Non-linear mechanical behaviour of oriented polypropylene

J. Me&. Phys. Solids, 1963, Vol. II. pp. 217 to 229. Pergamon PressLtd. Printedin Great Britain.

NON-LINEAR

MECHANICAL

ORIENTED By

I. 3%. Brown

BEHAVIOUR

OF

POLYPROPYLENE* WARD-~

University,

and E. T. Providence,

ONAT R.I.

(Receiued Sfh March, 1963)

THIS paper is concerned with the study of isothermal extensional deformations of an oriented polypropylene monofilament under time-dependent loading. The monofilament has been subjected to various loading programmes, each containing a number of loading steps. It is found that the monofilament exhibits non-linear memory dependent behaviour. The question of mathematical representation of the observed non-linear behaviour is considered. It is assumed that the elongaCon at a given time is a non-linear functional of the stress-rate history to which the filament has been subjected prior to this instance. It is then shown that the experimental results can be represented, with reasonable accuracy, by approximating this functional by the sum of a linear It is seen that the loading programmes employed in the and third order hereditary functional. study provide substantial information for the construction of the kernels defming the two functionals.

1.

INTRODUCTION

IN GENE~~AL, amorphous polymers in the temperature range of the glass-rubber transit,ion display linear viscoelastic behaviour at low strain levels (- l per cent) (FERRY 1961). This is also true for crystalline polymers within a similar but usually lower range of strain levels [e.g. polythene (KOLSKY 1960)]. For such materials there are many comprehensive studies of viscoelastic behaviour. A good example of this type of investigation is the work on pofyisobutylene reported by MARVIN (1954). In materials of high molecular orientation such as fibres, on the other hand, it is well known that the viscoelastic behaviour can be non-linear. In view of the complexity of the situat.ion a variety of different types of measurements have been made with such materials and the results have, in general, been represented either by empirical power laws relating stress and strain with time (PEIRCE 1923; PRESS and MARK 1943) or by postulating specific non-linear behaviour in terms of mechanical models with non-linear springs and ‘ activated dashpots ’ (HALSEY, WHITE and EYKING 1945). An alternative approach, proposed by Leaderman, was to attempt to derive a modified Boltzmann superposition principle (LEADERMAN 1943,

1961).

*The work reported in this paper was supported in part by Material Hewarch Program, Advanced Research Projects Agency, Department of Defense. ton secondmentto the Division of Applied Mathematics,Brown University from I.C.I. Ltd., Fibres Division, Harrogate, England. 217

I. M. WARD and E. ‘I’. ONAT

218

The

present

deformations loading.

paper

is concerned

of an oriented

The monofilament

with

the

polypropylene

study

of isothermal

monofilament

has been subjected

under

to various

extensional

time-dependent

loading

programmes;

each containing a number of loading steps, and the resulting time dependent elongations have been measured. The discussion of experimental results given in Section 4 shows that the monofilament exhibits non-linear memory-dependent mechanical behaviour. The question of a rational mathematical representation of the observed non-linear

behaviour

is considered

in Section

5.

It is assumed in this section that

the elongation at a given time is a non-linear functional of the stress-rate history to which the filament has been subjected prior to this instance. It is then shown that

the experimental

approximating functional. substantial

results

this functional

can be represented

with

reasonable

accuracy

by

by the sum of a linear and a third order hereditary

It is seen that the loading programmes information for the construction of

employed in the study provide the kernels defining the two

functions. 2.

In this section and the response

LINEAR

VISCOELASTIC

the loading

programmes

of a linear

viscoelastic

BEI~AVIOUH

employed solid

in the study

to these loading

discussed. As will be seen later, this discussion is of importance the non-linear nature of the material considered. There elastic

exist

several

behaviour

for

mathematical

representations

in bringing of linear

is out

visco-

example,

GROSS (1953) and KOLSKY (1959)]. We here with the integral representation appropriate to the uniaxial

shall be concerned stressing

equivalent

[see,

are described programmes

considered

in this paper :

e(t)

= --m

where e (t) is the elongation is the time dependent

uniaxial

I

J(t

-

T)?&

(per unit length)

of the specimen

at time t and (J (T) This represen-

stress, and J is the creep compliance.

tation can be considered to arise as a consequence of the Boltzmann superposition principle. It may also be noted that according to (2.1) the strain is a linear, continuous

hereditary

functional

Let e, (t, u,,) denote programme

of the stress history.

the strain response

U(T)==O, For a linear viscoelastic

(T(7) = ~7~= const.,

7<0; material

in a creep test defined

it follows

by the loading

7 > 0.

(2.2)

from (2.1) and (2.2) that

ec(t, ao)= J (1)uo.

(2.3)

We note from (2.3) that for a linear material the creep compliance e,/a, is independent of go and is given by the function J (t). A creep and rerooery test is characterized by the loading programme (Fig. la) C = 0, The following

7 < 0;

expression

(5=oo,

is denoted

o
u=o, response

T
(2.4)

Non-linear

mechanical 6

=

behaviour

6 (t, oO) -

where e (2, oo, 2,) is the elongation test (2.4)

of oriented

t >

e (t, oo, Q,

measured

219

polypropylene L

(2.5)

for t > t, in the creep and recovery

and e, (t, uo) is the creep elongation

which

would

be obtained

in the

creep test (2.2) (see Fig. la).

RESPONSE

TIME

FIG.

la.

Creep and recovery

(t)

programme.

KT,,.

LOADING PROGRAMM~

I

I

I

I

I

MME

, EXTENS,ON RESPONSE

0 FIG. lb.

Creep programme

For a linear viscoelastic

for superposition of two identical loads.

solid (2.1), (2.3) and (2.5) yield e, = J (t -

A third type of test employed (Fig.

lb)

(t)

11

tl) oo.

(2.6)

in the paper may be called a two step loading test

: lJ = 0,

7 (0;

We define the additional

(5 = (To, 0 < 7 < t,; creep response

e,’ = e’ (4 uo, 4) -

u = 2uo,

7 > t,.

(2.7)

as ec (t, uo),

t > t,,

where e’ is the elongation measured after the application of the second loading and e, has the previously defined meaning (Fig. lb). It is obvious that for a linear solid %’ == J (t -

tl) us.

(2.3) step of

(2.9)

I. M.

220

WORD

and

II:.

In Section j?, the measured creep, recovery material

considered

and f2.9) predicted

in the paper

T.

ON.\T

and additional

a.rc compared

with

creep responsc~ uf the

the responses (2.3),

(2.6)

by the linear theory.

A single monofilament of pol~prop~leue of dismetcr 3.7~ :, l(+ cnl was u~l in the csljcrime&s. This monofilament possessed a high dcgrce of molrcular orierttation and was fairly highly crystalline. Before making quantitatirc x~~casurements of creep and rrco\-try under various loading eonditions, a conditioning procedure was undertaken [SW hx~xn~l.w (1943)]. Tlic saniple was subjected to successive creep and rccovcry cycles, each cycle consisting of application of the load of 587 g for 2 hr 35 min (the maximum load and thr nlarinlutn period of loading for all the experiments reported here) follow-cd by a rcrorery 1:criod of 21 hr 25 min. The total cycle is then the convcnicnt period of onr da\-. This coI~ditioning procedure has two major ef’fccts on the crccl~ ant1 rccovcry tteha\+our. First, subsequent creep and recovery rcsponscs uudcr a given load arc then identical, i.e. the sample has lost its “ long-time memory ” and now only rcmcrubcrs loads applied in its immcdiatc past history. Secondly, aftor the conditioning proccdurc the dcformntion produced by any loading programme is almost complctcly recovcrablc provided that the recover>- period is at least eight times the period during which loads are applied. The response during the conditioning procedure is shown in Fig. 2. This diagram shows the creep and recovery bchaviour in ehronological order on a linear scale, with dotted gaps indicating portions where very slow rcrovery is taking place.

-. 2% EXTENSION

APPROXIMATE

FIG. 2.

Extensional

TIME

SCALE

(I IJNIT

creep and recovery of polyprolyvlcnc load of 587 $.

7

IO4 SECS)

initial cycling

under maximum

The defamation of the sample was mrasurrtl using a travclling mirroscope, the vernier scale of which could he read to ICY3 cm the sample length being 30.2 cm. The maximum &formation produced was about 0.7 cm, corresponding to about 2 per rent extension. After con~tioning the sample, an initial sure-ey was made of creep for several Icvt~IsoZ load. At least two sets of measurements WWF made al each lcvcl of load and the load leveis were chosen in a random order. These preliminary experiments wcrc followed by creep and rccoverv tests at various load intensities. The response to two step loading was also investigated, initial application of 281 g being followed by a further 281 g nftcr a spccificd tilne interval. The conditioning cycles and preliminary mcasurcmcnts wcrc not made in a tcnq)craturc controlled atmosphere, although measurements of the temperature were made during each run. A’ maximum variation of 4°F was observed between diffcrcnt runs, and the data showtd a small but significant dependence on temperature, the general pattern of hchaviour remaining unaffected. Subsequent measurements were therefore undcrtakcn in a room where the temperature was controlled to 73°F j, l”F, and only results obtained at temperatures falling within this range are used In drawing up the diagrams which analyse the non-linear behariour in detail (Fig. 4 d seq. below).

Non-linear rnechartical beheviouz of oriented ~oiypropy~ene

+,A“’

w+----1% EXTENSION

~&_-_

------~-~_-I@

TiME

---__a--_.,_--I

IO”

(SECS)

FIG. 4. Creep eorrtpliancecurves for different kvels of toad o.

w

Creep eompliances obtained from the experimtnts ~o~~d~~~tedunder five di&rcnt load levels are shown in Fig. 4 on a logarithmic time scale. It is see11from this figure that, except for short. times (t N 10~ set) and for only low intensities of loading (oO =: 67.6 and 12943 g), the creep complinnces do not coincide. This ohser\-ation is in contradiction

with the predictiort (2.3) of the lirtcar theory and it indica.tes

that the material considered is non-linear. The non-linear nature of the material may be mu& strikinglp risible by plotting e, (1, ao)/uoagniust “O for various fixed valncs oft (Fig. 5). According to the linear theory, each srrvh curve shoultl be a horizontal

i

cl

9300

;nx 0v

3000 1000 300 i 40 15 00

TIME IN SECS i

/’

i

‘1

a”

Non-linear

mechanical

behaviour

of oriented

polypropylene

223

LOAD LEVELS

~__

I

I

I

102 TIME

IO" (SECS)

FIG. 6. Comparison of recovery curves for different levels of loads (time of loading 9.3 x lo3 set).

5r

x A

TIME IN SECS

i

+ q 0

1000 300 100 40 I5

k 9

-

8 I

0. 10

I 100

I 200

I 300

I 400 LOAD

I 500 (G MS)co

I 600

FIG. ‘7. Recovery compliance e, (t - t,)/u,, against load q, (time of loading 9.3 x IO3 set) for various times (set).

I

I. 111. WARM

22-t creep

and recovery

and E. ‘J?.

ONAT

curves in Fig. 3 should coincide for a given intensity

of load.

A4s the figure shows, this is not the case except, for the low load int,ensities. It is interesting to observe that. the short-time recovery or ‘ instal~tat~e~)~~srecovery ' as it is sometimes termed is larger than the init.ial creep response. This effect increases with increasing applied load, as illustrated in Fig. 11 where the difference

between

the creep and reco\-cry’ cvmpliances

for

15 SW is plotted

against

the

load aO. Recovery respect&elf.

fompiinnces

and

~745~

IWSIIS

5*

curves are given in Figs. G and 7

The prediction of the linear theory co~~~erl~i~~grecta\-cry can be contrasted with the obserred behaviour in a manner similar to that employed in the above discussion of creep data. The effect of the time of loading

i, on the IWO\-cry bchaviour

has also been

studied and the results of these studies are given in Figs. x and 9.

i .

i

102

l-ME

(SECS)

The additional creep response is plotted For purposes of comparison the corresponding shown in this figure. The figure provides yet since cc’ (t - tl) does not coincide with e, (t)

103

.___._~~_.__-_

I

104

as a function of t - t, in Fig. 10. creep and recovery curves are also another indication of non-linearity as the linear theory predicts.

Non-linear

5.

mechanical

behaviour

of oriented

225

polypropylene

MATHEMATICAL DESCRIPTION OF OBSERVED BEEIAVIOWR

The results obtained

for the extensional

pylene

are clearly

at variance

elastic

behaviour

discussed

creep and recovery

of oriented polypro-

with the three simple implications of linear viscoin Section 2. The material considered is non-linear

and the relation (2.1) is not adequate for the mathematical observed memory dependent non-linear behaviour.

description

of the

0.3

Id TIME

lk.

I

I

I

IO'

IO

04

(SECS)

Comparison of creep and recovery curves for load of 281 g with additional creep due to addition of 281 g after 3000 see and IO00 sec.

10.

3000 sees

In the development behaviour

we adopt

of the specimen

q

of an adequate

the following

1000 sees fI

description of the observed non-linear We assume that the elongation

point of view.

at time t depends on all the previous values of the rate of loading In other words the elongation is assumed

to which the specimen has been subjected. to be a function

of the history

of rate of loading.

(5.1) It is well-knowl~ represented

that if the functional

by the Boltzmann

F is l~r~~~rand continuous

integral

(2.1) which

constitutes

then it can be the basis of the

theory of linear viscoelasticity. FrCchet has shown [see VOLTERRA and PiaBs (1936); also NAKADA (1960), GREEN and RIVLIN (1957)] that, where F is continuous and aon-linear, the functional F can be represented manner

:

to any desired

degree of accuracy

in the following

t

e (t) =

t

+ --m

s

J, (t -

TV) y

&I 1

t

ss . . .

-co

JN (t -

rl, . . . , t -

lb (71) TN) T. 1

‘a (T.v) & 1 . . . dTN . . dw

(5.2)

I. M. WARD and E. T. ONAT

226

where it is assumed, without loss of generality, symmetric functions of their arguments. An

interpretation

given loading The integrand individual elongation.

of this represetltation

that the kernels J,, . . . , JN are

can be obtained

by

considering

a

programme (z (7) as a superposition of finitesimal loading steps. of the first term in (5.2) can then be interpreted as representing the

and independent contribution of the loading step da (TV) to the final The integrand of the second term, on the other hand, can be thought

as representing

the joit~l contribution

to the final elongation, According programme

of the loading

to (5.2)

the time

dependent

of the creep test (2.2) is given ec (4 4

=

steps

du (TV) and

drr (TJ

etc. elongation

produced

by

the loading

by the expression

J, (0 m. + J, (4 t) go2+ . . . + JN (4 . . . , t) o.N.

W-9

Let us now recall that each curve in Fig. 5 describes the experimentally observed dependence of e, (t, u~‘)/G~on CT~for a fixed t. Moreover, let us observe that each curve in Fig. 5 can be represented, with reasonable accuracy, by an equation of the type ec/uo =- c1 + c2 uo2

(5.4)

where cI and cs are constants. This observation suggests; together with (5.3), that in a first attempt

at describ-

ing the present test results, only the first and third terms may be retained expression

(5.2).

e (t) =

Accordingly

J, (t -

we confine our attention

T& $

to the following

in the

functional

d~~ 1

s

The creep elongation

predicted

by (5.5) for the loading

programme

defined

by

(2.2) is

e, (t, a,,) = For the recovery integration,

3, (t) o. + J, (t, t, t) oo3.

test defined by the loading programme

(5.6)

(2.4), (5.6) yields by simple

and for t > t,,

e 0, go, tll =

J, (Qgo - JI (t - G) m. + [J, (6 6 t) - 3J, (t, t, t - 4)

+ 3J, (t, t -

t,, t -

We obtain the recovery response tracting (5.7) from (5.6) :

F,. = J, (t -

tl) u. -+ J, (t -

5,) -. J, (t (defined

t,, t -

+ 3 [J, (t, t, t If we now let S denote

in Fig. la),

t,, t -

t,)] a,$.

predicted

(5.7)

by (5.5), by sub-

tl) uo3

tl) - J, (t, t -

the time difference

help of (5.4) in the following

t,, t - t,, t -

t -

t,, t -

t,)] LQ,~.

(5.3)

t,, (5.8) can be written with the

form :

e, (S, o‘s, tl) = e, (S, ran) + 3 [J, (S + t,, S + t,, S) -

J, (S --t t,, S, S,)] uo3.

(5.9)

Non-linear mechanical behaviour of oriented polypropylene

227

We now want to show that the experimental recovery behaviour can be adequately described by (5.9). For this purpose we consider the difference A = e, -

ec = 3 [J,

(S + 1,, S + ti, S) -

and enquire first whether the experimentally for fixed S and t, obeys the cubic law (5.10).

2

APPLIED

J, (S + t,, S, S) ] oas,

(5.10)

observed dependence of A on a,, In Fig. 11 experimental values of

LOAD

(GM)

5 ilI

FIG. 11.

Additional

elastic recovery, i.e. difference between 15 see creep compliance as a function of applied load.

and recovery

A/U@ are plotted against o. for the, case of S = 15 see and t, = 9.3 x lo3 sec. Fig. 11 shows that, to a reasonable degree of approximation, the resulting experimental curve of A/u, against CQ,is a parabola as predicted by (5.10). Similar curves can be constructed from Fig. 13 for other values of S with the same afirmative results, It is clear that A depends on both S and t, as well as on a,,. Experimental information concerning this dependence is available in Figs. 3, 8 and 9. It will be noted that the time dependence of A is related to two functions of time. In fact, the comparison of the creep and recovery tests provide information for the experimental determination of the expression

J, (S + t,, S + t,, 8) - J, (8 + t,, 8, 8). The tests in Figs. 8 and 9 contain information on the dependence of the above expression on t,. For instance they show that this expression decreases with decreasing t,. We next consider the two-step loading programme defined by (2.7) and determine the elongation predicted by (5.5) for t > t, : e’ (4 00, G) =

J, (t) “0 + J, (t - 2,) co + [J, (t, t, t) + 3J, (t, t, t - t,) t%(t,t-tt,,t-t,)+J,(t-tt,,t-tl,t-tt,)]q,3.

(5.11)

The additional creep response as defined in Fig. lb is therefore given by eC’ 6% gal G) = ec 6% go) + 3 [J, where, as before, S denotes t -

(8 + t,, S + 4,s) + J, (S + t,, 8, NJ (ro3, (5.12) t,.

I. nr.

228 Comparison

and E. T. %acr

WAnD

of (5.12) with (5.9) shows that e,’ (S, o,,, tl) -

e, (S, oO, t) =

The test results shown in Fig. 10 indicate,

6J3

(S

+

t,,

S,

5”)

uo3.

in view of (5.13), that

(5.13) J,

(S+tl,

S,

S)

>

0

which is not surprising. We have shown that the present test results can be described,

with reasonable

accuracy, by the functional (5.5). We have also established, with the help of (5.6) (5.9) and (5.13), that the creep, creep and recovery and two-step loading tests provide, J,

(t

J3

(71,

in principle, +

t,,

t, t).

full

However,

information a ‘ complete

on J, (t), J, (t, ’ experimental

t, t),

J,

(t

+

t,,

determination

t +

will require the performance of other types of tests. 72, 73) It will be noted that in principle it might also be possible to obtain

experimental

data by including

we might have included

To test the advantages whether

other terms in the expression

the second

of alternative

t,,

t)

and

of the kernel

(5.2).

a fit to the

In particular

term

representations,

even terms in (5.2) should be included

be desirable to make more comprehensive sive as well as extensional deformation.

and in particular

to decide

as well as the odd terms, it would

experimentations

to include

comprcs-

Before closing this section, we want to enquire whether a simpler representation of non-linear behaviour suggested by LEBDERMAN (1943) is adequate for the present

material.

Leaderman’s

representation

has the form

where I< is the elasticity modulus of the material and K and.fare empirical functions of time and stress, respect.ively. It will be noticed that (5.14)constitutes a generalization of the Boltzmann’s superposition principle (2.1). For the creep test performed

under constant

stress oO (5.14)

yields

(5.15) where it is assumed

that *f(9) = 0.

For the recovery

e (t)= I< @)f(qJ - K (t- &)f(uJ, and hence,

test w-e obtain

from

(5.14) (5.16)

t > t,,

in view of (5.15),

(5.17)

er(J‘, go,&) = 2 + K (W.f(qJ, t > h. It is seen from

(5.15)and (5.17)that, according ep (S, uo, tl) -

As Fig. 3 and earlier discussions

indicate

to Leaderman’s

representation,

ec (S, uo) = 9.

(5.18)is clearly

(5.15) at variance

with ex-

Non-linear

mechanical

behaviour

of oriented

~OIy~ror~ylene

‘JZD

perimental evidence. It would seem, therefore, that Leadermnn’s representation is inadequate for the present mater&l. ACKNOWLEDGMENT The authors are indebted the course of this work.

to Professor

H. Kolsky

for his advice and encouragement

during

REFEREXCES FERRY, J. D. GREEN, A. K. and RIVLI~, R. S. Gnoss, B. G., WnrIX, H. J. and ;‘:YRING, H. KOLSKE’, H.

1961

V~~eoelast~c P,ropertiesof Po~~9~ers (Wiley, New York).

1957

AT&. Rat. Mech. 1, 1.

1953

HALSEY

1954 1960 1923 1943

Text. Ztes. 15, 295. The Mwhanical Testirzg of ZZigh Poly9ners. Progress in Non-Destructive Testing, Vol. 2. (Heywood, London). International Symposiu9n 092 Stress Wave Propagution Materials (Interscience, New York). Elastic and Creep Praperties of Filamentous and Other ZQh Polymers (The Textile Foundation, Washington, H.C.). Large Longitudinal Retarded Elastic Defo~~~at~on of Z&bberlike Natwal Polymers. Presented at the Thirty Second Annual Meeting of the American Society of Rheology, Madison. Proc. Second Int. Cmg. Rheol., p. 156. J. Phys. Sac. Japan 15, 2280. J. Text. Inst. 14 T, 390. Rayon Text. Mon. 24, 207, 339, 405.

1936

Theorie Generale des Ponctionnelles,

1945 1959 1960

LEADEI~XIAN,H.

1943 1961

MARVIN, R. S. NAKADA, 0. PEIRCE, F. T. PRESS, J. J. and MARK, H. VOLTERRA, V. and l’fR&S, J.

p. 61 (Gauthier-Villars).