Methods of investigation method of forecasting polymer durability

0032-~950/80/10261r~06 $07.50/0 O 1981PergamonPress Ltd.

PolymerScienceU.S.S.R.Vol.22, lqo. 10, pp. 2615--2620,1980 Printed in Poland

METHODS OF INVESTIGATION METHOD OF FORECASTING POLYMER DURABILITY* 1V~. N. ]3OKSH1TSKIIand N. F. LAPSHI~A Scientific Industrial Association "Neftekhimavtomatika" (Received

11 J u n e

1979)

Experimental verification is given of a method of forecasting long-term strength, which is based on the phenomenological theory of dispersion damage. TEE problem of forecasting the long-term strength of polymers remains highly urgent. This study therefore contains results of experimental verification of an extrapolation method [1]. I t was derived phenomenologically from the following kinetic equation [2]:

( ~ = - - A w n ~,,.

(1)

where a~>0 is the equivalent stress (strength criterion); 1 > ~ (0/>0 is the continuity of the solid [3]; A and ~ -- empirical parameters dependent on temperature a n d other external factors; n = 0 . 1 -- order of the breakdown process; ~ -- rate of variation of continuity. Normally [4] when a > 0 n = l . Continuity is approximately determined as follows

(2)

~=OlVo,

where C and (70 are the current and initial values of controllable material characteristics, e.g. tensile stress causing failure. I t is assumed [3] that V~(0)= 1

(3)

~(0 = 0 ,

(4)

where v is durability. I n uniaxial elongation for al----a=const integration of (1), when n = 1 and under initial condition (3), results in the equation = ( 1 - - A~ae'~t) I I*°,

(5)

form which using the condition (4) the time dependence of brittle strength follows (6)

v=e-"/A~o

For the practical implementation of this formula we t r y to determine parameters A a n d ~ b y a rapid method: from binomial (5) which can, within a fairly long time interval (Fig. 1), be substituted b y a straight line W= 1 --it',

* V y s o k o m o l . soyed. A22: No. 10, 2380-2384, 2615

(7)

1980.

M. N. BOKSHITSKII and N. F. L ~ S K ~ A

2616 where

i=l/aq, t'=t/r.

We determine the error involved in this substitution as

a=

(1--it')-- (1--t') ~

(1-t')~

x

100% ~
(8)

I n engineering practice it is normally assumed that 5n g 10%. Figure 2 shows the dependence of error $ on parameter 0 < i ~<1 for various dimensionless time values. The curves pass through maxima. Coordinations of the m a x i m u m (i.,, am) axe found from the condition ~$/~i=0. As a result 1

1

i,~= -F + In (1-t'-----~

O)

and &'=

[1--imt"

]

(l-r)'-

1

x

100%

(I0)

Analysis of Fig. 2 indicates that when t' ~<0.57 condition (10) holds good for a n y stress and when i~<0-07 $ < $ n , in practically the entire time interval.

d,,~;,I qJ

~

I

1

I

i

30

f 4

zo

0

O.q

0.8 t'

a.q [m

FIo. 1

0.8

i

FIG. 2

FIG. 1. Curves of mechanical breakdown of solids: /----0.08 (1); 0.I (2); 0.14 (3); 0.24 (4); 1.0 (5); 1.8 (6); 3.57 (7); 7.15 (8). FIG. 2. Dependence of the relative error $ on parameter ~: t ' ~ 0 - 9 (•); 0.8 (2); 0-7 (3); 0.6 (d); 0.5 (5); 0.4 (6); 0.3 (7). Let us further examine the mechanical breakdown [2] of two series of samples b y the action of constant stress ~1<~2. After exposing them to stress for t~ ~0.6r~ period of time, using equation (2) we determine experimentally continuities ~1 and ~ . Bearing in m i n d eqns. (6) and (7) rates of brittle breakage axe calculated as

vl=Ae ~ _ _t~,

vs~Ae~%_~

t~

(11)

and therefore, v_~l _ea(o.l_¢,)= vs

I--~i 1 -- ys

(12)

Methods of forecasting polymer durability

261T

As a result, from eqn. (12) parameter 2- 3 ~------

log

1--~1

(13)

and from eqn. (11) parameter t~

e

-

-

t~

e

(14)

The procedure examined for determining parameters of the initial kinetic eqn. (1), which is the continuation of the method of brittle rupture [4] was verified experimentally. The form of samples corresponded to GOST 11262-76 (type 6). Their dimensions, however, were halved. Samples were moulded at 116-5 MPa and the mould t e m p e r a t u r e was 25 °. Moulding time was 15 sec. Polystyrene softening point was also determined experimentally, according to GOST 21553-76 (visually). Limiting characteristics of the material were determined at 20°: tensile stress causing failure ~t and elongation at break ab and long-term strength at 50, 60 and 70 °. E x p e r i m e n t s were carried out in air. MeehanieM tests were carried out according to GOST 11262-76 using a ZM-40 tensile testing machine at a rate of deformation of 10 mm/min. Long-term strength was examined in accordance with GOST 18197-72 by uniaxial elongation using a tests bench [5] u n d e r conditions of p = c o n s t . Values of testing stress are given in Table 1. T ~ B ~ 1. CONDrrmNs OF T E S ~ O LO~O-TER~t S~E~GTH OF POLYSTYreNE T° 50

Stress, MPa

Number ofsamples

22"7 20-1 18"9

25 27 23

T° 60

Strew, MPa

Number ofsamples

20"2 18"0 15.7

23 26 23

T° 70

Stress, MPa 23"5 20"9 18"3 15-7 13'0

Number ofsample~ 9

10 8 8 9

The method of brittle rupture was tested at 70 °. For this purpose the samples were exposed to two constant stress levels on a test bench: ~1=10-4 MPa (14 samples) and ~1~13.0 MPa (15 samples) for t~=600 sec, whereby t ~ 0 . 6 v~=728 sec. Stress was then relieved and after keeping the samples under normal conditions for 24 hr they were broken using a ZM-40 machine at a rate of deformation of 10 mm/min. Residual strength values a01 and ~02 corresponding to given stress values ~1 and az were calculated from results. Mechanical test results of samples are given in Table 2. Corresponding statistical evaluation (standard deviation, variation coefficient, confidence interval and relative error) is given in Table 3. Confidence intervals were calculated with an accuracy of 95%. E x p e r i m e n t a l durability values of samples obtained for different temperatures and stress values are shown in Table 4. Confidence intervals Ay of particular average durability values were calculated with an accuracy of 95~/o. Figure 3 shows durability curves p l o t t ed using d a t a of Table 4. Average values of yx arc indicated. Experiments of brittle rupture indicate t h a t with constant testing stress a~-----10.4 MPa and ~-----13.0 MPa, the average residual strength ao1=43"3 MPa and a01----37"9 MPa. Using formula (2) we calculate continuities ~ - ~ 0 ~ / ~ t = 0 - 9 1 1 and ~l=a01/at=0"798 and then using formula (13) and (14), parameters A and ~ are determined. As a result ~----0.31 1]MPa a n d A-----5.93× 10 -I sec -1.

M. N. Bo~sHr~sY~u a n d N. F. LAPSHINA

2618

TABLE 2. MECHANICAL CHARACTERISTICS OF POLYST~Z'RENE A T 2 0 ° Sample, No.

~rABLE

at, MPa

%, %

Sample, No.

33.5 38.9 40.0 40.0 43.0 43.3 44.8 45.3

3.3 3.75 3.75 3.95 3.95 3.95 4.16 4.16

9 10 11 12 13 14 15 16

1 i

45"7 46.0 46.0 46"8 47 "4 47.6 48"5 49"1

%,

Sample, No.

% 4.16 4.16 4"18 5.0 5"0 5"0 5.41 5"63

Ht, MPa

17 18 19 20 21 22 23 24 25

s b,

% 5"83

50.2 51.3 52-8 52 "8 52.8 48.5 55.1 55-5 57.4

5"83 6.25 6"25 6"25

6"46 6.67 7"80 7.67

3. STATISTICAL E V A L U A T I O N OF M E C H A N I C A L CHARACTERISTICS OF P O L Y S T Y R E N E *

Wo=~XlOO%

S~, M P a

~t' MPa

at

&,%

~,%

5.128

~,=~ x 8b

lOOI/o

5"2

11"0

1"542

28"3

47.5

*

at, MPa

A~ t ~/~=-at

Aat, M P a

z,; ,%

x 100%

~a="f'~ X 100% 8b

4.17

2-0

11.64

0.5974

The numerator a n d denominator give statistical evaluations for strength a n d breaking elongation, respectively.

"t',SeC 0

10 ~ 0 D

P,K

0 0

ol

0

i0 a -

7O00 -

"3

10:

-

5OOO I

I 10

I zO Fla. 3

"~ d,/"ma

1

I~

20

I---~ 40

o'¢, M P O Fzo. 4

F i e . 3. D u r a b i l i t y c u r v e s o f p o l y s t y r e n e a t 50 (1), 60 (2) a n d 70 ° (3). F I e . 4. D e p e n d e n c e o f t h e L a r s o n - M i l l e r p a r a m e t e r o n s t r e s s a t 70 (1), 60 (2) a n d 50 ° (3).

Methods o f forecasting p o l y m e r d u r a b i l i t y

2619

T a b l e 5 illustrates a comparison of calculated % a n d e x p e r i m e n t a l 3. values o f durability. T h e f o r m e r were d e r i v e d f r o m f o r m u l a (6) using e x p e r i m e n t a l p a r a m e t e r s A a n d a. Analysis of T a b l e 5 indicates t h a t t h e calculated c u r v e is p r a c t i c a l l y in t h e 9 5 % confidence i n t e r v a l of e x p e r i m e n t a l values o f d u r a b i l i t y , being n e a r e r to t h e lower confidence limit. TABLE 4.

DURABILITY

N u m b e r of samples

OF POLYSTYRENE

Yz = l o g r

T°

a, M P a

70

13'0 15"7 18"3 20"9 23"5

9 8 8 10 9

1214 283 114 50 12

3.0844 2.4522 2.0582 1.7042 1.0924

0.0888 0.0564 0.0516 0.0657 0.0600

60

15"7 18.0 20-2

23 26 23

4107 924 253

3.6136 2.9656 2.4035

0.0988 0.1103 0.0926

50

18"9 20"1 22-7

23 27 25

11,189 4940 1413

4.0488 3.6937 3.1501

0.0718 0-0977 0.1007

3, s e c

The error i n v o l v e d in t h e m e t h o d ( 6 = 3e--3e X 100~/O~ is fully permissible if we bear / \ Vc in m i n d the v a r i a t i o n of values t y p i c a l of brittle strength. The m e t h o d proposed is also suitable for a c o m p l e x state of stress, w h e n t h e criterion of brittle s t r e n g t h is used as s t r e n g t h f a c t o r (usually m a x i m u m n o r m a l stress). E x p e r i m e n t a l results o b t a i n e d were also used for the verification of o t h e r e x t r a p o l a t i o n m e t h o d s applicable to polystyrene. TABLE 5. VERIFICATIO~ OF ' r ~ a, M P a 13-0 15.7 18.3 20.9 23.5

ACCURACY OF THE METHOD OF BRITTLE RUP'I'u~E

3c, see

~e, sec

736 270 102 40 16

1215 283 114 50 12

5,

%

39.4 4.8 10.4 21.0 27.0

3e+Zi3e, sec 1490 323 129 59 14

30--zt¢0, sec 99O 249 102 44 11

I t was shown [5] t h a t w i t h b r i t t l e r u p t u r e s t r u c t u r a l p a r a m e t e r a is a p p r o x i m a t e l y calculated as

= (~rc ~b)-'

(15)

Using results in T a b l e 3 we find t h a t a t 20 ° ~--0.41 M P a - * a n d at 70 ° ~--0.35 M P a -*, i.e. differs o n l y b y 11.4% f r o m t h e v a l u e d e r i v e d b y t h e m e t h o d of brittle r u p t u r e (0.31 M P a - 9 . Conversion to a t e m p e r a t u r e o f 70 ° was carried o u t using t h e well-known f o r m u l a • ~7/RT, which results in ~,=~, (T,/T@.

2620

M.N.

BOKSH1TSKII and N. F. LAPSHINA

Verification of well-known [6] parameters of durability shows that the Larson-Miller parameter appears to be the most accurate for polystyrene P = T (20~-log ~ ) = F (a),

(16)

where v is durability in hours, T -- absolute temperature and F (a) is the function of stress. Figure 4 shows the dependence of the Larson-Miller parameter on stress. Experimental points corresponding to a temperature of 50, 60 and 70 ° axe fairly accurately situated on a generalized curve. The divergence derived for various temperatures is less t h a n 3.3%. Extrapolation of the generalized straight line to a point corresponding to at at 20 ° (47.5 MPa) produces the typical value of parameter Pt, which m a y be calculated from formula (16) ff we assume that ~-~ 10-5 hr [6]. Then, this equation takes the form PL----15T

(17)

The calculated parameter for 20 ° (293 K) is P t = 4 3 9 5 K. A stress of a = 4 8 . 0 MPa corresponds to it on the parametric straight line. I t can be seen from Table 2 that this value practically agrees with at. I t m a y be assumed [6] that the parametric curve of polystyrene is reproduced if the dependence of at on temperature is known [6]. I n this case the Larson-Miller parameter is calculated for all temperatures from formula (17) and the generalized curve obtained is recommended for forecasting long-term strength. A similar result m a y be obtained b y a somewhat simpler method. For this purpose we continue the curve in Fig. 4 until it intersects with the ordinate. This point determines parameter P0=7600 K, which corresponds to zero strength, i.e. to strength somewhere in the melting point region (Tmelt). I n fact, calculation according to formula (17) gives the value of Tmelt=P0/15=233 °, which differs only b y 8.4% from the value determined experimentally (Tmelt=215°). Finally, to plot a parametric curve of polystyrene it is sufficient to draw a straight line in coordinates a - P through two points: (0, Po) and (at, Pt). Values of Po and Pt are calculated b y substituting into formula (17) the value of Tmel~ determined independently and also T~-293 K.

Translated by E. SEMERE REFERENCES 1. M. N. BOKSHITSKH, Zavodsk. lab., No. 2, 226, 1971 2. N. F. LAPSHINA, Kandidatskaya dissertatsiya, Moskovskii tekhnologicheskii in-t myasomolochnoi prom-sti (Post-Graduate Thesis, Moscow Technological Institute of the Meat and Dairy Industry), 1973 3. L. M. KACHANOV, Osnovy mekhaniki razrusheniya (Principles of the Mechanism of Decomposition). Izd. "]Tauka", 1974 4. Yu. G. KORABEL'NIKOV, Mekhanika polimerov, No. 4, 663, 1971 5. M. N. BOKSHITSKII, Mekhanika polimerov, No. 4, 654, 1970 6. S. GOLDFEIN, Modern Plastics, No. 8, 149, 1964