WSRC-MS-2001-00301

Determination of Residual Stresses by Thermal Relaxation

and Speckle Correlation Interferometry

M. J. Pechersky

Westinghouse Savannah River Company

Aiken, South Carolina 29808

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

This report has been reproduced directly from the best available copy.

Available for sale to the public, in paper, from: U.S. Department of Commerce, National Technical Information Service, 5285 Port Royal Road, Springfield, VA 22161, phone: (800) 553-6847, fax: (703) 605-6900, email: orders@ntis.fedworld.gov online ordering: http://www.ntis.gov/support/ordering.htm

Available electronically at http://www.osti.gov/bridge/

Available for a processing fee to U.S. Department
of Energy and its contractors, in paper, from: U.S. Department of Energy, Office
of Scientific and Technical Information, P.O. Box 62, Oak Ridge, TN 37831-0062,
phone: (865 ) 576-8401, fax: (865) 576-5728, email: reports@adonis.osti.gov

** **

Abstract

A new technique for the measurement of residual stresses is presented. The technique is based on strain measurements following thermal stress relaxation. The heat input is supplied by a low power infrared laser and the strain is measured with speckle pattern correlation interferometry. This paper presents a comprehensive overview of the technique and an example of how it has been applied in a practical situation.

Notation

__Symbols__:

A = dimensionless empirical coefficient in equation 20.

B = dimensionless empirical coefficient in equation 20.

E = Young’s modulus (N/m^{2}).

F = force (N).

d = dimension of heated spot (m).

k = spring constant (N/m).

*l* = length of spring k_{i } (m)

L = gauge length (m).

t = thickness (m)

T = Temperature (C)

X = total displacement (m).

x = displacement of intermediate point (m).

a = thermal expansion coefficient (m/m-C).

e = strain (m/m).

l = wavelength of laser light (m).

q = angle of incidence (radian).

s = stress (N/m^{2}).

__Subscripts and Overscripts__:

^ indicates specific spring constant in Figures 1 and 2.

~ indicates specific spring constant in Figures 1 and 2.

C indicates cold state.

H indicates hot state.

i = H or C and indicates hot or cold state.

Y = conditions a yield.

1 indicates initial state.** **

Introduction

It is well known that residual stresses can effect the fatigue strength, the corrosion behavior and or the fracture toughness of many engineering materials. Residual stresses may be defined as "those stresses existing without (and generally prior to) the application of any intended or, unintended external loads". Residual stresses are induced by fabrication processes such as welding, rolling and machining. It is possible to modify residual stresses by, for example shot peening to reduce tensile stresses or to induce compressive stresses on material surfaces. Often times one would like to assess the residual stresses in a structure to determine if stress modification is required and to quantify the effectiveness of the residual stress mitigation technique after its application.

One of the more common residual stress measurement techniques that depends
on stress relaxation is blind hole drilling(BHD). In this approach, a strain
gauge rosette is affixed to a region of interest and a small whole is drilled
through the substrate containing the rosette and into the material. The resultant
strain is measured and applied to a semi-empirical model to determine the stresses
present prior to the hole drilling. Another stress relaxation technique is known
as the crack compliance method or slot drilling in which single slots^{ }
or multiple slots are cut into a specimen. Another variation of this technique
is the ring core method. Again the resultant strain relief is measured and used
to deduce the stress present prior to the material removal. Examples of methods
that do not rely on stress relaxation for determining residual stresses are
x-ray diffraction, and neutron diffraction, as well as magnetic and ultrasonic
techniques.

The measurement technique described in this paper is a noncontacting stress relaxation technique. Rather than removing material, heat is applied to the object and the stress is relieved by plastic flow. This differs from the stress relaxation techniques mentioned above in that elastic deformation is assumed to occur in those processes. This new measurement technique has the advantage of not requiring any material removal and involves low temperatures (~ 200C) and hence is essentially nondestructive. It is also fast compared to material removal approaches. Furthermore, there is probably much less energy imparted into the object from a small amount of heating as compared to the energy involved in material removal by drilling, cutting or abrasive techniques. These aforementioned factors may be attractive attributes for field measurements. On the other hand, interpreting the measured results requires knowledge of the thermal properties of the material and a better estimate of its yield stress than for the material removal techniques.

A discussion of the theoretical basis of the measurement technique will follow this introduction, including the mathematical development of the semi-empirical stress model that is used to analyze the measured strains. Then an example of how this measurement was applied as part of a weld failure analysis will be presented. Finally some general comments will be made with regard to the range of applicability of this measurement technique and potential improvements.

The Laser Stress Relief Technique

This new method, which is a thermo-optical analogue to BHD is described in the following paragraphs. This method uses local heating to relieve stresses in a small spot and then uses laser speckle interferometry to measure the resulting strains. This strain, which is measured in a somewhat larger region surrounding the heated spot, is used to determine the state of stress prior to heating. The peak temperatures are on the order of 200C so that for most materials there will be no changes in phase or other material properties except for a slight local reduction in yield stress and hardness. Preliminary experiments with type 304 stainless steel were performed using resistance heating. The experimental results were in excellent agreement with finite element model predictions of the process. Subsequently, the resistance heating was replaced with laser heating. The heat input (22.5 Watt peak) from a sealed radio frequency excited Carbon Dioxide laser was used. In order to both control the heating temperature and efficiently couple the infrared photons from the laser into the test specimen, a substance known as Liquid Temperature Indicating Liquid was used. Without this substance the laser power would be so large as to make this approach impractical. Furthermore the measurement and control for the heat input would be very complicated. Using this laser heating approach with the Temperature Indicating Liquid defines the peak temperature to within ± 1C with virtually no effort. The Temperature Indicating Liquid changes phase at a predetermined temperature and exposes the underlying metal surface which is reflective to the laser beam. This essentially shuts down the heat input to material at a well defined temperature without the need to tightly control the laser input.

Since this laser based technique is a thermo-optical analogue to blind hole drilling a simple stress model is required to interpret the measured results. This simple stress model is developed and presented below. As in BHD, the simple model must be modified by empirical coefficients to be useful. These empirical coefficients can be determined by experimentation and/or numerical analysis as can be done in blind hole drilling.

Although the approach for this laser method is similar to that used in BHD, there are important differences . The BHD model is based on the solution of linear elastic, plane stress field equations for the stress and strain in the vicinity of through hole in a flat plate. The BHD model calculates the change in the elastic strain field between a plate that is uniformly stressed at its boundary with and without a through hole in it. The stress model presented here is based on a lumped parameter model in which elastic/perfectly plastic deformation is assumed to occur as a result of the heating.

Consider the one dimensional lumped parameter model of a general solid as shown
in Figure 1. This figure represents an elastic body in plane stress as being
composed of four springs. The bulk of the body has a stiffness, **k**. The
region where this stress is being measured is comprised of a series/parallel
combination of springs which itself is attached in parallel to **k**. The
heat is applied to the spring designated as **k _{i}** which can take
on values of

In principle it is possible to solve this problem using conservation of energy
and statics. This solution method leads to a transcendental equation which is
not very useful for our purpose. Furthermore, it would be difficult to characterize
a general solid body in this way. Instead we make the assumption that the "residual
force," **F ** does not change during the annealing process. This eliminates
the need to invoke conservation of energy and leads to a simple result which
can easily be recast into the desired residual stress versus strain relief relationships.
As mentioned earlier, constant stress is also assumed in the BHD model. For
the sake of brevity the springs will be referred to by the symbolic representations.

Figure 1. Lumped Parameter Representation of
a Structure

for Residual Stress Measurements.

The constant force assumption leads to a the lumped parameter model as shown
in Figure 2. The system is shown in Figure 2A at its initial position with a
total displacement **X _{1}**, with a corresponding force

Figure 2. Displacement of the Three Spring System
Resulting From the Local Annealing. (A)

Prior to heating, (B) At the end of heating and (C) after cooling.

The mechanical properties of the heated spring,** k _{i}** is shown
in

Figure 3. Force versus displacement diagram for
*ki*. (a) prior to heating,

(b) at the end of heating, (c) after cooling.

Initially (Figure 2A) the system is in state 1, *k _{C} _{ }*is
extended to

where is the force on the series
spring combination containing and
*ki*, and is the force on .
In terms of the total displacement:

is the stiffness of the three spring system.

During the heating process the temperature of ** ki** is raised form
ambient temperature,

During the heating process *ki* deforms in three ways. First the spring
expands due to thermal expansion by an amount,

where a is the thermal expansion coefficient,
** l** is the total length of

Finally, the spring deforms plastically, so the total deformation of the spring, neglecting interactions between these three processes is:

where ** x_{p} **is the plastic deformation, and the total
deformation is

and the force on both *ki* and is

By eliminating X_{2} in equations 6 and 7 and remembering that the
total force, *F* is assumed to be constant, the plastic strain can be expressed
as follows:

where:

The above expression gives the plastic deformation in terms of the unknown
force and the system geometry and properties and can easily be rearranged to
give *F* in terms of *x _{p}*. But,

Finally,** ki **cools back to ambient temperature so that the thermal
expansion is relaxed to zero but the permanent deformation,

* *

* x_{3}* can be eliminated by combining equations 11 and
12, applying the assumption that the total force remains constant one more time,
and substituting the expression for

Finally, substitution of equations 2 and 2a for *F* in equation 13 yields:

This is the result that we wished to obtain. It expresses the unknown force, F as a function of: the applied temperature difference and the measured displacement, with the material properties and geometry as parameters.

Our next step is to express this result in terms of stress and strain. Figure
4 shows a square of dimension *L* with a smaller heated square of dimension
d at its center. The square is assumed to be in a state of uniaxial stress;
s . Where s
is a principle stress since it is the only stress on the body From this geometry
Equation 14 can be recast into an expression in terms of the Young's modulus
at the elevated and ambient temperatures and the strain by defining the spring
constants in terms of the geometry and properties of the heated and unheated
regions of the square. For example if the heated region is considered by itself,
then;

Figure 4. Region of Measurement of Residual Stresses.

where *t* is the plate thickness, and as before *i* can take on the
a value of *C* or *H*. The subscript *C* is suppressed when expressing
Young's modulus at the ambient temperature. That is *E _{C } *is
simply written as

where mirror symmetry is used about the vertical centerline and shearing forces have been neglected. Substitution of equations 15, 16 and 17 into equation 14 and combining with 14a and 14b, yields:

which is, of course the average strain. If the ** L** which the distance
over which the strain is measured is substantially larger than

Equation 20 is the final result. It is the semi-empirical equation that can
be used to for measuring the residual stress. The coefficients ** A**
and

Laser Speckle Correlation Interferometry

The strain for the residual stress measurements is measured with Laser Speckle
Correlation Interferometry. Similar holographic techniques ^{ }as well
as laser speckle have also used to determine strains in blind hole drilling.
Laser speckle is a well established measurement technique. In this application,
two symmetrical laser beams of wavelength, l
which are derived from the same laser source illuminate the surface at an angle**
q **relative to the surface normal. An
image is captured prior to the heating pulse from the infrared laser. After
the heat has been applied and the object has cooled a second image is captured.
When the images are differenced, as set of fringes becomes visible. The spacing
between the fringes, D

In our case D ** X** is averaged over
the length

Evaluation of Residual Stresses in a Closure Weld

The details of a measurement of residual stressed near a Gas Tungsten Arc Weld(GTAW) which was performed on a 304L stainless steel container are described in this section. These measurements were performed after a through weld defect was discovered. As a part of the failure analysis it was questioned whether residual stresses played a role in the failure. A typical measurement sequence consisted of the following steps:

- Choose a region of interest and apply a spot of temperature indicating liquid.
The diameter of the spot is
.*d* - Acquire a laser speckle pattern in a region surrounding the spot. The length
of the speckle pattern surrounding the spot in the measurement is approximately
equal to but slightly larger than
.*L* - Heat the spot
with a CO*d*_{2}laser until the temperature liquid indicating liquid undergoes its phase change. - Allow the region to cool. (approximately 15 minutes)
- Acquire a second speckle pattern and form an image by computing the absolute difference in the value of the corresponding pixels in each image.
- Determine the average fringe spacing , and compute the strain, e from Equation 21.
- Substitute this value of strain into Equation 20 along with the relevant
material properties and the empirical coefficients A and B to compute the
residual stress, s.

Experimental Apparatus

The electronic speckle pattern interferometer (ESPI) is shown photographically
in Figure 5 and schematically in Figure 6. The interferometer is based on the
usual in-plane configuration as described in reference 17. The light source
for the interferometer is a 10 mW helium neon laser. The heat is applied by
the RF excited CO_{2} laser {7}. * (The numbers in the brackets {#}
refer to the item numbers in Figure 6. ) * The schematic in this figure
shows a four point bend load frame {18,19,29,21} which was used for calibration
for the bending stresses mentioned earlier. The CO

A 2 arc minute wedge prism {23} is located in one of the two interferometer beams illuminating the specimen. The wedge prism is rotated 180 degrees about its optic axis after the reference specklegram is collected but prior to heating the specimen. The rotation of the wedge prism introduces a set of "carrier" fringes on the final speckle interferogram. When the reference specklegram and the specklegram of the stress relieved specimen are differenced, the carrier fringes are algebraically added to the to the fringes formed by the material deformation. Since the wedge angle is fixed, the carrier fringe frequency does not vary from one test to the next. The carrier fringe frequency was approximately 0.45 fringes/mm.

In Figure 5, the load frame for calibration is shown on the left and the carbon dioxide laser can be seen on the right. During the measurements on the 304L stainless steel container, the load frame was replaced with a jack stand on which the container was placed. For circumferential stresses the container is placed in a vertical position and for the stresses along the length of the canister, it was placed on its side. The change in the carrier fringe frequency indicates whether the stresses are tensile or compressive.

**Figure 5 Photograph of Laser Residual Stress
Measurement Apparatus**

**Figure 6. Laser Residual Stress Test Schematic**

Exerimental Results

Figure 7 shows some of the measurement locations on the container. There is masking tape wrapped around the circumference of the container with the major markings at approximately 25.4 mm (1 inch) intervals. The numbered spots on the container are approximately 3.5 mm in diameter. This image was collected subsequent to testing so that all of the spots have been heated with the infrared laser. Some remnants of the temperature indicating liquid can be seen around the periphery of the heated regions. The largest diameter of the container is approximately 115 mm and its height is about 120 mm. The wall thickness is approximately 3 mm. The weld is at the top of the container and is made by inserting a hollow plug, whose wall thickness is also about 3 mm into the opening at the top and then doing the GTA weld through the outside wall. A portion of the plug and container are then removed with a pipe cutter.

Figure 7. Photograph of Container

Figure 8 shows a speckle gram of the measurement region near the weld. The bright center spot is the Liquid Temperature Indicator. The dark horizontal line in the image is the shadow created from the gap between the top of the weld and a metal ring that is used for personal protection during handling of the canister. This gap can be seen near the top of the canister in Figure 7. For these measurements the canister was positioned with this metal ring on the bottom. The specklegram subsequent to the rotation of the wedge prism is not shown since it appears identical to Figure 8. The difference between the images is that the wedge prism that was mentioned earlier was rotated 180 degrees about its optical axis. This causes a slight angular shift in the laser beam that passes through the wedge to be deflected slightly causing phase shifts in the speckle pattern without any apparent change in intensity distribution. When these two specklegrams are differenced and contrast enhanced a fringe pattern results as seen in Figure 9. Note that the images are spatially calibrated during the data collection so that the values of L can be determined. The fringe order is determined by selecting a location on the fringe pattern (say the center of a dark fringe) and counting the number of subsequent equivalent points in the horizontal distance L. In practice L is chosen such that there are an integral number of fringes. These measurements are performed on a computer with image analysis software and proceed fairly quickly. In Figure 9 the strain corresponding to the fringes shown is about 0.0002.

Figure 8. Specklegram of region prior to heating

Figure 9. Carrier Fringes

The difference image after the container after the spot was heated and the allowed to cool is shown in Figure 10. The equivilant strain in Figure10 is about 0.00013. This is less than the strain indicated by the carrier fringes.

Figure 10. Speckle Interferogram subsequent to cooling

The net strain is simply the difference between these two strains which is
-0.000068. The corresponding residual stress at this point is then computed
from Equation 20. The values of the A and B taken from reference 19 and used
in Equation 20 were 3.02´ 10^{-4} and
0.733, respectively. The calculated value is about 15.5 ksi in compression,
based on an assumed compressive yield stress of 46,500 psi at room temperature.
Twenty seven points were analyzed in this way and the only significant residual
stresses were in the weld region. Both the circumfirential and longitudinal
residual stresses were compressive in the weld region and were too low to detect
elsewhere.** **

Concluding Remarks

A new method to measure residual stress using a thermo-optical analogue to
blind hole drilling has been presented. A practical application of this technique
as it applies to weld residual stress measurements was also described. There
is still much room for further development of this technique. As of now, it
has only been demonstrated on Austenitic stainless steel. While this type of
material is fairly common, it would be worthwhile to apply this technique to
other materials. Two dimensional residual stresses have also been measured with
this technique. The main limitations at present have to do with the thermal
diffusivity and thermal expansion coefficient of engineering material. Austenitic
stainless steels have fairly low thermal diffusivities as compared to aluminum
and carbon steels. This implies that higher power lasers and higher surface
temperatures will be required for many other materials. Fortunately higher power,
sealed tube, air cooled carbon dioxide lasers are now commercially available
so that extension of this technique to these materials appears to be practical.
Another important area of improvement is the ability to obtain depth profiles
of residual stress. The current approach integrates the residual stressed to
a depth of about 1 mm. Varying the laser energy and spot size are possible ways
to achieve this goal.** **

Acknowledgment

This work was sponsored by the United States Department of Energy, Contract
Number DE-AC09-963R18500.** **

References

- Nickola, W. E., "Weld induced residual stress measurements via the hole-drilling strain gauge method," Report 84-WA/DE-25, American Society of Mechanical Engineers, New York (1984).
- Determining residual stresses by the hole-drilling strain-gauge method," ASTM Standard E837-92, American Society for Testing and Materials, Philadelphia (1992).
- Measurement of residual stresses by the hole-drilling strain-gauge method." Technical Note TN 503-4, Measurements Group, Raleigh, NC, (1993).
- Prime, Michael B., "Residual stress measurement by successive extension of a slot: The crack compliance method," Applied Mechanics Reviews, Vol. 52, No. 2 pp. 75 –96, 1999.
- Petrucci, G. and B. Zuccarello, " Modification of the Rectilinear Groove Method for the Analysis of Uniform Residual Stresses," Experimental Techniques, November/December 1997.
- Schajer, G.S., George Roy, Machael T. Flaman, Jian Lu, "Hole-drilling
and Ring Core Methods, Chapter 2" in
*Handbook of Measurement of Residual Stresses*, by the Society for Experimental Mechanics, Inc. Ed. Jian Lu, Fairmont Press, Inc. Lilburn, GA, USA, (1996), pp. 5 – 34. - P.S. Prevey, "X-ray diffraction residual stress techniques," in
*Metals Handbook ®, Vol. 10, Materials Characterization, 9*, American Society for Metals, Metals Park, OH (1981).^{th}ed. - Holden, T. M., G. Roy, "The Application of Neutron Diffraction To the
Measurement of Residual Stress and Strain, Chapter 3" in
*Handbook of Measurement of Residual Stresses*, by the Society for Experimental Mechanics, Inc. Ed. Jian Lu, Fairmont Press, Inc. Lilburn, GA, USA, (1996), pp. 133-148. - C.O. Ruud, "A review of nondestructive methods for residual stress measurements," J. Metals, 33(4), pp. 35-40 (1981).
- Pechersky M. J., Robert F. Miller, Chandra S. Vikram, "Residual Stress Measurements with Laser Speckle Correlation Interferometry and Local Heat Treating," Optical Engineering, Vol. 10 No. 10, October 1995.
- Vikram, C. S., M.J. Pechersky, C. Feng and D. Engelhaupt, "Residual Stress Analysis by Local Laser Heating and Speckle Correlation Interferometry," Experimental Techniques, Vol. 20, No. 6, Nov./Dec. 1996, pp. 27-30.
- OMEGALAQ ® Temperature Indicating Liquid, Omega Engineering, Inc. Stamford, CT., USA
- Schajer, G. S., "Measurement of Non-uniform Residual Stresses Using the Hole Drilling Method, Part II – Practical Application of the Integral Method," Transactions of the ASME, Vol. 110, pp. 344 – 349.
- Nelson, D. V., J. J. McCrikerd, "Residual-stress determination through combined use of holographic interferometry and blind-hole drilling," Experimental Mechanics, 26(4), 1986, pp. 371 – 378.
- Hsieh, C. T., S. T. Lin, C. K. Lee, "In-Plane Residual Stress Measurement with One Axisymmetric Phase-Shifting Holographic Blind-Hole Fringe Pattern," Proceedings of the International Conference on Applications of Optical Holography, SPIE Vol. 2577, June 1995.
- Zhang, J., "Two-dimensional in-plane electronic speckle pattern interferometer and its application to residual stress determination," Opt. Eng. 37(8), pp. 2402 – 2409, 1998.
- Jones R. and C. Wykes, C.,
*Holographic and Speckle Interferometry, 2*Cambridge University Press, Cambridge, UK (1989)^{nd}Ed., *Speckle Metrology*, Ed. Sirohi, R. S., Marcel Dekker, Inc., New York, NY, (1993).- Pechersky, Martin J., Edwin Estochen and C. S. Vikram, "Measurement
of Residual Stresses with Through Thickness Gradients Using Laser Heating
and Speckle Interferometry,"
*Proceedings of the SEM Annual Conference on Theoretical, Experimental and Computational Mechanics*, Cincinnati, Ohio, June 7 -9, 1999, pp. 746 – 749. - Vikram, C. S. and M. J. Pechersky, "Wedge Prism for Direction Resolved Speckle Correlation Interferometry," Optical Engineering, Vol. 38, No. 10, Oct. 1999, pp. 1743-1747.
*Image Pro Plus*, Media Cybernetics, Inc. Silver Springs, MD, USA- Pechersky, M. J., P.S. Lam and C. S. Vikram, "Bi-directional Residual
Stress Measurements with Laser Annealing and Speckle Interferometry,"
*Proceedings of the SEM Spring Conference on Experimental and Applied Mechanics and Experimental/Numerical Mechanics in Electronic Packaging III*, Houston, Texas, June 1-3, 1998, pp. 184-186.