160 7. OPTICAL TECHNIQUES
Figure 7.14: Schematic structure of Equation (7.1) for an Integral Method calculation.
number of active pixels. e majority of the computation time is used in assembling the G
T
G
matrix because the G matrix in Figure 7.14 is so large. Solution of the resulting Equation (7.2)
is fast by comparison because the assembled G
T
G matrix is symmetric and relatively modest in
size.
If desired, regularization can be added to the solution using a procedure analogous to that
used for strain gauge measurements; see Equations (5.23) and (5.24). However, for optical mea-
surements it is difficult to use the Morozov criterion to determine the amount of regularization
needed because the least-squares solution gives a misfit (D residual) even without any regular-
ization. In this case, it is practical just to adjust the amount of regularization until a reasonably
but not excessively smooth solution is achieved. Fortunately, the data averaging that occurs dur-
ing the least-squares solution of the optical data tends to reduce overall noise so that often little
or no regularization is needed.
7.8 RESIDUAL STRESS COMPUTATION USING
INCREMENTAL DATA
All measurement techniques for hole-drilling residual stress evaluations are differential in char-
acter. at is, they measure the difference between a starting state and a subsequent state. is is
seen clearly when using strain gauges in that they must be “zeroed” after being installed and all
subsequent measurements are made relative to that initial state. For hole-drilling measurements
the starting state is the original uncut surface and the subsequent states occur after each incre-
7.8. RESIDUAL STRESS COMPUTATION USING INCREMENTAL DATA 161
ment of hole drilling. Optical measurements follow the same pattern, with an initial image made
of the uncut surface followed by subsequent images made of the surface after each hole-drilling
increment. e deformation of the imaged surface is then determined from the differences be-
tween the initial and subsequent images. is procedure works fairly well, but is not ideal for
incremental measurements because the overall drilling process can take a significant time, dur-
ing which time the optics may drift slightly, thereby causing a deterioration in the relationship
between the later images and the initial image. is is particularly an issue with interferometry
measurements such as ESPI and Moiré, where measurement noise tends to grow gradually with
time from the initial measurement. us, in such measurements, it is an advantage for the initial
image to be as “fresh” as possible.
e desire for a “fresh” initial image can be achieved during an incremental hole-drilling
measurement by referencing each measured image to the one immediately before it rather than
to the initially measured image. is practice significantly improves the quality of the measured
data and substantially reduces the noise in the overall measurement. is is an important advan-
tage given the noise sensitivity of the inverse solution required to determine the through-depth
profile of the residual stresses.
Incremental image referencing can be accommodated mathematically with just a minor
adjustment to the mathematical method described above. In Figure 7.14, starting from the sec-
ond row of blocks in matrix G , the w
1
, w
2
and w
3
contents of the blocks in the previous row
are subtracted from each row to form a modified matrix G . is matrix can then be used in
Equations (7.1) and (7.2), where ı takes the meaning of displacement during the previous hole
depth increment only. To make this process clear, Table 7.2 reproduces the analogous, but much
more compact strain gauge example
N
a matrix from Chapter 5. Table 7.3 shows the equivalent
N
a
matrix for incremental strain referencing.
It was noted in Chapter 5 that the diminishing sensitivity of hole-drilling surface defor-
mations to deep interior stresses causes the
N
a matrix to be badly conditioned numerically, and
thus to be very sensitive to measurement noise. is characteristic of the
N
a matrix is revealed by
the small numerical values of the elements along the diagonal relative to those away from the
diagonal. However, inspection of Table 7.3 shows the opposite situation, with the diagonal el-
ements within the incremental equivalent
N
a matrix being larger then the off-diagonal elements.
is makes the matrix much better conditioned numerically and therefore less sensitive to the
effects of noise. e same effect occurs with matrix G in optical measurements, and so incre-
mental referencing is advantageous, even with DIC, where temporal degradation of measured
images is much less significant than with ESPI or Moiré measurements.
Figure 7.15 shows the results of an example incremental depth ESPI measurement on a
shot-peened specimen. Figure 7.15a shows the stresses computed using optical data referenced
from the initial measurement on the uncut surface, while Figure 7.15b shows the corresponding
stresses computed using optical data referenced from the immediately previous image. It can be
seen that the incremental results are much smoother and less prone to local noise disturbances.
162 7. OPTICAL TECHNIQUES
Table 7.2: Matrix
N
a for a 1/16” Type A rosette with a 2-mm diameter hole. Depths h and H are
in mm. Multiply depths by 0.5 for a 1/32” rosette or by 2 for a 1/8” rosette.
h
[a]
0.10 -0.0144
0.20 -0.0189 -0.0152
0.30 -0.0224 -0.0192 -0.0147
0.40 -0.0250 -0.0222 -0.0182 -0.0132
0.50 -0.0269 -0.0241 -0.0206 -0.0164 -0.0113
0.60 -0.0285 -0.0256 -0.0221 -0.0184 -0.0141 -0.0093
0.70 -0.0296 -0.0267 -0.0233 -0.0196 -0.0158 -0.0117 -0.0074
0.80 -0.0305 -0.0276 -0.0241 -0.0205 -0.0168 -0.0131 -0.0093 -0.0056
0.90 -0.0310 -0.0281 -0.0247 -0.0211 -0.0175 -0.0138 -0.0105 -0.0073 -0.0041
1.00 -0.0315 -0.0286 -0.0252 -0.0216 -0.0179 -0.0144 -0.0111 -0.0082 -0.0054 -0.0028
H→ 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
Table 7.3: Incremental matrix
N
a for a 1/16” Type A rosette with a 2-mm diameter hole. Depths
h and H are in mm. Multiply depths by 0.5 for a 1/32” rosette or by 2 for a 1/8” rosette.
h
[a]
0.10 -0.0144
0.20 -0.0045 -0.0152
0.30 -0.0035 -0.0040 -0.0147
0.40 -0.0026 -0.0030 -0.0035 -0.0132
0.50 -0.0019 -0.0019 -0.0024 -0.0032 -0.0113
0.60 -0.0016 -0.0015 -0.0015 -0.0020 -0.0028 -0.0093
0.70 -0.0011 -0.0011 -0.0012 -0.0012 -0.0017 -0.0024 -0.0074
0.80 -0.0009 -0.0009 -0.0008 -0.0009 -0.0010 -0.0014 -0.0019 -0.0056
0.90 -0.0005 -0.0005 -0.0006 -0.0006 -0.0007 -0.0007 -0.0012 -0.0017 -0.0041
1.00 -0.0005 -0.0005 -0.0005 -0.0005 -0.0004 -0.0006 -0.0006 -0.0009 -0.0013 -0.0028
H→ 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
In addition, the x and y stresses are more consistently equal, which is a feature to be expected
from a shot-peened specimen. ese advances are due both to the improvement in quality of the
incrementally referenced data and to the better mathematical conditioning of the incremental
G matrix.
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