Home \ Browse Journal \ ASAP \ Time Dependences of the Resistance Coefficients of Polymeric Materials

2020 ASAP

инэос-open

INEOS OPEN, 2020, 3 (X), XX–XX 

Journal of Nesmeyanov Institute of Organoelement Compounds
of the Russian Academy of Sciences

Download PDF
DOI: 10.32931/io2016a

 

issue_cover_html_asap      

Time Dependences of the Resistance Coefficients
of Polymeric Materials

A. V. Matseevich and A. A. Askadskii*

Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28, Moscow, 119991 Russia
 

Corresponding author: A. A. Askadskii, e-mail: andrey@ineos.ac.ru
Received 11 June 2020; accepted 27 July 2020

Abstract

grab    

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 structure-sensitive 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 tr ≈ 2 (s).

Key words: durability of materials, stress, time to rupture, structure-sensitive 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 t0 is the preexponential factor, U0 is the initial activation energy of rupture, g is the so-called structure-sensitive 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 tr using the Bailey criterion which is valid in the case when the stress s is not constant:

(2)

where tr 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 tr 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 structure-sensitive 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 tr, 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 structure-sensitive 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 structure-sensitive parameter g(t).

If the stress s and temperature T are constant in time, and only the structure-sensitive 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 structure-sensitive parameter reduces linearly in time:

(5)

where g0 is the initial structure-sensitive parameter and a is the rate of reduction of the structure-sensitive parameter in time due to improvement of the material structure.

The linear reduction of the structure-sensitive parameter is not universal. The structure-sensitive parameter can change according to the exponential, logarithmic, and linear-stepwise 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 structure-sensitive 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 tr 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, U0 = 150 kJ/mol, T = 293 K, a = 1.7·10–8 s–1, and g0 = 1.6 kJ/mol·MPa. The constant R is equal to 8.314 J/mol·K, and the constant t0 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 tr should be expressed in years, then:

(11)

These parameters afford the material durability tr ≈ 0.6 year.

Let us calculate the dependence of s on tr.

Table 1. Durability tr at the rate of reduction of the structure-sensitive parameter a = 1.7·10–8 s–1

s, MPa

tr, s

lg tr

st/s0

st/s0

st/s0

st/s0

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 st/s0 on lg tr. 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 = tr

s, MPa

lg tr (s)

st/s0

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 st/s0 on lg tr obtained at different initial stresses s0 ranging from 33 to 42 MPa.

Figure 1. Dependence of st/s0 on lg tr at different initial stresses: 33 (1), 36 (2), 39 (3), and 42 (4) MPa. The time tr is expressed in seconds.

It is obvious that the higher the initial stress s0, the lower the value of st/s0 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 structure-sensitive 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 structure-sensitive 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 tr at the reduction rate of the structure-sensitive parameter a = 1.0·10–8 s–1

s, MPa

tr, s

lg tr

st/s0

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(s), lg tr, and st/s0 are presented in Table 4.

Table 4. Durabilty tr at the reduction rate of the structure-sensitivity coefficient a = 0.5·10–8 s–1

s, MPa

tr, s

lg tr

st/s0

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 tr on st/s0 at the same values of s differ insignificantly.

Finally, let us analyze the dependence of st/s0 on lg tr at different values of the initial activation energy U0. The results of the calculations performed using formula (9) are presented in Table 5.

Table 5. Durability tr at different values of the initial activation energy of rupture U0

U0, kJ/mol

s, MPa

tr, s

lg tr

st/s0

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 st/s0 . This is associated with the fact that the stresses s also increase. Figure 2 demonstrates the dependences of st/s0 on lg t.

Figure 2. Dependence of st/s0 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, sx, 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 long-term strength of PVC. The time tr 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

st/s0

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 = γ0 = 1.6 kJ/mol·MPa and U0 = 150 kJ/mol):

1)  4.6 = ln B  – α ln (49.6  σx) +29.0 · β 

2)  9.2 = ln B    α  ln (42.4 – σx+ 33.73 · β 

3)  11.5 = ln B   α ln (38.6 – σx+ 36.22 · β  

4)   16.1 =  ln α ln (31.05 – σx41.18 · β

Let us assume that sx = 30 MPa. The solution of this system of equations leads to the following values of the material parameters: sx = 30 MPa, a = 2 s/MPa, b = 0.6 s, and ln B = –6.0 (s).

The Bailey criterion (4) at the alternating structure-sensitive 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 tr at the constant stress s:

(17)

Plugging all the mentioned values of parameters B, sx,  a, g, and b into relation (17), we can obtain the values of lg tr at different stresses s as well as the values of the ratio st/s0. 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 st

lg t (s)

st, MPa

st/s0

1.567

110

1.0

3.340

100

0.909

5.201

90

0.818

6.256

85

0.773

The total dependence of st/s0 on lg tr is depicted in Fig. 4.

Figure 4. Dependence of st/s0 on lg tr. The time tr 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 structure-sensitive 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 structure-sensitive parameter in the durability equation. Of course, this is not the only one reason for the formation of the curved dependence of kr = ss0 on tr 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

  1. V. R. Regel', A. I. Slutsker, E. E. Tomashevskii, Kinetic Nature of the Strength of Solid Bodies, Nauka, Moscow, 1974 [in Russian].
  2. A. A. Askadskii, Polymer Deformation, Khimiya, Moscow, 1973 [in Russian].
  3. V. P. Tamuzh, V. S. Kuksenko, Decomposition Mechanics of Polymeric Materials, Zinatne, Riga, 1978 [in Russian].
  4. 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].
  5. A. Askadskii, T. Matseevich, Al. Askadskii, P. Moroz, E. Romanova, Structure and Properties of Wood-Polymer Composites (WPC), Cambridge Scholars Publ., Cambridge, 2019.
  6. P. A. Moroz, Al. A. Askadskii, T. A. Matseevich, A. A. Askadskii, E. I. Romanova, Wood-Polymer Composites: Structure, Properties, and Application, ASV, Moscow, 2020 [in Russian].
  7. V. E. Gul', Structure and Strength of Polymers, Khimiya, Moscow, 1978 [in Russian].
  8. 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].
  9. G. M. Bartenev, A. G. Barteneva, Relaxation Properties of Polymers, Khimiya, Moscow, 1992 [in Russian].
  10. G. M. Bartenev, C. Ya. Frenkel', Physics of Polymers, Khimiya, Leningrad, 1990 [in Russian].