EFFECT ON T H E 2.*

OF

TEMPERATURE

CREEP

OF

AND HUMIDITY

POLYMER

MATERIALS

UNIAXIAL ELONGATION UNDER VARIABLE

TE MPERATUR E - HUMIDITY CONDITIONS R . D. M a k s i m o v , E. a n d V. P . M o e h a l o v

A.

Sokolov,

UDC 539.376 : 678

Grade PN-3 p o l y e s t e r r e s in was tested for creep in uniaxial elongation within the linear r a n g e of the s t r e s s - s t r a i n c h a r a c t e r i s t i c s and under variable t e m p e r a t u r e - h u m i d i t y conditions. The r e s u l t i n g creep test curves a r e now c o m p a r e d with calculations. The latter are based on the c r e e p law and the creep c h a r a c t e r i s t i c s which have been derived e a r l i e r [1] f r o m tests unaer steady t e m p e r a t u r e - h u m i d i t y conditions. The validity of these calculations is estabIished by a c o m p a r i s o n between statistical e s t i m a t e s pertainine to the r e peatability of test data and to the a c c u r a c y of the description of r e s u l t s obtained in both original and control tests.

In the e a r l i e r r e p o r t [1] we have p r e s e n t e d the r e s u l t s of testin~ grade PN-3 p o l y e s t e r r e s i n f o r creep under uniaxial elongation, with the m a t e r i a I at various fixed t e m p e r a t u r e s and humidity levels. It has been shown that, within the range of p h y s i c a l l y linear s t r a i n s under constant s t r e s s , the creep m a y be described by the equation

o

where a are the coefficients of elastic compliance, K a r e the influence functions, T is the t e m p e r a t u r e , and w is the humidity e x p r e s s e d in p e r c e n t of the initial m a s s of the material. The compliance functions in the direction of loadin~ have been approximated here as follows: N

~

N

N

N

am1 (T, w) = ¢zl÷a2w + (¢~z-b~4w) T+ ( o~s+,aGw)T3,

(2)

where r~ a r e coefficients, ~ = T - T 0 (T0=20°C), and ~v = w - w 0 (w0= 0.7%); the influence functions in the d i r e c tion of loading have been approximated here as follows: n K I I l I ( I - - s , T, w)

a(T,w) V -

~

t-s

bi --

---.(T,,~,) e

'~o,

,

(3)

where ~0i is the base s p e c t r u m of relaxation times at T =T Oand w =w0; and the shift function a (T, w) of the relaxation s p e c t r u m has been approximated by the e x p r e s s i o n In a(r. w)=,~,T+~2r'+,~,w+~4~'+~w~

(4)

where fl a r e coefficients. * For communication 1, see [1]. Institute of P o l y m e r Mechanics, A c a d e m y of Sciences of the Latvian SSR, Riga. T r a n s l a t e d f r o m Mekhanika P o l i m e r o v , No. 6, pp. 976-982, N o v e m b e r - D e c e m b e r , 1975. Original article submitted October 18, 1974.

©19 76 Plenum Publishing Corporation, 22 7 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 form 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.

834

a ....

0

2000

~000

6000

~25 !

LOO

0

2*

#B

72

95

1 J I

~

Fig. 1

~

, ~ ....... ~,,,,

t - _. . . . . .

~

U__

_:

I J

$

Fig. 2

Fig. 1. Humidity v a r i a t i o n in the m a t e r i a l during h u m i d i f i c a t i o n - d e s i c c a t i o n t r e a t m e n t (a); c r e e p c u r v e s (b): 1) w=0.2%, 2) w=0.3%, 3) w=0.6%; dots r e p r e s e n t t e s t v a l u e s , the lines r e p r e s e n t calctflated values. Fig. 2. C r e e p c u r v e s and t e m p e r a t u r e - h u m i d i t y c y c l e s : 1, 2, 3) r e f e r to the r e s p e c t i v e t e s t No.; dots r e p r e s e n t t e s t values, lines r e p r e s e n t calculated values. The object of this study was to a s c e r t a i n whether r e l a t i o n s (1)-(4) and the e s t a b l i s h e d p a r a m e t e r s a r e a l s o suitable for describin~ the p h y s i c a l l y l i n e a r c r e e p under v a r i a b l e t e m p e r a t u r e - h u m i d i t y conditions. We solved the p r o b l e m b y p e r f o r m i n g control t e s t s for c r e e p under uniaxial elongation, v a r i a b l e t e m p e r a t u r e T and humidity w, and then c o m p a r i n g the t e s t data with calculations b a s e d on f o r m u l a s (1)-(4) and c r e e p c h a r a c t e r i s t i c s obtained in [1] under s t e a d y t e m p e r a t u r e - h u m i d i t y conditions. In a p r e l i m i n a r y study we checked the r e v e r s i b i l i t y of changes in the s t r a i n c h a r a c t e r i s t i c s of the m a t e r i a l durtn~ a h u m i d i f i c a t i o n - d e s i c c a t i o n cycle. This was done as follows. Specimens w e r e held in w a t e r without loading, wherupon they w e r e dried f i r s t under r o o m conditions and then for 24 h in a heat c h a m b e r at T=50°C. The changes in the humidity of the m a t e r i a l during this t r e a t m e n t a r e shown in Fig. l a . T h r e e batches of s p e c i m e n s w e r e subjected to such a h u m i d i f i c a t i o n - d e s i c c a t i o n cycle, f o r 1500, 2800, and 5250 h, with the m a x i m u m humidity during this cycle being 2.0, 2.5, and 2.9%, and with the humidity a f t e r d e s i c cation being 0.2, 0.3, and 0.6%, r e s p e c t i v e l y . Followin~ this t r e a t m e n t , the s p e c i m e n s w e r e t e s t e d for c r e e p under uniaxial elongation at T =40°C and a s t r e s s equal to 25% of t h e i r s h o r t - t e r m s t r e n g t h . . A v e r a g e c r e e p t e s t c a r v e s a r e shown in Fig. lb. On the s a m e d i a g r a m a r e a l s o shown curves calculated a c c o r d i n ~ to f o r m u l a s (1)-(4), the p a r a m e t e r s h e r e havin~ been d e t e r m i n e d in e a r l i e r t e s t s [1] with s p e c i m e n s which had not been s u b j e c t e d to h u m i d i f i c a t i o n - d e s i c c a t i o n cycles. A c c o r d i n g to Fig. 110, the c r e e p s t r a i n in the m a t e r i a l tends to d e c r e a s e a f t e r h u m i d i f i c a t i o n - d e s i c c a t i o n t r e a t m e n t , but the o v e r a l l difference between calculated and m e a s u r e d values hardly exceeds the confidence interval which c h a r a c t e r i z e s the t e s t data. We conclude, t h e r e f o r e , that the c r e e p c h a r a c t e r i s t i c of the n ~ t e r i a l has changed only insi~nificantIy durin~ humidification-desiccation treatment. We will now analyze the r e s u l t s of c r e e p t e s t s p e r f o r m e d under v a r i a b l e t e m p e r a t u r e - h u m i d i t y condltions. In all, s i x control t e s t s w e r e p e r f o r m e d . T e s t No. 1 was p e r f o r m e d at a constant humidity w =0.7% of the s p e c i m e n s . The t e m p e r a t u r e was v a r i e d as follows: for the f i r s t 24 h it was maintained at 30°C, then it was r a i s e d to 40°C within 10-.15 rain, and maintained at that level for another 24 h, whereupon it was dropped to 30°C and r a i s e d to 40°C within the next 24 h. The avera~_e c r e e p curve for this cycle is shown in FiR. 2. The s t r a i n s a r e calculated on the a s s u m p t i o n of stepwise t e m p e r a t u r e chan~es (Fi~. 3). E x p r e s s i n g the total s t r a i n as a s u m ~ ll = ~ H ° + ~ HC + ~ HT (of the c o n d i t i o n a l - m o m e n t a r y s t r a i n ~ H °, the c r e e p s t r a i n eHT), and the s t r a i n due to t h e r m a l expansion e T } , w e f i n d e x p r e s s [ o n s f o r ~1~°, e11C, and EH T at {:~;=0 and a s t e p w i s e t e m p e r a t u r e chan~e. 835

0

r,i

I

r,

I

I

t

0ol-~ . . . . .

• / J°l"

t

et,I

~

= ....

17" ..-----'~r

Fig. 3

3

...~.5

t/

-~'-': -~-

Fig. 4

F i g . 3. D i a g r a m of t e m p e r a t u r e

changes.

F i g . 4. C r e e p c u r v e s a n d t e m p e r a t u r e - h u m i d i t y c y c l e s : 1) t e s t I%. 4, 2) t e s t No. 5, 3) t e s t No. 6; d o t s r e p r e s e n t t e s t v a l u e s , s o l i d l i n e s r e p r e sent calculated values. F r o m (1) and (2) w e h a v e

£:110~

a , m ( T t ) o l i = (cq+o~3T,+aJt3)o'H [ a , li,(T2)aH= (c~+c~'T2+czsT2Z)~ll

at

t'~
(5)

at t~
T h e c r e e p s t r a i n is found f o r t h e (s + l ) - t h i n t e r v a l (t s _
(6) i=l

TOi

s=O

eli " = ~

_

"g0i

,s=g

T01

C o m p o n e n t ~ l I T is s t i p u l a t e d in l he f o r m

t7)

~

[~T(TI-T2) at

t'~
w h e r e a T is t h e c o e f f i c i e n t l i n e a r t h e r m a l e x p a n s i o n . M e a s u r e m e n t s h a v e s h o w n t h a t a T r e m a i n s c o n s t a n t and e q u a l to 1.1 • 10 -4 °C - I t h r o u g h o u t t h e ~[ven t e m p e r a t u r e r a n g e . T h e c o e f f i c i e n t s in (5)-(7) h a d b e e n d e t e r m i n e d in [1] f r o m t e s t s a t c o n s t a n t t e m p e r a t u r e a n d h u m i d try, T h e c u r v e c a l c u l a t e d a c c o r d i n g to (5)-(7), w i t h the v a l u e s o f c o e f f i c i e n t s a s d e t e r m i n e d in [1], is s h o w n in F i ~ . 2. T h e r m s r e l a t i v e e r r o r r w h i c h c h a r a c t e r i z e s the d i f f e r e n c e b e t w e e n t h e t e s t c u r v e and t h e c a l c u l a t e d c u r v e i s 7.5%. T e s t No. 2 w a s p e r f o r m e d a t c o n s t a n t t e m p e r a t u r e T =30 ° a n d a h u m i d i t y c o n t i n u o u s l y d e c r e a s i n g ( f r o m 3.7 t o 2.4%) d u r i n ~ d e s i c c a t i o n of t h e m a t e r i a l . T h e c r e e p t e s t c u r v e i s s h o w n in F i g . 2. F o r t h e p u r p o s e of s t r a i n c a l c u l a t i o n s , t h e t e m p e r a t u r e i m a t e d b y t h e followin~ r e l a t i o n s :

and the humidity during this test are approx-

w (t) = w (t) - Wo= 3,02e-°'°t2t; T = T - To = 10° C.

(8)

The t o t a l s t r a i n is found a s the s u m e 11 = £ 110 + ~ 1IC" In o r d e r t o d e t e r m i n e e 110, w e i n s e r t into t h e e q u a l i t y 11° = a l l l l ( T , W)all e x p r e s s i o n (2), c o e f f i c i e n t s a d e t e r m i n e d in [1], a n d c o n d i t i o n s (8). A s a r e s u l t , w e o b tain

etl°(t) = (0.363+0.36t .e-O.ol2t) • 10-4~i1. 836

(9)

In the case of continuously varyin~ T and w, e siC is m o r e conveniently calculated to a fictitious time s c a l e z: the inteeTal of the product of function a (T, w) by the differential of r e a l time t:

z = S niT(a),

~,(s)]ds.

00)

0

Such an approach, namely, the ehan~e to this fictitious time s c a l e , has been s u c c e s s f u l l y used in [2] for takin~ into account the effect of variable humidity in the solution of the mixed boundary-value p r o b l e m for a v i s c o e l a s t i c material. When mapped to this z - s c a l e , the c r e e p s t r a i n becomes independent of T and w. Consequently, the equation

e~c(/)= ~r'__TL ~ , bi[ l--exp( - -~o~) ]

01)

i=1

derived f r o m (1) and (3) with T = T Oand w =w 0 may be used for calculatin~ ~ it C at T # const and w # const; it is n e c e s s a r y now to r e p l a c e t by z in (11). In order to establish the c o r r e s p o n d e n c e between z and t under conditions (8) and (10), we insert (4), (8), and the values of coefficients fi as determined in [1]. The :result is t

z = S exp ( 1.97+ 9.43e -°.°t~- 4.83e -°.°~5~)ds.

(12)

0

The intearral (12) has been evaluated n u m e r i c a l l y on a computer by the trapezoid method. In this way, strains under test conditions No. 2 can be calculated accordin~ to (9), (11) with (12). The resultin~ test curve and calculated curve are compared in Fig. 2; the e r r o r in the description of this test curve is r = 11.6%. Test No. 3 was p e r f o r m e d under the followin¢ t e m p e r a t u r e - h u m i d i t y treatment. The t e m p e r a t u r e was varied stepwise f r o m 30°C (interval I) to 40°C (interval II) to 30°C (interval III). The humidity of s p e c imens was varied continuously by desiccation f r o m 2.1 to 1.7% (FIe. 2). No abrupt change in the drying rate o c c u r r e d durin~ this t e m p e r a t u r e cycltn~ and, t h e r e f o r e , the humidity of the material as a function of time could be approximated by a single equation. For the purpose of s t r a i n calculations, the t e m p e r a t u r e and the humidity durin¢ this test a r e e x p r e s s e d as follows: "~' (i) = ~:'(0 -- ~'o= 1.4- 0.56- iO--~t; ~= T-- To=

lO°C-intervaI ° " v 20 C-irlter al

i, Iit, II.

We will now derive e x p r e s s i o n s for calculattn~ the components of total strain. terval we f i r s t define the fictitious time scale accordin~ to (13): e5,3

el, m = ~

(i3)

F o r e ttC in each in-

6~8,4

(t -- e--°-°°99 ; zii = 0-.0--[6- (0.785-- e-°°l°l),

04)

where the s u b s c r i p t s I, II, III denote the intervals where these e x p r e s s i o n s apply. We next intewrate over the intervals, to the fictitious time scale, the second t e r m on the right-.hand side of Eq. (1) with T = T 0 and w = w 0. The resultin~ equation is analogous to Eq. (6), the difference bein~ that t has been r e p l a c e d by z and that we have h e r e the equalities a (T~ =To) =- 1, a (T 2 =T 0) ~ 1. On the basis of the foregoing, ~ 11C is calculated accordin~ to Eq. (6) in the fictitious time scale [with a (T 1) =a (T 2) = 1] as p e r (14). Expressions for calculatin~ ~ i1° are obtained by insertin~ into the equality ell ° = am1 (T, w) (hi Eq. (2), conditions (13), and the values of coefficients a as d e t e r m i n e d in [1]. The r e s u l t s c11°i,iii = (54 0.059t)" 10-6ffl 1 and eil°Ii = (73 - 0.1070 • 10-6~1° The strain component ~llT is calculated according ~o (7). Adding the individual components, calculated in the manner shown here, yields the total strain. calculated curve is shown in Fig. 2. The e r r o r in this case is r =8.25%.

The

Test No. 4 was p e r f o r m e d at constant humidity of the m a t e r i a l w = 0.8% and a t e m p e r a t u r e r i s i n g at a constant r a t e f r o m 20 to 50°C. The test time was 5 h. The r e s u l t i n g creep curve is shown in Fig. 4.

837

For the purpose of s t r a i n calculations, the t e m p e r a t u r e and the humidity during this test a r e exp r e s s e d as follows: "~ (t) = T ( t) -- To=6t;

(15)

-~(t) =w(t)- wo=O.1%. In o r d e r to determine ~ 1C , we f i r s t establish the fictitious time scale z by inserting (4) and (15) in (10), t

z = ~exp (O.15s~+O.94s+O.29)ds.

(16)

0

Now e l l C is calculated a c c o r d i n g to (11), to the fictitious time s c a l e , with the correspondence between z and t b a s e d on (16). The s t r a i n component e ti ° is found by inserting into the equality e l i ° =a Iill(T, w) fill e x p r e s s i o n s (2) and (15): el io(t) = (32.3 + 2.55t + 0.205t a). 10-6(;:j.

(17)

T h e r m a l expansion under conditions (15) is ~,i T

(t) = czTT"(t) =6aTt.

(18)

The s t r a i n curve calculated for the m a t e r i a l on the basis of (16)-(18) is shown in Fig. 4. The e r r o r in the description of test data is in this case r =25.4%. Test No. 5 was p e r f o r m e d under the following t e m p e r a t u r e - h u m i d i t y conditions: T'(I) =6t;

w(t) =2.03%.

(19)

Accordingly, this test differed f r o m the p r e c e d i n g test only in the humidity level. The average test curve is shown in Fig. 4. E x p r e s s i o n s for calculating the s t r a i n s under conditions (19) a r e derived on the basis of the same concepts as those for the p r e c e d i n g case. We thus have t

z = j exp (0.15s=+ 1.17s+3.74)ds; l)

(20)

e:10(t) = (52.8+2.87t+0,657ta) • 10-6~1. The creep test curve and the c r e e p curve calculated a c c o r d i n g to the e a r l i e r p r o c e d u r e with both (18) and (20) are c o m p a r e d in Fig. 4. The e r r o r in the description of test data is in this case r =11.5%. Test No. 6 was p e r f o r m e d at a r i s i n g t e m p e r a t u r e and d e c r e a s i n g humidity of the material (Fig. 4): T'(t) =6t;

w (t) = 1.98-0.08t.

(21)

The e x p r e s s i o n s needed for calculating the total s t r a i n s under conditions (21) a r e t

z = J exp ( O . 1 4 s 2 + l . l s + 3 . 7 ) d s ;

(22)

0

ell 0(t) = (52 + 2.02t + 0.645t ~ --0.01814) • 10-6(Ji~. The creep curve calculated a c c o r d i n g to (22) and (18) is shown in Fig. 4. The e r r o r in the description of test data is in this case r =t7.3%. We will now c o m p a r e the r m s relative e r r o r s which c h a r a c t e r i z e the repeatability and the a c c u r a c y of the description of the original and the control tests. The s t a t i s t i c a l estimates pertaining to the original tests under steady t e m p e r a t u r e - h u m i d i t y conditions a r e as follows: e r r o r c h a r a c t e r i z i n g the repeatability of test data r r s =11.5% and e r r o r c h a r a c t e r i z i n g the approximation of average test data r a s =7.8% ([1]). Next we consider the s t a t i s t i c a l estimates p e r t a i n i n g to the control test under variable t e m p e r a t u r e humidity conditions. The e r r o r c h a r a c t e r i z i n g the r e p e a t a b i l i t y of test data r r v is not much different for 838

all ~ix t e s t s , its a v e r a g e magnitude being 16.6%. The e r r o r c h a r a c t e r i z i n g the difference between c a l c u iated and a v e r a g e t e s t curves r d v is significantly different in each of the s i x c a s e s and its magnitude is, in the o r d e r of the t e s t s , 7.5, 11.6, 8.2, 25.4, 11.5, and 17.3%. A c o m p a r a t i v e evaluation of t h e s e e r r o r s leads us to the following conclusions. The a p p r o x i m a t i o n of the original t e s t s is sufficiently a c c u r a t e (ras < r r s ) . The r e p e a t a b i l i t y of t e s t data under v a r i a b l e t e m p e r a t u r e - h u m i d i t y conditions is w o r s e than under s t e a d y conditions ( t r y > r r s ) . The difference between calculated c r e e p curves and c r e e p t e s t curves c o r r e s p o n d i n g to v a r i a b l e t e m p e r a t u r e - h u m i d i t y conditions is, on the a v e r a g e , l a r g e r than that between a p p r o x i m a t i n g curves and t e s t curves c o r r e s p o n d i n g to steady conditions (rdv > r a s ) . In four out of s i x controls we find that r d v < t r y , i.e., the calculated curves lie within the confidence intervals for the t e s t values. As to the i n c r e a s i n g e r r o r in the description of v a r i a b l e - h u midity t e s t data, it m a y be s u g g e s t e d that this is due to the nonuniform distribution of m o i s t u r e over the s p e c i m e n volume in this case. In o r d e r to r e d u c e the calculation e r r o r , t h e r e f o r e , it is n e c e s s a r y to take into account the r a t e of w a t e r diffusion in a m a t e r i a l , e s p e c i a l l y when the s p e c i m e n s b e c o m e thicker. CONCLUSIONS On the basis of a comparison between the statistical estimates of repeatability and accuracy in the description of creep tests under steady and variable temperature-humidity conditions, respectively, it has been established that the creep law and characteristics determined from tests under steady temperaturehumidity conditions may, after appropriate mathematical transformations, be used for calculating the creep of a material under variable temperature-humidity conditions within the ranges and the rates of their variation covered in this study. LITERATURE 1.

2.

R. D. Maksimov, E. A. Sokolov, and V. P. Mochalov, "Effect of t e m p e r a t u r e and humidity on the c r e e p of p o l y m e r m a t e r i a l s . 1. Uniaxial elongation under s t e a d y t e m p e r a t u r e - h u m i d i t y conditions," Mekh. P o l i m . , No. 3, 393-399 (1975). M. A. Koltunov and I. E. T r o y a n o v s k i i , "Method of e l a s t i c solutions to p r o b l e m s in t h e r m o v i s c o e l a s ticity," Mekh. P o l i m . , No. 4, 603-614 (1970).

EQUILIBRIUM OF

CITED

GLASS-PLASTIC

MINIMUM A.

MASS B.

Mitkevich

WITH and

PRESSURE A V.

NONGEODESIC D.

Protasov

VESSELS WINDING

UDC 678..067.5

A t e n s - s h a p e d v e s s e l having various c e n t r a l a p e r t u r e s made by a continuous winding p r o c e s s is considered. Using the methods of variational calculus, the o p t i m u m winding t r a j e c t o r y (yielding a s t r u c t u r e of m i n i m u m m a s s ) is e s t a b l i s h e d , subject to c e r t a i n limitations r e g a r d ing the angle of geodesic deviation and the s t r e s s level in the f i l a m e n t s . The shape of the b a s e is d e t e r m i n e d with due allowance for the t r a j e c t o r y found in this way, which c o m p r i s e s sections of geodesic lines and lines of limiting deviation.

A n u m b e r of technical p r o b l e m s concerning, for e x a m p l e , the o p t i m u m d e s i ~ of p r e s s u r e v e s s e l s with various central a p e r t u r e s (Fig. 1) involve a nongeodesic mode of r e i n f o r c i n g the shell. T r a n s l a t e d f r o m Mekhanika P o l i m e r o v , No. 6, pp. 983-987, N o v e m b e r - D e c e m b e r , 1975. Original a r ticle s u b m i t t e d J a n u a r y 28, 1974.

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