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A typical experimental setup for shearography is shown in Fig. 1. It consists of a light source, typically a laser, an imaging system, a shear device placed either between the object and the lens or between the lens and the recording medium, and a recording medium such as photo-emulsion or a CCD/CMOS imager. The object is illuminated by a laser beam, and its image is formed on the detector via the shear device. Shearing causes a point on the object to be imaged as two points. Alternatively, two points on the object are imaged as a single point. A wave from one point acts as a reference wave to the wave from the second point; hence, there is no need for a reference wave. Between exposures, the object is subjected to an external loading that deforms the object.
Fig. 1 Schematic of a speckle pattern shear interferometer: DL-diverging lens; L-imaging lens; I-image sensor.
The theoretical framework of shearography is based on the same phase difference equation that is valid for both holographic interferometry and speckle pattern interferometry2,3. The optical phase difference equation is given by7:
$$ \phi =\left({\boldsymbol{k}}_{o}-{\boldsymbol{k}}_{i}\right) \cdot {\boldsymbol{L}}={\boldsymbol{k}} \cdot {\boldsymbol{L}} $$ (1) where ko and ki are the propagation vectors in the directions of observation and illumination, respectively, and L is the deformation vector at a point on the object. The vector k [=
$ \left({\boldsymbol{k}}_{o}-{\boldsymbol{k}}_{i}\right) $ ] is called the sensitivity vector. The deformation vector is expressed as$ {\boldsymbol{L}}=u\hat{\boldsymbol{i}}+v\hat{\boldsymbol{j}}+w\hat{\boldsymbol{k}} $ with components u, v, and w along the x-, y-, and z-axes.In shearography, there are two points on the object under consideration: a point P and its sheared counterpart Q. On loading the object, these two points undergo different displacements, as shown in Fig. 2. The point
$ P\left(x,y,z\right) $ is on the undeformed surface, and the point$ Q\left(x+\Delta x,y,z\right) $ is on the sheared surface. These are very close to each other on the two surfaces with a lateral shear$ \Delta x $ along the x-axis. When the object is deformed, these points are displaced to their new locations$ P'\left(x+u,y+v,z+w\right) $ and$ Q'(x+\Delta x+ $ $ u+\Delta u,y+v+\Delta v,z+w+\Delta w) $ . Point P undergoes a displacement$ {\boldsymbol{L}}=u\hat{\boldsymbol{i}}+v\hat{\boldsymbol{j}}+w\hat{\boldsymbol{k}} $ and point Q undergoes a displacement$ {{\boldsymbol{L}}}'=\left(u+\Delta u\right)\hat{\boldsymbol{i}}+\left(v+\Delta v\right)\hat{\boldsymbol{j}}+\left(w+\Delta w\right)\hat{\boldsymbol{k}} $ . Therefore,$ \Delta {\boldsymbol{L}} $ , the change in the displacement vector, is given by$ \Delta {\boldsymbol{L}}={{\boldsymbol{L}}}'-{\boldsymbol{L}}=\Delta u\hat{\boldsymbol{i}}+\Delta v\hat{\boldsymbol{j}}+\Delta w\hat{\boldsymbol{k}} $ .Fig. 2 Original and deformed states of the object showing the location of sheared points on deformation: P is a point on the surface and Q is its sheared counterpart.
On loading, these points displace to P' and Q'. Deformation is very small and hence sensitivity vectors remain unchanged.Following Eq. 1, the phase difference
$ {\phi }_{P} $ between the waves reaching any point on the detector plane when point P displaces to point P' is given by$$ {\phi }_{P}=\left({\boldsymbol{k}}_{o}-{\boldsymbol{k}}_{i}\right) \cdot \left(u\hat{\boldsymbol{i}}+v\hat{\boldsymbol{j}}+w\hat{k}\right)={k}_{x}u+{k}_{y}v+{k}_{z}w $$ (2) where kx, ky, and kz are the components of the vector
$ {\boldsymbol{k}}=\left({\boldsymbol{k}}_{o}-{\boldsymbol{k}}_{i}\right) $ , i.e.,$ {\boldsymbol{k}}={k}_{x}\hat{\boldsymbol{i}}+{k}_{y}\hat{\boldsymbol{j}}+{k}_{z}\hat{\boldsymbol{k}} $ .Similarly, the phase difference between the waves reaching the same point on the detector plane when point Q moves to point Q' is given by
$$ \begin{split}{\phi }_{Q}=&{\boldsymbol{k}} \cdot \left[\left(u+\Delta u\right)\hat{\boldsymbol{i}}+\left(v+\Delta v\right)\hat{\boldsymbol{j}}+\left(w+\Delta w\right)\hat{k}\right]\\=&{k}_{x}\left(u+\Delta u\right)+{k}_{y}\left(v+\Delta v\right)+{k}_{z}\left(w+\Delta w\right) \end{split}$$ (3) Calculations of the phases
$ {\phi }_{P} $ and$ {\phi }_{Q} $ in terms of the coordinates of the source and a point on the detector, and their respective distances, can be found in books237,353 and in several references19,265,314.Because the wave from the sheared point acts as a reference wave to the one from the other point on the surface or vice-versa, there is no need for a reference wave. These two waves produce an interference pattern. Indeed, waves from the scattering points within the resolution element around points P and Q interfere, producing a speckle at the image point. When the entire object is considered, its image is a speckle pattern. A record of the speckle pattern is called a shearogram. Let the amplitude of the waves from point P and point Q at a point
$ \left({x}_{i},{y}_{i}\right) $ at the image plane be$$ {a}_{P}\left({x}_{i},{y}_{i}\right)\propto {a}_{0}\left(x,y\right){e}^{i{\theta }_{P}} $$ $$ {a}_{Q}\left({x}_{i},{y}_{i}\right)\propto {a}_{0}\left(x+\Delta x,y\right){e}^{i{\theta }_{Q}} $$ Both the amplitudes [
$ {a}_{0}\left(x,y\right) $ and$ {a}_{0}\left(x+\Delta x,y\right)] $ and phases [$ {\theta }_{P} $ and$ {\theta }_{Q} $ ] are random variables. The irradiance distribution at the image point can be expressed as$$ \begin{split} &I\left({x}_{i},{y}_{i}\right)\propto {\left|{a}_{0}\left(x,y\right)\right|}^{2}+{\left|{a}_{0}\left(x+\Delta x,y\right)\right|}^{2}+\\&2{a}_{0}\left(x,y\right){a}_{0}\left(x+\Delta x,y\right) \cos\phi ;\phi =\left({\theta }_{Q}-{\theta }_{P}\right) \\I(x_i,y_i)=& {\cal{I}}_{1}\left({x}_{i},{y}_{i}\right)+{\cal{I}}_{2}\left({x}_{i},{y}_{i}\right)+2\sqrt{{\cal{I}}_{1}\left({x}_{i},{y}_{i}\right){\cal{I}}_{2}\left({x}_{i},{y}_{i}\right)} \cos\phi \\ =&{I}_{0}\left(1+\gamma \cos\phi \right) \\[-10pt]\end{split} $$ (4) where
$ {\cal{I}}_{1}\left({x}_{i},{y}_{i}\right) $ and$ {\cal{I}}_{2}\left({x}_{i},{y}_{i}\right) $ are the irradiances of the waves from points P and Q at the image point$ \left({x}_{i},{y}_{i}\right) $ , I0 is the total irradiance (sum of irradiances of both waves), and$ \gamma $ is the modulation. Because the two points P and Q are very close to each other, the amplitudes of the waves from these points can be assumed to be equal. In this case, the irradiance distribution can be expressed as$$ I\left({x}_{i},{y}_{i}\right)={I}_{0}\left(1+ \cos\phi \right) $$ Usually, two exposures are made in shearography: the first exposure when the object is in its undeformed state and the second exposure after the application of load, that is, when the object is in its deformed state.
The irradiance distribution in the first exposure can be expressed as
$$ {I}_{1}\left({x}_{i},{y}_{i}\right)={I}_{0}\left(1+ \cos\phi \right) $$ (5) On loading, the surface deforms, and points P and Q respectively move to points P' and Q'. The waves from points P' and Q' acquire additional phases
$ {\phi }_{P} $ and$ {\phi }_{Q} $ respectively.The irradiance distribution recorded now is given by
$$\begin{split} {I}_{2}\left({x}_{i},{y}_{i}\right)=&2{a}_{0}^{2}\left[1+ \cos\left\{\left({\theta }_{Q}+{\phi }_{Q}\right)-\left({\theta }_{P}+{\phi }_{P}\right)\right\}\right]\\=&{I}_{0}\left[1+ \cos\left(\phi +{\Delta \phi }_{QP}\right)\right]\end{split}$$ (6) where
$ {\Delta \phi }_{QP}=\left({\phi }_{Q}-{\phi }_{P}\right) $ is the phase difference between the two waves from points Q and P at a point on the recording plane. By rewriting the phase difference$ {\Delta \phi }_{QP} $ , we obtain$$\begin{split}& {k}_{x}\left(u+\Delta u\right)+{k}_{y}\left(v+\Delta v\right)+{k}_{z}\left(w+\Delta w\right)-\\&({k}_{x}u+{k}_{y}v+{k}_{z}w)={k}_{x}\Delta u+{k}_{y}\Delta v+{k}_{z}\Delta w \end{split}$$ (7) Assuming that the magnitude of the shear is very small, the phase difference can be expressed as
$$ \begin{split}{\Delta \phi }_{QP}=&\left({\phi }_{Q}-{\phi }_{P}\right)=\left({k}_{x}\frac{\partial u}{\partial x}+{k}_{y}\frac{\partial v}{\partial x}+{k}_{z}\frac{\partial w}{\partial x}\right)\Delta x\\=&\left({\boldsymbol{k}}_{o}-{\boldsymbol{k}}_{i}\right) \cdot \frac{\partial {\boldsymbol{L}}}{\partial x}\Delta x={\boldsymbol{k}} \cdot \frac{\partial {\boldsymbol{L}}}{\partial x}\Delta x \end{split}$$ (8) For a very small shear,
$ {\Delta {\boldsymbol{L}}}/{\Delta x} $ is expressed as$ {\partial {\boldsymbol{L}}}/{\partial x} $ . The phase difference depends on the derivatives of the components of the deformation vector if an exceedingly small lateral shear is used. In practice, the difference quotient is measured rather than the derivative. If the magnitude of the shear is made very small such that it approximates a derivative, the deformation phase$ {\Delta \phi }_{QP} $ tends to become very small, and hence the sensitivity of the technique becomes poor. It is possible to employ other shear types, e.g., radial shear and theta shear, and obtain the radial derivative$ \left({\partial w}/{\partial r}\right) $ 28,29 and the theta derivative$ \left({\partial w}/{\partial \theta }\right) $ 28,29.Although almost all publications discuss the theory of shearography along with other aspects, including applications, several papers are devoted only to the theoretical aspects of the technique70,71,76,166,168,224.
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In photographic recording, the irradiance distributions I1(xi, yi) and I2(xi, yi) are recorded sequentially over a time period T. The recorded exposure E is expressed as
$$ E={E}_{1}+{E}_{2}=2{I}_{0}T\left[1+ \cos\left(\varnothing +\frac{{\Delta \phi }_{QP}}{2}\right) \cos\left(\frac{{\Delta \phi }_{QP}}{2}\right)\right] $$ (9) where T is the time of exposure. Thus, the record contains a speckle pattern modulated by
$ \cos\left({{\Delta \phi }_{QP}}/{2}\right) $ . The term$ \cos\left({{\Delta \phi }_{QP}}/{2}\right) $ carries the information of the derivative of the deformation vector that an object has suffered due to loading. This information, which is in the form of a fringe pattern, is extracted by Fourier filtering9,23,55. Usually, laser light is used for Fourier filtering, but in some cases, filtering can also be performed with white light55. The fringe pattern is speckled, and the fringes have a low contrast with a cos2-type distribution. The fringes can be sharpened by multiple exposures or by making use of the nonlinearity of the recording process31,33,36,48. The contrast of the fringes is improved using apertures in front of the lens, albeit at the expense of loss of light13,29,34,36.If the shearogram corresponding to the undeformed state of an object recorded on photo-emulsion is repositioned exactly after development, the deformation derivatives can be observed in real time as the object is deformed74,93. The theoretical framework of real-time shearography is similar to that of real-time holographic interferometry: there is a phase change of
$ \pi $ in the fringe pattern as compared to double-exposure shearography168. Real-time shearography has been used to determine fractional fringe order by the translation of a wedge plate110. Instead of recording a shearogram on a photographic plate, the use of photorefractive crystals is suggested for real-time shearography282. To extract information pertaining to$ {\Delta \phi }_{QP} $ , a doubly exposed shearogram is Fourier filtered9,13, resulting in a fringe pattern. These fringes can be sharpened by making multiple exposures or by making use of the nonlinearity of the recording process. -
In electronic recording, the two exposures corresponding to I1(xi, yi) and I2(xi, yi) are handled independently. In one processing method, the exposure corresponding to I2(xi, yi) is subtracted pixel-by-pixel from I1(xi, yi). The voltage output of the detector is proportional to [I1(xi, yi)-I2(xi, yi)]. The brightness on the monitor is proportional to the voltage output of the detector, and therefore it can be expressed as
$$\begin{split} B\propto \left[{I}_{1}\left({x}_{i},{y}_{i}\right)-{I}_{2}\left({x}_{i},{y}_{i}\right)\right]=&{I}_{0}\left[ \cos\phi - \cos\left(\phi +{\Delta \phi }_{QP}\right)\right]\\=&2{I}_{0} \sin \left(\phi +\frac{\Delta {\phi }_{QP}}{2}\right) \sin \frac{{\Delta \phi }_{QP}}{2}\end{split} $$ (10) For negative values of
$ \sin {{\Delta \phi }_{QP}}/{2} $ , the brightness on the monitor will be zero. To avoid this loss of signal, the output signal is squared before being displayed, and therefore the brightness B on the display monitor is expressed as$$ B\propto 4\;{I}_{0}^{2}{ \sin }^{2}\left(\phi +\frac{{\Delta \phi }_{QP}}{2}\right){ \sin }^{2}\left(\frac{{\Delta \phi }_{QP}}{2}\right) $$ (11) The term
$ { \sin }^{2}\left(\phi +{{\Delta \phi }_{QP}}/{2}\right) $ contains the random phase, and therefore it represents the speckled portion of the brightness distribution on the monitor, which is modulated by the term$ { \sin }^{2}\left({{\Delta \phi }_{QP}}/{2}\right) $ . There would be dark regions, called fringes, wherever$ {{\phi }_{QP}}/{2}=m\pi $ for integer values of m. Substituting for$ \Delta {\phi }_{QP} $ , we obtain the condition for the formation of dark fringes as$$ \begin{split}\Delta {\phi }_{QP}=&\left({k}_{x}\frac{\partial u}{\partial x}+{k}_{y}\frac{\partial v}{\partial x}+{k}_{z}\frac{\partial w}{\partial x}\right)\Delta x\\=&\left({\boldsymbol{k}}_{o}-{\boldsymbol{k}}_{i}\right) \cdot \frac{\partial {\boldsymbol{L}}}{\partial x}\Delta x={\boldsymbol{k}} \cdot \frac{\partial {\boldsymbol{L}}}{\partial x}\Delta x=2m\pi \end{split}$$ (12) The fringe pattern arises because of the gradient of the deformation vector. The fringe visibility depends on the correlation between the two interfering waves20,61,256.
It can be seen from Eq. 10 that the DC term is eliminated by the subtraction process. In another recording scheme, two speckle patterns with a phase difference of
$ \pi /2 $ are obtained simultaneously using polarized beam illumination for the deformed and undeformed states of the object122. Because the phase difference is fixed, a fast Fourier transform (FFT) can be used to eliminate the DC terms of the speckle patterns. In the early 1980s, TV cameras were used to record shearograms, but were followed by CCD and CMOS cameras. Ng and Chau examined the noise-related performance of CCD cameras used for shearography84. Osten et al.232 demonstrated that CMOS cameras produce better shearograms than CCD cameras, even for non-cooperative surfaces under examination. -
Let an object be vibrating sinusoidally with frequency
$ \omega $ and amplitude w. The average irradiance distribution Iav recorded over a period much longer than the period of vibration is expressed as9,63,123,155,226,265,348,366$$ {I}_{av}\left({x}_{i},{y}_{i}\right)={I}_{0}\left(1+\gamma \cos\phi {J}_{0}\left(\mathrm{\Omega }\right)\right) $$ (13) where
$ \mathrm{\Omega }=({4\pi }/{\lambda })({\partial w}/{\partial x})\Delta x $ when the observation and illumination directions are anti-parallel and the object is illuminated normally,$ \Delta x $ is the shear, and J0(x) is the Bessel function of zero order. The output from the CCD camera will be proportional to the irradiance distribution, and therefore the fringe pattern on the monitor will have a strong speckle background. Furthermore, the zero-order Bessel function causes only a few fringes to be observed. However, stroboscopic illumination circumvents this problem: the fringes are now cosinusoidal and the background can be removed by subtraction, as is done in the static case117,123,183. With stroboscopic illumination, the shearogram can be frozen at any instant of illumination by short stroboscopic pulses synchronized with the frequency of the vibrating object, and the phase-shift technique can be applied to obtain the derivative of the vibration amplitude155,197,251. Shearography with stroboscopic illumination and a large shear is used to measure small out-of-plane vibration amplitudes over a large range251. To measure transient vibrations, Hung et al.125 used a high-speed camera, and the images were stored in the memory of the high speed acquisition system.In another method to improve the visibility of time-averaged fringes, time-averaged shearograms are captured sequentially. The processor outputs a signal that produces the brightness on the monitor, which is proportional to86,120,162,226,314
$$ B\left(x,y\right)\propto {J}_{0}^{2}\left(\mathrm{\Omega }\right)\propto {J}_{0}^{2}\left(\frac{4\pi }{\lambda }\frac{\partial w\left(x,y\right)}{\partial x}\Delta x\right) $$ (14) The visibility of the fringes will be maximum where
$ {\partial w\left(x,y\right)}/{\partial x}=0 $ and will decrease rapidly as$ {\partial w\left(x,y\right)}/{\partial x} $ increases. Hence, a zero-order Bessel fringe will occur where the amplitude is maximum. The zero-order Bessel fringe can be shifted if one of the mirrors of the shearing Michelson interferometer is excited with the same frequency as that of the object86 or by using the scheme suggested by Valera and Jones120, Valera et al.124, and Chatters et al.107. The zero-order Bessel fringe can also be shifted by wavelength modulation135,162,314. However, this requires an unbalanced Michelson interferometer for shearing. It has been shown that a single Bessel fringe pattern obtained under subtraction operation could be sufficient to calculate the phase using a phase recovery method, based on genetic algorithms280. Methods for improving the visibility of fringes were described by Chen et al.226. For the study of transient vibrations such as those arising from impact, the use of a double-pulse laser is suggested185,220. Steichen et al.185 also mention the use of a double-flash CCD camera that may record two shearograms separated in the range of 100 ns to 50 ms. Procedures to use single-pulse shearography and double-pulse shearography for the study of vibrating objects were described by Spooren et al.80. Another study reported the use of a double-pulse laser for shearographic recording in which a carrier frequency is generated by changing the curvature of the illumination beam between the impact stressing of the metallic plates194,196,220. A procedure to measure damping using shearography was described by Wong and Chan195. A method that produces binary phase patterns where the phase changes are related to the zeros of the Bessel function is presented for estimating the vibration amplitudes from the shearographic fringe patterns366. -
As pointed out in the introduction, a speckle pattern interferometer can be configured to be sensitive either to an in-plane displacement component or an out-of-plane displacement component. The same is true for a speckle pattern shear interferometer, i.e., it can be configured to yield fringes pertaining either to the in-plane derivative (strain) or out-of-plane derivative (slope).
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There are two distinct possibilities, as shown in Fig. 3a, 3b. In one arrangement, the object is illuminated by collimated beams lying either in the y-z plane or x-z plane, which are symmetric to the normal at a point on the object surface, and the observation can be along any direction but preferably along the normal to the surface.
The formation of dark fringes in digital shearography is described by Eq. 12, which has been rewritten as
$$ \left({\boldsymbol{k}}_{o}-{\boldsymbol{k}}_{i}\right) \cdot \frac{\partial {\boldsymbol{L}}}{\partial x}\Delta x=2m\pi $$ (15) When two symmetric directions of illumination are used, as shown in Fig. 3a, the fringe formation is governed by
$$ \left({\boldsymbol{k}}_{o}-{\boldsymbol{k}}_{i2}\right) \cdot \frac{\partial {\boldsymbol{L}}}{\partial x}\Delta x-\left({\boldsymbol{k}}_{o}-{\boldsymbol{k}}_{i1}\right) \cdot \frac{\partial {\boldsymbol{L}}}{\partial x}\Delta x=2{m}_{1}\pi -2{m}_{2}\pi =2m'\pi $$ $$ \left({\boldsymbol{k}}_{i1}-{\boldsymbol{k}}_{i2}\right) \cdot \frac{\partial {\boldsymbol{L}}}{\partial x}\Delta x=2m'\pi $$ (16) where m' is an integer that takes values 0, ±1, ±2, ±3, ···. Assuming that the beams lie in the x-z plane and make an angle of
$ \pm \theta $ with the surface normal, then$ {\boldsymbol{k}}_{i1}={2\pi }/{\lambda } $ $ \left( \sin \theta \hat{\boldsymbol{i}}- \cos\theta \hat{\boldsymbol{k}}\right) $ and$ {\boldsymbol{k}}_{i2}={2\pi }/{\lambda }\left(- \sin \theta \hat{\boldsymbol{i}}- \cos\theta \hat{\boldsymbol{k}}\right) $ . Hence,$$\begin{split} \left({\boldsymbol{k}}_{i1}-{\boldsymbol{k}}_{i2}\right) \cdot \frac{\partial {\boldsymbol{L}}}{\partial x}\Delta x=&\frac{4\pi }{\lambda } \sin \theta \frac{\partial u}{\partial x}\Delta x=2m'\pi \to \to\\ \frac{\partial u}{\partial x}=&\frac{m'\lambda }{2 \sin \theta \Delta x}=\frac{\lambda {\Delta \phi }_{QP}}{4\pi \sin \theta \Delta x}\end{split} $$ (17) The sensitivity of the principal strain fringe pattern depends on both the magnitude of the shear and the angle of illumination. This method has been implemented experimentally in several ways, including using a Michelson interferometer136,152,164,187,206,236,345,351, a birefringent crystal133,211,265, a birefringent wedge160, and a Wollaston prism379 for shearing. When a wedge plate covers half the aperture of an imaging lens or when a two-aperture or multi-aperture mask with wedges is placed in front of an imaging lens, the recording contains information about the in-plane component along with the derivatives of the displacement. This has been studied by Ng and Chau75, Mohan et al.91, Mohan and Sirohi166, and Wang et al.224.
Similarly, if for a given direction of illumination, two directions symmetric to the surface normal are used for observation, the configuration is in-plane gradient sensitive. Assuming that the directions of observation lie in the x-z plane and make an angle of
$ \pm \alpha $ with the local normal, the fringe formation is governed by$$\begin{split} \left({\boldsymbol{k}}_{01}-{\boldsymbol{k}}_{02}\right) \cdot \frac{\partial {\boldsymbol{L}}}{\partial x}\Delta x=&\frac{4\pi }{\lambda } \sin \alpha \frac{\partial u}{\partial x}\Delta x=2m\pi \to \to \\\frac{\partial u}{\partial x}=&\frac{m\lambda }{2 \sin \alpha \Delta x}=\frac{\lambda {\Delta \phi }_{QP}}{4\pi \sin \alpha \Delta x} \end{split}$$ (18) In practice,
$ \theta $ can take values approaching 900, but the angle$ \alpha $ usually takes much smaller values limited by the lens aperture. However, the experimental arrangement can be configured so that angle$ \alpha $ can also take large values169,379,392.When the shear is along the y-direction and the beams are confined to the y-z plane, the y-gradient of the in-plane component under these configurations is given by
$$ \frac{\partial v}{\partial y}=\frac{m\lambda }{2 \sin \theta \Delta y}=\frac{\lambda {\Delta \phi }_{QP}}{4\pi \sin \theta \Delta y} $$ (19) and
$$ \frac{\partial v}{\partial y}=\frac{m\lambda }{2 \sin \alpha \Delta y}=\frac{\lambda {\Delta \phi }_{QP}}{4\pi \sin \alpha \Delta y} $$ (20) where
$ \Delta y $ is the shear. To realize this experimentally, an opaque plate consisting of several openings is placed before the imaging lens. Each opening carries either a shear plate, or a ground glass plate, or just a plate for compensation27,32,34,39,169. This arrangement generates the carrier frequency and is well adopted for Fourier filtering whether the recording is photographic or electronic. A three-aperture arrangement with one aperture carrying a wedge plate produces an in-plane fringe pattern and a combination of in-plane and derivative fringes166. The influence of in-plane displacement on slope fringes was examined by Mohan et al.91,166. It has also been pointed out that a single illumination direction will also give the in-plane derivative when loading is such that there is no out-of-plane deformation or there is a sequential illumination from two directions163. It has been mentioned that an in-plane sensitive configuration can be devised that yields all four in-plane strains independent of out-of-plane strains163. -
It is obvious from the phase difference Eq. 12 that the configuration will be sensitive to the gradient of out-of-plane deformation if the directions of illumination and observation are anti-parallel
$ \left({\boldsymbol{k}}_{i}=-{\boldsymbol{k}}_{o}\right) $ and are along the normal to the surface. In that situation,$$ \left({\boldsymbol{k}}_{o}-{\boldsymbol{k}}_{i}\right) \cdot \frac{\partial {\boldsymbol{L}}}{\partial x}\Delta x=\frac{4\pi }{\lambda }\hat{\boldsymbol{k}} \cdot \frac{\partial {\boldsymbol{L}}}{\partial x}\Delta x=\frac{4\pi }{\lambda }\frac{\partial w}{\partial x}\Delta x $$ (21) Hence
$$ \frac{\partial w}{\partial x}=\frac{m\lambda }{2 \Delta x}=\frac{\lambda {\Delta \phi }_{QP}}{4\pi \Delta x} $$ (22) When the shear is along the y-direction, we have
$$ \frac{\partial w}{\partial y}=\frac{m\lambda }{2 \Delta y}=\frac{\lambda {\Delta \phi }_{QP}}{4\pi \Delta y} $$ (23) Orthogonal slopes, or a slope with double the sensitivity and curvature, can be obtained using a four-aperture arrangement in which two wedges are appropriately placed at the two opposite apertures and compensating plates on the remaining apertures34. Sirohi and Mohan170 described a two-aperture configuration that is insensitive to in-plane deformation. This configuration is used to obtain the slope fringes. Using a similar configuration, the sensitivity to slope change was increased two-fold by coating the object with a retro-reflective paint181. It was shown that the visibility of the correlation fringes increases when the object surface is covered by a microsphere of 10.2μm mean diameter230. Using two CCD cameras in a Michelson interferometer for shearing, both
$ {\partial w}/{\partial x} $ and$ {\partial w}/{\partial y} $ are obtained for the same load on the object188. -
Phase shifting, either temporal or spatial, is employed when quantitative data are required about the strain field or defects. In many cases, phase maps are presented even for the visualization of gradients of the deformation vector. Temporal phase-shifting was first introduced in shearography by Owner-Petersen61, followed by Kadono et al.62. Present-day shearographic instruments are equipped with phase shifting. To obtain the phase values at each pixel a minimum of three irradiance values are required, with different phase values in a
$ 0\;{\rm{to}}\;2\pi $ interval. Some algorithms use more than three irradiance values. In temporal phase shifting, these irradiance values are captured sequentially by changing the phase difference between the interfering waves. If three phase steps are used in the interval$ 0\;{\rm{to}}\;2\pi $ such that the phase is stepped by$ 2\pi /3 $ , that is, the irradiance distribution in the interference pattern is captured at 0,$ 2\pi /3 $ and$ 4\pi /3 $ , then the phase$ \phi $ at each pixel is obtained from62,97,112,141,248,281,324$$ \phi ={\tan}^{-1}\frac{\sqrt{3}\left({I}_{3}-{I}_{2}\right)}{\left({2I}_{1}-{I}_{2}-{I}_{3}\right)} $$ (24) where I1, I2, and I3 are the irradiance distributions captured at the 0,
$ 2\pi /3 $ and$ 4\pi /3 $ phase steps, respectively. When four phase steps are used in the interval$ 0\;{\rm{to}}\;2\pi $ with a phase step of$ \pi /2 $ , the phase$ \phi $ at each pixel is obtained from61,105,179,197,202,265$$ \phi ={\tan}^{-1}\frac{\left({I}_{4}-{I}_{2}\right)}{\left({I}_{1}-{I}_{3}\right)} $$ (25) where I1, I2, I3, and I4 are the irradiance distributions captured at the 0,
$ \pi /2,\;\pi $ and$ 3\pi /2 $ phase steps, respectively. The phase$ \phi $ is a wrapped phase and must be unwrapped. Unwrapping poses no problem when the shear is small. However, there are situations in which variable shear is required to obtain a derivative map. Brug198,202 presented a method to obtain derivatives in real time using a Michelson interferometer for shearing and two CCD cameras, where shear varies with time.The five-step algorithm is also used because it is robust to noise and insensitive to calibration errors214. Wu et al.320 described a method of in situ calibration of phase shift by employing an additional optical element in the Michelson interferometer. Common phase-shift algorithms used in digital shearography were enumerated by Zhao et al.360.
The phase steps can be obtained by translating a PZT-mounted mirror97,111,176,338,339, or using a polarization-based phase-shifter160,202,209,275,284,295,305, or a liquid crystal cell62,265,273, or a Wollaston prism along with a liquid crystal cell305, or a Wollaston prism along with HiBi fiber wrapped around a PZT cylinder96,120,124,162,179, or a combination of a quarter-wave and a rotating half-wave plate275, or by translation of a diffractive optical element146,361 or by source wavelength modulation141,152,223. A lateral shift of the source with a PZT provides a phase shift that has been incorporated in phase-shift algorithms149. It has been shown that rotation of an object introduces an additional phase that varies linearly with the angle of rotation262. The rotation of the object has been used for phase shifting.
The temporal phase-shift method is susceptible to external disturbances, such as vibration, temperature fluctuation, or rapid motion of the test object itself. The spatial phase shifting (SPS) technique is a simple method to eliminate external disturbances. In spatial phase shifting (SPS), a carrier frequency is introduced such that there are three or four pixels between the fringes depending on whether a three-step or four-step phase-shift algorithm is used150,271,283. In one publication, a four-step error-compensating algorithm was used to generate the phase map271. Alternately, the Fourier transform method can be applied to obtain the phase difference between the undeformed and deformed states of an object325,326,345,351,372,377,380,382. It is desirable that the zeroth order and the desired first-order spectra should be separated. This determines the spatial frequency of carrier fringes. A shearographic setup in which a Michelson interferometer is embedded for shearing usually employs temporal phase shifting. However, it can be used for spatial phase shifting if a CCD camera with a sufficiently large number of pixels is employed325.
A study of the mode shapes of a turbine blade excited by a piezoelectric shaker at different frequencies was carried out by DSPI at a wavelength of 10.6 μm and by shearography at a wavelength of 532 nm to assess which of the two techniques could be used in an industrial environment348. Some interesting results were presented in this paper348. Measurements of the modal rotation fields of an engineering object, e.g., a beam with single and multiple damages, are obtained with shearography, and an optimal sampling technique is used to improve damage localization330,352,374. Because the sensitivity of shearography depends on the magnitude of shear, a numerical study of damage localization as a function of shear has also been reported355. A new method for structural damage identification using cubic spline interpolation has been described376. The method is based on the interpolation of modal rotations measured using speckle shearography. The signal-to-noise ratio (SNR) in shearography as a function of the magnitude of shear on the modal rotation fields has also been investigated356.
To study time-dependent deformations, it is better to capture information in a single frame. For this reason, the Fourier transform method for processing data is preferred. However, it requires a carrier frequency, so that various Fourier spectra are separated. Several methods have been proposed to provide carrier frequency in the shearogram47,53,82,126,194,196,222,301,330,370,372,382. Carrier frequency fringes are generated by changing the curvature of the illuminating beam before the second exposure or the second frame47,167,221 or by a small rotation of the object171. When a Mach-Zehnder interferometer is used as a shearing device, a parallel shift of the mirror introduces shear, and the rotation of the mirror produces the carrier frequency246,330. The use of multi-aperture with wedge plates in front of the imaging lens automatically creates a carrier frequency. Bhaduri et al.291,294 employed three-aperture shearography with two wedge plates to obtain a curvature phase map, and with a single wedge plate both the displacement and the displacement gradient using the Fourier transform method. Carrier frequency fringes have also been used for data reduction44,53, to locate and size the debonds in GRP plates69, to obtain surface coordinates and slopes73,159, to obtain flexural strains40, and to determine the order of a fringe47.
Joenathan et al.190 developed a method in which the object is deformed continuously, and a large number of sheared images of the object deformation are acquired using a high-speed CCD camera. The derivative of the object deformation is then retrieved from this large set of data using Fourier transformation190,272. This method is capable of obtaining information for object displacements over 500 μm. In another study, a continuous wavelet transform was applied to extract the phase change from a series of shearograms276. In addition to the Fourier transform and the windowed-Fourier transform methods of phase recovery from a shearogram, some researchers have demonstrated other methods such as the curvelet transform for edge detection328, Hilbert-Huang transform340, and wavelet transform358. A heterodyne shearographic system in which orthogonally polarized beams are frequency shifted by
$ \pm {\omega }_{s} $ is described for obtaining derivative information in real time341,358. Frequency-shifted beams illuminate the object, which is imaged through a Wollaston prism sandwiched between two polarizers on the CCD camera. The output of the CCD is a heterodyne signal over which the difference phase rides. The difference phase information is extracted using wavelet transformation and a proper filter. Andhee et al.260 carried out a comparative study of conventional and phase-stepped shearography. Conventional shearography is good for making qualitative observations in real time, while phase-stepped shearography delivers quantitative information161. A procedure to maintain sub-pixel alignment between shearograms obtained with a single camera in a polarization-based phase-stepped two-bucket shearing interferometer has been described267. The procedure was based on cross-correlating the two shearograms. A more recent paper discusses an improvement of the two-bucket shearing interferometer when dealing with the unfavorable polarization states that arise when a rough metallic surface is illuminated with a linearly polarized light284. Recently, a theory of surface phase-resolved shearography has been presented that considers speckle statistics and delivers less noisy specklegrams373.
Shearography and its applications – a chronological review
- Light: Advanced Manufacturing 3, Article number: (2022)
- Received: 16 July 2021
- Revised: 21 December 2021
- Accepted: 25 December 2021 Published online: 14 January 2022
doi: https://doi.org/10.37188/lam.2022.001
Abstract: This paper presents the activities in the field of shearography in chronological order and highlights the great potential of this holographic measurement technology. After a brief introduction, the basic theory of shearography is presented. Shear devices, phase-shift arrangements, and multiplexed shearography systems are described. Finally, the application areas where shearography has been accepted and successfully used as a tool are presented.
Research Summary
Shearography: A technique for non-destructive testing
Shearography is a displacement gradient sensitive, full-field optical technique that is resilient to environmental disturbances and vibrations and is capable to examine large structures. It can be used on shopfloor as well in field. The major application of the technique is the non-destructive inspection of laminates. It has been applied to examine components and systems in aerospace and automobile industries, and to inspect art objects like paintings for conservation. The review article presents the evolution of the technique, various optical configurations, recording procedures, and applications.
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