Determination of Residual Stresses by Thermal Relaxation
and Speckle Correlation Interferometry

M. J. Pechersky
Westinghouse Savannah River Company
Aiken, South Carolina 29808


This document was prepared in conjunction with work accomplished under Contract No. DE-AC09-96SR18500 with the U.S. Department of Energy.


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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.



A = dimensionless empirical coefficient in equation 20.
B = dimensionless empirical coefficient in equation 20.
E = Young’s modulus (N/m2).
F = force (N).
d = dimension of heated spot (m).
k = spring constant (N/m).
l = length of spring ki (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/m2).

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.


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 ki which can take on values of k C or kH depending on whether the spring is cold or has been heated, respectively.

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 X1, with a corresponding force F. F and hence X1 are unknown. In Figure 2B, ki is heated and deforms due to: thermal expansion (xth), and elastic (xy) and plastic (xp) deformation. When the system has cooled, some permanent deformation remains due to plastic flow. Spring ki, has a permanent deformation x3 and, the total deformation is (X3 - X1), in Figure 2C.

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, ki is shown in Figure 3. The spring is assumed to consist of an elastic/perfectly plastic material. This assumption is reasonable even for materials that work harden since the total plastic deformation is comparable to the elastic portion. Curve (a) in Figure 3 shows the force displacement curve for prior to heating. Curve (b) shows the force/displacement curve for ki when it is at the elevated temperature TH. The force on the spring must drop to the yield force and its displacement is restrained only as a result of its interaction with the parallel spring, . Curve (c) shows the force/displacement diagram after the spring has cooled. The elastic portion of Curve (c) has the same slope as curve (a) but it is translated due to the plastic deformation. The object of the analysis based on this simple model is to express the unknown force, F in terms of the measurable displacement (X3 - X1). F will then be transformed to a stress in the final steps of the derivation.

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, kC is extended to x1 and the entire system is stretched to a point X1. The total force, F is:

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, TC to a higher temperature, TH. The heating process is assumed to be slow enough so that no pressure waves or other dynamic effects occur, but fast enough so that the other springs are not heated. When this heating process is completed, the force versus displacement diagram for ki is altered as shown in Figure 3, curve (b). In state 1, ki is stiffer than in state 2 and its spring constant is kC. It also has a higher yield stress (force). In state 2 the spring constant kH is lower. The force at yield in state 2 is designated as FY.

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 ki and, D T = THTC. Second, ki is elastically deformed until the yield stress corresponding to FY is reached. This deformation is given by:

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

where xp is the plastic deformation, and the total deformation is X2. Now the force on


and the force on both ki and is

By eliminating X2 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:


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 xp. But, xp is not a measurable quantity. The measurable quantity is:

Finally, ki cools back to ambient temperature so that the thermal expansion is relaxed to zero but the permanent deformation, xp, remains. Thus, the final the final deformation of the system is X3 and the final deformation of ki is x3. This situation is shown in Figure 2C and in Figure 3 by the curve labeled (c). Now the forces on the springs are:


x3 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 xp from equation 8 so that:

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 EC is simply written as E. Similarly;

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 d, the dimension of the heated spot and empirical coefficients A and B are included then equation 18 becomes:

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 B are determined by experiment or analysis as previously mentioned.

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 X can be measured and substituted into the following equation:

In our case D X is averaged over the length L so that e is the average strain over L. Thus using equations 20 and 21, a noncontacting measurement can be obtained. One of course must know the yield stress, thermal expansion coefficient, and Young’s modulus as well as A and B to measure an unknown residual stress. The values of A and B for uniaxial test specimens have been obtained earlier in references 10 and 11. A and B values for compressive bending have also been measured. A specific example of such a measurement is provided below.

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:

  1. Choose a region of interest and apply a spot of temperature indicating liquid. The diameter of the spot is d.
  2. 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.
  3. Heat the spot d with a CO2 laser until the temperature liquid indicating liquid undergoes its phase change.
  4. Allow the region to cool. (approximately 15 minutes)
  5. Acquire a second speckle pattern and form an image by computing the absolute difference in the value of the corresponding pixels in each image.
  6. Determine the average fringe spacing , and compute the strain, e from Equation 21.
  7. 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 CO2 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 CO2 laser beam is directed by way of a pair of turning mirrors {16 } and {17} onto the spot which is painted on the specimen. The CO2 laser power is monitored with a pyroelectric detector {5} at the rear of the laser. One of the mirrors {16} is flipped out of the way when specklegrams are being collected. Depending on the desired heated spot size, focusing optics may be placed in the optical path of the CO2 laser beam.

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.


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


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