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Optics and Lasers in Engineering 27 (1997) 61-13 Copyright 0 1996 Elsevier Science Limited

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Measurement of Residual Stress by Phase Shift Shearography

M. Y. Y. Hung, K. W. Long & J. Q. Wang

Department of Mechanical Engineering, Oakland University, Rochester, MI 48309, USA

ABSTRACT

This paper presents a method for evaluating residual stresses. The approach is based on measuring the deformation due to the relief of stress produced by a drilled-hole or a ball indentation, and the deformation is rapidly measured by digital shearography. This method does not require mounting strain gages/transducers. Unlike holography, shearography does not require special vibration isolation. These features make the method practical for evaluation of residual stresses in a productionlfield environment. Copyright 0 1996 Elsevier Science Limited

1 INTRODUCTION

The presence of residual stresses in load-resisting structures/components can be detrimental. These stresses are often caused by the fabrication and/or installation process; it also may be induced during the service life of the structure. When the service stress is superimposed onto the already present residual stress, the total stress may exceed the design stress limits of the strucutre. Many structural failures are the consequence of this type of combined stress; therefore the determination of residual stress is an important task in engineering. One common method’” uses strain gages to measure the radial strain caused by the relieved stress around a drilled hole. There are several disadvantages of this method. The mounting of special strain gages is costly and time consuming. In addition, a special precision guide is needed for locating the center of the hole and experimental results show that a small misalignment can cause significant

61

62 Y. Y. Hung, K. W. Long, J. Q. Wang

errors4 Second, the strain gages are located at a distance from the edge of the hole. The sensitivity of this method is reduced because the change in strain decreases rapidly as the distance from the edge increases. As a result of these drawbacks, this method is not very suitable for mass inspection in industry.

Optical methods have also been investigated for residual stress meas- urement. Underwood5 suggested optical-interferometry to measure the surface displacements around a shallow indentation. The non-symmetrical fringe pattern indicates the presence of residual stress. This method is not practical, as the indented area has to be polished to optical quality. Nelson & McCrickerd6 used the combination of holography and blind hole drilling to measure residual stress. This method eliminates the disadvan- tages of the two previously mentioned methods. However, the strenuous requirement on vibration isolation still limits the wide use of this method in a production environment. The use of shearography for measuring residual stresses was demonstrated by the authors in Ref. 7. Shearography alleviates the stringent vibration isolation requirement of holography. However, the previous shearographic measurement was based on the photographic technique that not only requires photographic processing but also the subsequent Fourier optical filtering to display fringe patterns. Moreover, human interpretation of the fringe pattern is needed.

This paper demonstrates the use of digital shearography to detect residual stress. The change in the out-of-plane displacement caused by a drilled hole is recorded by digital shearography.* Digital shearography employs a video camera as a recording medium and digital image processing is used to process and form the fringe pattern. The fringe pattern depicts the surface displacement around the vicinity of a hole region as a result of the relieved residual stress. The phase distribution in the fringe pattern is automatically determined by a phase shift procedure.

2 DESCRIPTION OF DIGITAL SHEAROGRAPHY

Contrary to the conventional film-based shearography’ which uses a wedge as an image shearing device, digital shearography employs a birefringent crystal to produce the shearing effect. A schematic diagram of digital shearography is illustrated in Fig. 1. The test object is illuminated with laser light via a single mode optical fiber and it is imaged by a video image-shearing camera. A frame grabber installed in a microcomputer is used to digitize and process the acquired images. By digitizing and comparing the two speckle images of the test object before and after deformation, a fringe pattern depicting the displacement derivative

Measurement of residual stress

Diverging Laser Beam

Device Polaroid

Fig. 1. Schematic diagram of phase shift shearography.

distribution is generated by the computer in real-time and it may be

63

instantly displayed on a video monitor. The illumination optics and the camera can be an integrated piece detached from the laser, the camera control circuit and the micro-computer. This arrangement provides easy mobility and flexibility desirable in an industrial environment. A VCR can be incorporated into the system to allow continuous recording of test data as well as recording of experimental details through voice input.

3 PRINCIPLES OF DIGITAL SHEAROGRAPHY

For completeness, a review of digital shearography is given. Since the object is coherently illuminated, the two images interfere with each other producing a resultant image intensity distribution given by:

I = Z,(l + cos 4) (I)

where Z, is the object image and I$ is a random phase function. Thus, the object’s image is modulated by (1-t cos 4) which represents a random interference pattern commonly referred to as speckle pattern.

When the object is deformed, an optical path change occurs due to the surface displacement in the object. This optical path change produces a

64 Y. Y. IIun~, K. W. Long, J. Q. Wmg

relative phase change between the two interfering points. Thus, the intensity distribution of the speckled-image is slightly altered to:

I’ = I,,[ 1 + cos (4 -I A)] (2)

where I’ is the intensity distribution after deformation, and A is the relative phase change due to the relative displacement between two neighboring points on the object surface.

The undeformed speckled image I is first digitized into the computer via a frame grabber. Digital subtraction of the two speckle images yields:

I, = IJcos (4 + A) - cos 41 (3)

where Z, is the intensity of the subtracted image. Since the intensity of an image cannot be negative, absolute values of the subtraction will be displayed. The above equation shows a fringe pattern is formed, with the dark fringe lines corresponding to:

A=Nlr (4)

where N = 0, 2, 4, 6, 8, . . . are the dark fringe orders, and odd integers correspond to bright fringes.

The above subtraction operation can be performed by the hardware built-into a real-time frame grabber which requires less than &s. Therefore, the resultant fringe pattern can be generated and displayed in real-time (i.e. at video rate). Frame grabbers with real-time subtraction capability are commercially available.

It can be shown that A is related to the derivatives of displacement by:

(5)

where (u, u, w) are the components of the displacement vector; A is the wavelength of the laser light; 6x is the amount of shearing which is assumed to be parallel to the x-direction. A, B and C are sensitivity factors which are related to the position of the illumination point S(xs,y5,x,), the camera position (x,,y,,, z,,) and the point P(x, y, z) on the test object by:

(6)

where R?, = (x - xJ2 + ( y - ycJ2 + (z - z,,)‘, and R? = (x - x$ + (y -- y,)* +

(z - zJ2.

Merrsuremenl of residltnl swesss 65

The shearing direction determines the direction of the displacement derivatives. The derivatives in eqn (5) become the derivatives with respect to y should the shearing direction be parallel to the y-axis. It should be noted that if the magnitude of shearing is large, the derivatives in eqn (5) become the relative displacements between two neighboring points on the object separated by a distance equal to the amount of shearing. The latter arrangement will be used in the determination of residual stresses.

4 PHASE SHIFT TECHNIQUE

The phase shift technique is used to determine the phase distribution in fringe patterns. This technique eliminates the need of human interpreta- tion of the fringe pattern. A fringe pattern of shearography may be generalized as:

Z, = u + h cos (A) (7)

In general, there are three unknowns at each point in a fringe pattern. The unknowns are: a-the DC intensity, b-the modulation of the inter- ference fringes and the fringe phase A. Therefore, three measurements are generally needed to determine the phase. Phase measurement is based on superimposing a uniform phase on the original fringe pattern, thus producing a phase shift in the fringe pattern. Digitizing three fringe patterns with different amounts of phase shift provides three equations for the solution of the three unknowns, and the phase distribution is thus determined. Multiple phase shift algorithms are also available for phase determination. A summary of the various phase determination algorithms can be found in Ref. 10.

The ability to apply the phase determination algorithms in digital shearography relies on a phase shift technique. With a Michelson shearing interferometer, phase shifting is performed by translation of the mirrors of the interferometer. In this paper, a novel phase shift technique” is employed by simply translating the shearing crystal in a direction perpendicular to line of sight. To simplify the analysis, it is assumed that a collimated beam is used to illuminate the object. The amount of phase shift can be controlled at will by the amount of crystal translation. Within a small phase shift range of 0-27~ the amount of the phase shift is proportional to the translation of the shearing crystal. The proportionality is determined by a calibration procedure. In the calibration, the birefrin- gent crystal is precisely translated by a micro-translation stage to produce a phase shift of 2n which can be easily observed on a real-time shearographic fringe pattern.

66 y. y. flung, K. W. Long, J. Q. Wafl~

Here, the four-frame method is used. By digitizing four fringe patterns with

Z, = a + b cos (A)

I2 = a + b cos

I3 = a + b cos (A + x)

I4 = a + b cos

The phase distribution of the fringe pattern can be determined by:

12 - I, A = arctan -

[ 1 4 - 4

(8)

(9

The computed phase is wrapped into the range of -x and x. An unwrapped phase distribution is then determined by the phase unwrap- ping algorithm proposed by Macy.”

In Fig. 2(a), a fringe pattern of a plate with clamped edges loaded by pressure is shown. Figure 2(b) shows the 3D plot of the phase distribution of Fig. 2(a) determined by the phase measurement technique.

5. RESIDUAL STRESS DETECTION

A schematic diagram for the residual stress measurement is shown in Fig. 3. A relatively large shearing is employed in the residual stress detection. In this case, the technique basically compares the deformation of two different regions on the object surface. In essence, one region acts as a reference beam for the other. The fringe pattern produced thus depicts the relative displacement between the two regions. Since both the reference beam and the object beam are derived from the same object, this set-up still enjoys the advantage of not requiring special vibration isolation. Furthermore, the technique is insensitive to rigid body tilting of the object. This is because the reference region and the test region rotate the same amount thus compensating each other. Note that rigid body tilting produces a set of parallel linear fringes. Since the deformation due to stress relief by hole drilling is very localized, the other region is hardly affected and thus it serves as a reference beam. The technique therefore measures the absolute out-of-plane displacements around the neighbour- hood of the hole. With coincident illumination and viewing directions

Measurement of residual stress 67

1

._ a0

-1

-2 40 -

50

Fig. 2. (a) Fringe pattern of a plate with clamped edges loaded by pressure. (b) 3D plot of phase distribution.

which are parallel to the z-axis, the fringe formed is related to the out-of-plane displacement w, of the hole region by:

A = y [wc, + 6x, y) - w(x, y)] (10)

where SX is the amount of shearing.

hX Y. Y. Ih~g, K. W. Longv J. Q. Wang

Ret&me Area

Object

Diverging Laser Beam

Shearing Device Polaroid

cl- Frame

Grabber

Computer

Fig. 3. Schematic diagram of residual stress measurement.

When a hole is drilled into a loaded structure, the stresses at that location are relieved because the radial and shear stress at the boundary become zero. The change in stresses will result in a change in the in-plane and out-of-plane displacement. Referring to Fig. 4, and by using the stress function approach,” the change in out-of-plane displacement can be expressed as:

CT, CLt w=----- I 2 g cos 29 E r’ 1

(11)

where W = out-of-plane displacement, CL = Poisson’s ratio, u1 = uniaxial stress, r = thickness of the specimen, ~1 = radius of the hole, K = radial dimension to the point of interest, r = R/a, and 8 = angular rotation from stress direction to point of interest.

The out-of-plane produced by the relief of a two-dimensional state of residual stress may be obtained by the principle of superposition. That is:

w = tfra - a.)Pf [

2Cos28 E r.7 J

(12)

In this case U, and u’, arc principle stresses. Therefore, by measuring MI at any point in the vicinity of the hole, ((rA - CT,) can be determined.

Merrswement of residunl stress 69

Fig. 4. Thin plate with hole under biaxial stresses.

In the testing, the object illuminated with laser light and imaged by the video image-shearing camera is first digitized and stored in the memory of a frame grabber. A hole is then drilled in the object. Subtracting the deformed image from the stored image produces a fringe pattern depicting the relative displacement described by eqn (10). The presence of residual stresses may be revealed from the fringe pattern. Figure 5 shows a fringe pattern due to a hole drilled in a sample with tensile stress. With a real-time frame grabber, the fringe pattern can be formed and displayed

Fig. 5. Fringe pattern by hole drilling.

70 Y. Y. Hung, K. W. Long, J. Q. Wang

Fig. 6. Phase distribution of fringe pattern at selected segment.

on a monitor instantly at video rate. Thus the speed is high enough to meet the high inspection rate encountered in a typical industry setting.

The phase distribution of the fringe pattern can be automatically determined by the phase shift technique. By digitizing four fringe patterns with the known uniform phases superimposed on the original fringe pattern, the phase distribution and hence the out-of-plane displacement can be readily determined. Figure 6 shows the phase distribution determined by the phase shift technique. The overlay plot on the figure displays the out-of-plane displacement along a selected line segment.

For fast and qualitative analysis of residual stresses, a micro-indentation can be used instead of the blind hole. The fringe pattern due to an indentation will be disturbed by the presence of residual stress. Figure 7(a) shows a fringe pattern caused by an indentation without the presence of residual stress whereas Fig. 7(b) shows the fringe pattern due to indentation and the relief of residual stresses. It should be noted that the fringe pattern is more or less axisymmetric in the absence of residual stress. The principal directions of the residual stresses are indicated by the axes of symmetry while the stress magnitudes are given by the degree of deviation. The method serves as a fast but qualitative means of detecting residual stresses. Quantification of the stresses relieved by the indentation is a difficult task due to the complexity of the mechanics of the plastic indentation. Nevertheless, the hole-drilling stress relief method may be used to quantify residual stresses.

Measurement of residual stress

(4

Fig. 7. (a) Fringe pattern of indentation without residual stress. (b) Fringe pattern of indentation with residual stress.

6 COMPARISON WITH THE STRAIN GAGE METHOD

Even if the method uses a blind-hole to measure residual stress, the optical method has several advantages over the strain gage method. These advantages can be summarized as follows:

72 Y. Y. Ilung, K. W. Long. J. Q. Wmg

l full-field measurement can be obtained by the optical method: l the alignment of hole drilling is not critical; l less cost and time required because no special preparation and tools are

needed; l the optical method is more sensitive, as it allows the displacement

measurement very close to the edge of the hole.

7 CONCLUSION

Digital shearography eliminates photographic recording and processing: it also eliminates the subsequent Fourier filtering process to view the fringe pattern. The application of the phase shift technique to digital shearog- raphy allows the phase distribution to be determined automatically and without need of human interpretation of the fringe pattern. The prelimi- nary results of this optical method for detecting residual stress show great potential for employment in an industrial environment. The residual stress test method is closer to being non-destructive with the use of a ball indentation as the indentation can be very small. However, the quantita- tive determination of residual stresses by the ball indentation is difficult. The blind-hole method should be used whenever quantitative analysis is needed. The main advantages of this method are summarized as follows:

l simple optical set-up, . relaxed vibration isolation requirement, l does not require special surface preparation, l electronic recording allows the result to be obtained instantly, without

consumable materials, and l does not require highly skilled operator since the process can be

automated.

With the above mentioned merits, it can be concluded that is feasible to employ phase shift shearography for evaluating residual stresses in an industrial environment.

REFERENCES

1. Rendler, N. J. & Vigness, I., Hole-drilling strain-gage method of measuring residual stress. Exp. Mech. 6 (1966) 57746.

2. Stcmdard Method for De[ermining Residual Srrm hp the Hole-drilling Strain (;crge Method, Stand. E837-81. American Society of Test and Material, Philadelphia (19X1 ).

Merrsrrrcwwnt 01 rcChtI .strcw 73

3. Mtwwremwts of Resirhrtrl Strtw by the Blind Hole Drillirtg Mrthod. Technical Data Bulletin T-503, Photoelastic Division, Measurements Group, Inc., Raleigh, NC, 1YYO.

4. Wang, H.-P., The alignment error of the hole-drilling method. EXII. Mech. 19 (1979) 23-7.

5. Underwood, J. H., Residual-stress measurement using surface displacements around an indentation. Exp. Me&. (1973) 373-80.

6. Nelson, D. V. & McCrickerd, J. T., Residual-stress determination through combined use of holographic interferometry and blind-hole drilling. Exp. Mech. ( 1 YX6) 371-37X.

7. Hung, Y. Y., Long, K. W. & Hovanesian, J. D., Nondestructive detection of residual stresses by shearography. In Proc. VI Int. Congress on Experimentul Mechunics, June 1988.

8. Hung, Y. Y., Nondestructive evaluation by electronic shearography. In 16th Synlposiw~l on Nondestructive Evuluution, San Antonio, Texas, 21-23 April 10x7.

9. Hung, Y. Y., Shearography: a new optical method for strain measurement and nondestructive testing. Opt. Engng 21 (lYX2) 391-5.

10. Creath, K., Phase-measurement techniques for nondestructive testing. In Proceedings of SEM Conf~wnce on Hologrum Inte~feronzetry and Speckle Metrology, Baltimore, Maryland, 5-8 Nov 1990, pp. 473-8.

11. Hung, Y. Y. & Wang, J. Q., Dual-beam digital shearography for full-field measurement of in-plane displacement derivatives. In Proceedings of thr IVY_5 SEM Spring Conference. Grand Rapids, Michigan, 12-14 June 1YYS. pp. 770-90.

12. Macy, W. W.. Two-dimensional fringe pattern analysis. Appi. Optics 22 ( I YX3) 3X08.

1.3. Timoshenko, S. P. & Goodier, J. N., Thtwr\~ of f<lrr.stic~it>~, 2nd cdn., (‘hapter 4. McGraw-Hill, New York, (IYSI).