## Abstract

In this study, we propose a second-order moiré method by performing digital sampling at two stages to realize high-accuracy deformation measurement in a wide field of view, where a grid image is recorded at a low magnification. Simulations have verified that this method has high strain measurement accuracy when the grid pitch is close to or even smaller than two pixels for both parallel and oblique grids with random noise. As an application, the two-dimensional microscale strain distributions of a carbon fiber reinforced plastic specimen when the grid pitch was about 2.1 pixels were presented. Shear strain concentration was detected before an interlaminar crack emerged, and tensile strain concentration was found prior to the occurrence of a transverse crack. The proposed method makes the two-step phase-shifting technique achieved indirectly, not only enlarging the field of view, but also maintaining the measurement accuracy.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Measurement of the deformation of an object is extremely important for evaluating mechanical properties, structural instability, crack propagation, and residual stress. With miniaturization and integration of products, measurement of deformation at millimeter, micron, and nanoscale levels has attracted much attention in the field of materials science.

Techniques generally used for measurement of minute deformation of an object at present include the digital image correlation (DIC) method [1], electronic speckle pattern interferometry (ESPI) [2], geometric phase analysis (GPA) [3] using fast Fourier transformation (FFT) [4] or windowed Fourier transform (WFT) [5], the grid method [6,7], the moiré methods [8], and the phase-shifting analysis methods [9] including the temporal phase-shifting method [10], the spatiotemporal phase-shifting method [11], and the like as optical full-field measurement methods.

Although DIC is simple and has high measurement accuracy, it is disadvantageous in that the deformation measurement is easily influenced by the disturbance of noise. ESPI has high measurement sensitivity, but is significantly sensitive to vibration. GPA using FFT analyzes the global frequency spectrum, so that it is weak in phase analysis in regions when the grating frequency has an abrupt change, and the phase measurement error is great in areas other than the central region. GPA using WFT can analyze the local frequency spectrum, however, the calculation speed is low, and the analyzable grating pitch is usually greater than 3 pixels that makes the field of view narrow.

The moiré methods are effective for performing non-destructive measurement of the deformation distribution from visual changes in moiré fringes before and after deformation [12]. Various moiré methods have been applied to strain measurement of materials such as metals, polymer, and composite materials [13], displacement measurement of structures such as bridges and buildings [14], and shape or height measurement [15]. However, in the traditional moiré methods, it is difficult to balance the field of view with measurement accuracy.

In the microscope scanning moiré [16,17], the CCD moiré [18], moiré interferometry [19], the digital moiré [20], the secondary moiré [21] and the 2-pixel sampling moiré [22] methods, the fringe centering technique [23] is usually adopted for deformation measurement. Since only the center lines of the moiré fringes are used for calculation of the deformation distribution, the measurement accuracy of deformation is not high.

When the temporal phase-shifting method is used, it is possible to improve measurement accuracy in moiré methods [24]. Plenty of phase-shifting or phase-stepping algorithms [25,26], including two-step phase-shifting [27,28], have been reported for extracting the phase of a fringe pattern. However, the operation of recording a plurality of phase-shifted moiré fringes or grating images is time-consuming and is not suitable for dynamic measurement easily influenced by vibration.

In the sampling moiré method [29–33], measurement accuracy can be improved using the spatial phase-shifting method, but it is impossible to realize both high accuracy and a wide field of view at the same time. A specimen having a larger grating pitch imparts high measurement accuracy, but has a decreased field of view. A specimen having a smaller grating pitch (less than 3 pixels) can be set to have a wide field of view, but the spatial phase-shifting method for calculating deformation is difficult to be executed.

Recently, the spatiotemporal phase-shifting method [34] in which the temporal phase-shifting and the sampling moiré methods are combined has been reported. The phase of a fringe pattern can be accurately extracted even in the presence of strong noise, intensity saturation, vibration and phase-shifting error. Nevertheless, the fringe pitch should be greater than 3 pixels narrowing the field of view, and a series of phase-shifted fringe images should be recorded in each state limiting its application to dynamic measurement.

In micro/nano-scale deformation measurement, high precision and a large analysis area are two aspects of various technical pursuits. In this study, we propose a moiré method to achieve high deformation measurement accuracy and a wide field of view concurrently. A grid image is first processed by 2-pixel or 3-pixel down-sampling and the first sampling moiré fringes are treated as gratings to generate the phase-shifted second-order moiré fringes. The displacement and strain distributions can be measured from the second-order moiré phase, and this method is called the second-order moiré method. The measurement principle is introduced, and simulation verification is presented to show the effectiveness of this method. As an example, the strain distributions of a carbon fiber reinforced plastic (CFRP) specimen measured by this method are illustrated.

## 2. Principle of second-order moiré method

#### 2.1 Calculation principle of second-order moiré phase

The principle of phase measurement in the second-order moiré method is shown in Fig. 1. The pitch of a one-dimensional (1D) grating attached to the specimen surface is assumed to be *P _{y}*. The intensity distribution of the grating image captured in this case is expressed by

*a*(

*x*,

*y*),

*b*(

*x*,

*y*), and

*φ*

_{0}are the amplitude, the background intensity, and the initial phase of the grating, respectively.

When down-sampling and intensity interpolation are performed on the captured grating image at a sampling pitch *T* close to the grating pitch in the *y* direction, multiple phase-shifted sampling moiré fringes are obtainable at the same time. Here the phase expressions of the specimen grating and the sampling (reference) grating are *φ _{s}*=2

*πy*/

*P*and

_{y}*φ*=2

_{r}*πy*/

*T*, respectively.

The intensities of the first sampling moiré fringes which are obtained by the sampling moiré method can be represented by Eq. (2) on the assumption that *A*(*x*,*y*) denotes the amplitude of the moiré fringes, *φ*_{0} is the initial phase of the grating, and *B*(*x*,*y*) includes the background intensity and high-order frequencies (2*πy/P _{y}*, 2

*πy*(1/

*P*+1/

_{y}*T*), and the like).

*P*used for generation of the second-order moiré fringes. When a down-sampling pitch is assumed to be

_{m}*T*

^{(2)}as down-sampling at the second stage with respect to each of the first moiré fringes,

*T*×

*T*

^{(2)}phase-shifted second-order moiré fringes are obtainable. Here, the superscript “(2)” indicates that the variable is related to the second-order moiré (similarly hereinafter). To ensure high measurement accuracy, the second sampling pitch

*T*

^{(2)}is usually determined by

The intensity distributions of the second-order moiré fringes can be expressed by Eq. (4) on the assumption that *A*^{(2)}(*x*,*y*) represents the amplitude of the second-order moiré fringes, *k* and *t* denote the numbers of the phase-shifting procedures, and *B*^{(2)}(*x*,*y*) stands for the background intensity of the second-order moiré fringes.

*W*is defined as in Eq. (6) where

*j*is the imaginary unit.

#### 2.2 Deformation measurement principle

Based on the phase of the second-order moiré fringes, the first sampling moiré fringe phase *φ _{m}* and the original grating phase

*φ*can be respectively obtained by

_{s}*T*and the second sampling pitch

*T*

^{(2)}do not change before and after deformation, the grating phase difference Δ

*φ*is equal to the phase difference of the first sampling moiré fringes Δ

_{s}*φ*, and is also equal to the phase difference of the second-order moiré fringes Δ

_{m}*φ*

_{m}^{(2)}before and after deformation. If the first sampling process is performed in the

*y*direction, but the principal direction of the first sampling moiré fringes is close to the

*x*direction, the second sampling process will be executed in the

*x*direction. In this case, the phase difference of the second-order moiré fringes is also equivalent to the phase difference of the first sampling moiré fringes, as well as the grating phase difference, as long as the first and second sampling pitches remain unchanged.

Suppose the specimen grating pitch changes from *P _{y}* to

*P*’ in the

_{y}*y*direction, when an external force is loaded to deform the specimen. The phase of the specimen grating will change to

*φ*'=2

_{s}*πy*/

*P*. The grating phase difference before and after deformation can be represented by

_{y}’*u*expresses the displacement of the specimen grating in the

_{y}*y*direction, the phase of the deformed specimen grating can be expressed as

*φ*'(

_{s}*y*) = 2

*π*(

*y*-

*u*)/

_{y}*p*. Here,

_{y}*p*is the actual pitch of the specimen grating with the unit of µm or nm or the like, corresponding to

_{y}*P*with the unit of a pixel.

_{y}The phase difference of the specimen gratings before and after deformation can be calculated by *Δφ _{s}*(

*y*)=

*φ*'(

_{s}*y*)-

*φ*(

_{s}*y*)=-2

*πu*/

_{y}*p*on the basis of the phase

_{y}*φ*(

_{s}*y*) = 2

*πy*/

*p*of the specimen grating before deformation. In consequence, the displacement is proportional to the phase difference of the specimen gratings due to deformation.

_{y}The displacement of the specimen in the *y* direction is determinable by

*P*instead of

_{y}*p*, the unit of the displacement will be a pixel.

_{y}Similarly, when the pitch of the grating in the *x* direction is *p _{x}*, measurement of the displacement in the

*x*direction can be performed by replacing

*y*with

*x*in the Eqs. (7)–(11).

The 2D strains of the specimen in the *x* and *y* directions and the shear strain can be measured from the first derivatives of the displacements or phase differences. If there are some cracks or defects on the grid images, the derivatives of the phase differences can be corrected using a local phase unwrapping algorithm [13].

#### 2.3 Measurement principle in case of an oblique grid

If the specimen grating is an oblique 2D grid, the oblique angle should be considered when calculating the deformation. Assume the clockwise rotation angle from the positive *x* direction to the direction of the grating lines is *θ* (Fig. 2) when the positive *y* direction is downward (*θ* means the anticlockwise rotation angle when the *y* axis is positive in the upward direction). If the first sampling moiré fringes are very dense depending on *θ* and the ratio of the grid pitch to the first sampling pitch, the grid image can be rotated by *θ* to calculate the second-order moiré phase, and then rotated back to the original orientation and cropped to the original size.

As is well-known, the phase difference of a grating before and after deformation does not change with the analysis direction. Since the phase difference of the second-order moiré fringes is equal to the grating phase difference as seen in Eq. (9), it can be deduced that the phase difference of the second-order moiré fringes before and after deformation in any direction is the same. Therefore, even if the grid image is rotated to calculate the second-order moiré phase and then rotated back, the phase difference of the second-order moiré fringes can be used for deformation measurement.

The 2D grid can be separated to two oblique gratings *X* and *Y* (Fig. 2). For each grating, the second-order moiré phase can be calculated using Eqs. (3) and (4) by replacing the term “1/*P*” with “1/*P _{Xx}*+1/

*P*” or “1/

_{Xy}*P*+1/

_{Yx}*P*”, where

_{Yy}*P*and

_{Xx}*P*are the grating pitches of the grating

_{Xy}*X*in the

*x*and

*y*directions, respectively, and

*P*and

_{Yx}*P*mean the grating pitches of the grating

_{Yy}*Y*in the two analysis directions, respectively. The unit of these pitch components is pixel.

The deformation measurement process of the second-order moiré method from an oblique 2D grid is shown in Fig. 2. Suppose the second-order moiré phases of the gratings *X* and *Y* are $\varphi _m^{(2)}(X)$ and $\varphi _m^{(2)}(Y)$, respectively. Based on the deformation measurement using 2D moiré phase analysis [30], the displacements in the *x* and *y* directions is obtainable by

*X*and

*Y*, respectively. The grating pitch components

*p*,

_{Xx}*p*,

_{Xy}*p*,

_{Yx}*p*are in µm or nm or the like, corresponding to

_{Yy}*P*,

_{Xx}*P*,

_{Xy}*P*, P

_{Yx}*in pixel, respectively.*

_{Yy}Then the 2D strain distributions are measurable using the following equation

*p*, and oblique angle of

*θ*, the matrix

*M*in Eqs. (14) and (15) can be expressed as

## 3. Simulation verification and discussion

#### 3.1 Comparison of different methods from a 1D grating

### 3.1.1 Strain measurement error with change of theoretical strain

A 1D grating pattern with a pitch of 2.1 pixels and an intensity distribution of cosine waves was generated in a region of 500×500 pixels, as shown in Fig. 3(a). The grating was digitally stretched until the strain was 0.015 at 0.001 intervals. Normally distributed random noise with amplitude of 10% the grid amplitude (*σ*=10%) was added to the grating patterns before and after deformation to simulate an experimental environment.

The strain distributions were obtained using the conventional Fourier transform method, the sampling moiré method and the developed second-order moiré method for comparation. When calculating the strain distribution using the sampling moiré method, the sampling pitch was chosen as 3 pixels, because the spatial phase-shifting step number should be greater than or equal to 3. For the strain calculation using the second-order moiré method, the first sampling pitch was 2 pixels close to the grid pitch in the first down-sampling process, and the second sampling pitch was determined using Eq. (3) in the second down-sampling process.

The evaluation area was 300×300 pixels in the image center. The root mean square (RMS) errors of the strains measured by different methods with the increase of the theoretical strain were acquired, as seen in Fig. 3(a). It can be seen that the measurement accuracies of Fourier transform and the second-order moiré method are much higher than that of the conventional sampling moiré method when the grating pitch is 2.1 pixels.

To compare the strain measurement accuracies of Fourier transform and the second-order moiré method, the RSM errors within the range of 0 to 2.5×10^{−3} are displayed in Fig. 3(b). We can see that Fourier transform causes a periodic error with respect to the strain magnitude. Although the error tends to increase in proportion to the strain in the second-order moiré method, the RMS error is lower than 2×10^{−3}, indicating the effectiveness of the strain measurement of this method.

### 3.1.2 Strain measurement error with change of grating pitch

A parallel grating pattern with an intensity distribution of cosine wave and a pitch of 1.5 to 4.0 pixels was produced in a region of 500×500 pixels, similar to Fig. 3(a). The theoretical tensile strain exerted to the grating was 0.01. To simulate an experimental environment, random noise with amplitude of 10% the grating amplitude (*σ*=10%) was added.

The strain distributions were measured using Fourier transform, the sampling moiré method and the developed second-order moiré method. The sampling pitch in the sampling moiré method was set to 3 pixels when the grating pitch was less than 3.5 pixels, and set to 4 pixels when the grating pitch was equal to or greater than 3.5 pixels. In the second-order moiré method, the first sampling pitch was 2 pixels when the grating pitch was equal to or less than 2.4 pixels, 3 pixels when the grating pitch was greater than 2.4 pixels and less than 3.5 pixels, and 4 pixels when the grating pitch was equal to or greater than 3.5 pixels. The second sampling pitch was calculated using Eq. (3) in the second down-sampling process.

The evaluation area was also 300×300 pixels in the central region. The relationship between the RMS errors in the strain measurement using different methods and the grating pitch in the evaluation area is described in Fig. 4(a). The sampling moiré method has a great RMS error overall compared to Fourier transform and the second-order moiré method, and the error was greatest at a grating pitch of 2 pixels.

The reason why the RMS errors are not plotted when the grating pitch is 2, 3, or 4 pixels in the second-order moiré method, lies in that the first sampling pitch and the grating pitch are completely the same and the analyzed direction is exactly parallel to the principal direction of the grating, resulting that the spacing of the first-order moiré fringes is infinitely great and the second down-sampling process cannot be performed. This is one limitation of the second-order moiré method. However, it is almost not a problem in actual experiments because the grating direction usually does not exactly coincide with the specimen axial direction or the analyzed direction, and the specimen grating pitch is usually not identical to an integral multiplication of pixel.

Anyhow, the greatest RMS error of the strain was less than 2.6×10^{−3} when the grating pitch was between 2 pixels and 4 pixels in the second-order moiré method, suggesting this method is accurate in strain measurement.

### 3.1.3 Strain measurement error when the grating is inclined by 2°

To study the strain measurement accuracy for a grating with different pitches and an oblique angle of 2° from simulation, the grating parameters and deformation conditions as well as the noise level are the same as in Section 3.1.2, with the exception of the grating angle. The relationship between the RMS errors of the measured strain and the grating pitch of an oblique grating with angle of 2° using the above mentioned three methods, is shown in Fig. 4(b).

When the grating is slightly rotated, the strain can be measured when the grating pitch is equal to 2, 3 or 4 pixels in the second-order moiré method. The RMS error of the strain was lower than 2.7×10^{−3} when the grating pitch was in the range of 2 pixels to 4 pixels for this oblique grating, demonstrating that the strain can be accurately measured by the second-order moiré method.

#### 3.2 Strain results from a 2D grid

### 3.2.1 Strain measurement error with change of theoretical strain

A 2D grid pattern with a cosine wave intensity distribution and a grid pitch of 2.1 pixels in either of the *x* and *y* directions was generated in 500×500 pixels, shown in Fig. 5(a). The grid was digitally compressed until the strain was -0.01 at -0.001 intervals in the *x* direction, and was stretched until 0.1 at 0.01 strain intervals in the *y* direction. A random noise with amplitude of 10% the grid amplitude (*σ*=10%) was added to the grid patterns before and after deformation, respectively.

The grid was first separated to two parallel gratings in the *x* and *y* directions using a smoothing filter with size of 1×3 pixels or 3×1 pixels. For each grating, the strain distributions under different loading conditions were measured using the second-order moiré method. The first sampling pitch was 2 pixels in the first down-sampling process, and the second sampling pitch was decided using Eq. (3) in the second down-sampling process.

The evaluation area was 300×300 pixels in the center of the image. The RMS errors of the strain distributions with the increase of the theoretical strains in the *x* and *y* directions in the evaluation area was obtained, depicted in Fig. 5. Examples of the 2D strain distributions were plotted in Figs. 5(b) and 5(c) when the theoretical strains are -0.005 in the *x* direction and 0.05 in the *y* direction, respectively.

The RMS errors of the strains in both the *x* and *y* directions tended to increase as the absolute values of the preset strains increase. The greatest RMS error was less than 4.5×10^{−4} when the theoretical strain was -0.01 in the *x* direction, and around 0.009 when the theoretical strain was 0.1 in the *y* direction. It can be said that the 2D strains can be calculated with high accuracy using the second-order moiré method.

In particular, compared to the RMS error (4×10^{−4}) when the theoretical strain was 0 for a 1D grating with a pitch of 2.1 pixels shown in Fig. 3(b), the RMS error (1.4×10^{−4}) when the theoretical strain was 0 in the *x* direction displayed in Fig. 5(b) for a 2D grid was smaller. This was because a smoothing filter was used when decomposing from a 2D grid to 1D gratings, and the effect of random noise was reduced by the smoothing process.

### 3.2.2 Strain measurement error along cross section

From a series of strain distributions measured in Section 3.2.1, a specific situation was selected to investigate the strain fluctuations along two cross-section lines AA’ and BB’ marked in Fig. 5, when the theoretical strains are -0.005 in the *x* direction and 0.05 in the *y* direction, respectively. AA’ and BB’ with a length of 300 pixels were located at the centers of the 250th-row and 250th-column cross sections, respectively.

The relative errors of the strains measured by the second-order moiré method relative to the theoretical strains in the *x* direction along AA’ and *y* direction along BB’ were plotted in Fig. 6. The scattered points show the original relative error data of the measured strains without any filter when the noise level is *σ*=10%, and the wavy lines are the relative error results after using a smoothing filter with a size of 21×21 pixels in the *x* direction or 19×19 pixels in the *y* direction. The relative error of the strain in the *x* direction varies between -0.04 to 0.06 (-4% to 6%), and the average relative error is around 0.01 (1%), when the theoretical strain is -0.005. The relative error of the strain in the *y* direction is in the range of -0.06 to -0.03 (-6% to -3%), and the average relative error is approximately -0.047 (-4.7%), when the theoretical strain is 0.05.

Although the average relative error in the *y* direction is slightly greater (-4.7%), the relative error of the strain in the *y* direction is acceptable, because the greatest absolute value of the relative error is less than 6% without any filter even when the noise level is 10% for a large strain (*ɛ _{yy}*=0.05). In the actual experiment, the strain results will be smoothed more or less, and the relative error will be close to the average value.

Both the relative errors of the measured strains in the *x* and *y* directions demonstrates the high strain measurement accuracy of the proposed second-order moiré method.

#### 3.3. Discussion of this method

The proposed method prominently improves the measurement accuracy through phase-shifting analysis as compared to the microscope scanning moiré method and the 2-pixel sampling moiré method [22]. The area of the analysis region in this method can be expanded by several to several hundred times as compared to the sampling moiré method and the digital moiré method, because the grid pitch is usually 1.6∼3 pixels in the second-order moiré method, and 4∼30 pixels in the latter two methods. Besides, only a single grating image is required to be recorded for deformation measurement and this method is suitable for dynamic tests in which the temporal phase-shifting method is difficult to be applied. Moreover, this technique has good noise-resistance characteristics and can be used for measurement of non-destructive deformation at the atomic scale to the meter scale in the case in which a grating image can be recorded. It can also be expanded to measure the out-of-plane deformation by analyzing a projected fringe pattern.

As with any other method, the second-order moiré method also has limitations. Since the digital sampling is performed at two stages, both the spatial resolution of deformation measurement and the calculation speed are reduced compared to the traditional sampling moiré method and the temporal phase-shifting moiré method. However, in the same field of view as the other methods, the small grid spacing and the adopted multi-step filtering process [35] make this method retain a certain amount of spatial resolution of deformation measurement. This can be seen from the following experimental results that the microscale strain concentration areas can be successfully detected. Besides, since the phase distribution of the second moiré fringes can be independently calculated at each pixel using the same algorithm, this method is suitable for implementing parallel processing. The current parallel computing using a multi-core central processing unit (CPU) or graphics processing unit (GPU) makes computing speed no longer a matter of concern.

## 4. Strain measurement of CFRP under three-point bending

#### 4.1 Specimen preparation and grid fabrication

The test specimen was laminated CFRP with the length of 50 mm, width of 11.8 mm and thickness of 1.5 mm [Fig. 7(a)]. The laminate configuration was [± 45°]_{4}, where the prepregs (Q1110-2500) were manufactured by Toho Tenax Co., Ltd [Fig. 7(b)]. The fibers were polyacrylonitrile (PAN)-based carbon fibers with the diameter of about 5-7 µm. The specimen was polished in an automatic polishing machine using #800 sand papers followed by 15 µm diamond spray. The reason for grinding with coarser diamond particles was to weaken the Newton’s ring under the laser microscope.

A 2D grid with a pitch of 3 µm was then fabricated on the 50×1.5 mm^{2} surface using ultraviolet (UV) nanoimprint lithography (EUN-4200, Engineering System Co., Ltd). The nanoimprint resist was PAK01 which was coated on the specimen at 1500 rpm for 150 s. The air pressure was 0.2 MPa, the UV wavelength was 375 nm and the UV exposure time was 30s. After grid fabrication, a thin platinum layer was coated on the grid surface. The grid pattern was in a square distribution, and one principal direction of the grid was at an angle of about 45° to the length (axial) direction of the specimen [Fig. 7(c)]. In theory, a grid including gratings in any two directions can be used to calculate the 2D deformation. Although the calculation is simplest when the grid angle is near 0°/90°, this example shows the deformation calculation process from a general grid.

#### 4.2 Three-point bending test

The CFRP specimen was loaded using a self-developed three-point bending device with a step motor (CRK543AKD-H100, Oriental Motor Co., Ltd.) and a load cell (TCLB-200L, Tokyo Measuring Instruments Laboratory Co, Ltd.). The loading configuration is sketched in Fig. 7(a) and the photograph of the experimental setup is shown in Fig. 8. The support span was 32 mm, and the radius of the support heads was 3.2 mm. The minimum movement distance of the loading head was 80 nm.

The three-point bending test was performed until the total displacement was 2.4 mm and the maximum bending stress at the bottom surface was 205 MPa. The displacement rate of the load head was 3.2 µm/s. The grid image was taken every 0.2 mm displacement when the bending stress was less than 180 MPa, and every 0.08 mm displacement when the bending stress was greater than 180 MPa.

The grid image was recorded using a laser scanning microscope (Lasertec Optelics Hybrid) during the bending test. The standard image resolution was 1024×1024 pixels, and the high image resolution was 2048×2048 pixels. In this study, to increase the field of view for strain measurement, the images were recorded at the lowest magnification of the objective lens of 5× and at the resolution of 2048×2048 pixels. In this case, the calibrated image scale was 1.429 µm/pixel. Therefore, the grid pitch 3 µm before deformation corresponded to 2.1 pixels in the grid image.

#### 4.3 Calculation conditions

The grid images were rotated anticlockwise by 45° to calculate the phase differences of the two gratings separated from the grid using the second-order moiré method, because the angle between one principle direction of the grid and the axial direction of the specimen was near 45°. Then the phase images were rotated back to the original orientation and cropped to the original size of the grid image. After that, the strain distributions in the analysis direction were calculated using Eqs. (14) and (15), under the assumption that the longitudinal direction of the specimen is the *x* direction, and the perpendicular direction is the *y* direction [Fig. 7(a)].

In the process of phase calculation using the second-order moiré method, since the grid pitch was 2.1 pixels before deformation and would not be greater than 2.3 pixels even under the assumption of a large deformation of 10%, the first sampling pitch was set to 2 pixels, the integer closest to the grid pitch during the bending test. The second sampling pitch was calculated using Eq. (3). For instance, the second sampling pitch was 42 pixels before deformation.

To weaken the influence of noise on strain measurement, spatial filtering was performed in multiple stages. The multi-step filtering process with multiple small-size filters has been proved to act better than the one-step filtering process for strain mapping [35]. First, two smoothing filters with sizes of 1×3 pixels and 3×1 pixels were used to divide the 2D grid into two 1D gratings by repeating once, respectively. Besides, the second-order moiré phases were smoothed by a Sine/Cosine filter with size of 5×5 pixels by repeating 4 times. The moiré phase differences were smoothed 5 times by a Sine/Cosine filter with size of 11×11 pixels. Finally, an averaging filter with size of 7×7 pixels was used twice to reduce the variation in the calculated strain distributions. The whole strain measurement process was performed in MATLAB.

#### 4.4 Strain distributions of CFRP and discussion

The region of interest (ROI) of the strain distribution is shown in the red square area (2.9×0.79 mm^{2}) in Fig. 7(b), because the damage appeared in the lower parts at 201 MPa which is the maximum tensile stress at the bottom surface of the specimen (similarly hereinafter). The grid images at 0, 50, 162, 191, and 201 MPa are presented in Fig. 9. As an example, the phase distributions of the second-order moiré fringes at 0 and 197 MPa, as well as the corresponding phase differences in the *x* and *y* directions, are illustrated in Fig. 10. From the phase differences, the shear behavior at the layer interfaces and the phase disturbance phenomenon in the crack A area can be visually observed. The displacement distribution features are the same as the phase differences after unwrapping.

Since there are some spots and cracks on the grid images (Fig. 9), the strains were calculated using Eq. (15) combined with a local phase unwrapping algorithm [13]. The measured distributions of the strains in the *x* and *y* directions, the shear strain and the maximum principal strain at 50, 162, 191, 197 and 201 MPa are listed in Fig. 11. Note that in some spot areas where the grid pattern still exists, the strain error is small, whereas in some other spot areas where no grid pattern exists, the strain error cannot be ignored (Fig. 11). Fortunately, these spots are seldom located in the lower layers, and will not influence the detected strain concentrations introduced below.

From the strain distribution in the *x* direction in Fig. 11(a), the tensile strain in the longitudinal direction of the specimen increases as approaching the bottom layer of the specimen. From the strain distribution in the *y* direction in Fig. 11(b), it is observed that great compressive strain zones are generated at the layer interfaces. It is considered that there are many resin rich regions with low rigidity at interfaces compared to other areas. With regard to the shear strain distribution in Fig. 11(c), the shear strain concentrations occurred at interlayer interfaces. The maximum principal strain in Fig. 11(d) also concentrated at interlaminar interfaces.

From strain distributions, the crack occurrence locations can be predicted. The occurrence of an interlaminar crack, i.e., the crack A marked in Fig. 9, was evaluated from shear strain distributions as seen in Fig. 12(a). The shear strain concentration (11.9%) was detected at the right side of the interface between the bottom -45° and 45° layers at 191 MPa, and the crack A was observed at this concentration location from the grid image at 197 MPa. It suggests that the interlaminar crack A (delamination) is caused by the shear stress at the layer interface. The corresponding transverse crack connected to the crack A, which can be observed in the grid image at 197 MPa in Fig. 12(a), originates from this interlaminar crack and arises from the tensile strain in the *x* direction presented in Fig. 11(a).

The occurrence of a transverse crack, i.e., the crack B labeled in Fig. 9, was evaluated from the strain distributions in the *x* direction as seen in Fig. 12(b). The *x*-direction tensile strain concentrations were detected at a bottom location at the bottommost 45° layer, i.e., 7.2% at 191 MPa and 8.0% at 197 MPa, respectively. Correspondingly, the crack B was found in the grid image at 201 MPa at this strain concentration position. It indicates the transverse crack B is resulted from the tensile stress in the bottommost layer.

## 5. Conclusions

A second-order moiré method was proposed for deformation measurement with high accuracy and a wide field of view. A grid image with a pitch of close to or even less than two pixels can be processed to generate second-order moiré fringes. The second order moiré phase can be accurately measured using a developed spatial phase-shifting technique, and then the displacement and strain distributions can be determined in a large field of view. This method allows the phase-shifting technique to be applied to the phase analysis of a grid image with a pitch of only about two pixels, which is impossible to be observed by the naked eye. Simulations were performed to verify the strain measurement accuracies from 1D horizontal and oblique gratings as well as a 2D grid. Using the proposed method, the 2D strain distributions of a CFRP specimen under three-point bending were measured, and the crack and delamination occurrence behaviors were quantitatively evaluated.

## Funding

Japan Society for the Promotion of Science (KAKENHI, 18K13665).

## Disclosures

The authors declare no conflicts of interest.

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