STRENGTH

OF

OF

PREDICTION OF

POLYMER R.

MATERIALS

THE

LONG-TERM

REVIEW*

MATERIALS. D.

Maksimov

DURABILITY

and

Yu.

S.

Urzhumtsev

UDC 539.4: 539.376:678

The state of the development of methods for predicting the l o n g - t e r m durability of p o l y m e r materials was discussed at the Second All-Union Conference on P o l y m e r Mechanics in 1971 [1, 2]. It was resolved at this c o n f e r e n c e to e,ontinue this work with special attention toward the experimental checking of prediction t e c h niques. In the f i v e - y e a r period since this c o n f e r e n c e , p r o g r e s s was noted, especially in techniques for predicting the deformation p r o p e r t i e s which had been put forward in the 1960's. The basis of this work is the study of the a c c e l e r a t i o n o1" inhibition of relaxation p r o c e s s e s in m a t e r i a l s under the action of various factors. These f a c t o r s include t e m p e r a t u r e , s t r e s s , intense vibration, m o i s t u r e , and concentration of p l a s t i c i z e r in the material. Study of the effect of small vibrations on c r e e p and the relaxation of strains was continued and new data w e r e obtained. Thus, the authors of [3, 4] studied the effect of low-amplitude vibrations in the region of sound f r e quencies on the relaxation of s t r a t a for r e p r e s e n t a t i v e s of v a r i o u s c l a s s e s of p o l y m e r s including polyethylene, p o l y propylene, p o l y a c r y l a t e s , and aromatiepoly~mides. I t w a s s h o w n t h a t w i t h i n e r e a s i n g frequency of the a p p l i e d v i b r a tions, the s t r a i n - r e l a x a t i o n p r o c e s s is a c c e l e r a t e d . The f r e q u e n c y - t i m e analogy is obeyed upon the action of vibrations of various frequencies, which p e r m i t s us to consider the effect of vibration as an introduction to the v i s c o e l a s t i c relationships of the displacement function of the relaxation s p e c t r u m . The dependence of the displacement function on frequency is shown in Fig. 1: Curve 1 r e p r e s e n t s f i n e - s p h e r u l i t e potypropylene and c u r v e 2 r e p r e s e n t s l a r g e - s p h e r u l i t e polypropylene. It is seen f r o m this figure that the effect of vibration depends on the s t r u c t u r e of the m a t e r i a l and is c o n s i d e r a b l y g r e a t e r for t a r g e - s p h e r u l i t e samples than for samples with f i n e - s p h e r u l i t e s t r u c t u r e . We note that the data obtained in this work on the vibrational relaxation of s t r a i n in polyethylene a g r e e with the results of work on the vibrational c r e e p of polyethylene c a r r i e d out at the Institute of P o l y m e r Mechanics, Academy of Sciences of the Latvian SSR in the t 9 6 0 ' s [5-7]. Bartenev and Shelkovnikova [8] studied the vibrational relaxation of s t r a i n in c a r b o n - b l a c k reinforced rubbers. A significant effect of relatively l o w - f r e q u e n c y vibrations (up to 25 Hz) on the relaxation of s t r a i n was found experimentally. The hypothesis that the vibrational effect in this case is related to the thixotropic p r o p e r t i e s of ~he reinforced r u b b e r s was proposed. Kania and Zawadzki [9] continued the .study of the i s o t h e r m i c vibrational c r e e p of polymethyl m e t h a e r y l a t e in the linear region of v i s c o e l a s t i c deformation and found that with i n c r e a s i n g amplitude of the vibrational load, the rate of deformation of this m a t e r i a t i n c r e a s e s . E x p e r i m e n t a l studies were c a r r i e d out on the vibrational c r e e p of woven epoxy-phenol fiberglass [10-13]. Tubular fiberglass s a m p l e s were tested with two types of s t r e s s i n g : twisting with additional vibrational load of the torque and twisting with longitudinal static load and additional vibrational load. The frequency of the v a r i a b l e component of the load was 20 Hz and the amplitude did not exceed 5% of the s h o r t - t e r m ultimate strength. In these t e s t s , no difference between the static and vibrational c r e e p was found. Contrasting behavior was observed upon i n c r e a s i n g the vibrational frequency to 20 kHz. These tests were p e r f o r m e d on samples cut at a 45 o angle toward the directions of the reinforcement. The application of such high-frequency vibrations leads to an a c c e l e r a t i o n of creep. This effect cannot be explained only by the heating of the m a t e r i a l using the data of the static tests. Thus, the vibrational c r e e p was described by the relationship of the viscoelasticity, in *Report presented at the Third All-Union Conference on P o l y m e r Mechanics, Riga N o v e m b e r 10-12, 1976.

Institute of P o l y m e r Mechanics, A c a d e m y of Sciences of the Latvian SSR, Riga. T r a n s l a t e d f r o m Mekhanika P o l i m e r o v , No. 4, pp. 631-645, July-August, 1977. Original a r t i c l e submitted F e b r u a r y 21, 1977. T~isl material ~ protected by copyright registered in the name o f Plenum Publishing Corporation 227 West 17th Street Ne*v York N. Y IOOI L N o part this publication m a y be reproduced, stored in a retrieval system, or transmitted, in any form' or by any means, electronic, mec'hanical, photocopying, recording or otherwise, w i t h o u t written permission o f the publisher. A copy o f this article is available f r o m tire publisher for $ 7. 50.

[of

]m~rofilming,

531

\

f'fo kHz

"~A~ ~

.o.° Fig. 1

~,,. 2

\a

1_I I°°I :i° ,"J 2

0

6

~

tO

12

Fig. 2

Fig. 1. F r e q u e n c y function of t h e r e l a x a t i o n s p e c t r u m d i s p l a c e ment [4]. Fig. 2. T i m e dependence of the bulk modulus of a p o l y e s t e r c o m posite [18] : a) d i s p l a c e m e n t of the r e l a x a t i o n s p e c t r u m depending on the h y d r o s t a t i c p r e s s u r e ; b) g e n e r a l i z e d c u r v e of the bulk

modulus; p=50 kgf/cm2 (Y), 75 (0), i00 (i), 125 (A), 150 ([~), 175 (A), 200 (0), 225 (~), and 250 kgf/cm2 (0). which the difference nucleus K ( t - t *, a T , av) depends on the t e m p e r a t u r e aT and the v i b r a t i o n a l av d i s p l a c e merit functions of the r e l a x a t i o n a l s p e c t r u m . We note that the i m p o r t a n c e of the study of the r e s i s t a n c e of m a t e r i a l s to the c o n c u r r e n t action of static s t r e s s e s and v i b r a t i o n s is without doubt, since the p r e d i c t i o n of the b e h a v i o r of a m a t e r i a l under such c o m m o n s t r e s s conditions is i m p o s s i b l e in s o m e c a s e s on the b a s i s of purely static t e s t s . New e x p e r i m e n t a l evidence on the validity of the s t r e s s - t i m e analogy has been obtained. Savitskii [14] and, independently, Daugste et al. [13, 15] showed that in the monaxial s t r e t c h i n g of g l a s s fiber, the s t r e s s t i m e analogy holds in a b r o a d r a n g e of s t r e s s e s and t i m e p e r i o d s of s t r e s s i n g . The s t r e s s - t i m e analogy was also supported by analysis of the bulk c r e e p under conditions of hydrostatic p r e s s u r e . Ainbinder at al. [16, 17] showed that for m o s t t h e r m o s e t t i n g p l a s t i c s , the r e l a t i o n s h i p between h y d r o s t a t i c p r e s s u r e and bulk c r e e p d e f o r m a t i o n is substantially nonlinear and that the p r e s s u r e - t i m e analogy holds. This conclusion was e x p e r i m e n t a l l y supported in a study of the b e h a v i o r of a p o l y e s t e r c o m p o s i t e under conditions of hydrostatic p r e s s u r e [18] ; a significant physical nonlinearity was found for the r e l a t i o n s h i p between the s p h e r i c a l s t r e s s t e n s o r and c r e e p deformation. This nonlinearity is a consequence of the dependence of r e l a x a t i o n t i m e s on the s p h e r i c a l s t r e s s t e n s o r . It was found that this dependence, s i m i l a r to the t e m p e r a t u r e dependence, is e x p r e s s e d through one d i s p l a c e m e n t function of the relaxation s p e c t r u m . This p e r m i t t e d obtaining a g e n e r a l i z e d c u r v e for the t i m e dependence of the bulk r e l a x a t i o n modulus at a l a r g e t i m e i n t e r v a l r e l a t i v e to s h o r t - t e r m bulk c r e e p at different levels of hydrostatic c o m p r e s s i o n . T h e s e data for v a r i o u s values of h y d r o s t a t i c c o m p r e s s i o n a r e shown in Fig. 2. The dependence of the l o g a r i t h m of the d i s p l a c e m e n t of the relaxation s p e c t r u m as a function oi the h y d r o s t a t i c c o m p r e s s i o n is also shown in this figure. Thus, Fig. 2 i l l u s t r a t e s , in e s s e n c e , the validity of the s t r e s s - t i m e analogy in the bulk d e f o r m a t i o n of a m a t e r i a l under conditions of h y d r o s t a t i c c o m p r e s s i o n , which is v e r y i m p o r t a n t in predicting the v i s c o e l a s t i c p r o p e r t i e s of those p o l y m e r s , the bulk c r e e p of which cannot be ignored. Studies in which new possibilities of the method of reduced v a r i a b l e s a r e r e v e a l e d have appeared. Thus, Lipatov et al. [19, 20] and Bunin and Irgen [21] showed that the introduction of a d i s p e r s e filler into a p o l y m e r or change in the content of p l a s t i c i z e r , s i m i l a r to t e m p e r a t u r e , shifts the r e l a x a t i o n s p e c t r u m along the t i m e scale. A c o n c e n t r a t i o n - t i m e analogy is found in the m e c h a n i c a l b e h a v i o r of the m a t e r i a l , which m a y be used for predicting the v i s c o e l a s t i c p r o p e r t i e s of m a t e r i a l s with varying content of p l a s t i c i z e r or f i l l e r . Thus, the validity of a n u m b e r of analogies which m a y be used as a means of predicting v i s c o e l a s t i c p r o p e r t i e s of p o l y m e r m a t e r i a l s has been established. N e v e r t h e l e s s , studies devoted to the development of methods of p r e d i c t i o n a r e b a s e d p r i m a r i l y on the t e m p e r a t u r e method. This is also understandable as change in t e m p e r a t u r e ; specifically, it is a r a t h e r available m e a n s of a c c e l e r a t i n g or inhibiting r e l a x a t i o n p r o c e s s e s . We m a y delineate s e v e r a l basic directions in the r e c e n t development of the t e m p e r a t u r e method: 1) broadening of the c l a s s of p o l y m e r s , for which e x p e r i m e n t a l evidence of the validity of the t e m p e r a t u r e - t i m e analogy has

532

~b'~Gr--

0

Fig. 3

tO

/ ~-

30

50

Fig. 4

Fig. 3. Creep of F t o r o p l a s t - 4 Teflon [22]: a) initial c r e e p curves at v a r i o u s t e m p e r a t u r e s (°C) ; b) t e m p e r a t u r e displacement function; c) generalized c r e e p curve. Fig. 4. T e m p e r a t u r e displacement function for polyvinyl chloride [27] : The points c o r r e s p o n d to experimental values with ~ constant and the line is derived f r o m tests for c r e e p with constant or. been obtained; 2) a s e a r c h for means of operational checking of the validity of the t e m p e r a t u r e - t i m e analogy; 3) study of the t e m p e r a t u r e - t i m e dependence of the deformation p r o p e r t i e s under conditions of a complex s t r e s s e d state; 4) study of the features of the t e m p e r a t u r e - t i m e analogy in the region of physical nonlinearity of the v i s c o e l a s t i c p r o p e r t i e s of a material. Let us examine the basic results obtained in each of these d i r e c tions of r e s e a r c h . 1. With r e g a r d to broadening of the c l a s s of t h e r m o r h e o l o g i c a l l y simple m a t e r i a l s , major attention,has been given to the study of the t e m p e r a t u r e - - t i m e dependence of the deformation p r o p e r t i e s of c r o s s - l i n k e d and c r y s t a l l i z i n g p o l y m e r s . The heightened i n t e r e s t in these m a t e r i a l s is related to the p r o p o s a l made in the first works on the t e m p e r a t u r e - t i m e analogy that this analogy will not hold for c r o s s - l i n k e d and crystallizing polym e r s . Subsequently, studies c a r r i e d out both within and outside the Soviet Union have shown that the t e m p e r a t u r e - t i m e analogy may hold for the v i s c o e l a s t i c p r o p e r t i e s of these p o l y m e r s . Of the latter, we shall consider the work of Cherskii et al. [22], who obtained c r e e p c u r v e s for F t o r o p l a s t - 4 Teflon at v a r i o u s t e m p e r a t u r e s , including negative t e m p e r a t u r e s down t o - 60°C (shown in Fig. 3). This figure shows that it is possible to cons t r u c t a generalized dependence of the c r e e p deforn'lation on time in the p r o l o n g e d - t i m e interval on the basis of the r e s u l t s of s h o r t - t e r m testing. The generalization of the data at various t e m p e r a t u r e s was possible as a consequence cf the validity of the t e m p e r a t u r e - t i m e analogy. The results of this work have direct practical application for the p r o b l e m of using Teflon in aviation technology under a r c t i c conditions. The i n c r e a s e in the number of v a r i e t i e s of p o l y m e r m a t e r i a l s , the modification of their p r o p e r t i e s , and the c r e a t i o n of new types of c o p o l y m e r s inevitably will lead to the appearance of m a t e r i a l s with p r o p e r t i e s which a r e not t h e r m o r h e o l o g i c a l l y simple. Both within the Soviet Union and abroad [23, 24], works have appeared in which the concept of t h e r m o r h e o l o g i c a l l y complex behavior of p o l y m e r s is introduced. The m a j o r feature which unites the v a r i o u s interpretations of this concept is the dependence of the displacement of the relaxation s p e c t r u m not only on t e m p e r a t u r e , but also on time. An attempt to determine such a dependence was undertaken in our l a b o r a t o r y [25]. Analogous results f r o m experiments on a b r o a d e r scale were obtained in studying the v i s c o e l a s t i c p r o p e r t i e s of mixtures of t h e r m o p l a s t i c and t h e r m o s e t t i n g p o l y m e r s [26]. On the whole, it should be pointed out that the value of studying t h e r m o r h e o l o g i e a l l y complex behavior of p o l y m e r s for the p u r p o s e s of prediction r e m a i n s unclear. 2. The check of the validity of the t e m p e r a t u r e - t i m e analogy in the quasistatic deformation of p o l y m e r m a t e r i a l s is usually c a r r i e d out by testing of c r e e p or the relaxation of s t r e s s e s at various t e m p e r a t u r e s . Such testing is v e r y laborious and special equipment is required. One of these methods was t r e a t e d by Efimova and Maksimov [27], who studied and d e m o n s t r a t e d tbc feasibility of determining the t e m p e r a t u r e displacement function on the b a s i s of the family of cr v e r s u s e curves obtained at a fixed deformation rate and various t e m p e r atures. A coincidence of t e m p e r a t u r e displacement functions found f r o m testing with different types of s t r e s s ing was obtained: c r e e p with constant a and quasistatic deformation with constant ~ (Fig. 4). This r e c o m m e n d s the above-mentioned method for factory l a b o r a t o r i e s and engineering and design bureaus for the operative determination and c o m p a r i s o n of the v i s c o e l a s t i c p r o p e r t i e s of new m a t e r i a l s ; the realization of this method is achieved on relatively c o m m o n f a c t o r y - m a d e testing machines.

533

3. The application of the analogy method in the complex s t r e s s e d state was examined by Mochalov et al. [28-33]. The effect of the t e m p e r a t u r e and m o i s t u r e content on the deformation p r o p e r t i e s of a p o l y e s t e r r e s i n was studied. We note that m o i s t u r e content, in addition to t e m p e r a t u r e , has a significant effect onthe d e f o r m a tion p r o p e r t i e s of hydrophilic p o l y m e r s , and the p r o b l e m of predicting the behavior of a m a t e r i a l in r e a l s t r u c t u r e s should be solved, as a rule, considering the effect of both f a c t o r s . Two m a j o r p r o b l e m s were posed in these studies: 1) the c o n s t r u c t i o n of an analytical model of the d e f o r m a t i o n of a m a t e r i a l taking into account the effects of t e m p e r a t u r e and m o i s t u r e ; 2) testing the feasibility of using the effect of a c c e l e r a t i n g c r e e p by i n c r e a s i n g the t e m p e r a t u r e and m o i s t u r e in o r d e r to predict the l o n g - t e r m c r e e p of a m a t e r i a l according to rapid tests. The test p r o g r a m was the following. The s h o r t - t e r m (up to 5 h) fundamental tests for c r e e p were initially conducted at constant s t r e s s e s and various t e m p e r a t u r e and m o i s t u r e levels held constant over time. The s t r e s s e s did not exceed the linearity limit of the v i s c o e l a s t i c p r o p e r t i e s . The s t r e t c h tests were conducted at four t e m p e r a t u r e levels, and six m o i s t u r e values were selected for each t e m p e r a t u r e level. Thus, t e s t s were c a r r i e d out f o r 24 combined values of t e m p e r a t u r e and moisture. The s h e a r tests w e r e c a r r i e d out at four t e m p e r a t u r e levels and six m o i s t u r e levels. Finally, t e s t s for combined s h e a r and s t r e t c h were c a r r i e d out at one t e m p e r a t u r e level and six m o i s t u r e l e v e l s . T h e s e w e r e the basic t e s t s which yielded data a c c o r d i n g to which the relationships of the linear v i s c o e l a s t i c i t y were approximated. To check these equations, s h o r t - t e r m control tests were c a r r i e d out at v a r y i n g t e m p e r a t u r e s and m o i s t u r e s . This check was conducted by c o m p a r i n g the e x p e r i m e n t a l and calculated curves obtained using the p a r a m e t e r s found in the fundamental tests. The feasibility of prediction was actually not tested in this c a s e , since the duration of the s h o r t - t e r m control tests did not exceed the t i m e s of the fundamental tests. The n e c e s s i t y for checking the a c c u r a c y of prediction a r o s e , since in analyzing the results of the s h o r t - t e r m t e s t s , the validity of the t e m p e r a t u r e - m o i s t u r e - t i m e analogy was d e m o n s t r a t e d and the feasibility of determining the theological c h a r a c t e r i s t i c s of a m a t e r i a l over a time interval considerably g r e a t e r than the duration of d i r e c t t e s t s a r o s e . Thus, prolonged control t e s t s w e r e c a r r i e d out. The duration of the t e s t s for c r e e p upon s t r e t c h i n g was 6 y e a r s ; the s h e a r testing lasted 3 y e a r s , and the combined s h e a r and s t r e t c h testing lasted 2 y e a r s ° The tests were c a r r i e d out on tubular s a m p l e s of PN-3 resin. Let us now briefly examine the r e s u l t s of these tests. In Fig. 5a, the yield curves depending on In t for various combinations of t e m p e r a t u r e and m o i s t u r e upon one-dimensional s t r e t c h i n g are shown. These curves m a y be reduced to one generalized dependence (Fig. 5c) c o r r e c t e d to given values of t e m p e r a t u r e and moisture. In this c a s e , the yield c u r v e s were c o r r e c t e d to 20°C and 0.7% m o i s t u r e content of the m a t e r i a l (relative to mass). The relaxation s p e c t r u m displacement function found according to displacements between neighboring yield c u r v e s is shown to be dependent on t e m p e r a t u r e and m o i s t u r e in Fig. 5b. Analogous data were obtained in t e s t s on c r e e p upon s h e a r and combined s h e a r and stretching. Analysis of the results obtained showed that the t e m p e r a t u r e - m o i s t u r e relaxation s p e c t r u m displacement function at negligible bulk c r e e p as seen in the tests is independent of the type of s t r e s s state. The displacement function at 20°C for t h r e e types of s t r e s s states (stretching, shearing, and combined shearing and stretching) is shown in Fig. 3. The function In a(T, w), as seen from the figure, is s a t i s f a c t o r i l y approximated with a c c u r a c y sufficient for p r a c t i c a l application by one equation for all tLle types of s t r e s s states listed above. All this p e r m i t s us to adopt a relatively simple analytical model taking into account the effect of t e m p e r a t u r e and moisture. The observation of linearity of the v i s c o e l a s t i c p r o p e r t i e s and the experimentally d e m o n s t r a t e d feasibility of neglecting the bulk c r e e p when the change in volume o c c u r s only elastically provides a b a s i s for using the following relationships as an analytical s t a r t i n g point: t

e,j= ~-~s~j(t) + ~K(t-t')sij(l') dff; O=cJtKo,

(1)

0

where eij and sij a r e components of the deformation deviator and s t r e s s deviator, a is the mean n o r m a t s t r e s s , 0 is the relative volume change, G is the modulus of s h e a r elasticity, and K0 is the modulus of bulk elasticity. Generalization of Eqs. (1) for the effect of uniform t e m p e r a t u r e fields and m o i s t u r e content of the m a t e r i a l was taken in the f o r m of the equations 1

eli

¢

1

sij(t) + J K(t-t', T, w)s(t')dt'; O - Ko(T, w) × 2G(T, ~ ) 0 X(~+3CzT(T)(T--To) +3aw(w) (W--Wo),

where ~T is the coefficient of linear t h e r m a l expansion and ~w is the coefficient of linear expansion for m o i s ture swelling. The p a r a m e t e r s and functions c h a r a c t e r i z i n g the creep of a m a t e r i a l were determined f r o m the

534

6

2'

8

9

t0

6

8

fO

t2

t4

t6

f8

20

22

Fig. 5. Yield c u r v e s of PN-3 at v a r i o u s combinations of t e m p e r a t u r e and m o i s t u r e [30]: a) initial c u r v e s ; b) functions of the t e m p e r a t u r e - m o i s t u r e displacement; c) g e n e r a l i z e d yield c u r v e c o r r e c t e d to 20°C and 0.7% moisture. T= 20°C (1-5), 40 ° (6-11), 45 ° (12-16), and 50 ° (17-22); w=0.5% (1, 6, 12, 17),0.7%(2, 7, 13, 18), 1.0% (8, 19), 1.25% (3, 9, 14, 20), 1.5% (4, 10, 15, 21), and 2.0% (5, 11, 16, 22). /D U~

0,3

-~5

1.3

Fig. 6. Relaxation s p e c t r u m displacement function found in t e s t s with various types of the s t r e s s e d state; f i l l e d t r i a n g l e s ) o n e - d i m e n s i o n a l stretching; open circles) pure s h e a r ; filled circles) shift with o n e - d i m e n sional stretching. fundamental s h o r t - t e r m s h e a r and s t r e t c h t e s t s . The c r e e p c u r v e s at s t r e s s , t e m p e r a t u r e , and m o i s t u r e constant over t i m e but v a r y i n g f r o m run to run were approximated by the equation

s~ Fbs~F(t, T, w), ei~=2G(T,w )

(2)

in which the t:ime function F was taken in the f o r m

h=l

T0h

where a(T, w) is the function of t e m p e r a t u r e and m o i s t u r e and T0k(k= 1,..., m) is the d i s c r e t e relaxation time spectrum. The determination of the p a r a m e t e r s appearing in Eqs. (2) and (3) was c a r r i e d out on a computer by minimizing the approximation e r r o r of the experimental data by an algorithm searching for the minimum of the

535

.~.

C

,Y

2

a

t" t 0 "~ h

f5

0,6 0,~ Fig. 7. L o n g - t e r m c r e e p upon stretching [331: a) experimental c r e e p c u r v e (open circles) and prediction r e s u l t s (line); b) e x p e r i mental t e m p e r a t u r e values; c) experimental values (open circles) and approximation (line) of m o i s t u r e content of material.

-

~12.#n~ cmVkgf

m

~

°

2,O:

- - "

~,0

a o

I

20 o

l

,as

,

,~

,

t40"*h ,.~

,

2,0 '

~

..........

,

2,5yearn '

]

Fig. 8. L o n g - t e r m c r e e p upon s h e a r [33]: a} e x p e r i m e n t (open circles) and c u r v e s calculated taking into account the effect of t e m p e r a t u r e and m o i s t u r e (1} and taking into account only t e m p e r a t u r e (2) ; b) experimental values (open circles) and approximation (dashed line) of t e m p e r a t u r e v a r i a t i o n ; c) the s a m e . f o r m o i s t u r e content of material. function of many variables using the slope method [34]. It was found that the dependence of the time function on t e m p e r a t u r e and moisture {reflecting the validity of the t e m p e r a t u r e - m o i s t u r e - - t i m e analogy} taken in the f o r m of Eq. (3) p e r m i t s us to d e s c r i b e all the s t a r t i n g s h o r t - t e r m t e s t s : ia s t r e t c h , s h e a r , and combined s h e a r and stretch. Thus, the function F(t~ T, w) may be written on an a r b i t r a r y scale z equal at constant values of t e m p e r a t u r e and m o i s t u r e to the product of the actual time by the function A (T, w) or integral for variable t

values of T and w: z = .[a[T(t'), wU')]dt'. 0

As a r e s u l t of this analysis, creep c h a r a c t e r i s t i c s were determined for a time interval exceeding the duration of the initial s h o r t - t e r m tests by a factor of approximately 105. What is the reliability of such a p r o g n o s i s ? This question must be answered by c o m p a r i n g the results of the prediction with the tests of given longt e r m control t e s t s . It was specifically with this aim that the l o n g - t e r m experiments, the r e s u l t s of which a r e given in Figs. 7-9, were c a r r i e d out. In Fig. 7, the curve for l o n g - t e r m c r e e p upon s t r e t c h is shown. The test lasted 6 y e a r s and the t e m p e r a t u r e was varied in the range 15 ° to 21°C. The m o i s t u r e content of the m a t e r i a l , as a result of seasonal f l u c tuations of the relative a t m o s p h e r i c humidity in the test site, v a r i e d cyclically. Most apparent is the nonmonotonic nature of the experimental c r e e p c u r v e : I n c r e a s i n g r a t e of c r e e p coincides in time with i n c r e a s i n g

536

i

2 !

t'lO -3u

a ............

~

t

tO. . . .

~__.~,_~',~ ¢.5

t~ ye#rs 2,0

.-Z=~-%---~

,.o w F - 6 - ~ ; ~ - 7 - ~

Fig. 9. L o n g - t e r m c r e e p upon c o n c u r r e n t s h e a r and s t r e t c h [33]: a) experimental (open circles) and calculated (line) creep curves ~C (1) and ~C2 (2); b) e x p e r i m e n t a l values of t e m p e r a t u r e ; c) exp e r i m e n t a l (open c i r c l e s ) and approximation (line) of the r e e l s t u r e content of a m a t e r i a l . .q,Co~.o

:

~'....... ;..... ,~ " : - .......

t./O "~ h 5

fO

t5

2O

Fig. I0

23

30

3

Fig. ii

Fig. 10. L o n g - t e r m Creep of PN-3 upon s t r e t c h : mean c u r v e of l o n g - t e r m control tests (1), the prediction according to the data of s h o r t - t e r m t e s t s for the m a t h e m a t i c a l expectation of c r e e p d e f o r mation (2), and the confidence range of the prediction curve at p = 0.95 (3). Fig. 11. The t e m p e r a t u r e dependence of the s h o r t - t e r m strength of polyvinyl chloride (a) and the limit of linearity of v i s c o e l a s t i c p r o p e r t i e s (b) [37]: 1, 2) c o u r s e of the t e m p e r a t u r e - s t r e s s action in control t e s t s ; the points are values of a and T in the fundamental tests for creep. m o i s t u r e content of the m a t e r i a l . With d e c r e a s i n g m o i s t u r e content, the deformation is sharply inhibited and even changes sign (the s a m p l e shrinks), which is explained by the p r e d o m i n a n c e in absolute value of the shrinkage of the m a t e r i a l o v e r the " s t r e s s " c r e e p deformation. We note that this o c c u r s as a consequence of fluctuations o[ the relative a t m o s p h e r i c humidity in the range f r o m 40 to 90%. Thus, this situation should be c o n s i d e r e d in developing methods and the c a r r y i n g out of l o n g - t e r m tests of p o l y m e r m a t e r i a l s . The a c c u r a c y of the prediction of l o n g - t e r m creep upon s t r e t c h may be evaluated from a c o m p a r i s o n of the experimental c u r v e and the analytical c u r v e obtained on the basis of the r e s u l t s of s h o r t - t e r m 5-h tests. The l o n g - t e r m c r e e p upon s h e a r is shown in Fig. 8 for a test duration of 3 y e a r s . The prediction on t;ne basis of the data f r o m s h o r t - t e r m tests is also presented in this figure. The seasonal fluctuations of t e m p e r a t u r e and m o i s t u r e content of the m a t e r i a l were analytically approximated by sinusoidal functions. Control t e s t s with the c o n c u r r e n t twisting and stretching of tubular samples were c a r r i e d out for 2 y e a r s . The e x p e r i m e n t a l c u r v e s and prediction r e s u l t s obtained a r e shown in Fig. 9. F r o m a c o m p a r i s o n of the c a l culated and e x p e r i m e n t a l control data (see Figs. 7-9), it follows that for all t h r e e types of s t r e s s e d state of a m a t e r i a l , the prediction of t o n g - t e r m c r e e p with few exceptions falls within the confidence intervals of the

537

[¢~

...............

a

t

° kgf/mmZ 0

2,o '='~

20

' ~

*0

b

b

(a ~Iga''

$0

2/2

o,8

t min Fig. 12

..... i Fig. 13

Fig. 12. C r e e p of polyvinyl chloride [37]: a, b) for the t e m p e r a t u r e - s t r e s s actions c o r r e s p o n d i n g to 1 and 2 in Fig. 11; the points c o r r e s p o n d to experimental testing and the line c o r r e s p o n d s to a calculation. Fig. 13. T e m p e r a t u r e displacement function (a) and generalized w e a r c u r v e of polymethyl m e t h a c r y l a t e (b) [47]: T=-- 38°C (o), - 2 0 ° ([])~ 20 (A), 35 (O), 50 (~), and S0°C (I). experimental control curves. The d i s c r e p a n c i e s between experiment and calculation found in s e v e r a l individual c a s e s may a r i s e as a consequence of the assumptions made in the calculation: The statistic s c a t t e r i n g of t e m p e r a t u r e s and m o i s t u r e contents which exists relative to the determined approximations T(t) and w(t) and the possible nonuniformity of the m o i s t u r e content over the bulk of a m a t e r i a l were not considered. Taking account of the r a n d o m fluctuations of the t e m p e r a t u r e and m o i s t u r e factors might i n c r e a s e the a c c u r a c y of prediction. Andrikson et al. [35, 36] p r o p o s e d c a r r y i n g out an evaluation of the statistical moments of the n o n - s t e a d y - s t a t e r a n d o m fluctuations of t e m p e r a t u r e and moisture on the basis of exponential smoothing. The feasibility of applying methods of the t h e o r y of n o n - s t e a d y - s t a t e r a n d o m p r o c e s s e s for predicting the c r e e p of p o l y m e r m a t e r i a l s in the n o n - s t e a d y - s t a t e random change of the t e m p e r a t u r e and m o i s t u r e conditions was d e m o n s t r a t e d (Fig. 10). These r e s u l t s r e l a t e to the region in which the linearity of v i s c o e l a s t i c p r o p e r t i e s a r e valid within p e r m i s s i b l e e r r o r . However, the relationships of linear v i s c o e l a s t i c i t y for many p o t y m e r s are valid in r e l a tively s m a l l r a n g e s of s t r e s s e s , and t h e s e r a n g e s n a r r o w with i n c r e a s i n g t e m p e r s ture. The t e m p e r a t u r e dependence of the limits of linearity of v i s c o e l a s t i c p r o p e r t i e s is s o m e t i m e s not considered in c a r r y i n g out a c c e l e r a t e d t e m p e r a t u r e tests. The p r o b l e m s thus entailed are shown by Efimova and Maksimov [37]. In Fig. 11, values for ~ and T at which p r e l i m i n a r y testing for c r e e p of potyvinyl chloride was c a r r i e d out a r e p r e sented in s t r e s s - t e m p e r a t u r e coordinates. The analysis of the data obtained indicated that the region in which the relationship between ~ and e may be taken as linear with an e r r o r not exceeding the experimental e r r o r is limited to c u r v e b; for c o m p a r i s o n , the t e m p e r a t u r e dependence of the s h o r t - t e r m ultimate strength is shown by curve a. If the limit of linearity is determined only at one value of t e m p e r a t u r e and the testing of the validity of the t e m p e r a t u r e - t i m e analogy is conducted at a s t r e s s level somewhat below the lowest found limit of linearity, then an i n c r e a s e of t e m p e r a t u r e only may lead to significant nonlinearity, which results in e r r o r in the check of the validity of the t e m p e r a t u r e - t i m e analogy. This is c l e a r l y shown by Efimova and Maksimov [37] in control tests for c r e e p with v a r y i n g t e m p e r a t u r e . The s t r e s s e s were constant, and the changes in t e m p e r a t u r e a r e shown in curves 1 and 2 (see Fig. 11); in the c a s e of c u r v e 1, the t e m p e r a t u r e change remained in the region of linearity, while in the case of curve 2, nonlinearity was reached. In Fig. 12, c r e e p curves are shown with points obtained f r o m testing and lines r e p r e s e n t i n g curves calculated according to relationships of linear t h e r m o v i s c o e l a s t i c i t y . In the first c a s e (see Fig. 12a), the experimental and analytical c u r v e s coincide, while in the second case (Fig. 12b), only a t e m p e r a t u r e change led to the violation of linear t h e r m o v i s c o e l a s ticity, which follows f r o m a d i s c r e p a n c y of the analytical and experimental c a r v e s . Thus, in organizing a testing p r o g r a m for checking the t e m p e r a t u r e - t i m e analogy, the t e m p e r a t u r e dependence of the limits of physical nonlinearity should be c o n s i d e r e d and this should find reflection in the techniques of a c c e l e r a t e d testing. We c o n s i d e r that such detailed attention is r e q u i r e d for this question because

538

the linear relationships of t h e r m o r h e o l o g i c a t l y simple bodies are used s o m e t i m e s without experimental checking sufficient for this purpose. 4. T h e r e is no n e c e s s i t y to elaborate on the question of the importance of predicting deformation p r o p e r ties in the region of nonlinear v i s c o e l a s t i c i t y . To this end, the relationships of physically nonlinear t h e r m o v i s c o e l a s t i c i t y obtained by II'yushin et al. [38, 39] may be used. These relationships were obtained on the basis of a broadened interpretation of the principle of the t e m p e r a t u r e - t i m e analogy on the assumption that all the t i m e p a r a m e t e r s included in the nonlinear relationships depend uniformly on t e m p e r a t u r e . In the most g e n e r a l f o r m of the relationship between d e f o r m a t i o n and s t r e s s in the f o r m of the F r e c h e t - V o l t e r r a equation t

t

t

e(t)=a(y(t)+ [K,(t-V)o(t'ldt'+ ~ [.K2(t-t', t-t")(~(t')(J(t")dt'dt"+ 0 t

0

t

0 0 t

0 0

the a r b i t r a r y time may be determined as an integral of the product of the t e m p e r a t u r e function and the differential of the actual t i m e t

t"

0

0

t"

0

as a result of which we obtain a time scale in which the deformation p r o p e r t i e s of a nonlinear v i s c o e l a s t i c m a t e r i a l will be independent of t e m p e r a t u r e . After substituting z, z', z",.., for t, t', t'~,.., according to Eq. (5), Eq. (4), which appears invariant for different constant and variable t e m p e r a t u r e s , in fact, will reflect a strong t e m p e r a t u r e effect. E x p e r i m e n t a l checking of this approach relates to the case of the nonlinear theory of one-dimensional v i s c o e l a s t i c i t y in which the relationship between s t r e s s and deformation is given in the f o r m of a sum of single i n t e g r a l s , i.e., a t h e o r y which is a special case of the m a j o r quasilinear theory of viscoelasticity. Such an equation in which linear and cubic t e r m s are retained has the f o r m t

t

~(t) --,~, (0 +b, j" .,'q(t-~', ,~.,) o(~') ~l'+ b~j" ,~(t-~', ,~,.~)[o (t')]~dt'. 0

(6)

0

The test results and technique for determining the p a r a m e t e r s and functions in Eq. (6) are given by Maksimov et ait. [40]. Maksimov et al. [40] and U r z h u m t s e v [41] showed in the case of various materials that the functions a'r~ and aT3, which r e f l e c t the t e m p e r a t u r e dependence of the c r e e p nuclei K i and K3, may coinride or, on the o t h e r hand, differ sharply. The c a u s e of this behavior is not yet c l e a r . Thus, these works p e r m i t us to make at least one conclusion: At p r e s e n t , having determined the t e m p e r a t u r e displacement function in the linear region, t h e r e is no basis for c a r r y i n g it into the region of significant nonlinearity without careful experimental checking. One of the m a j o r p r o b I e m s of predicting the r e s i s t a n c e of s t r u c t u r a l pIastics is the development of methods for the a c c e l e r a t e d determination of the l o n g - t e r m strength of these m a t e r i a l s . As e a r l y as 1966, on the basis of the dimension relations, I I ' y u s h i n a n d O g i b a l o v [42] gave a generalization of the phenomenological evaluation of the durability of p o l y m e r s , in which the destruction t e m p e r a t u r e and time were related by one universal function; i.e., the t e m p e r a t u r e - t i m e equivalence was assumed. Thus, P e k a r s k a s and Rayatskas [43, 44] showed this for the e a s e of the I o n g - t e r m s t r e n g t h of adhesive compounds of p o l y m e r films and GoI'dman and Grinman [45, 46] d e m o n s t r a t e d this behavior in the ease of high-density polyethylene. The t e m p e r a t u r e - t i m e dependence of strength may be described by the p a r a m e t r i c method using the L a r s o n - M i I l e r equation as well as the Zhurkov equation. However, c a r r y i n g out quantitative calculations of durability on the basis of extrapolating experimental curves according to the Zhurkov equation should be considered a v e r y r i s k y operation, because durability c u r v e s in s e m i l o g a r i t h m i c coordinates in the region of small and large s t r e s s e s deviate f r o m a linear dependence. This behavior is seen well in Fig. 13, in which the generalized durability c u r v e obtained by E r m o l o v and Potapov [47] for polymethyl m e t h a c r y l a t e was obtained; the method of reduced v a r i a b l e s was used in constructing the curve. The use of physical methods for studying the accumulation of damage in a m a t e r i a l upon l o n g - t e r m tests and a c c e l e r a t e d testing c a r r i e d out under " r i g o r o u s " conditions should facilitate the finding and development of methods for predicting l o n g - t e r m strength. An example of this approach is the work of Korsukov et al. [48],

539

....

5

0

7.,D;i

~0

r5

Fig. 14. Kinetics of the c o n c e n t r a t i o n of c h e m i c a l bond b r e a k a g e s of oriented polyethylene in c r e e p at v a r i o u s t e m p e r a t u r e s [48]; 40 k g f / m m 2. =

]

2.5 c • 10 "19 c m "3

2.0

In a t

a

f.5 o

ol 12

i O

ol

i

T-T~ "C 10

20

30

%.ol

#

5

o,5

......rto/i 2

3

~

Fig. 15

5

5

7

i

3

~

7

Fig. 16

Fig. 15. T e m p e r a t u r e d i s p l a c e m e n t s of c u r v e s for the c o n c e n t r a tion of c h e m i c a l bond b r e a k a g e s obtained under c r e e p conditions at 20 (I), 30 (2), and 50°C (3). Fig. 16. Temperature displacement function (a) and generalized curve for change in the chemical bond breakage concentration obtained from creep tests at various temperatures (b) : T = 20 (©), 30 (A), and 50°C (0); t0=l sec. who obtained c u r v e s for the concentration of c h e m i c a l bond b r e a k a g e s of oriented polyethylene in c r e e p up to d e s t r u c t i o n at v a r i o u s t e m p e r a t u r e s ; the r e c o r d i n g of the b r e a k a g e s was c a r r i e d out using i n f r a r e d s p e c t r o s c o p y (Fig. 14). The t i m e dependence of the b r e a k a g e concentration was d e s c r i b e d by these authors [48] by the f o r m u l a c (t) = c* ( 1 -- e-kd t), (7) w h e r e kd is the r a t e constant of destruction, the t e m p e r a t u r e dependence of which m a y be r e p r e s e n t e d as follows : kd(T) =k ~a(T- To). (8) It follows from Eqs. (7) and (8) that, in this case, the temperature-time equivalence holds for the concentration of defeats; the c u r v e s f o r e v e r s u s In (t/t0) at v a r i o u s t e m p e r a t u r e s should be displaced along the In (t/t0) axis (Fig. 15). Then, by d e t e r m i n i n g the t e m p e r a t u r e d i s p l a c e m e n t function, we m a y c o n s t r u c t a g e n e r a l i z e d c u r v e for the accumulation of c h e m i c a l bond b r e a k a g e s (Fig. 16). The expansion of studies on the p r o b l e m of predicting the r e s i s t a n c e of p o l y m e r s has stimulated i n t e r e s t in using a c c e l e r a t e d testing methods in industrial l a b o r a t o r y p r a c t i c e . The lack of standardized, regulated techniques and the d i s p a r a t e n e s s of the publications on prediction methods a r e among the r e a s o n s accounting for the difficulty in introducing t h e s e methods. Studies c a r r i e d out at the Institute of P o l y m e r Mechanics, A c a d e m y of Sciences of the Lativan SSR w e r e devoted to solving t h e s e p r o b l e m s . Thus, in 1973, methodological r e c o m m e n d a t i o n s for rapid t e m p e r a t u r e and s t r e s s t e s t s for d e t e r m i n i n g the l o n g - t e r m c r e e p of p o l y m e r m a t e r i a l s w e r e made [49]. Integration of the a c c u m u l a t e d data should facilitate the extensive v e r i f i c a t i o n of rapid testing methods [49]. Sections on t h e r m o v i s c o e l a s t i e i t y have a p p e a r e d in a n u m b e r of r e c e n t m o n o g r a p h s , in which the t h e r m o rheologically s i m p l e b e h a v i o r of m a n y p o l y m e r m a t e r i a l s established in n u m e r o u s e x p e r i m e n t a l studies [39, 50, 51] have found reflection.

540

Thus~ in the period f r o m 1971 up to 1976, the development of the methods for predicting the l o n g - t e r m r e s i s t a n c e of p o l y m e r m a t e r i a l s p r o g r e s s e d : The c l a s s of p o l y m e r m a t e r i a l s for which the feasibility of a c c e l e r a t e d testing using the analogy method was e s t a b l i s h e d was extended, concepts on planning a c c e l e r a t e d t e s t i n g in the l i n e a r and nonlinear regions of v i s c o e l a s t i c p r o p e r t i e s w e r e refined, and e x p e r i m e n t a l v e r i f i e a tion for the feasibility of p r e d i c t i n g d e f o r m a t i o n p r o p e r t i e s under conditions of a complex s t r e s s e d state taking into account changes in the t e m p e r a t u r e and m o i s t u r e e n v i r o n m e n t a l conditions, including the r a n d o m change of t h e s e f a c t o r s , was obtained. The i n t e g r a t i o n of the r e s u I t s obtained have p e r m i t t e d the development of r e c o m m e n d e d methods for use in industry for an o p e r a t i v e c o m p a r i s o n of the d e f o r m a t i o n p r o p e r t i e s of new materials. In coming y e a r s , it will be n e c e s s a r y to continue the study and development of methods of predicting the l o n g - t e r m d e f o r m a t i o n p r o p e r t i e s p r i m a r i l y in the nonlinear region of v i s c o e l a s t i c d e f o r m a t i o n at high s t r e s s leveIs. Speciai attention should be given to the development of p r e d i c t i v e methods for the l o n g - t e r m strength of s t r u c t u r a l p o l y m e r m a t e r i a l s f r o m rapid t e s t i n g techniques c a r r i e d out under " r i g o r o u s " conditions that r e s u l t in the a c c e l e r a t e d development of d e s t r u c t i v e p r o c e s s e s within the m a t e r i a I . LITERATURE i. 2. 3.

4.

5. 6. 7. 8. 9.

10. 11. 12. t3. 14.

15. 16. 17. 18.

CITED

Yu. S. Urzhumtsev, "Prediction of the deformability and destruction processes of polymer materials," Mekh. Polim., No. 3, 498-514 (1972). Yu. S. Urzhumtsev, ~The Second All-Union Conference on Polymer Mechanics," Mekh. Polim., No. 2, 378-380 (1972). A . A . Askadskii, Yu. I. Matveev, F. N. Nurmukhametov, G. L. Slonimskii, and B. D. T a r t a k o v s k i i , "The i n t e r r e l a t i o n s h i p of s t r e s s - r e l a x a t i o n p r o c e s s e s and a u d i t o r y - r a n g e v i b r a t i o n s , " Mekh. Polim., No. 2, 340-347 (1975). F . N . N u r m u k h a m e t o v , V. G. Dashevskii, A. A. Askadskii, and G. L. Slonimskii, "A study of the effect of v i b r a t i o n in the sonic f r e q u e n c y r a n g e on the s t r e s s - r e l a x a t i o n p r o c e s s in p o l y m e r s , " Mekh. Polim., No. 4, 579-590 (1976). R . D . Maksimov and Yu. S. U r z h u m t s e v , ~Wibrational c r e e p of p o l y m e r m a t e r i a l s . 2. Polyethylene. I s o t h e r m a l d e f o r m a t i o n r e g i m e , " Mekh. Polim., No. 2, 246-254 (1968). R . D . M a k s i m o v and Yu. S. U r z h u m t s e v , ~Vibrational c r e e p of p o l y m e r m a t e r i a l s . 3. Polyethylene. N o n i s o t h e r m a l d e f o r m a t i o n r e g i m e , ~ Mekh. Polim., No. 3 , 4 1 3 - 4 2 0 (1968). Yu. S. U r z h u m t s e v and R. D. M a k s i m o v , '~The v i b r a t i o n - t i m e analogy," Mekh. Polim., No. 4, 793-798 (1968). G . M . B a r t e n e v and L. A. Shelkovnikova, "The v i b r a t i o n a l relaxation of s t r e s s in r u b b e r s , " Mekh. Polim., No. 4, 720-723 (1972). A. Kania and I. Zawadzki, "Badania wibroptzania tworzyw t e r m o p l a s t y c z n y c h . B. P o l i m e t a k r y l a n metylu (Metapleks NO)," P r . Nauk. Inst. Materialoznaw. i Mech. Techn. PWr. Set. Stud. i M a t e r . , No. 20, 99-136 (1974). R . P . T u r t s i n y s h , Yu. S. U r z h u m t s e v , and R. D. Maksimov, ~Statie and vibrational c r e e p of f i b e r g l a s s upon s h e a r in the plane of r e i n f o r c e m e n t , " Mekh. Polim., No. 4, 605-610 (1971). R . P . T u r t s i n y s h , " E x p e r i m e n t a l study of the effect of vibration on the c r e e p of f i b e r g l a s s in a complex s t r e s s e d state, ~' Mekh. P o l i m . , No. 2, 358-360 (1973). Ch. L. Daugste and Yu. S. U r z h u m t s e v , "The sonic c r e e p of f i b e r g l a s s . 1," Mekh. Potim., No. 3, 397-404 (1973). Ch. L. Daugste, "Joint application of the t e m p e r a t u r e - t i m e and s t r e s s - t i m e analogies to cons t r u c t i n g g e n e r a l i z e d c u r v e s , " Mekh. P o l i m . , No. 3 , 4 2 7 - 4 3 1 (1974). G . M . Savitskii, "The conditions for the e x i s t e n c e of an analogy in d e f o r m a t i o n p r o p e r t i e s and a v e r i f i c a tion of the s t r e s s - t i m e analogy in the monaxiat s t r e t c h i n g of f i b e r g l a s s , ~ in: I m p r o v e m e n t of the R e l i ability and Durability of S t r u c t u r e s . An I n t e r u n i v e r s i t y Topical Science and Technology Review of the Leningrad Construction Engineering Institute [in Russian], No. 2 (1972), pp. 67-73. Yu. S. U r z h u m t s e v , Ch. L. Daugste, and A. V. K a l n r o z e , "The nonlinear t h e r m o v i s c o e l a s t i c i t y of p o l y m e r m a t e r i a l s belonging to the c l a s s o f r h e o l o g i c a l l y s i m p l e bodies," Mekh. Polim., No. 5, 778-784 (1974). S . B . Ainbinder, K. I. Alksne, 1~. L. Tyunina, and M. G. Laka, P o l y m e r P r o p e r t i e s at High P r e s s u r e s [in Russian], Moscow (1973). S . B . Ainbinder, V. A. Belyi, A. I. Sviridenok, M. I. P e t r o k o v e t s , and V. G. Savkin, " P o l y m e r friction," Mekh. P o l i m . , No. 2, 1135-1136 (1973). O . E . Ol'khovik, A. Ya. Gol'dman, and E. S. G r i g o r y a n , "The t e m p e r a t u r e - t i m e analogy in bulk c r e e p of c e r t a i n r i g i d p o l y m e r s , " Mekh. P o l i m . , No. 5, 930-936 (1974).

541

19. 20. 21. 22.

23. 24. 25. 26.

27. 28. 29. 30.

31.

32.

33. 34. 35. 36.

37.

38. 39. 40. 41. 42. 43. 44.

542

Yu. S. Lipatov, V. F. Babich, and N. I. Korzhuk, "The existence of a frequency-concentration superposition of filler in filled polymers," Vysokomolek. Soedin., Ser. A, 14, No. 7~ 1629-1635 (1974). Yu. S. Lipatov, V. F. Rosovitskii, and V. F. Babich, "The frequency-concentration superposition in the p o l y m e r - p o l y m e r filler system," Vysokomotek. Soedin., Ser. B, 16, No. 7, 512-514 (1974). V . G . Btmin and L. A. Irgen, "The effect of plasticizers on the viscoelastic properties of polyvinyl chloride and the c o n c e n t r a t i o n - t i m e analogy," Mekh. Polim. (in press). I.N. Cherskii, Yu. P. Kozyrev, V. A. Morov, A. A. Gerasimov, and N. A. Kovalenko, "The prediction of the viscoelastic behavior of Ftoroplast-4 Teflon at low t e m p e r a t u r e s , " Mekh. Polim., No. 4, 735-737 (1977). D.G. Fesko and N. W. Tschoegl, " T i m e - t e m p e r a t u r e superposition in styrene (butadiene)styrene block copolymers," Intern. J. Polym. Mater., 3, No. 1, 51-79 (1974). D. Kaplan and W. Tschoegl, " T i m e - t e m p e r a t u r e superposition in two -phase polyblends," Polym. Eng. Sci., 14, No. 1, 43-49 (1974). Yu. S. Urzhumtsev, " T e m p e r a t u r e - t i m e superposition for thermorheologically complex materials," Mekh. Polim., No. 2, 209-215 (1974). A. Ya. Gol'dman, G. Kh. Murzakhahov, and O. A. Soshina, "The t e m p e r a t u r e - t i m e analogy for thermorheologically complex polymer materials. 1. Mixtures of partially crystalline polymers with elastomers," Mekh. Polim., No. 4, 614-620 (i977). V.N. Efimova and R. Do Maksimov, "The determination of the t e m p e r a t u r e displacement function in the region of thermoviscoelasticity," Mekh. Polim., No. 4, 718-720 (1977). V . P . Mochalov, N. I. Ankisevich, and R. D. Maksimov, "The creep of PN-3 polyester resin under conditions of moisture absorption~ ~ Mekh. Polim., No. 4, 579-584 (1972). R . D . Maksimov, V. P. Mochalov, and Yu. S. Urzhumtsev, "The m o i s t u r e - t i m e analogy," Mekh. Polim., No. 5, 780-786 (1972). R . D . Maksimov, E. A. Sokolov, and V. P. Mochalov, "The effect of temperature and moisture content on the creep of polymer materials. 1. One-dimensional stretch under steady-state temperature and m(~isture conditions," Mekh. Polimo, No. 3, 393-399 (1975). R . D . Maksimov, E. A. Sokolov, and V. P. Mochalov, "The effect of temperature and moisture on the creep of polymer materials. 2. One-dimensional stretch under non-steady-state temperature and moisture conditions," Mekh. Polim., No. 6~ 976-982 (1975). R . D . Maksimov, V. P. Mochalov, and E. A° Sokolov, "The effect of temperature and moisture content on the creep of polymer materials. 3. Shear and concurrent shear and stretching," Mekh. Polim., No. 4, 627-632 (1976). R . D . Maksimov, V. P. Mochalov, and E. A. Sokolov, VThe effect of temperature and moisture content on the creep of polymer materials. 4. Prediction from rapid testing," Mekh. Polim., No. 6,982-987 (1976). A . F . Kregers, "Algorithm for finding the minimum of a function of many variables by the slope method," Algoritmy i P r o g r a m m y (Inform. Byull.), No. 2, 9 (t974). G.A. Andrikson and Z. V. Kalnroze, "Predicting creep of a polyester resin with the random action of temperature and moisture factors. 1. Stationary random factors," Mekh. Polim., No. 2, 219-226 (1973). G.A. Andrikson, Z. V. Kalnroze, and Yu. S. Urzhumtsev, "Predicting creep of polymer materials with random change of stress and t e m p e r a t u r e - m o i s t u r e environmental conditions. 2. Random nonstationary temperature and moisture factors," Mekh. Polim., No. 4, 616-621 (1976). V.N. Efimova and R. D. Maksimov, "A comparative analysis of the t e m p e r a t u r e - t i m e dependence of the deformation properties of polyvinyl chloride in the linear and nonlinear viscoelasticity regions," Mekh. Potim., No. 2, 213-219 (1977). A.A. II'yushin and P. M. Ogibalov, "Several basic questions in polymer mechanics," Mekh. Poiim., No. 3, 33-42 (1965). A.A. II'yushin and B. E. Pobedrya, The Basis of the Mathematical Theory of Thermoviscoelasticity [in Russian], Moscow (1970). R . D . Maksimov, Ch. L. Daugste, and E. A. Sokolov, "Features of the validity of the t e m p e r a t u r e - t i m e analogy in physically nonlinear creep of polymer materials," Mekh. Polim., No. 3,415-426 (1974). Yu. S. Urzhumtsev, "The temperature--time analogy. Review," Mekh. Polim., No. 1, 66-83 (1975). A.A. II'yushin and P. M. Ogibalov, "Criteria of the long-term strength of polymers," Mekh. Polim., No. 6, 828-832 (1966). V.-P. V. Pekarskas and V. L. Rayatskas, "Application of the t e m p e r a t u r e - c o n c e n t r a t i o n analogy for predicting the strength of adhesive polymer connections," Mekh. Polim., No. 168-~70 (1970). V.-P. V. Pekarskas and V. L. Rayatskas, "Predicting the durability of adhesive connections under stress by the method of the t e m p e r a t u r e - t i m e analogy," Mekh. Polim., No. 5, 937-940 (1974).

45. 46.

47. 48. 49. 50. 51.

A. Ya. G o l ' d m a n and A. M. Grinman, "A v a r i a n t of the t e m p e r a t u r e - t i m e analogy for p a r t i a l l y c r y s t a l line p o l y m e r s (high-density polyethylene)," Mekh. P o l i m . , No. 2 , 2 6 1 - 2 6 9 (t974). A. Ya. G o l ' d m a n and A. M. G r i n m a n , "The effect of t e m p e r a t u r e - t i m e f a e t o r s on the ultimate r e s i s t a n c e of c r y s t a l l i n e p o l y m e r s upon s i m p l e p l a n a r s t r e s s i n g (polytetrafluoroethyleae)," Mekh. P o l i m . , No. 6, 988-1001 (1975). S. 13. E r m o l o v and Yu. N. Potapbv, " E s t i m a t i n g the d u r a b i l i t y of p o l y m e r s in a wide t e m p e r a t u r e - t i m e r a n g e , " Mekh. P o l i m . , No. 4, 746-749 (1976}. V . E . K o r s u k o v , I. I. Novak, and P. M. Pakhomov, "The r o l e of b r e a k a g e s of c h e m i c a l bonds in the m e c h a n i s m of the d e f o r m a t i o n of p o l y m e r s , " Mekh. Polim. (in p r e s s ) . Yu. S. U r z h u m t s e v and R. D. M a k s i m o v , P r e d i c t i n g the D e f o r m a b i l i t y of P o l y m e r M a t e r i a l s [in Russian], R i g a (1975). P . M . Ogibalov, N. I. Malinin, V. P. Netrebko, and B. P. Kishkin, S t r u c t u r a l P o l y m e r s [in Russian], Books 1 and 2, Moscow (1972). V . V . Moskvitin, The R e s i s t a n c e of V i s c o e l a s t i c M a t e r i a l s . Applications to C h a r g e s of Solid-State Rocket Engines [in R u s s i a n ] , Moscow (1972).

TEMPERATURE

DEPENDENCE

OF

FILLED

OF

PLASTICIZATION

V.

POLYVINYL

G.

Bunin

OF CHLORIDE

IN

and

THE

L. A .

HIGHLY

Irgen

DEFORMATIVE OF

VARIOUS ELASTIC

PROPERTIES DEGREES STATE

UDC 539.374:678.01

Two b a s i c a p p r o a c h e s to the d e s c r i p t i o n of the i n c r e a s e in the stiffness of a p o l y m e r upon the introduction of a d i s p e r s e d m i n e r a l filler, which depend on two physical s t a t e s of the p o l y m e r - the v i t r e o u s and the highly e l a s t i c - exist. F o r a p o l y m e r in the v i t r e o u s state, the o r d e r of magnitude of the modulus of e l a s t i c i t y is c o m p a r a b l e with the magnitude of the modulus of e l a s t i c i t y of the f i l l e r ; t h e r e f o r e , during calculation of the s t i f f n e s s e s of c o m p o s i t i o n s , a s t a r t is often made f r o m the additivity of the pliabitities of s t r e s s e d components [1]. In the highly e l a s t i c state, the modulus of e l a s t i c i t y of a p o l y m e r is l e s s by four o r d e r s of mag~itude than the modulus of e l a s t i c i t y of the d i s p e r s e d m i n e r a l filler. The application of a load c a u s e s d e f o r m a t i o n of the m a t r i x , but the; filler r e m a i n s p r a c t i c a l l y undeformed. In this c a s e , during the calculation of i n c r e a s e in stiffn e s s , u s e is m a d e of equations which a r e b a s e d on hydrodynamic t h e o r i e s , at the b a s i s of which lies the Einstein equati:on, d e r i v e d for the v i s c o s i t y of a liquid containing d i s p e r s e d p a r t i c l e s [2]. A s e r i e s of equations which give s i m i l a r values and well d e s c r i b e the i n c r e a s e in the modulus of e l a s t i c i t y of a p o l y m e r upon the i n s e r t i o n of a filler exist. The m o s t widespread of t h e m is the G u t h - G o l d equation [2]. However, m a n y p o l y m e r i c c o m p o s i t i o n s in which the addition of a filler c a u s e s a m o r e effective i n c r e a s e in the modulus of e l a s t i c i t y than is p r e d i c t e d by t h e o r y a r e known. The c a u s e s of strong d i s a g r e e m e n t s of e x p e r i m e n t a l r e s u l t s with t h e o r e t i c a l equations should be sought in the f e a t u r e s of the s t r u c t u r e of a p o l y m e r in the p r e s e n c e of a filler. In [3, 4] it was shown that in p o l y m e r i c compositions the p o l y m e r displays o t h e r p r o p e r ties at the p h a s e boundary than the p o l y m e r in the m a t r i x , which is not subjected to the influence of the filler. The mobility of m a c r o m o l e c u l e s in the boundary l a y e r is d e c r e a s e d as the r e s u l t both of energetic interaction of the p o l y m e r with the s u r f a c e of the filler and of s t e r i c hindrances, as the r e s u l t of which the stiffness of the p o l y m e r in the boundary l a y e r p r o v e s to be of the s a m e o r d e r as the stiffness of the vitreous p o l y m e r . On the s t r e n g t h of this, the boundary l a y e r m a k e s an additional contribution to i n c r e a s e in the stiffness. T h e r e f o r e , a filled c o m p o s i t e should be looked upon as a t h r e e - c o m p o n e n t s y s t e m consisting of filler, boundary layer, and p o l y m e r i c m a t r i x , although, in the final a n a l y s i s , a distinct boundary between the boundary l a y e r and the p o l y m e r i c mat:rix, whieh is not s u b j e c t to the influence of the filler, does not exist. All-Union S c i e n t i f i c - R e s e a r c h Institute of the Use of P o l y m e r i c M a t e r i a l s in R e c l a m a t i o n and Water Management, Elgava. Institute of P o l y m e r Mechanics, A c a d e m y of Sciences of the Latvian SSR, Riga. T r a n s lated f r o m Mekhanika P o l i m e r o v , No. 4, pp. 646-650, J u l y - A u g u s t , 1977. Original a r t i c l e submitted May 27, 1976. This material is protected b y copyright registered in the name o f Plenum Publishing Corporation, 227 West 17th Street, N e w York. N.Y. l OOl l. N o part ] o f this publication may be reproduced, stored in a retrieval system or transraittecl in alzy form or by an), means electronic mechanical photocopying, microfilming, r*'.cording or otherwise w i t h o u t written permisMon o f the publisher A copy o f this article is available fro,n the publisher for $ 7.50.

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