*4.2. PhotoStress Method*

The relieved strain analysis performed in the near vicinity of the cut through-hole was performed by studying isochromatic fringes distribution observed on the specimen loaded by uniaxial tension. The isochromatic fringes captured after the milling of the through-hole lying on the longitudinal axis of the specimen loaded by the tension force of 250 N are shown in Figure 15a.

**Figure 15.** Relieved strain intensity analysis: (**a**) field of isochromatic fringes (PhotoStress); (**b**) finite element analysis (FEA).

Validation of the experimentally obtained results was done by comparison of the isochromatic fringes (interpreting the principal strain difference/strain intensity) with the strain intensity field obtained in Matlab (Figure 15b) as a consequence of stress-relieving (caused by the milling of the through-hole). The results obtained by the numerical analysis are symmetrical along the longitudinal axis of the specimen. However, the numerical analysis is carried out on an ideal model and with ideal boundary conditions. By comparing the results, the influence of the accuracy of the drilled through-hole on the measured values can be confirmed.

Four points (marked as 1, 2, 3 and 4) were chosen at the edge of the drilled throughhole (Figure 15a), in which the fringe order was assessed using digital compensator model 832 (Figure 16). Values of the relieved strain intensity in the aforementioned four points given in Table 3 were calculated according to Equation. (1).

**Figure 16.** The use of the digital compensator: (**a**) isochromatic fringes observed after cutting the through-hole using the compensator; (**b**) fringe order assessment.

**Table 3.** Comparison of the values of relieved strain intensity obtained in four selected points located at the circumference of the drilled through-hole with a diameter of 3.2 mm.


As the process of fringe order assessment is not automatized, correct determination of the fringe order value is significantly dependent on the practical skills and experience of the experimenter. According to Table 3, it is evident that the differences in the obtained results are significant. As the highest gradient of relieved strain intensity occurs at 0.1D distance to the edge of the drilled through-hole (D = 3.2 mm is the diameter of the drilled through-hole), the difference in results can also be caused by the assessment of the fringe order near the edge of the hole. To obtain more precise results, it would be necessary to arrange the elements allowing micro-scaled measurement into the measuring device or to use the digital photoelasticity method, which results of application are described in detail, e.g., in the paper of Ramesh and Sasikumar [40]. As the authors' workplace does not dispose of any of the aforementioned possibilities, the quantitative results obtained by the PhotoStress method need to be considered only as additional.

The second (comparative) analysis in the surrounding of two other through-holes drilled to the same specimen loaded by the identical uniaxial loading was realized to qualify the sensitivity of the PhotoStress method. According to Figure 17, it is evident that the through-hole denoted as 1 was drilled on the longitudinal axis of the specimen loaded by the uniaxial tension. The center of the through-hole denoted as 2 was moderately biased from the axis of the specimen (see Figure 17b). At first glance, the isochromatic fringe patterns observed in the vicinity of both through-holes seem to be identical. The moderate differences in color patterns (mainly in areas near the edges) can be observed through a thorough visual analysis. In the vicinity of the through-holes, the differences are not significant.

**Figure 17.** Demonstration of the PhotoStress method sensitivity—the through-hole denoted as 2 is not drilled at the axis of the specimen: (**a**) the overall view; (**b**) the detailed view.

Regular evaluation of the strain/stress analysis performed in the vicinity of the stress concentrator (close to of the drilled hole) is necessary to investigate residual stresses using the PhotoStress method. It can be stated that the results obtained from the optical method in combination with the hole-drilling method are directly dependent on the technical factors of both experimental techniques.

#### *4.3. Digital Image Correlation (DIC) Method*

The relieved strain fields obtained from the analysis performed by Q-400 Dantec Dynamics and described in part 3.2 are shown in Figure 18a. Concerning the principle of Dantec Dynamics correlation systems, which evaluate the data at the centers of facets, it is not possible to reconstruct the contour of the drilled hole having a diameter 3.2 mm. The hole obtained by the correlation of the images using the correlation parameters described above is of a diameter of approximately 4 mm. To compare the results obtained experimentally and numerically both in qualitative and quantitative way, it was necessary to remove the corresponding elements lying on the circles concentric to the edge of the hole from the resulting strain fields and to visualize them again (Figure 18b). As can be seen in Figure 18, the strain fields' distribution obtained by DIC corresponds with the results obtained numerically.

As the sensitivity of the DIC method is not as high as the PhotoStress method, the smaller the level of strain relieved is the higher differences in the results can be expected. In the case of small strains, as the drilled through-hole causes the stress concentration, it is convenient to perform the analysis as close as possible to the edge of the through-hole, where the highest levels of strains are relieved. On the other hand, the full-field comparison of the results obtained by FEA and DIC (Figure 19c), respectively, shows that the highest absolute difference between the results occurs just at the vicinity of the reconstructed through-hole edge.

For the quantitative analysis of the results obtained by DIC and FEA, 4 points (marked as 1, 2, 3, and 4) lying on the circumference of the circle of radius 4.5 mm were chosen (Figure 20).

The strain intensity values obtained in selected points for 4 different levels of smoothing (see Figure 14) are listed in Table 4. As can be seen, the difference in the resulting values is significant. However, using the local regression smoothing with the kernel size set up to 15 × 15 (according to the previous investigations of the authors, such an adjusted level of smoothing should be the optimal one for the facet size of 23 × 23 px and their overlapping of 6 px) the relative differences between the results obtained in four selected points by FEA and DIC achieved values of approximately 14%.

**Table 4.** Comparison of the relieved strain intensity obtained in four chosen points using DIC by 4 different levels of smoothing and FEA.


**Figure 18.** Qualitative comparison of the results obtained by (**a**) DIC; (**b**) FEA.

**Figure 19.** Relieved strain intensity fields obtained by: (**a**) FEA; (**b**) DIC; (**c**) absolute difference between the results obtained by FEA and DIC.

**Figure 20.** Location of 4 chosen (comparator) points and the comparison of the relieved strain intensity fields obtained by: (**a**) DIC, (**b**) FEA.

Another possibility to minimize correlation errors is to take into account the displacement fields (Figure 21), which are not so significantly affected by noise occurring by small deformation levels as the strains. The possibility of filtering out the effect of rigid-body motion is also one of the advantages of work with the displacement fields of the DIC method. While 2D DIC allows obtaining two displacement components in the analyzed object surface plane, 3D DIC provides displacements in three mutually perpendicular directions. One of the methodologies for calculation of the residual stresses from the displacement fields was developed by Makino and Nelson in 1994 [41].

**Figure 21.** Relieved displacement fields obtained after filtering out the effect of the rigid-body motion: (**a**) displacement X, (**b**) displacement Y, (**c**) displacement Z.

The full-field comparison of the results obtained by FEA and DIC was carried out on the relieved displacement total fields (Figure 22). Figure 23a shows their absolute difference field. As most of the developed methodologies are based on quantifying residual stresses from displacements, it was necessary to determine the relative deviation of the measured data from the reference data, i.e., FEA data. For that reason, the mean relative differences between the relieved total displacements obtained in points located at the area of concentric annuli of width 0.5 mm (see Figure 23a) were calculated as follows:

$$diff\_{relative}^{mean} = mean \left( abs \left( \frac{disp \, T\_{FEA}(i, j) - disp \, T\_{DIC}(i, j)}{disp \, T\_{FEA}(i, j)} \right) \cdot 100\% \right) \tag{3}$$

where *i*, *j* are the coordinates of the corresponding points, in which the data were compared.

**Figure 22.** Relieved displacement total fields obtained by: (**a**) FEA, (**b**) DIC.

**Figure 23.** Difference between the relieved displacement total fields obtained by FEA and DIC: (**a**) absolute difference, (**b**) mean relative difference.

To avoid the results being influenced, the significant relative difference occurred (mainly in two areas, where the results obtained by FEA approached zero) were removed from the calculation (see the empty areas in Figure 23b). It can be stated that the relative difference increases with the distance from the center of the cut through-hole. However, for the area of diameter approximately 9 mm, the maximum relative difference was 21% corresponding to the accuracy of DIC achieved by residual stress quantification [42].

In some measurements (mainly by the repeated use of the milling cutter) performed by the authors, a small part of the speckle-pattern located near the edge of the drilled through-hole was corrupted (see Figure 24).

As this phenomenon was not visible with the naked eye (the authors observed it only in the digital images obtained by the DIC system), the cause was analyzed using scanning electron microscope FE SEM MIRA 3 (TESCAN, Brno, Czech) by 60× magnification (Figure 24c). It was found that the worn cutting miller caused small indents at the edge of the through-hole cut into the coating made from PS-1D material and damaged the background (white) color layers. Correlating such problem areas can lead to correlation problems/errors. For this case, the authors recommend that the evaluation mask is defined outside the damaged area.

**Figure 24.** Milling the through-hole to the coating: (**a**) correctly milled through-hole without damage of the speckle-pattern; (**b**) damage of the speckle-pattern near the edge of the milled through-hole caused by the worn cutting miller; (**c**) analysis of the through-hole surrounding using an electron microscope.

#### **5. Discussion**

In this paper, the development of the device designed for the strain and stress analysis performed on the specimen made from various kinds of material (e.g., plastics, metals, composites, etc.) using the hole-drilling technique and combining the full-field optical methods (PhotoStress, DIC) is described. As the technique of digital image correlation is based on the correlation of digital images captured during the measurement process, high positioning accuracy is required. Through the experimental testing, the measurement series were performed and showed the following results:


As most of the authors do not provide any detailed information about their devices, to compare our device with the devices developed and described in the introduction to this paper is a relatively complex issue. Despite it, the pros and cons of the designed device can be mentioned. The main pros are:


The dimensions of the designed device can be considered as its disadvantage. Compared with the commercially produced devices, which have been developed and optimized for many years, our prototype is more massive and, thus, portable and manipulable with more difficulty. The reason for this is that our drilling device's design has considered the minimum required dimensions of the working space. The measuring systems (single- or stereo DIC system, PhotoStress—reflective polariscope) work on optical principles and, thus, the analyzed specimen needs to be illuminated properly. The proposed prototype of the hole-drilling device was optimized so that the negative influence of the mechanical clearances was eliminated. In the future, the optimization of its dimensions in terms of weight minimization (by maintaining the sufficient stiffness requirement) is planned. To minimize the other components (servomotors, linear guides, etc.), it is necessary to consider their utility by operating load.

The methodology of evaluating the results in the experimental part by comparing the relieved strain fields obtained by the optical methods and numerical modeling is described in the second part of the paper. According to the results obtained, it can be stated that the PhotoStress method provides immediate information about the strain distribution with the sensitivity similar the other interference-based full-field methods, e.g., ESPI and moiré interferometry that allows its using also for the quantification of the residual stresses of smaller magnitudes. Comparison of the relieved strain intensity fields obtained by FEA and the PhotoStress method showed that using this optical method for full-field qualitative analysis is possible. However, the quantitative analysis carried out in four selected points located at the edge of the drilled through-hole suggests the need for farther improvement of the measurement methodology. To eliminate the measurement error, the authors plan to perform the sensitivity analysis of the results obtained in the area located close to the edge of a drilled hole.

The results obtained from the digital image correlation method are in the form of displacement and strain fields. The software Istra4D ver. 4.3.0 provided with the Q-400 correlation system Dantec Dynamics allows correlation parameters to be set up, i.e., facet size and overlapping, which need to be adapted to the size of speckles created on the analyzed specimen surface. According to the manufacturer, the strains are calculated from the local curvatures of the facets or deformation gradient depending on the smoothing adjustment level. In the paper, the effect of various levels of local regression smoothing is presented (Figure 13). With default settings, i.e., without smoothing, the results obtained in four selected points compared with numerical ones differ by approximately 400%. However, with the optimal level of smoothing determined by the authors the results differ by approximately 14% (Table 4). It has to be noted that the aforementioned relative difference cannot be achieved in the entire strain field, but only in the locations of such strain levels, which can be evaluated by the correlation system with sufficient accuracy. For that reason, it is convenient to work with the displacement fields, which are not as influenced by the smoothing used. A full-field comparison of relative differences between the total displacement fields obtained by FEA and DIC shows that the highest accuracy of the results is achieved in the vicinity of the cut through-hole (approximately to the 5 mm distance from the center of the hole). In such a case, DIC results differ from the results obtained numerically maximally by 20%, which is the expected accuracy of DIC in residual stresses analysis.

There is no known universal procedure applicable to set up all the evaluating software parameters used to determine stress components from strains/displacements. For that reason, before the analysis is made on the real structures, it is convenient to take measurements in the laboratory. To realize a series of laboratory measurements leading to the optimization of the methodology for quantifying residual stresses determined from the strains obtained by the PhotoStress method or by displacements/strains evaluated by the DIC method is one of the authors' future aims. For their validation, the commercially produced devices RS200 and SINT MTS 3000 will be used, with which the authors have a lot of experience gained by solving technical problems in practice. Finally, the authors plan to use the developed prototype for the residual stresses analysis outside the laboratory.

**Author Contributions:** Conceptualization, M.P., M.H. and A.S.; methodology, M.P., M.H. and L.H.; software, I.V. and A.K.; validation, M.P., M.H. and A.K.; formal analysis, M.P., M.H. and A.S.; investigation, M.P., M.H. and L.H.; resources, M.H., A.K.; writing—original draft preparation, M.P., A.S. and M.H.; visualization, M.H., M.P., L.H. and I.V.; project administration, M.P. and M.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Slovak Grant Agency APVV 15-0435, KEGA 030TUKE-4/2020, VEGA 1/0141/20, VEGA 1/0355/18 and ITMS 26220220141.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing is not applicable to this article.

**Acknowledgments:** The authors would like to thank the Slovak Grant Agency APVV 15-0435, KEGA 030TUKE-4/2020, VEGA 1/0141/20, VEGA 1/0355/18 and ITMS 26220220141.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

#### **Appendix A**

1. Compute the combination strains *p*, *q* and *t* for the relieved strains *εa*, *ε<sup>b</sup>* and *ε<sup>c</sup>* measured by strain gauge rosette using:

$$\begin{array}{l} p = \frac{\underline{\varepsilon\_c + \varepsilon\_a}}{2}, \\ q = \frac{\underline{\varepsilon\_c - \varepsilon\_a}}{2}, \\ t = \frac{\underline{\varepsilon\_c + \varepsilon\_g - 2 \cdot \varepsilon\_k}}{2} \end{array} \tag{A1}$$

2. Determine the calibration constants values *a*, *b* corresponding to the hole diameter and type of strain gauge rosette used. Compute the three combination stresses *P*, *Q* and *T* corresponding to the three combination strains *p*, *q* and *t* using:

$$\begin{array}{l} P = \frac{\sigma\_{\text{y}} + \sigma\_{x}}{2} = -\frac{E \cdot p}{\mathbb{R} \cdot (1 + \mu)},\\ Q = \frac{\sigma\_{\text{y}} - \sigma\_{x}}{2} = -\frac{E \cdot q}{\frac{\mathbb{R}}{\mathbb{B}}},\\ T = \tau\_{xy} = -\frac{E \cdot t}{\mathbb{B}}.\end{array} \tag{A2}$$

where *P* = isotropic (equi-biaxial) stress, *Q* = 45◦ shear stress, *T* = xy shear stress. The calibration parameters are determined either by experimental investigation of the calibration specimen or using numerical modeling.

3. Compute the principal stresses *σ*max and *σ*min using:

$$
\sigma\_{\text{max}\prime} \,\, \sigma\_{\text{min}} = P \pm \sqrt{Q^2 + T^2} . \tag{A3}
$$

#### **Appendix B**

The 3D measuring system used for the analysis consists of two Allied Stingray F504G cameras with a CCD sensor to capture images in maximum 5 Mpx resolution (2452 (H) × 2056 (V) px). The process of camera calibration, measurement as well as the evaluation was performed in Istra4D ver. 4.3.0 control software, provided with correlation systems Dantec Dynamics. Other technical parameters of the Q-400 correlation system are given in Table A1.



#### **Appendix C**

The numerical determination of strains relieved is divided into several steps:

1. Creation of the analyzed specimen model (with dimensions according to Figure 9a) without the net cross-section in the specimen's analyzed area.

Note: the holes were created in the ending part of the model (at sufficient distance to the area of interest) for definition of boundary conditions, i.e., loading tension force of 250 N, and fixed support simulated the constrain of the specimen used in experimental analysis.

2. Convergence analysis of the solutions (Figure A1) performed by the stress analysis of the model with the net cross-section using different types of elements (e.g., CPS4R and CPS8R) with varying mesh refinement (e.g., 1/element size = 2, 1, 0.5, 0.25, 0.125, etc.). Selection of the proper element type (CPS8R) and sizing (0.5 mm) in the area of interest.

Note: the through-hole in the area of interest is created by removing of the finite elements which locations correspond to the hole with a diameter of 3.2 mm.

**Figure A1.** Convergence analysis of the solutions using two types of finite elements (CPS4R and CPS8R).

3. Creation of the finite element mesh using the linear quadrilateral elements of type CPS8R and its sizing (0.5 mm) defined in the analyzed area of interest A0 (with the size of 25 (X) × 30 (Y) mm2) with the total number of finite elements creating the area of interest 3560, and the number of nodes 10791 (Figure A2); definition of boundary conditions and material properties.

**Figure A2.** Meshed model and boundary conditions of the specimen without the net cross-section; detailed view on the finer mesh created in the area of interest.


Note: the finite element model with the area of interest A1 (size of 25 (X) × 30 (Y) mm2) was obtained, where the elements and the nodes have the same locations as in the case of the model without the net of the cross-section.

**Figure A3.** Meshed model and boundary conditions of the specimen with the through-hole created in the area of interest; detailed view on the finer mesh created in the area of interest.


**Figure A4.** Schematic visualization of strains relieved determination using FEA performed in Abaqus/CAE 2020 and post-processing of the exported data in Matlab R2020a.
