Quantifying the Effectiveness of Patterning, Test Conditions, and DIC Parameters for Characterization of Plastic Strain Localization

In total, three Cu-OFP tensile specimens were cut with electro-discharge machining (EDM) across an electron beam weld (EBW) and across a friction stir weld (FSW), see Fig. 1. Two of the specimens were cut next to each other from an electron beam welded copper plate. One of them was used as a reference specimen with the EDM-cut surface used as a natural DIC pattern (Fig. 1a) and the other one was patterned with the copper oxide pattern (Fig. 1b). From hereafter, the specimens are named EBW-EDM-S and EBW-P-S, respectively. P stands for patterned and S for spot lighting. A third specimen was cut across an FSW copper canister weld and patterned similarly to EBW-P-S, but in this instance indirect diffuse lighting was used instead of spot lighting (Fig. 1c). The specimen is designated as FSW-P-D, where D stands for diffuse lighting. The dimensions of the gauge length of the EBW specimens were 3.5x25x40 mm, and the dimensions of the FSW specimen were 3.0x30x53.6 mm.

The pattern bitmap was generated by specifying the spatial frequency content, with randomized phase, and calculating the inverse Fourier transformation as described in ref. [11]. The specified frequency content is directly related to the desired size of the pattern features, which was optimized by taking into consideration the imaging area of the specimen throughout the deformation. In other words, the camera had to be at a right distance from the specimen in order to see the whole specimen after deformation with the lens that was used. The resolution of the camera and lens system was evaluated at this set distance and taken into consideration during pattern generation so that the physical feature size on the specimen surface was close to 3 pixels in the image.

According to iDICs’s Good Practice guide for DIC, a feature of a traditional speckle pattern should appear as 3-5 pixels on the CCD of the camera [19]. This is to minimize aliasing from smaller features (<3 pixels), adding up as noise, and loss of spatial resolution from larger features (>5 pixels). The pattern used herein solves the problem associated with speckle patterns by introducing evenly spaced features with varying shape, that are all within the desired size range, enabling consistent information from the whole field-of-view.

Before patterning, the EDM cut surface of the tensile specimens was polished with wet grinding and diamond paste polishing up to 1 μm to allow an evenly thick copper oxide layer to be formed. An opaque layer of black copper(II)oxide (CuO) was formed by treatment in 100 g/l of 50/50 mixture of sodium hydroxide (NaOH) and sodium chlorite (NaClO₂) at 85 °C for about 10 min. AZ5214E photoresist was then spin-coated on top of the copper oxide layer. The spinning parameters were carefully adjusted for slow acceleration and deceleration, because the specimen was heavy and had an elongated shape. Even so it had to be well attached to the spinner. Maximum spin rate was about 800 RPM resulting in resist thickness of about 4 μm.

The photomask for pattern exposure was created with a Microtech LW405 laser writer using bitmap-patterning mode. The pattern was inscribed with the laser writer on a soda-lime glass plate which had a thin chromium layer. The pattern was transferred to the photoresist by placing the photomask directly on top of the copper specimen, onto which the photoresist was spin-coated, and the pattern was exposed with UV light from a mercury lamp. This method introduces a slight loss in optical resolution, but given the size of the features (>150 μm), the loss is irrelevant. After exposure, the photoresist was developed in AZ351B solution; a NaOH-based developer, and finally the cupric oxide was patterned by etching in 2% HCl solution for a short time. The remaining photoresist was rinsed away with acetone.

Figure 2 shows optical micrographs of the EDM cut surface and the copper oxide patterned surface with the same magnification. The feature size of the pattern is relatively large when compared to the roughness of the EDM cut surface, but the EDM cut surface still works as a simple DIC pattern in 2D measurement, even if it is not optimal. The grains of copper are visible in the etched regions of the patterned surface.

Tensile testing with 2D optical displacement measurement through digital image correlation (DIC) was performed with a Zwick/Roell Z020 tensile testing machine and a Strain-Master DIC system by LaVision. The camera was a LaVision Imager pro X equipped with a Nikon Micro-Nikkor 105 mm lens. Dynamic range of the camera is 14 bits. The size of the camera sensor is 1600 × 1200 pixels, but the test images were taken with a narrowed field of view of roughly 600 × 1600 pixels. Physical size of the pixels was about 50 μm/pixel for the EBW tests and 60 μm/pixel for the FSW test due to slightly different camera distance from the specimen. Images were taken every 250 s, but only every 20th image was used for the digital image correlation. This is equal to frame rate of one image per 1.4 h and cross-head displacement of 0.08 mm of the tensile testing machine. The tests were performed with constant extension rate of 0.001 mm/min, which corresponds to a strain rate of 4.17 × 10⁻⁷ 1/s for gauge length of 40 mm and 3.1 × 10⁻⁷ 1/s for gauge length of 53.6 mm. The very low extension rate led to test duration of about two to three weeks depending on the specimen.

Two different lighting configurations were used in different experiments; 1) a direct spotlight from in front of the specimen (EBW-EDM-S, EBW-P-S) and 2) indirect diffused light (FSW-P-D), where the spotlight was placed behind the specimen and the light was reflected from two curved curtains, made of projector screen canvas, located in front of the specimen (details can be found in ref. [20]). The monitoring of the specimen was performed from an opening between the curtains.

Digital image correlation was performed repeatedly with LaVision DaVis software version 8.4.0 with varying calculation settings (Table 1). Since the trade-off between displacement precision and spatial resolution in DIC measurements is especially relevant, the subset size was systematically varied from 13 to 37 in a logarithmic series, roughly doubling the number of pixels per subset at each step in the series. Gaussian weighted round subsets were used, which affects the characteristics of the spatial resolution of the measurement.

Displacement data obtained with DaVis software was transferred to Matlab for further analysis and visualization. The vertical engineering strain components were calculated by central finite differences on raw displacement field data. The edges of the displacement data were omitted to avoid erroneous strains from edge effects. No out-of-plane compensation was performed, which results in small errors in strain when necking of the specimen occurs. Since the plastic strains in these experiments are on the order of 50% or larger, this error is acceptable (0.05% estimated maximum apparent strain from out of plane component of the displacement field at final fracture zone).

Data accuracy and precision were assessed by calculating statistics for a section of the displacement field data that could be reasonably approximated as having uniform strain. Since the strain field was not perfectly uniform, the underlying trend was approximated by local regression smoothing (Lowess smoothing) over an area much larger than the subset size, and subtracted from the calculated strain field. A typical result of this procedure is shown in Fig. 3a). The remaining strain field thus can be considered as the noise in the DIC strain measurement. Its standard deviation quantifies the strain noise level, or the strain detection limit for the given experiment conditions and DIC calculation settings.

Quantifying the spatial resolution of the measurement is not as straightforward. There is clearly an intrinsic length scale to the noise in the DIC strain measurement, obtained by the above procedure. How well that length scale corresponds to the concept of spatial resolution, with which the displacement field or strain field is measured, is still somewhat of an open question, but in the absence of an imposed displacement field with variations in strain or displacement of well-defined length scale, the autocorrelation length of the noise in the measurement may serve as an upper bound on the spatial resolution of the measurement. There may be extraneous noise that is not subject to the smoothing effect inherent to DIC calculations with subsets, but if so the measurement can trivially be improved by smoothing the displacement field. The intrinsic length scale of the noise includes the combined effects of the subset size and strain calculation method, as well as those of the feature size distribution of the pattern. If the point spread function for the displacement field, calculated using Gaussian weights to emulate “round” subsets on a vertically stretched grid, is a 2-dimensional Gaussian function, with characteristic width a horizontally and b vertically, the expected point spread function for the vertical component of the strain field calculated from this displacement field is:, where., with z a dimensionless number between −1 and + 1, and grid points as the unit of measurement for x, y, a, and b in Lagrangian coordinates.

This theoretical function shape corresponds well with the experimental autocorrelation data calculated from the noise in the DIC strain measurement, as shown in Fig. 3b). Therefore, in addition to the standard deviation of the noise in the DIC strain measurement, the autocorrelation length was calculated from the noise component of the measured strain field. The two-parameter fit using equation (1) gives the least-squares fit coefficient b for vertical point spread. This value was then multiplied by the elongated grid spacing in pixels at that specific point in time to obtain a length characterizing the spatial resolution of the DIC measurement in pixels in the vertical direction.

The DIC Challenge [18] provides test images for comparing the spatial resolution of different DIC codes. In particular, for Sample 14, sinusoidally varying “commanded” displacements with known constant amplitude and decreasing wavelength in horizontal direction (which results in increasing strain as the wavelength decreases) are imposed on the images. In the DIC Challenge paper, a relatively small attenuation of the measured strain variation amplitude is used as the threshold for considering the strain variations below the spatial resolution of the DIC measurement. Thus the DIC Challenge paper approached the concept of spatial resolution as answering the question “Does this DIC measurement accurately measure the strain amplitude, for strain variations at this wavelength?” This is not quite the same as answering the question “Are strain variations of this wavelength and amplitude detectable with this DIC measurement?” The combination of noise amplitude and autocorrelation length is directly suitable for answering the latter question, but not the first one.

However, for a rough comparison, the attenuation of the commanded displacement variations may be compared with the effect of filtering the commanded displacements by convolution with a Gaussian kernel of the appropriate width to result in the same autocorrelation length for originally uncorrelated noise. Thus, DIC was repeatedly performed on Sample 14 L5 of the DIC Challenge, with the same settings as for the experimental images, varying the subset size from 13 to 55 pixels. The larger the subset size and the smaller the wavelength of the imposed displacements, the more the measured displacements or strains will be attenuated from the imposed ones due to spatial filtering. For each of the DIC measurements with different subset sizes, the noise amplitude and autocorrelation length were extracted in horizontal direction from the horizontal component of the measured strain field, from the left side of Sample 14 L5 (x-position 50-550 pixels), which contains no or very small strains. The small strains were compensated by the local regression smoothing. The commanded displacements were then convolved with a Gaussian kernel given in equation (2), where the kernel width is defined by the autocorrelation length. The simultaneous attenuation of the commanded displacements by the DIC measurement and by convolution with the autocorrelation length were then plotted with the Matlab script of the DIC Challenge [21]., where.

For the case of convolving sinusoidally varying commanded displacements with a Gaussian kernel, when the sine wavelength λ = 2π/k and amplitude α are constant:, the attenuation may be calculated analytically:

In the DIC Challenge paper, the spatial resolution is characterized by the wavelength of the sinusoidally varying displacement field for which the measured displacement variation amplitude or strain variation amplitude reaches an arbitrarily chosen attenuation threshold. The attenuation ratio R for both displacements and strains equals to the exponential in equation (4):

Thus, for this specific case there is an exact relationship between the metric of spatial resolution in the DIC Challenge paper and the autocorrelation length used in this paper. Although this relationship does not necessarily hold for DIC measurements in general, solving equation (5) for the wavelength results in a conversion factor that can be used to estimate the spatial resolution obtained using the procedure of the DIC Challenge from the autocorrelation length obtained using the procedure in this paper, or vice versa:

For the threshold of R = 0.95 proposed in the DIC challenge paper for displacements, this results in λ_est = 9.809w, and for the value of R = 0.90 used for strains in λ_est = 6.844w.

It is well established, that there is a trade-off between spatial resolution and precision, which governs and which is affected by the choice of subset size in relation to quality of the pattern and test conditions when analysing DIC measurements. Therefore, to assess the impact of methods to improve the spatial resolution or reduce the noise level in a DIC measurement, it is necessary to characterize that trade-off. Figure 4 shows the reduction in noise of a strain measurement when the test conditions are improved or the subset size is increased. In addition to the amplitude of the noise, the wavelength of the noise is affected by the measurement conditions and the subset size. In the absence of an imposed displacement field with variations in strain or displacement of well-defined length scale, the length scale of the noise in the measurement was determined, and used as a proxy for the spatial resolution of the measurement.

Figure 5 plots the extracted autocorrelation length versus the noise level for the three different specimens. The autocorrelation length correlates inversely with the noise level showing a logarithmic increase in noise when the subset size is reduced. The entire trade-off curves are shifted towards less noise by patterning and by diffuse lighting, enabling detection of smaller variations in strain. However, a simultaneous shift of the autocorrelation values to the right is observed. This is most likely explained by extraneous noise components in the worse measurements, with smaller wavelength than the autocorrelation length, which will shift the autocorrelation length towards smaller values. This shift is more visible with noisier data. The different length scale of the trade-off curve for specimen FSW-P-D, when compared to the EBW specimens, is most likely explained by the slightly different camera distance from the specimen.

A comparison of the sum-of-differential and relative-to-first correlation methods is presented in Fig. 6 with data for frames 100, 200, and 300 for the specimen FSW-P-D. The respective points in time are 134.5 h, 270.3 h, and 406.2 h, and the mean strain over the gauge length is 6%, 15%, and 26% without significant strain localization. Signal-to-noise ratios for the mean strain, when using sum-of-differential and subset size 27, are 49.5, 60.5, and 61.5, respectively. It is worth pointing out that if the signal of interest is the localization of deformation, with onset after some time and increasing amplitude, the signal to noise ratio improves even though the noise level increases. It is evident from Fig. 6 that relative-to-first accumulates less noise, but still the noise level increases similarly with added images in the series regardless of the correlation method. The deterioration of noise level is probably explained by increasing residual differences between deformed and initial images after the deformation is accounted for by the subset shape function, and how those affect the numerical minimisation of the residual by varying the deformation within the DIC algorithm. The deterioration of the spatial resolution as the experiment progresses is a result of elongation of the grid spacing, since the vertical point spread coefficient of the autocorrelation peak was multiplied by the elongated grid spacing.

It should be noted, that due to the non-linearity of the DIC calculation, its spatial filtering effect may be significantly different from the linear approximation of convolution with a point spread function. As a result, especially for unreasonably large subset sizes, the results of a DIC calculation may contain noise components with smaller wavelength than the autocorrelation length, and the spatial resolution may be smaller than expected from the subset size. For example, it can be seen in open implementations of DIC algorithms that under certain conditions with discontinuous displacement fields the cost function is bimodal, and the measured displacement discontinuously jumps when the smaller peak becomes the larger one. Likewise, the minimum in the cost function becomes less sharp when it is further away from zero —for example by not normalizing the image intensities— allowing spurious noise-dominated terms in the correlation to contribute more. These noise components will shift the autocorrelation length towards smaller values, and it seems that the summation of small displacements amplifies this effect, since the autocorrelation lengths obtained for sum-of-differential measurements, when compared to relative-to-first measurements, are smaller when using the same subset size (Fig. 6). To investigate this in detail is beyond the scope of this paper and it would require low-level access to the DIC algorithm and intermediate results. However, if such noise deteriorates the data, the measurement can trivially be improved by smoothing the displacement field with filters smaller than the spatial resolution of the measurement. Thus, the autocorrelation length is an approximate measure of all of the noise components, including any extraneous noise which might not be dependent on the spatial resolution of the measurement. Therefore, the autocorrelation length should be taken as an upper bound of the spatial resolution, with the given test conditions and computational factors.

It is a choice of the experimentalist to aim for better noise level or better spatial resolution of a DIC measurement. The DIC software often provides filters for the improvement of the noise level, but this happens at the expense of the spatial resolution. The effect of the filters on the autocorrelation length is shown in Fig. 7. The filters clearly compress the trade-off curve towards larger autocorrelation lengths while the noise is reduced. This is especially evident with small subsets, which suffer from high level of noise initially. Larger subsets, on the other hand, do not benefit from the filters, since they already perform as initial smoothing filters in the DIC calculation process.

The best data quality was achieved by using the optimized DIC pattern and indirect diffuse lighting, which minimized specular reflections from the specimen surface. Alternatively, a white pattern background would have eliminated point-like specular reflections from facets on the grains of copper or the EDM-roughened surface, but possibly not a glare from the spotlight. Thus, using the diffuse lighting or a white coating would nearly eliminate contrast from the specimen with the as-cut EDM surface. However, copper oxide was used for patterning because it adheres to the copper surface better than a paint or foil would have, and this process is incompatible with applying a white background.

Since the metric of spatial resolution in the DIC Challenge paper [18] is different than the metric proposed in this paper, a direct comparison of the obtained values is not generally valid. In addition, the images provided in the DIC Challenge did not include strain variations with sufficiently short wavelength to really challenge the DIC codes with respect to the detectability of the commanded strain variations. However, for the special case of spatial filtering by convolution with a Gaussian function, the relationship between the two metrics is well-defined by the shape of the Gaussian function, and each of the methodologies can be compared with that common reference. Therefore, we compared the measured strains with both the commanded strains and the commanded strains smoothed by a convolution that for random noise results in the same autocorrelation length as the noise autocorrelation length of the measured strains. The obtained curves are shown in Fig. 8. DIC measurement with increasing subset size results in increasing autocorrelation length of the measured strains, as shown in Fig. 8a), as well as increasing attenuation of the measurement from the commanded strains at shorter wavelengths (Fig. 8b-f). Convolution of the commanded strains in Fig. 8b-f), with increasingly broad point spread functions, corresponding to the same strain noise autocorrelation lengths from Fig. 8a), results in good agreement for the attenuation of the strain variations. Thus it seems that for this DIC code with these patterns and these parameters, the relationship that holds between these two metrics of spatial resolution that holds for convolution with a Gaussian point spread function, approximately holds for the spatial filtering that happens in the DIC measurement, as well. Table 2 compares these two metrics, with numerical values for the autocorrelation length w, the corresponding attenuation threshold wavelength λ_est = 6.844w from equation (6), and the spatial resolution that the Matlab script of the DIC Challenge returns for the strains measured in this paper.