Experimental study of the volume compliance of polymeric systems in the glassy state

2440

O. YE. Ov.'KItOVIKa n d V. G. BARANOV

6. L. M. STRIGUN, L. S. VARTANYAN and N. M. FAKANUEL, Uspekhi khimii 37: 6, 969, 1968 7. T. J. STONE a n d S. A. WATERS, J. Chem. See., 213, 1964 8. L. R. MAHONEY a n d M. DAROOGE, J. Amer'. Chem. See., 92: 4, 890, 1970 9. A. V. RAGIMOV, B. A. MAMEDOVA, A. A. MEDZHIDOV and B. I. LIOGON'KII, Kinetika i kataliz 20: 2, 352, 1979 10. V. M. KOBRYANSKII, A. I. KAZANTSEVA a n d A. A. BERLIN, Vysokomol. soyed. B22: 666, 1980 (l~ot translated in Polymer Sci. U.S.S.R.) l l . A. V. RAGIMOV, B. A. MAMEDOV, V. Yu. ALIYEV, A. A. MEDZHIDOV, B. I. LIOGON'and C h . O . ISMAILOVA, Azerb. khim. zh., 4, 63, 1980

Polymer Science U.S.S.I~. Vol. 24, 1~o. 10. pp. 2440-2451, 1982 Printed in Poland

0032-3950/82 $7.50+.00 © 1983 Pergamon Press Ltd.

EXPERIMENTAL STUDY OF THE VOLUME COMPLIANCE OF POLYMERIC SYSTEMS IN THE GLASSY STATE* O. Y~. OL'K~OVrK and V. G. B ~ A ~ o v High Polymer Institute, U.S.S.R. Academy of Sciences

(Receiv~l 14 June 1981) A n experimental study has been made of volume strains for some polymers u n d e r uniaxial drawing, compression and hydrostatic pressure. The volume compliance in the case of a positive average stress is shown to be higher by almost one order t h a n the corresponding value under compression from all sides. The fracture process depends solely on the rupture of m a i n chains, and takes place as soon as the volume increase attains some definite value. The volume deformation is the main factor responsible for the physical nonlinearity of the creep process and b y introducing it into hereditary equations of viscoelasticity it is possible to describe nonstationary regimes of deformation (loading) in the complex stressed state.

RV.LATIONS~_rrS between pressure P, temperature T and volume V in polymers are a major source of information regarding properties and structural transitions occurring in polymers. The data are also of interest from the standpoint of optimization of technological processes in view of the marked tendency for pressure to rise during the synthesis of polymers and their fabrication into articles. In particular, it is of interest to determine relations between P, V and T with time while the polymer is simultaneously under the influence of shear and volume deformations. Kinetic features of volume changes under such con* Vysokomol. soyed. A24: No. 10, 2130-2138, 1982.

Volume compliance of polymeric systems in glassy state

2441

ditions m a y be used b y authors investigation physico-chemical aspects of the failure (rupture) of polymers. I t should be noted t h a t volume measurement:s of polymers in the synthesis of high polymers are used [1] as a measure of degrees of conversion in direct reactions. Reductions in polymer volumes under hydrostatic compression and she observed changes in polymer properties, e.g. increased strength [2] or redu(:ed creep [3] m a y be regarded as a continuation of direct reactions initiate(! b)' hydrostatic pressure. In addition, a reaction after the removal of pressure m~y be either reversible or irreversible. On the one hand a volume increase und~:' a positive average stress [4, 5] m a y be regarded as being a reversible rea(:~i,:)n whose course results in a reversible effect, i.e. in reduced strength or incrc~sed creep. Thus it is fair to assume t h a t important properties of high polymers such as strength and deformation m a y be analyzed as a function of the degree= ~)[" completeness of a reaction monitored in volume terms, allowing for ratio)us factors affecting the reaction kinetics, which will include the mechanical st~',:,ss field initiating direct or reversible reactions. From a practical standpoint the value of a study of volume deformation lies in the fact t h a t it could open the way to direct measurement of degrees of impairment of materials under a ]c)~(:[, and be used as a means of measuring the density of a material as a functi();~ of polymer properties. Despite these considerations information available in literature is not :~.ttogether adequate [4-7] and, in particular, there is a lack of data and results obtain(.'d by authors undertaking kinetic investigations of features of volume cha~ges with time, particularly in the case of materials under positive average Jew, is of stress. Our aim in the present instance was to investigate nonequilibrium volume values for a number of polymers (PVC, PMMA, resin for GIgPs based on E l ) - t 3 resin hardened with triethanolaminetitanate, and some composite materi~ds) subjected to simple deformations (stretching and compression strains). The to,st procedures and the equipment and specimens were described in [5, 7]. In Fig. 1 we have nonequilibrium values for the volume deformatiotz of PVC as a function of pressure at different temperatures under a load, the loading rate being 10 MPa/min. I n the range 293-355 K the isotherms draw ch,ser together as the pressure rises, but do not intersect. The isotherms conv(;rge up to a pressure of 100 MPa. At the same time the effective thermal expal~:,io~ coefficient is reduced 2.5-fold at 100 MPa compared with atmospheric pres~re. Above 100 MPa the volume coefficient of thermal expansion decreases 1.7-f(~ld in the temperature interval under consideration as the pressure l'ises to 250 5l['a.. Intersection of the compression isotherms starts at 355 K, as was found ~t.t pressures of 100 and 180 MPa. I t is clear from the intersection of the isothc':ms t h a t there is a range of temperature and pressure (above the intersection p(~int) where the effective value of the thermal expansion coefficient takes on a u e g ~ i v e value, i.e. a volume reduction occurs upon heating. This anomaly bears a rel~xa-

2442

O. Y~,. OL'XHOV~ and V. G. BABANOV

tion c h a r a c t e r a n d is caused b y a n increase in the v o l u m e creep w h e n t e m p e r a t u r e is reached, since the t i m e required for the equilibrium be established a t t h e lower t e m p e r a t u r e is too long. I t m a y well be first intersection, occurring a t 355 K, is due to a n g-transition, whose ~ture for PVC is 357 [8].

a certain state to that the tempera-

p, HPa x ~

0! • 2 03 xq +5 A6

200

/ , x ae,te A x~e

A

~pA •

IO0 A d Z~ A ~_

+

+ X O&o

+

X

×

o

O&o Ao

Io,~

~1

•

+xO&

I

I

o

-2

I

z

I

q e, o

FIC. 1. Compression isotherms for PVC at 293 (1), 313 (2), 339 (3), 355 (4), 372 (5) and 383 K (6).

~,MPo

!

GO o~, A,,

o-*. o~ V V v

LIO ~v i

20

~'vVv

0

T

v

3 VVVVV

vvvO v v ~-Vvv

eV .Voo °t°

Z

A ~AA~ A

V~¢ °e

°

1

•

ooo °

2o,%

Fro. 2. Tensile isotherms for PVC at 293 (1), 303 (2), 313 (3), 323 (4), 333 (5), 343 (6) and 355 K (7). The d a t a in Fig. 2 illustrate the v o l u m e increase for P V C u n d e r u n i a x i a l s t r e t c h i n g , wl~en t h e i r is a positive average stress am ( h y d r o s t a t i c pressure) in a specimen, a t different t e m p e r a t u r e s . F e a t u r e s o f t h e v o l u m e increase when t h e r e is a positive a v e r a g e stress, equal to one-third of the tensile stress value

Volume compliance of polymeric systems in glassy state

2443

[9, 10], differ m a r k e d l y f r o m the compression on all sides (see Fig. 1). l~or instance, the v o l u m e d e f o r m a t i o n is considerably larger, the h y d r o s t a t i c pressure levels being identical, p a r t i c u l a r l y w h e n t h e t e m p e r a t u r e is raised. PVC testing u n d e r conditions o f uniaxial e x t e n s i o n e n d e d in f r a c t u r e a t all t e m p e r a t u r e s , t h e f r a c t u r e occurring a t a relative v o l u m e increase of 1.8-1.9% (with the exc e p t i o n o f 355 K). I t appears to be correct t h a t v o l u m e d e f o r m a t i o n a t t h e m o m e n t o f f r a c t u r e is c o n s t a n t for a single physical substate. I n t h e case of p o l y m e r i c materials u n d e r a mechanical load a v o l u m e increase occurs as a result of a free v o l u m e increase, a n d on the o t h e r h a n d a similar p a t t e r n occurs also u p o n heating. F o r the l a t t e r case it is k n o w n [9] t h a t softening (a-transition) occurs as soon as t h e free v o l u m e fraction a m o u n t s to 2-3O/o. A q u a n t i t a t i v e e s t i m a t e of t h e t h e r m a l a n d mechanical softening, t a k i n g PVC (Fig. 2) a n d P M M A [7] as examples, shows t h a t mechanical softening occurs ¢~t a free v o l u m e level a p p r o x i m a t i n g to 2.3°/0 (after d e d u c t i n g the free v o l u m e in the glasslike state). T h u s for PVC in the glasslike s t a t e mechanical softening occurs w h e n there is a free v o l u m e increase of 2%, i.e. in the absence of m e c h a n ical loads the free v o l u m e f r a c t i o n for the glasslike state is 0.3%. I n a n d a b o v e t h e t e m p e r a t u r e region close to t h e a-transition the fraction of the free v o l u m e increase obtainable b y mechanical m e a n s begins to decrease to zero in t h e highelastic state, confirmation o f which is p r o v i d e d b y the d a t a in Fig. 2 (curve 7) a n d b y tests carried o u t on unfilled p o l y u r e t h a n e s . T h e influence of the a v e r a g e stress sign a n d o f t e m p e r a t u r e on the degree o f v o l u m e d e f o r m a t i o n is i l l u s t r a t e d b y the compressibility d a t a (see Table 1). I t is clear t h a t with a m > 0 the v o l u m e compressibility is considerably h i g h e r t h a n in the case of compression f r o m all sides, as is e v i d e n t in p a r t i c u l a r in t h e v i c i n i t y of the a-transition. Our results show t h a t u n d e r compression from all sides t h e v o l u m e modulus is m u c h higher t h a n the shear modulus, a n d accordingly,

TABLE

p, MPa 20 15 10 7.5 5.0 2.0 --2.5 10 --20 100 --200 -- 300 -

-

-

-

I. ISOTI~E~R~AL

COM.PI%ESSIBILITY

OF

PVC

fix 10 6 P a -I

293

fix 10~ (Pa -1) at temperatures, K 313 336 355 372

3"5 3"2 2"9 2"8 2"7 2"5 2"3 2'2 1"9 1"9 1'9 1"8

4"7 3"8 3"5 3"2 2"8 2"4 2"3 2"2 2'0 1"9 1"9

M

6"7 3'8 3"4 3"0 2"7 2'6 2"2 2"1 2"0

15.0 4.5 4.1 3.7 2.8 2.4 2.2

383 m

5"0 4-5 4"1 3'5 2"9 2.7

5'3 4.6 4.2 3.4 3"1 2"8

0 . YE. 0L'KHO~IK and V. G. BARANOV

2444

it is possible to fully accept that in the pressure region in question the incompressibility hypothesis used in engineering calculations is quite valid. Quite a different picture appears under a positive average stress. Thus under conditions o f uniaxial stretching of PVC the ratio of volume compressibility to tensile compressibility changes as the stress is increased under a creep regime from 15 to 35 MPa within ~he limits 0-69-1.63 respectively. Thus resistance to a volume change at ¢m~ 0 is much lower, and m a y be of a magnitude slightly exceeding resistance to a change in shape. o'm ~MPa

×c

f l /

I

1

~

t / - Y.X .,~,/,~

f/"

,

i

,,

-

,"

"Ix"w I

_

0"8

/

/

/

f

4o

0.*

O

-0.0

I

I

-0.8

0,%

Fia. 3. Compression isotherms for PVC (a), PMMA (b), epoxy resin for glass plastics (c), fabric composite on polyester resin (d) at 303 K; /--hydrostatic compression, 2--uniaxia[: compression (first cycle), 3--first loading cycle, 4--uniaxial compression second cycle, and 5--second loading cycle. Let us now turn to volume changes under uniaxial compression, the results of which appear in Fig. 3 on the coordinates average stress-volume deformation for four materials. Let us a t t e m p t a detailed analysis with PVC as an example. In the first cycle of uniaxial compression (up to ~-~6"5~o) the degree of volume deformation (curve 2) is slightly less than under compression from all sides (curve 1). When the polymer is under load, the uniaxial compression isotherm differs from the original mode of deformation; this is particularly noticeable in the range of low average stress. The degree of residual deformation (vohime)~ after the first cycle was ~-0-20~o. During the second uniaxial compression cycle the deformation was up to s----10~o. Up to a 6% linear deformation the PVC volume decreases under uniaxial compression, and then, as the linear deformatiol~,

Volume compliance of polymeric systems in glassy state

2445

increases, a volume increase occurs under a practically constant average stress. The residual volume increase for PVC after two uniaxial compression cycles was 0-82% . A qualitatively similar picture is observed for PMMA as well. The epoxy resin (binder) for glass reinforced plastics and the fabric component on a polyester base ruptured during the first cycle of compression. R u p t u r e of the resin and the composite under uaiaxial stretching is accompanied by volume increases of 0.62 and 0.53°/~) respectively. These two materials broke do~m under uniaxial compression with a volume increase similar to t h a t occurring under tension (taking the isotherms of compression from all sides as a frame ()f reference (curves 1), i.e. the safe state). To account for divergence of the isotherms for uniaxial compression from hose for compression from all sides we must consider a loosening of the material ~?,s a result of the action of tangential (shear) stress. I t should be noted t h a t under uniaxial compression over an area located at an angle of 45 ° to the axis ~)f the specimen we found t h a t the tangential stresses were highest and were equal, in the direction indicated, to one-half of the compressive stress. We will show t h a t under conditions of pure shear, equivalent to stretching of a small ~,ube i~l one direction and compression of the cube in a perpendicular direction, ~.~ volume change takes place. Applying a generalized form of Hooke's law to ~n elastic element t h a t is subjected to pure shear [ll] and considering the fact o f the inequality of elastic moduli [9, 11], we obtain under tension E~tr and compression E¢omp 0=(1--2z)

r

Ecom ~ '

(1)

where 1~ is Poisson's coefficient, z is the tangential stress: and ~ is the relative volume change. F o r viscoelastic media the elastic moduli in equation (1) must be replaced by integral operators. Direct tests were carried out using a device described in [6] and confirmed t h a t equation (1) is valid for torsion of PMMA and P T P E specimens. Quantitative relations of volume strains under shear call for separate investigation and are outside the scope of the present work. An a t t e m p t was made to investigate volume deformations occurring in clastomers (polyurethanes). I t was found t h a t for polyurethanes under compression from all sides the value of the volume elastic modulus is (3-4) × 103 Pa. In uniaxial stretching and compression tests with a relative linear deformation of up to 20~o we were unable to determine the extent of volume deformations because they were so small, since here the latter are outside the sensitivity limit for the instrument [6]. Apparently, elastomers have sufficient free volume to allow harmonization of molecular movements and displacements, whereas molecular motion is inhibited for polymers in the glass-like state. To shed fight on the problem of the relationship between breakdown processes and deformation we carried out creep tests for PVC under tension and

O. YE. OL'KHOVIK and V. G. BARANOV

2446

measured the volume deformation. The results are seen in Fig. 4. These d a t a were used to calculate the effective activation energy A U and the effective activation volume AV*, which were equal to 168 k J/mole and 2800 cmS/mole respectively. Such a high value of AU means t h a t the volume increase is accompanied by chemical bond scission. The value obtained for the activation volume shows t h a t when the molar volume of the PVC repeat unit is 62.5 cm3/mole, the volume of the kinetic unit is ~ 50 monomer units. Using the data in Fig. 4b together with the creep tests under hydrostatic compression we calculated volume compliance values for PVC over a 100 min period and a wide range of average stress (Fig. 5). I t can be seen t h a t with a negative average stress the volume

8,% 2.0-

~ a

•5

*4

b

2-01'7. G **5

..3

4,,

P •

° o

2

f'2•

I"2.

L°

o

°

°

• e

e oee

_

Do

°

•

•

0"4 ~*.

•

I 2O

1 •

•

•

•

•

e • e

lee

•

•

2

•

"

O#

I

I

I

GO

I

I

/O0

20

l

I

60

T/me, min

I

/

I

IO0

FIQ. 4. Volume creep of PVC under tensile stress at 30 MPa (a) and temperature 323 K (b); a: 1--303, 2--313, 3--318, 4--323, 5--328 K; b: 1--15, 2--20, 3--25, 4--27.5; 5--30, 6--32.5, 7--35 MPa.

)3 ~I 0~ Pa "1 2.9

,7

70. •

7

v

~• • °• ,Q•o p°

•8

•

•

•

•

•

•

4 •

°

•

•

•

°

•

"

-

;

-

i

:

:

I

17

5O

100

"/'[me, rain FIG. 5. Volume compliance of PVC at 303 K, average stress --250 (1), - - 5 0 (2), - - 1 0 (3), - - 5 (4), 6.6 (5), 8.3 (6), 9.2 (7), 10 (8), 10.8 (9) and 11.7 MPa (10).

Volume compliance of polymeric systems izt glassy state TABLE

2.

VOLUME

DEFORMATIONS

OF

P T F E ~N STRESS R E L A X A T I O N

TENSION

Time, rain

or, MPa

0, °/o

AND

1

2 3 5 7 10 20 30 40 60

0 8.0 7.70 7.54 7.37 7.20 6.98 6.64 6.15 5.80 5.65 5-6O

K x 10-~, Pa

0 1-06 1.07

1.06 1.04 1,02 1.02 1.01 1.00 1.00

0.99 1.00

PROCESSES

UND~,;R

COMPRESSIO~

o', MPa

2.52 2.41 2-37 2.36 2-35 2.28 2.19 2.05 1.93

1.90 1.87

[ t

0 18.6 15-6 14.5 14-3 13.4 12.8 12.3 11-9 ll-7 11.6 11.3

-----Kx 0, o~ i

It)

:(

Pa z ......... compression

. . . . . . . . . . . . . . I. . .

tensiou 0 0.5

2447

o o.102 .... 0.072 t).065 - 0.061 --0.059 -0.056 -- 0.053 - 0.051 --0.049 -- 0.048 -- 0.047

i

6.~b

7 ":t 7.4 7.6 7.6 7.7 7-~ 7.8 7.9 8.~ 8-1

c o m p l i a n c e fl is ( 2 - 3 ) × 10 -9 P a -1, a n d changes o n l y to a n insignificant e x t e n t w i t h t i m e . I n t h e p o s i t i v e a v e r a g e stress r a n g e (curves 4-10) t h e c o m p l i a n c e is higher b y one order, c o m p a r e d w i t h compression, a n d changes v e r y m a r k e d l y with time. A n a l y z i n g Fig. 4 it should be n o t e d t h a t P V C b r e a k d o w n u n d e r a creep regime, as u n d e r d e f o r m a t i o n conditions (Fig. 2) occurs a t a m o m e n t w h e n t h e v o l u m e increase is u p to 1.8-2-0~/o, a n d c o n s t a n c y of t h e v o l u m e d e f o r m a t i o n a t t h e m o m e n t of f r a c t u r e holds, e v e n w h e n t h e t e m p e r a t u r e is changed. O u r results show t h a t degrees of v o l u m e d e f o r m a t i o n m a y be used as a f a c t o r r e l a t i n g to t h e b r e a k d o w n o f materials. F r o m a p h y s i c a l s t a n d p o i n t this a p p r o a c h m o s t fully reflects processes t a k i n g place in a m a t e r i a l t h a t is u n d e r m e c h a n i c a l stress. T h e process of diffusion b r e a k d o w n is one of i n t e r n a l b o n d scission w h e r e e a c h act, of p h y s i c a l or chemical b o n d scission signals t h a t a v o l u m e increase h a s occurred. T h e use of v o l u m e d e f o r m a t i o n as a m e a n s o f m e a s u r i n g i m p a i r m e n t o f a m a t e r i a l w h i c h is responsible for b r e a k d o w n a n d d e f o r m a t i o n has s o m e a d v a n t a g e s (despite t h e difficulty e x p e r i e n c e d in m e a s u r i n g v o l u m e deform a t i o n s ) o v e r o t h e r p a r a m e t e r s , viz. stress or d e f o r m a t i o n , since t h e r e are situations where, if we t a k e stress or d e f o r m a t i o n (shear) as a m e a n s of investigation, we will be u n a b l e to deal w i t h s o m e p h y s i c a l aspects of t h e m a t t e r , or explain t h e r e a s o n for b r e a k d o w n in some cases. F o r instance, it is well k n o w n t h a t elongations o f h u n d r e d s o f p e r c e n t a g e s are o b t a i n a b l e for p o l y m e r s u n d e r a c t i v e d e f o r m a t i o n , w h e r e a s p o l y m e r s s u b j e c t to a creep r e g i m e h a v e b r e a k i n g elongations of several p e r c e n t a g e s , i.e. t h e c o n s t a n c y o f so no longer holds w h e n changes w i t h i n wide limits, or w h e n one stressed s t a t e is r e p l a c e d b y a n o t h e r (on going f r o m tension to c o m p r e s s i o n or torsion). As a n o t h e r e x a m p l e t h a t g r a p h i c a l l y illustrates t h e s e c o n d a r y role o f stress we m a y t a k e t h e r e l a x a t i o n

2448

O. Y~.. OL'KHOWXand V. G. Bn_~ANOV

breakdown of materials. In this case the deformation of a specimen remains constant, b u t the stress decreases continuously with time. Nevertheless break.down takes place after a certain period of time has elapsed. Direct measurement of the volume deformation of P T F E under a regime of stress relaxation under uniaxial tension and compression (see Table 2) showed that a loosening of the material takes place with time under these conditions. A quantitative analysis of degrees of loosening of materials m a y be illustrated b y plotting the volume elastic moduli against time. The data in Table 2 do not substantiate a widely held opinion about the role of initial stresses, e.g. in composite materials, when it is shown (sometimes .even experimentally) that the foregoing stresses decrease almost to zero, and m a y therefore be neglected. Actually, stresses that decrease with time m a y be y e t more dangerous since these, on relaxing, cause a loosening of the material. An analysis of the experimental data relating to volume creep shows t h a t an equation of state expressing volume deformation as a function of average stress m a y be written in the form of three integral equations for subregions L1, L2 and L a. Where the average stress for L 1 is positive, the volume creep is largely nonlinear, and corresponds to a hereditary viscoelasticity equation o f the t y p e associated with cubic theory [12]

O (am,t)= ~ am(r)-~ f lll(t--r)am(r)d 0

-~b f //a(t--r )

dr

0

Subregion L2 comprises a region of small negative stresses (up to 10-20 !YIPa). Here the volume deformations are linear as a function of pressure, and their magnitude is lower b y one order than in subregion L~. A volume change in subregion L 2 obeys the latter equation, retaining the linear term only. Subregion L a is in the range of high hydrostatic pressures (from 20 to 250 MPa) and is characterized both b y nonlinear behaviour, and b y the fact t h a t here the rate of volume creep and the degree of inelastic deformation are decreasing functions of average stress. For some polymers (PTFE, PMMA, PVC, high- and low-density PE, Caprolon, epoxy and polyester resins) the volume creep is negligible when the pressure exceeds 250 MPa, and is outside the limits o f operation of existing measuring methods [7]. An analytical representation for the volume deformation in L 3 obeys a high-elasticity equation proposed in [13] t

KO:fl(am) am+ f//(t--r) A(am) am (r) dr 0

On the basis of the volume creep data obtained in the present instance it is possible to explain some features of creep under shear. For instance, i t is well known that experimental points on creep curves on the coordinates stress intensity-time, plotted in torsion experiments, are located below those plotted in tensile tests and above those for uniaxial compression. This divergence

Volume compliance of polymeric systems in glassy state

2449

of creep carves has not yet been explained from a physical standpoint. On the basis of the present investigation we would conclude t h a t the reason for the divergence is a failure to allow for concomitant volume deformations. A volume increase during the stretching of a polymer accelerates creep in tension compared with uniaxial compression; on the other hand, the polymer volume decreases during compression (uniaxial), with the result t h a t the creep process in this ease is less rapid. Thus the results of our study of the role of volume deformation in shear creep show t h a t a deformation tensor first invariaut has to be introduced into traditional hereditary viscoelasticity equations [13]. Since a volume change has a direct influence on creep rates, it must be introduced into the kernel of a hereditary equation, which in our case will be written as t

2GE~j=S~j+ .{ H[(t--~), f~(0)].f~(a~, a~) S~I(~) dv

(2)

0

Since the kernel includes a deformation tensor first invariant we can now describe the creep in the case of an arbitrary stressed state, allowing for prehistory defined as volume deformation for a current moment of time. Equation (2) will describe various conditions of strain of materials whatever the form of the stressed state. Representing function f l as f~-----A~° (A, ~--being coefficients to be determined by experiments) it proved possible to describe the creep of a number of polymers under tension, compression and shear (and/or under pressure) using a single equation. Equation (2) has an important advantage in t h a t it m a y be used to predict creep in a third area, where accelerated deformation is due to a loosening o f the material. Let us now t u r n to the influence of volume deformation on the process of breakdo~m, observing t h a t the stable state of a polymer (absence of degradation, or the appearance of large deformations) under a mechanical load is definitely related to the magnitude of the specific volume, i.e. the stable state of an originally isotropic polymeric medium is disturbed and lost as soon as a volume increase reaches a certain level relative to the safe state. This holds in the case o f three simple deformations, viz. under tension, compression and shear. The relations in question are illustrated in Fig. 6 on the coordinates average stressvolume deformation. I n the case of uniaxial drawing the stable state is lost as soon as the volume increase reaches the level of 0er (curve 1). Under uniaxial compression a critical situation arises when there is an apparent volume increase (see Fig. 3, curves 2), though constancy of a volume increase at the moment o f breakdown or of flow is also realized under compression, ff we deduct the volume increase from the safe state, i.e. from the isothermal compression obtained under hydrostatic pressure. Under shear conditions breakdown takes place at zero average stress with volume increase up to the level of 0or (Fig. 6, curve 4).

2450

O. Y~-. O T . ' ~ o v , ~ and V. G. B~aA~ov

On the basis of the results of our s t u d y of volume deformation as a function of average stress and stress intensity, and their relations to time, it is now possible t o formulate the condition for long-term strength in the arbitrary stressed state as follows: Ol(ara, T, t)-FO2(~u, T, t)=Ocr=Const (3)

0comp~,o,

Ttm~

FIG. 7

FIG. 6

Fro. 6. Influence of volume deformation on the breakdown of polymers: 1--uniaxial extension, 2--hydrostatic compression, 3--uniaxial compression, d--shear. FIG. 7. Durability curves calculated by equation (5) under tension (1), shear (2) and uniaxial compression (3). Replacing volume deformations b y hereditary equations we can now determine the durability of a material in the arbitrary stressed state including nonstationary regimes of loading or deforn~ation. Let us consider the condition of long-term strength (3) on the assumption that a volume change with time is in line with the simplest relations of the type of

Now, taking equations (3) and (4) into account we m a y derive an expression for the time prior to breakdown •

I-0er-- (B1 aM + B2 au 7

~b = t o e x p I

L

_-=---~

vl aM+

J

(5)

Let us now make a qualitative comparison of durability under tension, compression and shear, using equation (5). We will assume in this case t h a t constants B I = B 2 = C~= 0 2 = 1. The relationship between normal and tangentiM stress m a y (let us say) be written as a = 2 v . This correlation is based on t h e experimental data on short-term strength. With the above assumptions t h e durability curves under tension and shear, calculated b y equation (5), practically coincide (Fig. 7), as was also found in the experiments [14, 15]. As regards uniaxial compression it should be noted that the stress level according to equation

Volume compliance of polymeric systems in glassy state

24fit

(5) m u s t b e higher b y a f a c t o r o f ~ 4 t h a n in t h e case o f tension. H o w e v e r , t h e r e a r e no e x p e r i m e n t a l d a t a on l o n g - t e r m s t r e n g t h u n d e r u n i a x i a l c o m p r e s s i o n , it is o n l y k n o w n t h a t w h e n stress levels u n d e r t e n s i o n a n d c o m p r e s s i o n a r e i d e n t i c a l no b r e a k d o w n occurs in t h e l a t t e r case [14]. T h u s t h e results o f this i n v e s t i g a t i o n s u p p o r t a n u n a m b i g u o u s conclusion t h a t v o l u m e d e f o r m a t i o n does h a v e a m a j o r influence on t h e s t r e n g t h a n d deformation of polymeric materials. By measuring the density of a material under l o a d it is possible to d e t e r m i n e t h e residual s t r e n g t h o f f a b r i c a t e d p r o d u c t s .

Translated by R. J. A. HENDRY REFERENCES

1. A. A. BERLIN, T. Ya. KEFELI and {~. V. KOROLEV, Poliefirakrilaty (Polyester Acrylates), p. 276, Nauka, Moscow, 1967 2. S.B. AINBINDER, K. I. ALKSNE, E. L. TYUNINA and M, G. LAKA, Svoistva polimerov pri vysokikh davleniyakh (Properties of Polymers under High Pressures). p. 190, Khlmiya, Moscow, 1973 3. O. Ye. OL'KHOVIK, Vysokomol. soyed. A19: 129, 1977 (Translated in Polymer Sci. U.S.S.R. 19: 1, 151, 1977) 4. O. Ye. OL'KHOVI'K, Mekhanika kompositnykh materialov, 4, 712, 1979 5. Ye. S. OS1P0VA, Z. V. MIKHAILOVA, L. V. BYKOVA and V. V. I~OVRIGA, Mekhanika polimerov, 2, 236, 1978 6. O. Ye. OL'KHOVIK, Vysokomol. soyed. A22: 611. 1980 (Translated in Polymer Sci. U.S.S.R. 22: 3, 674, 1980) 7. 0. Ye. OL'KItOVIK, Vysokomol. soyed. A18: 1012, 1976 (Translated in Polymer SoL U.S.S.R. 18: 5, 1158, 1976) 8. I. I. PEREPECHKO, Akusticheskiye metody issledovm~iya polimerov (Acoustic Methods of Polymer Research). p. 296, Khiraiya, Moscow, 1973 9. A. A. ASKADSKII, Deformatsiya polimerov (Deformation of Polymers). p. 448, Khimiya, Moscow, 1973 10. G. V. VINOGRADOV and A. Ya. MAX,KIN, Reologiya polimerov, p. 440, Khimiya, Moscow, 1977 11. A. K. MALMEISTER, V. P. TAMUZH and G. A. TETERS, Soprotivleniye zhestkikh polimernykh materialov (Resistance of Rigid Polymeric Materials). p. 500, Zinatiye, Riga, 1972 12. A. A. ILYUSHIN and B. Ye. POBEDRYA, Osnovy matematieheskoi teorii vyazkouprugosti (Mathematical Basis of the Theory of Viscoelasticity). 13. 280, Nauka, Mescow~ 1970 13. V. V. MOSKVITIN, Mekhanika polimerov, 6, 994, 1969 14. V. A. STEPANOV, Mekhanika polimerov, 1, 95, 1975 15. V. R. REGEL, A. I. SLUTSKER and E. Ye. TOMASHEVSKII, Kineticheskaya priroda prochnosti tverdykh tel (Kinetic Nature of the Strength of Solid Bodies). p. 560, Naukat Moscow, 1974