Quartz glass, with high temperature resistance, excellent shock resistance, and ultralow expansion, is widely applied in optics for astronomy and aerospace applications [1, 2]. The elastic modulus, reflecting deformation resistance, is the material parameter of chief concern in structural design with quartz glass. Once the elastic modulus of quartz glass can be accurately measured, a structure with imaging lenses can be precisely simulated, analyzed, and evaluated. Therefore, improving the measurement of the elastic modulus of quartz glass is advantageous.

Three-point bending, with deflection found after a certain load is applied to the center of the specimen, is a typical method for measuring elastic modulus. Within the elastic limit, the elastic modulus is evaluated by analyzing load-deflection relations. Generally, deflection is observed using a displacement transducer. For example, a strain gauge is used to measure deflection [3]. However, the accuracy of a strain gauge is low, especially when the deflection of quartz glass is tiny. The optical lever is used to measure deflection because it has an amplifying effect on displacement. However, the setup must maintain the parallelism between observed ruler and mirror, and undesirable reading error is introduced by the eyepiece [4]. The Hall transducer is also used to calculate elastic modulus, by converting deflection into an electric signal [5]. This method has high accuracy for common materials, but the error caused by deformation of the Hall transducer coupling to deflection is introduced. Furthermore, the Bragg transducer is used to measure the Bragg wavelength shift and calculate deflection [6]. According to its theoretical model, this method is not affected by the position of the Bragg transducer.

All of these deflection-measuring methods are essentially single-point measurements. The location of the transducer is usually chosen to be the center of the plate. Under ideal conditions, the loading point is in the middle, and the measured data from the transducer is the maximum deflection. However, when the load is eccentric the load point and the maximum deflection point are not in the center and do not coincide; continuing to use the center deflection to calculate elastic modulus brings measurement error. Moreover, the additional displacement due to shear force cannot be ignored when the span-depth ratio of sample is small, and it also brings measurement error. Therefore, errors from eccentric load and shear force need to be corrected.

Digital speckle-pattern interferometry (DSPI) is a full-field optical measurement method with high sensitivity to small displacements [7, 8]. In our previous work, the feasibility of measuring the elastic modulus of quartz glass by DSPI was verified, and the contact-deformation problem of contact-displacement-sensor coupling during deflection was effectively solved [9]. In addition, a model for correcting the error caused by eccentric load was previously proposed, but this correction model ignores the change in bending moment caused by the eccentric load. When the load’s eccentricity is large, the correction effect of the model is not ideal. In this study, a more accurate measurement model with eccentric load correction is proposed, by analyzing the positional misalignment between maximum deflection and load. Based on the advantage of DSPI full-field measurement, the influence of shear force on measurement is considered and the measurement model with shear-force correction is given by measuring two points. In the simulation analysis, the feasibility of the two error-correction models is verified respectively. In the test, the two errors caused by eccentric load and shear force are comprehensively corrected. The experimental results show that the proposed model can effectively correct the measurement error of elastic modulus, and using DSPI to measure the elastic modulus can correct the errors based on the deflection information for multiple points in a single measurement, which will help to improve the accuracy of measuring the elastic modulus.

2.1. Deflection Measurement Using Digital Speckle-pattern Interferometry

A typical optical path of DSPI for measuring out-of-plane displacement is illustrated in Fig. 1. The coherent beam emitted by a laser is divided into two beams, one of which is reflected by mirror M4 to illuminate the surface of quartz glass, after expansion to form a speckle field. Another beam strikes mirror M3 driven by a piezoelectric transducer (PZT) (after being reflected by mirrors M1 and M2), and forms a reference light field after expansion. The two beams interfere on the surface of the camera sensor to form an interference field. The phase of the interference field will change due to the bending deformation of the quartz glass after loading. After extracting the phase of the interference field, the full-field deflection information of the quartz glass can be calculated by processing the phase-difference map. When the illumination angle of the optical path is tiny, the out-of-plane deformation can be expressed as

Figure 1. Optical setup of digital speckle-pattern interferometry (DSPI) for deflection measurement.

(1)$$w=\frac{\lambda }{4\pi }\delta$$

where δ is the relative phase difference, λ is the laser’s wavelength, and w is the out-of-plane component of the displacement. Based on the flat assumption in three-point bending tests, each cross section of the quartz glass plate is still flat and perpendicular to the central axis after bending deformation. Therefore, the in-plane displacement of the quartz glass can be ignored, and the out-of-plane displacement w represents the deflection. After the phase-difference map is obtained, the deflection of the quartz glass is determined. Four-step phase shift is a common phase-extraction method used for static measurement [10, 11]. A phase shift is introduced four times by driving mirror M3 via the PZT, and then phase extraction is achieved by acquiring four interference images. The phase of the speckle interferogram is extracted before and after loading, and the phase difference is obtained by subtraction.

The resolution of deformation measurement in DSPI primarily relies on the phase measurement resolution, which is subject to the impact of system noise and phase-processing algorithms. Typically, the phase resolution of a DSPI measurement system falls between π⁄10 and π⁄25 [12, 13]. Based on a phase resolution of π⁄10 and a laser wavelength of 532 nm, the mathematical model for DSPI measurement of out-of-plane deformation shows that the achievable deformation measurement resolution can be as low as 13.3 nm. In contrast, engineering applications commonly measure the bending deflection of specimens using a dial gauge, which offers a measurement resolution of merely 0.001 mm. Thus, assuming other conditions are equal, DSPI offers superior resolution for measuring the elastic modulus.

2.2. Error Correction for an Eccentric Load

The measurement principle of three-point bending is shown in Fig. 2. A specimen of thickness h and width b is placed on two supporting rollers with span of L, and an upper roller is used to apply pressure to the specimen. In Fig. 2, O is the origin of the coordinate system, x_F is the coordinate of the loading point, and F is the loading pressure. w_max is the maximum deflection of the quartz glass, and x_wmax is the coordinate of the point of maximum deflection. The elasticity modulus can be calculated from the load-deflection relations as

Figure 2. Eccentric mode of the three-point bending test.

(2)$$E=\frac{3\text{F}{\text{L}}^2}{4\text{b}{\text{h}}^{3}w}x-\frac{F}{\text{b}{\text{h}}^{3}w}{x}^3$$

where E is the elastic modulus, and w is the deflection at coordinate x caused by a bending moment. When the load is eccentric, the bending moment of each cross-section of the specimen varies. With increasing eccentricity, the load point and maximum deflection point do not coincide with each other. Therefore, an eccentric-load-correction measurement model for elastic modulus E_e is expressed as

(3)$$E_e=\frac{2F\left(L-x_F\right)x}{\text{Lb}{\text{h}}^{3}w}\left(2x_{F}L-x_F^2-x^2\right)$$

Because load position x_F cannot be measured directly, the positional relationship between maximum deflection and load can be analyzed as

(4)$$x_{\text{w}\text{max}}=\sqrt{\frac{2x_{F}L-x_F^2}{3}}$$

It is convenient to locate the point of maximum deflection from the full-field deflection information measured by DSPI. By calculating the elastic modulus with x_wmax instead of x_F, Eq. (3) can be derived as

(5)$$E_e=\frac{2\text{Fx}\sqrt{L^2-3x_{\text{w}\text{max}}^2}}{\text{Lb}{\text{h}}^{3}w}\left(3x_{\text{w}\text{max}}^2-x^2\right)$$

Due to the structural limitations of the support rollers, there is a blind area on the sample surface that cannot be captured by the camera, as shown in Fig. 2. The deflection of point S is not zero obviously. However, when calculating the deflection based on speckle interferometry, it is usually assumed that the deflection of edge point S is zero [14]. To overcome this problem, the point S is taken as the reference point, and the relative deflection is used for elastic modulus calculation within the elastic limit. The elastic modulus measurement model with eccentric load correction can be expressed as

(6)$$E_e=\frac{2F\sqrt{L^2-3x_{\text{w}\text{max}}^2}}{\text{Lb}{\text{h}}^3\Delta w}\times \left[3x_{\text{w}\text{max}}^2\left(x-x_S\right)-\left(x^2-x_S^2\right)\right]$$

where x_S is the coordinate of point S and ∆w is the relative deflection between the point at x and point S.

2.3. Error Correction for Shear Force

Generally, the elastic modulus is determined by three-point bending tests of specimens with suitable span-depth ratios. When the span-depth ratio is small, the additional deflection caused by the shear force cannot be ignored. Eq. (2) is derived from theoretical elementary beam theory, which only considers deflection caused by bending moments, but the overall measured deflection by DSPI includes shear deformation [15]. If the shear deformation cannot be separated from the overall deflection, the measurement accuracy decreases. The additional deflection caused by shear force can be expressed as

(7)$$w_{\delta }=\frac{\text{sFx}}{2\text{Gbh}}$$

where G is the shear modulus in the length-depth plane of the specimen, and s is the shear factor.

The deflection measured by a DSPI system is actually the sum of bending deflection and additional deflection. Combining Eqs. (2) and (7), the total deflection w_t can be expressed as

(8)$$w_t=\left(\frac{3\text{F}{\text{L}}^2}{4\text{b}{\text{h}}^3}x-\frac{F}{\text{b}{\text{h}}^3}{x}^3\right)\times \left[\frac{1}{E_s}+\frac{2\text{s}{\text{h}}^2}{G\left(3L^2-4x^2\right)}\right]\left(\frac{3\text{F}{\text{L}}^2}{4\text{b}{\text{h}}^3}x-\frac{F}{\text{b}{\text{h}}^3}{x}^3\right)\frac{1}{E}$$

where E_s is the elastic modulus corrected for the shear force. The relationship between E_s and E can be represented as

(9)$$\frac{1}{E}=\frac{1}{E_s}+\frac{2\text{s}{\text{h}}^2}{G\left(3L^2-4x^2\right)}$$

Because the shear force causes different additional deflections at different points, it results in unequal values of E at different positions. According to Eq. (2), the elastic modulus E directly measured at points A and B, whose locations are x = x_A and x = x_B, are indicated as E^A and E^B respectively. The relationship between E^A, E^B, and E_s can be represented as

(10)$$E_s=\frac{4\left(x_B^2-x_A^2\right)E^A{E}^B}{3L^2\left(E^B-E^A\right)+4\left(x_B^2{E}^A-x_B^2{E}^B\right)}$$

Eq. (10) shows that the measurement error caused by shear force can be effectively reduced by measuring the deflection at two points, but when considering both shear force and eccentric load at the same time, the superimposed deflection w^t needs to be modified based on Eq. (5). Then the elastic modulus E^e–s corrected for eccentric load and shear force can be represented as

(11)$$E_{e-s}=\frac{4\left(x_B^2-x_A^2\right)E_e^A{E}_e^B}{3x_{\text{w}\text{max}}^2\left(E_e^B-E_e^A\right)+\left(x_B^2{E}_e^A-x_A^2{E}_e^B\right)}$$

Because DSPI has the advantage of full-field measurement, the deflection at two points can be calculated directly from the interferogram instead of using multiple contact transducers, and thus measurement accuracy is further improved.

3.1. Finite-element Simulation

In a real measurement process, the measurement results are affected by many factors concurrently. To examine the impression of the modified model under the influence of a single error, finite-element simulations are carried out. Glass fiber with elastic modulus of 73 GPa and Poisson’s ratio of 0.22 is used as the simulated sample. The span is 40 mm, the sample is 60 mm wide and 1 mm thick, and the load is 2 N. To make the simulation results reliable, the hexahedron element is used to divide the mesh, and the mesh size is 0.05 mm. Figure 3 shows the deformation nephogram of the sample surface when the load is eccentric. When the eccentricity value of the load is set to 1, 2, 3, 5, and 10 mm respectively, the calculated elastic modulus E_e is listed in Table 1. With increasing eccentricity, the elastic modulus corrected for eccentric load does not change significantly, which proves that the proposed eccentric model is effective in correcting the error caused by eccentric load.

TABLE 1. Error correction for an eccentric load.

Eccentricity Value (mm)	Ee (GPa)
0	72.87
1	72.86
2	72.82
3	72.84
5	72.77
10	72.52

Figure 3. Deformation nephogram.

To test the effect of shear force on the elastic modulus measurement, apply 2 N load at the center of the specimen. When the sample thickness is set to 1, 2, 3, 5, and 10 mm respectively, the eccentric-corrected elastic modulus of two points at the locations x_A = L/4 and x_B = L/2 are calculated, the results are defined as E^A and E^B. The corrected elastic modulus calculated according to Eq. (10) is recorded as E_s. The simulation results are shown in Table 2.

TABLE 2. Error correction for shear force.

Thickness (mm)	EA (GPa)	EB (GPa)	Es (GPa)
1	72.36	72.24	72.84
2	72.43	72.39	72.17
3	71.67	71.62	72.24
5	69.92	69.77	71.03
10	62.57	62.36	64.06

Table 2 shows that as the specimen thickness increases while keeping the span fixed, the additional displacement caused by shear force also increases. When the specimen thickness is small, the impact of shear force can be effectively mitigated by concurrently calculating the elastic modulus of two points. When the thickness is large, shear force has a more obvious influence on the measurement results, and although the effect of the correction model is more significant, it still cannot meet the measurement requirements. Therefore, to obtain accurate measurement results an appropriate sample size should be selected, to ensure a sufficient span-depth ratio.

3.2. Experiment and Analysis

Since the dial gauge, which is needed in the traditional test method for measuring the elastic modulus of glass, will interfere the image acquisition of our DSPI system on the sample surface, it is difficult to perform synchronous comparative experiments. Additionally, manual loading cannot guarantee complete consistency of loading speed and load size for each measurement, duplicate comparative test cannot also be carried out between traditional method and DSPI method here. Given the above two points, comparison between traditional method and DSPI method are not conducted, but comparison about error corrections using DSPI to measure elastic modulus is discussed in this paper. With no doubt, DSPI in measuring displacement has absolute superiority in high precision.

The measurement system, which includes the optical path for DSPI measurement and the loading device, is shown in Fig. 4. The three-point bending load is applied to the quartz glass by turning a screw micrometer. The load-force data are recorded by a pressure transducer in the loading device, which has a range of 0–50 N and a sensitivity of 2 mv/V. The deflection-measurement system uses a laser with a central wavelength of 532 nm and an output power of 400 mw as illumination source. The interference field is captured using a black-and-white industrial camera with a resolution of 2,464 × 2,056 pixels. Since quartz glass is transparent, the specimen used in the experiment is painted to enhance diffuse reflection. Because the sample is placed vertically in the loading device, the specimen may be displaced rather than being bent merely during the loading process. Therefore, the specimen is preloaded appropriately before loading to avoid additional measurement errors.

Figure 4. The measurement system.

The quartz glass plate is fixed in the loading device, whose span L is 40 mm. Extra phase differences of the interference field are introduced by driving the PZT four times continuously, and the phase map of the speckle interference field is extracted using a phase-shifting algorithm. The phase-difference map of the specimen resulting from loading is obtained by subtracting the phase distributions before and after deformation. A typical phase-difference map is shown in Fig. 5(a). The region of interest in the phase-difference distribution is filtered for further calculation; the result is shown in Fig. 5(b). The unwrapped phase-difference map and three-dimensional representation of deflection are shown in Figs. 5(c) and 5(d) respectively.

Figure 5. Deflection-measurement images: (a) Phase-difference map, (b) the filtered map, (c) the unwrapped map, and (d) three-dimensional deformation map.

A quartz glass plate with length of 60 mm and width of 40 mm is selected as the specimen. To avoid excessive influence of shear on the measurement, a specimen with thickness of 2 mm is selected. Loads of 3, 5, and 7 N are applied to the specimen of 2-mm thickness; The phase maps are shown in Fig. 6. Based on the deflection obtained from the phase map and the accurate load value obtained from the pressure transducer, the elastic modulus can be calculated according to section II. The standard elastic modulus E and eccentric elastic modulus E_e are directly calculated at the position x = x_wmax. As shown in Fig. 2, the values of the eccentric elastic modulus at points A with x_A = x_wmax/2 and B with x_B = x_wmax are calculated according to Eq. (6), and then the total corrected elastic modulus E_e–s can be calculated according to Eq. (11). The measured results are shown in Table 3.

TABLE 3. Measured results for a 2-mm specimen.

Number	Force (N)	E (GPa)	Ee (GPa)	Ee-s (GPa)
1	3.04	68.02	69.26	73.33
2	2.94	66.82	67.82	70.42
3	3.12	69.12	70.32	70.39
4	4.99	67.93	69.92	71.15
5	5.09	67.93	69.39	70.32
6	4.90	71.29	72.25	71.35
7	7.25	71.44	72.22	73.13
8	6.76	71.43	72.37	73.29
9	6.66	69.19	69.98	70.04
Average	-	69.24	70.39	71.49
Standard Deviation	-	1.65	1.49	1.30

Figure 6. Phase-difference maps under different loads: (a) 3 N, (b) 5 N, (c) 7 N.

Figure 6 shows that under different loads the fringe density of the phase-difference map is different. With increasing load, the interference fringe is more intense. If the load is small, the phase diagram will be more seriously affected by noise. On the contrary, if the load is large the interference fringe will be difficult to identify, as it will exceed the measuring range of the system. Under the same load, the smaller the thickness of the specimen, the greater its bending deflection; Therefore, load values within the appropriate range should be selected for specimens of different sizes, for reliable measurement results. The resolution of the pressure sensor in the experimental system is 0.1 N, while the resolution of DSPI measurement of deflection can reach 13.3 nm. Assuming that the measurement of the sample size is accurate and error-free, for a sample with a thickness of 2 mm in the current experimental environment, the theoretical resolution of the experimental system for measuring the elastic modulus is 2.86, 1.71, and 1.23 GPa for loads of 3, 5, and 7 N respectively. In comparison, the theoretical resolution for measuring the elastic modulus using a micrometer is 36.96, 22.18, and 15.84 GPa for the same loads. The resolution of measuring the elastic modulus is related to the load size: The greater the load, or the larger the deformation, the higher the measurement resolution. This is also the reason why relevant measurement standards impose restrictions on sample size, to select a suitable size to obtain a larger deformation, while the high precision of DSPI reduces the requirements for the sample. The resolution of the DSPI system is much higher than that of the pressure sensor, so the resolution of the elastic modulus measurement is mainly determined by the resolution of the load. Therefore, selecting the largest possible load within the measuring range of the system, or selecting a pressure sensor with higher resolution, can improve the accuracy of the measurement results.

Table 3 shows measured results for a 2-mm specimen under different loads. The average value of E is 69.24 Gpa, and the standard deviation is 1.65 Gpa; The average value of E_e is 70.39 Gpa, and the standard deviation E_e is 1.49 Gpa; And the average value of E_e–_s is 71.49 Gpa, and the standard deviation E_e–_s is 1.30 GPa. The standard value of the elastic modulus of quartz glass is around 72 GPa. Compared to the standard elastic modulus E, the eccentric elastic modulus E_e obtained at the same location is close to the true value. This shows that the proposed model to correct for eccentric load is effective. In addition, the final elastic modulus corrected for both eccentric load and shear force is more accurate than the eccentric elastic modulus, and the standard deviation of E_e–_s is smaller than that of E_e. This indicates that the proposed comprehensive error correction for both eccentric load and shear force is accurate and reliable.

To further verify that the error caused by shear force is affected by the span-depth ratio, a specimen of 1-mm thickness is selected as the test object to repeat the above experiments. Because the 1-mm sample is easier to bend than the 2-mm sample, smaller loads up to 1 N are applied, to avoid exceeding the measurement range. Table 4 shows measured results for a 1-mm specimen under different loads. The average value of E is 69.85 Gpa, and the standard deviation is 1.18 Gpa; The average value of E_e is 71.34 Gpa, and the standard deviation E_e is 1.05 Gpa; And the average value of E_e–_s is 71.87 Gpa, and the standard deviation E_e–_s is 0.97 GPa. Compared to Table 3, the difference of average value between E_e and E_e–_s for the 1-mm specimen is smaller. This verifies that the greater the span-depth ratio, the smaller the influence of shear force on the measurement results, which is consistent with the simulation results. The correction of various errors depends on the deflection data for multiple points, and full-field measurement is a prominent feature of DSPI, so the deflection data for multiple points can be calculated simultaneously according to the phase diagram, which is beneficial for correcting errors.

TABLE 4. Measured results for a 1-mm specimen.

Number	Force (N)	E (GPa)	Ee (GPa)	Ee–s (GPa)
1	0.39	69.54	70.86	72.49
2	0.49	69.51	70.38	73.62
3	0.58	68.60	69.87	71.44
4	0.86	68.59	71.99	70.80
5	0.95	71.52	72.11	70.63
6	0.98	71.33	72.86	72.24
Average	-	69.85	71.34	71.87
Standard Deviation	-	1.18	1.05	0.97

The fluctuation in the measurement of elastic modulus E can be attributed to various factors. First, the resolution of elastic modulus measurement is influenced by the size of the applied load; the largest possible load should be selected (within the measuring range of the system), or a pressure sensor with higher resolution should be chosen. Second, a manually applied load during the experimental procedure cannot precisely control the loading rate, and the loading rate of the three-point-bending load may affect the measurement of the elastic modulus. In addition, the residual noise after phase-map filtering can also cause differences in repeated measurements. Adopting an automatically controlled loading method and choosing a suitable adaptive-filtering algorithm can help to obtain more stable measurements.

Error correction for eccentric load and shear force in measuring the elastic modulus of quartz glass using digital speckle interferometry is modeled and evaluated. Compared to traditional displacement sensors, DSPI not only offers higher deflection-measurement accuracy, but also can determine the full-field deflection distribution of the quartz-glass plate. The error caused by an eccentric load is eliminated by analyzing the positional relationship between the maximum deflection and the loading point. The effect of shear deformation can be corrected by measuring the deflection at two points. Instead of taking multiple measurements or arranging multiple sensors to eliminate errors, the deflection at multiple points can be obtained directly from the phase-difference diagram in a single measurement. Simulations show that the sample size is important in the measurement of elastic modulus, and the experimental results show that measuring elastic modulus using DSPI is simple and has the advantage of full-field measurement, which is of great significance for accomplishing high-precision measurement of the elastic modulus.

The authors declare no conflict of interest.

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.

National Natural Science Foundation of China (NSFC) (No. 52075044, 52075045); Natural Science Foundation of Beijing Municipality (No. 4212047).