## Abstract

Aimed at the low accuracy problem of shear strain measurement in Moiré methods, a two-dimensional (2D) Moiré phase analysis method is proposed for full-field deformation measurement with high accuracy. A grid image is first processed by the spatial phase-shifting sampling Moiré technique to get the Moiré phases in two directions, which are then conjointly analyzed for measuring 2D displacement and strain distributions. The strain especially the shear strain measurement accuracy is remarkably improved, and dynamic deformation is measurable from automatic batch processing of single-shot grid images. As an application, the 2D microscale strain distributions of a titanium alloy were measured, and the crack occurrence location was successfully predicted from strain concentration.

© 2017 Optical Society of America

## 1. Introduction

High strength metallic materials with a high ductility has been developed by designing multi-phase microstructures [1, 2]. However, local damage formation occurs due to heterogeneous deformation associated with complexity of the multi-phase microstructures [1]. This fact has been concerned for a further improvement of the ductility. For instance, a Titanium 6-Aluminium 4-Vanadium (Ti-6Al-4V) alloy with an hcp/bcc laminated microstructure shows intensive strain localization along a microstructure interface, which causes failure probably due to local accumulation of dislocations [3]. Therefore, it is vitally necessary to measure the micro and nano-scale strain distributions coupled with an analysis on strain components such as shear strain related to slip deformation.

In recent decades, optical techniques for deformation measurement are gaining more and more attention due to their non-contact, non-destructive and high-sensitivity features. Among various optical techniques, the grid-based techniques including geometric phase analysis (GPA) and the moiré technique also get rapid development and have been widely used in deformation measurement of various materials and structures. GPA usually uses fast Fourier transform (FFT) to get the grating phase for shortening the computation time, and then obtain the deformation from the grating phase difference [4, 5]. However, when the grating has strong local distortion or the deformation is highly nonuniform, a non-negligible error will occur in the phase calculation, because it is difficult for FFT to accurately acquire the fundamental frequency component in the case of a wide frequency band.

For the moiré technique, one of the greatest strengths is deformation visualization as a moiré pattern can be considered as an amplification phenomenon of a specimen grating. Moiré fringes are caused by the interference of two gratings or grids. Since the moiré technique was first reported in 1948 [6], various moiré methods have been successively developed for in-plane and out-of-plane deformation measurement. The geometric moiré [7] and the overlapping moiré [8] method visually show moiré fringes generated from superposition of two gratings, and are suitable for measuring large deformation. The charge coupled device (CCD) or complementary metal oxide semiconductor (CMOS) moiré method treats the pixel array of an imaging sensor as the reference grating. As the CCD or CMOS moiré fringes are unclear when the light intensity is weak or when there is a glass cover or when the grating pitch is not suitable, this method is constrained in applications for deformation measurement. Moiré interferometry [9] uses a coherence laser beam to serve as the reference grating in a large field of view with one side of dozens of millimeters. This method has high deformation measurement sensitive, i.e., the displacement sensitivity is usually 0.417 μm/fringe, and the strain sensitivity can reach several micro strains. However, the complicated operation and the weakness in noise resistance limit its pervasive application.

In the microscope scanning moiré method such as the scanning electron microscope (SEM) moiré [10–12], the transmission electron microscope (TEM) moiré [13], the atomic force microscope (AFM) moiré [14], the laser scanning microscope (LSM) [8] or laser scanning confocal microscope (LSCM) moiré [15, 16] method, the scanning lines or scanning dots are considered as the reference grating. As commercial microscopes are easy to operate and have high resolutions, the scanning moiré method has been extensively used for micro- and nano-scale deformation measurement of various materials. A three-dimensional deformation method has been reported based on SEM moiré fringes [17]. The digital moiré method [18] employs a computer to generate moiré fringes, and has great flexibility in the specimen grating pitch.

The aforementioned moiré methods mainly use the fringe-centering technique to process moiré fringes. Only the information of centerlines of moiré fringes is used, and thus the deformation measurement accuracy is low. Besides, it is difficult to automatically batch deformation analysis, because the centerlines of moiré fringes often need to be manually repaired. Although the temporal phase-shifting moiré technique [19] can improve the deformation measurement accuracy, a phase-shifting device is necessary, and it takes time to record several images making this technique not suitable for dynamic analysis.

In the recent years, a sampling moiré method [20] has been proposed and developed for deformation measurement. The displacement can be accurately measured from a single-shot grid image using the spatial phase-shifting technique. At present, the sampling moiré method is mainly applied in structural health monitoring by measuring the displacement of infrastructures such as long crane [21], thermal power plant [22] and bridges, etc. Furthermore, by use of multiple frequencies information of the sampling moiré fringe, arbitrary repeated pattern can be used to measure the displacement and strain measurement [23]. In addition, to develop this method from large scale monitoring to small scale inspection, a reconstructed multiplication moiré method using two-pixel sampling moiré fringes [24] and a strain distribution measurement method using multi-pixel sampling moiré fringes [25] have been developed. The 2D full-field normal strains at the micro and nano- scales can be accurately measured, when the principal directions (perpendicular to the grating or grid lines) of the 2D grid are exactly parallel and perpendicular to the measuring direction.

However, there are usually angles between the principal directions of the 2D grid and the measuring direction in the practical experiments, because it is difficult to precisely control the grid direction in many grid fabrication techniques. In this case, the accuracy of the shear strain measurement in the present sampling moiré method is low, because the displacement measurement in either direction only considers a one-dimensional (1D) grating with the principal direction of close to this direction. Actually, both the two 1D gratings are necessary to be considered for displacement measurement in either direction [26].

For high-accuracy shear strain measurement, in this study, a 2D moiré phase analysis method is proposed by combining the sampling moiré (spatial phase-shifting) method and the conjoint analysis of 2D phases. The spatial phase shifting technique adopts a local discrete Fourier transform (DFT) algorithm to get the moiré phase by extracting the fundamental frequency component directly. Simulations are performed to verify the measurement accuracy of this method. The normal and shear strain distributions of a titanium (Ti) alloy in a tensile test are measured and successfully used for predicting the crack occurrence location.

## 2. Deformation measurement principle

To accurately measure the strain distributions from a single-shot periodic pattern nondestructively, a moiré technique is proposed by combining the spatial phase-shifting (sampling moiré) method and a 2D phase analysis. The spatial phase-shifting (sampling moiré) method can determine the accurate phase distributions in the *x* and *y* directions, and the 2D phase analysis provides accurate displacement and strain distributions from phase differences.

#### 2.1 Measurement principle for Moiré phase

A 2D periodic pattern (grid for short) can be considered as a combination of two 1D gratings, i.e., grating *X* and grating *Y*, as seen in Fig. 1. At beginning, when the specimen grating is fabricated, suppose the pitches of grating *X* in the *x* (horizontal rightwards) and *y* (vertical upwards) directions are *p _{Xx}* and

*p*, and the pitches of grating

_{Xy}*Y*in the

*x*(horizontal rightwards) and

*y*(vertical upwards) directions are

*p*and

_{Yx}*p*, respectively. The intensity of the 2D grid before deformation can be expressed as

_{Yy}*A*and

_{X}*A*are the modulated amplitudes of grating

_{Y}*X*and grating

*Y*, respectively, and

*B*includes the background and high-order intensities.

After using a low pass filter or Fourier transform, the 2D grid can be separated to grating *X* and grating *Y.* The intensities of grating *X* and grating *Y* can be repetitively indicated by

*B*means the background and high-order intensities of grating

_{X}*X*,

*B*gives the background and high-order intensities of grating

_{Y}*Y*, and

*φ*and

_{X}*φ*stand for the phases of grating

_{Y}*X*and grating

*Y*, respectively.

After the specimen deformed, grating *X* and grating *Y* will change to grating *X*’ and grating *Y*’, as shown in Fig. 1. Suppose the pitches of grating *X’* in the *x* and *y* directions are *p’ _{Xx}* and

*p’*, and the pitches of grating

_{Xy}*Y*in the

*x*and

*y*directions are

*p’*and

_{Yx}*p’*, respectively. The intensities of the 2D grid, grating

_{Yy}*X*’ and grating

*Y*’ after deformation can be respectively represented by

*A’*and

_{X}*A’*are the modulated amplitudes of grating

_{Y}*X*’ and grating

*Y*’, respectively, and

*B*’,

*B*’

*and*

_{X}*B*’

*show the background and high-order intensities of the 2D grid, grating*

_{Y}*X*’ and grating

*Y*’, respectively, and

*φ’*and

_{X}*φ’*indicate the phases of grating

_{Y}*X*’ and grating

*Y*’, respectively.

From Eqs. (2), (3), (5) and (6), the phase differences of grating *X* and grating *Y* due to temperature change are respectively obtainable from

For grating *X* and grating *X*’, spatial phase-shifting moiré fringes in the *x* direction can be generated from *Nx*-pixels down-sampling and intensity interpolation shown in Fig. 2. The intensities of *Nx*–step phase-shifting moiré fringes before and after deformation can be respectively expressed by:

*φ*and

_{X,mx}*φ’*mean the phases of moiré fringes when

_{X,mx}*k*= 0 generated from grating

_{x}*X*and grating

*X*’ in the

*x*direction, respectively.

For grating *Y* and grating *Y*’, spatial phase-shifting moiré fringes in the y direction can be formed from *Ny*-pixel down-sampling and intensity interpolation shown in Fig. 2. The intensities of *Ny*–step phase-shifting moiré fringes before and after deformation can be respectively represented by:

*φ*and

_{Y,my}*φ’*signify the phases of moiré fringes when

_{Y,my}*k*= 0 generated from grating

_{y}*Y*and grating

*Y*’ in the

*y*direction, respectively.

The phases *φ _{X,mx}*,

*φ’*,

_{X,mx}*φ*and

_{Y,my}*φ’*of moiré fringes in Eqs. (9)-(12) can be calculated from the phase-shifting method using a discrete Fourier transform algorithm, as shown in

_{Y,my}#### 2.2 Measurement principle for displacement and strain distributions

From Eqs. (9), (10) and (7), the phase difference of moiré fringes in the *x* direction is equal to the phase difference of grating *X*, which is determinable from

From Eqs. (11), (12) and (8), the phase difference of moiré fringes in the *y* direction is equivalent to the phase difference of grating *Y*, which is determinable from

Suppose the displacements of the specimen in the *x* and *y* directions are *u _{x}* and

*u*, respectively, the phases of grating

_{y}*X*and grating

*Y*after deformation can also be expressed as

From Eqs. (2), (3), (16) and (17), the phase differences of grating *X* and grating *Y* due to loading change can also be determined by

As known from Eqs. (14) and (15), the phase differences of moiré fringes in the *x* and *y* directions are equal to the phase differences of grating *X* and grating *Y*, respectively. Consequently, based on Eqs. (18) and (19), the relationship among the phase differences of moiré fringes and the displacements of the specimen can be obtained from

Therefore, the displacements of the specimen in the *x* and *y* directions are measurable from

*M*stands for the matrix comprised of four pitch components of grating

*X*and grating

*Y*in the

*x*and

*y*directions.

Since the strains in different directions are the partial differentials of the displacements, the *x*-direction strain, the *y*-direction strain and the shear strain can be measured by the following equation

The relationship between the deformation and the phase difference shown in Eqs. (16)-(22) is essentially same to that in GPA. The difference between the proposed method and GPA lies in the phase extraction method represented in Eqs. (1)-(15). Instead of calculating the grating phase by FFT, the proposed method computes the moiré phase by local DFT which is fast and effective even when the grating pitch is highly nonuniform. Besides, as the moiré spacing is greater than the grating spacing, the moiré phase is easier to be smoothed compared with the grating phase using a sine/cosine filter [27] to reduce the influence of noise.

#### 2.3 Deformation measurement from an orthogonal grid

Suppose the pitch and the angle of grating *X* are *p _{X}* and

*θ*, respectively, and the pitch and the angle of grating

_{X}*Y*are

*p*and

_{Y}*θ*, respectively, as seen in Fig. 1. The angles

_{Y}*θ*and

_{X}*θ*are defined as positive when anticlockwise rotation from the

_{Y}*x*direction. The matrix

*M*in Eqs. (21) and (22) can also be expressed as

If the fabricated grating at room temperature is a standard cross grating with a uniform pitch, i.e., *p _{X}* =

*p*=

_{Y}*p*and

*θ*-π/2 =

_{X}*θ*=

_{Y}*θ*, the matrix

*M*in Eqs. (21) and (22) can be transformed into

As the fabricated grating is a standard cross grating in most cases, the most common calculation equations for displacement and strain measurement are as follows

#### 2.4 Deformation measurement procedure

The flow chart and measurement process of the 2D phase analysis moiré technique for deformation measurement is presented in Fig. 3. A grating was first fabricated on a specimen if there is no periodic structure on its surface. Then, the grating image was recorded by an image recorder (such as a microscope, a CCD/CMOS image sensor, etc.). Next, the grating images was processed by down-sampling with the pitch of close to (or integral multiple or fractional multiple) the specimen grating pitch, and intensity interpolation to generate sampling moiré fringes. After that, the phase of the moiré fringes before deformation is calculated by the spatial phase-shifting using a Fourier transform algorism.

According to the same procedure, the phase of the moiré fringes after deformation can also be obtained, after the specimen is deformed under an external load (such as mechanical, electrical, magnetic, mechanical-heating, electrical-heating, magnetic-heating, mechanical-electrical, etc.). After the phase differences of the moiré fringes before and after deformation are determined, the displacement and strain distributions can be measured using a 2D phase analysis.

## 3. Simulation verification of strain measurement accuracy

#### 3.1 Comparison of measurement accuracy with conventional methods

This section is used to compare the proposed technique with the traditional sampling moiré technique, and verify the 2D deformation measurement accuracy of the proposed technique from simulation. In the traditional sampling moiré method using 1D phase analysis [25], the displacement in one direction is only calculated using the moiré phase difference in that direction, i.e., ${u}_{x}\text{=}-{p}_{Xx}\cdot \Delta {\phi}_{X,mx}/2\pi $, ${u}_{y}\text{=}-{p}_{Yy}\cdot \Delta {\phi}_{Y,my}/2\pi $. The strain components are obtained from the partial differentials of the displacements, i.e., ${\epsilon}_{xx}\text{=}\partial {u}_{x}/\partial x$, ${\epsilon}_{yy}\text{=}\partial {u}_{y}/\partial y$ and ${\gamma}_{xy}\text{=}\partial {u}_{x}/\partial y+\partial {u}_{y}/\partial x$. However, in the proposed 2D moiré phase analysis method, the displacement in one direction is obtained from the joint analysis of the moiré phase differences in two directions as shown in Eq. (21) or (25). The strain components are also obtained from the partial differentials of the displacements as shown in Eq. (22) or (26).

A cross grating with the pitches of 10 pixels in both the *x* and *y* directions was first rotated by different angles to become a series of inclined gratings. The size of the grating image was 480 × 400 pixels. These inclined gratings were then deformed by exerting theoretical *x*-direction, *y*-direction, and shear strains. Next, these deformations were measured using the proposed technique.

Figure 4(a) shows an example of deformation measurement when the grating oblique angle is 10 degrees. For the 2D grid before deformation in Fig. 4(b), the grating image was first processed to two 1D gratings in Fig. 4(c) by two low pass filters in the *x* and *y* directions, respectively. Spatial phase-shifted sampling moiré fringes in Fig. 4(d) were then generated, and two phase maps in Fig. 4(e) of moiré fringes before deformation in the *x* and *y* directions were calculated. Similarly, two phase maps of moiré fringes after deformation in the two directions were also obtained. Next, phase differences of moiré fringes due to deformation were determined in the *x* and *y* directions, as shown in Fig. 4(f).

Based on 2D phase analysis, the 2D displacement and strain distributions were measured as represented in Fig. 4(g). The average *x*-direction, *y*-direction and shear strains were 0.01138, −0.00354 and 0.00543, respectively, almost equal to the preset theoretical *x*-direction, *y*-direction and shear strains which were 0.01152, −0.00352 and 0.00547, respectively. However, if using the traditional 1D phase analysis, the average *x*-direction, *y*-direction and shear strains were 0.01186, −0.00402 and 0.00280, respectively, deviating from the theoretical strains especially for the shear strain.

When the oblique angles of the 2D grid in Fig. 4(a) were 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 16, 19, 22, 25 degrees, the average strains measured from the proposed 2D phase analysis and the traditional 1D phase analysis moiré techniques were compared with the theoretical strains. Figure 5(a) gives the comparison of the *x*-direction and *y*-direction strains, and Fig. 5(b) presents the comparison results of the strain under different oblique angles of the 2D grid. It can be seen that, the measurement results from the proposed technique are much closer to the theoretical values than those from the traditional technique.

Figure 6 gives the relative errors of the measured strains relative to the theoretical strains. It is obvious that, the measurement accuracies for all the *x*-direction, *y*-direction and shear strains of the proposed technique are much higher than those of the traditional technique.

#### 3.2 Measurement accuracy from an oblique grating with random noise

This section illustrates the strain measurement from an inclined grating with random noise. The inclined grating can be separated to grating *X* and grating *Y*, which are perpendicular to each other. The oblique angles of grating *Y* from the *x* direction and grating *X* from the *y* direction are 10 degrees shown in Fig. 7(a). The grating pitches of grating *X* in the *x* and *y* directions were 10.1543 and 57.5877 pixels, respectively, and the grating pitches of grating *Y* in the *x* and *y* directions were 57.5877 and 10.1543 pixels, respectively. The size of the grating image was 480 × 400 pixels. A random noise with the amplitude of σ = 2% of the grating amplitude was added to the oblique grating. The grating at load N1 was deformed to the grating at load N2 by exerting *x*-direction, *y*-direction and shear strains.

The strains of this grating were measured using the proposed technique. The absolute errors, the relative errors and the standard deviations of the measured strain along with the theoretical strains were listed in Fig. 7(b). All the absolute errors were within −0.00014 to 0, the relative errors range from −1.2% to 0.6%, and the standard deviations were less than 0.0015, suggesting the high accuracy of the proposed technique for strain measurement when the oblique grating has a random noise.

## 4. Strain distribution measurement of Ti alloy

#### 4.1 Grid fabrication and tensile experiment

The specimen was a Ti-6.29Al-4.35V-0.155O-0.225Fe alloy [28]. Figure 8 presents the specimen geometry and the used mechanical loading device. The thickness and the minimum width of the specimen were 1 mm and 1.8 mm, respectively. A grid with the pitches of 3 μm in two perpendicular directions was produced by UV nanoimprint lithography (EUN-4200 device) in an area of 1.8 × 15 mm^{2}. Since it is difficult to control the exact direction of the fabricated grid in nanoimprint lithography [29, 30], the grid oblique angle relative to the axial direction of the specimen (*x* direction) were necessary to be measured. In this experiment, the included angle between one grating line and the *x* direction was measured to be 2 degrees by MB-ruler. The tensile test was performed under a laser scanning microscope (Lasertec OPTELICS HYBRID). In the tensile test, an area of 219 × 204 μm^{2} containing a prior β grain boundary at one edge of the specimen was chosen as the region of interest, as indicated in Fig. 9.

#### 4.2 Strain distributions under different loads

The grating images in this region of interest under different tensile loads were recorded, and the corresponding strain distributions were measured using the proposed technique. The sampling pitch was 4 pixels to generate 4-step phase-shifting moiré fringes. A sine/cosine filter [27] with size of 5 × 5 pixels was used to smooth the moiré phases, and an average filter with size of 13 × 13 pixels was used to smooth the strain data. Examples of the 4-pixel sampling moiré fringes and moiré phases in the *x* and *y* directions are shown in Fig. 10 when the tensile load is 604 MPa. Some typical grating images, and the 4-pixel down-sampling moiré fringes in the *x* and *y* directions were illustrated in Fig. 11 when the nominal stresses are 0, 225, 604 and 660 MPa, respectively. The corresponding phases of these moiré fringes in the *x* and *y* directions were then calculated. From the phase differences relative to the phase under 0 MPa, the distributions of *x*-direction, *y*-direction and shear strains were measured as seen in Fig. 11 under 225, 604 and 660 MPa, respectively.

From the grating images, no obvious deformation could be observed under 604 MPa. However, from the distributions of the *x*-direction and shear strains in Fig. 11, obvious microscale strain concentration could be observed under 604 MPa. Finally, microscale initial crack was found at the strain concentration region from the grating image under 710 MPa in Fig. 12. It indicates that the proposed technique can provide accurate strain distributions which enable precise prediction of crack occurrence location.

Besides, oblique slip lines with different orientations were observed in different grains under greater tensile loads, as shown in Fig. 13. The slip lines were able to be visualized from the distribution of a part of shear strain *ε _{yx}* =

*∂u*/

_{y}*∂x*, as seen in Fig. 13. Since nature of dislocation slip is plastic shear deformation, the appearance of shear strain corresponding to the slip lines is reasonable. The slip-resolved strain mapping will be helpful information to clarify an underlying mechanism of damage initiation associated with local dislocation accumulation.

## 5. Summary

A two-dimensional moiré phase analysis method was proposed to accurately measure the in-plane strain distributions even if the specimen grid is inclined at a large angle. The relative error of the shear strain measurement relative to the theoretical value was significantly reduced from 50% to 0.8% when there was an angle between the grating principal direction and the measuring direction. The measured normal and shear strain distributions of a Ti alloy was successfully used to predict the crack occurrence location from the strain concentration. Dynamic test and automatic batch processing are realizable for accurate strain distribution measurement in the proposed method. This method is useful for failure analysis, residual stress measurement, strengthening and toughening of materials ranging from nanoscale to meter scale, optimal design of interfaces, production quality control, structural health monitoring, etc.

## Funding

Cross-ministerial Strategic Innovation Promotion Program - Innovative Measurement and Analysis for Structural Materials (SIP-IMASM) (Unit D66); Japan Society for the Promotion of Science (JSPS) KAKENHI (JP16K17988, JP16K05996).

## Acknowledgments

The authors are grateful to Dr. Masataka OHKUBO in AIST, and Drs. Yoshihisa TANAKA and Kimiyoshi NAITO in NIMS for discussion on experimental design.

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