Estimation of Internal Mechanical Stresses in Insulation Based on Experimental Results

V. S. Dmitrevskii, V. F. Grafov, V. M. Taiurskii

(Presented by the scientific seminar of the Department of Electrical Insulation and Cable Engineering)

In polymer insulation of electrical-engineering structures, internal mechanical stresses arise in the course of its manufacture (for example, orientational stresses), and also due to differences in the coefficients of linear expansion of the polymer and the insulated elements. The presence of internal mechanical stresses leads to the formation of microcracks in the material. The time to breakdown of insulation that contains microcracks in its volume is reduced.

It is shown in [1] that annealing of polyethylene leads to an increase in the lifetime under impulse voltage. A study of the long-term electrical strength of cables with rubber insulation [2] under alternating voltage also showed that the action of mechanical loads sharply reduces the time to breakdown of the insulation.

To determine the magnitude of internal mechanical stresses, various methods are given in the literature: optical, cantilever [3], photoelasticity [4] (the most accurate, but requiring rather complex apparatus), and others. Yu. S. Pervushin [5] solved the problem of determining internal stresses in cast insulation of compounded cylindrical electrical elements.

The purpose of the present work is a quantitative estimate of internal mechanical stresses in electrical-insulation structures from the results of testing them for service life. The solution of the problem posed is based on the equation of the “life curve” given in [6]:

\[ \tau = \tau_0 \exp \left\{ \frac{ D - \sqrt{(Ae^{-bT}\beta fE)^2 + (\gamma\sigma)^2} \ln \frac{2De}{\sqrt{(Ae^{-bT}\beta fE)^2 + (\gamma\sigma)^2}} }{ 2KT } \right\}, \tag{1} \]

where
\(D\) is the energy of rupture (dissociation) of a chemical bond;
\(A, b, \gamma\) are constants determined by the structure and properties of the material;
\(2KT\) is the average energy of two atoms forming one chemical bond;
\(\beta\) is a coefficient taking into account the concentration of electric field strengths due to inhomogeneities in the structure of the material;
\(\tau_0\) is the relaxation time of the vibrations of an atom in a molecule, equal to \(10^{-13}\) sec;
\(f\) is the coefficient of inhomogeneity of the electric field;

\(\sigma, E\) are the mechanical stress and the electric field strength;

\(\tau\) is the time to breakdown of the insulation.

We determine the parameters of equation (1) on the basis of experimental data on failures of three batches of specimens under the action of an electric field with strengths \(E_1\), \(E_2\), and \(E_3\) at normal temperature. The value of the bond-rupture energy was determined from the equation

\[ D = 2KT\, \frac{ \left(\ln \frac{E_2}{E_1}\right)\ln \frac{\tau_3}{\tau_0} + \frac{E_3}{E_2}\left(\ln \frac{E_1}{E_3}\right)\ln \frac{\tau_2}{\tau_0} + \frac{E_3}{E_1}\left(\ln \frac{E_3}{E_2}\right)\ln \frac{\tau_1}{\tau_0} }{ \ln \frac{E_2}{E_1} + \frac{E_3}{E_2}\ln \frac{E_1}{E_3} + \frac{E_3}{E_1}\ln \frac{E_3}{E_2} }. \tag{2} \]

Expression (2) was obtained under the assumption that

\[ \sqrt{(Ae^{-bT}\beta f)^2+\left(\frac{\gamma\sigma}{E}\right)^2} \]

remains constant in the investigated range of electric-field strengths. Indeed, as analysis showed, the error in this case does not exceed 13.5%. We reduce equation (1) to the form

\[ BE\ln \frac{2e}{BE}=1-\frac{2KT}{D}\ln \frac{\tau}{\tau_0}. \tag{3} \]

\[ B^2=B_0^2+\left(\frac{\gamma\sigma}{D}\right)^2\cdot \frac{1}{E^2}. \tag{4} \]

\[ B_0=\frac{Ae^{-bT}\beta f}{D}. \tag{5} \]

From equation (3), by the method of iterations, we determine \(B\) for each probability of insulation breakdown, and from the mean value \(\overline{B}\) we construct the dependence \(\overline{B}^2=f\left(\frac{1}{E^2}\right)\). From the straight line obtained it is not difficult to determine \(Ae^{-bT}\beta f\) and \((\gamma\sigma)\).

The tests were carried out on specimens (Table 1) of EZK-10 epoxy compound and polyethylene terephthalate film (PETF) under prolonged exposure to alternating voltage of 400 and 50 Hz, respectively.

Table 1

The specimens were tested in batches of 50 each. The loading regimes are given in Table 2.

The results of calculations of the parameters of equation (1) by the above-described procedure are presented in Fig. 1 and in Table 3. As can be seen from Fig. 1, the values of \(\overline{B}^2\) fit well on a straight line.

Table 2

Table 3

For most polymer dielectrics \(\gamma\) is equal to \(10^{-27}\ \dfrac{\text{J}}{\text{bond}}\cdot\dfrac{\text{m}^2}{\text{N}}\).

Apparently, for epoxy compounds and PETF, \(\gamma\) will be close to the indicated value. For an exact determination of \(\gamma\), special experiments on mechanical testing of the materials must be carried out. The obtained values of the internal mechanical stresses are comparable with the value \(\sigma_{\text{rupt.}}\) of the materials.

It is known that relaxation of internal mechanical stresses occurs during storage. In this connection, we tested an additional batch of specimens made of epoxy compound that had been stored at room temperature for 2.5 months after manufacture. The tests were carried out at an electric-field strength of \(48 \cdot 10^6\) V/m. Assuming that \(B_0\) does not change during storage, we obtain \(\gamma \sigma_{\text{st}} = 2.92 \cdot 10^{20}\) J/bond, and \(\sigma_{\text{st}} = 2.92 \cdot 10^7\) N/m\(^2\). The experiment performed indicates relaxation of the internal mechanical stresses with time.

Fig. 1. Dependence \(\overline{B}^{\,2} = f\left(\dfrac{1}{E^2}\right)\) for epoxy compound EZK-10 (1) and PETF film (2)

Conclusions

1. A method is proposed for an approximate estimate of internal mechanical stresses in polymer insulation from the results of testing it for service life.

2. The estimated calculated values of \(\sigma\) for the epoxy compound EZK-10 and PETF are comparable with the value \(\sigma_{\text{rupt.}}\) of the material.

3. During prolonged storage of epoxy-compound specimens, a decrease in \(\sigma\) occurs.

References

1. V. S. Dmitrevskii, V. F. Vazhov, I. I. Skvirskaya, V. G. Sotnikov. Effect of annealing of polyethylene specimens on the number of pulses to breakdown. — In the collection: “EP. Electrotechnical Materials,” division of VNII EM (Informelektro), Moscow, 1972 (in press).

2. V. S. Dmitrevskii, V. A. Stryzhkov. Combined effect of electric stress and static deformations on the reliability of the insulation of a flexible cable of the KShVG type. — Collection of papers of NII KP, issue 1, “Energiya,” 1969.

3. A. T. Sanzharovskii. Internal stresses in coatings. — “High-Molecular Compounds,” 1960, No. 11.

4. M. Frocht. Photoelasticity, Moscow, 1948.

5. Yu. S. Pervushin. On the determination of internal stresses in cast epoxy insulation of compounded electrical elements of cylindrical shape. — Collection of papers of the Ufa Aviation Institute, 1970, issue 15, pp. 95–101.

6. A. A. Vorob’ev, V. F. Grafov, V. S. Dmitrevskii, V. A. Lkhamazhapov, V. I. Merkulov. Aging of solid dielectrics and reliability of electrical insulation. Proceedings of the meeting of the IV Scientific Council of the Academy of Sciences of the USSR on theoretical and electrophysical problems of electric power engineering, Novosibirsk, 1970.