Russian Physics Journal, Vol. 50, No. 7, 2007

PHYSICS OF SEMICONDUCTORS AND DIELECTRICS THE EFFECT OF TECHNOLOGICAL MICRODEFECTS OF STRUCTURE ON SERVICE LIFE OF POLYMER MATERIALS V. A. Skripnyak and A. A. Kozulin

UDC 539.3

Within the framework of the kinetic theory of strength, relations for estimation of durability of thermoplastic materials under stationary and non-stationary thermomechanical impacts are obtained It is shown that durability decreases linearly with increase in the natural (Napierian) logarithm of the mean size of microcracks formed during manufacturing of polymer products.

Currently used techniques offer a means to estimate durability of polymer materials and structures made of them under creep conditions, i.e., under constant stress, constant temperature, and low strain rates [1–4]. The estimation of service life of polymer materials under non-stationary thermomechanical impacts is, however, complicated [2–4]. The limited available experimental data on long-term strength of polymer materials makes the development of models and strength estimation techniques relying on the physical concepts of fracture mechanisms quite urgent. The purpose of this work is to investigate the process of rupture of thermoplastic polymer materials under varying thermal and mechanical loadings. Within the framework of mechanics of damaged media, it is assumed that durability of structure units, safety factors, and deformability are controlled by peculiarities of buildup of microdamages. The non-linear summation principle together with the hypothesis of independence of fracture mechanisms require that the fracture surface be constructed in a multi-dimensional space, whose coordinates are time, temperature, strain rate tensor components, stress tensor components, and internal parameters characterizing the structure and damageability of a material [1–5]. It is reasonable to use the dependences representing projections of the fracture surface onto the time–stress plane. The formation of dissipative microdamage structures is represented as a kink in the dependence of fracture stress on time. In this work, in order to estimate service life of structures we propose to use the condition of microscopic fracture of materials. The kinetics of buildup of microdamage is prescribed by an equation for mean size of microcracks [6]. It is assumed that in the course of processing of polymer materials and manufacture of articles from them, microcracks of mean size r0 and concentration N are formed. The size of initial microcracks and their concentration depend on the process of polymer material recycling. Durability of a polymer material t f in a stressed condition is controlled by the time during which the mean size of microcracks r increases to critical size r*, which would result in fragmentation of the material [6] tf

r = ∫ r dt .

(1)

t =0

The time of applying a load is taken as a reference. The time to failure t f in the course of loading of a material in a non-stationary temperature field was derived within the framework of a kinetic approach including the principle of a temperature – time superposition [2]. In this case, the fracture criterion has the form

Tomsk State University, e-mail: [email protected]. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 3–6, July, 2007. Original article submitted October 20, 2006. 1064-8887/07/5007-0633 ©2007 Springer Science+Business Media, Inc.

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TABLE 1. Values of U0, m and log(t0) of Polymer Materials Versus Loading Type [2, 7] Polymer material

U0, kJ/Mol 130 307 181 260 90

High-density polyethylene Low-density polyethylene Polypropylene

m ⋅ 106, kJ/(Mol⋅MPa) 1.09 4.39 3.60 2.47 0.9–2.13

log(t0, s) –11 –39 –19 –12 –11.8

r = r *,

(2)

where r* is the critical crack size for which the material is taken to fail entirely. The critical crack size r* is derived using the formula [6] 2

r *(T ) = ( K1c (T ) / σ1 ) / 2π ,

(3)

where K1c(T) is the fracture toughness of a polymer material at the temperature T and σ1 is the main component of tensile stresses. The rate of microcrack growth is determined by the equation

(

)

r = A exp [( −U 0 + mσ1 ) RT ] π2 8 ( σ1 σ т ) r ( t ) , 2

(4)

where r(t) is the mean microcrack size, A is the model coefficient, U0 is the activation energy of the fracture process corresponding to the activation energy of rupture of chemical bonding in an unloaded state (σ1 = 0), Т(t) is the temperature, К, m is the structure-sensitive parameter representing polymer material properties, σ1(t) is the main stress tensor component, σт(T) is the material yield stress, and R is the absolute gas constant. The value of A to a first approximation is taken to be equal to the thermal vibration frequency ~1012 s–1. The activation energy U0 of the crack growth is determined by the polymer molecular structure. Parameter m is strongly affected by the material microstructure formed in the course of manufacturing. Table 1 presents the values of certain parameters for several thermoplastic materials. Equation (4) implies that the size of microcracks in a polymer material increases even if the material is in an elastic strained state. Upon reaching the yield stress, the rate of crack growth is sharply increased and durability decreases. Under conditions of non-stationary thermomechanical loading, durability is controlled by the loading history Т(t) and σ1(t). By integrating Eq. (4) under the prescribed conditions and known mean microcrack size r0 in the initial material, we can estimate durability of polymer materials using Eqs. (2), (3) 2

t

⎛ r * ⎞ f π2 ⎛ −U + m σ1 (t ) ⎞ ⎛ σ1 (t ) ⎞ ln ⎜ ⎟ = ∫ A exp ⎜ 0 ⎟ ⎜ σ ⎟ dt . RT (t ) ⎝ ⎠⎝ т ⎠ ⎝ r0 ⎠ 0 8

(5)

In a special case, at constant temperature and pressure, its durability will be determined by the formula ⎛ U − mσ1 ⎞ t f = t0 exp ⎜ 0 ⎟, ⎝ RT ⎠

⎛ r * ⎞⎛ σ ⎞ t0 = 2 ln ⎜ ⎟⎜ 1 ⎟ π ⎝ r0 ⎠ ⎝ σ т ⎠ 8

−2

,

where tf is the time to failure of a polymer material, r* is the critical microcrack size and r0 is the initial microdefect size.

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(6)

Fig. 1

Fig. 2

Equation (6) suggests that with increasing size of technological defects r0 durability of thermoplastic materials is linearly decreased with increasing natural logarithm of the mean size of cracks formed in the course of polymer material manufacturing. The above relation (6) is similar to the S. N. Zhurkov equation [8] describing a linear dependence σ1 − log(t f ) (dashed line in Fig. 1). Figure 1 presents the calculated values of durability of polypropylene. Curves 1–7 correspond to the temperatures 363, 353, 343, 313, 293, 283 and 273 K The calculation for polypropylene structures was made for A = 1012 s, U0 = 125 kJ/Mol, m = 0.9 kJ/(Mol⋅MPa), and K1с = 2.7–4.3 MPa⋅m1/2. A change in the tilt of the curves σ1 − log(t f ) in point A indicates a considerable decrease in polymer material durability with increasing stress. Under creep conditions, where stresses are comparatively small ( σ1 / σ т < 0.15 ), polypropylene durability decreases exponentially with increasing stresses. Polypropylene PPRC “Random Co-polymer” under normal conditions has the yield stress ~25–35 MPa [4]. The dependence of its yield stress on temperature could be approximated by the formula σ т (T ) = σ т(293 K) [1 − 0.18 (T − 293 K )] . The approximating curve is shown in comparison with the experimental data [7] in Fig. 2. At the temperature 270 K under the action of stresses approximating the yield stress polypropylene undergoes brittle fracture, which is manifested in a sharp decrease of its durability. In order to prolong service life of articles made from polypropylene up to 50 years at a standard temperature of 350 K it is necessary to ensure stresses less than 0.5– 0.7 MPa. The calculated values of polypropylene durability versus stress within the temperature interval from 293 to 353 K are shown in Fig. 3. The values of r0 were prescribed taking into account the experimental data from [2–4]. The calculations make use of the values of r0 and ln(r*/r0) that are equal to 0.1 mm and 4.6, respectively. The resulting estimates demonstrate that polypropylene service life in the structures subjected to the temperature 293 K approximates 100 years at the maximum stresses less than 1 MPa. At low values of σ1 the value of exp ( −mσ1 / RT ) ≈ 1 , and Eq. (6) can be re-arranged to the form

635

Fig. 3

−2

⎛σ ⎞ 8 ⎛ r *⎞ ⎛U ⎞ t f = B ( r*, r0 , T ) ⎜ 1 ⎟ , B ( r*, r0 , T ) = 2 ln ⎜ ⎟ exp ⎜ 0 ⎟ . ⎝ RT ⎠ π ⎝ σт ⎠ ⎝ r0 ⎠

(7)

Equation (7) suggests that at low stresses the relation ln(tf) ~ ln(σ1) is linear

ln ( t f ) = [ ln ( B ) + 2ln ( σ т )] − 2ln ( σ1 ) . In order to estimate service life of polymer materials in the case where maximum tensile stresses σ1 are comparable to σ т , it is reasonable to use a linear dependence ln(t f ) ~ σ1 resulting from Eq. (6): tf =

⎛ r *⎞ ⎛U ln ⎜ ⎟ exp ⎜ 0 ⎝ RT π ⎝ r0 ⎠ 8

2

⎞ ⎛ mσ1 ⎞ ⎛ mσ1 ⎞ ⎟ = B ( r*, r0 , T ) exp ⎜ − ⎟, ⎟ exp ⎜ − ⎝ RT ⎠ ⎝ RT ⎠ ⎠

m ⎞ ln t f = ln ( B ) − ⎜⎛ ⎟ σ1 . RT ⎝ ⎠

(8)

Thus for estimation of service life of polypropylene, polyethylene and other thermoplastics we can recommend to use Eqs. (2), (3), and (5). The calculated values of durability of thermoplastic materials under stationary thermomechanical action are asymptotically close to the estimates resulting from the S. N. Zhurkov formula. The relations make use of mean size of technological defects formed in the course of manufacturing of polymer materials. Durability under stationary thermomechanical action can be estimated using Eq. (6) and at low stresses – using an approximated formula (7). REFERENCES

1. 2. 3. 4.

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G. M. Bartenev, Polymer Toughness and Fracture Mechanisms [in Russian], Moscow, Khimiya (1984). I. Narisava, Toughness of Polymer Materials [Russian Translation], Moscow, Khimiya (1987). V. R. Regel, Mekhanika Polimerov, No. 1, 98–112, (1971). V. A. Skripnyak, V. A. Ustugov, and N. L. Saveliev, Simulation of processes in synergetic systems, in: Proc. Int. Conf. “Baikal readings II on simulation of processes in synergetic systems”, Tomsk, TUP, 137–140 (2002).

5. 6. 7. 8.

V. A. Skripnyak, V. A. Ustugov, and N. L. Saveliev, Simulation of processes in synergetic systems, in: Proc. Int. Conf. “Baikal readings II on simulation of processes in synergetic systems”, Tomsk, TUP, 134–137 (2002). Guy Pluvinage, Mechanique Elastoplastique de la Rupture (cruteres d’amoecage), Toulouse (France) (1989). E. L. Kalinicheva and M. B. Sakovtseva, Selection of Plastics for Manufacture and Use of Articles. Reference book, Leningran, Khimiya (1987). S. N. Zhurkov and V. A. Petrov, Dokl. Akad. Nauk, 239, No. 6, 1316–1319 (1978).

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