EFFECT

OF AGGRESSIVE

RELAXATION

MEDIA

ON STRESS

IN POLYMERS

T . M. D z h u n i s b e k o v , a n d G. K . S t r o g a n o v

N.

I.

Malinin,

UDC 539.388.620.169

Following the a c c e p t e d t e r m i n o l o g y , given p a r t i c u l a r l y in [1], we can divide a g g r e s s i v e r e s p e c t to a c o n s i d e r e d solid into wstrong" and t w e a k ' . When ~strong" s u r f a c e - a c t i v e media a r a p i d failure of the s a m p l e [2] o c c u r s in the a b s e n c e of s o m e other c a u s e s which a c c e l e r a t e p r o c e s s (high s t r e s s level, high t e m p e r a t u r e ) , wWeak~ s u r f a c e - a c t i v e m e d i a acting on solids to an i m m e d i a t e failure of it, but reduce the t i m e - t o - f a i l u r e in the s t r e s s field [1, 3].

m e d i a with act on solids, the failure do not lead

In t u r n "weak m s u r f a c e - a c t i v e m e d i a can be divided with r e s p e c t to the solid i m m e r s e d in t h e m into solvent and nonsolvent. Molecules of solvent a g g r e s s i v e m e d i a p e n e t r a t e into an i n t e r i o r l a y e r of the solid, plastifying it and producing swelling. Solvent a g g r e s s i v e m e d i a b r i n g about not only plastffication and swelling of the m a t e r i a l s p e c i m e n but also dissolution of the s u r f a c e l a y e r s r e s u l t i n g in a d e c r e a s e , for e x a m p l e , of its t r a n s v e r s e c r o s s section. As shown by G. M. B a r t e n e v and c o - w o r k e r s [1-3] mweakm a g g r e s s i v e media acting on a p o l y m e r s a m p l e substantially reduce its l o n g - t e r m strength (mainly in consequence of a reduction of the activation e n e r g y U in the known S. N. Zhurkov relationship) so that in the usual calculation f o r m u l a f o r the strength 1o] = ~_~.

(i)

n

(where [cr] is a safe s t r e s s for a given t i m e - t o - r u p t u r e t; n is the strength safety factor) the ultimate s t r e n g t h ~b for a given t i m e - t o - r u p t u r e t m u s t be d e t e r m i n e d by e x p e r i m e n t s in the s a m e a g g r e s i v e m e d i u m in Which the p o l y m e r p e r f o r m s under r e a l working conditions. The magnitude of the s t r e s s e s acting in p o l y m e r c o m p o n e n t s is d e t e r m i n e d l a r g e l y by the r e l a x a t i o n p r o p e r t i e s of the m a t e r i a l [4] which in p o l y m e r s a r e m a n i f e s t e d to a c o n s i d e r a b l y g r e a t e r d e g r e e than in m e t a l s , f o r e x a m p l e [4, 5]. An a g g r e s s i v e m e d i u m , p e n e t r a t i n g into the i n t e r i o r l a y e r of a p o l y m e r s a m p l e a s a consequence of diffusion, filtration, etc., type phenomena, p l a s t i c i z e s it and a c c e l e r a t e s the r e l a x a t i o n p r o c e s s e s in it. Thus, i n o n e study [6] it has been e s t a b l i s h e d that an i n c r e a s e in the m o i s t u r e content of Nylon 66 by 1% a c c e l e r a t e s the c r e e p p r o c e s s by a f a c t o r of 101"45. Also, an active medium p e n e t r a t i n g into the i n t e r i o r l a y e r of a m a t e r i a l s a m p l e c a u s e s it to swell and if the p o l y m e r s a m p l e w o r k s under conditions c l o s e to those for r e l a x a t i o n of s t r e s s e s then the swelling r e s u l t s in a change in the initial s t r e s s field. The a i m of the p r e s e n t work is to obtain a s y s t e m of r e l a t i o n s h i p s f o r l i n e a r v i s c o e l a s t i c i t y (for the c a s e of o n e - d i m e n s i o n a l extension o r c o m p r e s s i o n ) which t a k e s account of the effect of plastification of the p o l y m e r by the active medium, p e n e t r a t i o n into the i n t e r i o r l a y e r of the p o l y m e r and the effect of swelling of the p o l y m e r during this, as well as to i l l u s t r a t e the applicability of the s y s t e m of r e l a t i o n s h i p s to a s i m p l e e x a m p l e . It is a s s u m e d that the active m e d i u m does not r e a c t c h e m i c a l l y with the p o l y m e r i c m a t e r i a l nor leach component p a r t s f r o m it. We a s s u m e f i r s t of all that p e n e t r a t i o n of the active m e d i u m into the i n t e r i o r l a y e r of the p o l y m e r s a m p l e o c c u r s as a r e s u l t of diffusion. The content of m e d i u m in unit volume of s a m p l e d e t e r m i n e s its c o n c e n t r a t i o n C. The differential equation of the diffusion p r o c e s s has the f o r m : M. V. L o m o n o s o v Institute of Mechanics, Moscow State University. T r a n s l a t e d f r o m F i z i k o K h i m i c h e s k a y a Mekhanika Materialov, Vol. 10, No. 5, pp. 55-59, S e p t e m b e r - O c t o b e r , 1974. Original a r t i c l e s u b m i t t e d May 6, 1973.

01976 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any fbrm or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15.00.

532

OC Ot

i

....

t h

~ D d i v grad C,

(2)

where t is the time, D is the coefficient of diffusion which can be a s s u m e d to be independent of the s t r a i n s and s t r e s s e s of the m a t e r i a l for not too high s t r e s s values of o- (for example, for (r < [cr]). F o r s t r e s s e s close to the ultimate strength ~rb strong c r a c k f o r m a t i o n o c c u r s in the m a t e r i a l [7, 8] resulting in disintegration; with this the value of D can substantially increase.

The relationship (2) does not differ f r o m the known equation for t h e r m a l conductivity; solution of a number of t h e o r e t i c a l boundary p r o b l e m s based on the use of this relationship will be found in the monograph [9]. Thus in the case of the p r o b l e m of axially s y m m e t r i c diffusion of a medium into a c i r c u l a r sample of radius a for initial values of C (r, 0) =0 and limiting values of C (a, t) = C . , t > 0 ( C , is the c o n c e n t r a tion of medium in a m a x i m a l l y swollen sample) the d e t e r m i n i n g conditions p o s s e s s the f o r m Fig. 1. Kinetic curve for the swelling in w a t e r of polyamide P-12 and.diag r a m of the swelling p r o c e s s .

C(r,t) = C,

2C, V~ e-~~

-

T

10(r~,)

~o [1 ( a s , )

rt~l

(3)

'

where I0 and I1 a r e z e r o and first o r d e r B e s s e l functions, and a n are the positive roots of the equation

I0 (a s 0 =0. The V o l t e r r a integral equation in the t h e o r y of linear v i s c o e l a s t i c i t y t

(4)

0

(R is the relaxation kernel, E 0 is the "instantaneous n modulus) for a solid plastified and swollen by a liquid medium has to be g e n e r a l i z e d by taking account of the following. The deformation ~ in equation (4) is part of the total d e f o r m a t i o n a , due to m e c h a n i c a l s t r e s s e s . The other p a r t of the deformation, due to swelling (~s) can, a c c o r d i n g to the data of [10], be taken as tic where/3 is a coefficient analogous to the coefficient of linear expansion which makes allowance for swelling of the m a t e r i a l under the action of a liquid. A c c e l e r a t i o n of the relaxation p r o c e s s when a p l a s t i f t e r is p r e s e n t by analogy with s i m i l a r p r o c e s s e s due to the effect of t e m p e r a t u r e [6, 11] can be allowed for by using the s o - c a l l e d reduced time [5] [

":t

-.

ae

o

'

(5)

I:

o

where the coefficient 1/a c d e t e r m i n e s how much m o r e rapidly the relaxation p r o c e s s o c c u r s at c o n c e n t r a tion G than at C =0. On the basis of this d i s c u s s i o n we have instead of the relationship (4) t'

o(t')=

E,, (.~ o - y "" ) -

t R ( t ' - ~') [ ~ . ( - . ' ) - - ~ C ( ~ ' ) ] d ~ '

.

(6)

? P e n e t r a t i o n of a liquid medium into a p o l y m e r can o c c u r not a c c o r d i n g to the laws of diffusion but a c c o r d i n g to the laws of filtration as through a porous material. It is known that during the p r o g r e s s i o n of a liquid in a porous m a t e r i a l (for example p e t r o l e u m through a l a y e r of sand) in the region where it has just r e a c h e d t h e r e is o b s e r v e d a distinct liquid front [12]. If penetration of a n ' a g g r e s s i v e liquid medium into a p o l y m e r sample o c c u r s with a front and a large e x t r a p r e s s u r e is not applied to the liquid then the sole t h e r m o d y n a m i c f o r c e driving the liquid into the i n t e r i o r l a y e r of the sample is the force resulting from the affinity of the liquid for the given p o l y m e r . This f o r c e does not depend on the path t r a v e r s e d by the liquid into the sample, and consequently the rate of spreading of the liquid front in the sample can be a s s u m e d to be a constant value.

533

The r e s u l t s of a determination of the rate of movement of water (u) in a sample of polyamide P-12 of d i a m e t e r d =5 m m and height h =10 mm are given in Fig. 1. In the same figure a d i a g r a m is given illustrating the p r o c e s s of advancement Of water into the depth of the sample. At time t =0 the d r y sample was i m m e r s e d in water. It was a s s u m e d that the advancement of water o c c u r s at the same rate in both the radial and the axial directions. The volume AV wetted with water (hatched a r e a in the diagram) is made up as follows:

p

P

V = 2 [ut (d"- "Jr"2dh) -- u" t" (4d -~- 2h)-[-4it :~t3].

Fig. 2. D i a g r a m used for r e s o l v i n g the p r o b l e m of the relaxation of s t r e s s e s in a rod of m a t e r i a l in the p r e s e n c e of an a g g r e s s i v e medium.

(7)

The quantity of liquid passing into the sample is AG=C,AV.

(8)

Equation (7) is valid up to the moment of complete saturation of the sample with liquid, i.e., for t <_ t , , subsequently instead of AV it iS n e c e s s a r y to use v, the total volume of the sample. ]?or the s a m p l e s in the d e s c r i b e d e x p e r i m e n t s the following relationship is valid (9)

t, = d . 2u

The rate of m o v e m e n t of w a t e r u is determined f r o m f o r m u l a (9), t , is the time f r o m which f u r t h e r i n c r e a s e in AG does not occur. The w a t e r content C , is the quotient of the division of the maximally attained value of AG by the volume of the sample V. In Fig. 1 the continuous line - points calculated a c c o r d i n g to Eqs. (7) and (8) - gives the experimental r e s u l t s . The s a m p l e s were p e r i o d i c a l l y withdrawn f r o m the v e s s e l of water, dried on filter p a p e r and weighed. At the same time the height of the sample was m e a s u r e d to determine the change in its g e o m e t r i cal dimensions during swelling. F r o m Fig. 1 it is evident that the experimental points s a t i s f a c t o r i l y coincide with the calculated curve. The hypothesis of frontal propagation of an a g g r e s s i v e liquid medium within a sample p e r m i t s considerable simplification in solving the p r o b l e m of relaxation of s t r e s s e s in components made f r o m polym e r i c m a t e r i a l s when acted upon by a g g r e s s i v e media. In the volume of the sample during this p r o c e s s t h e r e exist two regions of which one is free f r o m liquid and the other is occupied by liquid. As only some of the volume is occupied by the liquid it either i n c r e a s e s or s t r e s s e s develop in it due to swelling and t h e r e is a sudden d e c r e a s e in the relaxation time. As an example we c o n s i d e r the p r o b l e m of the strength of a linear v i s c o e l a s t i c c i r c u l a r rod of radius a (Fig. 2) which is d e f o r m e d at t =0; the deformation e , at t -> 0 r e m a i n s constant. Simultaneously at t =0 the region surrounding the rod is flooded with a nonsolvent a g g r e s s i v e liquid. The full load P(t) acting on the rod, can be d e t e r m i n e d if the tensile s t r e s s a(r) is integrated o v e r the whole c r o s s section of the sample and the p r o p e r t y of additivity of the integral (6) is used, both with r e s p e c t to time and with r e s p e c t to the deformation. We obtain: p (t) ---- ~

i ~ (r) dr = ~. (a -- ~,t)'-" .:, E (t) i t,I 0

(lo)

--[-2r162

rE

9 -ut

dr -t- 2~z--,

r E K--~--r

a--ut

t

Here

e s =tiC,,

K = 1 / a c , , the relaxation modulus E(t) = E0[- ~ R(t-~-)dT].

The relationship (10) is applied with 0 ~ t ~ a / u . F o r t > a / u the f i r s t t e r m on the right hand side of equation (10) d i s a p p e a r s ; the lower limits of the integrals of the other t h r e e t e r m s are zero. In another study [I3] e x p e r i m e n t s a r e d e s c r i b e d of the c o m p r e s s i o n of polyamide P-12 in water under conditions c o r r e s p o n d i n g to the stated p r o b l e m d e s c r i b e d above. With these for c o m p a r i s o n , the r e s u l t s

534

w e r e given of e x p e r i m e n t s f o r s t r e s s r e l a x a t i o n in a i r which is a s s u m e d to be an inactive medium. On the b a s i s of this work [13] and the data p r e s e n t e d in Fig. 1 as well as the e x p e r i m e n t a l d e t e r m i n a t i o n of the value r found to be 0.2% f r o m f o r m u l a (10), the calculated dependence P(t) was plotted. The value of K in a c c o r d a n c e with [6] was t a k e n to be [101"4~]~ =7.41 (0.6% was the w a t e r content of the m a x i m a l l y swollen sample). The i n t e g r a l s in f o r m u l a (10) w e r e d e t e r m i n e d by calculation f r o m S i m p s o n ' s f o r m u l a . Qualitative and quantitative a g r e e m e n t of the e x p e r i m e n t a l and c a l c u l a t e d dependence P(t) was obtained. In p a r t i c u l a r it w a s e s t a b l i s h e d that f o r c o m p r e s s i o n in w a t e r the c u r v e P(t) must be situated higher than that for c o m p r e s s i o n in a i r ; this was c o n f i r m e d by e x p e r i m e n t . All the e x p e r i m e n t s d e s c r i b e d above w e r e c a r r i e d out at a t e m p e r a t u r e of 20~ In the c a s e of the action of a solvent m e d i u m a solution front a r i s e s during the advance of which the t r a n s v e r s e c r o s s - s e c t i o n of the s a m p l e is d e c r e a s e d . LITERATURE 1o 2. -% 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

CITED

G~ Mo B a r t e n e v , I. V. R a z u m o v s k a y a , and E. M. K a r t a s h o v , Fiz.-Khim~ Mekhan. Mater., No. 5 (1967). V. Io Likhtman, E. D. Shchukin, and P. A. Rebinder, P h y s i c o c h e m i c a l Mechanics of Metals [inRussian], Izd. Akad. Nauk SSSR (1962). G . M . B a r t e n e v and I. V. R a z u m o v s k a y a , Doklo Akad. Nauk SSSR, 150, No. 4 (1963). Yu. N. Rabotnov, C r e e p of Construction Components [in Russian], Nauka (1966). P . M . Ogibalov, N. I. Malinin, et a l . , C o n s t r u c t i o n a l P o l y m e r s , Book I [in Russian], Izd. MGU (1972). W . H . Howard and M. L. Williams, Text. Res. J . , 3.66, No. 8 (1966). V . E . Gul', Structure and Strength of P o l y m e r s [in Russian], Khimiya (1971). E . E . Z i e g l e r , SPE J., 10, No. 4 (1954). H . S . C a r s l a w and J. C. J a e g e r , Conduction of Heat in Solids, Oxford Univ. P r e s s (1959). O . F . Shlenskii and V. V. L a v r e n t ' e v , P l a s t . Massy, No. 7 (1970). J . D . F e r r y , in: Die P h y s i k d e r H o e h p o l y m e r e n , Bd. IV, T h e o r i e und Molekulare Deutung T e c h n o l o g i s c h e r Eigenshaften von H o c h p o l y m e r e n Werkstoffen, S p r i n g e r - V e r l a g (1956). J. Buckley and M. C. L e v e r e t t , T r a n s . AIME, 14__.66 (1942). N . I . Malinin, V. N. K e s t e l ' m a n , and T. M. Dzhunisbekov, in: Relaxation P h e n o m e n a in P o l y m e r s [in Russian], Izd. AzGU, Vol. II(1974).

535