*3.1. PhotoStress Method*

The PhotoStress method is a non-contact full-field measurement technique used to determine surface strains and transform them into stresses occurring in a structure. To use this method, a special strain-sensitive plastic coating needs to be bonded to the analyzed structure. Subsequently, after applying the loading to the structure, its strains are transmitted to the coating assuming the same strain condition as the part of the structure, which it is bonded to. After illuminating the coating by polarized light emitted from a reflection polariscope, a colorful pattern that is viewed through the polariscope occurs due to strain. There are two types of color pattern investigated known as isoclinic and isochromatic fringes. At every point on an isoclinic, the directions of principal strains are parallel to the direction of the analyzer and polarizer polarization. The photoelastic strain pattern appears as a series of successive and contiguous, variously colored bands known as isochromatic fringes representing a different level of the principal stresses difference [29,30]. When the loading is applied to the structure, the coated part color changes from black characterizing the no-loading state of the structure and first colors appear in the areas of the highest stress. After increasing the loading, the color fringes spread throughout the coated part (Figure 10), additional fringes are generated in the highly stressed areas of the investigated structure and move towards the regions of zero or low stress until the maximum load is achieved.

**Figure 10.** Characteristic change of the isochromatic fringes observed in PhotoStress method during increasing uniaxial loading.

The quantitative analysis of principal strain difference (strain intensity) occurring at any point of the coating can be quickly performed using a digital compensator attached to the polariscope. Strain intensity analysis depends only on the recognition of the fringe order defined by a color and understanding of the relationship between the fringe order and strain intensity as follows:

$$
\varepsilon\_1 - \varepsilon\_2 = N \cdot \frac{\lambda}{2 \cdot t\_c \cdot K} = N \cdot f \tag{1}
$$

where *ε*1,*ε*<sup>2</sup> are the principal strains, *N* is the fringe order, *λ* is the wavelength of the light emitted by polariscope, *tc* is the thickness of the coating, *K* is the strain optical coefficient of the coating and *f* is the fringe value of coating.

The advantage of the PhotoStress method is its high sensitivity for small strain/stress levels. On the other hand, in the areas with high strain/stress gradient, a problem can occur by quantifying the values (for the higher-order fringe recognition, the microscope is required to be used).

### *3.2. Digital Image Correlation Method*

The deformation analysis was also carried out by a low-speed digital image correlation system Q-400 (Dantec Dynamics A/S, Skovlunde, Denmark) working on the principle of the DIC method (some detailed information are presented in Appendix B). The digital image correlation principle is based on the correlation of digital images captured during loading of the analyzed object, where the images are not compared as whole units, but as small image elements called facets. The shape of the facet used by correlation systems Dantec Dynamics is squared, and each facet commonly comprises a group of pixels ranging in size from 15 × 15 px to 30 × 30 px. Depending on the type of analysis, the facet size can be adjusted (reduced or enlarged). The facets may touch or overlap, but there must not be an empty area between them. Since the information about the displacements of the analyzed object is obtained at the nodes of the virtual grid, which correspond in position to the centers of the facets, the overlap of the facets (Figure 11) is one way to increase the data resolution, i.e., to obtain a more considerable amount of data and, thus, to better reconstruct the surface of the analyzed object (especially around the edges). The manufacturer of correlation devices, Dantec Dynamics, recommends overlapping the facets up to 1/3 of the facet size because with such an overlap the data points are still independent.

**Figure 11.** Illustrative example of the increase of the data resolution caused by the facet overlap: (**a**) defined area of interest; (**b**) measuring points (green dots) when the facets are in touch, (**c**) measuring points (green dots) when the facets are overlapped.

Dantec Dynamics correlation devices use an algorithm based on a pseudo-affine transformation to obtain information on the transformation coordinates of the analyzed object surface points. If transformation parameters of possible displacement, elongation, shear, and distortion of the facet *a*<sup>0</sup> − *a*<sup>7</sup> as shown in Figure 12 are considered, using the aforementioned algorithm the transformation coordinates (*u*, *v*) can be calculated as follows:

$$\begin{array}{l} u(a\_0, a\_1, a\_2, a\_3, \widetilde{\mathbf{x}}, \widetilde{\mathbf{y}}) = a\_0 + a\_1 \cdot \widetilde{\mathbf{x}} + a\_2 \cdot \widetilde{\mathbf{y}} + a\_3 \cdot \widetilde{\mathbf{x}} \cdot \widetilde{\mathbf{y}},\\ v(a\_4, a\_5, a\_6, a\_7, \widetilde{\mathbf{x}}, \widetilde{\mathbf{y}}) = a\_4 + a\_5 \cdot \widetilde{\mathbf{x}} + a\_6 \cdot \widetilde{\mathbf{y}} + a\_7 \cdot \widetilde{\mathbf{x}} \cdot \widetilde{\mathbf{y}},\end{array} \tag{2}$$

where (*x*, *<sup>y</sup>*) *<sup>T</sup>* are the lens-distorted 2D coordinates of the point in the normalized image plane [31].

**Figure 12.** Transformation parameters used in the algorithm based on pseudo-affine transformation.

There are some relevant differences between the measurements of relieved deformations performed by 2D DIC and 3D DIC, respectively. Firstly, if the hole-drilling causes normal displacement of the analyzed specimen surface (this displacement component cannot be analyzed by a single-camera system), it leads to a correlation error and affects

the analysis results. Another advantage of the 3D correlation system is the possibility to measure all the three displacement components not only of a flat but also curved specimen. Higher distortion of the lenses, which are not directed perpendicularly to the analyzed object surface, can be considered a small drawback of the 3D systems. Also, the requirement for a proper arrangement of the cameras towards the analyzed object means that the analysis performed by the 3D correlation system is done from a greater distance to the specimen. However, this limitation can be reduced using quality CCD cameras with sufficient image resolution of the sensors.

The accuracy of the results of the deformation analysis for both types of the analysis (2D or 3D) can be affected, for example, by:

• The quality of the speckle-pattern created on the analyzed object surface and illumination

For correct image correlation, it is essential to make sure that each facet is unique, i.e., it contains a unique pixel distribution with different levels of intensity of grey color. As the aim of the analysis was to use image resolution of the sensors as well as possible and to perform evaluation of the results near the edge of the drilled hole, it was necessary to create the black-and-white speckle-pattern with very fine speckles (Figure 13). The illumination of the specimen surface should be homogenous and reach high sharpness and sufficient contrast of speckles. Compliance with this requirement was ensured using the Dedolight DLH400DT (Dedo Weigert Film GmbH, München, Germany) halogen reflector with white light.

**Figure 13.** Pairs of digital images captured by 3D Q-400 Dantec Dynamics: (**a**) reference image; (**b**) evaluated image with a defined mask of evaluation.

• Calibration parameters

The correctness of the coordinates transformation of the object points from 3D world coordinate system into 2D sensor system depends on the accuracy of the obtained external (i.e., mutual position and rotation of the cameras) as well as internal (focal length, principal point coordinates, tangential and radial distortion of the lenses) calibration parameters of the cameras. The calibration of Dantec Dynamics correlation systems is automatized, and based on the Zhang algorithm [32]. Information about the calibration accuracy is provided by the so-called calibration residuum, whose value should not exceed 0.5 px. The size of the calibration target should approximately correspond to the size of the analyzed specimen. Therefore, calibration of the stereo-camera correlation system was done using a calibration target comprising 9 × 9 checkboard fields with a precisely defined distance of 3 mm. The residuum obtained by the cameras calibration reached the value of 0.279 px, which is, according to the aforementioned criterion, considered as accurate calibration.

• Correlation parameters and the related levels of smoothing

Both Q-400 Dantec Dynamics cameras captured the digital images with image resolution of 1800 × 2056 px. The pixel density was approximately 64 px/mm. During the

analysis a couple of reference images (Figure 13a) captured by the maximum loading force of 250 N were taken in the Istra4D ver. 4.3.0 control software. The pair of images capturing the deformation of the analyzed specimen surface area after drilling of the hole (Figure 13b), was correlated with the reference one according to the following aspects. The facet size as one of the correlation parameters set up by the evaluation of the measurement was adapted to the fact that the drilled hole's edge would be reconstructed as accurately as possible if the facet was as small as possible. However, facets are required to be unique, i.e., all the facets have to comprise randomly distributed pixels with a high range of gray values. Therefore, the measurement was evaluated with the facet size set to 23 × 23 px, and their overlapping of 6 px ensured the increase of the data points (points in which the displacements and strains were evaluated) resolution.

The results of the deformation analysis performed by the Dantec Dynamics correlation systems are due largely to the properly set smoothing level. Istra4D ver. 4.3.0 contains two types of smoothing. The first known as *local regression* should be used mainly in cases if a high deformation gradient is expected in the deformation analysis results. The second is called *smoothing spline* and is used to smooth the deformation field with approximately homogeneously distributed deformation levels. As with the hole-drilling method the stress concentrator occurs in the specimen, the authors used a filter of local regression. Several studies described, e.g., in [33,34] were conducted on the proper setting of the local regression level. For the illustration, the authors point to the effect of local regression in Figure 14, which shows the relieved strain intensity fields obtained by different levels of smoothing. Figure 14a shows the relieved strain intensity field obtained by default settings of smoothing, i.e., without smoothing. The other three relieved strain intensity fields correspond to the results obtained by the settings of kernel size to 7 × 7 (Figure 14b), 15 × 15 (Figure 14c), and 31 × 31 (Figure 14d).

**Figure 14.** Influence of smoothing on the strain intensity field obtained by Q-400: (**a**) without loading; (**b**) kernel size of 7 × 7; (**c**) kernel size of 15 × 15; (**d**) kernel size of 31 × 31.

According to the manufacturers of Dantec Dynamics correlation systems, the strains are computed only from the local curvatures of facets when the kernel size is set to 3 × 3. The higher the kernel size, the higher the influence of the deformation gradient on the strains. For the kernel size set to 9 × 9 up to 31 × 31 (i.e., the highest level of smoothing based on local regression in Istra4D ver. 4.3.0), the strains are computed only from the deformation gradients. According to the authors' analysis, the kernel size of 15 × 15 for which the results obtained are presented in Figure 14c corresponds to the optimal setting of smoothing for the described type of evaluation. The effect of oversize smoothing can be observed in Figure 14d.

#### **4. Results**

Although the aforementioned measurement was conducted by two different noncontact optical methods providing the full-field information about the strains, the quantitative comparison of the results obtained from both techniques is not simple. While the results of the measurement by the DIC method provide quantitative information about

the displacements as well as strains in the center of each facet, for the quantification of the results obtained using PhotoStress method, the use of a digital compensator is necessarily required in each evaluated location of the specimen.

In many cases, experimental testing results serve to verify the results obtained numerically, e.g., using software based on the finite element method (FEM) [35]. The abovedescribed strain analysis in the vicinity of the hole drilled in the flat specimen loaded by uniaxial tension, is a typical analysis performed by the finite element analysis (FEA). For that reason, the authors reversed the standard approach and, thus, verified the experimentally obtained results by numerical analysis. Such an approach has already been used by the authors, and the results obtained were published in several scientific publications, e.g., [36–39]. Moreover, in conventional quantification of residual stresses by the hole-drilling technique the FEA is usually used, e.g., the correlation parameters *a*, *b* used in the formulas for the computation of stress components from the relieved strains are determined mainly in a numerical way.

#### *4.1. Finite Element Analysis*

The numerical model of the analyzed specimen was created in Abaqus/CAE 2020 (SIMULIA, Johnston, RI, USA) software. The analysis was focused on the determination of the relieved displacements/strains occurring in the specimen (with dimensions, mechanical properties, constraints and loading described in head 3) after milling a through-hole lying on its longitudinal axis. Detailed information on the procedure of numerical analysis is presented in Appendix C.
