Stress relaxation in oriented polypropylene

Stress relaxation in oriented polypropylene

Materials Science and Engineering, 28 ( 1 9 7 7 ) 119 - 126 119 © Elsevier S e q u o i a S.A., L a u s a n n e -- P r i n t e d in t h e N e t h e r...

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Materials Science and Engineering, 28 ( 1 9 7 7 ) 119 - 126

119

© Elsevier S e q u o i a S.A., L a u s a n n e -- P r i n t e d in t h e N e t h e r l a n d s

Stress Relaxation in Oriented Polypropylene

D. M. S H I N O Z A K I

Departmenl o f Mechanical Engineering, University o f Manitoba, Winnipeg, Manitoba (Canada) G. W. G R O V E S

Department o f Metallurgy and Seienee o f Materials, University o f Oxford, Oxford (Gt. Britain) R. G. C. A R R I D G E

H. H. Wills Physics Laboratory, University o f Bristol, Bristol (Gt. Britain) (Received in revised f o r m N o v e m b e r 9, 1 9 7 6 )

SUMMARY

The stress relaxation of polypropylene, oriented by h o t drawing, has been studied after applying tensile strains of 5% and 15% at various angles to the draw direction. The stress relaxation experiments were performed at temperatures of 23, 36.5, 55 and 85 °C. At the three lower temperatures, the tensile stress, o, was a linear function of ln(t + c) where t is the time of relaxation and c is a constant, thus following a relationship which has been observed to apply repeatedly to stress relaxation in metals [3]. The results were interpreted in terms of plastic deformation in the form of shear parallel to the molecular axis arising from a single thermallyactivated rate process. In order to account for the orientation dependence of the rate of stress relaxation it was necessary to postulate that tensile stress normal to the molecular axis as well as the shear stress parallel to the molecular axis increased the rate of the process, and thus, both a normal stress activation volume and a shear stress activation volume were introduced. These were f o u n d to be of similar magnitudes, of the order of the volume of the crystalline unit cell.

INTRODUCTION

Recent work has examined the role of internal stress in the relaxation of spherulitic, cold-drawn and injection moulded polyethylene [1, 2]. The approach utilized the so-called power law derived for metallic systems:

0 - - 0 i = k ( t + a) -n

where o is the applied stress, oi is the internal stress which is assumed independent of time, t is the time and k, a, n are constants. By using a somewhat different approach in analyzing the relaxation data, again derived for metals, it is possible to gain some information about the mechanism of plastic deformation [3, 4]. The various modes of plastic deformation in oriented crystalline polymers have been examined recently [5 - 10]. The mechanisms are those involving shear processes (intermoleeular, interfibrillar and interlamellar), normal processes (interlamellar and interfibrillar separation), and twinning processes within the crystallites. In oriented polypropylene, X-ray diffraction studies and the anisotropy of yield behaviour have shown the predominance of the intermolecular shear process in room temperature deformation [11, 12]. The nature of this shear mechanism has not been examined in detail. Stress relaxation experiments have provided some insight into the discrete processes occurring during plastic deformation in metals, and a similar analysis was t h o u g h t to be applicable in oriented polymers. Furthermore, a recent examination of the deformation processes occurring at elevated temperatures has shown that a combination of intermolecular and interfibrillar shear account for the relaxation when specimens were annealed under fixed grip conditions after being deformed [ 13]. The conditions of testing were similar to those used in a stress relaxation experiment. It was expected,

120 therefore, that intermolecular and/or interfibrillar shear would be the dominant mechanism by which this material plastically relaxed.

EXPERIMENTAL Polypropylene, obtained from I.C.I. (Welwyn Garden City), designated PP1864, was compression moulded and slow cooled under vacuum. (This material was characterized by RAPRA* and was found to have a number average molecular weight, Mn = 3.1 X 105, and a weight average molecular weight, Mw = 1.97 X 106.) This moulded material was h o t drawn in air at 150 °C in an Instron tensile testing machine using the Instron environmental chamber. The total strain after drawing was approximately 600%, as measured by the decrease in cross sectional area. The hot-drawing was performed on specimens similar in size and geometry to those used in previous publications [11, 12]. Tensile specimens of the standard "dumb-bell" shape were cut from the drawn, cooled material at various angles (0 o) to the initial draw direction. These were ground and polished flat to approximately 0.50 mm thickness using water-cooled emery papers (Fig. 1}. The low angle and wide angle X-ray diffraction photographs show that the lamellae are oriented perpendicular, and the molecules are oriented, on average, parallel to the h o t drawing axis. Previous studies of yield anisotropy and deformation mechanisms have shown that oriented polypropylene tensile tested in the approximate range 20 ° < Oo < 70 ° deforms predominantly by intermolecular shear [12]. This was the range of orientations over which the stress relaxation experiments were conducted. At room temperature, a complete spectrum of orientations, 0 ° ~< Oo ~< 90 °, was examined. It was expected that the specimens outside the range 20 ° < Oo < 70 ° would exhibit somewhat different relaxation behaviour since the plastic flow mechanisms would be different. The stress relaxation experiments were carried out in an Instron with specimens and grips enclosed in the environmental chamber both at elevated temperatures and at room *Rubber and Plastics Research Association, U.K.

i

I.Ocml I

0.25cm

7.005c~ J

Fig. 1. Dimensions of tensile specimens. temperature. In the latter case, the chamber isolated the specimen from the surroundings. All relaxation experiments at room temperature were carried out over approximately 50 000 s. Linearity in the data plotted as applied stress (o) vs. logi0 (time) was observed. Later tests at larger strains and at higher temperatures were carried out over about 5000 s. The data were collected in the form of applied stress (o) as a function of time (s) at a given temperature (T), strain (eo) and initial orientation (O0). Temperatures of 23, 36.5, 55.0 and 85.0 °C were used. Two strains were considered, 5% and 15% elongation. The first is below the nominal yield strain of this material, while the second is close to the yield point. At each temperature and strain a number of orientations were tested (00 = 20 °, 30 °, 55 °, 70°).

RESULTS AND DISCUSSION The single activated rate process and stress relaxation

Previous investigations [9, 10] have shown that in h o t drawn polypropylene the mechanism of plastic deformation for 20 ° < O0 70 ° was predominantly shear parallel to the molecular axis. The stress relaxation experim e n t is a m e t h o d of gaining insight into the exact molecular mechanism which is responsible for the plastic deformation. An analysis of the data may be based on the following argument. In general, the total strain rate imposed upon a specimen during a tensile test is the sum of the elastic and plastic components of the strain rate, plus a term accounting for the machine deformation.

121 If the stiffness o f the m a c h i n e is large, t h e n the m a c h i n e strain rate is negligible. This a p p r o x i m a t i o n is particularly valid where the m a c h i n e is m u c h stiffer t h a n the s p e c i m e n (which is generally true when p o l y m e r s are tested on an Instron). In this case eT

=

ep

+

c~.

T = O sin 0 cos 0.

F o r a stress r e l a x a t i o n e x p e r i m e n t ~T = 0 = ~

is necessary t h a t T remains sufficiently large for the rate of the f o r w a r d process t o domihate t h a t o f the reverse process, which is neglected in eqn. (2)). The shear stress, r, is t h e r e f o r e the shear stress resolved in the d i r e c t i o n o f the molecular axis due to an applied tensile stress, a, at an angle, 0, to the m o l e c u l a r axis (4)

T h e n o r m a l stress c o m p o n e n t , On, is given by

+~.

On

Hence,

= O

sin 2 0.

(5)

E q u a t i o n (2) t h e n b e c o m e s

c , = --ee.

÷ H o G - A e x p [-- k--~ ] exp [~-~

7P -

Also, d c-e = ~ (E = m o d u l u s o f material).

(w sin 0 cos 0 + m sin 2 0)].

(6)

We can integrate eqn. (6) to give Hence,

~p -

d E

or qp -

÷ G

o-

(in shear, with G = shear m o d u l u s ) .

(1)

It is assumed t h a t the d e f o r m a t i o n can be described in t e r m s o f a single activated rate process, p r o d u c i n g shear d e f o r m a t i o n parallel to t h e m o l e c u l a r axis. Now, it has b e e n observed in yield stress studies o f o r i e n t e d polymers t h a t b o t h the resolved shear stress parallel to the m o l e c u l a r axis, T, and the n o r m a l stress c o m p o n e n t acting on planes parallel to the m o l e c u l a r axis, an, play a p a r t in t h e yield criterion [11, 1 6 ] . By analogy, we m a y e x p e c t b o t h these stress c o m p o n e n t s t o influence the rate o f the r e l a x a t i o n process, which we t a k e i n t o a c c o u n t b y writing ~p = A e x p - -

( H - - wT -- m ° n ) kT

(2)

w h e r e w and m are a c t i v a t i o n v o l u m e s f o r shear and n o r m a l stress c o m p o n e n t s , respectively. T h e stress c o m p o n e n t s in eqn. (2) are effective stresses and, f o r e x a m p l e , T = Tapplie d --

Tin t

(3)

w h e r e Tint is an " i n t e r n a l stress". Because o f the high t e m p e r a t u r e o f drawing in the preparation o f o u r s p e c i m e n s we will assume Tint = 0(Tint m a y also be a c o n s t a n t q u a n t i t y w i t h o u t a f f e c t i n g t h e analysis significantly, m e r e l y leading t o an e f f e c t i v e a c t i v a t i o n energy, H ' = H + w r i , t , b u t in o r d e r t o a p p l y eqn. (2) it

H K(O )

kT --ln(t+c)+ K(O )

kT kT + K(O---))ln[ A a ( w + m tan 0) 1'

(7)

where

K(O) = w sin 0 cos 0 + m sin 2 0.

(8)

T h e integration c o n s t a n t in eqn. (7) is d e f i n e d by C=

kT G A ( w + m tan 0)

H-exp ( --

WT0

kT

--

mano

),

(9)

where To and Ono are the shear and n o r m a l stress c o m p o n e n t s at t = 0. T h e e x p e r i m e n t a l time d e p e n d e n c e o f o follows the f o r m o f eqn. (7). T h e e x p e r i m e n tal p l o t o f a vs. l n ( t ) thus shows a deviation f r o m a straight line such t h a t each p o i n t is displaced along the t i m e scale b y a fixed a m o u n t , c. A t large values o f t, the c o r r e c t i o n term, c, is c o m p a r a t i v e l y small, h e n c e the o vs. l n ( t ) curve is linear at large t. T h e value for c can be o b t a i n e d f r o m the deviation f r o m linearity at t = 0, and linear plots o f o versus ln(t + c) are t h e n o b t a i n e d , as in Fig. 2. This value for c d e p e n d s u p o n T, w, m and ao (eqn. (9)). Thus, t h e r e is an analytic s o l u t i o n describing the stress r e l a x a t i o n b e h a v i o u r o f o r i e n t e d p o l y p r o p y l e n e , assuming (1) the p r e d o m i nance o f a single activated rate process; (2) the m a c h i n e hardness >> the stiffness o f the

122

35-

25

v

~15

I

101

I

I;2

'03 (t.c)

I~

105

Seconds

Fig. 2. Typical stress relaxation curves for oriented polypropylene deformed initially to 5% and relaxed at room temperature. Other curves at higher temperatures are similarly linear. solid. This a p p r o a c h , which has been used for metals with an activation v o l u m e for effective shear stress o n l y [ 3 ] , can be continued f u r t h e r b y e x a m i n i n g the o r i e n t a t i o n d e p e n d e n c e o f the relaxation. At the start o f the stress r e l a x a t i o n test, the s p e c i m e n has been s u b j e c t e d t o a strain which is significant (i.e. 5% or 15%). This results in a new o r i e n t a t i o n o f the m o l e c u l a r axis, 01, d i f f e r e n t f r o m the initial o r i e n t a t i o n , 0o [12] lo 01 = sin -1 ( ~ sin 00) where l0 and 11 are t h e initial and final specim e n lengths, respectively. T h e r e l a x a t i o n f u n c t i o n is t h e r e f o r e : o = a(T, 01) - - ~ ( T , 0 1 ) l n ( t + c)

(10)

where a(T, 01 ) c o n t a i n s all the time-independ e n t t e r m s and 6(T, 01) =

kT .

w s i n 0 1 cos 01 + r o s i n 2 01

(11)

In this w a y it is possible t o describe the r e l a x a t i o n o f stress as a f u n c t i o n o f t i m e and o r i e n t a t i o n . T h e basic a s s u m p t i o n s are: (1) a single activated rate process d o m i n a t e s the plastic d e f o r m a t i o n ; (2) the plastic strain is p r e d o m i n a n t l y interm o l e c u l a r shear; (3) t h e relevant stress c o m p o n e n t s are t h e resolved effective shear stress, r, and the n o r m a l stress, a n .

T h e e x p e r i m e n t a l stress r e l a x a t i o n curves were p l o t t e d as applied stress (o) vs. ln(t + c). F o r T = 21 °C, 36.5 °C and 55 °C, over the o r i e n t a t i o n ranges used, it was observed t h a t the o vs. ln(t + c) curves were linear. 9, the c o e f f i c i e n t o f ln(t + c) is t h e n o b t a i n e d f r o m the o vs. ln(t + c) plots (Figs. 2 and 3). Values for the d e f o r m a t i o n process p a r a m e t e r s , w and m, m a y be o b t a i n e d as follows. By plotting (9 sin 01 cos 01) -1 against tan 01, straight lines o f slope m / k T and i n t e r c e p t w / k T should be o b t a i n e d . When this is d o n e , the plots are a p p a r e n t l y straight lines, a p a r t f r o m the highest t e m p e r a t u r e ( T = 85 °C) (Figs. 4 - 7). At 85 °C, the o vs. ln(t + c) curve departs f r o m linearity at large times and the slope o f the curve at s h o r t e r times was used in p l o t t i n g Fig. 7. T h e m e a s u r e d values o f w and m are given in Table 1. T h e deviation f r o m linearity in v vs. ln(t + c) at T = 85 °C at long times is n o t surprising (Fig. 8). T h e applied stresses involved in these tests are v e r y low and m a y reach values c o m p a r a b l e w i t h an internal stress during the test. General

T h e application o f rate t h e o r y in a general f o r m t o the plastic d e f o r m a t i o n o f solids does rmt require a specific m o l e c u l a r mechanism. Initially it is assumed t h a t the plastic strain is a c c o m m o d a t e d b y the m o v e m e n t of molecules from one position to another, passing over an i n t e r m e d i a t e activated state.

123

t "--ob -15

I

I

101

-[02 (t+c)

l

103 SecOnds --.-

104

105

F i g . 3. T y p i c a l s t r e s s r e l a x a t i o n c u r v e s f o r o r i e n t e d p o l y p r o p y l e n e d e f o r m e d initially t o 15% a n d r e l a x e d at r o o m t e m p e r a t u r e . O t h e r c u r v e s at h i g h e r t e m p e r a t u r e s are s i m i l a r l y linear.

_1¸

-10 -2-

I 1+



2:I

I~ -2.0

-30

o I

-2-~

0.5

0

I

1.0

1-0

1.5

2.0

Fig. 6. P l o t s d e r i v e d f r o m T = 55 °C r e l a x a t i o n d a t a t o d e t e r m i n e m a n d w.

-3.( I

I

Tane I

5°/°

I

I

I

2.0

Tan OI F i g . 4. P l o t s d e r i v e d f r o m t h e r o o m t e m p e r a t u r e l a x a t i o n d a t a t o d e t e r m i n e m a n d w.

re-

0to

o

\

i

~-4-0

~-.5

-

2.0[

° -

~ = 5°/o~"

£o:e5°Io

4.5

o!~

~!o Tan e 1

I

0.5

I

10 Tan {91

I

1.5

~!s

2!o

i

2.0

Fig. 5. P l o t s d e r i v e d f r o m t h e T = 37 °C r e l a x a t i o n d a t a t o d e t e r m i n e m a n d w.

Fig. 7. P l o t s d e r i v e d f r o m T = 85 °C r e l a x a t i o n d a t a t o d e t e r m i n e m a n d w. T h e e 0 = 5% p o i n t s are w i d e l y s c a t t e r e d b e c a u s e t h e c u r v e s in o v s . l n ( t + c) are non-linear.

124

"5 30"

t b~ -2

-1

,01

192

,o3

(uc)--~ Fig. 8. Typical I" v s . l n ( t + c) c u r v e f o r T = 85 °C s h o w i n g n o n - l i n e a r i t y at l o n g e r times. TABLE 1 T (°K)

Pre-strain (%)

m ((M 3) x 1027)

w ((M 3) x 1027)

298 298 310 310 328 328 358

5 15 5 15 5 15 15

3.0 3.6 2.3 3.0 3.3 3.8 1.9

3.0 1.9 4.2 2.4 6.2 3.5 1.2

This process o f moving through an activated state may involve any one of a n u m b e r of molecular m o v e m e n t s or combinations of them. But in the simplest case, there is only one kind of molecular process, and this gives rise to th e strain rate being limited by a single activated rate process. In an u n o r i e n t e d p o l y m e r , where the organization of the molecules and larger morphological features such as lamellae is r a n d o m over a bulk specimen, the relationship between an applied stress and the local aggregations varies, depending u p o n the orientation of the lamellae. If the plastic strain occurs by a stress activated molecular process, a range of relaxation rates will be observable in the u n o r i e n t e d material since the different lamellae are under the influence o f different resolved shear stresses. The use o f an oriented crystalline p o l y m e r such as p o l y p r o p y l e n e simplifies the analysis, since previous work has shown, i ndependent l y,

the p r e d o m i n a n c e of one shear m ode of plastic d e f o r m a t i o n . The observation that the stress relaxation is linear when plotted vs. ln(t + c) simply reinforces the previous conclusion t hat some form of plastic shear or slip occurs within the crystalline phase. The single activated rate process seems reasonable, therefore, in the light of the possibility of some discrete crystallographic mechanism. The orientation d e p e n d e n c e of the slopes of the stress relaxation lines (Fig. 2) is reminiscent of the orientation dependence of the tensile yield stress in oriented polymers [11, 16] in t h a t a reduction is observed as 0 increases from 0 t o 45 ° and a low value is maintained as 0 increases b e y o n d 45 °. In either case, this orientation dependence canno t be explained solely on the basis of the variation with 0 of the resolved shear stress in the molecular axis direction, a sin 0 cos 0, which is symmetrical a b o u t 0 = 45 o. A role must be assigned to the normal stress c o m p o n e n t , a sin 2 0, and in the case of the relaxation data, this is most simply done by introducing a normal stress activation volume in addition to the shear stress activation volume. The magnitudes of w, the shear stress activation volume, measured in these experiments are reasonable, being of the order of the volume of the unit cell in the p o l y p r o p y l e n e crystal. T h e y are in agreement with the magnitudes r e p o r t e d by Li e t al. [ 1 5 ] . The structural model of a defect mechanism of plastic flow is consistent with Li e t al.'s observations

125 and previous X-ray observations on this material [ 12 ]. Similar values for the shear stress activation volume derived from analysis of yield in various polymers by assumption of an activated rate process are given by Haward and Thackray [17]. They compare these values with the volume of the statistical link in solution, finding the "Eyring v o l u m e " to be 2 - 10 times that of the statistical link. Duckett, Rabinowitz and Ward [18] using Robertson's molecular theory of yield modified for pressure effects found activation volumes for yield in p o l y m e t h y l methacrylate and polyethylene terephthalate of 109 × 10 -3° m 3 and 234 X 10 -30 m 3, respectively, an order of magnitude less than the "Eyring volumes" derived here and by Haward and Thackray. It would clearly be of value to perform stress relaxation experiments on several different polymers for comparison with studies of yielding. The magnitudes of m, the normal stress activation volume, are similar to those of w. These values of m may seem to be surprisingly high in view of the fact that in other studies the observed macroscopic mechanism of plastic flow has been a shear deformation parallel to the molecular axis [12]. However, it may be noted that where shear and normal stress components have been combined in a Coulomb criterion for the yield stress of oriented polyethylene [16], the coefficient of the normal stress term is as high as 0.45 times that of the shear stress term. While yield stress and stress relaxation observations are not directly related to one another it is interesting to note t h a t in both cases the shear stress and normal stress components can be assigned roles of comparable importance in order to account for the orientation dependence of the measurements. One assumption which may give rise to difficulty when related to polymers is t h a t the total strain rate is separable into elastic and plastic components, i T = ~p + ~e. In metals, this arises from the accepted theory that dislocation motion is responsible for the plastic strain (~p), while truly elastic behaviour {movement of atoms within their potential wells) is responsible for the elastic strain (ee). In a polymer, the separation between elastic and plastic components of flow is n o t so clear. The motion of a molec-

ular segment from one low energy configuration to another may n o t be a sufficient condition for the flow to be permanent in the metallic sense. The constraint of adjoining segments of the same molecule may result in the reverse movement of the segment into its former position after the stress is removed. In this sense, such "elastic" strain may be an activated rate process. This concept of "elastic-plastic m e m o r y " is consistent with the recent application of Asaro's model of memory solids to crystalline polymers [14]. In these experiments, the use of a crystalline polymer presumably reduces this problem since the elastic constraint discussed must come only from the fold surfaces of the lamellae or interlamellar material.

CONCLUSIONS It has been shown that stress relaxation experiments can provide useful insight into the process of plastic flow in crystalline polymers. In particular, plots of log (t + c) v s . a for highly oriented polypropylene have been found to be linear over a wide range of relaxation time, temperature and orientation, and in this the polymer may be compared with metals where linearity of this plot has been repeatedly verified [3]. The orientation dependence of the stress relaxation could be explained on the basis of thermally activated shear parallel to the molecular axis, provided that an activation volume for the component of stress normal to the molecular axis, of similar magnitude to the resolved shear stress activation volume, was introduced. The two activation volumes have magnitudes, on the order of the volume of the crystalline unit cell.

ACKNOWLEDGEMENTS This work was performed while one of the authors (D.M.S.) was supported by a National Research Council of Canada post-doctoral fellowship at the H. H. Wills Physics Laboratory. Thanks are due also to Professor Keller for his provision of laboratory facilities and • his help in the course of these experiments.

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10 D. P. Pope and A. Keller, J. Mater. Sci., 10 (1975) 747. 11 D. Shinozaki and G. W. Groves, J. Mater. Sci., 8 (1973) 71. 12 D. Shinozaki and G. W. Groves, J. Mater. Sci., 8 (1973) 1012. 13 D. Shinozaki and G. W. Groves, to be published in J. Mater. Sci. 14 C. M. Sargent and D. M. Shinozaki, to be published. 15 J. C. M. Li, C. A. Pampillo and L. A. Davis, in H. H. Kausch, J. A. Hassell and R. I. Jaffee, (eds.), Deformation and Fracture of High Polymers, Plenum Press, New York, 1973. 16 A. Keller and J. G. Rider, J. Mater. Sci., 1 {1966) 389. 17 R. N. Haward and G. Thackray, Proc. R. Soc. London, Ser. A, 302 (1968) 453. 18 R. A. Duckett, S. Rabinowitz and I. M. Ward, J. Mater. Sci., 5 (1970) 909.