith Cr-K
a
radiation (l¼ 2.2897 A
), generator power 35 kV/50 mA, and 0.5 mm collimator. The
stress measurement parameters are E¼210000MPa, n ¼0.28, A
RX
¼1.49. For each
stress measurement, seven frames were taken at c¼45
, 30
, 15
,0,15
,30
,
and 45
(v¼57
- 147
with 15
steps). The (211) rings on the frame were used for
stress calculation. Figure 9.11 shows the biaxial stress tensor measured with different
input d
0
in the range of 1.165 - 1.175 A
. The measured stress tensor is independent
of the input d
0
(s
11
¼623MPa, s
12
¼638MPa, s
22
¼80MPa). All measured stress
components are horizontal lines in the plot. The lines for s
11
and s
22
are almost
overlapping due to very close results, and the shear stress s
12
is very small. The stress
state can be considered equibiaxial. The pseudohydrostatic terms
ph
changes with the
input d
0
.
