Digital method for analysing speckle-interferometric images of material deformation

Residual stresses arising from the technological processes of manufacturing aircraft structures can considerably affect the bearing capacity of the structure, which justifies the need to assess the level of these stresses. The study analysed the speckle interferometric images in terms of the parameter image intensity. This method allows assessing the picture of the change in this parameter both in a particular band, and performing a comparative analysis in a number of bands. Due to the presence of a correlation between the material deformation and the intensity of the image, the dynamics of the variation of the stress state (residual stresses) is estimated. When decoding the holograms, the authors used the Aleksandrov-BonchBruevich vector equation. Double exposure of speckle holograms was applied and photo processing of the hologram was performed. It is shown that, depending on the position of the beam from the hole, there is a change in the image intensity. It was proved that the method employed in the study does not depend on external factors. The proposed approach allows taking a fresh look at the picture of stress state analysis and evaluating the qualitative and quantitative processes in the deformation zone, namely around the drilled hole in the plate.


Introduction
In composites currently used in aircraft structures, as a result of manufacturing processes, residual stresses occur rather frequently, which can considerably affect the bearing capacity of the structure [1][2][3][4][5][6][7][8][9][10][11]. This is especially important for composite structures with various nanomodifications [12]. As a consequence, it is extremely important to assess the level of these stresses. In these studies, a method for determining residual stresses in orthotropic composite plates made of various materials in spatiotemporal observation [13] was developed and verified, based on the measurement of local displacements by the method of electron speckle interferometry. In [14; 15], the analysis of the stress state in plates with holes of different diameters was analysed in minute detail. The study [16] developed and approbated a new method for determining the main components of residual stresses in composite plates. It is based on through-hole drilling and further measurements of hole diameter increments in the main stress directions using electron speckle interferometry. The theoretical and metrological foundations of the approach follow from the analytical solution of S.G. Lekhnitskiy, describing the stress concentration along the edge of the central open hole in a rectangular orthotropic plate under tension in the main directions of anisotropy. In [17][18][19][20][21][22][23], a new experimental method was presented for determining the parameters of both singular and non-singular fracture mechanics using a modified version of the crack compliance method. Its essence lies in measuring the local deformation response in terms of displacements at a small increment in the crack length by the method of electron speckle interferometry. Studies [24][25][26][27] presented the speckle-interferometry analysis of the stress-strain state around the hole at the stages of loading up to fracture. Methods of off-bench registration and decoding of speckle holograms were proposed for measuring individual plane and bending normal spatial components of deformations and displacements of full-scale structures when tested in industrial conditions. Study [28] presented the results of deformation of a sample with different hole sizes. Paper [9] described a simple technique for measuring the elastic moduli and Poisson's ratio. The papers used in this study contributed to a detailed analysis of the operational parameters of the investigated design solutions. It is believed that the technical parameters of materials are determined by interference fringes, which constitute the locus of points of equal displacement during deformation. The sensitivity of the interferometric bands significantly differs in various components of the strain tensor. For example, when sufficiently thick specimens are stretched, flat components are observed in the interferograms, and when thin specimens are stretched, materials bending out of the plane of deformation are observed. In this paper, to study such problems, it was proposed to apply a digital method of processing photographs by the parameter intensity (brightness) of images. This parameter was used to investigate the microstructure and surface of materials upon mechanical deformation tests [29][30][31][32][33][34][35]. Study [13] demonstrated that the quantitative composition of the material structure can be estimated by a given parameter. There is also a correlation between the intensity of the image and the surface roughness of the product [36]. Investigation of micrographs of polymer nanoparticles by the parameter of image intensity allows accurately identifying the shape and size of particles, the dynamics of the polymer deposition under study [37]. By analysing speckle-interferometric images in detail, considering the dynamics of the change in the image intensity, one can analyse its change both in a separate band, in a number of adjacent bands, and presented in different images. For example, when analysing the stress state with different hole diameters or with different composite materials. This method allows for a deeper understanding of the deformation processes of the material, because there is certainly a correlation between material deformation (shear) and the intensity of the images. Various tasks related to material deformation are described in [38][39][40][41].

Material and Methods
At present, when decoding holograms, the vector equation of Aleksandrov-Bonch-Bruevich is mainly used [42].
where U is the vector of displacement of the point of the object, ρi, ρ o are the vectors of illumination and observation, λ is the laser wavelength, n is the order of bands. In this equation, the vectors of displacements of the points of the object recorded on the holographic interferogram are associated with the vectors of illumination and observation, which explain the phenomenon of mobility of the patterns of bands on the surface of 3D diffusely reflecting objects. Its use in measuring deformations encounters difficulties due to the poor definiteness of the obtained equations of spatial measurements. Therefore, at present, double exposure of speckle holograms is widely used, one before applying a load or at some stage of loading an object, the other after applying an additional load. After photo processing of such a hologram, while viewing it on the reconstructed image, it is possible to study the changes in the object, its stress-strain state. The stress  and deformation of the surface  from the interferogram when filtering the double exposure speckle holograms are determined as follows [43]: where, is the modulus of elasticity, is the price of bands on the speckle interferogram, A is the band pitch.
where: is the wavelength of the used illumination of the speckle hologram, is the focal length of the lens that performs the Fourier transform during filtering. sin = tg (6) at small angles of filtration of spatial frequencies, where a is the diffraction filtration angle. Thus, mathematically, a speckle-interferometric image is obtained in a comparative analysis of the matrices of two images, where the first matrix characterises the "undeformed" image, the second -the deformed one. The matrices themselves consist of МN cells with data on the image intensity L. Since the images are presented in grayscale (Grayscale model), we have L values with non-negative numbers in the range 0-255. Here: where k is the sampling rate, and  ph ,  r is the fraction of the size of the elements in a digital photograph and its meaning in reality, respectively. Subtracting the resulting matrices, we obtain a matrix with МN cells with data L = f(x,y). Here x and y are spatial coordinates (on a plane). It is this one that determines the regularities of changes in surface deformations by parameter L. Image processing is carried out using the graphic application of the Image Processing Toolbox (IPT) package of the MatLab programme [44][45][46][47][48][49][50].
Here k is the number of measurements L in a given section of the band. Figure 2b demonstrates a scheme for analysing the change in L depending on the shape of the band. It is advisable to analyse the change in L depending on the curvature of the band along the ray (radial method) or in longitudinal (or cross) sections. The use of the ray method is necessary when the curvature of the band boundary is large, i.e., in the variant with an approximate position of the beam normal to the band boundary.
In the second casein versions with low curvature and complexity of the beam in the "combined" versions [65][66][67][68].

Results and Discussion
Shown in Figure 3, b is a speckle-interferometric image of a plate made of D-16T material 6 mm thick with a hole 6 mm in diameter at the stage of loading with a differential load of 1.7 tf. [69][70][71][72][73][74][75][76]. Analysis of the law of variation of L, the width of the band H in the given cross-sections along band 1 from 20 to 180 mm with a step of 40 mm are presented in Figures 3a, 3b. The dynamics of the change in the width of the H band from the minimum value in section 20 and a gradual increase to section 180 is clearly visible. In this case, the shape of the change L in the sequence of sections is deformed from a single-vertex view to a two-vertex and threevertex views. And the value of L max -L min (the difference in bands 1 and 2) has a slight tendency to fall. There is a certain "defect" in the region of sections 140-180, which makes its own corrections [77][78][79][80][81][82][83][84].  In the next speckle image [85][86][87][88][89][90] (Figure 4), we have almost circular bands. For this variant of the study, it is advisable to analyse the change in the intensity of the image using the ray method. Here, the rays are located at different angles. In this case, for the cases of 0 о , 90 о , the ray method is transformed into a measurement method along the longitudinal section. Figure 5 demonstrates the sequence of changes in the diagrams in the sequence of rays in the range of 15 o -165 o . A forced "increasing" process is clearly seen, based on a harmonious change in the amplitude L max -L min . and the value of the harmonic width l. Here, the energy from the drilling mechanism in the material is converted into mechanical three-amplitude plastic deformation. In band 7, almost complete attenuation occurs. It should be noted that the cross section of 90° and after it (105°-165°), the amplitude vibration differs in shape from the oscillatory processes in the sections of 15°-75°. The values of L max in the bands P2, P4, P6 are practically the same, and in the case of cross sections 15 о -75 о it has a growth tendency [91][92][93].  (Figure 6, a). Using the apparatus of the graphic application of the Image Processing Toolbox (IPT) package of the Matlab programme, the data array L, proceeding from the matrices before and after drilling the plate, was obtained. The law of variation in the B band was investigated (Figure 6, b, c), where L hi, L 0i is the intensity of the image in a given i-th cell in a plate with and without a hole was investigated. Next, the upper part of the image is divided into 13 rays at an angle of 15 o [94]. a) b) c) Figure 6. Plate before drilling (a), plates with a hole (b), the layout of the rays on the display in MatLab (c) On the band B on the j-th ray, the segment l j , was chosen (Figure 4), which characterises its width. Further the diagrams L hj , L 0j , L j = f(x,y) and estimates of their average values depending on the direction were built on it ( Figure 7). The data obtained are displayed in Figure 8, a.  Figure 8, b). This is reflected in the dynamics of change  . On rays 1 and 13, we have values opposite in sign ( are, respectively, 1.72 and -2.97). Their difference is more than 2.7 times. Thus, we can assume that at these poles there is a stress state of different sign. On beams 2-8, the value of  has a downward trend from beam 1 to beam 13. In this case, the assessment  , S L on rays 1, 13 was carried out at a distance of one diameter from the beginning of the hole (Figure 8, c), since these rays are at the beginning of the nucleation of the B band. It should be noted that on beams 1 and 13 there is a maximum spread S L . It can be considered that the points on the hole 1 and 2 (Figure 8, c) (rays 1, 13) are, as it were, the generators of wave processes in the bands. Based on Figure 9, it can be seen that at points 1 and 2 there is a difference in the values of L, which is characterised by the value L 1-2 . At the same time, for beam 13 we have at the beginning a decreasing tendency of L change, for beam 1 it is increasing (antiphase movement of deformation). In the longitudinal section, the amplitude fluctuation of the image intensity with the l х wave is clearly visible. Inside this wave, there are "internal" oscillations li. In this case, l х , li change both in the direction of increasing (stretching) and in the direction of decreasing (compression). The l х value and the L max -L min amplitude in this wave determine the "force" of plastic deformation, and therefore contribute to the formation of the "geometry" of the fringe and the total number of interference fringes [95; 96].

Conclusion
In this study, the authors developed a method for assessing the dynamics of changes in the image intensity in the investigated band (bands) of the speckle-interferometric image. It was demonstrated that, depending on the position of the beam on the hole (the point of the onset of deformation), a wave process of changing the image intensity occurs. The criterion for evaluating the comparison of the dynamics of change in the image intensity in the cross sections of the band(s) can be the maximum value at the vertex (one-vertex case) or the average value of the change in intensity (with its standard deviation). The section in a given band, depending on its curvature, must be determined by the ray method or by transverse or longitudinal sections. It was proved that the applied method does not depend on "external" factors: material structure, defectiveness, etc. It allows drawing quantitative and qualitative comparative conclusions for different variants of research objects. It was proved that the applied method allows estimating the boundaries of the bands with a given probability, and hence their area, which is critical in comparative studies of different speckle-interferometric images. The presented algorithm allows evaluating the dynamics of the change in the process in the given bands, which in turn enables the evaluation of the uniformity of plastic deformation in the sample.