or a few μm. Additionally, the literature shows how to solve the problem of the kinematics of the roughness of the material for the triangular asperities of the surface crushed (burnished) with a flat punch under various friction conditions [36–38]. Kukiełka [10,39,40] stated that the kinematics of the smoothing of the asperities depends only on the apex angle of the rough surface and determined the angle boundaries for which there is a change in the flow mechanism.

The author of Reference [39] qualitatively distinguished three different cases of flow of the surface layer of material. The considerations were carried out with the assumption that the surface asperity after the previous treatments is triangular, symmetrical, and regular. The research shows that in the burnishing process, these cases depend only on the vertical angle θ of the asperity (Figure 1):

**Figure 1.** *Cont*.

**Figure 1.** The influence of the vertical angle of a symmetrical triangle asperity on the depth of strain zone and surface roughness profile after burnishing: θ ≤ 80◦ (**a**), 80◦ < θ < 145◦ (**b**), θ ≥ 145◦ (**c**) [39].

A significant problem is to ensure the regularity of the outline and the appropriate vertical angle θ of the shaped irregularities in the treatment preceding burnishing. Due to the type of the machined part and its intended use, the value of the feed, *ft*, in the preceding machining and the value of the vertical angle θ of the asperity should be selected each time.

In previous paper [12] on the initial stages of the experiment related to elastic recovery issues, we analyzed the influence of the deviations of the height and distance of the asperities after turning on the deviation of the height of the asperities of the surface as well as elastic recovery after burnishing. Research was performed for different burnishing depths. The calculations were performed for apex angles θ = 90◦ and θ = 120◦. The received results showed that the elastic recovery increases as the burnishing depth increases. Each of the asperities were analyzed separately. The analysis of the state of knowledge shows that none of the researchers analyzed the problem of elastic recovery, which is a novelty and originality of the presented article. In the paper, FEM (Finite Element Method) analysis with using the dynamic explicit method for the roller burnishing process was established. The surface state after turning (as previous treatment) was taken into account. The results of dependencies of elastic recovery of the asperities and the deviation of height, Δ*ht*, for exemplary burnishing depth is presented. The influence of the surface after previous treatment on the burnished product quality can be analyzed by developed numerical algorithms. It allows for better understanding of the phenomena which occur in the tool–workpiece contact zones. The basis for the development of guidelines for the selection of the conditions of rolling and burnishing processes considering the required technological quality of the product can be formulated. This paper consists of numerical simulations and analysis, model test, and the research on the steel C45, with the aim to understand elastic recovery as well as analysis of elastic recovery of the asperities with different vertical angles.

#### **2. Idea of Elastic Recovery**

It is assumed that during the burnishing process, deformations of the material occur in terms of elastic and visco-plastic strains [41]. It is known that visco-plastic strains will remain in the material, while elastic strains change [41–43]. After the burnishing tool has passed, the phenomenon of elastic recovery occurs and visco-plastic deformations remain in the processed material, while the elastic deformations change and settle on the level corresponding to the new state of equilibrium of the object. Thus, the states of displacements, stresses, and strains in the tested object during and after the end of the process differ significantly, because after the process is completed, the nodes are displaced in the elastic range. The value of elastic deformations will decrease by the value Δφ*(E)*, while the elastic deformations will remain at the value φ*(E)* in the surface layer of the product. This is shown in Figure 2.

**Figure 2.** Elastic recovery during the roller burnishing process.

During the process, the burnishing tool is plunged to the depth *a*. It can be noticed that the contact point of the tool and the workpiece material *A*, as a result of this depression, moves to the point *A* . Then, after the rotation of the workpiece and the burnishing tool, the load is unloaded and point *A* is moved to position *A"*. The radial distance between points *A* and *A"* is marked as Δ*ui* and represents the elastic recovery of the material after burnishing, and ϕ is the actual deformation of the material.

The process of roller burnishing is treated in this study as a process with unknown boundary conditions in the tool–workpiece contact area. It is a geometrical and physical boundary and initial value problem. During experimental research and simulations of the process, special attention was drawn to the elastic strain of the material, which takes place after moving away the burnishing element, called elastic recovery, Δ*ui*. The state of the surface layer after previous treatment and after roller burnishing are considered together. They take into account two asperities, *A1* and *A2*, after the turning process, which are burnished. Asperity *A1* after turning was characterized by constant height (*h*) and distance (*s*). Asperity *A2* possesses different values of the height or distance. The difference between the asperities was called height deviation after turning (Δ*ht*) and between distances: distance deviation after turning (Δ*st*). Similarly, deviations were defined after burnishing and marked as Δ*hb* and Δ*sb*. The schematic diagram of elastic recovery in joint analysis of the surface after turning and burnishing is presented in Figure 3.

**Figure 3.** Schematic diagram of the elastic recovery of the material after burnishing: *A1*—first asperity, *A2*—second asperity after previous machining (turning), *h*—asperity's height, Δ*ht*—height deviation of asperity after turning, Δ*hb*—asperity's height deviation after burnishing, *ab*—burnishing depth, Δ*u*01—elastic recovery of the material of first asperity *A1*, Δ*u*02—elastic recovery of the material of second asperity *A2*.

#### **3. Researches of the Influence of Height and Distance Deviations on the Elastic Recovery**

#### *3.1. Experimental Researches—Materials and Methods, Discussion*

The experiment was conducted for the following input factors: height deviation (Δ*ht*) and distance deviation (Δ*st*) of the surface asperities after rolling, determined in compliance with Table 1, and the vertical angle of asperities was constant, θ = 60◦. The five-level rotatable experiment plan was used to set up the values of the deviations.

**Table 1.** Experimental plan with real and coded values.

The values of the input factors were changed in the following ranges: Δ*ht* = 0.01–0.04 mm, Δ*st* = 0.05–0.2 mm. The tests were repeated three times for each point of the experiment plan. Tests were performed at room temperature (T = 19 ◦C). The external surfaces of the samples made of steel C45 were prepared in the turning process on a NEF 520 numerical lathe in "TEPRO" Vacuous Technology Plant in Koszalin, with a VCMT 160402 machining plate. The following machining parameters were used: velocity *vt* = 200 m/min and feed *ft* = 0.15 mm/rev in rough machining and *vt* = 200 m/min, *ft* = 0.1 mm/rev in finishing. Six samples were made on the shaft (Figure 4). Each sample included two asperities, which differed in the value of the height and distance deviations, according to the experiment plan. The accuracy of the shaping of the outline was verified on a Werth optoelectronic microscope. Figure 5 shows an example of a shaft with samples prepared in accordance with the experiment plan, Table 1 (first 6 implementations). The values of the height and distance deviations obtained in measurements were not significantly different (at the significance level α = 0.05) from those planned on the basis of single factor analysis of variance.

**Figure 4.** General view of the shaft with the samples after turning process and zoom of the sample (nb 3) after turning process, where *Rt* is the asperity height.

**Figure 5.** Burnishing position (**a**), general view of the shaft with the samples after burnishing (**b**), zoom of the asperities after roller burnishing (**c**).

Next, the samples were burnished using a Fette T27 tangent burnishing head with high stiffness to reduce the influence of dimensional changes of the head on results. The burnishing depth for each sample was calculated separately, while assuming that it is *a* = 0.5 *Rt*, where *Rt* is the asperity height without a deviation (constant one).

After the burnishing process, the values of parameters Δ*u*0 were calculated and profiles with profilometer T8000 were obtained. Exemplary profile of burnished asperities is presented in Figure 6.

The results of the measurements were developed statistically in the Experiment Planner program developed by Kukiełka L. and Kukiełka S. [44–50]. The following regression equation is obtained of the dependence of the material elastic recovery after burnishing (Δ*u*0) from the deviations of height (Δ*ht*) and distance (Δ*st*) of asperities after turning, for correlation coefficient *R* = 0.985:

$$\begin{array}{l} \Delta \mu\_0 = 0.0007 \Delta \text{s}\_t - 0.013 \Delta h\_l - 2.423 \times 10^{-14} \Delta \text{s}\_t \Delta h\_l + \\ + 0.0048 \Delta \text{s}\_t^2 + 0.12 \Delta h\_l^2 \end{array} \tag{1}$$

In order to determine the form of Equation (1), appropriate analytical and experimental tests were carried out, in the following stages: determination of the sets of tested, constants, disturbing and resulting factors, determining the range of variability (research area) of the examined factors, adoption of the class of the mathematical model of the research object, coding of researched factors, implementation of the actual research, experiment plan, the results of the experiment, elimination of results with a gross error, calculating the inter-row variance, checking the homogeneity of variance in the sample, calculating the coefficients as a function of the regression, statistical analysis of the regression function, study of the significance of the multidimensional correlation coefficient, checking the adequacy of the mathematical model, and decoding the regression function. The detailed procedure

is given in References [39,41,44]. The graph of the function according to Equation (1) with the correlation coefficient *R* = 0.86 is presented in Figure 7.

**Figure 7.** Dependence of the material elastic recovery after burnishing (Δ*u*0) from the deviations of height (Δ*ht*) and distance (Δ*st*).

It is visible that an increase of the deviation of the asperities' height causes a decrease in the value of elastic recovery, while the opposite situation occurs when the deviation of the distance is analyzed. Knowledge about influence of determined profile parameters on the physical phenomena that occur in the burnishing process is unknown. It is important because they impact on the quality of the formed product's surface layer, tribological properties, and fatigue issues. Many physical phenomena whose observation or measurement is very difficult or impossible take place during processes of burnishing, such as the pressure in contact zones, the friction forces, strain and stresses, elastic recovery, the slip and contact zones, and states of displacement. After passing the burnishing tool, visco-plastic deformations remain in the workpiece, while the elastic deformations change and settle at the level corresponding to the new equilibrium state of the object. Thus, the states of displacements, stresses, and strains in the tested object during and after the end of the process differ significantly, because after the process is completed, the nodes are displaced in the elastic range. In the case of higher asperities, the displacement is greater than in the case of those with a height deviation. The opposite is the case for a distance deviation. These issues are very complex and require further analysis.

#### *3.2. FEM Modeling and Numerical Results*

The computer models (2D and 3D) of the process in the real scale were characterized by the possibility of changing the value of the height deviation (Δ*ht*) and the distance deviation (Δ*st*). Exemplary calculations were performed for the apex angle θ = 60◦ and the asperities feed *f* = 2.7 mm, and the deviations were changed according to Table 1. Numerical analyses were conducted with the aid of Ansys Structural, LS-Dyna and LS-Prepost programs. In the analyzed case, the object was treated as an elastic body (in terms of reversible deformations) and visco-plastic (in terms of irreversible deformations), while the tool (roller) was treated as a perfectly rigid body. Based on the results obtained, various analyses can be carried out. In the discussed case, only some of the results were presented in order to determine the impact of the burnishing process of the rough surface (with regular triangular asperities) on the value of elastic material return after the burnishing process. Moreover, in incremental material models, it was assumed that the object was made of C45 steel, for which the cumulative value of the yield stress in 2D models is described by the regression equation according to Equation (2):

$$
\sigma\_p = 976.95 \left( 0.0667 + \dot{\varphi}\_i^{(\text{VP})} \right)^{0.2052} \left[ 2.33 \times 10^{-6} \left( \dot{\varphi}\_i^{(\text{VP})} \right)^2 + 8.83 \times 10^{-5} \dot{\varphi}\_i^{(\text{VP})} + 1 \right]^{1.337} \tag{2}
$$

where: yield stress is σ*p*, intensity of the real strain is ϕ*i*, and strain rate is . ϕ*i* .

The necessary parameters for this material are Young's modulus *E* = 210 GPa, Poisson ratio *҃* = 0.29, yield stress *Re* = 425 MPa, and hardening modulus *ET* = 1024 MPa. In 3D models, object discretization was performed using finite elements of the object with a linear shape function, shell elements (tool), and contact elements (contact of the tool with the object). The nonlinear, anisotropic Coulomb friction model and the coefficient of static friction μ*s* = 0.1 and dynamic friction μ*d* = 0.05 were adopted. Then, adequate boundary conditions were set—mainly concerning the displacements: tool cavity (up to the value of 0.5 of the roughness height) and the degrees of freedom for the workpiece, as well as the set rotational speed ensuring the required burnishing speed *vb* = 0.52 m/s. In this case, the shaft with surface asperities was rotated at *n* = 477 rpm, then the burnishing element (roll) was moved to the burnished surface. The contact of the roller with the machined surface caused the tool to rotate. In the next step, methods of solving the discrete equation of motion were introduced. For this purpose, in accordance with the adopted algorithm of explicit solution, at each analysis step, an approximation of the column vector of acceleration and velocity increment as a function of the column vector of displacement increment was introduced, all with the following dimension integration and were given the required number of steps, as well as the total calculation time *t* = 3.5 s. The geometrical model and mesh grid are shown in Figure 8. The 3D FEM model was developed according to Hill plasticity condition and low strain rates. Thermal strains for this model were approximately 2%.

**Figure 8.** Geometry of the object, tool, and finite element mesh.

The temporary yield stress in the 3D model was described with the aid of the Cowper–Symonds model. A Cowper–Symonds model allows for linear isotropic (β = 1), kinematic (β = 0), or mixed (used in this study) (0 < β < 1) plastic strain hardening, and the effect of the plastic strain velocity is given by the following power relation:

$$\sigma\_{\mathcal{P}} = \left[1 + \left(\dot{\phi}\_{i}^{(p)} / \mathbb{C}\right)\right]^{\mathrm{m}} \left(\mathbb{R}\_{\varepsilon} + \beta E\_{\mathcal{P}} \phi\_{i}^{(p)}\right) \tag{3}$$

where β is the plastic strain hardening parameter, *Re* (MPa) is the initial static yield point, . ϕ(*p*) *i* (s−1) is the plastic strain velocity, C (s−1) is the material parameter defining the effects of the plastic strain velocity, m = 1/P is the material constant defining the sensitivity of the material to the plastic strain velocity, ϕ(*p*) *i* is the plastic strain intensity, and *EP* = *ETE E*−*ET* is a material parameter dependent on both the plastic strain hardening modulus, *ET* = ∂σ*p*/∂ϕ(*p*) *i* , and Young's modulus, *E*.

To find the value of material elastic recovery in performed computer simulations, one had to select areas when the tool affected the subject, and after it moved away. Two kinds of analysis were performed. The first, when Δ*ht* (height deviation) values after the turning process were Δ*ht* = 0–0.04 mm and Δ*st* (distance deviation) values after the turning process were Δ*st* = 0, and the second, when Δ*st* = 0–0.2 mm but Δ*ht* = 0. On the basis of computer simulations, the graphs of dependencies of elastic recovery of the first Δ*u*1 and the second Δ*u*2 asperity and the deviation of height, Δ*ht*, for the sample after burnishing were elaborated. The graph of dependencies of elastic recovery Δ*u*1 and Δ*u*2 in the asperities *A1* and *A2* from distance deviation Δ*st* is presented in Figure 9.

**Figure 9.** The graphs of dependencies of elastic recovery Δ*u*1 and Δ*u*2 accordingly in the asperities *A1* and *A2*, from height deviation, Δ*ht* (**a**), and dependencies of elastic recovery Δ*u*1 and Δ*u*2 accordingly in the asperity *A1* and *A2* from distance deviation, Δ*st* (**b**).

It has been observed that together with the increase of the deviation of surface asperities' height, the elastic recovery decreases. Comparing the values of elastic recovery for an asperity with constant geometry *A1* and changeable geometry *A2*, it can be seen that the value of elastic recovery was higher for asperity *A1*.

Computer simulations were also performed to validate the experimental studies according to the experiment plan presented in Table 1. An example of the results obtained from the simulation is shown in Figure 10, including a geometric model with a mesh of finite elements (a), mesh deformation (b), elastic recovery of selected asperity nodes with variable geometry (A) and constant (B).

**Figure 10.** Geometric model with a mesh of finite elements and mesh deformation (**a**), elastic return of selected inequality nodes with variable geometry—A and constant—B (**b**).

On the basis of the obtained results, an Equation (4) was statistically developed, describing the influence of height deviation and distance deviation on the elastic recovery of the material. Multidimensional correlation coefficient *R* = 0.8332.

$$
\Delta \hat{\overline{u}}\_0 = 0.0048 - 0.0032 \Delta \text{s}\_l - 0.026 \Delta h\_l - 3.526 \times 10^{-14} \Delta \text{s}\_l \Delta h\_l + 0.024 \Delta \text{s}\_l^2 + 0.358 \Delta h\_l^2 \tag{4}$$

As in the case of experimental studies, it was found that an increase of the asperities' height deviation causes a decrease in the value of elastic recovery, while the opposite situation occurs when the distance deviation is analyzed. The results of experimental studies and computer simulations differ slightly from each other. This may result from simplifications and assumptions made when developing the computer model (e.g., refinement of the finite element mesh, shape of elements). Based on Equation (4), a spatial diagram (Figure 11) was prepared.

**Figure 11.** Relation of elastic material recovery after burnishing—Δ*u*0 = *f*(Δ*ht*, Δ*st*).

#### **4. Researches of the Phenomenon of Elastic Recovery Depending on the Vertical Angle of the Asperities**

#### *4.1. Modelling Researches of the Phenomenon of Elastic Recovery*

Experimental studies with the use of a model material were carried out in order to assess the effect of the flow blocking phenomenon through adjacent irregularities on the elastic return of the surface after burnishing. The tests were carried out in accordance with the rheological, geometric, and time scales [4,36,44]. Modeling consists in replacing a real research object with a model object. The real and model objects form a system of equivalent objects whose similarity is determined by mathematical relationships. Then, the modeling results can be translated into a real research object. Physical modeling on substitute material is used in experimental qualitative and quantitative analysis. The most frequently used model material replacing metal is a non-metallic material, such as plasticine, wax, putty, etc. These are materials with much lower plastic flow resistance. Physical modeling on plasticine is used for experimental qualitative and quantitative analysis of the plastic working processes. Qualitative analysis mainly concerns the kinematics of the material flow process, while the physical analysis covers the study of force and energy parameters (force, unit pressure on the contact surface of the tool—model material). In recent years, the technique of visualizing the flow trajectory of material particles of the shaped material, the so-called method of visioplasticity, has been used. It is based on the study of plastic flow kinematics based on the observation and measurements of the coordination grid. Performing experimental tests of visualization on real metal materials is laborious and costly and requires a station with very high-pressure forces. Therefore, often in experiments, the real material is replaced by the so-called model material or numerical analyses are performed [39,42,45]. In order to ensure the condition of rheological similarity, the characteristics for the model material—plasticine—were determined. This is due to its availability, low price, easy workability, plasticity (plasticizing stresses are 100–1000 times lower than the appropriate metal stresses), the possibility of its modification and regeneration, and therefore, multiple use. The material model was developed based on the static compression test. The results of this test are approximate data due to the presence of disturbances in the measurement of the compression force. The factor

disturbing the measurement of the compressive force is the friction occurring on the faces of the cylindrical sample.

Under ideal conditions, during a static compression test, the diameter of a cylindrical specimen would change uniformly over its entire height. However, for real conditions, the effect of friction causes the sample material to flow unevenly (a barrel is formed), which is a result of blocking the flow of material on the front contact surfaces of the sample with the substrate and the punch. The static tensile test does not have this disadvantage, however, due to the nature of the material, it is impossible to carry out.

The compression test was carried out on the stand shown in Figure 12a. A cylindrical sample with the initial height *h*0 = 50 mm and the initial diameter *d*0 = 2*r*0 = 40 mm was used for the tests. On the basis of the compression test, the dependence of the force as a function of the change of sample height *P* = *f* (Δ*h*) was developed (Figure 12b). Using the results of the compression test and appropriate dependencies from the works of Dawidenkow and Spiridonowa [39,43,46], a material model was developed in the form of an actual compression diagram (Figure 12c). The obtained material characteristics for the model material are similar to the material characteristics of C 45 steel.

**Figure 12.** Stand for static compression test on the model material: 1—bow dynamometer, 2—dial displacement sensor, 3—compressed sample, and 4—dial gauge calibrated in force units (**a**), change of the compressive force *P* as a function of the height change Δ*h* of the cylindrical sample (**b**), the actual compression diagram for the model material (**c**).

The aim of the model research was to determine the relationship between the vertical angle of triangular asperities and the elastic recovery of the model material. The burnishing element in the test was a flat punch mounted on a hydraulic press. The tests allowed for visual assessment of material displacement, numerical determination of the size of deformation, and elastic recovery of the material. It is also possible to determine the magnitude of stresses and the direction of material flow. The samples made of model material (plasticine) were characterized by the vertical angles of asperities from θ = 60◦ to θ = 150◦, every 15◦. Each sample had three asperities with the same vertical angles and was cut along the longer sides to apply a coordination grid. Photos of samples before, during, and after the burnishing process are presented in Table 2.

**Table 2.** View of the exemplary samples.

Then, the samples were placed in the matrix, which task was to properly position the sample in relation to the punch and to ensure appropriate boundary and initial conditions. The stamp was displaced, causing the deformation of the vertices of the triangular asperities and the deformation of the coordination grid. The sample and stamp were covered with a layer of talcum powder to reduce friction between them. Using a coordination grid applied to the samples, it was possible to perform appropriate measurements of the elastic recovery of the material.

The obtained results are presented in the form of a graph in Figure 13. The model tests of the elastic recovery of the model material using the visioplasticity method show that with the increase of the value of the vertical angle of the surface asperities, the value of the elastic recovery of the material decreases. The results were confirmed by numerical analysis. In some areas, the results are very close. The differences may be due to simplifications and assumptions made during creating a computer model. The accuracy of FEM calculations is influenced, among others, by the ratio of the side length of the finite elements used to discretize the object (object and tool), mesh refinement (mainly in areas of strong geometric or physical nonlinearity), and the finite element shape function. Although, in order to determine the significance of the influence of these factors and to determine an effective discrete model for the case of roller burnishing of a rough surface, sensitivity analysis was performed. As a result, acceptable gaps between the results were obtained.

#### *4.2. Computer Simulation of the Phenomenon of Elastic Recovery Depending on the Vertical Angle of the Asperities*

Computer simulations were developed for the same values of the asperities' peak angles as in the research on the model material. The calculations were made in two stages. In the first stage, triangular irregularities were burnished, and in the second stage, the burnishing tool was moved away. It was burnished to a depth of *a* = 0.5 h, where h is the height of the triangular asperity. This approach to the problem made it possible to calculate the value of the material elastic recovery by

determining the displacement of the nodal point of the peaks of the asperity. The difference between the node displacement modules at the moment of contact with the burnishing tool and after the tool is fully retracted is the desired result (Figure 14). Table 3 presents selected results from 2D computer simulations before, during, and after the burnishing process, showing displacement maps.

**Table 3.** Results of sample computer simulations.

Figures 14 and 15 show diagrams of the displacement of the node lying on the contact surface (the highest point of asperity) along the *Y* axis.

In the initial phase of the process, the node was slowly displaced due to the impact of the tool on the burnished material. The asperity was deforming. At the moment when the burnishing tool shifted to half of the roughness, the observed node reached the highest value of displacement. The first stage of numerical analysis has been completed. In the second step, when the time exceeded 1 s, the crushing tool was moved back.

**Figure 14.** Diagram of displacement along the *Y* axis of selected nodes of asperity with a vertical angle in the process of crushing with a flat punch: A—top of asperity, B—bottom between asperities.

The phenomenon of elastic recovery of the material is visible and then the stabilization of the position of the node (A) (the peak of the roughness), which has stopped moving along the *Y* axis. Similar results were obtained for the cases of crushing the remaining asperities with different vertical angles. When analyzing the node located between the asperities (B) (on the bottom), no elastic recovery was found for the case of asperities with a constant vertical angle and height.

The difference between the individual simulation results concerns the displacement value, which is related to the burnishing depth. The burnishing depth resulted from the geometry of the outline, i.e., the vertical angle of the asperities, and thus its height. The distance of the asperity was the same in each case. The elastic recovery also changed its value—the greater the vertical angle of the roughness, the smaller the value of the elastic recovery.

Computer simulations made it possible to observe and quantify the amount of material elastic recovery during the burnishing process. It is possible to check the results at any time during the process. It was confirmed that with the increase of the value of the vertical angle of the surface roughness, the

value of the elastic recovery of the material is smaller. Moreover, using the developed numerical model, it is possible to qualitatively and quantitatively determine how much the bottoms between surface asperities rise in the process of their crushing. In cases where the vertical angles of the asperities were equal to θ = 60◦ and θ = 75◦, the indentations of the asperities (bottom) did not rise, and the material core remained undeformed, and with complete deformation, deformed asperities are visible, separated from each other by gaps (planes of discontinuity) at depth *a* = 0.5 *Rt*. In the range of vertical angle θ = 90–135◦, the plastic deformation zone increased to encompass the material core. The indentations between the asperities (bottom) rise (Figure 15), and with complete deformation, gaps are visible, but of less depth. For the angle θ = 150◦, deformations of the roughness and the core of the material occur, and the cavities between the asperities (bottoms) rise completely.

**Figure 15.** Diagram of displacement along the Y axis of selected asperity nodes with the vertical angle θ = 75◦ (**a**), θ = 90◦ (**b**), θ = 105◦ (**c**), θ = 120◦ (**d**), θ = 135◦ (**e**), θ = 150◦ (**f**), crushed with a flat punch: A—peak of asperity, B—bottom between asperities.

#### **5. Conclusions**

The phenomenon of elastic recovery of the material during burnishing is a very complicated issue. The presented results of model tests, simulations, and analyses do not reflect the complexity of the problem. They confirm that this phenomenon occurs and there are methods by which it is possible to try to quantify the dependence of the elastic recovery on, for example, the apex angle of the inequality or deviations of the inequality outline. In the conducted analyses, only the elastic recovery of the burnished material was taken into account. The spring-return occurring in the tool material was omitted, treating the tool as perfectly rigid. It should also be noted that the phenomenon of elastic recovery probably also takes place during the treatments preceding burnishing. It is not possible to continuously measure the elastic recovery during modeling or experimental studies. This possibility is provided by computer simulations using the finite element method. However, the history of the material was not taken into account in the simulations performed. Model tests and computer simulations are a cheaper way than carrying out experimental tests each time. They give a guideline to consider the material elastic recovery when designing the burnishing process, especially in the case of fine machining. Then, when calculating the burnishing depth, its value should be increased by the elastic recovery of the material. Based on the research, the following conclusions can be drawn:

**Author Contributions:** The individual contributions of the authors: conceptualization, A.K.; methodology, A.K. and Ł.B.; investigation, A.K. and Ł.B.; software, A.K.; data curation, Ł.B.; validation, A.K.; writing—original draft preparation, A.K. and Ł.B.; writing—review and editing, A.K. and Ł.B.; visualization, Ł.B.; supervision, A.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors gratefully acknowledge the financial support from the Koszalin University of Technology, funding No. 524.02.23, entitled "Analysis of the deformation state of burnished surface with taking into account surface layer state and nonlinearity of the process".

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Nomenclature**

#### **References**

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### *Article* **Tensile Deformation Behavior of Typical Porous Laminate Structure at Di**ff**erent Temperatures**

#### **Ping Wang, Ye-Da Lian \* and Zhi-Xun Wen**

School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi'an 710072, China<EMAIL_ADDRESS>(P.W<EMAIL_ADDRESS>(Z.-X.W.)

**\*** Correspondence<EMAIL_ADDRESS>
Received: 8 November 2020; Accepted: 25 November 2020; Published: 26 November 2020

**Abstract:** In this study, the Ni-Cr-W superalloy GH3230 is used as the test material. According to the actual structure of the flame tube, a porous laminate structure specimen is designed. The structure consists of impact holes, overflow holes and pin fins. High-temperature tensile tests at 650 ◦C, 750 ◦C and 850 ◦C were carried out to study the high-temperature mechanical properties and fracture mechanism of the specimens of porous laminate structure, and the strain nephogram of the specimens were obtained by digital image correlation (DIC) technique. Due to the large number and dense arrangement of overflow holes, an obvious hole interference effect can be found from the strain nephogram. The stress concentration around the pore and the interference between the pores provide priority places and paths for the initiation and propagation of microcracks. The test found that the microcracks of the porous laminate structure first occurred around the hole, the overflow surface fractured first, after which the impact surface fractured. The strength of the alloy exhibits a significant temperature sensitivity to temperature. From 650 ◦C to 750 ◦C, the ultimate strength (σb) and yield strength (σ0.2) decrease slightly, but they decrease significantly at 850 ◦C. The microstructure of the fracture surface shows that all microcracks occur at the interface between the matrix and the carbides but that the fracture mode of the specimens gradually changes from intergranular fracture to transgranular fracture as the temperature increases. Due to the pinning effect of the intracrystalline diffusive solute atoms on the dislocations, the stress-strain curves of the high-temperature tensile tests at 650 ◦C and 750 ◦C showed zigzag characteristic fluctuations during the strengthening stage.

**Keywords:** digital image correlation technology; Ni-Cr-W superalloy; microstructure; fracture mechanism

#### **1. Introduction**

In general, in order to make superalloys suitable for long-term use under critical temperature and stress conditions, a large number of alloying elements such as chromium, molybdenum, boron and C are added to the alloys [1,2]. Among them, the Ni-Cr-W superalloy is a kind of nickel based deformation superalloy with solid solution strengthening and carbide dispersion strengthening. A large amount of refractory elements such as W and Cr are added to the alloy to improve the strength of the matrix, and a small amount of C element is added to form dispersed carbides to hinder the grain growth and strengthen the grain boundary [3–5]. Therefore, it has excellent fatigue creep mechanical properties, corrosion resistance and oxidation resistance, and is widely used in gas turbine and aero-engine components, such as blades, flame tubes, etc. [6–9].

At present, some research has been done on the microstructure and mechanical properties of Ni-Cr-W superalloys. Han y et al. [10] studied the effect of carbides formed by grain boundary element segregation on the mechanical properties and microstructure of Ni-Cr-W superalloys. It is found that, with the increase of the heat exposure time, the morphology of M6C carbide changes from grain to strip, which reduces the mechanical properties of the material. Hu and Tang et al. [11,12] studied the

precipitation behavior of M23C6-type carbides at the grain boundary of a Ni-Cr-W superalloy and their effect on the mechanical properties. The results show that the large interfacial energy at the large angle grain boundary is conducive to the formation of M23C6-type carbides and that the decrease of the tensile strength and yield strength is mainly caused by the fracture of the M23C6-type carbide. Hao et al. [13] studied the effect of the C element content on the microstructure and fracture properties of Ni-Cr-W-Fe superalloys and found that a high C element content was prone to forming M23C6 carbides, which accelerated the crack along the grain boundary extension. MQ et al. [14] studied the effect of grain refinement on the microstructure and mechanical properties of Ni-Cr-W superalloys. Bai et al. [15] studied the effect of temperature on the tensile mechanical properties of Ni-Cr-W superalloys. In summary, although there have been some studies on the mechanical properties of this type of superalloy, most research work has not considered the impact of the real structure in engineering practice on the material properties, especially the high-temperature mechanical properties. With the increase of the turbine inlet temperature and the increase of engine efficiency, higher requirements are put on the operating temperature of hot components such as blades and combustion chambers. At present, film-cooling hole cooling methods are mostly used. However, the existence of dense vent membrane holes makes the structural stress state complex, and cracks are more likely to occur around the holes [16,17]. Therefore, it is necessary to understand the mechanical properties of the film-cooling holes' structure.

In this paper, according to the actual structure of the combustion chamber, the porous laminate structure was designed with a Ni-Cr-W superalloy GH3230, and its high temperature tensile mechanical properties were studied at different temperatures. The surface strain field of the structure was obtained by Digital image correlation (DIC) technology.

#### **2. Experimental Material and Process**

#### *2.1. Material and Specimens*

The deformed superalloy GH3230 was used in the present experiment. The experimental GH3230 superalloy materials are all from Beijing Institute of Aerial Materials (Beijing, China). The nominal chemical composition is given in Table 1. The ingot was prepared by vacuum induction furnace + electroslag remelting, and then the microstructure was homogenized by standard heat treatment (1310 ◦C/1 h + 1320 ◦C/2 h + 1330 ◦C/2 h + 1340 ◦C/4 h, 1120 ◦C/4 h (AC), 870 ◦C/32 h (AC) (AC: air cooling)). Finally, the original material was prepared by a cold rolling process. The test alloy is a Ni-Cr-based solid solution strengthened deformation superalloy, which is an isotropic polycrystalline material. After being grinded, polished and electrochemically etched in a solution of 95 mL HCL + 10 mL CH3COOH (by using a voltage of 4 V and corrosion time of 10 s), the grain distribution and microstructure of the specimen were observed through a scanning electron microscope (SEM), and the results are shown in Figure 1. The microstructure of the alloy is mainly composed of γ matrix, M6C carbide and M23C6 carbide (Figure 1a). Granular M6C carbide is the primary carbide, which is uniformly distributed in the alloy matrix (Figure 1c). The microstructure of M23C6 carbide is linear or lamellar, mainly distributing at the grain boundary (Figure 1d). The carbide is enriched in some regions of the grain boundary (Figure 1b). According to the results of the EDS analysis, M6C carbide mainly contains Ni, W, Co, W, etc., while M23C6 type carbide is mainly enriched in Ni, W, Cr, etc., as shown in Figure 1e,f.

**Table 1.** Chemical composition of the GH3230 alloy (wt.%).

**Figure 1.** Scanning electron microscope (SEM) images of the microstructure of the GH3230 superalloy: (**a**) microstructure and morphology; (**b**) segregation and aggregation of intergranular carbides; (**c**) morphology and distribution of M6C-type carbides in grains; (**d**) morphology and distribution of M23C6 type between the grains; (**e**,**f**) the results of the EDS analysis at positions 1 and 2 in the figure, respectively.

In order to simulate the structure of the flame tube in actual engineering and in combination with its structural characteristics, a double-layer thin wall plate structure is designed. The structure is composed of pin fins and two kinds of film-cooling holes: overflow holes and impact holes. The number and arrangement of the two kinds of holes are different. In order to eliminate the influence of the recast layer on crack initiation around the holes, picosecond laser drilling technology is used for punching. The diameter of the overflow hole is 1.5 mm, the arrangement is a triangle symmetry, and the plate thickness is 0.5 mm.  The pin fin connecting the two layers is a cylinder with a diameter of 0.15 mm and a height of 0.25 mm. The actual picture and detailed size of the specimen are shown in Figure 2.

**Figure 2.** Physical picture and geometry schematic of the specimens (unit: mm).

#### *2.2. Experimental Procedure*

In order to study the mechanical behavior and microstructure evolution of porous laminates under a high-temperature, uniaxial tensile condition, the experiments at 650 ◦C, 750 ◦C and 850 ◦C were carried out on a high-temperature electronic creep testing machine (RDL100, Changchun Machinery Experimental Equipment Co., Ltd., Changchun, China) according to the actual working conditions, and each condition was tested twice. The test process is controlled by displacement, and the tensile rate is 0.2 mm/min. During the test, a K-type thermocouple was attached to the upper, middle and lower sections of the specimen to monitor the temperature in real time. In order to study the interference effect between film-cooling holes and the thermal strength of the structure, digital image correlation (DIC) (XTDIC Version 8.2.1, Xi'an Jiaotong University, Xi'an, China) was used to measure the strain field on the surface of the specimen. Before the test, high-temperature paint was sprayed on the surface of the test piece to make the speckle area. During the whole experiment, a high-temperature lens was used to collect images every 10 s. The images are processed by the DIC system, and the changes of the gray value in the speckle region are calculated. The strain data of the specimen surface is obtained, and

the strain nephogram is finally outputted. The schematic diagram of the test process and the preformed speckle pattern of the test piece surface are shown in Figure 3. After the test, the yield strength, ultimate tensile strength and elongation were obtained. In order to observe the special microstructure of the grain boundary, carbides and matrix, a metallographic analysis was carried out by grinding, polishing and electrochemical corrosion of the test alloy. The electrochemical corrosion was carried out in 95 mL HCl (w% ≥ 99.5%) + 10 mL CH3COOH (w% ≥ 99.5%) with a voltage of 4 V and corrosion time of 10 s. The microstructure and microcracks at the fracture surface were observed by optical microscope (OM) (VHX-6000, KEYENCE, Osaka, Japan) and scanning electron microscope (SEM) (ZEISS Gemini300, Carl Zeiss AG, Oberkochen, Germany). The high temperature failure mechanism is explained.

**Figure 3.** Schematic diagram of the test process and preformed speckle pattern on the surface of the specimens.

#### **3. Experimental Results and Discussion**

#### *3.1. Tensile Mechanical Behavior at High Temperature*

Since the cross-sectional area of the specimen varies along the tensile direction, the minimum cross-sectional area is used to calculate the stress, and the strain is calculated based on the distance between the two most distant holes along the tensile direction of the specimen. The high-temperature tensile stress-strain curves are shown at different temperatures in Figure 4. The specific high-temperature mechanical parameters of the tested alloy are shown in Table 2. The results show that the specimen has no obvious yield stage under all temperature conditions, so that the stress value corresponding to the 0.2% plastic deformation of the test piece is used as its yield strength (σ0.2).

From the stress-strain curve, it can be seen that the mechanical properties of the specimen at 650 ◦C are the most excellent, with the best values for the yield strength (σ0.2) and ultimate strength (σb) but with a lower elongation. The mechanical properties at 750 ◦C are slightly lower than those at 650 ◦C, but at 850 ◦C the yield strength (σ0.2) and ultimate strength (σb) of the material are significantly reduced, and there is no strengthening stage. Compared with 650 ◦C and 750 ◦C, the yield strength (σ0.2) and ultimate strength (σb) at 850 ◦C are reduced by 20.6% and 52.4%, respectively; however, the elongation (δ) is 1.18 and 1.06 times that at 650 ◦C and 750 ◦C, respectively. The variation trends of σ0.2, σb and δ at different temperatures are shown in Figure 4b. It can be inferred that the temperature has no significant effect on the elastic deformation ability of materials. In addition, the stress-strain curves show obvious zigzag fluctuations in the strengthening stage at 650 ◦C and 750 ◦C. This phenomenon

is usually related to the portevin Le Chatelier effect based on dynamic strain aging (DSA), that is, the pinning effect of diffused solute atoms on dislocations [18–20].

**Figure 4.** Tensile test results for different temperatures: 650 ◦C/750 ◦C/850 ◦C. (**a**) stress-strain curve of the high-temperature tensile experiment; (**b**) the change trend of high-temperature tensile mechanical properties parameters (σ0.2, σb and δ) with the test temperature.

**Table 2.** Mechanical characteristics of the alloys determined at different testing temperatures.

The studies in [15,21] found that below 800 ◦C the yield strength (σ0.2) and ultimate tensile strength (σb) of the Ni-Cr-W deformed superalloy decreased slightly as the temperature increased, but that above 800 ◦C the ultimate tensile strength (σb) dropped drastically, which is consistent with the high-temperature tensile results in our study. Therefore, being the same kinds of material, the micromechanism should be the same. At a high temperature, the main reason for this phenomenon is the existence of the secondary phase M23C6 at the grain boundary, which reduces the strength of the grain boundary. As a result, cracks initiate at the junction of M23C6 and the grain boundary, and then propagate, resulting in alloy failure. In addition, the high dislocation density and cross slip of the internal structure of the alloy at a high temperature are also responsible for the phenomenon. Therefore, this kind of material shows a certain sensitivity to temperature. When the temperature is too high, this will lead to the rapid deterioration of its thermal strength.

#### *3.2. Mode of Crack Propagation and Strain Field around Holes*

Macroscopic fracture paths are similar at three temperatures. From Figure 5a–c, it can be found that both sides of the test pieces are broken from the middle part of the test piece and that the initial cracks are generated at the middle film-cooling holes. However, due to the differences in the size and arrangement of the impact holes and overflow holes, the crack propagation paths on both sides of the specimen are different. Cracks on the surface of the overflow holes are first generated on the middle overflow hole and expand outward along the direction perpendicular to the loading axis. When the cracks develop toward the edge of the specimen, the edge breaks. The inwardly expanding cracks are in a zigzag shape, which first expands along the direction perpendicular to the load axis, then expands along the 45◦ direction connecting the two holes, and then returns to the direction perpendicular to the load axis. The cracks on the plate with impact holes are first generated on the middle hole before expanding along the direction perpendicular to the loading axis until they extend to the edge, breaking the entire impact surface. The "zigzag" fracture mode of the plate with overflow holes is mainly caused by the serious stress concentration around the hole at the beginning, resulting in crack initiation. Then, due to the interference effect between the holes, the stress level in the direction 45◦ to the loading direction is higher, and the crack propagates along the 45◦ direction. When the crack propagates for a certain distance, the axial stress of the specimen begins to dominate the fracture process, and the crack will propagate along the path that is perpendicular to the loading axis until the plate with overflow holes breaks. Due to the small diameter and sparse layout of the impact holes, the interference effect between the holes is not obvious, causing the cracks to expand along the direction perpendicular to the loading axis until the plate with impact holes fractures. The micromorphology of the overflow hole after the tensile test at 850 ◦C is shown in Figure 5d–f. It can be seen from the figure that the circular overflow hole has been elongated into an ellipse. Compared with the original test piece, the edge of the overflow hole parallel to the loading direction was complete, but a large number of microcracks were generated at the edge of the hole perpendicular to the direction of the loading axis. Furthermore, the crack initiation position was at the grain boundary. This is mainly due to the machining defects around the pores and the presence of intergranular carbides, which provide a place for microcracks and the nucleation of micropores [13].

The images taken by the high-temperature camera were processed by the DIC system, and the strain nephograms at each temperature were obtained, as shown in Figure 6. Through the strain field nephogram, it can be found that there are obvious hole interference effects between the overflow holes under the three temperature conditions, which shows that there is an obvious high strain band area between the adjacent holes. The results show that the banded zone is 45◦, −45◦ and 90◦ to the tensile direction and that the maximum strain occurs around the hole where the strain band intersects. It can be seen in Figure 6 that when σ = σ0.2, the strain bands are clear. When the ultimate strength is reached, the width of the strain band increases and the high strain area becomes larger. Among these areas, the high strain region between the two big holes in the middle expands the fastest, which makes the middle part of the specimen become a high strain area. Due to the rapid propagation of cracks, the stress drops sharply, but the strain increases slowly. From the strain nephogram, it can also be found that the cracks first occur at the edge of the two holes in the middle before extending to both ends. The propagation of the cracks leads to a sharp increase in the strain in the region between the two holes, which is obviously higher than that in the other regions. When the cracks between the two holes are connected, the plate with overflow holes breaks, and then, 2 to 3 s later, the plate with impact holes fractures, and the whole specimen finally breaks.

**Figure 5.** Images of test pieces during testing: (**a**) 650 ◦C; (**b**) 750 ◦C; (**c**) 850 ◦C; (**d**–**f**) microcracks around the hole at 850 ◦C.

By observing the strain nephogram of the plate with impact holes, it is found that the hole interference effect also exists between the impact holes but that it is not as obvious as that between the overflow holes. This is related to the size, density and layout of the impact holes. The strain nephogram has obvious differences at different temperatures. At 750 ◦C, because the distance between the impact holes arranged in a diamond shape at the upper and lower ends of the specimen is small, strain bands of ±45◦ and 90◦ to the tensile direction are formed between the holes when yielding. However, at 650 ◦C, due to the low temperature and high strength of the material, there is no obvious difference in the strain of each part of the whole face, while at 850 ◦C the strain of the whole surface is relatively large, and there is also no obvious difference in strain due to the low strength of the material. Because the isolated impact hole is far away from other impact holes, there is no strain band between the isolated hole and other holes under the three temperature conditions. However, the strain around the isolated hole is the greatest. As time increases, the strain in the middle part is obviously higher than for the other parts. The crack propagates from the periphery of the hole along the high strain region to both ends. The fracture mode shown by the strain nephogram is consistent with the actual fracture mode of the specimen in the test.

**Figure 6.** *Cont*.

**Figure 6.** DIC strain nephograms of the specimen plates with overflow holes (d = 1.5 mm) and impact holes (d = 0.9 mm) at three different stress state stages (σ = 0/σ = σ0.2/σ = σb) and, before fracture, under three temperature conditions.

#### *3.3. Fracture Analysis*

Figure 7b,e,g shows the fracture morphology seen under the optical microscope under different temperature conditions. It can be seen that the macromorphology corresponds to the fracture mode described in Section 3.2. Under the interference effect between the holes, a crack propagation path appears along the strain band on the plate with overflow holes, while there is no obvious hole interference effect on the plate with impact holes. The fracture morphologies at 650 ◦C and 750 ◦C are similar, with the fracture being straight and the Diameter Shrinkage phenomenon not being obvious. However, the fracture surface at 850 ◦C shows an obvious Diameter Shrinkage phenomenon, and the fracture deformation is serious, which indicates that the plasticity of the material is significantly improved at a high temperature. The fracture surface is observed after electrochemical corrosion, as shown in Figure 7c,f,i, where it can be clearly seen that the fracture at 650 ◦C is a typical intergranular fracture, the fracture at 750 ◦C shows the mixed fracture characteristics of an intergranular fracture and transgranular fracture, and the fracture at 850 ◦C is almost all transgranular fracture. Figure 8 shows the microstructure of the fracture surface and fracture side under SEM. The fracture mode can also be identified by Figure 8a–f, which is consistent with the results observed by optical microscope. In addition, by comparing the microstructure of the fractured test pieces with the microstructure of the material before the test, it can be found that, after the high-temperature tensile test, the amount of carbide on the alloy surface increases when compared with the original structure, but the size decreases slightly. At all temperature conditions, there are no significant differences between the carbides near the fracture surface and those around the holes, but the grains near the fracture surface are obviously elongated. Figure 8g–i shows the micromorphology of the fracture surface after the high-temperature tensile test. It can be seen that all the fractures are ductile dimple fractures, and a large number of second-phase particles are uniformly distributed on the fracture surface. The EDS analysis confirmed

that the second-phase particles were granular M6C carbides. At the same temperature, the grain size of the plate with overflow holes is larger than that of the plate with impact holes. This is due to the large number and dense arrangement of overflow holes, which makes the stress on the plate with overflow holes larger under the same loading conditions.

**Figure 7.** Fracture morphology and side morphology after electrochemical corrosion. (**a**–**c**) 650 ◦C; (**d**–**f**) 750 ◦C and (**g**–**i**) 850 ◦C.

**Figure 8.** Fracture morphology at 650 ◦C/750 ◦C/850 ◦C. (**a**–**c**) Inclined fracture view of the surface of the plate with overflow holes and (**d**–**f**) the surface of the plate with impact holes; (**g**–**i**) fracture surface morphology and (**j**–**l**) microstructure near the fracture surface.

#### *3.4. High-Temperature Fracture Mechanism of Porous Laminates*

The unique structure, which can simulate actual engineering, has a significant influence on the mechanical properties of the test pieces. The effective cross-sectional area is reduced by 73.2–84.6% due to the lamellar and porous structure, resulting in a significant loss of tensile mechanical properties. According to the strain field measured by DIC in Section 3.2, it can be seen that there is an obvious stress concentration area around the holes. Moreover, due to the large diameter and dense arrangement of the overflow holes, there is a significant pore interference effect between the overflow holes under the three temperature conditions. It is shown that there are obvious high strain banded regions between adjacent holes. The direction of the strain band is ±45◦ and 90◦ to the tensile direction, and the distribution is symmetrical. The strain around the holes where the deformation bands intersect is the largest, which provides a place for the initiation of cracks on the surface with overflow holes. When the initial microcrack is generated around the holes, the propagation direction of the microcrack is consistent with that of the interference band. When the crack propagates for a certain distance, the effective cross-sectional area decreases, and the stress on the surface with impact holes increases rapidly. The cracks are produced at both ends of the impact hole in the center of the test piece and propagate rapidly to both ends. Then, the cracks on both sides extend outward at the same time until they penetrate the whole test piece, resulting in its fracture failure. The pin fins between the laminates

did not bear a direct axial force and did not undergo significant deformation, but they still played a certain role in the strength of the structure itself, which was mainly manifested by bonding the plate with overflow holes and impact holes in order to improve the stiffness of the laminate structure.

According to the high-temperature tensile stress-strain curve of the alloy in Figure 4, it can be seen that the test alloy shows a significant sensitivity to temperature. The ultimate strength (σb) and yield strength (σ0.2) of the specimens are higher at 650 ◦C and 750 ◦C but decrease significantly at 850 ◦C, while the elongation (δ) increases significantly. When the temperature exceeds a certain critical value, the thermal strength of the material will deteriorate rapidly, and the fracture mode of the material will change from intergranular fracture to transgranular fracture with the increase of the test temperature. Under the three temperature conditions, the crack is generated at the grain boundary around the homogeneous hole, as shown in Figure 5f. The reason for this is that the strength of M23C6 carbide at the grain boundary is high, so that it is difficult for dislocation to cut or bypass it, resulting in the dislocation stacking at the M23C6-type carbide and forming a dislocation nail. However, when the stress reaches a certain value, the local stress concentration occurs in the γ/M23C6 bonding area, which makes the cracks appear in the γ/M23C6 bonding area [20]. From the previous article, one can see that the test material is mainly composed of the matrix phase and two kinds of carbide phases, belonging to the typical carbide dispersion strengthened superalloy. Therefore, the strength of carbides plays an important role in the mechanical properties of the alloy, especially M23C6 carbide, which can strengthen the grain boundary. At a lower temperature (650 ◦C), the strength of the matrix phase is greater than that of the grain boundary, so that the crack will propagate along the grain boundary of the interference band between the holes, resulting in an intergranular fracture of the alloy. In addition, after the strengthening stage, the stress-strain curve appears to zigzag due to the cross slip of matrix deformation in the crystal and the pinning effect of diffusion solute atoms on the dislocation motion. With the increase of temperature, both the grain boundary strength and matrix phase strength decrease, but the decrease rate of the matrix strength is faster than that of the grain boundary [22]; when the temperature reaches 850 ◦C, the strength of the grain boundary will be greater than that of the matrix, and the crack will propagate along the inter-hole strain band. Because the strength of the matrix phase is greatly reduced, the dislocation can easily pass through the M6C-type carbide by the climbing mechanism and Orowan bypass mechanism [23]. The crack will propagate through the grain until the specimen breaks. At the same time, due to the decrease of the material strength, the plasticity of the specimen will increase significantly, resulting in a significant increase in the elongation (δ) of the test piece, and the polygonal grains will be elongated into long strips during the tensile test. The microstructure evolution and fracture mechanism of the alloy under high-temperature tensile testing are drawn for different temperatures according to the experimental phenomena, as shown in Figure 9.

**Figure 9.** The schematic of the microstructure evolution and fracture mechanism of the tested alloy at different temperatures.

#### **4. Conclusions**

In this study, the Ni-Cr-W superalloy GH3230 was used as the experimental material, and porous laminate specimens were designed in combination with an actual engineering structure. High-temperature tensile tests at 650 ◦C, 750 ◦C and 850 ◦C were carried out. The high temperature mechanical properties and fracture mechanism of deformed superalloy porous laminate specimens were studied by means of digital image correlation (DIC), an ultra depth of field optical microscope, a scanning electron microscope (SEM) and EDS element analysis technology. The main conclusions are as follows:

1. Through the strain field results measured by DIC, it is found that an obvious stress concentration area is formed around the pores of the film-cooling holes, Moreover, due to the large diameter and dense arrangement of the overflow holes, there is a significant pore interference effect between the overflow holes under three temperature conditions. It is shown that there are obvious high strain banded regions between adjacent holes. The direction of the strain band is ±45◦ and 90◦ to the tensile direction, and the distribution is symmetrical. On the contrary, no obvious interference effect was found on the side with impact holes, and the crack initiation was mainly due to the stress concentration around the holes. The pin fins between the laminates have no obvious effect on the structural strength, and their function is mainly to bond the laminates on both sides.

**Author Contributions:** Design study, Y.-D.L. and Z.-X.W.; Literature search, P.W.; Formal analysis, P.W.; Investigation, Y.-D.L. and P.W.; Data curation, P.W.; Writing—Original draft preparation, Y.-D.L. and P.W.; Writing—Review and editing, Y.-D.L.; Funding acquisition, Z.-X.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (Project No.: 51875462), the Natural Science Basic Research Plan in Shaanxi Province of China (Project No.: 2020JC-16), the National Science Technology Major Project of China (Project No.: 2017-IV-0003-0040, 2017-V-0003-0052) and Taicang Scientific Research Institute Innovation leading special plan (Project No.: tc2019dyds11).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

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*Article*

### **Investigation of the Impact of Load on the Magnetic Field Strength of the Crane by the Magnetic Metal Memory Technique**

### **Agnieszka Koson-Schab \* and Janusz Szpytko**

Faculty of Mechanical Engineering and Robotics, AGH University of Science and Technology,

A. Mickiewicza Av. 30, PL30-059 Krakow, Poland<EMAIL_ADDRESS>
**\*** Correspondence<EMAIL_ADDRESS>
Received: 24 October 2020; Accepted: 4 December 2020; Published: 6 December 2020

**Abstract:** The paper deals with the problem of applicability of the metal magnetic memory (MMM) technique in the crane structural inspection and monitoring. The MMM method does not require the external magnetization of a structure that results in reduction of downtime of maintenance operations. Measurement of the intensity of the self-magnetic leakage signal can be an alternative to other non-destructive methods used for inspection of a large crane's structure and equipment. However, the complexity of the residual magnetization effect in the MMM technique is the problem with its application. Thus, the magnetic flux leakage behavior on the crane girder surface under different measurements and the crane's load conditions is analyzed based on the results obtained during experiments carried out on the overhead traveling crane.

**Keywords:** metal magnetic memory; magnetic field non-destructive evaluation; diagnostics; gantry crane

#### **1. Introduction**

Currently, there is a growing interest in obtaining information from the research of operating structures in terms of their degradation processes for the purpose of improving their construction, manufacturing, and operation. Construction safety requirements are related to the development of new diagnostic methods enabling the detection and localization of structural degradation.

Structures of large-scale industrial transport devices, subjected to loads resulting from the type of their work, are exposed to high stressors. These stressors affect the fatigue strength of the structure as well as damage the material structure. In the interest of safety, you should perform frequent diagnostic tests of these devices.

Nowadays, technical diagnostics increasingly use non-invasive diagnostic methods for diagnosing or monitoring (continuous control) without interrupting the normal operation of the object [1–3]. One of the least invasive methods of diagnosis is magnetic metal memory (MMM). This is used for inspection of welds, structural elements such as loaded cyclically on rotors, gears, steel ropes, and much more. The MMM method is one of the fastest and cheapest for detecting and assessing early caused damage based on the natural stress-induced magnetic flux leakage (MFL) [4,5].

The MMM method, based on detecting the impact of changes in the residual magnetization of the examined object on its magnetic flux leakage, is increasingly used in diagnostics. It allows indicating places or dangerous areas, cracks, or other defects of elements made of ferromagnetic materials [6,7].

The MMM method is based on the magneto elasticity Villari effect. Local disturbances in the earth's magnetic field are measured at the surface of ferromagnetic materials. Ferromagnetic materials in the presence of the geomagnetic field are subjected to stress, which causes the induction of a magnetic field [8–11].

The analysis of the test results consists of comparing the normal and tangential components of the magnetic flux leakage along the measuring path.

It is based on the known phenomenon of magnetic flux leakage of material with different magnetic permeability. Changes in the sign and leakage magnetic field *Hp* value are indications on the basis of which the assessment of stress and strain in the material of the tested element is made, define *Hp* by the Formula (1) [12]:

$$H\_p = \frac{1}{\mu\_0} B\_p \tag{1}$$

where:

*B*0 is the vertical value of the magnetic induction component [T]; μ0 is vacuum magnetic permeability 4π × 10−7 [H/m].

Areas of stress concentration are detected in which processes leading to a reduction in material strength can occur [11,13–15].

The MMM method can detect the residual magnetic field due to mechanical stress and geomagnetic field and can be performed without artificial magnetization. The magnetometer using the MMM method is based on magnetoresistance effects. It is characterized by a small volume, high sensitivity, and reliability. This makes it easier to detect the fatigue of the tested elements [2,16].

It is widely accepted that the magnetic state of ferromagnetic material may be irreversibly altered by mechanical loading due to magneto-elastic effects [17]. On the macroscopic scale, there are close relationships between the mechanical and magnetic properties of ferromagnetic material. Under the influence of mechanical stress in the ferromagnetic material, deformation occurs and changes the internal magnetic field. In general, the MMM is an after effect which is directly correlated with the distribution of natural magnetic fields and the accumulated stress-strain which has been experienced by the ferrous or paramagnetic material.

The heterogeneity of the structure, mechanical properties, and the magneto-plastic effect are the result of the leakage magnetic field, which will lead to an irreversible increase in the magnetic induction under a weak magnetic field. The ferromagnetic element is subjected to the external stresses in a certain direction [18].

The MMM signals are a weak magnetic signal, which consists of vulnerable factors such as lift-off height, chemical composition, the specimen shape, the notch shape, the heat-treating technology, the initial magnetization state of ferromagnetic materials, and environmental magnetic field [10,14,19,20].

The objective of the paper is to determine the experimental impact of the structure load change on the level of the own magnetic field, using the MMM method and measuring the magnetic field strength in the elastic deformation zone, for the purpose of inferring changes in the technical condition of the structure, on the example of the girder of the overhead crane.

#### **2. Crane Girder Bridge Testing Approach with Use MMM Technique**

The aim of the experiment was to determine the impact of the load change on the level of self-magnetic leakage field (SMLF)of a double girder bridge crane in terms of analyzing the possibility of using its residual magnetic flux.

As a result of stresses and plastic deformations, changes in the structure of the material caused by fatigue, local irregularities are observed as local changes in the degree of the material magnetization.

The value measured in the test is the value of the selected component of the magnetic field strength *H* measured near the diagnosed object, represented by Formula (2):

$$H = \frac{B}{\mu\_r \cdot \mu\_0},\tag{2}$$

where [21]:

*H* is magnetic field strength [A/m],

*B* is magnetic induction [T], μ*r* is relative magnetic permeability, μ0 is vacuum magnetic permeability 4π × 10−7 [H/m].

After analyzing the available literature on the research of objects using the MMM method, our own test plan was developed to reduce the cost and time of the experiment. The research plan cover: object model, the duration of individual experiments, identifying confounding factors and constants. Figure 1 shows the test object model.

**Figure 1.** Block diagram of the research subject.

The model research object (the girder of a double-girder overhead crane) can be represented by the function defined by Equation (3):

$$y(t) = f\left(\mathbf{x}, z, c, t\right),\tag{3}$$

where:

*y*—output values: magnetic field strength *H* in the tested girder section in [A/m];

*x*—input values: *x* = {*x*1*, x*2};

*x*1—beam load time in [s],

*x*/1—beam deflection in [mm],

*c*—solid factors: *c* = *{c*/1*, c*/2*,* ... *ci}*;

*c*1—the high of raising in [mm], *c*2—load lifting speed in [m/s], *c*3—ambient temperature of crane in [◦C], *c*4—crane girder load in [kg], *c*5—the place on the girder where the load was applied in [m], *ci*—other that did not affect the results of the tests carried out,

*z*—interference: *z* = {*z*1*, z*2*,* ... *zi*};

*z*1—measurement errors, *z*2—magnetic field disturbances caused by the device's operation in [A/m], *z*3—material structure, *zi*—factors unnoticed affecting the measurement.

The SMFL signal measured using the MMM method is influenced by static and dynamic loads, movements of driving and lifting mechanisms, and transient states of the drive systems.

The frequency inventers used in the crane drive system generate the external disturbances that affect the measurement signal. This drawback was eliminated by carrying out the measurement experiments when crane motors and frequency inverters were turned out. The other exogenous conditions, such as temperature (if the measurement is performed in the temperature range −20–60 ◦C), have minor impact on the measurement results.

Research assumptions:

The experiments were carried out on the double-girder overhead traveling crane with hoisting capacity *m* = 1000 kg, a span of the girders *l* = 8 m, and trolley wheelbase *p* = 1 m. Figures 2 and 3 present the crane subjected to examine, and girder beam cross-section, respectively. The girder is made of S235JR structural non-alloy steel [22]. The beam, supported on both sides at its ends, was subjected to elastic deformation, giving an elastic force in its center of 1000 kg, resulting in deflection of the beam f = 12.17 mm. The loading conditions of the tested girder are shown in Figures 3 and 4.

**Figure 2.** The tested double-girder overhead traveling crane with a trolley and a hoist.

**Figure 3.** Crane's girder cross-section, where: F is tested plane.

The *r* = 1.4 m length section between points G and H (Figure 4) of the crane's girder bottom surface was selected for examination using (where G is the starting point and *x* is the direction of the measurement). Points G and H are appointed symmetrically to the center of the crane beam.

The SMFL signal was measured using the Tester Stress Concentration TSC-4M-16, which is equipped with a scanning device. The scanning device is a four-wheeled cart with the flux-gate transducers, and the encoder for sensing position of the cart moved along a measuring element (Figure 5). The magnetometers installed in the scanning device allow to measure *Hx* and *Hy* distribution of the self-magnetic leakage field signal along the surface of an inspected structure.

The measurement device allows to measure the magnetic field intensity in the range ±2000 A/m. The basic relative error of *Hp* measurement is 5%, and the sensor movement accuracy is 0.9 mm. The magnetometer TSC-4M-16 is equipped with the software for data acquisition and analysis.

The measurements of the magnetic field strength in the designated spar section were carried out according to our own test plan.

During the inspection, the trolley with a winch was set in the middle of the crane's girder. The load was placed in the center of the trolley. Figure 4 presents the dimensions and position of the trolley with a load during the experiments. The crane's girder, where: *l* is crane span, *l* 2 is the distance of the suspended load from the beginning of the girder, *m* is load suspended on a rope, *k* is load-lifting height measured from the ground, *r* is the examined section between points G and H of the girder. In order to eliminate interference magnetic field strength, the tests were performed with the driving and lifting mechanisms turned off.

**Figure 4.** Experimental setup.

**Figure 5.** Tester of Stress Concentration TSC-4M-16 and scanning 4-channel sensor.

All measurements of the magnetic field strength were carried out in the same conditions:

The research was divided into two stages.

During the first stage, the intensity of the residual magnetic field in the spar loaded for 4 h was tested to the following scenario repeated every 24 h for 4 days, the results are presented in Table 1

During the second stage of the research, measurements of the magnetic field strength were carried out after 10 loads of 1000 kg for 5 s:

• measurement variant E—The first measurements were made according to the measurement variant A, where *m* = 0 kg, *t* = 0 s, in Table 2, these measurements were called the measurement cycle 0. Then, the load *m* = 1000 kg was raised to *k* = 10 cm, after *t* = 5 s the load was abandoned (*m* = 0 kg, *t* = 0 s). This operation was repeated 10 times. The movement mechanisms were turned off, then the magnetic field strength was measured (measurement cycle 10). Six cycles were performed, after each, the magnetic field intensity in a given beam section was examined.

Four repeated measurements were carried out for each case A, B, A, B, C, D and E.

#### **3. Results and Discussion**

The results of the measurements magnetic field strength were analyzed. During the analysis of the measurements, the focus was on the graphs based on the value of the magnetic field strength at a given section of the tested object. The results of measurement experiments carried out for each case A, B, C, D, and E are presented in Table 1 in the form of maximum, minimum, and average values of tangential *Hx* and normal *Hy* component of SMFL are listed as determined as the mean values of four measurements repeated for each case. Maximum, minimum, and average values of magnetic field strength are listed as *Hx* and *Hy* in Table 1.

Graphs weremade for*Hx*min andmax and*Hy*min andmax to compare themaximum, andminimum values of the magnetic field strength in the tested section of the crane girder during the experiment. These graphs are shown in Figure 6.

The minimal values of the tangential component *x* the strength of the magnetic field in the tested girder, after raising the load, reach higher values. With the passage of time under load, *Hxmin* values are increasing. After unloading the crane, the magnetic field strength in the tested section of the *Hxmin* girder decreases.

**Figure 6.** Minimum of tangential *Hx* and normal component *Hy* after repeatedly applying variants A, B, C, D, for 4 days long.

The lowest *Hx* value is obtained after several hours of relaxing the crane girder (Figure 7). The minimum normal component y of the magnetic field strength for each loading and unloading of the girder reacts with a change in value. The more measurement days, the smaller the range of change.

The same happens with max *Hx* and *Hy* values. Each operation related to the load causes a change in the value of the magnetic field strength for both components.

**Figure 7.** Maximum of tangential *Hx* and normal component *Hy* after repeatedly applying variants A, B, C, D, for 4 days long.

The maximum values of the magnetic field strength in the tested spar cross-section, after loading with the mass *m* = 1000 kg, decrease their value. After the load time *t* = 4 h, the measurement shows a decrease in the *Hmax* value for the *x* and *y* components. However, after removing the load where *m* = 0, *Hmax* reaches higher values.

Figures 8–11 show graphs of tangential *Hx* and normal *Hy* component of the magnetic field, for samples 1, 2 and 15, 16. On their basis, it is possible to notice the characteristics of changes in magnetic field strength waveforms *Hx* and *Hy* at the beginning of the experiment, then immediately after the first loading of the device, and before the last unloading and after the whole experiment.

**Figure 8.** Graphs of tangential *Hx* and normal *Hy* component of the magnetic field, for sample 1.

**Figure 9.** Graphs of tangential *Hx* and normal *Hy* component of the magnetic field, for sample 2.

**Figure 10.** Graphs of tangential *Hx* and normal *Hy* component of magnetic field, for sample 15.

**Figure 11.** Graphs of tangential *Hx* and normal *Hy* component of the magnetic field, for sample 16.

After comparing the charts of the *Hx* and *Hy* components, we notice slight differences in the course of the chart for *Hx*. The diagrams, regardless of the load and its length, are similar due to the curve. For *x* in the range 340–370 mm for unloaded girder, *Hxmin* is about 50 A/m. After the last experiment, unloading the girder, this value is about 120 A/m. Whereas *Hxmax* is in the range 550–600 mm is reduced from 620 to 570 A/m. The *Hy* changes its max and min and the course of the curve during each experiment. The graph shows tendencies to large changes in magnetic field strength for *Hy*. It can be stated that the magnetic field in the element subjected to stress reduces its amplitude vibration.

The next stage of the research included measurement variant E. The influence of dynamic loads on the intensity of the girder's own magnetic field on the section determined in accordance with Figure 3 was examined.

After unloading the device magnetometric measurements were carried out. A total of 6 series of 10 loads and unloading of the girder were made. The measurement results for one magnetometer are presented in Table 2. The maximum, minimum, and average values of *Hx* and *Hy* are shown in Figures 12 and 13.

**Table 2.** Minimum, maximum, and average values of tangential and normal component of the magnetic signals for experiments E.

After the first series of short loads on the crane girder, a change in the value of the magnetic field in the measured plane of the beam is noticeable. The minimum *Hx* values increase from 255 to 275 A/m, while the *Hy* decreases from 134 to 126 A/m. The second series of loads causes a slight decrease in the value of *H* for *x*. For *y*, *Hmin* achieves a value close to the measurement before the beam is loaded. The results of measurements of successive series show the independence of the magnetic field strength from the number of series made. The maximum values of *H* for *x* and *y* show a similar graph. After the first 10 loads, *Hmax* decreases, the next series increases the level of the magnetic field in the beam plane being tested for both components. In the third series, you can see a drop in value. After the last 6 trial, *Hmax* reaches the highest values *Hxmax* = 719 A/m and *Hymax* = 200 A/m.

**Figure 12.** Minimum of tangential component *Hx* and *Hy* for experiment E.

**Figure 13.** Maximum of tangential component *Hx* and *Hy* for experiment E.

Examples of magnetometric charts made on the basis of the first measurement taken without a girder load, and the last one is shown in Figures 14 and 15.

**Figure 14.** Graphs of tangential *Hx* and normal *Hy* component of the magnetic field, for sample 17.

**Figure 15.** Graphs of tangential *Hx* and normal *Hy* component of the magnetic field, for sample 23.

The curve illustrating the *x* component before and after the experiment involving a similar course, when the curve illustrating the component *Hy* contains a significantly different character. The minimum *Hx* reaches in the range of 350–400 mm while the maximum in the range 550–600 mm. In the case of the *y* component, it is impossible to clearly determine the fixed place where it reaches its maximum and minimum.

#### **4. Conclusions**

The paper addresses the problem of the crane structure inspection by applying the MMM method. The crane girder was the subject of inspections of the effect of load on changes in tangential and normal components of the magnetic field using MMM. When analyzing the results of measurement experiments, it can be seen that with each load change, the magnetic field strength changes. These changes are visible in Figures 5 and 6. A similar situation occurs when the load is applied dynamically. After each series of measurements, the change in *Hx* and *Hy* measurements is visible (Figures 11 and 12).

Based on the research, it can be concluded that short-range, high-value forces act between atoms of a given material with elastic properties. During elastic deformation of such material, interatomic distances and angles between atomic bonds change. As a result, it is associated with large energy changes because of the energy of elasticity increases. Elastic deformations are small and these small deformations are accompanied by relatively high forces. After the force causing the elastic deformation disappears, the material returns to its previous dimensions, the material atoms will take their previous positions again.

Microplastic deformations by dislocation development may cause changes in the magnetic flux leakage. However, according to our best knowledge, the relation between the microplastic deformation and magnetic flux leakage is not reported in the literature.

The tests may support the safety assessment of engineering structures subjected to loading. The knowledge of the influence of the load on the level of the magnetic field intensity allows us to predict the degradation time of the device structure. The use of the MMM method for monitoring can be helpful in the early detection of structural damage. This will result in increased work safety.

The next stage of the research will be to determine the intensity of the magnetic field during beam loads causing plastic deformation. These works are aimed at obtaining as much information as possible about the behavior of the magnetic field strength in the girders of the gantries working under load. Further research may be directed towards the challenging problem to obtain the damage indicator parameter and predict the absolute fatigue life of a crane's structure. It requires analysis of magnetic flux leakage behavior under various load conditions during elastic and plastic deformations. Future work may consider to carry out more experiments, to collect more data and use data driven techniques to identify the influence of elastic load and plastic deformation on the magnetic flux leakage signal and their coupling effects.

**Author Contributions:** Conceptualization, A.K.-S. and J.S.; methodology, J.S.; software, A.K.-S.; validation, J.S.; formal analysis, A.K.-S.; investigation, A.K.-S.; resources, A.K.-S., data curation, A.K.-S.; writing—original draft preparation, A.K.-S. and J.S.; writing—review and editing, A.K.-S., visualization, A.K.-S.; supervision, A.K.-S. and J.S.; project administration, J.S.; funding acquisition, J.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work has been financially supported by the Polish Ministry of Education and Science (Polish Ministry of Science and Higher Education).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

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### *Article* **Single-Sensor Vibration-Scanning Method for Assessing the Mechanical Properties of 3D Printed Elements**

**Ryszard Buchalik \* and Grzegorz Nowak \***

Department of Power Engineering and Turbomachinery, Silesian University of Technology, Konarskiego 18, 44-100 Gliwice, Poland

**\*** Correspondence<EMAIL_ADDRESS>(R.B<EMAIL_ADDRESS>(G.N.)

**Abstract:** This paper considers issues related to the assessment of the mechanical properties of elements made with 3D printing technology. To enable experimental testing, an automated test stand was built to perform amplitude and phase angle measurements of any point of the specimen. A contactless, optical measurement method was selected, as it is especially adequate when it comes to elements with small dimensions and masses. One innovative element of the test stand is the original method of phase angle measurement using a single vibration sensor fitted with a system forcing and ensuring full measurement synchronization and dynamic state repeatability. Additionally, numerical models of tested objects were produced and simulations of their oscillations were performed. Based on that, the properties of the tested material (PLA) were considered, with a special focus on the density, elastic modulus, and damping. The analyses were conducted for a few elements with different dimensions at different vibration frequencies.

**Keywords:** 3D printing; additive manufacturing; material properties; phase shift measurement; vibration scanning; resonance; vibration shapes

### **1. Introduction**

In recent years, there has been a huge increase in interest in Additive Manufacturing (AM) as a modern technology for the production of a wide range of elements. The method's advantages are the relatively fast production, the possibility of creating complex shapes and joining different materials, and the ease of obtaining the desired color or finish [1]. The use of varied internal structures in this technology makes it possible to shape the properties of produced elements, which is extensively presented in [2]. Gibson et al. [3] describe the application of bulk metallic glasses in AM, which behave like thermoplastics. Considering the recent development of 3D printing technology and the increasing potential of the technology's application in many different fields, it is important to establish the mechanical properties of elements made in this way [4,5]. There are works describing certain features and mechanical (material) properties of such materials [6–12], including product documentation of the raw material supplied by the manufacturer. However, most of them primarily concentrate on testing the material in steady states and on slow changes in essential parameters, such as the load. They offer very scarce information on the material behavior under fast-changing loads (a few dozen Hz and more), i.e., vibration. In this area, work focuses on the impact of geometric discontinuities (e.g., cracks) on the dynamic state of the element [13,14]. The technical data supplied by the manufacturers of devices, on the one hand, and filament producers, on the other hand, are not always confirmed by experimental testing results. Selected examples from the literature focus on the impact of the infill and the printing direction [15,16], taking account of fatigue [17] and creep [18,19] processes, the printing system vibrations [20] and damping [21]. Testing materials made with Additive Manufacturing technology is onerous due to the substantial impact of a great number of factors. The finished product parameters vary, depending on a number of parameters of the production process. The produced object's mechanical parameters

**Citation:** Buchalik, R.; Nowak, G. Single-Sensor Vibration-Scanning Method for Assessing the Mechanical Properties of 3D Printed Elements. *Materials* **2021**, *14*, 1072. https://doi.org/10.3390/ma14051072

Academic Editor: Giovanni Bruno

Received: 27 January 2021 Accepted: 19 February 2021 Published: 25 February 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

depend, in a complex way, on the temperature of the nozzle depositing the material, the method of the material feeding, the shape and diameter of the nozzle and the nozzle thermal conductivity, the ambient temperature, the cooling method and intensity, the layer thickness, the printing speed, the material storage conditions (the history of temperature, time, and humidity), the printer accuracy, and the whole printed element cooling rate. Another key component of the 3D printing system that has an essential impact on the final product parameters is the program creating the extruder movement trajectory [22]. In the case of fused filament fabrication technology, an enormous number of combinations of printing parameters are used on the shell and in the infill. The parameters of the first layer (in contact with the base platform) also differ from the other layers. An element made with this technology is usually characterized by strongly anisotropic properties, which is mainly due to the existence of layers and contact between them [15,16]. An essential part of the production process is to determine the spatial orientation according to which the object will be made. Material cooling also generates internal stresses and may result in bending of the finished product [23].

The behavior of a mechanical element when exposed to vibrations and the possibility of resonant frequency excitation within it can lead to serious problems in the design and operation of machines [24]. There are studies based on the vibration analysis of elements made with 3D printing technology [25–31]. Some of them focus on determining damping inside the material if oscillating motion occurs and make use of comparative analysis. An exhaustive theoretical description of the phenomenon is presented in [26]. Gietl et al. [21] presented the way in which the impact of the medium (air resistance) is eliminated from measurements using a fiber-optic sensor. Efforts aiming to establish material parameters seem especially important when it comes to design and prediction of the occurrence of potentially harmful resonances of elements made with 3D technology. They will make it possible to simulate this type of system at the prototype-making stage and avoid many operating problems, e.g., excessive wear, noise, fatigue-related damage, etc. It is worth mentioning that very technologically advanced elements, even including turbine blades, can also be successfully produced with AM technology [32]. Due to the progress in the field of searching for materials with high vibration-isolating capabilities [33], it is also important to search for new methods for determining the material's elasticity and ability to dissipate the energy (damping) of its vibrations (damping), characterized by simplicity and the possibility of application in various conditions. The description of material parameters is additionally complicated in the case of structures reinforced with fibers made of different materials and embedded within the base material [34]. Damping or elasticity calculations can be very complex for specimens composed of many different material layers characterized by different properties [35] and different joints or infill [36]. The destruction of material due to vibration shapes, among other factors, can be taken into account [37].

In connection with the above-described aspects of AM technology and its expected development, vibration testing methods should be widely used. Typically, vibration testing involves the use of a sensor rigidly and directly mounted on the tested object. The only more widespread type of non-contact vibration sensor is the inductive sensor; however, its use is practically only possible in the case of materials showing magnetic properties, with a significantly different electrical and magnetic permeability from a vacuum. The only rational choice in the case of AM materials with a relatively low density that are light, small, and non-magnetic seems to be the optical measurement method. Due to the significant technological progress and reduction of the costs of photosensitive and laser elements, it seems reasonable to develop optical methods for vibration measurement.

There are several basic methods of optical vibration measurement, which allow various physical quantities to be measured. The first may be displacement-triangulation sensors in many arrangements [38,39] or an analysis of the image of the object on which the appropriate pattern is displayed at defined angles [40–42]. Due to the differential-integral dependence, the mechanical quantities can be mutually converted. However, it should be remembered that the numerical accuracy of the result changes, depending on the frequency of vibrations.

The aim of this paper is to consider issues related to the mechanical properties of objects made with 3D printing technology. An analysis of selected mechanical parameters is presented based on dynamic testing and the use of resonance frequencies. Experiments were performed to this end using a novel method of vibration measurement on a purposemade measuring stand. The analysis covers the vibration amplitude and the phase shift by means of an unconventional method using only one displacement measuring sensor. Experimental data collected in this way can be coupled with a sophisticated numerical method [43] to determine the material properties and their uncertainty. The method proposed below can be applied to specimens made from different kinds of materials, even those which are naturally non-isotropic and non-uniform [44].

#### **2. Materials and Methods**

A purpose-designed test stand was constructed to perform measurements of dynamic states of specimens made with 3D printing technology.

#### *2.1. Test Stand*

The test stand (Figures 1 and 2) is composed of a vibration exciter and a measuring system; the whole stand is computer-controlled. The measuring system's fundamental element is a triangulation displacement sensor. Its operating principle is the emission of a laser beam illuminating the investigated point of the specimen; the beam is then subjected to diffusive reflection. The reflected signal returns to the sensor at an angle, depending on the reflection point distance from the sensor, which is recorded by a light-sensitive matrix (one-dimensional image sensor) placed in the instrument and fitted with an appropriate optical system (lenses). The optical sensor returns a digital string of data representing the signal sampled with the frequency of 48 kHz and resolution reaching a few micrometers in the operating range of 10 mm. The test stand also includes a robotic arm to which the above-mentioned sensor is fixed. The arm enables the full control of motion in three spatial directions.

**Figure 1.** Test stand general view.

**Figure 2.** Diagram of the test stand: 1—vibration exciter; 2—amplifier; 3—digital-to-analog converter; 4—computer; 5—mounting; 6—specimen under investigation; 7—robotic arm; 8—optical sensor (distance measurement); 9—synchronizing signal connection; and 10—laser beam.

The analysed specimen is fixed to the electromagnetic core of the vibration exciter powered by an analogue amplifier. The signal is generated using the first channel of the computer sound card. The second channel is used to generate a rectangular waveform (with the frequency of 48 kHz), which is the synchronizing signal for the optical sensor. In this case, each ascending slope triggers a recording of the observed point distance from the sensor. The entire system is operated, programmed, and controlled in the LabView® environment, including the readout and processing of measurement quantities (LabView® Sound and Vibration Toolkit). After the measurement points and the range of tested frequencies are determined, the measurement is fully automatic. The stand's basic operating mode is vibration scanning of the specimen. This process requires, firstly, an array containing tested frequencies (usually, a range of values with monotonic, arithmetic, or geometric progression is determined) and, secondly, the coordinates defining all points of measurement (packets of three numbers—x,y,z). The number of individual measurements is the product of the number of geometrical locations and the number of scanned frequencies. Such a single measurement consists of placing the sensor in a set location and starting to send a signal with a set frequency to the input of the electromagnetic exciter amplifier, which is thus set in motion. At the same time, the exciter core acts as the tested specimen support. The motion of the support is thus the base excitation with an amplitude and frequency set by the user. After the exciter is activated, it is necessary to bring the oscillating system to a quasi-steady state, which, as demonstrated by the tests, takes a few seconds, depending on the vibration frequency. At this stage, the synchronizing signal is flat—no data are collected. Then, the next stage of collecting data from the optical displacement sensor begins, at which point the synchronizing signal starts to oscillate at 48 kHz. The data collection time should be long enough to capture many full oscillation periods (being the effect of the input function frequency), including potential occurrence of acoustic beats. An appropriately long measurement time, i.e., a period producing an adequately large set of measuring data, enables correct data processing and, first of all, makes it possible to improve the accuracy by averaging data for many periods. Based on the testing results, the duration of the data acquisition stage adopted in this study was one second. After the measurements are completed, the signal forcing the exciter motion and the sensor synchronization is turned off.

The number of collected data defining the measurement point distance from the sensor should be equal to the product of the sampling frequency (48 kHz) and the measuring time. The obtained set of measurement results is then processed. First, the number of collected data is checked. It is also checked whether each data value is included within the measuring range and whether there are no transmission errors. In the next step, the data are filtered using a band-pass filter to capture the one and only frequency-dependent component of the currently tested quantity (equal to the input function supplied to the vibration exciter). Fourier analysis is then conducted and the main frequency is determined, together with its amplitude and phase. It is also confirmed whether it is equal to the input function frequency (with a small error margin taking account of the numerical character of the testing and the finite nature of the data set).

In addition, methods allowing compensation of the elasticity of the specimen holder may be used [45]. The proposed arrangement of the test stand may also be employed for imaging and determining significant characteristics for nonlinear elements [46]. After minor modifications, it can also be applied to investigate complicated models made on a small scale, especially damping in the structure [47].

#### *2.2. Phase Angle Determination Procedure*

The most innovative part of the constructed test stand is the possibility of measuring the phase angle on a test rig fitted with a single measuring sensor. The process makes use of the mutual relative repeatability of the input function signal and of the signal synchronizing the measuring sensor (and also of the signal triggering data acquisition) described above in Section 2.1. Data collection always begins precisely at the same instant of the vibration exciter cycle defined by the base excitation waveform, i.e., the measurement always begins at the same moment of the cycle duration (*t*0), at a set value of the input function frequency. Consequently, resultant signals (the time-dependent phase shift curve) collected at two points, or more specifically, their relative phase shift, represent the real phase shift between oscillating motion at these points at a set frequency. The idea of measurement procedure realization is illustrated in Figure 3. The top and bottom waveforms represent the measurement results for two different points on the specimen. The blue line marks the history of the actual time-dependent displacement of the tested point on the specimen. The red line illustrates the synchronizing signal, the beginning of which is synchronized with the input function—it always appears at the same moment of the cycle, and consequently, at the same position of the exciter core, and hence, at the same position of the entire specimen. The red markings denote collected measurement data, based on which, after they are connected, the phase shift *θ* can be determined. The quantity on the vertical axis in Figure 3 can be expressed as displacement, velocity, or acceleration, which is irrelevant for the phase angle measurement idea.

The phase angle determination method consists of setting the measuring sensor so that the measuring light beam falls onto the tested element. A two-channel signal is generated next, lasting long enough for the element vibrations to stabilize (usually from a fraction of a second to a few seconds). The signal from one of the channels consists of a sinusoidal waveform starting from zero; the other channel consists of a zero curve (no signal) for a set, precisely measured over time (vibration stabilization). A signal is then generated with a frequency from several to a few dozen kHz, determining the data sampling frequency, i.e., the accuracy of frequency and the phase shift measurement by the measuring sensor. The sampling frequency depends on the maximum frequency of the measuring sensor operation—the higher the sampling frequency, the more accurate the history of oscillations and thus phase angle determination accuracy. The procedure is repeated, varying the measuring sensor location so that the light beam falls onto another point on the tested element.

**Figure 3.** Schematic diagram of the phase shift measurement method.

The measurements result in two one-dimensional tables with numerical values, each of which includes recorded measurements. Plotting their (approximated) history in a chart as a function of time makes it possible to draw a comparison between signals from the two points and determine the phase angle shift between them (*θ*) Therefore, the essence of the measurement is the use (by controlling the exciter) of fully repeatable base excitation precisely synchronized in time with data acquisition. The key component here is the tested process repeatability—the same history of the input function force is applied to the same system, which results in identical specimen behavior. The measuring sensor synchronizing signal (the moment of the start of the measurement) always appears at the same instant of the period of the exciter vibrations. This makes it possible to subsequently collect data for any number of measuring sensor locations, with a separate measurement being performed for each of these locations. Combining the data collected in this way enables determination of their relative phase shift characteristics. It is assumed that the material properties do not change during the test.

#### **3. Beam Vibration Model**

The tests presented in the paper were carried out for a prismatic cantilever beam with a uniformly distributed mass fixed on one end and subjected to dynamic input functions (the Euler–Bernoulli model). Because, during the testing, the oscillating motion measurements are only possible in one (vertical) direction (due to the test bench design) and the beam in question is symmetric, the vibrations may be considered one-dimensional. Only oscillations occurring in this vertical direction are taken into account; torsional forms are ignored, even though they are also present, resulting in slight displacements in the analysed direction. Additionally, oscillations due to deviation from the assumed ideal and symmetrical dimensions of the system are also ignored. Under these assumptions, the analysed specimen reference geometry can be simplified. The analysis only covers the change in vibration values along the specimen length, because a change of the measurement coordinate in other spatial directions should not affect the results—the system can be

considered one-dimensional. The oscillating motion direction is perpendicular to the "length" coordinate.

The natural vibration form is determined using a model constructed based on Equation (1). In the determination of the vibration frequency of a prismatic cantilever beam with a constant density along the beam's entire length, the basic assumption is equilibrium between internal forces (the left side of the equation) and inertia (the right side of the equation) for each element along the beam length (*dx*):

$$\frac{\partial V\_{\mathbf{x}}}{\partial \mathbf{x}} d\mathbf{x} = m \frac{\partial^2 y}{\partial t^2} d\mathbf{x},\tag{1}$$

where *Vx* denotes the shearing force; *m* is the linear mass (mass per length); and *x* and *y* are axial and lateral coordinates, respectively.

Considering the relation between the bending moment *Mx* and the shearing force for the prismatic beam,

$$\frac{dM\_x}{dx} = V\_x \tag{2}$$

and taking account of the beam deflection equation

$$\frac{M\_{\mathbf{x}}}{EI} = \frac{\partial^2 y}{\partial \mathbf{x}^{2'}}\tag{3}$$

the fourth-order differential equation with respect to time (*t*) is obtained (4):

$$
\frac{
\partial^2 y
}{
\partial x^2
} + \frac{EI}{m} \frac{
\partial^4 y
}{
\partial t^4
} = 0,
\tag{4}
$$

where *E* and *I* are the elastic modulus and the second moment of the area, respectively. In terms of the considerations presented herein, the essential solutions are only those corresponding to standing waves, i.e., cases where natural vibration forms are created. The solution of Equation (4) can therefore be presented as a function of separable variables a product of two functions where one depends on time and the other only depends on space.

Consequently, vibration forms can be expressed using relation (5), which is a general solution of differential Equation (4) [48]:

$$X(x) = C\_1 \sin(ax) + C\_2 \cos(ax) + C\_3 \sinh(ax) + C\_4 \cosh(ax),\tag{5}$$

where *X* denotes deflection at axial coordinate *x*, and *α* = 4 *mω*2 *EI* . Constants *Cn* are determined using boundary conditions. Since one end of the beam is fixed (*x* = 0), no motion is possible there (neither translation nor rotation occurs).

Moreover, due to free motion of the other end, the forces acting at this point (*x* = *l*) are zero, *l* is the length of the beam. The solution expressing frequencies in this situation is

$$f = \frac{\beta}{I^2} \sqrt{\frac{EI}{m}}.\tag{6}$$

where *β* = 0.560 and *β* = 3.507 for the first and second bending form under analysis, respectively. The analysed forms of the prismatic cantilever beam natural vibration are presented in Figure 4.

**Figure 4.** First (**left**) and second (**right**) vibration form in the direction perpendicular to the beam; top: schematic diagram of vibration forms; bottom: FEM results (blue- smallest amplitude, red biggest amplitude).

#### **4. Experimental Testing**

The tests were performed on specimens in the form of a prismatic cantilever beam with a rectangular cross-section made with Additive Manufacturing technology. The specimens were printed using the Makerbot Replicator 2X device with the following settings: Infill of 100% (printing degree: 45◦, moved with every layer by 90◦) and two shells. The nozzle temperature was 210 ◦C, and the printing proceeded horizontally (the shortest side upwards). The Basf Innofil 3D PLA WHITE filament was used. Several specimens with different dimensions in the form of rectangular prisms were made for the purpose of testing. The ones selected for the measurements were as follows: I—197 × 20 × 6 mm; II—197 × 20 × 4 mm; and III—117 × 20 × 4 mm. The outermost 19-mm part of the specimen was slotted into the fixing system made of two flat steel elements. The specimen was clamped in between using screws and the whole specimen was attached to the electromagnetic exciter core. The fixing system (cf. Figure 2) was a compact structure with a much higher stiffness than the tested specimens. The first step was to establish the specimen's true density using laboratory pan scales. The calculations were performed by adopting the specimen dimensions defined in the geometry source file. The actual measurement of geometry indicated slightly higher values due to the surface texture being an effect of the manufacturing technology (excessive extrusion, material swelling, and excessive looseness). The filament's technical data sheet [49] lists selected data characterizing its raw state and the printed product material. The density is given as 1.26 g/cm3. The measurement results are listed in Table 1. The determined density value resulted from the specimen's geometrical dimensions and mass.

The specimen refers to the tested beam part protruding from the fixture. The first to be tested was the 197 × 20 × 6 mm specimen. It is hereinafter referred to as specimen I. To make a rough estimate of its dynamic characteristic, the specimen was modeled using the Ansys Mechanical package. The model only comprises the part protruding from the support, i.e., the tested dimensions are 178 × 20 × 6 mm. The beam was fixed on the 6 × 20 front plane to obtain the cantilever beam model. The established value of density (1.24 g/cm3) and the elastic modulus value read from the material specification sheet data (2852 MPa) were applied. Because the geometrical dimensions and mass were measured

with the accuracy of 0.1 mm and 0.01 g, respectively, the density determination uncertainty was estimated at the level of 0.03 g/cm3.

**Table 1.** Specimen data.

#### *4.1. Determination of Resonance Frequencies Using the Displacement Amplitude*

Following the procedure described in Section 2.1 above, the tested specimens' displacement amplitude was measured in frequency ranges determined through numerical simulations. The obtained values of frequency (numerical simulation) for subsequent vibration modes are listed in Table 2. The vibration forms for which the motion of every point only occurs in the direction possible to record due to the test bench design (vertical) are marked with an asterisk (\*). This direction remains the same, irrespective of changes in the horizontal coordinate normal to the specimen axis. The selected vibration forms therefore represent motion that can be defined using a single coordinate and are a solution of the Euler–Lagrange equation. Mode 2 (Table 2) represents transverse vibration in the second plane (left-right, not analysed); modes 4, 6, and 8 are related to the rotation of cross-sections (specimen torsion); and mode 9 involves simple compression and axial tension of the specimen (longitudinal wave mode).

In the case of modes used for further analysis (modes 1 and 3 from Table 2), the selection of the Poisson ratio, and consequently, of the shear stiffness (the Kirchhoff modulus), for the analysed geometry does not have an impact on the change in the frequency or form of these natural vibrations (assuming the defined Young modulus).

In the next step, the specimen was attached to the vibration exciter and the abovedescribed (Section 2) vibration scanning was carried out. The location of measurement points on the tested system's geometry is presented in Figure 5. Measuring frequencies were in the range of 43–48 Hz with the scanning step of 0.1 Hz and 284–300 Hz with the scanning step of 0.2 Hz. An ambient temperature was maintained at the level of 23 ± 1 ◦C; air from the surroundings was gently blown onto the specimen by a fan.

Measurements were performed at 10 points located 20 mm apart along the specimen's center line (points 1–10) and at two points on the element holding the specimen (Point 11 and 12). Point 10 on the specimen and Point 11 on the fixture were as close to each other as possible, and close to the fixing element edge (about 2 mm). The measurements performed in the fixing system enabled an indirect assessment of stiffness through observations of the vibration phase angle between Point 11 and 12.

Table 3 presents the results of the displacement amplitude measurement for all 12 points under analysis. The coordinates of the points defined in relation to the fixing screw (D) are given in the second row of the table. It can be seen that the first resonance frequency is included in the investigated range. However, a deeper analysis of the results indicates that precise determination of the resonant frequency is debatable. For the investigated vibration form (mode 1), the highest vibration amplitude occurs at the specimen's

far-end (which is the most distant from the support (Point 1)). The simplest pattern of finding the resonance frequency value (by finding the maximum amplitude in this place) gives the result of 45.5 Hz.

**Figure 5.** Diagram of the measurement: A—tested specimen; B—fixing metal element; C—fixing screws; D—exciter fixing screw; and 1–12—points of measurement.

**Table 3.** Measurement results of specimen I.

Alternative approaches, i.e., finding the maximum vibration ratio between Point 1 and Point 12, 11, and 10, result in the respective values of 44.5, 45.5, and 45.8 Hz (marked). The curves illustrating the vibration forms for these three values are presented in Figure 6.

**Figure 6.** Amplitude at frequencies close to the first resonance frequency as a function of location.

It can be seen that the shape of the forms changes; the nodal point is shifted towards the specimen center with a rise in the base excitation frequency. Negative values represent a shift of 180◦ in relation to the other points (the value is rounded down or up to 0◦ (360◦) or 180◦). Because, under the adopted theoretical assumptions and using the constructed numerical model only the specimen's protruding part is simulated, the most appropriate approach in this situation is to adopt a frequency for which the waveform node is located on the interface between the specimen's tested part and the support as the natural frequency of mode 1. Considering the discrete character of the measurements, it is assumed that the point is located half way between measuring points 10 and 11. It follows from Table 3 that the lowest amplitudes for these two points, which at the same time result in the occurrence of the minimum of the approximating function between them (built on all 12 points of measurement), correspond to 45.6 Hz. This value is adopted as the first resonance frequency adequate for further consideration.

A similar procedure can be applied for the next resonance amplitude. The value determined in this way is 290.6 Hz if the criterion is the maximum amplitude of the end, which is the most distant point from the fixing screw, and 296.2 Hz if the waveform node is assumed on the specimen and fixture interface.

Figure 7 illustrates the natural vibration forms for these frequencies. It should be noted that for the two frequencies, the point for which the approximated amplitude is zero, i.e., the point corresponding to the specimen cross-section that does not move vertically, but only rotates, is located at a position close to *x* = 158 mm.

**Figure 7.** Amplitude at frequencies close to the second resonance frequency as a function of location.

The difference between the node location for the two frequencies (Figure 7) is less than 0.5 mm and, considering the measurement accuracy, is of no practical significance. A higher vibration amplitude for the lower frequency does not necessarily mean a different location of the excited wave nodal point. It may just be due to the way in which energy is supplied to the oscillating system or characteristics of the exciter and the amplifier system. The value of 290.6 Hz was adopted for further consideration.

Figure 8 presents the curves illustrating changes in the beam end amplitude, depending on the frequency in the resonance area of the first two bending modes. It can be observed that the curve representing the second mode is more asymmetric in relation to the extremum. This problem may be due to an unknown relation between the exciting force and frequency. It is unknown because, firstly, the characteristic of the sound card amplification, the amplifier, and the exciter is not perfectly linear, and secondly, because the specimen mass and motion affect the motion of the exciter. This characteristic has been corrected to a certain extent, but only for an exciter without a specimen attached to it. If the specimen is attached, especially if there is motion near resonance, additional inertia forces are introduced into the system. In turn, this has a significant impact on the motion of the exciter core and the forces generated within it. Modeling this phenomenon would require knowledge on the exciter's internal structure (core geometry, number of cores, magnetic and electromagnetic field distribution, etc.) and the amplifier's response to induced currents. The same phenomenon has, among others, an impact on the shape of the curves in Figure 8 and the amplitudes in Figures 6 and 7.

**Figure 8.** Amplitude depending on the vibration frequency for the first (**left**) and second (**right**) vibration form of specimen I.

Apart from the measurements mentioned above, tests on the analysed element free vibration were also performed. For this purpose, the beam was statically bent by about 3.5 mm at its free end. The end was then released and the specimen motion was recorded using the optical displacement sensor employed previously. The sensor was placed at Point 1 (cf. Figure 5). The recorded changes in the vibration signal for the 197 × 20 × 6 mm beam are illustrated in Figure 9. The Fourier analysis performed for this data set indicates the frequency value of 44.77 Hz.

**Figure 9.** Displacement (distance) depending on selected time points for free vibrations with a decreasing amplitude for specimen I.

#### *4.2. Evaluation of Vibration Damping*

The measurement results presented in Figure 9 were used to evaluate vibration damping of the tested specimens. To this end, an algorithm that detects subsequent maxima of the history of changes in values of displacement in subsequent periods and then calculates the ratio between energy (*V*) in the oscillating element in a given cycle and in the cycle preceding it was created. Based on that and using slight transformations, it was possible to determine the loss factor in relation to a single cycle.

$$
\eta = \frac{\Delta V}{V} = \frac{V\_{i-1} - V\_i}{V\_{i-1}} \tag{7}
$$

It was assumed that stored energy is proportional to the square of the amplitude, as in the case of a harmonic oscillator. The obtained results are shown in Figure 10 (left). The same results, but calculated on a different time scale and taking account of the division into values established using minima and maxima (for subsequent periods), are presented in Figure 10 (right). The average value of *η* for the first second of the measurements totals 8.4% for the function maxima shown in Figure 9, and 8.2% for the minima. With time, and as the amplitude decreases, the relative error, and consequently the spread of points around the average value, increases. For this reason, measurements should be limited to the initial phase of oscillations right after they are induced. Figure 10 illustrates changes in the relative energy drop in subsequent cycles.

The presented procedure is equivalent to matching a curve described by equation *Ae*−*ζωnt* to measurement data [50] and establishing a specific damping ratio ζ as a loss factor per radian:

$$
\zeta = \frac{\eta}{4\,\pi}.\tag{8}
$$

Factor 4 results from the fact that, at resonance, the loss factor is twice the loss damping ratio. Loss factor values lower than zero contradict the laws of physics. If such a value is found, it is an effect of measuring errors and indicates that the apparatus has an inadequate accuracy. This is particularly visible and significant at small amplitudes, where the relative error becomes unacceptably large. The red line in Figure 10 represents a straight line matched to measuring data. It can be seen that as the displacement

amplitude decreases, so does the loss factor, which is due to the material's smaller strains. Additionally, the aerodynamic interaction between the oscillating element and the environment decreases substantially.

**Figure 10.** Loss factor depending on time (**left**) and a detailed view of the loss factor, depending on the time (**right**).

Figure 9 also shows the displacement amplitude history of Point 1 (red curve). It can be seen that at a free-end displacement of the order of 3.5 mm, after about 1 s, the amplitude falls below 1 mm. This explains the rising relative error mentioned above. The presented methods deliver loss at every one period, contrary to matching the *Ae*−*ζωnt* curve. This also enables an analysis of the change in the loss factor during each period, together with the change in the amplitude (and time). It has to be mentioned that the samples are collected at a much higher frequency compared to the vibrations. Even in an extreme situation, where the real extremum is exactly in the middle of the collected samples, the deviation of the determined maximum is not bigger than the sensor resolution.

#### *4.3. Phase Angle Analysis*