Marker Identification Technique for Deformation Measurement

Abstract

A technique for noncontact optical measurement of in-plane displacement based on correlation analysis is presented. This approach can be used to identify the position of a marker before and after deformation of a specimen or plane test object. Some mechanical parameters of materials and strains can be calculated using information from several markers' displacement. The method is simpler than other optical techniques in experimental processing and can be efficiently used to measure large deformations. Two applications of this method are illustrated. One application of this method is to test the mechanical behavior of polymer film material. Another application of this method is to measure the displacement field in a strata movement simulation experiment.

1. Introduction

Surface deformation measurement is one of the key factors in the experimental mechanics. Many optical methods have been developed for this aim, such as Moiré interferometry [1 –3], shearography [4], electronic speckle pattern interferometry (ESPI) [5], and digital speckle correlation method (DSCM) [6 –9]. Each method has its advantages and disadvantages in different fields of application. The DSCM enables direct measurement of the surface deformation field to subpixel accuracy. Due to its advantages of noncontact, whole field, wide measuring range, and so forth, some of its applications in the field of experimental mechanics have been achieved in recent years [10 –12]. Publications in a wide range of journals have attracted the attention of both researchers and practitioners with backgrounds in the mechanics of solids, applied physics, mechanical engineering, and materials science [13 –16].

In this paper, a marker identification technique is developed, which can be used to recognize the position of a marker before and after deformation. The method is based on the correlation theory and has some special assets. Its experimental procedure is simpler than that of the DSCM. For illustration the marker identification method (MIM) is used to test some material parameters of polymer film and measure the displacement fields in a strata movement simulation experiment.

2. Marker Identification Method (MIM)

Correlation theory has been widely used in the fields of physics, chemistry, mathematics, mechanics, image processing, and so forth. A correlation coefficient is a quantitative assessment of the strength of relationship between two variables in a set of (x, y) pairs [8, 17, 18]. In experimental mechanics field, correlation coefficient is an important tool in particle image velocimetry (PIV) techniques [19] and digital speckle correlation method [20, 21]. For this method, an image of a plane specimen surface is recorded before loading and defined as an undeformed pattern. Other images of the deformed pattern are taken during loading the specimen. These images show a deformed pattern relative to the undeformed pattern. From the difference between both these patterns the deformations can be calculated. Then the experimental measurement becomes a process of mathematical calculation, in which the correlation criterion plays the function of bridge [22 –24].

In the marker identification process the information carriers are dots marked on the surface of the specimen or the object, respectively. Several dots were marked on the specimen and sometimes the natural texture on the specimen surface also can be used as those dots. Before and after loading, two patterns of the specimen were recorded by a charge coupled device (CCD) camera and stored in computer as data files. The measurement system used in the tests is shown in Figure 1.

Figure 1:

Schematic of computer vision system.

A subset, whose center is one marked dot, is selected as the reference subset in the undeformed image. The next step is to find the target subset, that is, finding the location where the reference subset moves to. The coordinate system and deformation of the markers are illustrated in Figure 2. In order to find the best possible estimation of the deformation values a criterion must be established mathematically. Here we use a cross-correlation coefficient for the two subsets that can be written as [12]

C = \frac{\sum (f_{i} (x, y) - \overset{—}{f}) \cdot (g_{i} (x^{*}, y^{*}) - \overset{—}{g})}{\sqrt{\sum {(f_{i} (x, y) - \overset{—}{f})}^{2} \cdot \sum {(g_{i} (x^{*}, y^{*}) - \overset{—}{g})}^{2}}},

(1)

where (x, y) and (x^*, y^*) are Cartesian coordinates of a material point in the two subsets. f_i(x, y) and g_i(x^*, y^*) are gray levels of that point in the corresponding subsets, and $\overset{—}{f}$ and $\overset{—}{g}$ are the average values of f_i(x, y) and g_i(x^*, y^*), respectively. Each of these subset points (x, y) is mapped on to (x^*, y^*) using

\begin{matrix} x^{*} = x + u, \\ y^{*} = y + v, \end{matrix}

(2)

where u and v are the in-plane displacement components of each subset's center point.

Figure 2:

Markers and their deformation kinematics.

The correlation coefficient C represents how close the related two subsets are, with C = 1 corresponding to perfect correlation. Owing to the systematic errors, random errors, and the distortion of image, the correlation coefficient C cannot equal 1 in practice generally. The maximum of C is considered as the coincidence of the assumed displacement with the actual deformation components.

The digital pattern obtained by a digitizer is given as discrete data. To accurately determine the deformation at arbitrary locations, an interpolation method should be used. In this present work, a bicubic polynomial interpolation method is used which can be expressed as a convolution operator with the kernel function

Cubic (d) = {\begin{cases} (d - 2) d d + 1, & (d \leq 1), \\ [(5 - d) d - 8.0] d + 4.0, & (1 < d \leq 2), \end{cases}

(3)

where d is the x or y component of the distance between the sampled pixels and the subpixel location. With this method, the gray value of an arbitrary location (x, y) is calculated by the formula

g (x, y) = \sum_{i = 1}^{4} \sum_{j = 1}^{4} g (x_{i}, y_{j}) Cubic (d x_{i}) Cubic (d y_{j}),

(4)

where g(x_i, y_j) is the gray value of the nearest 4 × 4 sampled pixels around the subpixel location and dx_i and dy_j are the x and y components of the distance between the sampled pixels (x_i, y_j) and the subpixel location, respectively. The flow chart for the marker identification routine is shown in Figure 3.

Figure 3:

Flow chart for the marker identification routine.

Considering a point P on the object surface and two small segments PA and PB along x-axis and y-axis, as shown in Figure 2, the normal strain and the shear strain can be determined according to the theory of linear elasticity. Linear elastic material response and small deformations provided

\begin{matrix} ∊_{x} = \frac{(u_{A} - u_{P})}{\overset{—}{P A}}, \\ ∊_{y} = \frac{(v_{B} - v_{P})}{\overset{—}{P B}}, \\ γ_{x y} = α + β \approx tan α + tan β = \frac{(v_{A} - v_{P})}{\overset{—}{P A}} + \frac{(u_{B} - u_{P})}{\overset{—}{P B}}, \end{matrix}

(5)

where u_i = x_i′ – x_i, v_i = y_i′ – y_i (i = P, A, B), $\overset{—}{P A} = x_{A} - x_{P}$ , and $\overset{—}{P B} = y_{B} - y_{P}$ .

The aim of material testing is to determine the characteristic material parameters, that is, Young's modulus E and Poisson's ratio ν or the shear-modulus G, respectively. As the three quantities are depending on each other, if the loading force F_x and the cross-section area A₀ in the unloaded state are given, assuming linear elastic material response, the stress state is known; then (6) yields Young's modulus E and either Poisson's ratio ν or the shear-modulus G:

\begin{matrix} ν = - \frac{∊_{y}}{∊_{x}}, \\ A (∊) = A_{0} {(1 - ν \cdot ∊_{x})}^{2} = A_{0} {(1 + ∊_{y})}^{2}, \\ σ_{x} = \frac{F_{x}}{A (∊)} = \frac{F_{x}}{A_{0}} {(1 + ∊_{y})}^{- 2}, \\ E = \frac{F_{x}}{A_{0}} \times ∊_{x}^{- 1} \times {(1 + ∊_{y})}^{- 2}, \\ (G = \frac{E}{2 (1 + ν)}) . \end{matrix}

(6)

3. Analysis and Discussion of Marker Identification Method

The precision of displacement measurement of the marker identification method is similar to that of the DSCM in small deformation region, that is, accuracy up to ±0.01 pixels in theory (it may be up to ±0.05 pixels or even more in practice owing to the effect of systematic errors, random errors, and the quality of the computer vision system). And in large deformation region, the error will be higher than that in small deformation. The precision of the strain measurement is directly related to the displacement identification of the marker and the distance between markers, so it can be improved by increasing the distance between markers and employing high quality CCD camera and digitizing board.

The MIM has some special advantages than other optical techniques in experimental mechanics. Firstly, the preparative procedure of the experiment is simpler than that of the DSCM. No speckle field is needed on the specimen surface and the information carrier of deformation is marked by dots. Making several artificial markers or selecting several natural markers on the specimen's surface will be easy to realize. Secondly, the distribution of correlation coefficient is unimodal, as shown in Figure 4(a). In digital speckle correlation method the correlation coefficient distribution in an area of calculation is single-peaked in a small area with some small peaks in the surrounding area, as shown in Figure 4(b). The unimodality of correlation coefficient broadens the measuring region and the proposed method can be used to measure small deformation as well as large deformation. Thirdly, owing to the unimodality of correlation coefficients, the chance of misidentification is reduced in the progress of calculation, and the conditions and environment of the testing are not stringent. Fourthly, this method can be used from microscale to macroscale. The size and color of the markers can be chosen for different testing objects. The actual size of the marker relies on the enlargement factor of the image capture system. This size of the marker on the captured pictures must be smaller than the subset for correlation calculation. The color of the marker should have high-contrast to the specimen's surface. MIM can be used in some special testing and environment conditions, such as engineering testing, long-term testing, temperature, moist, and humid environment.

Figure 4:

Distribution of the correlation coefficient.

4. Application of Marker Identification Method in Mesoscale

In this section, the measurement range and the precision of the method have been discussed by a simulated test. And then a uniaxial tension test and a creep test of polymer film material also have been performed to demonstrate the application of the MIM.

4.1. Discussion for the Measurement Range and the Precision

The measurement range and the precision of the method are discussed by a simulated test. In this test, first, an image was captured as the reference image. And deformed images with 0%, 25%, and 50% actual deformations were created, as shown in Figure 5, by an image processing software. Here we assumed that the Poisson ratio was a constant and was equal to 0.4. Then the deformed images including reference image were all compared with the reference image by the marker identification method. The exact strain for case 1 should be ∊_1x = 0, ∊_1y = 0, for case 2 should be ∊_1x = 0.25, ∊_1y = – 0.1, and for case 3 should be ∊_1x = 0.5, ∊_1y = – 0.2, respectively. The corresponding strains were calculated to be ∊_1x = 0, ∊_1y = 0 for case 1, ∊_1x = 0.2510, ∊_1y = – 0.1005 for case 2, and ∊_1x = 0.5119, ∊_1y = – 0.1965 for case 3. Here the distance between the selected makers is 276 pixels and 268 pixels in x and y direction, respectively.

Figure 5:

Images for simulated test: (a) undeformed image, (b) deformed image with 20% actual deformation, and (c) deformed image with 50% actual deformation.

The results are summarized in Table 1. It is shown that the marker identification method is precise and can be effectually used to measure the strain in wide measurement range.

Table 1:

Results of the simulated strain.

	Exact values	Measured values	Difference	Error (%)

Case 1
∊_x	0	0	0
∊_y	0	0	0
Case 2
∊_x	0.25	0.2510	0.0010	0.4%
∊_y	−0.1	−0.1005	−0.0005	0.5%
Case 3
∊_x	0.5	0.5119	0.0119	2.38%
∊_y	−0.2	−0.1965	0.0035	−1.75%

4.2. Uniaxial Tension Test

The experiment involved a uniaxial tension test of Nylon 6 polymer film material. The specimen's shape and dimensions are shown in Figure 6. Dots are marked on its surface as shown in Figure 7. During the measurement the ambient temperature has been kept constant at 20°C. The experiment has been performed on a CSS-44100 electronic universal testing machine. The crossbeam speed has been set to 1 mm/min.

Figure 6:

Dimension of the specimen.

Figure 7:

Marked dots on the specimen (i stands for the ith dot, i = 1, 2, …, 9).

The load-time curve is shown in Figure 8(a). The computer vision system was working parallel to the loading process and the images were taken every 15 seconds. The image board digitizes each video line into 768 pixels while counting 576 video lines through each frame, and the discrete intensity values are recorded in 256 gray levels in the vision system.

Figure 8:

Experimental results of the marker identification method for Nylon 6 polymer film material. (a) Load versus time curve, (b) strain versus time curve, (c) stress versus strain curve of the material, and (d) stress versus strain curve in cross-direction.

By calculating the elongation between points 1 and 3, 4 and 6, and 7 and 9, respectively (see Figure 6), three values ∊_x1, ∊_x2, and ∊_x3 at time t_i were obtained, the mean value of which $\bar{∊_{x}} = (1 / 3) (∊_{x 1}, ∊_{x 2}, ∊_{x 3})$ yields the longitudinal strains of the specimen. So the strain-time curve can be obtained by calculating the longitudinal strains at different times, as shown in Figure 8(b).

The load-time curve and the strain-time curves are connected over the time. Regarding (5) the stress-strain curve in longitudinal direction can be achieved as shown in Figure 8(c). The transversal strains are calculated in the same way and the stress-strain curve in cross-direction was also obtained as shown in Figure 8(d).

In the elastic deformation state Young's modulus $\overset{—}{E} = 1.04$ GPa and Poisson's ratio $\overset{—}{ν} = 0.406$ are calculated according to (6).

4.3. Creep Test

The third experiment is related to testing the time-depending response of macromolecular film material. The strain state was determined by measuring the deformations over time by means of the MIM. Based on the obtained data the material response, that is, the bulk-modulus K(t) as a function of time, and Poisson's ration were calculated. The specimen of size 60 × 3.83 × 0.5 mm was loaded by constant load F = 40 N. The measurements were performed in time intervals ΔT = 60 min. Taking into account the lateral contraction of the cross-section, the stress σ(x) as a function of time holds

σ_{x} (t) = \frac{F}{b \cdot d} {[1 - ν (t) \cdot ∊_{x} (t)]}^{- 2} .

(7)

At time t(0⁺) = 2 min. ∊_x and ∊_y were determined, yielding a mean value of Poisson's ratio of ν(0⁺) = 0.406. Further measurements had proven that the changes of Poisson's ratio over time could be neglected:

ν (t) \approx ν (0^{+}) \approx const .

(8)

Based on Boltzmann's superposition principle [14] the relation between the uniaxial stress state and the strain state runs in discrete formulation [15, 16], and linear viscoelastic material response provided

σ_{x} (t) = 3 (1 - 2 ν) \times {K (0^{+}) \cdot ∊_{x} (t_{n}) - \frac{1}{2} \sum_{ν = 1}^{n} [K (t_{n} - t_{ν}) - K (t_{n} - t_{ν - 1})] \times [∊_{x} (t_{ν - 1}) + ∊_{x} (t_{ν})]},

(9)

where K(t) denotes the bulk-modulus. The result of the evaluation according to (9) is given in Table 2 and Figure 9, respectively.

Table 2:

Results of the strain and bulk-modulus in creep test.

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

Time (min)	2	60	120	180	240	300	360	420	480	540	600	660	720	780	840
∊_x(t) (%)	2.37	4.0	4.95	5.50	5.75	5.9	6.0	6.1	6.2	6.27	6.35	6.41	6.47	6.53	6.60
K(t) (N/mm²)	1500.6	745	612	607	605	599	595	581	571	563	561	558	554	548	543

Figure 9:

Results of the strain and bulk-modulus in creep test.

5. Application of Marker Identification Method in Macroscale

To carry out simulative experiment of similar materials is a common way to investigate the law of movement of the mined rock under the condition of mining in depth [25 –27]. In this section, the MIM is used to measure the strata movement displacement fields.

5.1. Simulative Experiment of Similar Materials in Mining Progress

According to the actual status of a working face in Huainan Coal Group Co., Ltd., we made a similar material model. The different strata were simulated by similar materials with different proportioning of sand, gypsum, and white lime. An artificial faultage was made by mica powder. In order to measure the deformation of the model, some markers (15 × 17 = 255) were made on the surface of the model, as shown in Figure 10. The model and the actual status of the working face must meet geometric similarity, kinematics similarity, and boundary condition similarity.

Figure 10:

General picture of the experimental model.

In order to capture the patterns of the model under different deformation states, an optical image capture system was employed. The image capture device used here was a digital camera, instead of CCD camera with image board. Before exploitation, a pattern was captured as the undeformed pattern. Then, with the process of exploitation, patterns under different advances were captured as deformed patterns.

The deformation fields of the model under different advances can be obtained by the MIM. For example, the displacement fields in vertical direction with advance length equal to 1.836 m and 2.232 m are shown in Figure 11. The stability and subsidence rule of strata can be achieved by the analysis of these deformation fields.

Figure 11:

Displacement contour map in vertical direction with different advances (unit: mm).

5.2. Precision Analysis in This Experiment

As a noncontacted optical measurement technique, the precision of the MIM may reach subpixel level. The magnification factor used here is 1 mm = 0.75 pixels. Thus, the error of method is less than 1 mm. To quantitatively evaluate the error of the MIM in this paper, a comparison between the MIM and the theodolite was made, as shown in Figure 12. We can see that the measurement results by the MIM agree well with the results by the theodolite.

Figure 12:

Results of comparison between MIM and theodolite.

6. Conclusion

An optical experimental technique, the “marker identification method” (MIM), has been presented, which shows special advantages. The measurement setup is quite simple and does not need special equipment, so MIM can be regarded as a low-cost method, which is easy to handle in practice. The accuracy of the method is enough to practical applications, material properties' test, and structural analyses. It has been shown that the method can be used effectively not only in the range of linear elasticity but also in the analysis of viscoelastic problems. Further on it is possible to measure finite displacement as well as to take into account nonlinear and elastic-plastic material response. Illustrations indicate that this method not only can be used in mesoscale but also can be used in macroscale.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11002100) and Tianjin Natural Science Foundation (no. 11JCYBJC26800). The support is gratefully acknowledged.

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