Materials Science, Vol. 45, No. 2, 2009
AUTOMATED EVALUATION OF STRAIN FIELDS BY THE COORDINATE-GRID METHOD P. V. Yasnii,1, 2 I. V. Konovalenko,1 and P. O. Marushchak1
UDC 670.191.33
We develop an automated coordinate-grid method (CGM) aimed at the evaluation of surface displacements and strains. Points of marking of the surface are used as bases. The principles of application of the CGM are described in detail. By using the proposed method, we find the elastoplastic strains in the material. The possibilities of the CGM are experimentally verified and its applicability to the evaluation of strains and displacements is demonstrated. Keywords: digital photographic image, vector of displacement, strain, stress.
There exist numerous approaches to the determination of local strains in structural elements among which one can mention the well-known coordinate-grid method [1–6]. The application of digital technologies makes it possible to perform mathematical processing of the obtained photographic images and substantially decrease the laboriousness of data processing [5, 6]. To increase the accuracy of measurements, it is customary to use highresolution optical microscopes and photocameras [7]. However, there exist methodical difficulties connected with the identification of points of the coordination grid, taking into account the gradient of illumination of the specimen surface, and other specific features of getting and processing the images. In what follows, we study the strains formed on the surface of a steel specimen under static loading and determine the vectors of displacements of mesovolumes of the material of the specimen. Experimental Procedure The discrete monitoring of the process of deformation of the specimen surface was performed by recording and processing digital photographic images of a preliminarily plotted coordinate grid. We used flat specimens 200 mm in length, 25 mm in width, and 5 mm in thickness. On the lateral side of the specimens, we made concentrators 2 mm in depth and grew fatigue cracks to analyze plastic zones for different values of the stress intensity factors. The specimens were loaded in an STM-100 universal servohydrailic tensile-testing machine. The specimens were first marked by the method of electrochemical etching by making a grid of dimples. The original of the grid was printed on a transparent film and put on a polished surface of the specimen (with roughness not higher than Ra = 0.63 ) covered with a photoresist lacquer. The specimen surface was irradiated with the light of a mercury-quartz lamp and, as a result, the image of the dividing grid was obtained on the photosensitive protective lacquer coating [8]. The grid of points was developed in a bath with a weak NaOH solution, and the specimen was washed in distilled water and dried. Then it was dipped in a sodium-chloride–sodium-phosphate mixture for electrochemical etching of the unprotected regions. The dimples of the dividing grid were obtained with a depth of 2–3 μm, a diameter of 0.02 mm, and steps of up to 0.1 mm. 1 Pulyui Ternopil State Technical University, Ternopil, Ukraine. 2 Corresponding author; e-mail:
[email protected].
Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 45, No. 2, pp. 130–136, March–April, 2009. Original article submitted August 6, 2008. 1068–820X/09/4502–0291
© 2009
Springer Science+Business Media, Inc.
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AND
P. O. MARUSHCHAK
In the process of loading, we recorded the images of the specimen prior to loading and the required number of images after tension with a photocamera and entered the data into a computer. The images of the specimens with plotted coordinate grid were made in an installation containing an MBS-10 binocular microscope with an X70 magnification, a digital camera with a resolution of 6 MPix, and lighting equipment. The photographs of the studied regions in the intact state and after plastic deformation were analyzed according to a specially developed algorithm including the following steps: getting the matrix of brightness, program doubling of the resolution, Gaussian filtration, adaptive binary transformation, clusterization, formation of the chains of spots, and computation of strains. Preparation of Photographic Images for the Quantitative Analysis We studied the deformation of the surface of flat specimens made of 25Kh1M1F steel under the conditions of static uniaxial tension. The load varied from 0 to 20 kN with steps of 2 kN. We first recorded the image of the region in the unloaded state and then make the images for each stage of loading. The initial multigrade images M = { mi } were used to obtain the images G = { gi } each pixel of which corresponds to the intensity of a certain pixel in the initial image. The intensity of a pixel is proportional to the intensity of light coming to this pixel from the photographed object. This parameter most often varies from 0 to 255. It was measured according to the approaches of the US National Television Standards Committee (NTSC) [9]: gi = 0.30 mi R + 0.59 mi G + 0.11 mi B,
(1)
where mi R, m i G, and m i B are the intensities of the red, green, and blue components for the i th pixel of the image M. After doubling the resolution of the image, we managed to detect the spots applied to the specimen more clearly and form sharp images of their edges. For this purpose, we performed bilinear interpolation taking into account the intensity of four neighboring pixels [10]. As a result of this transformation, we get the image GD = { g D i}, where the level of brightness of each pixel is given by the formula g D i = a x + b y + c x y + d,
(2)
where, in turn, a, b, c, and d are the coefficients of bilinear interpolation found from the system of four equations with four unknowns according to the known values of brightness for the nearest four points of the image G. In this way, we form the image whose geometric sizes are twice larger than the sizes of the original image. Gaussian Filtration As a result of possible gradients of illumination, the physicochemical specific features of etching of the spots, their deformation distortion, and the action other factors, the image contains a large number of fine elements, which make the detection of the location of spots more complicated and introduce additional errors in the determination of the coordinates of their centers. In order to partially remove these elements and decrease their influence on the results of numerical analysis, we used the low-frequency filter with Gaussian kernel [11] K F ( x, y ) =
⎛ x 2 + y2 ⎞ exp ⎜− ⎟, 2 πσ 2 2σ 2 ⎠ ⎝ 1
where σ is the mean-square deviation of the Gaussian measured in pixels.
(3)
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Fig. 1. Fragment of the nondeformed surface of the specimen: (a) multigrade image, (b) binary image. The elements of the discrete kernel of the filter (2hk + 1 × 2hk + 1) in size were computed as follows:
H ij =
⎛ (i − h − 1)2 + ( j − h − 1)2 ⎞ k k exp ⎜− ⎟, 2 2 2πσ 2σ ⎝ ⎠ 1
(4)
where hk is the half size of the kernel. At the center of the kernel, the weight coefficients of the pixels are much larger than on its boundary. This is why the intensity of neighboring elements of the image is leveled. The key parameter of this filter is the mean-square deviation σ. If σ is very small (e.g., smaller than a pixel), the effect of leveling is practically absent because the weight coefficients of all pixels, except the central pixel, approach zero. If σ is too large (for different images, the values of this quantity are different), then a large part of the image disappears together with the noise. Since the spots sought in the image are relatively small (~10 pix), the procedure of filtering was based on the assumption that σ = 2 and Hk = 5. In this way, we get the minimum distortion of the shape of objects with reasonably good removal of weak noise. As a result of this procedure, we get a filtered halftone image GF = { gFi }. Adaptive Binary Transformation To determine the location of a spot, for each pixel, it is necessary to clarify whether it is an element of the spot or belongs to the background. For this purpose, we used a binary transformation. On the binary image, white pixels correspond to the background and black pixels probably belong to the sought object. However, due to the nonuniformity of illumination in the process of taking photographic images, the standard methods of binary transformations form distorted images of the spots. This is why we used the method of binarization in which the parameters of transformation were found individually for each part of the image. For each pixel of the image, we consider its vicinity with size bR , determine the minimum IB min and maximum IB max values of the intensity, and use these values to compute the limit of transformation:
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LB =
AND
LA (I − I B min ) , 100 B max
P. O. MARUSHCHAK
(5)
where LA is the relative limit of transformation, %. We set LA = 20%, and the radius of adaptive window b R was equal to 100 pix. As a distinctive feature of the described approach, we can mention the fact that bR is larger than the distance between the spots. As a result of the binary transformation, we get a matrix B = { bi } with elements ⎪⎧ 0, bi = ⎨ ⎩⎪ 1,
for
gFi < L B ,
for
gFi ≥ L B .
(6)
The indicated algorithm was applied to a fragment of the image (see Fig. 1). Agglomerative Clusterization To gather the sets of points of the binary image into interrelated groups, we used the procedure of agglomerative clusterization. First, it was assumed that each point is an individual cluster but then the clusters were merged if the distance between them was smaller than or equal to the limiting distance Llim , namely, l ≤ Llim .
(7)
In practice, in processing the accumulated images, it was assumed that the clusters are merged if they have at least one neighboring point, i.e., Llim =
2 (the distance between the neighboring pixels along the diagonal
is equal to 2 pix). In the procedure of clusterization, an important role is played by the sizes of the minimum Smin and maximum Smax clusters. Since the sizes of the spots made on the specimen surface are approximately equal, the clusters whose sizes lie outside the admissible range are discarded, as well as the clusters adjacent to the edges of the image. After clusterization, we determine the coordinates of the centers of each cluster:
XC
1 = N
N
∑ xi , 1
YC
1 = N
N
∑ yi ,
(8)
1
where N is the number of points in the cluster, xi and yi are the coordinates of the i th point. The set of the centers of clusters forms the following array: C = { ci } = { ( xci , yci )}.
(9)
Each cluster in an element of the image different from the background. The collection of elements of this sort includes the required spots and noise elements that were not removed in the previous stages of the algorithm.
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Fig. 2. Ordered distances (a) and the angles of inclination (b) between the centers of clusters a di (1, 3); the differentials of the curve of distances (2) and the angles of inclination of chains (4).
Formation of the Chains of Spots and Evaluation of Their Parameters After the construction of the set of clusters, they were gathered into interrelated chains in the following order: first, the parameters of the set of spots were computed and then the chains of spots were detected. In the first stage, we computed the mean distance between the spots LSmid and the mean values of the angles of inclination of the chains of spots LA1 and LA2 . For each cluster in (9), we found two neighboring clusters and determined the distance from them to the center of the cluster c i and the slope of the straight line passing through the center of the cluster and the point ( x ci , y ci ): A d = { a di}
and
Aa = { a ai },
(10)
where adi and aai are, respectively, the distance from the center of the cluster ci to the center of the neighboring cluster and the slope of the straight line passing through their centers. Although the array of clusters (9) contains noise elements, their number is insignificant as compared with the number of correctly identified clusters. Thus, the array A d contains mainly the distances between spots applied to the surface of the specimens with constant steps. To determine the range of values corresponding to the distances between the actual spots in the array A d, we arranged this array in the order of increasing and performed the operation of digital differentiation to construct the array of derivatives Ad′ (Fig. 2a). In the differentiated array, we selected the range of values close to zero and corresponding to the range of distances between the actual spots in the image according to the boundary condition adi ′ ≤
Alim ( max( Ad′ ) − min( Ad′ )) , 100
(11)
where Alim is the relative limit below which the values of the derivative are regarded as close to zero. The range of values below Alim corresponds to the range of distances between the spots in the image. Thus, we get the set of distances between the clusters corresponding to the applied spots. As a result of averaging these values, we find LSmid . The angles of inclination of the chains of spots are found as a result of similar transformations performed for the array Aa with the only difference that there are two mutually perpendicular directions of chains.
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Fig. 3. Distributions of the normalized coordinates of the centers of clusters (1) and their differential (2) along the X-axis (a) and along the Y-axis (b).
This is why, instead of a single range, it is necessary to select two largest ranges with values of the derivative close to zero in the differential array Ad′ (Fig. 2b). As a result of averaging of each of these ranges, we obtain the mean values of the directions of the chains of spots LA1 and LA2 . The procedure of detection of the chains of interrelated clusters includes the following stages: rotation of the coordinate axes and determination of new coordinates of the centers of spots, detection of horizontal and vertical strips containing the chains of spots, and removal of the erroneously detected chains. The coordinates of the centers of detected clusters were normalized by rotation the coordinate system of the image about its center as a result of which the X-axis coincided with the direction of LA1 : R xCi = xCi cos( L A1 ) + yCi sin( L A1 ) ,
(12) R yCi
= − xCi sin( L A1 ) + yCi cos( L A1 ) .
R R The obtained arrays X CR = { xCi } and YCR = { yCi } are arranged in the order of increasing and differentiated. In the plot of the derivative, we select the ranges of values close to zero each of which corresponds to a strip, most likely containing a chain of points. Within the limits of each of these strips, the coordinates of points R vary smoothly and insignificantly. The plots of typical distributions of the normalized coordinates xCi
and
R yCi are presented in Fig. 3. The outlined method of selection of strips containing the chains of points can be used until the distance between the centers of spots is much larger that the displacements of the centers of spots as a result of deformation. Since, for the investigated images, the ratio of these quantities varies from 15 to 30, the proposed algorithm has a significant reliability margin. Within the boundaries of each detected strip, the points were ordered to form a chain. In this way, the chains of interrelated points were obtained in the directions L A1 and L A2 . After this, the chains containing a single point and the chains whose direction differed from the principal direction ( LA1 or LA2 ) were discarded. To determine the direction of the chain, its components were approximated by the method of least squares. The slope of the straight line obtained as a result determined the direction of the chain. As a result of these actions, the required chains of spots were localized in the image (Fig. 3).
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Fig. 4. Results of the analysis of deformation of the surface: (a) chains of spots in a fragment of the image, (b) vector field of displacements.
The displacements of the corresponding centers of clusters used to find the values and direction of strains were computed by comparing the images obtained prior to and after deformation. In Fig. 4, we show the vector strain field for a fragment of the image presented in Fig. 1. It was established that, in the investigated zone, the level of strains decreases as the distance from the concentrator increases. The picture obtained by comparing the images of the surface shows that the zones of maximum concentration correspond to the zones of fracture of the specimen. Thus, it is possible predict the region of most probable crack propagation according to the strain fields computed by using the automated coordinate-grid method [13]. The following specific features of the proposed method should be emphasized: — the plastic strains formed in the material are assessed for each elementary point of the grid, and the system of equations for the description of the behavior of the material is solved on the mesoscopic level to analyze the displacements of the nodes (common vertices) of the neighboring elements of the grid; — the results of our analysis enable us to write a system of equations for displacements in the entire studied region and determine the total strain. CONCLUSIONS We propose a method for the analysis of the fields of displacements and mesostrains based on the determination of the coordinates of the centers of points of the grid, identification of the corresponding points in the images of the intact and deformed material, and automated evaluation of their displacements. The method is based on the application of elements of the theory of identification of images to the images of the surfaces of investigated specimens as a result of which the location of the preliminarily applied markers is determined. By comparing the images obtained prior to and after loading, one can find the strain fields and the vectors of displacements. By analyzing the displacements and the stress–strain diagram of the material, we can also determine the stress–strain states of local regions on the surface of the specimen. The present work was financially supported by the Ukrainian State Foundation for Fundamental Research, No. 0107U004213.
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