2020 ASAP

INEOS OPEN, 2020, 3 (X), XX–XX Journal of Nesmeyanov Institute of Organoelement Compounds Download PDF


Time Dependences of the Resistance Coefficients
of Polymeric Materials
Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28, Moscow, 119991 Russia
Corresponding author: A. A. Askadskii, email: andrey@ineos.ac.ru
Received 11 June 2020; accepted 27 July 2020
Abstract
One of the possible approaches to the analysis of a physical mechanism of time dependence for the resistance coefficients of materials is suggested. The material durability at the constant stress is described using the Zhurkov and Gul' equations and the durability at the alternating stress—using the Bailey criterion. The low strains lead to structuring of a material that is reflected in a reduction of the structuresensitive coefficient in these equations. This affords 20% increase in the durability. The dependence of the resistance coefficient assumes an extremal character; the maximum is observed at the time to rupture lg t_{r} ≈ 2 (s).
Key words: durability of materials, stress, time to rupture, structuresensitive coefficient.
Introduction
The resistance coefficient of a material is defined as a ratio of the current strength to its initial value. The resistance coefficient is often used to evaluate the material durability and predict the service life under certain conditions. Of particular importance is the analysis of a physical meaning of the factors that define the rate of material aging as well as the rate of structure recovery in time, which can occur simultaneously with aging. Let us consider one of the possible reasons for a peculiar shape of a kinetic curve for the time dependence of the resistance coefficient that is connected with the polymer strength. In turn, the polymer strength is connected with its durability t, which, according to the thermofluctuation concept of the rupture mechanism rationalized by S. N. Zhurkov, is determined by the following relation:
(1) 
where t_{0} is the preexponential factor, U_{0} is the initial activation energy of rupture, g is the socalled structuresensitive parameter of a material, s is the constant stress, T is the absolute temperature, and R is the universal gas constant.
The Zhurkov equation was discussed in detail in several monographs [1–6].
The time dependence of the resistance coefficients of polymeric materials was studied in a multitude of reports which are summarized in the following monographs [1, 3, 4, 7, 8]. These works analyzed the dependence of durability on stress rather than the resistance coefficient itself. The stress growth always afforded a reduction in the durability. The question about an increase of the durability with strain and stress growth was not raised and studied in these reports. The investigations were concerned with cellulose, polyamide, and a range of other fibers.
Calculations
Let us describe the durability t_{r} using the Bailey criterion which is valid in the case when the stress s is not constant:

(2) 
where t_{r} is the time from loading to sample rupture, t[s(t)] is the durability described by equation (1) at the constant stress equal to the instantaneous value s(t).
The stress s(t) can change in time according to any law. Using criterion (2), one can calculate the sample life span when the function s(t) and parameters of equation (1) are known. For the calculation of t_{r} using equation (2), these parameters must remain unchanged during the sample loading. The temperature must also be constant. Actually, the sample loading can change its structure (which changes the structuresensitive parameter g) and temperature. Even if the ambient temperature is maintained strictly constant, the cyclic impacts on the sample can lead to its heating. Then, the application of the Bailey criterion can afford overestimated values of the durability t_{r}, since we would plug the lower temperature than the real temperature of the sample.
In general, the Bailey criterion at the alternating stress, temperature and structuresensitive parameter should be written in the following form:

(3) 
where t[s(t); T(t); g(t)] is the durability described by equation (1) at the constant stress equal to the instantaneous value s(t), at the constant temperature equal to the instantaneous value T(t), and corresponding to the unchanged material structure which is defined by the instantaneous value of the structuresensitive parameter g(t).
If the stress s and temperature T are constant in time, and only the structuresensitive parameter g changes, then the Bailey criterion should be written as follows:

(4) 
Let us assume that, due to the constant stress on the sample and arising deformation, the structuresensitive parameter reduces linearly in time:

(5) 
where g_{0} is the initial structuresensitive parameter and a is the rate of reduction of the structuresensitive parameter in time due to improvement of the material structure.
The linear reduction of the structuresensitive parameter is not universal. The structuresensitive parameter can change according to the exponential, logarithmic, and linearstepwise dependences, but all this must be analyzed subsequently based on the coincidence of the calculated and experimental data. The linear dependence is fulfilled better for isotropic polymers since, during the action of stresses and arising deformation, they can orient. Note that the structuresensitive parameter of the oriented polymers always reduces and their durability increases compared to those of the isotropic polymers.
Then, equation (1) can be written in the following form:
(6) 
The Bailey criterion will be as follows:

(7) 
The solution of this integral equation leads to the following relation:

(8) 
Relation (8) suggests that the time to rupture t_{r} at the constant stress s is equal to:

(9) 
Let us choose the following typical values of parameters in equation (9) which are characteristic of many polymers: s = 30 MPa, U_{0} = 150 kJ/mol, T = 293 K, a = 1.7·10^{–8} s^{–1}, and g_{0} = 1.6 kJ/mol·MPa. The constant R is equal to 8.314 J/mol·K, and the constant t_{0} is equal to 10^{–12} s.
The polymers used for durability measurement represented fibers and bars based on poly(methyl methacrylate), poly(ethylene terephthalate), polystyrene, polycarbonate, polyethylene, cellulose, polyamide 6, etc. The vulcanizates of different rubbers were also used. The polymers were in the glassy, crystalline and rubbery states.
Relation (9) with these parameters will be as follows:
(10) 
If t_{r} should be expressed in years, then:
(11) 
These parameters afford the material durability t_{r} ≈ 0.6 year.
Let us calculate the dependence of s on t_{r}.
Table 1. Durability t_{r} at the rate of reduction of the structuresensitive parameter a = 1.7·10^{–8} s^{–1}
s, MPa 
t_{r}, s 
lg t_{r} 
s_{t}/s_{0} 
s_{t}/s_{0} 
s_{t}/s_{0} 
s_{t}/s_{0} 
29 
4605176 
6.663 
0.879 
0.805 
0.744 
0.690 
30 
1903576 
6.279 
0.909 
0.833 
0.769 
0.714 
31 
895390 
5.952 
0.939 
0.861 
0.795 
0.738 
32 
442132 
5.645 
0.967 
0.889 
0.820 
0.762 
33 
224715 
5.352 
1 
0.917 
0.846 
0.786 
36 
30516 
4.484 

1 
0.923 
0.857 
39 
4212 
3.624 


1 
0.929 
42 
590 
2.771 



1 
The data presented in Table 1 describe a descending branch of the dependence of s_{t}/s_{0} on lg t_{r}. To describe an ascending branch of this curve, let us use equation (1). According to this equation, the dependence of the stress s on the durability τ is determined by the following relation:

(12) 
The results of the calculations are summarized in Table 2.
Table 2. Values of the stress s leading to different values of the durability t = t_{r}
s, MPa 
lg t_{r} (s) 
s_{t}/s_{0} 
51.73 
0 
1.0 
55.23 
1.0 
1.068 
58.73 
2.0 
1.135 
62.23 
3.0 
1.203 
Figure 1 depicts the dependences of s_{t}/s_{0} on lg t_{r} obtained at different initial stresses s_{0} ranging from 33 to 42 MPa.
Figure 1. Dependence of s_{t}/s_{0} on lg t_{r} at different initial stresses: 33 (1), 36 (2), 39 (3), and 42 (4) MPa. The time t_{r} is expressed in seconds.
It is obvious that the higher the initial stress s_{0}, the lower the value of s_{t}/s_{0} that provides a transition from the ascending branch of the curve to the descending one. Hence, the suggested mechanism of a peculiar course of the kinetic curve for the time dependence of the resistance coefficient, which is connected with a reduction in the structuresensitive coefficient in the durability equation, allows one to describe adequately the curve under consideration, which resembles the Weller curve.
Now let us change the reduction rate of the structuresensitive parameter in time a. We assume that a = 1.0·10^{–8} s^{–1}. The values of the time to rupture at different constant stresses are presented in Table 3.
Table 3. Durability t_{r} at the reduction rate of the structuresensitive parameter a = 1.0·10^{–8} s^{–1}
s, MPa 
t_{r}, s 
lg t_{r} 
s_{t}/s_{0} 
28 
9708842 
6.987 
0.848 
29 
3039574 
6.483 
0.879 
30 
1745901 
6.242 
0.909 
31 
859319 
5.934 
0.939 
32 
433079 
5.637 
0.970 
33 
222301 
5.347 
1 
Let us assume even the lower value of a equal to 0.5·10^{–8} s^{–1}. The resulting values of t_{r }(s), lg t_{r}, and s_{t}/s_{0} are presented in Table 4.
Table 4. Durabilty t_{r} at the reduction rate of the structuresensitivity coefficient a = 0.5·10^{–8} s^{–1}
s, MPa 
t_{r}, s 
lg t_{r} 
s_{t}/s_{0} 
27 
17705141 
7.248 
0.844 
28 
7124936 
6.852 
0.875 
29 
4848072 
6.686 
0.906 
30 
1652375 
6.218 
0.937 
31 
836063 
5.922 
0.969 
32 
427846 
5.631 
1 
The data presented in Tables 3 and 4 suggest that the dependences of lg t_{r} on s_{t}/s_{0} at the same values of s differ insignificantly.
Finally, let us analyze the dependence of s_{t}/s_{0} on lg t_{r} at different values of the initial activation energy U_{0}. The results of the calculations performed using formula (9) are presented in Table 5.
Table 5. Durability t_{r} at different values of the initial activation energy of rupture U_{0}
U_{0}, kJ/mol 
s, MPa 
t_{r}, s 
lg t_{r} 
s_{t}/s_{0} 
150 
30 
1903576 
6.279 
0.545 
155 
35 
485719 
5.686 
0.636 
160 
40 
135534 
5.132 
0.727 
165 
45 
39190 
4.593 
0.818 
170 
50 
1869 
2.272 
0.909 
175 
55 
6.990 
0.844 
1 
It is obvious that an increase in the initial activation energy leads to a rapid increase in the ratio s_{t}/s_{0} . This is associated with the fact that the stresses s also increase. Figure 2 demonstrates the dependences of s_{t}/s_{0} on lg t.
Figure 2. Dependence of s_{t}/s_{0} on lg t at different values of the initial activation energy and stress. The time t is expressed in seconds.
This curve characterizes the descending branch of the durability dependence of the resistance coefficient.
It should be taken into account that the concept of S. N. Zhurkov is not the only one. Many researchers exploring the strength and durability of polymers give special attention not only to chemical but also to intermolecular bonds. The most comprehensive studies in this field were performed by V. E. Gul'. According to his concept, in any polymeric body, the loading falls, first of all, on the intermolecular bonds. Because the intermolecular bonds are weak (compared to the chemical ones), they undergo cleavage first, and, as the intermolecular bonds break, the loading on chemical bonds increases. If we deal with an oriented system (for example, fibers), then, the total energy of intermolecular bonds appears to be essentially higher than that of the chemical bonds in a main polymer chain. Therefore, in the oriented systems, the chemical bonds are disrupted first, and the whole rupture mechanism is connected almost only with the cleavage of these bonds. In the nonoriented systems, the intermolecular bonds play a great role. Introducing the possibility of their cleavage, V. E. Gul' deduced the following equation that describes the polymer durability:
(13) 
where B, s_{x}, a, g, and b are the material constants, s is the constant stress.
Let us take a logarithm of equation (13):
(14) 
We measured the durability of poly(vinyl chloride) under the action of a range of constant stresses s. The experimental temperature was 293 K. The resulting dependence of the durability on the stress is depicted in Fig. 3.
Figure 3. Dependence of the longterm strength of PVC. The time t_{r} is expressed in seconds.
Let us choose four durability values which correspond to four stresses. These values are listed in Table 6.
Table 6. Values of lg t and s for a poly(vinyl chloride) film obtained at different constant stresses s
lg t (s) 
s, MPa 
s_{t}/s_{0} 
0 
57.24 
1.0 
2.0 
49.6 
1.154 
4.0 
42.4 
1.350 
5.0 
38.6 
1.483 
7.0 
31.05 
1.843 
Using these data, we obtain the following system of equations (given that γ = γ_{0} = 1.6 kJ/mol·MPa and U_{0} = 150 kJ/mol):
1)
2)
3)
4)
Let us assume that s_{x} = 30 MPa. The solution of this system of equations leads to the following values of the material parameters: s_{x} = 30 MPa, a = 2 s/MPa, b = 0.6 s, and ln B = –6.0 (s).
The Bailey criterion (4) at the alternating structuresensitive parameter γ takes the following form:

(15) 
The solution of definite integral (15) leads to the following relation:

(16) 
The latter can be used to define the time to rupture t_{r} at the constant stress s:

(17) 
Plugging all the mentioned values of parameters B, s_{x}, a, g, and b into relation (17), we can obtain the values of lg t_{r} at different stresses s as well as the values of the ratio s_{t}/s_{0}. These data are listed in Table 7.
Table 7. Values of lg τ and s for a poly(vinyl chloride) film obtained under the action of the alternating stresses s_{t}
lg t (s) 
s_{t}, MPa 
s_{t}/s_{0} 
1.567 
110 
1.0 
3.340 
100 
0.909 
5.201 
90 
0.818 
6.256 
85 
0.773 
The total dependence of s_{t}/s_{0} on lg t_{r} is depicted in Fig. 4.
Figure 4. Dependence of s_{t}/s_{0} on lg t_{r}. The time t_{r} is expressed in seconds.
This dependence looks like the Weller curve since it has a maximum. This maximum is connected with the improvement of the material structure during deformation, which is reflected in a reduction in the structuresensitive parameter γ with the stress growth. Further investigations in this field will be connected with the analysis of the effect of creep and stress relaxation processes [7–10] at the alternating stresses and strains on the durability of polymeric materials.
Conclusions
The suggested procedure for the construction of the time dependence of the resistance coefficient allows one to establish the physical meaning of its variation. The resistance coefficient is connected with the improvement of the material structure during deformation, which leads to a reduction in the structuresensitive parameter in the durability equation. Of course, this is not the only one reason for the formation of the curved dependence of k_{r} = s_{t }/ s_{0} on t_{r} featuring a maximum, which resembles the Weller curve. Further investigations in this field will deal with the consideration of relaxation processes (creep and stress relaxation), which occur during polymer deformation upon loading.
Acknowledgements
This work was supported by the Ministry of Science and Higher Education of the Russian Federation.
References
 V. R. Regel', A. I. Slutsker, E. E. Tomashevskii, Kinetic Nature of the Strength of Solid Bodies, Nauka, Moscow, 1974 [in Russian].
 A. A. Askadskii, Polymer Deformation, Khimiya, Moscow, 1973 [in Russian].
 V. P. Tamuzh, V. S. Kuksenko, Decomposition Mechanics of Polymeric Materials, Zinatne, Riga, 1978 [in Russian].
 A. A. Askadskii, M. N. Popova, V. I. Kondrashchenko, Physics and Chemistry of Polymeric Materials and the Methods for Their Investigation, ASV, Moscow, 2015 [in Russian].
 A. Askadskii, T. Matseevich, Al. Askadskii, P. Moroz, E. Romanova, Structure and Properties of WoodPolymer Composites (WPC), Cambridge Scholars Publ., Cambridge, 2019.
 P. A. Moroz, Al. A. Askadskii, T. A. Matseevich, A. A. Askadskii, E. I. Romanova, WoodPolymer Composites: Structure, Properties, and Application, ASV, Moscow, 2020 [in Russian].
 V. E. Gul', Structure and Strength of Polymers, Khimiya, Moscow, 1978 [in Russian].
 A. A. Askadskii, T. A. Matseevich, M. N. Popova, Secondary Polymeric Materials. Mechanical and Barrier Properties, Plasticization, Blends and Nanocomposites, ASV, Moscow, 2017 [in Russian].
 G. M. Bartenev, A. G. Barteneva, Relaxation Properties of Polymers, Khimiya, Moscow, 1992 [in Russian].
 G. M. Bartenev, C. Ya. Frenkel', Physics of Polymers, Khimiya, Leningrad, 1990 [in Russian].