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Abstract
Three-dimensional (3D) deformation distribution measurement is of great interest in applications of materials evaluation. In this study, we propose a stereo sampling moiré method for full-field 3D deformation measurement based on a stereomicroscope. From the phase analysis on the left and right image planes as well as the relationship between displacements in the world and image coordinate systems, the 3D displacements and the in-plane strains of the specimen can be acquired. We also propose a method to calibrate the microscope convergence angle using the grid pitch variation. The validation experiment shows that the difference between the out-of-plane displacement measured by the proposed method and the movement of the sample stage is less than 0.2 µm. The microscopic 3D displacements and the in-plane strain distributions of a carbon fiber reinforced plastic specimen in a three-point bending test are investigated.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Three-dimensional (3D) deformation measurement at the microscale of materials and structures is essential to evaluate their mechanical properties, crack propagation and growth, as well as residual stress. At present, the fringe projection based methods [1] and the shadow moiré method can measure 3D shape, but they cannot measure the in-plane strain distribution especially in a micro region. Digital holography [2] combined with stereophotogrammetry [3] has been reported for 3D deformation measurement, however, four cameras and experience in building complex light paths are required. The sampling moiré method [4–7] has been used to measure 3D shape and strain simultaneously using two cameras [8], however, two reference planes are needed before and after the specimen, making this way unsuitable for micro-area deformation measurement.
Although the stereo digital image correlation (DIC) method [9,10] can easily measure 3D deformations in a full field of view, the measurement range is generally greater than the millimeter order. A few studies have been reported on 3D micro-deformation measurement using stereo DIC applied to microscope images [11,12], but they have a disadvantage of being susceptible to disturbance noise. Digital speckle pattern interferometry [13] and moiré interferometry [14] can measure 3D micro-deformation precisely in principle. However, the former is sensitive to external vibrations resulting in a large amount of speckle noise in the fringe image, which is difficult to be removed. The latter has disadvantages such as the complicated adjustment of the interference optical path and the extreme sensitivity to external vibration.
In recent years, methods of tilting the sample stage under a microscope for 3D height measurement [15] or 3D deformation analysis [16] have been proposed. These methods have strong noise immunity due to the use of grating. Nevertheless, tilting the sample stage makes loading experiments not easy to be performed and the rotation angle is subject to mechanical errors. As described above, there are still some problems to be solved for 3D measurement methods that can simultaneously measure the distributions of in-plane displacement/strain and out-of-plane displacement in a micro region.
For microscopic 3D measurements, stereomicroscopies are natural platforms [17] due to their low cost, simplicity and convenience although the magnification is not very high. Fluorescent stereomicroscopy has been reported for 3D shape and deformation measurement [18]. A stereomicroscopic system with phase matching has been used for 3D shape measurement [19]. Binocular stereomicroscopy has been used for 3D deformation measurement of moving objects [20]. If the stereomicroscope can be combined with the digital sampling moiré method which is already widely used for strain measurements [7,21,22], a new method of measuring microscopic 3D deformation will emerge.
In this study, we propose a stereo sampling moiré method for 3D deformation measurement under a stereomicroscope. Specimen grid images are captured from two directions and the gird phases are analyzed by a spatial phase-shifting technique for 3D deformation measurement. In addition, we have found through multiple experiments that the traditional Zhang’s method [23] is unsuitable for the calibration of the convergence angle in the stereomicroscope, perhaps due to the complex combination of lenses inside the stereomicroscope. This study also proposes a method to calibrate the convergence angle of the stereomicroscope using the grating pitch variation. The reliability of the stereo sampling moiré method is verified experimentally, and the microscopic 3D displacements and strain distributions of a carbon fiber reinforced plastic (CFRP) specimen under a three-point bending test are measured.
2. Measurement principle of stereo sampling moiré method
2.1 Optical path of 3D displacement measurement system
The optical path of the used Galilean-type stereomicroscope is depicted in Fig. 1(a). A stereo camera or two identical cameras are placed in the binocular position. The angle between the imaging direction of the left or right camera and the vertical direction is the left or right convergence angle. The optical path can be simplified to two rotated cameras taking images from two different directions as shown in Fig. 1(b). The coordinate system of the specimen is defined as the world coordinate system (x, y, z). The grid pattern is placed on the xy plane and the y axis is the rotary axis when the cameras are rotated. The z direction is positive when it approaches the cameras.
Fig. 1. Principle of 3D deformation measurement: (a) Diagram of optical path of a stereomicroscope and (b) geometric relationship among displacements on the specimen plane and the image planes.
The image coordinates of the imaging surfaces of the left and right cameras are defined as (i_L, j_L) and (i_R, j_R), respectively. The image coordinates i_L and i_R are close to the x direction. The image coordinates j_L and j_R are made approximately parallel to the y direction, which can be achieved by adjusting the positions of the specimen and the cameras to make the vertical grid lines on images roughly parallel to the vertical screen edges. The proposal of this study is to acquire the 3D deformation distribution by performing sampling moiré phase analysis on the left and right grid images.
2.2 Relationship among displacements on different planes
Since the displacements on the image planes can be measured directly from the grid images acquired by the cameras before and after deformation, the key to measuring the 3D deformation of an object is to determine the relationship among the displacements in the world coordinate system (x, y, z) and the displacements reflected on the left (i_L, j_L) and right (i_R, j_R) image planes. The world and camera coordinate systems and the geometric relationship among displacements on different planes is shown in Fig. 1(b).
According to the trigonometric relationships OB = uzsinθL and BC = DO'=uxcosθL, and the relationship between the line segments OB + BC = OC = ui_L, we can get the relationship among the displacements (ux, uz) in the x and z directions, and the displacement (ui_L) reflected on the left image plane:
Similarly, according to the trigonometric relationships EF = uztanθR and FO'=OA = ui_R/cosθR, and the relationship between the line segments EF + FO'=EO'=ux, we can get the relationship among the displacements (ux, uz) in the x and z directions, and the displacement (ui_R) reflected on the right image plane:
The above equation can be transformed into
From Eqs. (1) and (3), we can find the formulas for the displacements in the x and z directions expressed in terms of the displacements on the image planes:
As the y direction is almost the same as the j_L and the j_R directions before the specimen is deformed, the displacement in the y direction can be obtained by averaging the displacements in the j_L and j_R directions:
That is, as long as the displacements on the image planes of the two cameras are obtained, the 3D displacements in the world coordinate system can be calculated.
2.3 Calculation of displacements on image planes by sampling moiré
On either camera's image plane, the displacements in two perpendicular directions can be accurately measured by phase analysis using the sampling moiré method. The method of 3D deformation measurement by sampling moiré phase analysis is called the stereo sampling moiré method. The measurement process of the proposed method is illustrated in Fig. 2.
A grid image on either image plane can be separated into two grating images in two perpendicular directions by a low-pass or moving average filter. As an example, the calculation principle of the displacement in the j direction (abbreviation of j_L or j_R direction) will be introduced. The intensity of the grating image in the j direction can be written as ${I_j} = {A_j}\cos [{2{\rm{\pi }}j/{P_j} + {\varphi_0}} ]+ {B_j} = {A_j}\cos {\varphi _j} + {B_j}$, where Aj, Bj, φ0, φj represent the modulation amplitude, the background intensity, the initial phase, and the phase of the grating j, respectively. The uppercase Pj indicates the grid pitch in pixels on the image.
When the grating image is down-sampled with an interval Tj close to the grating pitch and the sampled intensity is linearly or higher-order interpolated in the j direction by image processing in a computer, a sampling moiré pattern can be generated (Fig. 1 in [24]). After shifting the starting position of the sampling by one pixel for Tj-1 times in sequence in the j direction, a series of phase-shifting sampling moiré patterns can be formed. The intensities of these phase-shifting sampling moiré fringes can be expressed by
The phase of the first sampling moiré fringes when kj = 0 can be determined using the spatial phase-shifting technique with a discrete Fourier transform algorithm [25]:
Likewise, if the grating pitch becomes P′j, the sampling moiré phase after deformation can also be acquired. The difference of the sampling moiré phase before and after deformation is equal to the grating phase difference:
Corresponding to the uppercase Pj in pixels, the lowercase pj in microns is used to denote the physical grid pitch. Suppose the displacement of the specimen in the j direction is uj, then the phase of the grating j after deformation can also be expressed as $\varphi {^{\prime}_j} = 2\pi ({j - {u_j}} )/{p_j} + {\varphi _0}$. Therefore, the relationship between the phase difference and the displacement in the j direction is $\varDelta {\varphi _j} = \varphi {^{\prime}_j} - {\varphi _j} ={-} 2\pi {u_j}/{p_j}$ [26]. Combining Eq. (9), we can get the relationship between the displacement and the sampling moiré phase difference:
Note that Eq. (10) refers to the case where the j direction is positive vertically upwards. If the coordinate j is positive vertically downwards in the used calculation software, the negative sign in Eq. (10) should be deleted.
For the left and right cameras, the sampling moiré phases and their differences in the i and j directions before and after deformation ($\varDelta {\varphi _{mi\_L}},\varDelta {\varphi _{mj\_L}},\varDelta {\varphi _{mi\_R}},\varDelta {\varphi _{mj\_R}}$) on the left and right image planes can be respectively obtained through Eqs. (7) and (8). Consequently, the displacements in the i and j directions on the left and right imaging planes can be derived from Eq. (10), respectively:
Fig. 3. Relationship among grid pitches: (a) Relationship among grid pitches on the specimen plane and the image planes, and (b) relationship among grid pitches when the sample stage is tilted at different angles.
2.4 Measurement of 3D displacements and strains
Substituting Eq. (11) into Eqs. (4) and (5), the displacement expressions in the x and z directions can be acquired:
The displacement in the y direction can be obtained by substituting Eq. (12) into Eq. (6):
Obviously, the 3D displacement distributions of the specimen in the world coordinate system can be measured by Eqs. (13)– (15) even when the two cameras are left-right asymmetrical. In the special case of left-right symmetry (θL=θR=θ), the displacement expressions in the x and z directions can be simplified as
After the displacement distributions in the world coordinate system are determined, the in-plane strain distributions can be measured by partial differentials of the displacements in the x and y directions based on Eqs. (13) and (15):
If there are non-negligible defects on the surface of the specimen, the strains can also be calculated directly from the phase differences by means of local phase unwrapping [27]. If the thickness of the specimen (h) in the z direction is known, the average strain in the z direction can be calculated using ɛz = uz/h based on Eq. (14).
In brief, the proposed method can be used for measuring the 3D displacement distributions in the x, y, and z directions, the in-plane strain distributions in the x and y directions as well as the shear strain, and the average strain in the z direction.
2.5 Discussion of stereo sampling moiré method
The stereo sampling moiré method promotes the sampling moiré method from two-dimensional (2D) deformation measurement to three-dimensional (3D) deformation measurement. Since the phase analysis is performed with a spatial phase-shifting algorithm, only a single grid image is recorded for each left and right camera under each load. There is no need to use a phase-shifting device to record multiple grid images.
Note that in the current method, the j_L and j_R directions of the left and right cameras should be adjusted to be parallel or at small angles to the y direction of the world coordinate system before deformation. If there is an obvious angle βL between the j_L and y directions or an obvious angle βR between the j_R and y directions, a multiplicative factor cosβL or cosβR should be taken into account for displacement measurement in the y direction. When βL≤5° and βR≤5°, the effect of these angles on the displacement measurement is almost negligible because cosβL > 0.996 and cosβR > 0.996. Since it is easy to adjust the angles among the vertical grid lines on the left and right images and the vertical screen edges to be within 5° before deformation just by eye in experiments, the effect of βL and βR on the deformation measurement is negligible.
3. Calibration of convergence angle of stereomicroscope
We have calibrated the convergence angles of the microscope many times by Zhang’s method [23], but the results are far from the reference value provided by the microscope manufacturer. In this work, we propose a method to calibrate the microscope convergence angle using the grating pitch variation.
3.1 Development of calibration method for convergence angle
For a grid pattern at a certain working distance, the number of pixels of the lattice spacing (grid pitch) on an image taken from the front side is maximum compared to images taken from tilt angles. Under the stereomicroscope, before the sample stage is tilted, both cameras acquire images from the inclined directions, where the angles between the left and right inclined directions and the vertical direction are the left and right convergence angles, respectively (Fig. 1(a)). If the sample stage is tilted so that the sample stage is parallel to the image plane of either camera at the time of image acquisition, the camera will acquire images from the front side of the grid (Fig. 3(b)). Therefore, we can calibrate either convergence angle of the stereomicroscope by gradually tilting the sample stage to find the position when the number of pixels of the grid pitch on the left or right image is maximum.
The calculation principle of the convergence angle will be illustrated taking the left camera as an example. For the left camera, the sample stage should be tilted counterclockwise, and the maximum tilt angle should ideally exceed the expected convergence angle. For each grid image at each tilt angle α, the sampling moiré phase in the i_L direction (close to the x direction) can be measured using Eqs. (7) and (8). Then, the grid pitch in pixels in the i_L direction can be measured from the sampling moiré phase:
To attenuate the effect of errors in the sample stage tilting on the convergence angle measurement, a series of measured grid pitches can be fitted to a quadratic curve to find the maximum grating pitch Pi_L_max. Consequently, the left convergence angle can be calculated by
Similarly, the right convergence angle is obtainable using the following equation:
where Pi_R(0) indicates the grid pitch in the i_R direction before tilting, and the maximum grid pitch Pi_R_max in the i_R direction is determined by quadratic curve fitting.3.2 Calibration experiment
All experiments in this study were conducted under a Galilean-type stereomicroscope (Nikon SMZ1270) with two parallel zoom optical paths. The stereomicroscope can be equipped with five types of objective lenses ranging from 0.5× to 2×. It has an optical built-in zoom mechanism capable of magnifying from 0.63× to 8× inside the microscope. The object image can be magnified by combining the magnifications of the objective lens and the built-in zoom.
Two complementary metal-oxide semiconductor (CMOS) cameras (Imaging Source, DFK 33UX249) were attached to the eyepieces via two C-mount zoom adapters (MICRONET, NY-CZ) to enable stereo photography under the stereomicroscope. A coaxial epi-illumination system was used for illumination to enable imaging of the object with sufficient light intensity.
The experimental setup for the convergence angle calibration is shown in Fig. 4(a). The used tilt stage (Kozu Seiki, SH10A-RL) can be tilted up to ±10° with a resolution of 0.1°. In this calibration experiment, two tilt stages were stacked together for testing so that the tilt angle can reach ±20°. The calibration grid was a 2D lattice with a pitch of 16.9 µm (Fig. 4(b)) supported by an aluminum frame with a 1mm pitch. The left and right convergence angles were investigated when the magnifications of the objective lens were 1.5× and 2× under different zoom ratios.
Fig. 4. Calibration results of convergence angles: (a) Experimental setup, (b) grid image on the left image plane, (c) and (d) grid pitches in the i_L and i_R directions versus the tilt angle of the sample stage, respectively, when the objective lens magnification is 1.5× and the zoom ratio is 4×, (e) and (f) grid pitches in the i_L and i_R directions versus the tilt angle of the sample stage, respectively, when the objective lens magnification is 2× and the zoom ratio is 4×.
The distance from the camera sensor to the intersection of the tilting eyepiece barrels and the vertical optical axis of the microscope is approximately 230 mm, and the distance from this intersection to the top surface of each objective lens is approximately 240 mm. The length and the working distance of the 1.5× objective lens are 84 mm and 44 mm, respectively, and those of the 2× objective lens are 93 mm and 35 mm, respectively.
3.3 Calibration results of convergence angle
During the tilting of the sample stage, the sampling moiré phase close to the x direction was calculated from each grid image using Eq. (8). To attenuate the effect of random noise, each moiré phase distribution was filtered using a sine/cosine filter [28] with size of 10×10 pixels repeated by 20 times. Afterwards, the grid pitch distribution in the i_L or i_R direction at each sample stage tilt angle was measured using Eq. (19). To avoid the influence of possible lens distortion on the pitch measurement, the average grid pitch was calculated within the evaluation area with size of 500 × 500 pixels in the center of each 1920×1080 pixels image.
After the grid pitch at each tilt angle was calculated, the maximum grid pitch was acquired by quadratic curve fitting. As an example, the variations of the grid pitches in the i_L and i_R directions during the tilt process and the fitted curves were illustrated in Figs. 4(c) and 4(d), when the magnification of the objective lens was 1.5× and the zoom ratio was 4×. As an additional example, the grid pitch variations and the fitted curves were displayed in Figs. 4(e) and 4(f) when the magnification of the objective lens was 2× and the zoom ratio was 4×. The maximum grid pitch in either direction at each magnification was determined at the top of each fitted curve. Subsequently, the left and right convergence angles were calculated using Eqs. (21) and (22) based on the grid pitches before tilting and the maximum grid pitches during tilting.
The measured convergence angles of the stereomicroscope at different objective lens magnifications and different zoom ratios were listed in Table 1. As can be seen, there is a significant change in the convergence angle of the stereomicroscope when the objective lens magnification is different, which is due to the change in working distance. Whereas there is a slight change in the convergence angle when the objective lens magnification is fixed and the zoom ratio is changed, which may be related to the slight change in the zoom lens system. To attenuate possible angular measurement errors, the average of the convergence angles at different zoom ratios was taken for the 3D deformation calculation when the objective lens magnification is fixed.
Table 1. Database of the calibrated convergence angles at different objective lens and zoom ratios.
From Table 1, the sum of the calibrated left and right convergence angles is around 18° at each zoom ratio when the objective lens magnification is 1.5×. This fits well with the microscope manufacturer's statement that the total angle of convergence is approximately 18° when the objective lens magnification is 1.5×, demonstrating the accuracy of the proposed angular calibration method.
4. Experimental verification of the developed method
From Eqs. (15) and (16), when the left and right convergence angles are equal, the formulas for the in-plane displacement measurement are actually taking the average values of the conventional 2D displacement measurement results for the left and right cameras. The accuracy of the in-plane displacements and strains measured directly using the conventional sampling moiré method and its derivative methods has been experimentally verified, even if the effect of the out-of-plane deformation is ignored. And the in-plane deformation measurement using the sampling moiré method and its derivative methods has been widely used in the evaluation of various composite materials. Therefore, this study mainly focuses on the experimental verification of the accuracy of the out-of-plane deformation measured by the proposed method.
4.1 Verification experiment
The experimental setup used to verify the accuracy of the out-of-plane deformation measurement is presented in Fig. 5(a). The stereomicroscope and the grid used for verification are the same as in Fig. 4 and Section 3. The grid is subjected to displacement in the z direction through the motorized Z-stage (Kohzu, ZA10A-W2C01). A motor controller (Kohzu, CRUX-A) with a built-in motor driver for programmed pulse control and a computer with Chamonix software installed are used to set and control the amount of rotation and rotation speed. The repeatable positioning accuracy of this Z-stage is ± 0.5 µm or less.
Fig. 5. (a) Experimental setup for out-of-plane displacement verification, (b) and (c) grid images on the left and right image planes of stereo cameras, respectively, observed with a 1.5× objective lens and an 8× zoom lens.
The verification experiment was conducted using a combination of an objective lens with a magnification of 1.5x and a built-in zoom with a magnification of 8x. The movement range of the Z-stage was set to 25 µm with an interval of 0.5 µm, and the amount of out-of-plane displacement was calculated for each movement amount. The grid images recorded by the left and right cameras before movement are shown in Figs. 5(b) and 5(c), respectively.
4.2 Calculation conditions
As seen from Eq. (14), the calculation of the out-of-plane displacement requires only the use of the sampling moiré phase differences in the i_L and i_R directions. Therefore, from the 2D grid images on the left and right image planes, a low-pass filter was used to separate the parallel gratings with the main directions in the i_L and i_R directions for phase analysis. As both the grid pitches on the left and right image planes were around 32 pixels, the sampling pitch was set to 32 pixels to calculate the sampling moiré phases in the i_L and i_R directions.
Each sampling moiré phase was smoothed by a sine/cosine filter with size of 11 × 11 pixels repeated by 5 times to reduce the influence of random noise. The out-of-plane displacement distribution in the z direction were calculated using Eq. (14) in the full area of 1920 × 1080 pixels. The average displacement in an area of 1022 × 781 pixels in the center of the image was used as the average out-of-plane displacement, which was compared to the displacement amount given to the motorized Z-stage.
4.3 Measurement accuracy of out-of-plane displacement
From Fig. 6(a), the out-of-plane displacement measured by the developed method is highly consistent with the set displacement of the Z-stage, and the fitted line of the measured displacement lies within the accuracy range of the Z-stage movement. Note that the measured displacement points are tightly wrapped around the fitted line in Fig. 6(a), indicating that there is a weak periodic error in the displacement measurement in the z direction, which is related to the unavoidable periodic error in the moiré phase measurement.
Fig. 6. Verification results of out-of-plane displacement: (a) Out-of-plane displacement measured by the proposed method compared to the set displacement of the motorized Z-stage, (b) and (c) profiles of the out-of-plane displacement along the horizontal line MM’ and the vertical line NN’ marked in (a), respectively, when the set displacement of the Z-stage is 5 µm.
In addition to the average displacement evaluation, to assess the measurement error of the distribution of the out-of-plane displacement, an example of the measured displacement distribution in the z direction is provided in Fig. 6(a) when the Z-stage is moved by 5 µm. The out-of-plane displacement profiles along the horizontal and vertical lines are presented in Figs. 6(b) and 6(c), respectively.
The out-of-plane displacement along the horizontal line MM’ has many randomly distributed weak periodic errors, while that along the vertical line NN’ have more pronounced periodic errors. The reason may lie in that Eq. (14) for calculating the out-of-plane displacement uses the sampling moiré phase differences in the i_L and i_R directions close to the horizontal x direction, and the periodic errors of the potential phase measurements in the two directions interfere and superimpose on each other.
Fortunately, the measurements of the out-of-plane displacements along the horizontal and vertical lines all lie within the accuracy error of the Z-stage movement, and all the absolute measurement errors compared to the set value of the Z-stage are less than 0.2 µm. The results shown in Fig. 6 demonstrate the accuracy of the proposed method in terms of out-of-plane displacement measurement.
5. 3D deformation distributions of CFRP under three-point bending
5.1 Specimen preparation and three-point bending test
The specimen used in this study was a [±45°]4 PAN-based CFRP laminate with size of 50.1 mm in length, 11.6 mm in width, and 2.1 mm in thickness, as shown in Figs. 7(a) and 7(c). One 50.1 × 11.6 mm2 surface was polished using a #800 sandpaper and 15 µm diamond spray on an automatic polishing machine (Struers, LabPol-30). On this polished surface, a 2D grid with pitch of 15 µm was fabricated using PAK01 resist in an ultraviolet nanoimprint device (Engineering System, EUN-4200), displayed in Fig. 7(b). The nanoimprint resist was exposed to ultraviolet light at a wavelength of 375 nm for 30s. Then, the specimen grid was coated with a thin layer of Pt in an auto fine coater (JEOL, JEC-3000FC) to increase the contrast. The colorful images of the sample surface and the grid morphology in Figs. 7(a) and 7(b) were recorded with a laser confocal microscope (Lasertec, Optelics Hybrid).
Fig. 7. CFRP specimen morphology: (a) Cross-sectional image of CFRP, (b) enlarged grid image on CFRP, (c) diagram of three-point bending test, (d) and (e) grid images recorded by the right camera of the stereomicroscope with a 1.5× objective lens and an 8× zoom lens, when the deflections are 0 and 4mm, respectively.
The CFRP specimen was subjected to a three-point bending load, as depicted in Figs. 7(c). The support span was 32 mm, and the span-to-depth ratio was around 16. The loading was implemented on a self-designed three-point bending device, the photographs of which were exhibited in Figs. 8(a) and 8(b). The movement of the loading head is driven by a stepping motor (Oriental Motor, CRK543AKD-H100) with one step corresponding to 80 nm of movement. The bending load was measured with a load cell (Tokyo Measuring Instruments Laboratory, TCLB-200L). The displacement speed of the loading head was set to 3.2 µm/sec until the displacement amount reached 5.6 mm. The displacement of the loading head is called bending deflection, and the load-deflection curve of CFRP is illustrated in Fig. 8(c). The interval of the imaging was every 0.08 mm deflection.
Fig. 8. Experimental setup: (a) Three-point bending device placed under the stereomicroscope, (b) CFRP specimen during the bending test, and (c) load-deflection curve of CFRP.
To measure the 3D deformation of CFRP, the three-point bending device was placed under the previously mentioned stereomicroscope (Fig. 8(a)), and on a 6-axis sample stage for easy microscope focusing. An area under the loading head was observed by the stereo cameras (Fig. 7(c)). The used magnification of the objective lens was 1.5×, and the magnification of the built-in zoom was 8×. As an example, Figs. 7(d) and 7(e) present the grid images recorded with the right camera before loading and when the deflection is 4 mm, respectively.
5.2 Calculation conditions of 3D deformation
Because the area of interest was out of the field of view during the loading head movement, the loading device was moved using the 6-axis stage in the x and y directions treating a stain as the marker point. Besides, since the out-of-plane displacement of CFRP was too large resulting in unclear focus of the grid, the loading device was shifted in the z direction using a 6-axis stage to focus clearly. Accordingly, instead of the absolute displacement, the relative displacement with respect to a reference point was calculated in the calculation of the 3D displacements. The strain calculation is not influenced by the rigid body translation.
As seen from Fig. 7(e), the upper part of the grid image is out of focus due to the uneven out-of-plane displacement of CFRP. Therefore, the lower region of the image (1.13 × 0.35 mm2) is selected as the region of interest (ROI), marked with rectangles in Figs. 7(d) and 7(e). The intersection of the loading line and the topmost point of the ROI rectangle is set as the zero-displacement point. i.e., the reference point when calculating the relative displacement.
In this study, the length direction of CFRP is defined as the x direction, close to the i_L and i_R directions on the image planes of cameras. The thickness direction of CFRP is treated as the y direction, parallel to the j_L and j_R directions on the image planes. The positive y direction is set to the opposite direction of the loading direction, and the judgment of the left and right cameras in Fig. 8(a) is performed by looking from the direction of the microscope support stand. The z direction is vertically upward which is parallel to the width direction of CFRP, labeled in Fig. 8(b).
Since the grid pitches on the left and right camera images before deformation were approximately 21.2 and 21.3 pixels, respectively, the sampling pitches were chosen to be 21 pixels in all four directions i_L, j_L, i_R and j_R. From the grid images under different loads, the sampling moiré phase distributions in the i_L and j_L directions were calculated using Eq. (8), shown in Fig. 9. Similarly, the sampling moiré phase distributions in the i_R and j_R directions were also measured on the image plane of the right camera (Fig. 10). To attenuate the effect of surface noise on deformation measurement of CFRP, each moiré phase map was filtered by a sine/cosine filter with size of 21 × 21 pixels repeated by 20 times. Since the spatial resolution of this method is one grid pitch and the moiré spacing is much greater than the grid pitch, the filtering operation basically does not reduce the spatial resolution of deformation measurement when the sine/cosine filter size is smaller than one grid pitch. The optimization of the filter parameters in practical experiments is a topic worthy of further research.
Fig. 9. Grid images recorded by the left camera and sampling moiré phase distributions in the i_L and j_L directions under different deflections of CFRP in the three-point bending test.
Fig. 10. Grid images recorded by the right camera and sampling moiré phase distributions in the i_R and j_R directions under different deflections of CFRP in the three-point bending test.
Note that the stress that appears in Figs. 9 and 10 means the maximum bending stress on the bottom surface of CFRP. The bending stress can be calculated using ${\sigma _b} = 3Fl/(2b{h^2})$, where F is the bending load, l = 32 mm stands for the support span, and b = 11.6 mm and h = 2.1 mm are the width and the thickness of the specimen, respectively.
After the sampling moiré phases at different deflections were measured, the moiré phase differences were calculated at different deflections in the i_L, j_L, i_R and j_R directions, presented in Fig. 11. These wrapped phase differences were unwrapped in the calculation of the displacements using a simple 2D phase unwrapping algorithm.
Fig. 11. Phase differences under different deflections of CFRP in the i_L and j_L directions on the left image plane, and in the i_R and j_R directions on the right image plane.
5.3 Displacement and strain distributions of CFRP and discussion
From Table 1, the average left and right convergence angles are 9.8° and 8.5°, respectively, when the magnification of the objective lens is 1.5×. Based on the convergence angles and the unwrapped moiré phase differences, the 3D displacements at different deflections were measured using Eqs. (13–(15), shown in Fig. 12. Here, the 3D displacements are the relative displacements relative to those at the intersection of the loading line and the top point of ROI.
Fig. 12. In-plane and out-of-plane relative displacement distributions of CFRP under different deflections in the three-point bending test, where the black point on the loading line is set as the zero-displacement point.
As seen in Fig. 12, the relative displacement in the x direction is negative on the left side of the loading line and positive on the right side, indicating that the specimen is subjected to tensile deformation in the length (x) direction. The relative displacement in the y direction is positive and gradually increases from top to bottom, expressing that the specimen suffers compressive deformation in the thickness (y) direction. Since the deformation measurement area is located at the lower part of the three-point bending specimen, the measured in-plane displacements are consistent with the theoretical prediction that the specimen is stretched in the x direction and compressed in the y direction.
The relative displacement in the z direction is negative in the whole region, suggesting that the surface of CFRP moves downward relative to the reference point (loading point in Fig. 12). Usually, for a specimen subjected to a three-point bending load, the compressive deformation in the y direction does not fluctuate much at the top and bottom regions, while the tensile deformation in the x direction is greater the closer to the bottom. Therefore, according to the law of volume conservation, the closer to the bottom, the more downward the surface in the z direction. The results of the displacement in the z direction in Fig. 12 are consistent with this analysis.
Based on the in-plane displacements, the strain distributions at different deflections were measured using Eq. (18), displayed in Fig. 13. The strains in the x direction are tensile, and those in the y direction are compressive, which agrees well with the above theoretical analysis. The shear strains are positive on the right side and are almost zero near the loading line, also conforming with the deformation characteristics of the specimen under a three-point bending load.
Fig. 13. Normal and shear strain distributions of CFRP under different deflections in the three-point bending test.
6. Conclusions
A stereo sampling moiré method was proposed for full-field 3D deformation measurement. From grid images recorded from two directions, the displacements on the image planes are obtainable by sampling moiré phase analysis using a spatial phase-shifting technique. Based on the geometric relationship between displacements in the world and image coordinates, the 3D displacement distributions and subsequently the strain distributions are measurable in the world coordinate system. When using this method to measure deformations under a stereomicroscope, it is necessary to know the left and right convergence angles of the microscope. We also proposed a method to calibrate the microscope convergence angle using the variation of the grating pitch, and the calibrated results are close to the reference value provided by the microscope manufacturer.
The validation experiment demonstrated that the out-of-plane displacement measured by the proposed method was in good agreement with the movement of the sample stage. Besides, the 3D displacements and the in-plane strain distributions at the microscales of a CFRP specimen in a three-point bending test were measured and discussed. The stereo sampling moiré method can be extended to macro-scale three-dimensional deformation measurements using only two cameras to record grid images.
Funding
Japan Society for the Promotion of Science (JP20H02038, JP20K04171).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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