rus
Issue #6/2025
A.P.Fedotkin, I.V.Laktionov, G.Kh.Sultanova, K.S.Kravchuk, V.N.Reshetov, A.S.Useinov
STUDY OF MICRO-SCALE SURFACE DEFORMATIONS BASED ON ATOMIC FORCE MICROSCOPY DATA ANALYSIS USING THE DIGITAL IMAGE CORRELATION METHOD
STUDY OF MICRO-SCALE SURFACE DEFORMATIONS BASED ON ATOMIC FORCE MICROSCOPY DATA ANALYSIS USING THE DIGITAL IMAGE CORRELATION METHOD
DOI: https://doi.org/10.22184/1993-8578.2025.18.6.326.334
The applicability of the digital image correlation (DIC) method to atomic force microscopy (AFM) topographies under local loading is demonstrated. Technically pure Grade 4 titanium after equal-channel angular pressing according to the Conform scheme was investigated. Local deformation was formed by a microspherical diamond indenter (R = 2.5 μm). AFM scanning of the pre-/post-indentation region was performed with registration using a square scratch pattern (~1 µm). Pre-processing included artifact suppression and field equalization. Subpixel DIC over local windows provided recovery of the radial and tangential components of the displacement field. Continuous maps were obtained, capturing the propagation of disturbances beyond the contact zone and the features of local plastic deformation. The results confirm the technical feasibility and informativeness of DIC for quantitative mapping of displacements during nanoindentation of ultrafine-grained titanium.
The applicability of the digital image correlation (DIC) method to atomic force microscopy (AFM) topographies under local loading is demonstrated. Technically pure Grade 4 titanium after equal-channel angular pressing according to the Conform scheme was investigated. Local deformation was formed by a microspherical diamond indenter (R = 2.5 μm). AFM scanning of the pre-/post-indentation region was performed with registration using a square scratch pattern (~1 µm). Pre-processing included artifact suppression and field equalization. Subpixel DIC over local windows provided recovery of the radial and tangential components of the displacement field. Continuous maps were obtained, capturing the propagation of disturbances beyond the contact zone and the features of local plastic deformation. The results confirm the technical feasibility and informativeness of DIC for quantitative mapping of displacements during nanoindentation of ultrafine-grained titanium.
Теги: digital image correlation displacement and deformation map nanoindentation наноиндентирование поля смещений и деформаций цифровая корреляция изображений
Received: 12.09.2025 | Accepted: 17.09.2025 | DOI: https://doi.org/10.22184/1993-8578.2025.18.6.326.334
Original paper
STUDY OF MICRO-SCALE SURFACE DEFORMATIONS BASED ON ATOMIC FORCE MICROSCOPY DATA ANALYSIS USING THE DIGITAL IMAGE CORRELATION METHOD
A.P.Fedotkin1, Engineer, ORCID: 0000-0003-3822-4811
I.V.Laktionov1, Engineer, ORCID: 0000-0002-8576-3669
G.Kh.Sultanova1, 2, Engineer, ORCID: 0000-0002-4770-5724
K.S.Kravchuk2, Cand. of Sci. (Physics and Mathematics), Researcher, ORCID: 0000-0002-9956-9939
V.N.Reshetov3, Doct. of Sci. (Physics and Mathematics), Prof., ORCID: 0000-0002-8426-5991
A.S.Useinov4, Cand. of Sci. (Physics and Mathematics), Deputy Director, ORCID: 0000-0002-9937-0954
Abstract. The applicability of the digital image correlation (DIC) method to atomic force microscopy (AFM) topographies under local loading is demonstrated. Technically pure Grade 4 titanium after equal-channel angular pressing according to the Conform scheme was investigated. Local deformation was formed by a microspherical diamond indenter (R = 2.5 μm). AFM scanning of the pre-/post-indentation region was performed with registration using a square scratch pattern (~1 µm). Pre-processing included artifact suppression and field equalization. Subpixel DIC over local windows provided recovery of the radial and tangential components of the displacement field. Continuous maps were obtained, capturing the propagation of disturbances beyond the contact zone and the features of local plastic deformation. The results confirm the technical feasibility and informativeness of DIC for quantitative mapping of displacements during nanoindentation of ultrafine-grained titanium.
Keywords: nanoindentation, digital image correlation, displacement and deformation map
For citation: A.P. Fedotkin, I.V. Laktionov, G.Kh. Sultanova, K.S. Kravchuk, V.N. Reshetov, A.S. Useinov. Study of micro-scale surface deformations based on atomic force microscopy data analysis using the digital image correlation method. NANOINDUSTRY. 2025. Vol. 18. No. 6. PP. 326–334. https://doi.org/
10.22184/1993-8578.2025.18.6.326.334.
INTRODUCTION
The Digital Image Correlation (DIC) method is one of the most versatile tools of modern experimental mechanics and materials science [1]. It is based on comparison of two images of the studied object – the initial one and the one obtained after a certain impact, for example, a mechanical load. Using mathematical correlation algorithms, it is possible to detect small displacements and deformations in the image structure, often at the level of subpixel accuracy [2]. This makes it possible to reconstruct the fields of displacements and deformations in the material and thereby gain a deeper understanding of the processes occurring during its loading.
At the macro level, the DIC method is traditionally used to analyse photographs of the specimens surface subjected to tension, bending or torsion. In this case, it becomes possible to record not only local deformations in the load zone, but also long-range disturbances, which significantly expands the understanding of deformation and fracture mechanisms. The method makes it possible to identify manifestations of anisotropy of mechanical properties, symmetry of deformation processes, and crack propagation mechanisms.
Atomic force microscopy (AFM) provides a fundamentally different experimental basis: images of surface topography with nanometre spatial resolution [3]. The application of the DIC method to AFM data opens up the possibility of deformation processes analysing at the nano- and microscale. The use of such an approach is particularly promising when studying the relief before and after local indentation of a microspherical indenter. This makes it possible to record displacement fields near the contact zone and far from it, which provides information on nanomechanical properties of the material, features of plastic deformation and residual stresses.
In this paper we study the possibility and conditions of applying the digital image correlation method to experimental AFM data obtained by local surface deformation on the micrometre scale of linear dimensions.
Basics of Digital Image Correlation Method
Digital image correlation is a computational processing method based on comparison of local fragments of images recorded before and after an external influence. The basic idea is as follows: an image is considered as a two-dimensional function of intensity distribution (brightness or height, in the case of AFM topography). This function is partitioned into small regions of interest (subsets or correlation windows), within which are the transformations (displacements, rotations, deformations) that minimise the discrepancies between the original and deformed images.
The most common approaches are:
correlation method (coordinate search), in which the offset is determined by the maximum of the correlation coefficient between image fragments [4];
a least squares method in which the bias parameters are estimated by minimising intensity difference in the neighbourhood of the fragments being compared[5];
frequency methods (based on fast Fourier transform) used for accelerated processing of large images [6]
Subpixel displacement detection is achieved by interpolating or approximating the similarity function extremum.
The application of DIC has a wide range of examples [7]:
macroscale mechanical testing – plotting tensile strain fields of metal, polymer and composite specimens;
micromechanics and materials science – study of local stress states in thin films and coatings;
biomechanics – tracking the tissues deformation and cellular structures under external influence;
geomechanics and geophysics – recording displacements in a rock mass in laboratory testing or fracture monitoring.
nanoscale studies – analysing changes in surface topography when subjected to localised loading, e.g. in nanoindentation experiments.
The key advantages of the method are:
no need to apply special markers (in case of AFM topography the role of markers is played by natural surface irregularities);
high sensitivity to small displacements and deformations;
possibility of automated and statistically reliable processing of large data sets.
Digital image correlation thus represents a versatile tool for strain analysis that can be adapted to specificity of atomic force microscopy data.
This method has been successfully applied to diamond-tip imprinted optical images as part of Vickers hardness measurements [8, 9].
EXPERIMENT
The sample used was technically pure Grade 4 titanium subjected to severe plastic deformation by the method of equal-channel angular pressing according to the Conform (RKUP-K) scheme with subsequent drawing (SRI FPM "UUNiT" and NanoMeT JSC). This method provides obtaining ultrafine-grained structure due to large plastic deformations accumulation without changing the external geometry of the sample.
According to the results of X-ray diffraction analysis, the average equivalent grain size was found to be about 41 ± 2 nm, which confirms the nanostructured state formation. The surface of the samples was mechanically polished to a roughness level of Ra < 5 nm.
The measurements were carried out on the NanoScan-4D nanohardness tester [10]. A tip made of single-crystalline diamond of spheroconical geometry was used as an indenter [11]. The radius of the spherical part of the tip was 2.5 μm, the height of the working part was 1 μm.
Images of the surface topography were obtained using the Ntegra Prima probe nanolab.
For the DIC method, it is important to match the images before and after exposure very precisely [9]. For this purpose, a 1 µm wide scratch pattern in the shape of a square was applied on the sample surface, within a microspherical tip was indented. This area was then rescanned by AFM to obtain an image of the deformed surface (Fig.1).
The obtained images were subjected to pre-processing in order to eliminate primary defects associated with the atomic force microscope operating mode. The slope subtraction, row subtraction and median filtering functions built into Ntegra Prima software were used. Slope subtraction is used to compare surface topography before and after imprinting. However, images may have point defects and outliers in the form of height-shifted lines caused by thermal drift or cantilever displacement during surface scanning. To remove such defects, string subtraction is applied to match neighbouring scanning strips in height and median filtering with a 3 × 3 grid is applied to remove point defects. The described processing functions result in partial data loss.
The prepared data were processed using digital image correlation algorithms.
The raw data were square scans of 1024 × 1024 pixels, corresponding to a physical size of 35 × 35 µm.
In the first step, rectangular regions in the corners of the scans were analysed, which were considered as undeformed reference regions. For each pair of corresponding corner regions, mutual displacements were calculated using the phaseCorrelate function from the OpenCV library. The obtained X and Y displacements were averaged to determine the total displacement between the scans, which allowed us to identify the area of their mutual overlap for further analysis.
At the second stage, a regular grid with fixed parameters was set on the scan: the grid spacing was 5 × 5 pixels with indentation of 50 points from the edges, the size of the correlation area was 100 × 100 pixels for each node. The local displacement was calculated for each grid cell using the phase correlation method.
In the third step, pixel displacements were converted into physical units using a scaling factor. The resulting displacement vector fields were manually corrected to compensate for scanning inaccuracies and to match by angular regions, including linear stretching and constant displacement.
In the final step, the results were visualised as maps of radial and tangential displacements relative to the centre of the print.
Figure 2 shows the displacement maps. The green circle indicates the indentation centre. The area with contrasting red-blue dots around the centre corresponds to the imprint zone and adjacent significantly deformed areas. In this area the algorithm is not able to detect deformations because the displacements differ significantly within the correlation window. In Fig.2, on the left (a) around this region there is a zone of radial extrusion of 0.05–0.15 µm adjacent to the print (mostly red dots on the graph). The regions away from the centre of the imprint show minor offsets with variable sign due to inaccuracy of the piezoelectric stage positioning and algorithm errors. In the scratch areas, the values of the calculated offsets undergo jumps. This may be related to the scanning mode.
On the right (b), the tangential displacement map does not show regions with reliably detectable displacements. This can be explained by the spherical indenter symmetry. The small areas with clockwise (red) and anti-clockwise (blue) displacements can be explained by the material structure, inaccurate positioning of the stage and the algorithm.
DISCUSSION
The quality and reliability of applying the DIC method to AFM data depend significantly on the characteristics of the original images [12]. In practice, the following factors should be taken into account when working with an atomic force microscope:
Scanning speed. If the speed is too high, the feedback does not have time to correctly work out terrain elements, which leads to artefacts in the obtained image. This can be expressed both in the form of individual spikes and in the form of systematic slope of scanning lines. The step between points in raster scanning is also significant: the smaller it is, the more informative the data is, but the longer the time of image acquisition and the higher the probability of drift;
Thermal insulation and seismic protection. Temperature drift and vibrations have a significant impact with long scan times. Even when the probe is positioned in the same surface area, repeat images may differ due to drifts. These distortions must be compensated for in the digital processing stage, otherwise they will be interpreted as false deformations;
Linearity of the scanner field. Nonlinearities of the piezoelectric scanner lead to distortion of the size and shape of objects when they are located in different areas of the field of view. For correct application of the DIC method it is necessary to carry out preliminary calibration of the scanner to exclude systematic errors;
Shape and condition of the tip. The AFM image is formed as a result of interaction of the probe with the surface. During long-term use, especially in contact and semi-contact modes on hard surfaces, the silicon tip wears out and changes its shape. This leads to systematic changes in the visible topography. Under the conditions of the DIC method, such effects may be misinterpreted as real topography changes. Therefore, it is necessary to control the probe condition and account for its wear.
These factors determine the applicability limits of the digital image correlation method to AFM data and should be taken into account both in the planning of the experiment and in the subsequent processing.
CONCLUSIONS
The applicability of subpixel DIC to AFM topographies is experimentally demonstrated to recover radial and tangential displacement fields for localised indentation in ultrafine-grained Ti Grade 4. The key practical conclusion: the correctness of DIC on AFM data is ensured only if metrological conditions of acquisition are met – use of a linear scanner with scale calibration, repeated scanning of the same field area, thermal stabilization and vibration isolation of the setup, and selection of the scanning speed/step "as fast as stability of the feedback loop allows", excluding under-adjustment. If these requirements are met, the resulting displacement maps are stable and physically consistent; if they are not met, false displacements due to drift, nonlinearities and probe artefacts dominate. The main limitations are related to temperature and time drift, scanner nonlinearities, and probe wear, which is particularly critical in the transition from displacements to deformations. Perspectives include regularised differentiation to compute strain tensors, deconvolution of the influence of probe shape, and coupling with finite element inverse analysis to identify local constitutive parameters.
ACKNOWLEDGEMENTS
The work was financially supported by Russian Science Foundation, grant No. 25-29-00291.
PEER REVIEW INFO
Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal’s website and SEL eLibrary eLIBRARY.RU.
Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Original paper
STUDY OF MICRO-SCALE SURFACE DEFORMATIONS BASED ON ATOMIC FORCE MICROSCOPY DATA ANALYSIS USING THE DIGITAL IMAGE CORRELATION METHOD
A.P.Fedotkin1, Engineer, ORCID: 0000-0003-3822-4811
I.V.Laktionov1, Engineer, ORCID: 0000-0002-8576-3669
G.Kh.Sultanova1, 2, Engineer, ORCID: 0000-0002-4770-5724
K.S.Kravchuk2, Cand. of Sci. (Physics and Mathematics), Researcher, ORCID: 0000-0002-9956-9939
V.N.Reshetov3, Doct. of Sci. (Physics and Mathematics), Prof., ORCID: 0000-0002-8426-5991
A.S.Useinov4, Cand. of Sci. (Physics and Mathematics), Deputy Director, ORCID: 0000-0002-9937-0954
Abstract. The applicability of the digital image correlation (DIC) method to atomic force microscopy (AFM) topographies under local loading is demonstrated. Technically pure Grade 4 titanium after equal-channel angular pressing according to the Conform scheme was investigated. Local deformation was formed by a microspherical diamond indenter (R = 2.5 μm). AFM scanning of the pre-/post-indentation region was performed with registration using a square scratch pattern (~1 µm). Pre-processing included artifact suppression and field equalization. Subpixel DIC over local windows provided recovery of the radial and tangential components of the displacement field. Continuous maps were obtained, capturing the propagation of disturbances beyond the contact zone and the features of local plastic deformation. The results confirm the technical feasibility and informativeness of DIC for quantitative mapping of displacements during nanoindentation of ultrafine-grained titanium.
Keywords: nanoindentation, digital image correlation, displacement and deformation map
For citation: A.P. Fedotkin, I.V. Laktionov, G.Kh. Sultanova, K.S. Kravchuk, V.N. Reshetov, A.S. Useinov. Study of micro-scale surface deformations based on atomic force microscopy data analysis using the digital image correlation method. NANOINDUSTRY. 2025. Vol. 18. No. 6. PP. 326–334. https://doi.org/
10.22184/1993-8578.2025.18.6.326.334.
INTRODUCTION
The Digital Image Correlation (DIC) method is one of the most versatile tools of modern experimental mechanics and materials science [1]. It is based on comparison of two images of the studied object – the initial one and the one obtained after a certain impact, for example, a mechanical load. Using mathematical correlation algorithms, it is possible to detect small displacements and deformations in the image structure, often at the level of subpixel accuracy [2]. This makes it possible to reconstruct the fields of displacements and deformations in the material and thereby gain a deeper understanding of the processes occurring during its loading.
At the macro level, the DIC method is traditionally used to analyse photographs of the specimens surface subjected to tension, bending or torsion. In this case, it becomes possible to record not only local deformations in the load zone, but also long-range disturbances, which significantly expands the understanding of deformation and fracture mechanisms. The method makes it possible to identify manifestations of anisotropy of mechanical properties, symmetry of deformation processes, and crack propagation mechanisms.
Atomic force microscopy (AFM) provides a fundamentally different experimental basis: images of surface topography with nanometre spatial resolution [3]. The application of the DIC method to AFM data opens up the possibility of deformation processes analysing at the nano- and microscale. The use of such an approach is particularly promising when studying the relief before and after local indentation of a microspherical indenter. This makes it possible to record displacement fields near the contact zone and far from it, which provides information on nanomechanical properties of the material, features of plastic deformation and residual stresses.
In this paper we study the possibility and conditions of applying the digital image correlation method to experimental AFM data obtained by local surface deformation on the micrometre scale of linear dimensions.
Basics of Digital Image Correlation Method
Digital image correlation is a computational processing method based on comparison of local fragments of images recorded before and after an external influence. The basic idea is as follows: an image is considered as a two-dimensional function of intensity distribution (brightness or height, in the case of AFM topography). This function is partitioned into small regions of interest (subsets or correlation windows), within which are the transformations (displacements, rotations, deformations) that minimise the discrepancies between the original and deformed images.
The most common approaches are:
correlation method (coordinate search), in which the offset is determined by the maximum of the correlation coefficient between image fragments [4];
a least squares method in which the bias parameters are estimated by minimising intensity difference in the neighbourhood of the fragments being compared[5];
frequency methods (based on fast Fourier transform) used for accelerated processing of large images [6]
Subpixel displacement detection is achieved by interpolating or approximating the similarity function extremum.
The application of DIC has a wide range of examples [7]:
macroscale mechanical testing – plotting tensile strain fields of metal, polymer and composite specimens;
micromechanics and materials science – study of local stress states in thin films and coatings;
biomechanics – tracking the tissues deformation and cellular structures under external influence;
geomechanics and geophysics – recording displacements in a rock mass in laboratory testing or fracture monitoring.
nanoscale studies – analysing changes in surface topography when subjected to localised loading, e.g. in nanoindentation experiments.
The key advantages of the method are:
no need to apply special markers (in case of AFM topography the role of markers is played by natural surface irregularities);
high sensitivity to small displacements and deformations;
possibility of automated and statistically reliable processing of large data sets.
Digital image correlation thus represents a versatile tool for strain analysis that can be adapted to specificity of atomic force microscopy data.
This method has been successfully applied to diamond-tip imprinted optical images as part of Vickers hardness measurements [8, 9].
EXPERIMENT
The sample used was technically pure Grade 4 titanium subjected to severe plastic deformation by the method of equal-channel angular pressing according to the Conform (RKUP-K) scheme with subsequent drawing (SRI FPM "UUNiT" and NanoMeT JSC). This method provides obtaining ultrafine-grained structure due to large plastic deformations accumulation without changing the external geometry of the sample.
According to the results of X-ray diffraction analysis, the average equivalent grain size was found to be about 41 ± 2 nm, which confirms the nanostructured state formation. The surface of the samples was mechanically polished to a roughness level of Ra < 5 nm.
The measurements were carried out on the NanoScan-4D nanohardness tester [10]. A tip made of single-crystalline diamond of spheroconical geometry was used as an indenter [11]. The radius of the spherical part of the tip was 2.5 μm, the height of the working part was 1 μm.
Images of the surface topography were obtained using the Ntegra Prima probe nanolab.
For the DIC method, it is important to match the images before and after exposure very precisely [9]. For this purpose, a 1 µm wide scratch pattern in the shape of a square was applied on the sample surface, within a microspherical tip was indented. This area was then rescanned by AFM to obtain an image of the deformed surface (Fig.1).
The obtained images were subjected to pre-processing in order to eliminate primary defects associated with the atomic force microscope operating mode. The slope subtraction, row subtraction and median filtering functions built into Ntegra Prima software were used. Slope subtraction is used to compare surface topography before and after imprinting. However, images may have point defects and outliers in the form of height-shifted lines caused by thermal drift or cantilever displacement during surface scanning. To remove such defects, string subtraction is applied to match neighbouring scanning strips in height and median filtering with a 3 × 3 grid is applied to remove point defects. The described processing functions result in partial data loss.
The prepared data were processed using digital image correlation algorithms.
The raw data were square scans of 1024 × 1024 pixels, corresponding to a physical size of 35 × 35 µm.
In the first step, rectangular regions in the corners of the scans were analysed, which were considered as undeformed reference regions. For each pair of corresponding corner regions, mutual displacements were calculated using the phaseCorrelate function from the OpenCV library. The obtained X and Y displacements were averaged to determine the total displacement between the scans, which allowed us to identify the area of their mutual overlap for further analysis.
At the second stage, a regular grid with fixed parameters was set on the scan: the grid spacing was 5 × 5 pixels with indentation of 50 points from the edges, the size of the correlation area was 100 × 100 pixels for each node. The local displacement was calculated for each grid cell using the phase correlation method.
In the third step, pixel displacements were converted into physical units using a scaling factor. The resulting displacement vector fields were manually corrected to compensate for scanning inaccuracies and to match by angular regions, including linear stretching and constant displacement.
In the final step, the results were visualised as maps of radial and tangential displacements relative to the centre of the print.
Figure 2 shows the displacement maps. The green circle indicates the indentation centre. The area with contrasting red-blue dots around the centre corresponds to the imprint zone and adjacent significantly deformed areas. In this area the algorithm is not able to detect deformations because the displacements differ significantly within the correlation window. In Fig.2, on the left (a) around this region there is a zone of radial extrusion of 0.05–0.15 µm adjacent to the print (mostly red dots on the graph). The regions away from the centre of the imprint show minor offsets with variable sign due to inaccuracy of the piezoelectric stage positioning and algorithm errors. In the scratch areas, the values of the calculated offsets undergo jumps. This may be related to the scanning mode.
On the right (b), the tangential displacement map does not show regions with reliably detectable displacements. This can be explained by the spherical indenter symmetry. The small areas with clockwise (red) and anti-clockwise (blue) displacements can be explained by the material structure, inaccurate positioning of the stage and the algorithm.
DISCUSSION
The quality and reliability of applying the DIC method to AFM data depend significantly on the characteristics of the original images [12]. In practice, the following factors should be taken into account when working with an atomic force microscope:
Scanning speed. If the speed is too high, the feedback does not have time to correctly work out terrain elements, which leads to artefacts in the obtained image. This can be expressed both in the form of individual spikes and in the form of systematic slope of scanning lines. The step between points in raster scanning is also significant: the smaller it is, the more informative the data is, but the longer the time of image acquisition and the higher the probability of drift;
Thermal insulation and seismic protection. Temperature drift and vibrations have a significant impact with long scan times. Even when the probe is positioned in the same surface area, repeat images may differ due to drifts. These distortions must be compensated for in the digital processing stage, otherwise they will be interpreted as false deformations;
Linearity of the scanner field. Nonlinearities of the piezoelectric scanner lead to distortion of the size and shape of objects when they are located in different areas of the field of view. For correct application of the DIC method it is necessary to carry out preliminary calibration of the scanner to exclude systematic errors;
Shape and condition of the tip. The AFM image is formed as a result of interaction of the probe with the surface. During long-term use, especially in contact and semi-contact modes on hard surfaces, the silicon tip wears out and changes its shape. This leads to systematic changes in the visible topography. Under the conditions of the DIC method, such effects may be misinterpreted as real topography changes. Therefore, it is necessary to control the probe condition and account for its wear.
These factors determine the applicability limits of the digital image correlation method to AFM data and should be taken into account both in the planning of the experiment and in the subsequent processing.
CONCLUSIONS
The applicability of subpixel DIC to AFM topographies is experimentally demonstrated to recover radial and tangential displacement fields for localised indentation in ultrafine-grained Ti Grade 4. The key practical conclusion: the correctness of DIC on AFM data is ensured only if metrological conditions of acquisition are met – use of a linear scanner with scale calibration, repeated scanning of the same field area, thermal stabilization and vibration isolation of the setup, and selection of the scanning speed/step "as fast as stability of the feedback loop allows", excluding under-adjustment. If these requirements are met, the resulting displacement maps are stable and physically consistent; if they are not met, false displacements due to drift, nonlinearities and probe artefacts dominate. The main limitations are related to temperature and time drift, scanner nonlinearities, and probe wear, which is particularly critical in the transition from displacements to deformations. Perspectives include regularised differentiation to compute strain tensors, deconvolution of the influence of probe shape, and coupling with finite element inverse analysis to identify local constitutive parameters.
ACKNOWLEDGEMENTS
The work was financially supported by Russian Science Foundation, grant No. 25-29-00291.
PEER REVIEW INFO
Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal’s website and SEL eLibrary eLIBRARY.RU.
Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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