Chosen Mechanical Properties of Electric Tape Fences

The quasi-static tensile deformation test or strip test was used for the determination of the breaking force and elongation of the electric tape fence samples and detailed survey of the load–extension behavior of the samples. The two types of the load–extension behavior along all nine samples can be distinguished. In the first case, the stress–strain curve is relatively smooth, indicating that both components (i.e., polymer MFs and metal wires) work similarly in terms of mechanical properties. This performance was observed for sample nos. 1, 2, 3, 4, 5, 7, and 9. In Fig. 5a, the stress–strain graphs for sample nos. 1, 3, and 5 are shown to demonstrate this type of behavior. A characteristic change can be observed between the elastic and plastic deformation, where the point is called the plasticity limit (marked by circle) and reversible deformation changes to the irreversible deformation beyond this point. The rupture of the first metal wire (marked by square) can be observed, which does not significantly affect the test because both components have similar rheological and mechanical properties. Thus, the metal component did not carry any significant force in this case. In the second case (sample nos. 6 and 8; Fig. 5b), and the load–extension curve contains a significant peak representing the rupture of a higher number of metallic conductors or the rupture of a significantly stronger metal component (sample no. 6) compared to other samples. In this case, the metal component transmits force during the initial loading of the sample accompanied by a steep increase, followed by a sharp drop of force and subsequent slight increase until the break.

Load–displacement graphs for a sample nos. 1, 3, and 5 and b sample nos. 6 and 8

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In Fig. 6, a comparison of the samples’ mean values of the breaking load and elongation together with 95% confidence intervals of means is displayed. As expected, the number of MFs, their diameters, and the width of the samples determine the breaking load of the samples. For example, sample no. 6, with the highest width w = 42 mm and total MF cross-sectional area S ÷ 8 mm2, reaches the highest breaking load F = 2769 N and the lowest elongation l/l0 = 14%.

Bar plots of mean values of the a breaking load [N] and b elongation [%] together with 95% confidence intervals of means for all electric tape fence samples

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This phenomenon was confirmed using multiple linear regression and hypothesis tests (F test and t test). The only statistically significant variable affecting the sample breaking load is the total MF cross-sectional area A [mm2], and a significant linear relationship exists between the response F and predictor variable A (the R2 value of 0.95 is close to 1, and the p value of 1.08E-5 is less than the significance level of 0.05). For further details, see Fig. 7.

Dependence of the breaking load [N] on the total MF cross-sectional area [mm2] for all electric tape fence samples

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As described above, plasticity limits (P) having coordinates [lP, FP] and stiffness coefficients (k1, k2) were determined for the three representative samples chosen from the electric tape fence sample set. The mean values together with the 95% confidence intervals of the means (in brackets) are shown in Table 3. The results show that the stiffness coefficients are directly connected with the breaking load of the samples. The plasticity limit P is a very interesting change point when observing the mechanical behavior of the product. For displacement l ≤ lP, elastic (or viscoelastic) deformation occurs, which means that the material comes back to its original size and shape when the load is no longer present. By contrast, for l > lP, the material no longer behaves elastically/viscoelastically, but it becomes permanently deformed. In the quasi-static survey, there is no statistically significant difference found between lP for all the three samples, whereas FP differs depending on the deformability and strength of the sample.

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The dynamic tensile deformation test provided information on the viscous and plastic behaviors of the electric fence tape samples. As shown in Fig. 5b, where the typical load–displacement curve is shown, the viscous behavior of the samples was confirmed by the fact that there is a relaxation of force in points R1 and R2 (R1’, R2’) and the fact that the load–displacement curve has a hysteresis character because of energy dissipation. The plastic behavior was proven by the fact that points C and C’ do not lie in the origin of the coordinate system.

The two-sample t test was used to check whether the deformation rate has a statistically significant effect on the mechanical behavior of the electric fence samples (H0: R1 = R1’, R2 = R2’, C = C’). The mean values and standard deviations of force F in points R1 and R2 for both rates are shown in Table 4, accompanied with the test statistics (p value). The rates (ranging from 100 to 200 mm/min) of the clamps have no statistically significant effect on the force in point R1 (R1’) because the p values are greater than 0.05. Unfortunately, the device does not offer higher rates. The effect of different deformation rates was confirmed for samples 1 and 5, where the force in the point R2 (R2’) was examined. A lower force was observed using a higher deformation rate for sample 1, which was not expected. This is probably because the fact that the time elapsed between points R1 and R2 was not enough to restore the state of tension. Sample 5 seems to show the greatest difference in the force behavior from the whole sample set and is the most sensitive to the change in rate. When compared with the quasi-static deformation test (above) and based on the relaxation tests (below), a viscous component in the rheological model of the electric fences is ultimately present.

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Table 4 displays the mean values of elongations and the standard deviations and results of the t tests (p value). This investigation also shows that there is no significance difference between elongations at points C and C’, and therefore, the statistically significant effect of the deformation rate on the plastic behavior of the electric tape fence was not confirmed. Thus, it is necessary to emphasize that the results of this study are limited by the use of only two speeds of clamps.

The relaxation curve represents another output of the dynamic tensile deformation test typically with a shape of an exponentially decreasing function at R1 (R1’) and R2 (R2’) points. This function can be approximated by the function F = F (t) (see Eq. (7)). The results are presented in Table 5, where the mean values of force constants FA, FB, and FC [N] and relaxation times τ3 and τ4 [s] and the standard deviations for sample no. 1 are shown. Moreover, in this case, the effect of the deformation rate on all the constants of function F = F (t) was examined using the two-sample t test for equal means. The results show that the different rates (in the range of 100 to 200 mm/min) of the clamps have no statistically significant effect on all the constants of all relaxation curves (points R1, R2; R1’, and R2’) of all the samples (sample nos. 1, 3, and 5) because all the p values are greater than 0.05. However, this difference would be significant over a wider range of deformation rates. The MSE values (see Table 5) ranged between relatively small values from 0.1 to 0.4 N for all fence samples, which means that the model effectively explains a given set of observation. In Fig. 8, the measured relaxation curve and approximated relaxation curve for one of the dynamic tensile deformation measurements of sample no. 1 in the R1 point are compared. The results clearly show that the model with two viscous components seems to be suitable, which is also confirmed by the MSE value of 0.21 N.

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Dependence of force on the time–measured relaxation curve and approximated relaxation curve for one of the dynamic tensile deformation measurements of sample no. 1 in the R1 point

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A general rheological model of electric fence was proposed according to the undertaken quasi-static tensile tests and stress–relaxation tests (see Fig. 4d). The spring element describing the elastic behavior and obeying Hooke’s law presents the relationship between a force and deformation. The dash-pot element describing the plastic behavior presents the relationship between a force and the velocity of the deformation. The sliding element as a slip function simply keeps the magnitude of the force below a given threshold value [15]. Here, ki is the stiffness coefficient of the spring elements, τi is the relaxation time of the dash-pot elements, and FP is a sliding force of the sliding (frictional or St. Venant) element. As is apparent from the figure, the proposed rheological model is comprised of spring elements, dash-pot elements, and a sliding element. These elements are used to allow the elastic deformation, viscous deformation, and internal changes in the bodies. The proposed model represents two parts: (a) elasto-plastic and (b) viscous. The elasto-plastic part consists of two spring elements and one sliding element, with the spring elements being connected in parallel. The viscous part consists of two spring elements and two dash-pot elements, with the spring elements being connected in parallel. The use of individual components in the model is in a good agreement with models published for plain woven fabrics made of synthetic materials [7, 16, 17].

This rheological model can be generally described by a second-order differential Eq. (8) using Eq. (3) or (4), which can then be overwritten in a stress–strain configuration (Eq. 9).

$$\ddot{F}+\left({\alpha }_{3}+{\alpha }_{4}\right)\dot{F}+{\alpha }_{3}{\alpha }_{4}F=\left({k}_{12}+{k}_{3}+{k}_{4}\right)\ddot{l}+\left[{k}_{12}\left({\alpha }_{3}+{\alpha }_{4}\right)+{k}_{3}{\alpha }_{4}+{k}_{4}{\alpha }_{3}\right]\dot{l}+{k}_{12}{\alpha }_{3}{\alpha }_{4}l+{C}_{0}{\alpha }_{3}{\alpha }_{4}$$

(8)

$$\ddot{\sigma }+\left({\alpha }_{3}+{\alpha }_{4}\right)\dot{\sigma }+{\alpha }_{3}{\alpha }_{4}\sigma =\left({E}_{12}+{E}_{3}+{E}_{4}\right)\ddot{\varepsilon }+\left[{E}_{12}\left({\alpha }_{3}+{\alpha }_{4}\right)+{E}_{3}{\alpha }_{4}+{E}_{4}{\alpha }_{3}\right]\dot{\varepsilon }+{E}_{12}{\alpha }_{3}{\alpha }_{4}\varepsilon +{C}_{0}{\alpha }_{3}{\alpha }_{4}$$

(9)

where \({k}_{12}={k}_{1}+{k}_{2}, {C}_{0}=0\) for \(l \le l_{P}\) and \({k}_{12}={k}_{1}, {C}_{0}={F}_{P}-{k}_{1}{l}_{P}\) for \(l > l_{P}\), \({E}_{12}={E}_{1}+{E}_{2}, {C}_{0}=0\) for \(l \le l_{P}\) and \({E}_{12}={E}_{1}, {C}_{0}={\sigma }_{P}-{E}_{1}{\varepsilon }_{P}\) for \(l > l_{P}\),

$${\alpha }_{i}=\frac{1}{{\tau }_{i}}.$$

The special shapes of the differential Eq. (8) can be analyzed, which correspond to the experiments performed above, i.e., the quasi-static tensile deformation and dynamic tensile deformation test. A static problem is a trivial case, where all time derivatives of both variables are zero. Equation (8), thus, changes to another form (i.e., Eq. (5) or (6)) depending on the displacement. Thus, the phenomenological constants k1, k2, and FP, and lP, can be determined by static and quasi-static deformation, respectively. The analysis of the time-dependent force on a fixed displacement lR, that is, the analysis of the relaxation test is the second case. F = F (t, lR), and Eq. (8) transforms into Eq. (10).

$$\ddot{F}+\left({\alpha }_{3}+{\alpha }_{4}\right)\dot{F}+{\alpha }_{3}{\alpha }_{4}F={k}_{12}{\alpha }_{3}{\alpha }_{4}{l}_{R}+{C}_{0}{\alpha }_{3}{\alpha }_{4}$$

(10)

On the right side of Eq. (10), there are time-independent constants. It is a second-order linear differential equation with constant coefficients and a special right side. Its general solution (Eq. 11) is given by the sum of the general solutions of the homogeneous equation (with zero in the right side) and the particular solution of the inhomogeneous equation.

$$F\left(t,{l}_{R}\right)={F}_{A}{e}^{-{\alpha }_{3}t}+{F}_{B}{e}^{-{\alpha }_{4}t}+{k}_{12}{l}_{R}+{C}_{0}$$

(11)

Equation (11) corresponds to the approximation function (Eq. 7) of the relaxation characteristics of the electric fence. The coefficients FA and FB (Eq. 11) have integration constants. The last two terms of Eq. (11) represent the constant FC of Eq. 7, giving it a clear physical meaning. It is the force generated in the first two branches of the rheological model (Fig. 5d). If the relaxation test lasts long enough, then Eq. (11) will transform into Eq. (12), which corresponds to relations (5) for \(l = l_{R} \le l_{P}\) and (6) for \(l = l_{R} > l_{P}\).

$$F\left( {t \to \infty ,l_{R} } \right) = k_{12} l_{R} + C_{0}$$

(12)

Using the constants FC, lR and with the knowledge of [lP, FP], the phenomenological constants k1, k2 can be determined using two relaxation tests (\(l_{R1} \le l_{P}\), \(l_{R2} > l_{P}\)). This knowledge can also be used to partially check whether the rheological model corresponds to experimental results. The values from Tables 3 and 4 can be substituted in the right sides of Eqs. (5) and (6). For sample 1, the forces FR1 = 120 N and FR2 = 433 N are obtained. These results agree well with the FC values within the confidence intervals. The displacement rates are shown in Table 5.

Attention remains to be paid on the integration constants FA and FB (Eq. 11) as their size will depend on the nature of the stress definition by which the displacement reaches the point lR, where the relaxation test will take place. If the deformation rate is large enough in the interval [0, lR], i.e., the strain time in this section for l will be an order of magnitude shorter than the respective relaxation times τ3, τ4, then the influence of viscous members in the rheological model can be neglected. The so-called blocking (freezing) of the damper will occur. In this case, the constants k3 and k4 can be directly determined from FA and FB by simply dividing them by the respective displacement lR. By doing this, the procedure for determining all parameters of the rheological model can be completed. It should be mentioned that the constants k3 and k4 were not calculated in this study unfortunately due to not meeting the technical condition of ensuring sufficiently large deformation rate during the experiment.