**3. Results**

As a result, the 'load-elongation' curves were obtained for samples of the Al-Mg alloy at different values of stiffness of the loading system: 120 MN/m (Figure 4), 50 MN/m (Figure 5), 18 MN/m (Figure 6), and 5 MN/m (Figure 7). To demonstrate the jerky type flow, enlarged fragments of these diagrams are additionally shown. The curves are plotted according to the built-in sensor of the testing system.

For the values of the loading system's stiffness of 120 MN/m (Figure 4b) and 50 MN/m (Figure 5b), the loading diagrams show single load drops (or 'teeth'), which are maintained by the initiation and propagation of a single PLC band. Up to a certain load level, uniform deformation of the sample is observed, which corresponds to a smooth section of elastic deformation and the initial stage of hardening. At the stage of supercritical deformation, the diagram is characterized by the presence of a large number of closely spaced teeth (Figures 4b and 5b).

A decrease in the level of the loading system's stiffness in relation to the sample leads to a change in the type of jerky flow of the Al-Mg alloy and an increase in the frequency and amplitude of the load jumps. Figure 6 shows the loading curve for a sample installed in a special gripping device, which realizes a stiffness of 18 MN/m. At a minimum stiffness level of 5 MN/m, we noted a decrease in the frequency of jumps on the diagram but an increase in their amplitude (Figure 7). An increase in the load leads to an increase in the amplitude of the load disruptions (Figure 7b).

**Figure 4.** The load-displacement curve for the loading system stiffness of 120 MN/m (**a**) and enlarged sections (blue boxes) of the curve illustrating the discontinuity of the plastic flow (**b**).

To determine the level of critical deformation (εcr), at which the effect of the jerky flow begins to appear, deformation diagrams were constructed by using an additional software module of the video system called the 'virtual' extensometer (Figure 8a). The 'virtual' extensometer allows for the non-contact determination of the deformation in the gauge length of the specimen based on the displacement fields' analysis. Thus, the 'stress-strain' curves for each group of the samples were obtained. As an example, Figure 8b shows a deformation diagram for a specimen tested at the maximum level of the loading system (120 MN/m).

**Figure 5.** *Cont*.

**Figure 5.** The load-displacement curve for the loading system's stiffness of 50 MN/m (**a**) and enlarged sections of the curve illustrating the discontinuity of the plastic flow (**b**).

**Figure 6.** The load-displacement curve for the loading system's stiffness of 18 MN/m (**a**) and the enlarged section (blue box) of the curve illustrating the discontinuity of the plastic flow (**b**).

**Figure 7.** The load-displacement curve for the loading system's stiffness of 5 MN/m (**a**) and the enlarged section (blue box) of the curve illustrating the discontinuity of the plastic flow (**b**).

This figure shows the moment of the onset of unstable plastic deformation at a result of the initiation of the effect of the jerky flow of the material. The level of critical deformation of the onset of the PLC effect is 2.36%. The εcr values for different levels of stiffness of the loading system are shown in Table 3.

**Figure 8.** The location of the 'virtual extensometer' of the *Vic-3D* measurement system (**a**); the stress-strain curve in the case of the loading system's stiffness of 120 MN/m (**b**).

**Table 3.** Influence of the loading system's stiffness on the critical strain value at the onset of the PLC effect.


## *Spatial-Time Inhomogeneity Depending on the Stiffness of Loading System*

The analysis of the evolution of inhomogeneous deformation fields and local rates of longitudinal deformation is carried out to study the kinetics of the development of the macroscopic localization of the plastic flow depending on the loading system's stiffness. At the moment of the breakdown of the load on the diagram, a sharp localization of the plastic flow occurs in the sample, and the front of the PLC strip is formed. Depending on the type of jerky flow (type A, B, or C), the deformation band begins to either move uniformly along the length of the sample (in the case of the discontinuous flow according to type A). A random occurrence of PLC bands is observed in the sample, causing frequent drops in the load of small amplitude (type B) or drops in the load of large amplitude (type C).

To visualize the formation and propagation of PLC bands, the authors proposed the presentation of the results in the form of a series of deformation profiles (ε*yy*) plotted along the length of the sample at equal time intervals. With the help of this representation, it is possible to characterize the basic laws and kinetics of the spatial-time inhomogeneous fields. As an example, Figure 9 shows a series of ε*yy* profiles for a specimen tested at the loading system stiffness of 120 MN/m. The time interval between the deformation profiles is 2 s. The origin of coordinates for the abscissa axis (Oy) is fixed in the center of the gauge length of the specimen. Under uniaxial tension of the aluminum-magnesium alloy, the effect of quasi-periodic homogenization of plastic deformation is observed, which consists of alternating stages of localization of the plastic flow when PLC bands appear and propagate, as well as stages of macroscopic leveling of longitudinal deformations on the surface of the cylindrical sample.

For the values of the loading system's stiffness of 120 MN/m and 50 MN/m, the jerky flow of type A is found. Figure 10 shows the development of longitudinal deformation fields at a stiffness of 120 MN/m, illustrating the formation and development of a single PLC band (marked with a white arrow).

**Figure 9.** Series of plots of longitudinal deformations plotted at equal time intervals along the specimen length in the case of a stiffness of 120 MN/m.

We consider a series of longitudinal strain profiles (Figure 11) for a single strain band shown in Figure 9. Numbers 1–6 (Figure 11) denote the profiles of longitudinal deformations, for which Figure 10 shows the corresponding strain fields. The propagation of a single strain band proceeds uniformly from one capture to the opposite one with a constant speed of the band's front movement.

**Figure 10.** Evolution of the longitudinal strain fields on the surface of an Al-Mg alloy specimen in the case of a stiffness of 120 MN/m.

**Figure 11.** Series of plots of longitudinal deformations plotted at equal time intervals along the specimen length during the single PLC band propagation.

The deformation of the material is interrupted after the crossing of the front of a single PLC band. The process of active plastic deformation is concentrated in a small area, in the area of the strain band. To illustrate this process, data on the evolution of local rates of longitudinal strain (dε*yy*) are given (Figure 12).

**Figure 12.** Evolution of local rates of the longitudinal strain fields on the surface of an Al-Mg alloy specimen in the case of a stiffness of 120 MN/m, numbers 1-6 corresponded to the plots of longitudinal deformations in the Figure 11.

In Figure 12, the white arrow indicates the direction of movement of the PLC band. Based on the obtained data, a series of profiles of local rates of longitudinal strain with equal time intervals was plotted as well (Figure 13).

**Figure 13.** Series of plots of the local rate of longitudinal strain plotted at equal time intervals along the specimen length in the case of 120 MN/m.

Figure 14 shows the change in the maximum and minimum local rates of longitudinal deformation depending on the frame number. The values of the rates are given in the interval of frames during which the initiation and propagation of the PLC band occurred (Figure 12). At the moment when a single band occurs, the deformation rate of the material in the region of the band is 0.66%/s, while in the rest of the sample, elastic unloading is recorded (negative values of the strain rates). The formed strain band flows at a constant rate of deformation of the order of 0.32%/s.

**Figure 14.** Frame dependence of maximum and minimum values of the local rate of longitudinal strain during the initiation and propagation of the single PLC band in the case of 120 MN/m.

A similar analysis of the spatio-temporal inhomogeneity and influence of the loading system's stiffness on the kinetics of PLC bands was carried out for the remaining samples from the test program presented in the work. When the stiffness is 50 MN/m, a series of longitudinal strain profiles (Figure 15) and a graph of maximum/minimum local rates of longitudinal strain (Figure 16) are similar to the results obtained when the stiffness is 120 MN/m. In the course of uniaxial tension, the manifestation of discontinuous flow type A is observed. At the moment when the single strain band occurs, the strain rate of the material in the region of the band front is slightly higher and amounts to about 0.74%/s. The advance of the formed band proceeds at a constant strain rate of 0.36%/s.

**Figure 15.** Series of plots of longitudinal deformations at equal time intervals along the specimen length in the case of stiffness 50 MN/m.

**Figure 16.** Frame dependence of the maximum and minimum values of the local rate of longitudinal strain during the initiation and propagation of the single PLC band in the case of 50 MN/m.

The reduction of the loading system's stiffness to 18 MN/m significantly affects the jerky flow. The frequency of load drops and their amplitude are higher. The effect of quasi-periodic homogenization of the plastic flow becomes less pronounced (Figure 17). The discontinuous flow type is mixed (A + B).

At times on the sample surface, the propagation of the single bands of the localized plastic flow can be recorded (Figure 18). Nevertheless, unlike the previous values of stiffness (120 MN/m and 50 MN/m), the movement of the strip along the length of the sample takes place at different speeds. It is of interest that the profiles designated by numbers 1–6 in Figure 19 bring inhomogeneous fields of local rates of longitudinal deformation (Figure 20) in order to estimate at what rate the material is deformed at the front of the band.

The deformation rate at the front of the strain band increases and takes values in the range from 0.39%/s to 1.33%/s (Figure 20). The deformation rate of the material when the PLC band appears exceeds the applied rate in the test (0.67%/s) due to an increase in the compliance of the loading system. Consequently, elastic unloading of the peripheral regions of the sample is observed at a rate of −0.14%/s.

**Figure 17.** Series of plots of longitudinal deformations at equal time intervals along the specimen length in the case of a stiffness of 18 MN/m.

**Figure 18.** Series of plots of longitudinal deformations at equal time intervals along the specimen length during the single PLC band propagation.

**Figure 19.** Evolution of local rates of the longitudinal strain fields on the surface of Al-Mg alloy specimen in case of stiffness 50 MN/m, numbers 1-6 corresponded to the plots of longitudinal deformations in the Figure 18.

**Figure 20.** Frame dependence of maximum and minimum values of the local rate of the longitudinal strain during the initiation and propagation of the single PLC band in the case of 18 MN/m.

At the minimum level of stiffness of the loading system, implemented in this work, RLS = 5 MN/m, load drops of a large amplitude are observed in the diagram, which corresponds to type C discontinuous flow. Figure 21 shows a series of longitudinal deformation profiles illustrating the kinetics of PLC bands. It should be noted that due to the high compliance of the loading system at the moment when the next PLC strip appears, a significant drop in the load is observed (Figure 7). Strain bands are formed randomly on the sample surface.

**Figure 21.** Series of plots of longitudinal deformations plotted at equal time intervals along the specimen length in the case of a stiffness of 5 MN/m.

For a more detailed analysis of the kinetics of the discontinuous flow at stiffness of 5 MN/n, a series of longitudinal deformation profiles was derived for a part of the loading (Figure 22). In this case, PLC stripes appear in the sample, but do not advance along the length. On the graph 'local rate of longitudinal deformation—frame' (Figure 23), one can distinguish a rather high intensity of the plastic flow at the front of the strip at the moment of its formation; values of the order of 3%/s are recorded.

**Figure 22.** Series of plots of longitudinal deformations plotted at equal time intervals along the specimen length during the several PLC bands propagation.

**Figure 23.** Frame dependence of maximum and minimum values of the local rate of longitudinal strain during the initiation and propagation of the single PLC band in the case of 5 MN/m.

#### **4. Discussion**

To estimate the degree of inhomogeneity of the process of macro-localization of the plastic flow under the Portevin-Le Chatelier effect, the coefficient of inhomogeneity of plastic deformation of the material (kPLC) during loading is considered. The coefficient is equal to the ratio of the maximum value of longitudinal deformation (ε*yy* (max)) to the average value of deformations (ε*yy* (mean)) for each frame recorded by the video system (1):

$$k\_{\rm PLC} = \varepsilon\_{yy} (\text{max}) / \varepsilon\_{yy} (\text{mean}),\tag{1}$$

Based on the analysis of the evolution of inhomogeneous fields of longitudinal deformations, the time dependence of the kPLC coefficient for the stage of material hardening, which is characterized by the presence of the effect of intermittent flow of the material, was constructed (Figure 24). For the stage of tooth formation and yield area, as well as for the stage of material softening, the change in kPLC was not considered.

**Figure 24.** Time dependence of the coefficient of inhomogeneity of plastic deformation due to the Portevin-Le Chatelier effect in the case of the loading system's stiffness: 120 MN/m (**a**), 50 MN/m (**b**), 18 MN/m (**c**), and 5 MN/m (**d**), blue dotted line corresponds to beginning and ending of the manifestation of the PLC effect.

#### **5. Conclusions**

Thus, this work shows the high efficiency of using the specialized equipment to reduce the stiffness of the loading system in order to experimentally study the spatial-time inhomogeneity of the plastic flow of the aluminum-magnesium alloy. Mechanical tests for uniaxial tension of standard cylindrical specimens in the range with the stiffness of the loading system equal to 120 MN/m, 50 MN/m, 18 MN/m, and 5 MN/m have been implemented. We carried out the analysis of the kinetics of the occurrence and propagation of the deformation bands of the localized plastic flow under the Portevin-Le Chatelier effect using the digital image correlation method. Experimental data have been obtained that illustrate how the properties of the loading system affect the unstable plastic flow.

It is shown that despite the constant chemical composition of the material, external loading factors (temperature, stretching rate), the type of discontinuous yield depends on the value of the loading chain compliance. At higher values of the loading system's stiffness (120 MN/m and 50 MN/m), the initiation of single PLC bands and their uniform distribution along the length of the sample (type A) are recorded on the material surface. With an increase in the compliance of the loading system, the intermittent fluidity passes to types B and C. On the loading curves, the amplitude of the load disruptions increases, which is accompanied by plastic deformation of the material in the area of the front of the PLC bands and the unloading of elastic deformations in the rest of the sample's volume (outside the front of the strip). Reducing the stiffness of the loading system leads to multiple

increases in the maximum values of the local rate of longitudinal strain during the initiation and propagation of the single PLC band. At high loading stiffness (120 MN/m), the maximum values of the local rate of longitudinal strain during the initiation of the single PLC band were 0.69%/s, at a loading stiffness of 50 MN/m–0.76%/s, at a loading stiffness of 18 MN/m–1.33%/s, and at a minimum loading stiffness (5 MN/m)–3.11%/s. In this case, the minimum values of the local rate of longitudinal strain during the propagation of the single PLC band decreased from 0.39%/s to 0.1%/s. These results are consistent with the experimental stretching diagrams obtained. Thus, at the maximum loading stiffness (120 MN/m), the diagrams show extended sections between single teeth, which correspond to the stage of propagation of the single PLC band. At the minimum loading stiffness (5 MN/m), the diagrams show successive teeth that correspond to the initiation of the single PLC band. This kind of research is relevant and requires further comprehensive theoretical and experimental studies and the implementation of a wider range of properties of the loading system.

The practical significance of the work is determined by the fact that the manifestation of the effects of intermittent flow in Al-Mg alloys leads to a significant decrease in the strength and plasticity of material in structures. The processes of strip formation and spontaneous macroscopic localization of plastic flow lead to the appearance of different thicknesses of structural elements, a decrease in the surface quality of parts, the appearance of concentrators, defects, and, as a result, to subsequent destruction or violation of the structural strength of parts. The study of the influence of the stiffness of the loading system on the of the Portevin-Le Chatelier effects has not been previously considered. The presented work shows that the stiffness of the loading system, as well as other parameters (deformation rate, temperature, chemical composition), has a direct effect on the kinetics of initiation and propagation of PLC deformation bands and the type of intermittent fluidity. Therefore, the loading stiffness must be considered along with other parameters when designing structures and planning experimental research programs that use materials that exhibit of the Portevin-Le Chatelier effects.

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

**Funding:** This research was funded by the Russian Science Foundation (No. 20-79-10235).

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The work was carried out by using the capabilities of a large unique scientific facility (UNU) 'A set of test and diagnostic equipment for studying the properties of structural and functional materials under complex thermomechanical influences', which is part of the Center of Experimental Mechanics, PNRPU (Perm, Russia).

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

#### **References**


**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
