Real-Time and Quantitative Measurement of Crack-Tip Stress Intensity Factors Using Digital Holographic Interferometry

Detection of the crack in an object is a critical problem for the health monitoring of a transparent object. *e real-time and quantitative measurement of the crack-tip stress intensity factor (SIF) remains an open issue. In this paper, an approach for realtime and quantitative measurement for the SIFs of a Mode I crack is presented based on digital holographic interferometry (DHI). A transmission digital holographic system is established to measure the phase difference of an object wave during loading. *e expression to achieve the SIF from the phase difference is formulated. To enhance the accuracy of measurement, calibrated phase unwrapping based on least-squares and iteration and median filtering is applied to retrieve the actual phase from the noisy wrapped one. *e SIFs of the Mode I crack in a transparent polymethyl methacrylate (PMMA) specimen are measured by this approach. *e results are compared with the theoretical ones to demonstrate the feasibility of the proposed approach.


Introduction
Transparent materials are widely used in many engineering applications such as conductive materials [1][2][3], electrode materials [4][5][6][7], thermoelectric materials [8], encapsulation materials [9], and bulletproof materials [10,11].e fracture and fatigue problems are critical for many applications of the transparent materials.e crack is the key reason of materials failure.Usually the stress intensity factor (SIF) is the main quantity to evaluate the influence of the crack in the materials in elasticity.In the last decades, many measurement approaches were used to detect the SIF near the crack in transparent materials including photoelasticity [12,13], coherent gradient sensing (CGS) [14][15][16], and digital coherent sensing (DGS) [17][18][19].However, these approaches need birefringent materials, identification of the fringes, phase shifting, or fabrication of speckles which affect the application and accuracy of the measurement of the SIF.
As a noninvasive and full-field measurement method, digital holographic interferometry (DHI) has been widely used in many measurements including deformation [20][21][22], contour [23,24], the field of fluid [25,26], vibration [27,28], and microscopy [29,30].By applying DHI in the measurement of transparent materials, one can get the stress-optic constant, stress field, and strain field [31,32].Especially, DHI can detect the damage and cracks in transparent materials which means it can be used in the health monitoring of a transparent structure and experimental study on the fracture and damage mechanics [33].Usually, DHI obtains physical quantities through the phase difference of an object wave.As it is obtained through numerical operations based on the arctangent function, the phase is wrapped into the interval [−π, π] and needed to be unwrapped to retrieve the actual nonwrapped phase [34].However, because stress near the crack tip is concentrated, the phase fringes are very close which results high speckle noise in the phase maps.e existence of high noise makes phase unwrapping very difficult and even to fail.Spatial filtering can be applied so as to reduce noise before phase unwrapping [35].However, filtering will smear the phase jumps which will make unwrapping to fail.So, detection of the crack by DHI remains a qualitative analysis and does not realize the quantitative measurement which limits the application of DHI in the measurement of transparent materials.
In this paper, we aim to realize unwrapping the phase near the crack tip by using the calibrated phase unwrapping method based on least-squares and iterations [36].Moreover, we present the scheme for the measurement of SIF near the crack tip in a transparent plate by DHI.In addition, the measured results are compared with the theoretical ones to validate the feasibility of the proposed approach.is paper is organized as follows: Section 2 rst presents the principles of the proposed approach; Section 3 introduces the experimental scheme of the measurement for K I in transparent materials by the proposed approach; in Section 4, the measured results are given and validated by the comparison to the theoretical ones; and some conclusions are given in Section 5 nally.

Digital Holographic Interferometry.
e principle diagram of transmission digital holographic interferometry [31] is shown in Figure 1. e input laser is divided into the object wave and reference wave by a beam splitter.e object wave and reference wave are both expanded and collimated by a spatial lter and a lens, respectively.e transparent object is placed in the collimated object wave and is imaged to the CCD target plane by an equivalent 4f system.Here, the negative lens in the system is used to adjust the eld of view of interest.
e collimated reference wave is refracted by a beam splitter and interfered with the object wave at the CCD plane.e principle axis of the reference wave forms a certain speci ed angle with that of the object wave which is normal to the CCD plane so that this system belongs to the image-plane o -axis holography.e CCD camera captures holograms at di erent loading states.e phase di erence related to the stress in the specimen is reconstructed according to the following principles shown in Figure 1.
Establishing x and y coordinates in the CCD plane and z coordinate normal to the plane, the object wave O(x, y) and the reference wave R(x, y) can be expressed as where j −1 √ , k 2π/λ, λ is the wavelength of the laser, θ x and θ y are the angles between z axis and the projection of the reference wave vector on the xz plane and yz plane, respectively.
e hologram captured by the CCD is described as Performing Fourier transform on both sides of the above equation, we can get the spectrum of the hologram: ( On the right side of this equation, the rst item is the spectrum of the zero-order di racted wave, the second is the spectrum of the object wave which is modulated by the reference wave and translated to the point (−θ x /λ, −θ y /λ) relative to the center of the spectrum plane, and the third is the spectrum of the conjugate object wave.So, by selecting the appropriate angles of θ x and θ y , we can separate the spectrum of the object wave G(f x , f y ) from the other spectrums.A rectangular window lter is used to extract en, the inverse Fourier transform of G(f x , f y ) is carried out to reconstruct the complex modulated object wave f(x, y) as follows: e phase di erence of object waves at two loading states, due to the specimen's deformation, is expressed as where ψ(x, y) is the wrapped phase di erence, f 1 (x, y) and f 2 (x, y) are the complex object waves at di erent loading states, and angle (.) is the function to calculate the phase angle of a complex value.

Phase Recovering Approach.
In this paper, we use calibrated phase unwrapping algorithm based on the leastsquares and iterations (CPULSI) [36] and median ltering to  Advances in Materials Science and Engineering recover the true phase from the noisy wrapped phase.Here, φ ij represents the true phase and ψ ij (∈ [−π, π], i.e., modulo 2π) represents the wrapped phase at the grid point (i, j) of a phase map.eir relationship can be expressed by the wrapping operator W(.) which is de ned as

Reference wave
where −π ≤ ψ ij ≤ π, k ij is an integer, M and N are, respectively, the number of grid points with respect to the i and j indexes.e 1st order spatial wrapped phase derivatives are de ned as where Δ x ij and Δ y ij are, respectively, the di erence with respect to the i and j indexes.
In practice, the presence of noise will generate errors between noisy and noise-free phase derivatives and make unwrapping di cult, even to fail.Here, we proposed a calibration approach in [36] to calibrate the phase derivatives exhibiting large errors: where sgn(. ..) is the signum function, T x and T y are the thresholds, and G x and G y are the calibrated phase derivatives.ese parameters are de ned as (E[. ..] means the statistical average) is calibration approach means that the phase derivatives whose values are larger than the standard deviation of phase derivatives are replaced by the average value of phase derivatives.
In the least-squares sense, the optimal solution φ ij can be obtained from the discrete Poisson equation with the Neumann boundary conditions: ρ ij, (10) where e discrete Poisson equation can be solved by many methods such as fast Fourier transform (FFT), discrete cosine transform (DCT), or the multigrid method.In this algorithm, the DCT method is selected to solve the leastsquares phase unwrapping problem.
eoretically, the solution obtained from ( 10) is the exact one.However, there exist errors between the unwrapped phase and the true phase due to noise and to the smoothing performance of the least-squares method.In the proposed approach, iterations of unwrapped phase errors are utilized to seek more accurate results [36].
After unwrapping, median ltering is used to reduce the noise in the phase map.Median ltering is one of the most commonly used spatial ltering methods.It uses the median value in the kernel to substitute the value of center of the kernel [37].So, it is easy to implement and e cient for impulsive noise and speckle noise.In addition, the phase    Advances in Materials Science and Engineering map must be padded out of the frontiers with the length of the kernels symmetrically.

Calculation of the Stress Intensity Factor.
For the uniform transparent plate specimen normal to the object wave principle axis, the relationship between the stress and phase di erence of the object wave in plane stress state can be expressed as [31] where σ x and σ y are the normal stresses in the plane, h is the thickness of the plate, c is the stress-optic constant of materials, and Δφ is the unwrapped denoised phase.
Taking into account the specimen with a Mode I through crack as shown in Figure 2, the stress near the crack tip can be expressed as [38] where K I is the stress intensity factor (SIF) of the Mode I crack and r and θ are the polar coordinates relative to the crack tip.By adding ( 13) and ( 14), we can get the equation to calculate K I from the stresses as So, from ( 12) and ( 15), one can calculate the stress sum and SIF near the crack tip.

Experiment
In order to validate its feasibility, the proposed approach is used to measure the SIF near the crack tip in a transparent specimen.
e specimen is fabricated with polymethyl methacrylate (PMMA).e shape and load of the specimen are shown in Figure 2.
e size of the specimen is 400 mm × 100 mm × 4 mm.A through sharp crack is fabricated in the center of the specimen by the fatigue testing machine, and its length is 22 mm.e specimen is subjected to the vertical uniform tensile load σ 00 increasing from 0 to 3 MPa which has been estimated to ensure linear elasticity.
e elastic modulus of PMMA is E 3.24 GPa, and Poisson's ratio is ] 0.35.e wavelength of the laser is λ 532 nm.A 1024 × 768 pixel CCD camera with a pixel size of 6.4 μm is used to record digital holograms.Advances in Materials Science and Engineering di erences at di erent load are shown in Figure 3. From these wrapped phase maps, it can be seen that there exists high speckle noise, the fringes near the crack tip are very close, and the phase data inside the crack region is invalid.So, these phase maps are very di cult to unwrap.In order to unwrap the phase data in Figure 3, we use a mask to shield the crack region; then, CPULSI is used to unwrap the masked phase maps.e unwrapped phase maps are shown in Figure 4, and we can see that all the phase maps are unwrapped successfully.

Experimental Results and Validation
In order to illustrate the necessity of CPULSI used in this paper, four other di erent classical phase unwrapping algorithms were selected: Goldstein branch cut algorithm (namely, "Gold"), quality-guided path following algorithm ("Quality"), Flynn's minimum-weight-discontinuity algorithm ("Flynn"), and minimum Lp norm algorithm ("Lp") [34].Figure 5 shows the unwrapped phase maps of the object wave as σ 00 2 MPa obtained from these four algorithms.Comparing Figure 4(c), we can see that the unwrapped result from CPULSI is better than those from the other algorithms obviously.In fact, the other algorithms cannot successfully unwrap the phase data in this study because there all exist discontinuities in the maps of Figure 5. So, it is necessary to use CPULSI in this study.

Calculation of the Stress Field and SIF near the Crack Tip.
In order to reduce the noise in the phase maps, median ltering with the kernel size of 21 × 21 pixels is applied on the unwrapped phase maps.e denoised unwrapped phase is used to calculate stress sum near the crack tip. Figure 6 shows the stress sum as σ 00 1 MPa, 1.5 MPa, 2 MPa, and 2.5 MPa.From these results, it can be seen that the maximum stress sums exist in the zone near the crack tip and those near the crack plane close to zero which is in accordance with the fracture theory.
According to (15), we can calculate SIFs from stress sums at di erent loads.Because ( 13) and ( 14) only express the stresses in nitely close to the crack tip in the range of linear elastic, and the K-dominate zone must be determined before calculation.Here, we select the dashed sectorial domain in Figure 6(a) which is 1.5 mm ≤ r ≤ 2.5 mm and −45 °≤ θ ≤ 45 °as the K-dominate zone is referring to some literatures [14,15,18,19].
e average value of the SIFs in the 6 Advances in Materials Science and Engineering K-dominate zone is set as the measured value.e measured SIFs at different loads are shown in Table 1.
In order to validate the accuracy of the measurement, the theoretical values of the SIFs are calculated by using the method in [39].For the crack in Figure 2, K I can be calculated as where σ 00 is the vertical uniform tensile load, a is the half length of the crack, and F is the calibration coefficient for the size of the specimen which can be expressed as where b is the half width of the specimen.e calculated theoretical values of SIFs at different loads are also shown in Table 1.We can obtain the relative errors by comparing the measured values and theoretical ones.From Table 1, it can be seen that the error of the measured SIF does not exceed 5% which means the results of the proposed approach are accurate.

Conclusions
In this paper, we address the real-time and quantitative measurement for the SIFs of the Mode I crack in the transparent plate by DHI.For the phase processing near the crack tip which is affected by the close fringes and high speckle noise, we use phase unwrapping algorithm CPULSI and median filtering to recover the phase maps.e calculation method of K I from the phase difference is formulated based on linear elastic fracture mechanics.e proposed approach is used to measure K I of the crack in the PMMA specimen.
e measured results are validated in comparison with the theoretical ones.
e proposed approach can be used for the real-time and quantitative test of transparent materials.

Figure 1 :
Figure 1: Principle diagram of the digital holographic system.BS 1 and BS 2 are the beam splitters; M 1 and M 2 are the mirrors; L 1 , L 2 , L 3 , and L 4 are the positive lenses; CL is the negative lens; SF 1 and SF 2 are the spatial lters.

Figure 2 :
Figure 2: Specimen shape and loading way.

Table 1 :
e comparison between the measured values and theoretical values of K I .