DEFORMATION V.

K.

MODEL

OF

TWINED

POLYMER

ROPES

Trincher

UDC 539.38

T h e r e l a t i o n s h i p b e t w e e n a f o r c e P and the d e f o r m a t i o n ~ = A l / l c a u s e d by it, a s e x p e r i m e n t a l l y d e t e r m i n e d f o r twined p o l y m e r r o p e s u n d e r v a r i o u s l o a d i n g r e g i m e s , t u r n s out to b e c o m p l i c a t e d ; h e r e , l d e n o t e s the i n i t i a l r o p e l e n g t h , A l d e n o t e s the e l o n g a t i o n . A c h a r a c t e r i s t i c f e a t u r e of the r e l a t i o n s h i p P(e) for c y c l i c l o a d i n g r e g i m e s f r o m l o a d P = 0 to s o m e l o a d P = P c is the e x i s t e n c e of a l a r g e h y s t e r e s i s loop w h o s e m a g n i t u d e and p o s i t i o n d e p e n d only s l i g h t l y on the d e f o r m a t i o n r a t e . Such a d e f o r m a t i o n p r o c e s s o c c u r r i n g in L T K - 2 5 - 1 0 0 0 r o p e s t r e a t e d in an MZI~IViV P - 5 e q u i p m e n t is r e p r e s e n t e d b y the d r a w n l i n e s in F i g . 1. In this c a s e the l i m i t i n g c y c l e r e m a i n e d u n a l t e r e d when the d e f o r m a t i o n r a t e w a s v a r i e d f r o m 2 9 10 -4 to 2 9 10 -3 s e c -1. The c h a n g e o v e r f r o m l o a d i n g to u n l o a d i n g w a s a c h i e v e d by f a s t r e v e r s a l of the l o a d . S a m p l e s of the s a m e r o p e w e r e l o a d e d 1-4 t i m e s in a p i l e d r i v e r t e s t i n g e q u i p m e n t e n s u r i n g an i n i t i a l d e f o r m a t i o n r a t e ~ equal to about 30 s e c -1. The f i r s t l o a d i n g at ~ = 28 s e c -1 and the f o u r t h l o a d i n g at E = 32 s e c - t ( i n d i c a t e d b y an open c i r c l e O and a r e c t a n g l e [3, r e s p e c t i v e l y ) t e r m i n a t e d in r u p t u r i n g of the s a m p l e . As can b e s e e n in F i g . 1, with l o a d i n g at s u c h d e f o r m a t i o n r a t e s we find a r e l a t i o n s h i p P(e) w h i c h c o m e s c l o s e to the r e l a t i o n s h i p f o r a s t a t i c l o a d . H o w e v e r , in s o m e c a s e s the i n f l u e n c e of the t i m e f a c t o r is e s s e n t i a l , f o r e x a m p l e , d u r i n g the o c c u r r e n c e of the r e v e r s e a f t e r e f f e c t . The l a t t e r m a n i f e s t t h e m s e l v e s in the p h e n o m e n o n that the d e f o r m a t i o n p r o d u c e d at the m o m e n t the u n l o a d i n g of the s a m p l e ends n o t i c e a b l y d e c r e a s e s with t i m e . F i g u r e 2 s h o w s the r e v e r s e a f t e r e f f e c t o v e r 30 h f o r the i n i t i a l d e f o r m a t i o n ~0 l e f t a f t e r a l o a d i n g - u n l o a d i n g c y c l e with m a x i m u m load P c = 800 kg and d e f o r m a tion r a t e ~ = 2 9 10 -4 s e c -1. H e r e , the r e v e r s e a f t e r e f f e c t a f t e r the f i r s t l o a d i n g - u n l o a d i n g c y c l e d o e s not 6

P D

P=O

~o

o 750

500

2~ 250

O

J

0,72

0,24

Fig. i

O,Z6

g

~O~ o 0

25 rain i0 h

Fig. 2

F i g . 1. P l o t s r e p r e s e n t i n g the e x p e r i m e n t a l d a t a on the d e f o r m a t i o n of L T K - 2 5 - 1 0 0 r o p e s d u r i n g s l o w (drawn l i n e s ) and f a s t (points) l o a d i n g : 1) f i r s t l o a d i n g - u n l o a d i n g c y c l e ; 2) t h i r d and n e x t c y c l e s ( l i m i t i n g c y c l e ) ; 3 ) l o a d i n g a f t e r i n c o m p l e t e u n l o a d i n g . F i g . 2. G r a p h i c a l r e p r e s e n t a t i o n of the r e v e r s e a f t e r e f f e c t (the s t a r t of the p r o c e s s is shown in d e t a i l b y t a k i n g a s m a l l e r s c a l e a l o n g the t - a x i s ) . I n s t i t u t e of M e c h a n i c s , M o s c o w S t a t e U n i v e r s i t y , M o s c o w . T r a n s l a t e d f r o m P r o b l e m y P r o c h n o s t i , No. 3, pp. 86-91, M a r c h , 1972. O r i g i n a l a r t i c l e s u b m i t t e d May 30, 1970.

9 1972 Consultants Bureau, a division o[ Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced [or any purpose whatsoever without permission o[ the publisher. A copy o[ this article is available [rom the publisher [or $15.00.

341

3

2

a

b

Fig. 3

c

Fig. 4

F i g . 3. D i a g r a m of the d r y f r i c t i o n e l e m e n t : 1) n o n e x t e n s i b l e f i l a m e n t ; 2) e l a s t i c e l e m e n t ; 3) r o u g h c y l i n d r i c a l s u r f a c e s . F i g . 4 . L o a d i n g and u n l o a d i n g p l o t s of the m o d e l T (a) and a f i l a m e n t c o m b i n e d w i t h a c h a i n of T e l e m e n t s (b, c). r e s t o r e the d e f o r m a t i o n to z e r o , b u t t e r m i n a t e s with a r e s i d u e d e f o r m a t i o n a R. A f t e r e a c h r e p e a t e d c y c l e with a m a x i m u m l o a d s m a l l e r than that a p p l i e d in the f i r s t c y c l e , the r e v e r s e a f t e r e f f e c t l e a d s to the s a m e r e s i d u a l d e f o r m a t i o n e R. We s h a l l c o n s i d e r the m a t h e m a t i c a l m o d e l w h i c h d e t e r m i n e s the r e l a t i o n s h i p b e t w e e n the p a r a m e t e r s P, a, t (time), and s h o w s the q u a l i t a t i v e f e a t u r e s noted a b o v e . B e f o r e the final f o r m u l a t i o n of the e q u a t i o n s we p e r f o r m a q u a l i t a t i v e a n a l y s i s of s o m e m e c h a n i c a l m o d e l s . The s u b s t a n t i a l h y s t e r e s i s , which d e p e n d s only s l i g h t l y on the r a t e of d e f o r m a t i o n , m a y b e a s c r i b e d to a p e c u l i a r i n f l u e n c e of the s l i d i n g f r i c t i o n on the o p e r a t i o n of r o p e s . L e t us c o n s i d e r a d r y f r i c t i o n e l e m e n t (Fig. 3). If the f o r c e a c t i n g on the m o d e l is d e n o t e d by P and the f o r c e a c t i n g on the e l a s t i c e l e m e n t ( i n t e r n a l f o r c e of the m o d e l ) by F, it is e a s i l y u n d e r s t o o d that d u r i n g the l o a d i n g p r o c e s s t h e s e f o r c e s o b e y the. r e l a t i o n s h i p F = aP,

(1)

p = ~F,

(2)

and d u r i n g u n l o a d i n g

w h e r e c~ d e p e n d s in a w e l l - k n o w n w a y on the f r i c t i o n c o e f f i c i e n t and on the a n g l e b e t w e e n the f i l a m e n t and the c y l i n d e r s , w h i l e 0 -< oL ~ 1. D e n o t i n g the r i g i d i t y of the e l a s t i c e l e m e n t by e, w e h a v e F =

(3)

cx,

w h e r e x d e n o t e s the e l o n g a t i o n of the e l a s t i c e l e m e n t and by d e f i n i t i o n e q u a l s the e l o n g a t i o n of the m o d e l T. U s i n g E q s . (1), (3), w e find f o r the l o a d i n g p r o c e s s 1

x = teA~ c-------. c

(4)

D u r i n g u n l o a d i n g f r o m f i n a l l o a d P c to P = oz2P c the p a r a m e t e r x r e m a i n s u n a l t e r e d , s i n c e the i n t e r n a l f o r c e F e of the m o d e l e q u a l s cx P c a t t h e s t a r t of u n l o a d i n g , and r e t a i n s that v a l u e b e c a u s e the i n e q u a l i t y P > c~F e h o l d s . A c c o r d i n g to E q s . (2), (3), d u r i n g f u r t h e r u n l o a d i n g , we h a v e x = --; v .

(s)

L e t us i n t r o d u c e a n o n l i n e a r l o a d i n g in m o d e l T by a s s u m i n g that the e l a s t i c e l e m e n t h a s l i m i t e d e x t e n s i b i l i t y , i . e . , t h a t , if p a r a m e t e r x a t t a i n s a g i v e n v a l u e , 5 and, c o n s e q u e n t l y , the i n t e r n a l f o r c e a t t a i n s the v a l u e F 0 = eS, a f u r t h e r i n c r e a s e of the l o a d d o e s not r e s u l t in f u r t h e r e l o n g a t i o n of the m o d e l T . F r o m E q s . (4) and (5), it f o l l o w s that in the c a s e of l i m i t e d e x t e n s i b i l i t y a c y c l e c o n s i s t i n g of l o a d i n g of m o d e l T to a l o a d P > ( ' t / o 0 F 0 and u n l o a d i n g to P = 0 is r e p r e s e n t e d b y the f i g u r e O a b a e O (Fig. da). L e t us now c o n s i d e r a u n i f o r m f i l a m e n t in which e a c h e l e m e n t of unit l e n g t h is m o d i f i e d b y N e l e m e n t s (T) with d i f f e r e n t c h a r a c t e r i s t i c s cq, c i , 6 i ( i -- 1 , 2 . . . . . N), and p a s s at o n c e to the l i m i t N ~ o o . The c o n tinuous p a r a m e t e r z, w h i c h v a r i e s in the r a n g e 0 -< z < 0% w i l l r e p l a c e the n u m b e r i of the e l e m e n t , and p a r a m e t e r s c~, e, 5 w i l l b e f u n c t i o n s of p a r a m e t e r z. T h e f u n c t i o n P(x) of this m o d e l is d e t e r m i n e d b y the foUowing r e l a t i o n s h i p s :

342

P

&

---7

R

"

~

Iii j

Fig. 5

I

Fig. 6

Fig. 5. Diagram of the complex model incorporating the elements T, M, and R, and used for describing the deformation of p o l y m e r ropes. Fig. 6. The c h a r a c t e r of slow (drawn lines) and fast (dashed lines) loading and unloading p r o c e s s e s . 1. The elongation dx of those elements (T) for which the z value lies in ~he interval (z, z + dz) reads

d x = c ( z ) F(z)dz,

if

~(z)F(z)~8(z);

dx = 8 (z) dz,

if

~ (z) F (z) ~, 8 (z).

(6)

2. The internal f o r c e F in an element c h a r a c t e r i z e d by a given value of z is given by

F(z) = a(z)P, 1 F(z) =--~-ffP,

if this element is in the loaded state;

(7)

if this element is in the unloaded state.

Relationships (6) and (7) define the functional relationship between f o r c e P and elongation x of the model, the latter depending on the loading history. Let us r e p r e s e n t the relationship in a m o r e explicit form; in doing so, we shall write down different relationships for Ioadingand unloading. We denote the value of the distribution F(z) Of the internal f o r c e s by the m o m e n t the sign of the loading rate is r e v e r s e d by F i (z), and the values of p a r a m e t e r s P and x at this moment by Pi and x i. In the case of loading it then follows f r o m Eqs. (6) and (7) that dx f'

~ a (Z)C(Z) dz:

x (Pi)= xi,

(8)

o

where the integration range co, which depends on the force P and the loading h i s t o r y via F i(z), is defined by the inequalities F (z) a (z) < p' c (z) r (z) P < 6 (z).

(9)

During unloading we shall have

d--L

dz, x (Pi)= xi,

dP

(10)

o

where the integration range oJ is defined by the inequalities 05 (z)

F (z)> P, ~ p < ~(z).

We shall illustrate the operation of the model by considering some actual c a s e s . c-(z) = c = c o n s t ,

o~ (z) = e - z , 8 (z) =

cPoe-2z.

(11) We take

(12)

343

In s o m e c a s e s we a l s o s u p p o s e that the d e f o r m a t i o n of the m o d e l e l e m e n t s is u n l i m i t e d ( 5 ~ o o ) . D u r i n g the f i r s t l o a d i n g of the m o d e l F i (z) -- 0 and, a c c o r d i n g to i n e q u a l i t i e s (9), the i n t e g r a t i o n r a n g e w is d e t e r m i n e d b y i n e q u a l i t y ~ e - Z P < ~P0 e - 2 z , f r o m w h i c h it f o l l o w s that the i n t e g r a t i o n r a n g e c o v e r s the i n t e r v a l (0, In Po/P). C o n s e q u e n t l y , u t i l i z i n g E q s . (8), w e can w r i t e down dx

(1

x(O)

P 70)

0

or

x(P)=cIP--

pZ

1

Po }"

2

(13)

The l a t t e r e q u a t i o n h o l d s up to l o a d s P -< P0- A t l o a d s f o r w h i c h P > P0, the m o d e l b e h a v e s a s a r i g i d b o d y , w h i c h f o l l o w s d i r e c t l y f r o m E q s . (9), (125. If (5 ~ co, the f o l l o w i n g r e l a t i o n s h i p w i l l hold for any v a l u e s of c(z), c~ (z): (14)

x (P) = kP, w h e r e k = ; c(z)c~ (z)dz = e o n s t .

F o r the e x a m p l e c o n s i d e r e d k = c .

0

F o r u n l o a d i n g f r o m a f i n a l v a l u e of l o a d P c w e h a v e

(

--'

Pi=Pc, xi = c Pc-- 2 Po]' Fi(z5 =e-'Pc T h e r a n g e co is d e f i n e d b y the i n e q u a l i t i e s <

e-2ZPc > P;

rp; 3

ce? < ce-2"Po

V.

o,

3 2 the r a n g e w c o v e r s the i n t e r v a l (0, In J P C - ~ ) , and we c a n C o n s e q u e n t l y , d u r i n g u n l o a d i n g to l o a d s P -> Pc/P0 write d"'fi" =

,

'

2

P.

or

x

=

c

2 V~c

-- P -- T"-~o

/ .

(15 ')

Upon further unloading, the range w covers the interval (0, [n ~r-P0/P). Consequently, dx

~p-=C

-

-- 1 , x\

(Pp~c =

Pc:

2

Po

P~o )

or

x = c \(~3-

If 6 - - %

3v'/po_

P) 9

(15"5

the f o l l o w i n g e q u a t i o n h o l d s f o r a l l P v a l u e s in s u c h a w a y that P < P c : X = C(2 I PP--c"-- P).

(16)

The loading-unloading cycle (13), (15) is shown in Fig.4b. Upon complete unloading F i (z) -= 0 and, consequently, any repeated loading cycle always coincides with the first cycle. W e shall consider the case of repeated loading after incomplete unloading to a value Pr < Pc; in the model (125 w e then restrict ourselves to the case that 6 4 ~ The internal force F i at the start of the repeated loading is given by

344

/Fi

'>'hi/

= ~ (z) P c = e-ZPc

Fi (z) _ -

~

(z) r = l____p,,

e'Pr

if

z41n

/,

{. Pc'

~.

The i n t e g r a t i o n r a n g e is d e f i n e d b y the i n e q u a l i t i e s

Pc <

Pr'

i . e . , the r a n g e w c o v e r s the i n t e r v a l (0, In 4 P - ~ r ) .

Consequently,

or x = c ( P - - 2]/PPr q- 2 [/PrPe).

(17)

Of c o u r s e , if P r = 0, Eq. (17) t r a n s f o r m s into Eq. (14). The p r o c e s s of l o a d i n g - u n l o a d i n g - l o a d i n g a f t e r i n c o m p l e t e u n l o a d i n g a s d e s c r i b e d b y E q s . (14), (16), (17) is r e p r e s e n t e d in F i g . 4e. We n o t e that the d e f o r m a t i o n of a r o p e and that of a u n i f o r m f i l a m e n t , both b e i n g d e s c r i b e d b y the m o d e l c o n s i d e r e d , h a v e s e v e r a l c h a r a c t e r i s t i c f e a t u r e s in c o m m o n , v i z . : the i n c r e a s e in r i g i d i t y upon l o a d i n g (at 6 < =o f o r the m o d e l ) , the p r e s e n c e of a v e r t i c a l t a n g e n t to the r e p r e s e n t i n g l i n e at the m o m e n t u n l o a d i n g s t a r t s , a h o r i z o n t a l t a n g e n t to t h e r e p r e s e n t i n g l i n e at the m o m e n t l o a d i n g s t a r t s a f t e r i n c o m p l e t e u n l o a d i n g , a v e r t i c a l t a n g e n t to the r e p r e s e n t i n g l i n e a t the s t a r t of l o a d i n g a f t e r i n c o m p l e t e u n l o a d i n g , and the c i r c u m s t a n c e that the f o l l o w i n g i n e q u a l i t i e s hold: P l > P2 at x < x c , Pl(xe) = P2(Xe), d P t ( X c ) / d x < d P 2 ( x c ) / d x (here, the i n d i c e s 1, 2 r e f e r to l o a d i n g a f t e r c o m p l e t e and i n c o m p l e t e l o a d i n g , r e s p e c t i v e l y ) . To d e s c r i b e the d e f o r m a t i o n of twined p o l y m e r r o p e s , we i n t r o d u c e a c o m p l e x m o d e l i n c o r p o r a t i n g a T e l e m e n t , a M a x w e l l - F e u c h t e l e m e n t M and an e l e m e n t R a s shown in F i g . 5. T h e e l e m e n t R m a k e s it p o s s i b l e to r e p r e s e n t the p r o p e r t y of r o p e s to a c q u i r e a r e s i d u a l d e f o r m a t i o n a R w h i c h d e p e n d s on the m a x i m u m l o a d . T h e r o u g h w e d g e of m o d e l R h a s a s l o p e s m a l l e r than the f r i c t i o n a n g l e ; the f o r c e in the e l a s t i c e l e m e n t 2 d e p e n d s on the e l o n g a t i o n eR of e l e m e n t R. We s h a l l now c o n s i d e r the m a t h e m a t i c a l m o d e l s u i t a b l e f o r d e s c r i b i n g the d e f o r m a t i o n of twined p o l y m e r r o p e s and d e r i v e d b y s t a r t i n g f r o m the a n a l o g of the m o d e l shown in F i g . 5. The f o r c e P and the d e f o r m a t i o n ~ a r e r e l a t e d b y the f o l l o w i n g e q u a t i o n s

P = P' q- 9 (~ -- Y);

'[

P' = F tel; dy ,, P"-a/- = p [~ - - Y};

} [

g = g - - e R,88=cp(Pmax)

I

(18)

The second equation of the set (18) represents the dependence of P' on ~ according to Eqs, (8)-(11) in w h i c h P h a s to b e r e p l a c e d by P ' , and x by a. The h m c t i o n s and c o n s t a n t s o c c u r r i n g in the s e t 0-8), v i z . , (z), c (z), 6(z), p, p a r e d e t e r m i n e d e x p e r i m e n t a l l y . In this p a p e r the d e t e r m i n a t i o n p r o c e d u r e is not r e ported. We a n a l y z e s o m e p r o p e r t i e s of m o d e l (18). We c o n s i d e r r e p e a t e d d e f o r m a t i o n b y l o a d s s m a l l e r than the m a x i m u m l o a d d u r i n g the f i r s t l o a d i n g , i . e . , e R = c o n s t . A f t e r the r o p e h a s b e e n k e p t in the u n l o a d e d s t a t e f o r s o long a t i m e that it c a n b e t a k e n that F(z) -= 0, l o a d i n g is d e s c r i b e d b y the e q u a t i o n s

P = f, (~') + p (g-- v): dy

09)

I,--a- =p~.--v); Y(0)=0, w h e r e ft ~) d e n o t e s the c u r v e r e p r e s e n t i n g l o a d i n g f r o m the z e r o s t a t e of m o d e l T. At low (infinitely tow) l o a d i n g r a t e s it f o l l o w s f r o m E q s . 0-9) that

345

"~=

;,

P

[1 (;'), Pc =

=

[, (~),

(20)

and at h i g h (infinitely high) l o a d i n g r a t e s the f o l l o w i n g e q u a t i o n s h o l d :

P=f,(-~)+o;, Pc =

v=0.

(21)

f,(;c)+p-~.

If the i n e q u a l i t y

a[, -~[>>P

(22)

' e cn in the p r o c e s s e s (20), (21) at one and the s a m e final i s f u l f i l l e d , t h e final v a l u e s of the d e f o r m a t i o n Sc' l o a d P c d i f f e r only s l i g h t l y . We n o t e that, owing to the i n c r e a s i n g c u r v a t u r e of the c u r v e f l ~ ) , the l e f t - h a n d t e r m in i n e q u a l i t y {22) e x c e e d s the r i g h t - h a n d t e r m e v e r m o r e a s the l o a d P c is r a i s e d . U s i n g the f o l l o w i n g a p p r o x i m a t i o n to fl (~')

I, (;) == r, (;i) + Ii (; dr,(~c~ fi==

-

O; (23)

d~

we find

The c r e e p at a c o n s t a n t l o a d P c f r o m the s t a t e with d e f o r m a t i o n -"e c p r o d u c e d b y f a s t l o a d i n g l e a d s to d e f o r m a t i o n e~; u t i l i z i n g the a p p r o x i m a t i o n to fl ~ ) , we c a n d e s c r i b e the c r e e p p r o c e s s by the e q u a t i o n s

(f; + I,) ; ~ m' + f~; dv =~,(;--v), ~'(o)

0

o r

s

e

1--e

;

(24)

m

U n l i k e l o a d i n g , the p r o c e s s e s of s l o w and f a s t u n l o a d i n g f r o m s t a t e P c , e~ m a y d i f f e r s u b s t a n t i a l l y , if Eq. (22) is f u l f i l l e d . Slow u n l o a d i n g p r o c e e d s in a c c o r d a n c e with t h e e q u a t i o n s v = ::, P = f~ ~ )

(2s)

w h e r e f2~) d e n o t e s the c u r v e r e p r e s e n t i n g c o m p l e t e u n l o a d i n g of m o d e l T f r o m the l o a d P = P c , and ends in s t a t e P = 0, ~ = 0. F a s t u n l o a d i n g l e a d s to the s t a t e y = s~, e = e0, w h e r e the r e s i d u a l d e f o r m a t i o n ~0 is d e t e r m i n e d b y the e q u a t i o n

h (;o) +

=

(26)

S i n c e the e - a x i s m a y b e a t a n g e n t of s e c o n d o r h i g h e r o r d e r to the c u r v e f2('~), the v a l u e of ~'0 m a y a l s o b e c o m p a r a b l e to ~ , if p is s m a l l . A q u a l i t a t i v e p i c t u r e of the p r o c e s s e s c o n s i d e r e d h e r e is s h o w n in F i g . 6. S t a t e P = 0, e = ~0, 3' = ~ equations

m a r k s the s t a r t of the p r o c e s s of i n v e r s e c r e e p w h i c h is d e s c r i b e d by the

(27)

dV dt = o ( ~ - - v). If in the i n t e r v a l (0, ~0) the function f2~) c a n b e a p p r o x i m a t e d b y a p a r a b o l a of the n - t h d e g r e e (f(~') = A~ n) and the a p p r o x i m a t i o n ~ = y - ( A / p ) y n is s u b s t i t u t e d in the f i r s t e q u a t i o n of s e t (27), the s o l u t i o n of t h i s s e t reads

346

;0

(2S)

Function (28) indicates that the i n v e r s e aftereffect proceeds v e r y slowly c o m p a r e d with an exponential p r o c e s s . We note that the inverse aftereffect for LTK-25-100 ropes (see Fig. 2) is described by equations of type (28) with n = 4; e R = 0.075; ~0 = 0.06, A//~ = 67 9103 h -1. Since for the model analyzed the c h a r a c t e r i s t i c time of c r e e p under load is much s h o r t e r than the c h a r a c t e r i s t i c time of the r e v e r s e aftereffect, it may be stated that there exists such an interval of d e f o r m a tion r a t e s , that, if a sample is exposed to consecutive l o a d i n g - u n l o a d i n g cycles with deformation r a t e s lying within this interval, the positions of the limiting closed curves (limiting cycles) by which these p r o c e s s e s are r e p r e s e n t e d in the P - e plane at any constant deformation rate will depend only slightly on the d e f o r m a tion rate. The actual boundaries of this interval depend on the values of the constants p, /1, the functions ft(e) and f2(e), the maximum load Pc in a cycle, and, of course, on the actual definition of the closeness of limiting c y c l e s . Consequently, the proposed mathematical model (18) b e a r s s e v e r a l qualitative r e s e m b l a n c e s to the experimentally established relationships, which makes it possible to use this model for describing the d e formation of twined p o l y m e r ropes. Finally, the author s i n c e r e l y thanks V.N. Kuznetsov.

347