VERIFICATION OF A TECHNIQUE FOR
HOLOGRAPHIC RESIDUAL STRESS MEASUREMENT
Michael Steinzig*
Gregory J. Hayman*
Michael B. Prime*
*HYTEC Incorporated, 110 Eastgate, Los Alamos, NM 87544
ABSTRACT
This paper describes a holographic hole-drilling system for
measuring residual stresses. The technique uses optically measured
surface displacements near a drilled hole to obtain a measure of the
specimen’s residual stress.
Unlike manual fringe counting
methods, the system described here uses automated software to
both acquire displacement data and convert it to residual stress.
We describe a verification of the software implementation where
the results from the holographic system are compared to a “known”
state of stress in a specially prepared ring and plug specimen. The
verification shows that the software implementation can give
residual stress results that compare well with predicted stress levels
over the full range of planar stress states: from equi-biaxial to
uniaxial to opposed-biaxial (one component tensile, one
compressive).
INTRODUCTION
The hole drilling method is a widely used technique for
experimentally determining residual stress. Typically, a strain-gage
rosette is used to measure surface strains that result when subsurface stress is relaxed due to the drilling of a small hole. These
strains are then related to the state of stress in the hole region prior
to the stress relaxation. In a similar manner, one can relate surface
displacements to sub-surface stress states.
The use of optical methods for measuring the surface
displacements encountered during hole-drilling was developed
during the 1980’s. The driving force for this development was the
inherent disadvantages of the strain gage method for measuring
residual stresses; that is, the strain gages must be applied to a plane
and smooth surface, the hole must be drilled very precisely in
relation to the strain gage, and significant time and expense are
required to apply the gages. In (McDonach, et al., 1983), the
authors developed the application of moiré interferometry in
conjunction with hole drilling. Holographic hole drilling was
explored independently by (Antonov, 1983), (Bass, et al., 1986),
and (Nelson and McCrickerd, 1986) during the mid 1980’s. In
these approaches, an interference fringe pattern relates the
displacements that have occurred as a result of the hole-drilling to
the sub-surface residual stress. Numerous authors published
variations of the holographic technique throughout the 1990’s,
(Hung and Hovanesian, 1990), (Lin, et al.,1994), (Zhang and
Chong, 1998). More recently, (Diaz, et al., 2000) incorporated an
in-plane sensitive electronic speckle pattern interferometer (ESPI)
system, and automated fringe analysis for rapid stress calculation.
INTERFEROMETRY OVERVIEW
The following is an introduction of the topic of ESPI. For a
more thorough review of the subject matter, refer to (Jones and
Wykes, 1989) or (Rastogi, et al., 1997).
In a classical interferometer, two beams of coherent light,
henceforth referred to as the illumination and reference beams, are
combined and the resulting wavefront is passed to a detector. In
speckle pattern interferometery, one or both beams exhibit speckle
patterns due the reflection of said beam(s) off of a diffusely
reflecting surface. When these two beams form an image on a
charge-coupled device (CCD), an interferogram results. The CCD
registers the intensity of the speckle pattern interferogram as:
I(x, y ) = A 2I (x, y ) + A 2R (x, y ) + 2A I (x, y )A R (x, y )cos[φ (x, y )]
(1)
where the (x,y) pair denotes a location on the test specimen as
imaged onto a particular sensor in the CCD array. I is the
interference intensity at the CCD array, AI and AR are the
amplitudes of the illumination and reference beams, respectively,
and φ is the effective phase angle of the interference. As the path
length of either light beam changes, the phase angle, φ, changes.
For a frequency-doubled Nd:YAG laser a 2π change in the phase
angle corresponds to 532 nanometers of overall path length change.
For a typical ESPI arrangement, a single coherent laser beam is
split to generate a reference beam and an illumination beam. The
illumination beam is reflected off of the test surface and is then
imaged onto a detector. The reference beam is sent directly to the
detector, as in Figure 1.
Published in Residual Stress Measurement and General Nondestructive Evaluation, PVP-Vol 429, The
ASME Pressure Vessels and Piping Conference; Atlanta, GA July 23-26, 2001. Ed. Don Bray.
Observation Direction
Zoom
Illumination Direction
CCD
Fiber
Test Surface
LASER
PZT
Framegrabber
PC
Monitor
Figure 1. Typical ESPI system layout
This interferometer can be used to measure shape change of the test
surface because surface displacements result in a path length
change in the illumination beam. The resulting interferogram can
be modeled as
I' (x, y ) = A 2I + A 2R + 2A I A R cos[φ + ∆P ]
(2)
where the (x,y) notation has been dropped to simplify the
equations. In order to quantify the surface displacements, the
optical path length change ∆P must be solved for in Equation (2).
The optical path length change must then be related to the
mechanical motion of the test surface. One method of solving for
∆P is to introduce known path length changes in the reference beam
using a phase shifter, and to acquire four interferograms before and
after surface displacement. The equations (before and after surface
displacements, respectively) are:
I i = A 2I + A 2R + 2A I A R cos[φ + ∆S i ] ,
(3)
I i' = A 2I + A 2R + 2A I A R cos[φ + ∆S i + ∆P ]
∆Si is the known phase shift and has values of 0,
π
2
, π , and
3π
2
for i = 1, 2, 3, and 4, respectively.
The path length change due to the surface displacements can now
be solved for using
(
(
)(
)(
)(
)(
)(
)(
 I' − I' I − I + I' − I' I − I
 1 3 1
2
4 2
4
3
∆P = tan −1 
'
'
'
'
 I 2 − I 4 I1 − I 3 − I1 − I 3 I 2 − I 4
)
)
(4)
This result must be passed through a phase unwrapping algorithm
to remove the modulo 2π effect caused by the arctangent function.
The path length change is related to the mechanical surface
displacements by
∆P =
4π
λ
K •u =
4π
λ
(K xux + K yu y + K zu z )
(5)
where K is the sensitivity direction as defined by the angle
bisector of the illumination and observation directions, shown in
Figure 1.
RESIDUAL STRESS MEASUREMENT SYSTEM
In the current work, a single-illumination-beam phase-shifting
ESPI has been developed for the measurement of residual stress.
The system’s software and related algorithms quickly and easily
transform surface displacement measurements into residual stress
values.
System Configuration
The residual stress measurement system consists of a laser
unit, optical fibers, a video head, an illumination head, and a
personal computer (PC). The laser unit houses a single frequency
20mW Nd:YAG laser with its associated power supplies and
controllers, beam splitting optics, laser-to-fiber couplers, and a
phase shifting device. Its purpose is to generate the coherent laser
light, split this light into a reference beam and an illumination
beam, and then deliver these two beams to the video head and
illumination heads, respectively, via fiber optic cables. The
illumination head collimates the light arriving from the laser unit
and is used to illuminate the test specimen’s surface. The video
head images the test surface and the reference beam onto a CCD
camera. The PC is used to control the phase shifting of the
reference beam, acquire video images from the CCD, save this data
Published in Residual Stress Measurement and General Nondestructive Evaluation, PVP-Vol 429, The
ASME Pressure Vessels and Piping Conference; Atlanta, GA July 23-26, 2001. Ed. Don Bray.
to the hard drive, and then analyze the data to obtain a residual
stress measurement.
Stress Analysis
The stress analysis algorithms are based on the work of
(Nelson, et al., 1997), but have been adapted to work well with an
ESPI system. A set of interferograms are acquired and saved to
memory. Then, the drill is placed in front of the test specimen and
a small hole is drilled to a given depth. Next the drill is removed
and more interferograms are acquired. These interferograms
provide the change in the illumination beam’s path length, ∆P,
which corresponds to the surface displacements caused by the
stress relaxation. This path length change is related to the residual
stress through the following equation,
 ∆P1   C11

 
∆P 2  = C 21
 ∆P 3  C

  31
C12
C 22
C 32
C13  σ xx 


C 23  σ yy 
C 33  τ xy 
(6)
in which Cij contain constitutive properties, geometric properties,
holographic sensitivity factors, and finite element derived
coefficients, and
∆Pi = ∆P (ri , θ i ) − ∆P (ri , θ i + π )
(7)
in which ∆Pi is the phase (path length) difference between two
surface points which are diametrically opposed to one another but
at the same radial location from the hole center.
to a solution heat treatment, cold working to improve strength,
followed by natural aging to ensure stable properties. The T351
designation refers to plate that has been stress relieved after
processing by uniaxial stretching to 1.5-3.0% strain. After this
stretching, residual stresses on the order of 1-2 ksi (6.8-13.6 MPa)
are expected to exist in the material (Prime, et al., 1999). The
modulus used for this material was E=73.08 GPa, Poisson’s Ratio
ν=0.33, and yield strength of 290 MPa.
The specimen was nominally a 4” (101.6 mm) diameter ring,
with a 2” (50.8 mm) diameter hole in the center and a
corresponding 2” plug. The ring/plug diametrical interference was
approximately 0.005” (0.127 mm). The components were 0.5”
(12.7 mm) thick.
For assembly, the plug was cooled to liquid nitrogen
temperature (-160 °C) leaving approximately 0.0038” (0.0965 mm)
clearance with the room temperature (20 °C) ring. Grease was used
to lubricate the interface and minimize any friction between the
surfaces. Prior to assembly, eight strain gages were mounted on the
ring and used to precisely determine the stress state after the
assembly. Figure 2 shows the locations of the gages, six of which
measured the radial strain component and two of which measured
hoop strains. The gages were placed on the ring near the plug
interface where the strains are the highest. One gage would be
sufficient to uniquely determine the stress state, however, using
multiple gages allow one to confirm that the stress state is
axisymmetric, and to establish uncertainty bounds on the stresses.
strain
gages
Therefore, three holographic pairs of data are sufficient to
determine the plane-stress state of the material being drilled into.
However, in ESPI, the CCD typically consists of 256,000 pixel
elements. Therefore, thousands of data points around the hole are
available to determine the stress state. Currently the software
samples thousands of phase change triads, with each triad
providing an independent stress calculation. The individual results
are averaged to provide the reported stress value.
RING AND PLUG SPECIMEN
An aluminum shrink-fit ring and plug was chosen as the
known residual stress specimen for several reasons. First, this
specimen has a closed form solution for the residual stresses, and
the stress distribution is relatively simple: the stresses are constant
in the plug, and only a function of radial position in the ring.
Second, the ring and plug specimen has appeared routinely in the
literature as a test specimen for evaluating residual stress
measurement techniques, e.g., (Gnaupel-Herold et al., 2000).
Third, this type of specimen provides the full range of biaxial stress
states for testing. In the ring near the interface, σθ is positive and σr
is negative. Near the outer edge of the ring the stress state is
nearly uniaxial since σr goes to zero. In the plug, the stress state is
equi-biaxial, σθ = σr. Therefore, with one specimen it is possible to
demonstrate the ability of the holographic system on three stress
states.
Aluminum 2024-T351 was chosen as the material for the ring
and plug. The 2024 alloy is readily available, well characterized,
and has good yield strength characteristics. The T3 temper refers
Figure 2. Strain gages locations on ring and plug
The stress state of the ring and plug can be calculated (Shigley
and Mitchell, 1998) by first calculating the pressure
p=
(
Eδ Ro2 − Ri2
2 Ri
Ro2
)
(8)
which is taken as positive, and δ is the radial interference between
the ring and plug. In the plug the stress is equal to the pressure in
the radial and angular directions. In the ring, the stress is:
Published in Residual Stress Measurement and General Nondestructive Evaluation, PVP-Vol 429, The
ASME Pressure Vessels and Piping Conference; Atlanta, GA July 23-26, 2001. Ed. Don Bray.
2 

1 − R o 
Ro2 − Ri2 
r 2 
pR 2 
R2 
σ θ = 2 i 2 1 + 2o 
Ro − Ri 
r 
σr =
pRi2
(9)
was added to the pressure calculated from the strain gage readings
to give the “known” pressure of 9.52 ksi, (65.64 MPa) and
calculate the residual stresses plotted in Figure 4.
EXPERIMENTAL SETUP AND PROCEDURE
Equation 10 can be combined with the usual elastic relations to
give ε (p,r), which can be inverted to give p(ε,r) and thus the strain
gages applied to the surface of the sample can be used to calculate
the pressure. The eight measured strains gave an average value for
the interference pressure of p=8.27±0.45 ksi (57.0 MPa ± 3.1). The
measured strains also confirmed that the stresses were
axisymmetric to within the precision of the measurements. Further
confirmation of the accuracy of the strain gage readings is provided
by the measured interference between the ring and the plug before
assembly of 0.0023” ± 0.0003” (0.058 mm ± .0076 mm), which
gives a calculated value of p=9.19 ksi ± 1.16 (63.4 Mpa ± 8.0).
Also, the change in outer diameter of the ring after assembly was
measured as 0.0019" ± 0.0006" (0.048 mm ± 0.015 mm), which
gives a calculated value of p=7.47 ksi ±2 (51.5 Mpa ±13.8). A
pressure of 8.27 ksi was used for comparison to the holography
system measurements, since that value is expected to be more
accurate than the dimensional measurements.
Several measurements of the as-received 2024-T351 plate
using the holography system indicated that the residual stresses
near the surface of the plate were 1.25 ± 0.42 ksi (8.6 Mpa ± 2.9),
consistent with the reported values for residual stress in 2024-T351
aluminum, as reported by ALCOA. The initial stress of 1.25 ksi
Ring & Plug Specimen
For these tests, a 3-foot by 2-foot granite table was used as the
workspace. The video head, illumination head, ring and plug
specimen, and drill assembly were each secured to a 2-foot by 2foot optical breadboard, which in turn was placed onto the granite
table. The laser unit was placed directly behind the optical
breadboard, as shown in Figure 3. The optical breadboard allowed
for easy alignment of the video head and drilling assembly relative
to the ring and plug specimen. The drilling axis and the video head
were placed perpendicular to the surface of the test specimen. The
illumination head was oriented such that the angle between the test
surface and the illumination beam was 25.5°.
The drilling assembly consists of a small air-driven turbine
drill that is mounted to a translator table. The drill operates
between 30,000 and 40,000 rpm. Such drills have been shown to
drill a hole without introducing significant stresses (Flaman and
Herring, 1985). The drilling depth is manually controlled via a
micrometer screw that has a resolution of 0.01 mm per increment.
The bit used was a 1/16” (1.59 mm) diameter square-edged end
mill with 2 flutes. Holes were spaced at least five hole diameters
apart, which means that neighboring holes would change the stress
distribution by less than 1%. Holes were drilled variously to depths
of 0.5 and 1.2 times the hole diameter to confirm that different hole
depths did not produce different results.
Laser Unit
Illumination Head
Drill
Video Head
Figure 3. Experimental setup of residual stress measurement system
and the test specimen
Published in Residual Stress Measurement and General Nondestructive Evaluation, PVP-Vol 429, The
ASME Pressure Vessels and Piping Conference; Atlanta, GA July 23-26, 2001. Ed. Don Bray.
There are several other possibilities for errors in this setup.
Two ratios of hole depth to diameter were used, 0.5 and 1.2. Since
the end mill had a nominal diameter of 0.0625” (1.588 mm), this
means the hole depths were nominally 0.03125” and 0.075” (.79
and 1.91 mm) deep. In practice, it is difficult to achieve this depth
exactly, and so each depth was measured. It was found that in most
cases the holes were within 0.001” (0.0254 mm) of the desired
depth, with a few holes being as far as 0.003” from the expected
depth. In addition, the hole diameters were measured, and found to
be consistently larger than the nominal diameter by about 0.008”
(.212 mm).
This difference between nominal and actual
measurements can have a significant impact on the results, but
since the software is “hard wired” for only specific h/D ratios, the
results in Figure 4 are based on the nominal h/D values The larger
than nominal hole sizes may indicate an eccentric alignment of the
bit head with the drilling axis, or that the drilling axis is not exactly
perpendicular to the axis of the ring and plug. Either of these
situations will contribute to errors. It might be expected that the
error would be completely fixed, but in practice these two
parameters (the eccentricity and perpendicularity) would probably
change every time the drill bit or the part position was changed.
The issue of the drill being not perpendicular to the part may have
error significance of its own, since the original equations were
derived assuming complete perpendicularity.
RESULTS AND DISCUSSION
In order to test the holographic system’s ability to measure
various residual stress states, holes were drilled in many locations
on the surface of the ring and plug. The results of the
measurements with the holographic system and the strain gage
results are plotted in Figure 4. The plug is in a state of equalbiaxial compression, shown by the horizontal line from a distance
of 0 to 1” (0-25.4 mm). The ring has opposed biaxial stress,
compressive in the radial direction and tensile in the hoop direction.
It should be noted that the radial stress must go to zero at the outer
edge of the ring, and therefore the initial 1.25 ksi stress added to the
calculated pressure was reduced exponentially to zero as a function
of radius. Although no error bars have been shown in Figure 4, we
can see that the values measured with the holography system span
the expected value fairly well. The worst discrepancy seems to
occur near the ring/plug interface, especially in the hoop stresses.
We speculate that this is caused by the gradient of stress as a
function of radial position. The original equations are developed
for a stress state with no gradient, and have not been adjusted to
account for this. Qualitatively, it makes sense that this would have
the largest effect on the hoop stresses in the ring near the interface,
since both the gradient and the magnitude are largest there.
Ring and Plug Holographic and "Known" stresses
20.0
18.0
Calculated-strain gage
16.0
14.0
H/D=1.2
12.0
H/D=0.5
Stress [ksi]
10.0
Plug
8.0
6.0
4.0
2.0
0.0
-2.0
-4.0
-6.0
-8.0
-10.0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Distance from plug center to hole center [in.]
Figure 4. Holographic hole-drilling results compared to strain gage
analysis of residual stress
Published in Residual Stress Measurement and General Nondestructive Evaluation, PVP-Vol 429, The
ASME Pressure Vessels and Piping Conference; Atlanta, GA July 23-26, 2001. Ed. Don Bray.
CONCLUSIONS
In general, residual stress measurement with any technique
available today can be expected to have uncertainties of at least ±
2-3 ksi (6.9-20.7 MPa). (Lu, 1996) The data from the holographic
instrument described here certainly seems to be within that range,
and actually seems better than that, although the “known” stress
used in this validation also has an uncertainty of at least ±0.5 ksi
(3.4 Mpa), making quantification of the accuracy of the
holographic method difficult.
We have manually adjusted the analysis to account for the
actual h/D values for 4 different holes (drilled in the ring near the
interface) and found that in each case both the hoop and radial
stresses dropped by about 10%. In future work, the software will
be modified to allow the actual h/D diameter as an input, to account
for variations in hole depth and diameter. The effect of stress
gradient and non-perpendicular holes will also be investigated. In
particular, it should be possible to utilize the holographic
displacement information to back-calculate a value for the stress
gradient.
Direct comparisons with other methods of stress measurement,
(x-ray diffraction and neutron scattering) are also planned for future
studies.
ACKNOWLEDGEMENTS
We would like to thank Professor Drew Nelson for his input during
the software development. Jack Hanlon contributed to the optical
development of the PRISM system during the time this project was
a Co-operative Research and Development Agreement with Los
Alamos National Laboratory.
REFERENCES
Antonov, A. A., 1983, “Inspecting the level of residual stresses
in welded joints by laser interferometry”. Weld. Prod. Vol. 30, pp.
29-31.
Bass, J. K., Schmitt, D., and Ahrens, T.J., 1986, “Holographic in
situ stress measurements”. Geophys. J. R. Astr. Soc. Vol. 85, pp.
13-41.
Diaz F.V., Kaufmann G.H., and Galizzi G.E., 2000,
“Determination of residual stresses using hole drilling and digital
speckle pattern interferometry with automated data analysis”,
Optics and Lasers in Engineering, Vol. 33, pp. 39-48.
Flaman, M. T., and Herring, J. A., 1985, “Comparison of Four
Hole-Producing Techniques for the Center-Hole Residual Stress
Measurement Method,” Experimental Techniques, Vol. 9, pp. 3032.
Gnaupel-Herold, T., Prask, H.J., Clark, A.V., Hehman, C.S.,
Nuygen, T.N., 2000, “A comparison of neutron and ultrasonic
determinations of residual stress”, Measurement Science and
Technology, Vol. 11, pp. 436-444.
Jones, R., and Wykes C., 1989, Holographic and Speckle
Interferometry: A discussion of the theory, practice and application
of the techniques, Cambridge University Press, Cambridge.
Lin S.T., Hsieh C.T., and Hu, C.P., 1994, “Two holographic
blind-hole methods for measuring residual stresses”, Experimenal
Mechanics, Vol. 34, pp. 141-147.
Hung YY, and Hovanesian JD, 1990, “Fast detection of residual
stresses in an industrial environment by thermoplastic-base
shearography”, Proceedings of the SEM Spring Conference on
Experimental Mechanics, Albuquerque, NM, pp. 769-775.
Lu, J., 1996, Handbook of Measurement of Residual Stresses,
Fairmont Press, Lilburn, GA.
McDonach, A., McKelvie, J., MacKenzie, P.M., and Walker,
C.A., 1983, “Improved moiré interferometry and applications in
fracture mechanics, residual stress and damaged composites”,
Experimental Techniques, Vol. 7, pp. 20-24.
Nelson, D. V., and McCrickerd, J. T., 1986, “Residual-stress
determination through combined use of holographic interferometry
and blind hole drilling”, Experimental Mechanics, Vol. 26, pp. 371378.
Nelson, D. V., Makino, A., and Fuchs, E. A., 1997, “The
Holographic-hole drilling method for residual stress
determination”, Optics and Lasers in Engineering, Vol. 27, pp. 323.
Prime, M.B., Jacobson, L., Pacheco, M. 1999, “Residual Stresses
Measured Before and After Stress Relief in Rolled Aluminum
Plate,” Book of Abstracts, 1999 ASME Mechanics and Materials
Conference, R. C. Batra and E. G. Henneke, ed., Blacksburg,
Virginia, pp. 241-242.
Rastogi, P.K., ed., 1997, Optical Measurement Techniques and
Applications, Artech House, Inc., Norwood, MA.
Shigley, J. E., and Mitchell, L. D., 1998, Mechanical
Engineering Design, McGraw Hill, pp. 76-77.
Zhang J., and Chong T.C., 1998, “Fiber electronic speckle
pattern interferometry and its applications in residual stress
measurements”, Applied Optics, Vol 37, No. 28, pp. 6707-6715.
Published in Residual Stress Measurement and General Nondestructive Evaluation, PVP-Vol 429, The
ASME Pressure Vessels and Piping Conference; Atlanta, GA July 23-26, 2001. Ed. Don Bray.