Effective Poisson's ratio of isotropic ice

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Effective Poisson's ratio of isotropic ice

Sinha, N. K.

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Effective

Poisson's

Ratio

_of

_Isotropic

Ice

by N.K. Sinha

Reprinted from

Proceedings of the Sixth (1987) International

Offshore Mechanics and Arctic Engineering Symposium

Houston, TX. March 1-5,1987

Vol. IV, p. 189- 195

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(IRC Paper No. 1472)

N R C

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CISTI

Price $3.00

NRCC 27998

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EFFECTIVE POISSON'S RATIO OF ISOTROPIC ICE

N. K. Sinha

Institute for Research in Construction National Research Council of Canada

Ottawa, Ontario, Canada

ABSTRACT

The r a t i o of l a t e r a l s t r a i n t o a x i a l s t r a i n i n a non-linear v i s c o e l a s t i c m a t e r i a l such a s i c e depends on f a b r i c , t e x t u r e , g r a i n s i z e , temperature, load and loading r a t e , s t r a i n and s t r a i n rate. R e l a t i v e c o n t r i b u t i o n s of e l a s t i c , delayed e l a s t i c , and viscous s t r a i n t o t h e t o t a l s t r a i n determine i t s numerical value. A micromechanically based r h e o l o g i c a l model has been used t o show t h a t m a t e r i a l response, however complex, i s reasonably p r e d i c t a b l e and agrees w i t h a v a i l a b l e experimental r e s u l t s .

INTRODUCTION

Under n a t u r a l c o n d i t i o n s i c e i s always a t a h i g h temperature thermally, i.e., a t homlogous temperatures g r e a t e r than about 0.4 Tm, where Tm i s melting p o i n t i n Kelvin. A t t h e s e temperatures deformation processes r e l a t e d t o g r a i n boundaries play a n important r o l e . For t h i s reason i c e e x h i b i t s mechanical p r o p e r t i e s t h a t appear t o be p e c u l i a r t o anyone u n f a m i l i a r w i t h t h e p r o p e r t i e s of m a t e r i a l s a t e l e v a t e d temperatures. The most n o t i c e a b l e e f f e c t s of such deformation processes a r e t h e p e c u l i a r i t i e s i n Young's modulus and Poisson's r a t i o . V a r i a b i l i t i e s observed i n a x i a l e l a s t i c response, o f t e n reported a s Young's modulus, a r e now understood ( 1 , 2 ) , but t h o s e of Poisson's r a t i o

-

t h e r a t i o of l a t e r a l s t r a i n t o a x i a l s t r a i n o r simply s t r a i n r a t i o , p

-

a r e not s o w e l l defined.

Dynanic values of Poisson's r a t i o i n t h e range 0.31 t o 0.37 f o r freshwater i c e were t a b u l a t e d some y e a r s ago by Gold (3). Lin'Kov's ( 4 ) i n - s i t u s e i s m i c determinations f o r s e a i c e ranged from 0.36 t o 0.39. Both Peschansky (5) and Langleben and Pounder (61, however, reported dynamic values of 0.29 f o r s e a ice. There i s l i t t l e discrepancy i n t h e d a t a , i r r e s p e c t i v e of i c e type. S t a t i c a l l y determined v a l u e s , on t h e o t h e r hand, show g r e a t v a r i a b i l i t y .

For t r a n s v e r s e l y i s o t r o p i c , columnar-grained, freshwater i c e loaded perpendicular t o t h e columns, Gold's experiments i n d i c a t e d values i n t h e range of 0.31 t o 0.54 f o r t h e r a t i o of t r a n s v e r s e t o

l o n g i t u d i n a l s t r a i n s (3). Corresponding t o t h i s , Wang

( 7 ) reported a range of 0.8 t o 1.2 f o r s e a ice. He obtained, however, values i n t h e range of 0 t o 0.2 i n t h e d i r e c t i o n p a r a l l e l t o t h e l e n g t h of t h e columns. Saeki e t a l . (8) a l s o s t u d i e d deformation along t h e l e n g t h of t h e columns i n s e a i c e but reported a complex response, t h e r a t i o i n c r e a s i n g from 0.02 t o 0.48 w i t h i n c r e a s e i n s t r e s s r a t e from 0.01 t o 0.5 M N * ~ - ~ s - ~ , then decreasing a t h i g h e r r a t e s . A d d i t i o n a l l y , Saeki e t a l . r e p o r t e d t h a t t h e r a t i o increased w i t h decrease i n temperature. T o t a l l y o p p o s i t e temperature and s t r e s s r a t e e f f e c t s were noted by Murat and Lainey

( 9 ) f o r laboratory-made s a l i n e ice. While examining t h e r a t e s e n s i t i v i t y of t h e compressive s t r e n g t h of congealed f r a z i l s e a i c e , Sinha (10) observed t h a t s t r a i n r a t i o depends not only on r a t e of loading o r loading h i s t o r y but a l s o on s t r e s s o r s t r a i n l e v e l and t h e (damage) s t a t e of t h e material.

I n d i s c u s s i n g Wang's (7) experimental work, Sinha (11) s t a t e d t h a t t h e c o n t r i b u t i o n of t h e grain-boundary o r i n t e r p l a t e l e t s l i d i n g s t r a i n t o t o t a l s t r a i n , i n a d d i t i o n t o t h a t of pure e l a s t i c and viscous flow, i n f l u e n c e s both e f f e c t i v e modulus and s t r a i n r a t i o . Unless low amplitude and very high frequency loading ( i n t h e o r d e r of MHz) a r e involved, i t i s almost impossible f o r only e l a s t i c deformation t o e x i s t without c o n t r i b u t i o n s from grain-boundary s l i d i n g o r delayed e l a s t i c and viscous d e f o r m a t i ~ n . This observation i s a p p l i c a b l e t o deformation p a r a l l e l a s w e l l a s normal t o t h e a x i s of loading. Consequently, t h e r a t i o between l a t e r a l s t r a i n and l o n g i t u d i n a l s t r a i n r e f l e c t s t h e c o n t r i b u t i o n of non-pure e l a s t i c deformation. The p r e s e n t paper compares t h e s e a s p e c t s of i c e behaviour w i t h experimental measurements made p r i m a r i l y on s a l i n e ice.

DEFINITION OF POISSON'S RATIO AND STRAIN RATIO P o l y c r y s t a l l i n e i c e i n n a t u r e i s g e n e r a l l y a n a n i s o t r o p i c , non-linear, v i s c o e l a s t i c s o l i d and i t s mechanical behaviour i s described by t h e g e n e r a l i z e d Hooke's law r e l a t i n g s t r a i n , E ~ ,t o s t r e s s , u ; t h a t

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where Sij d e n o t e s t h e compliances. G r a n u l a r snow i c e , f o r a l l p r a c t i c a l p u r p o s e s , may be c o n s i d e r e d t o be a n i s o t r o p i c m a t e r i a l . For commonly observed,

t r a n s v e r s e l y i s o t r o p i c ( o r o r t h o t r o p i c ) columnar-grained l a k e , r i v e r , o r s e a i c e ( c l a s s i f i e d a s S-2 i c e ) and c o n s i d e r i n g two d i m e n s i o n a l p l a n e - s t r e s s w i t h p r i n c i p a l e t r e a s e s i n XI and X p d i r e c t i o n , one i s u s u a l l y concerned w i t h t h e compliances S l l

-

SZ2 f S33, w i t h a x i s Xj a l o n g t h e axFs of t h e c o l u m a r g r a i n s , i . e . , t h e v e r t i c a l ( g r o v t h ) d i r e c t i o n , and a x i s XI and a x i s X2 i n t h e p l a n e of t h e i c e c o v e r p e r p e n d i c u l a r t o t h e growth d i r e c t i o n , i.e., i n the h o r i z o n t a l plane.

For s t r e s s e s a p p l i e d i n t h e h o r i z o n t a l p l a n e , s a y a x i s X

,

t h e c a s e i n many e n g i n e e r i n g s i t u a t i o n s , t h e l a t e r a l compliances of concern a r e Slland S31. ' h e major s t r a i n r a t i o s a r e g i v e n by P~~ = -S21

1

S l l and ~ 3 1

-

-Sgl

/

S l l ( 2 ) R e g a r d l e s s of l o a d i n g c o n d i t i o n s , d e f o r m a t i o n , e i , of any p o l y c r y s t a l l i n e m a t e r i a l ( i n c l u d i n g i c e ) a t h i g h homologous t e m p e r a t u r e s c a n b e d e s c r i b e d phenomenologically a s ( 1 ) '

i = 'ij 'j = ' i e + 'id + 'iv

where

tie

i s p u r e e l a s t i c and immediately r e v e r s i b l e , cid i s delayed e l a s t i c and recovers w i t h time, and cIV i s v i s c o u s o r permanent s t r a i n .

r'

For s i m p l i c i t y , c o n s i d e r i s o t r o p i c m a t e r i a l . With

'4- u n i a x i a l l o a d i n g under a s t r e s s of a l , t h e r a t i o

between l a t e r a l and a x i a l s t r a i n s , f o l l a w i n g Eq. 2 and 3, i s g i v e n by For p e r f e c t l y e l a s t i c l o a d i n g c o n d i t i o n s w i t h no c o n t r i b u t i o n from d e l a y e d e l a s t i c and v i s c o u s d e f o r m a t i o n , Eq. 4 g i v e s t h e P o i s s o n ' s r a t i o a s i n which s u b s c r i p t e i n d i c a t e s t h e e l a s t i c component of t h e deformation. P o i s s o n ' s r a t i o i s h i s t o r i c a l l y -synonymous w i t h e l a s t i c deformation. I n ice a t h i g h

homologous t e m p e r a t u r e s i t i s n e a r l y i m p o s s i b l e t o have only e l a s t i c d e f o r m a t i o n i n any s t a t i c measurements.

C n s e q u e n t l y , t h e s t a t i c a l l y determined s t r a i n r a t i o

C

r e f l e c t s t h e c o n t r i b u t i o n of non-pure e l a s t i c deformation. E q u a t i o n 4 shows t h e complexity of t h e s t r a i n ~ a t i o even f o r i s o t r o p i c m a t e r i a l . It may e a s i l y be 1 s e e n how complex a t r a n s v e r s e l y i s o t r o p i c m a t e r i a l \ would be, l e t a l o n e t h e u s u a l l y a n i s o t r o p i c s o l i d s t h a t L b o u n d i n n a t u r e . The q u e s t i o n a r i s e s whether p r e s e n t knowledge of i c e rheology and micromechanics can a s s i s t u n d e r s t a n d i n g of t h i s complex behaviour; i . e . , whether ,-.present knowledge c a n a s s i s t i n f o r m u l a t i n g t h e three-dimensional c o n s t i t u t i v e r e l a t i o n s t h a t a r e r e q u i r e d f o r e n g i n e e r i n g a p p l i c a t i o n s . k4 AXIAL STRAIN U n i a x i a l , c o n s t a n t - l o a d c r e e p of p o l y c r y s t a l l i n e i c e was u s e d by Sinha ( 1 ) t o d e m o n s t r a t e a method of examining t h e t h r e e s t r a i n components i n Eq. 3. He showed t h a t p u r e e l a s t i c d e f o r m a t i o n i s r e l a t e d t o l a t t i c e d e f o r m a t i o n and t h a t t h e v i s c o u s component can be a t t r i b u t e d t o i n t r a g r a n u l a r d e f o r m a t i o n p r o c e s s e s , p a r t i c u l a r l y t o t h e movement of d i s l o c a t i o n s f o r t h e

l e v e l s of s t r e s s and r a t e s of l o a d i n g e n c o u n t e r e d i n e n g i n e e r i n g s i t u a t i o n s . He h y p o t h e s i z e d t h a t d e l a y e d e l a s t i c i t y i s a s s o c i a t e d w i t h i n t e r g r a n u l a r s l i d i n g phenomena. These p h y s i c a l p r o c e s s e s were considerGd i n modifying t h e phenomenological c r e e p model, a l l o w i n g t h e i n c o r p o r a t i o n of t h e e f f e c t of g r a i n s i z e (2). k i a l s t r a i n , €1, a t t i m e , t , i n p u r e

randomly-oriented p o l y c r y s t a l l i n e m a t e r i a l of g r a i n s i z e , d , s u b j e c t e d t o u n i a x i a l s t r e s s , 01, a t t e m p e r a t u r e , T, was g i v e n by

where E i s Young's modulna;

E,

i s t h e v i s c o u s s t r a i n - 0

r a t e f o r u n i t o r r e f e r e n c e stress

ao;

c l is a c o n a t a n t c o r r e s p o n d i n g t o t h e u n i t o r r e f e r e n c e g r a i n s i r e , d l ; b, n, and s a r e c o n s t a n t a ; eT i s t h e i n v e r s e r e l a x s t i o n time. Both

;

and sT v a r y w i t h t e m p e r a t u r e , T, i n

" 0

K e l v i n and were shown t o h a v e t h e same v a l u e f o r a c t i v a t i o n e n e r g y , a s f o l l o w s :

i

(T2) ( T I ) F l P 2

vo vo

( 7 ) and a T ( T 2 ) a~ F1,2

where T1 and T2 a r e t e m p e r a t u r e s and F i s a s h i f t 1.2

f u n c t i o n g i v e n by F = exp

{g

(1

-

1 ) }

i n which Q

1,2 R T. T,

1 L

and R a r e a c t i v a t i o n energy and g a s c o n s t a n t s , r e s p e c t i v e l y .

I n r e a l i t y , t h e r h e o l o g i c a l e q u a t i o n a c t u a l l y a p p l i c a b l e t o i c e i s s i m p l e r than Eq. 6 because e x p e r i m e n t s o n ice have i n d i c a t e d t h a t s = 1 and b = l / n (1). Consequently, c r e e p of i s o t r o p i c i c e of a g i v e n g r a i n s i z e a t a g i v e n t e m p e r a t u r e c a n be

d e s c r i b e d i n terms of o n l y f i v e m a t e r i a l c o n s t a n t s , E, c1, aT,

kv-

and n. A s E c a n be e s t i m a t e d f a i r l y a c c u r a t e l y U f r o m a v a i l a b l e s i n g l e - c r y s t a l d a t a , t h e unknown m a t e r i a l c o n s t a n t s reduce t'o f o u r . For t h e system i n which g r a i n s i z e , d , i s e x p r e s s e d i n mm ( i . e . , d l a 1 nun), s t r e s s , 01, i n ~ N - r n - ~ o r MPa ( i . e . , uO = 1 MN*m-*), and t i m e , t , i n seconds, e x p e r i m e n t a l o b s e r v a t i o n s of pure. t r a n s v e r s e l y i s o t r o p i c i c e i n c o n j u n c t i o n w i t h micro-mechanical c o n s i d e r a t i o n s p r o v i d e d t h e f o l l o w i n g v a l u e s : E = 9.5 G N . I ~ - ~ = 9, a (T a 263 K) = 2.5 x 10-4 s'l,

;..

(T 5 263 K; E11,8 x 18-7 s-1 and n = 3 (2). The " 0

a c t i v a t i o n energy, Q, was found t o be 67 kJ mol-I (16 k c a l mol-I), t a k i n g R = 8.32 J mol-I K - ~ .

A s i m p l e numerical i n t e g r a t i o n method f o r

p r e d i c t i o n of s t r a i n , c o r r e s p o n d i n g t o a n a r b i t r a r y monotonically i n c r e a s i n g s t r e s s h i s t o r y , was developed by Sinha (12) on t h e b a s i s of Eq. 6. For t h e

p a r t i c u l a r c a s e of u n i a x i a l c o n s t a n t s t r e s s - r a t e ,

frl,

i t was shown (12) t h a t a x i a l s t r a i n ( f o r i c e ) i s g i v e n i n terms of t h e f o l l o w i n g t h r e e s t r a i n components a t t i m e , t , a f t e r i n i t i a t i o n of l o a d i n g ( s = 1 and b = l / n ) :

(6)

where ul = b l t = NAul =

~ f r ~ ~ t ;

N i s t h e number of s t e p s chosen t o d i v i d e t h e s t r e s s path.

LATERAL STRAIN

The u n i a x i a l e q u a t i o n s d e s c r i b e d i n t h e p r e v i o u s s e c t i o n were developed e s s e n t i a l l y f o r homogeneous and i s o t r o p i c i c e o r , a t t h e most, f o r l o a d a p p l i e d i n a p l a n e e x h i b i t i n g i s o t r o p y i n t e x t u r e and f a b r i c . A

s i m p l e method of p r e d i c t i n g l a t e r a l s t r a i n can be developed i f t h e micromechanics g o v e r n i n g t h e t h r e e s t r a i n components i n Eq. 6 o r 8 a r e examined. To c l a r i f y t h e b a s i c i d e a , only i s o t r o p i c i c e w i l l be d i s c u s s e d f o r l o a d i n g c o n d i t i o n s w i t h o u t microcracking. Behaviour of t r a n s v e r s e l y i s o t r o p i c and a n i s o t r o p i c i c e w i l l , because of t h e c o m p l e x i t i e s and p r e s e n t s p a c e l i m i t a t i o n s , be p r e s e n t e d elsewhere. Consider e l a s t i c r e s p o n s e f i r s t . It can be d e s c r i b e d i n a s i n g l e c r y s t a l of i c e by f i v e moduli. P o l y c r y s t a l l i n e m a t e r i a l s , on t h e o t h e r hand, e x h i b i t a mixed s i n g l e - c r y s t a l e l a s t i c r e s p o n s e , r e f e r r e d t o e a r l i e r a s "pure e l a s t i c . " The numerical v a l u e of 9.5 GPa f o r E, g i v e n e a r l i e r , r e p r e s e n t s an e s s e n t i a l l y average v a l u e of t h e f i v e e l a s t i c c o n s t a n t s f o r s i n g l e c r y s t a l s of i c e measured a t f r e q u e n c i e s of about 15-18 MHz (13, s e e Chapter 8 ) . T h i s a g r e e s w e l l w i t h Young's modulus of p o l y c r y s t a l l i n e i c e determined by dynamic methods (3). L a t e r a l e l a s t i c s t r a i n i n i s o t r o p i c g r a n u l a r i c e produced by d e f o r m a t i o n of t h e l a t t i c e i n v a r i o u s g r a i n s may, f o r l o a d a p p l i e d a l o n g t h e 1-axis, be g i v e n by

where pe i s t h e P o i s s o n ' s r a t i o .

The second term i n Eq. 6 o r 8 i s t h e

d e l a y e d - e l a s t i c term a s s o c i a t e d w i t h grain-boundary s l i d i n g s t r a i n . Although stress c o n c e n t r a t i o n s would o c c u r a t grain-boundary j o g s and i n c l u s i o n s and a t t r i p l e p o i n t s , t h e s l i d i n g p r o c e s s would be i s o t r o p i c and might n o t a f f e c t volume a p p r e c i a b l y provided t h e r e was no c r a c k i n g a c t i v i t y . Hence, m a t e r i a l behaviour c o u l d , f o r s i m p l i c i t y , be c o n s i d e r e d a s i n c o m p r e s s i b l e due t o t h i s p r o c e s s . The l a t e r a l component of t h e delayed e l a s t i c s t r a i n would t h e n be g i v e n by For i s o t r o p i c i c e , where pd i n d i c a t e s t h e s t r a i n r a t i o f o r d e l a y e d - e l a s t i c o r grain-boundary s l i d i n g e f f e c t . For v i s c o u s d e f o r m a t i o n w i t h i t s i n h e r e n t i n c o m p r e s s i b i l i t y t h e l a t e r a l component i s given by and i s o t r o p i c d e f o r m a t i o n g i v e s

where pv i s t h e s t r a i n r a t i o f o r viscous deformation. STRAIN RATIO

S u b s t i t u t i n g E ~ E~ ~ ,and E from Eq. 9 , 11 and 13, r e s p e c t i v e l y , i n

tq.

4 g f i e s t h e o v e r a l l s t r a i n r a t i o f o r i s o t r o p i c m a t e r i a l as

Equation 14 i n c o n j u n c t i o n w i t h Eq. 6 and 7 g i v e s t h e dependence of s t r a i n r a t i o , d u r i n g c o n s t a n t s t r e s s c r e e p , on s t r e s s , s t r a i n , t i m e , t e m p e r a t u r e and g r a i n s i z e . Equations 7 , 8 and 14, on t h e o t h e r hand, can be used t o examine t h e s t r e s s r a t e s e n s i t i v i t y of t h e s t r a i n r a t i o i n a d d i t i o n t o t h e i n f l u e n c e of g r a i n s i z e and temperature. A s pe # 0.5, Eq. 14 m a i n t a i n s t h e g e n e r a l c o m p r e s s i b i l i t y of t h e m a t e r i a l .

EXPERIMENT

So f a r , t h e most comprehensive measurements of t h e r a t e e f f e c t on s t r a i n r a t i o have been c a r r i e d o u t by Murat and Lainey ( 9 ) on l a b o r a t o r y - m a d e s a l i n e i c e w i t h a s a l i n i t y of 5 p p t ( p a r t s p e r thousand). They used beams i n f l e x u r e under f o u r - p o i n t l o a d i n g .

L o n g i t u d i n a l and t r a n s v e r s e s t r a i n s were measured w i t h f o i l - t y p e s t r a i n gauges mounted d i r e c t l y on t h e s u r f a c e on t h e compression s i d e . The beams were 850 mm l o n g and had c r o s s - s e c t i o n a l dimensions of 90 mm x 9 0 mm. These were c u t from a n i c e s h e e t grown i n a t a n k 2.1 m i n d i a m e t e r and 1 m deep a f t e r s e e d i n g w i t h snow. A s t h e f u l l t h i c k n e s s of t h e o r i g i n a l i c e s h e e t was used, t h e beams were complex i n t e x t u r e and f a b r i c . The t o p l a y e r , on which t h e s t r a i n gauges were mounted, was g r a n u l a r , i s o t r o p i c i c e w i t h a n average g r a i n s i z e of 1 mm. T h i s primary l a y e r was a b o u t 3 mm t h i c k . Under

i t was a t r a n s i t i o n zone of a b o u t 10 mm. The i c e below

t h i s zone, i . e . about 85% of t h e beam d e p t h , was r e p o r t e d t o be S-2 t y p e columnar-grained.

T e s t s were c a r r i e d o u t a t v a r i o u s r a t e s of l o a d i n g i n t h e range 0.01 ~ ~ - r n - ~ s-l t o 0.6 s - l a t d i f f e r e n t t e m p e r a t u r e s r a n g i n g from -5OC t o -40°C. To complement t h e s t r a i n gauge measurements t h e a u t h o r s measured t h e c e n t r a l d e f l e c t i o n of t h e beams d u r i n g t h e t e s t s .

RESULTS AND ANALYSIS

Murat and Lainey ( 9 ) adopted a method of r e p o r t i n g t h a t i s s u i t a b l e f o r m a t e r i a l s a t low homologous t e m p e r a t u r e s b u t r a t h e r awkward f o r t h e h i g h

t e m p e r a t u r e s i n v o l v e d i n i c e i n v e s t i g a t i o n s . S t r a i n used f o r c a l c u l a t i n g e f f e c t i v e modulus and s t r a i n r a t i o was between two normalized l i m i t s : 4% and 67% of t h e c o r r e s p o n d i n g f l e x u r a l s t r e n g t h . These l i m i t s were e s t a b l i s h e d from r e s u l t s of p r e v i o u s t e s t s .

Remembering t h a t Murat and Lainey used composite beams and t h a t s t r e n g t h depends on r a t e of l o a d i n g ,

t e m p e r a t u r e , and t e x t u r e and f a b r i c of t h e i c e , t h e method adopted f o r r e p o r t i n g t h e r e s u l t s makes comparative s t u d y d i f f i c u l t . S u f f i c i e n t i n f o r m a t i o n was provided i n t h e p a p e r , however, t o p e r m i t t h e a n a l y s i s r e q u i r e d f o r comparing t h e e x p e r i m e n t a l r e s u l t s w i t h t h e p r e s e n t t h e o r e t i c a l p r e d i c t i o n s .

Examination h a s shown t h a t t h e r e p o r t e d s t r e s s r a t e dependence of e f f e c t i v e moduli and s t r a i n r a t i o s c o r r e s p o n d s t o a n upper l i m i t s t r e s s of 0.365 M N O ~ - ~ ,

0.550 1.025 M N . ~ - ~ and 1.500 MN-m-2,

r e s p e c t i v e l y , a t -5, -20, -30 and -40°C. T h e o r e t i c a l c a l c u l a t i o n s were made s a t i s f y i n g t h e s e c o n d i t i o n s and t h e o b s e r v a t i o n t h a t t h e g r a i n s i z e of t h e s u r f a c e l a y e r of t h e beams was 1 mm. Equation 8 was used t o c a l c u l a t e t h e t h r e e components of s t r a i n r e q u i r e d f o r t o t a l s t r a i n , c l , e f f e c t i v e modulus, Eu = u l / ~ l ( s e c a n t modulus), and e s t i m a t i o n of t h e s t r a i n r a t i o from Eq. 14. It w a s a l s o assumed ( f o r reasons t o be e l a b o r a t e d l a t e r ) t h a t pe = 0.3 f o r p u r e e l a s t i c i t y . M a t e r i a l c o n s t a n t s were a s g i v e n e a r l i e r . A b s o l u t e l y no e f f o r t was made t o f i t t h e p r e s e n t e x p e r i m e n t a l r e s u l t s by m a n i p u l a t i n g t h e v a l u e s of t h e m a t e r i a l c o n s t a n t s . T h e o r e t i c a l p r e d i c t i o n s made h e r e a r e what i s known i n s o i l mechanics a s c l a s s A p r e d i c t i o n s . It

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should be kept in mind that theoretical predictions are It should be noted that in Fig. 1 the contribution for an isotropic material whereas the experimental of viscous strain to total strain is so small (3 or 4 results were obtained on ice that was composite in orders) that a logarithmic scale for the strain axis

nature. had to be used to show its real magnitude. These low

Experimental and theoretical values for total values for viscous strain indicate that the deformation strain are compared in Fig. 1. There seems to be process during the tests was not controlled by power little difference between theory and observations at law mechanisms, often called "plastic flow."

-30 and -40°C, at which all salts are essentially in Deformation was therefore primarily "elastodelayed solid state. Increase in brine volume should affect elastic" in character. Calculations of the effective deformation at -20 and -5OC, and since the modulus, E (the subscript refers to stress level), liquid-enhanced creep process has not yet been taken based on tRe above theory, should therefore agree with into account the present theory is expected to those estimated on the basis of measured strain and the underestimate strain. Larger deviation between theory assumption that stress remains linear and symmetric and experiment at -5OC than at -20°C supports this across the depth of the beam. Excellent agreement

hypothesis. between theory and experiment at -30 and -40°C

(Fig. 2c.d) supports this. Although theoretical overestimation of E is expected at higher temperatures, the ogserved deviations do not give any clear indication of whether it was due to

u - I , J ~ I ~ I ~ I , I ' I 'non-linearity in the stress distribution or to ~

: increased boundary sliding and viscous creep

-

associated with brine. Volumetric strain data are

C ~ = C ~ e + C ~ d + ~ ~ v

-

required for a better understanding of the deformation process. As _{the deviation is larger} at higher rates and the viscous component

decreases rapidly with increase in rate of I I

7 loading, it is quite possible that increased brine volume affects delayed elasticity significantly.

I

The dependence of calculated and measured

strain ratio, p 2 ~ , on stress rate is presented in

i

I

STRAIN GAUGE Fig. 3. Different stress levels were involved at

different temperatures and the results are

I

-

u " ' E ~ L E c l l o ~ " y t ~ specific for those conditions. Both theory and experiment show that the ratio decreases with increased rate of loading and decrease in

-

_temperature. _As_{both axial and lateral strains}

would be affected in a similar way by increased brine volume, there should be little noticeable difference between the predicted and the measured strain ratios at various temperatures. This OoO

a 4

On6 0e8 agreement is certainly evident in Fig. 3. In each S T R E S S RATE,bl. ~ ~ - r n - ' . s - l case, however, the predicted values are lower than _{the experimental observations. Such a discrepancy}

is not unexpected because of the location of the strain gauges and the composite nature of the tested beams, consisting mostly of S-2 type ice in which lateral strain is expected to be greater in the plane of isotropy than in the plane normal to

it (7). Theoretical predictions and experimental

observations carried out at the National Research Council of Canada on rate sensitivity and

anisotropy of strain ratio in columnar-grained

1

freshwater and Arctic sea ice will be reported

elsewhere. DISCUSSION

-THEORY, GRAIN SIZE, d = I mm

One major element of the theoretical prediction is the assumption that pe = 0.3 in Eq. 14. This was essentially based on the Poisson's ratio of 0.295 suggested by Langleben and Pounder (6) following their resonance

experiments, and was in good agreement also with a value of 0.29 reported by Peschansky (5).

Langleben and Pounder also reported that their value was relatively constant for sea ice, a p p a r ently independent of salinity in the range of 0.1

0.0 0.2 0.4 0.6 0.8 0.0 a 2 0.4 0.6 a 8 1.0 t o 6 - l p ~ t s n d o f t e a p e r a t u r e i n t h e r a n g e o f - 3 . 6

3

- 2 -1 to -15'~. This last statement i e important S T R E S S RATE.

kl,

M N - r n .s becauee Poieson's ratio, like Poung'a modulus of

ice, should not depend to any extent on teqera- tare in the range significant for engineering. Figure 1. Stress rate dependence of strain Weeks and Assur (13) proposed the following

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E F F E C T I V E M O D U L U S . E,, G N . ~ - *

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e q u a t i o n f o r what may be c a l l e d 'dynamic' s t r a i n r a t i o ,

%,

pD = 0.33.3

+

6.105 x exp (815.48) (15) where 0 i s temperature i n deg C. T h i s e q u a t i o n was developed by c u r v e f i t t i n g of t h e e x p e r i m e n t a l d a t a on s t r a i n r a t i o o b t a i n e d f o r s e a i c e by Lin'Kov ( 4 ) i n a narrow temperature range of -1 t o -4OC. Lin'Kov deduced p from t h e v e l o c i t i e s of t h e s h e a r , Vs, an! compressional, Vp, p l a t e waves u s i n g p ~ = 1-2

(vS/vp).

The q u e s t i o n is: how w e l l does Eq. 15 apply t o lower t e m p e r a t u r e s and what a r e i t s l i m i t a t i o n s ?

Equation 15 g i v e s pD v a l u e s of 0.358, 0.335, 0.333 and 0.333 a t -5, -20, -30 and -40°C, r e s p e c t i v e l y . I n f a c t , t h e s e a r e t h e v a l u e s used by Murat and Lainey ( 9 ) a s l i m i t i n g v a l u e s i n t h e i r e f f o r t t o develop e m p i r i c a l e q u a t i o n s g i v i n g t h e s t r e s s r a t e dependence of t h e i r d a t a . The p r e s e n t t h e o r e t i c a l p r e d i c t i o n s f o r s t r a i n r a t i o s a t a r e l a t i v e l y h i g h r a t e of l o a d i n g of 1 MN*mm2 s-l, a l s o shown i n Fig. 3, a r e 0.363, 0.344, 0.337 and 0.328 a t -5, -20, -30 and -40°C, r e s p e c t i v e l y . Agreement of t h e s e v a l u e s w i t h t h o s e g i v e n above i s r a t h e r remarkable. It may be concluded, t h e r e f o r e , t h a t Eq. 15 g i v e s t h e temperature dependence of t h e s t r a i n r a t i o a t a r a t e of l o a d i n g of about 1 M N o ~ - ~ s-I f o r i s o t r o p i c i c e w i t h g r a i n s i z e of 1 mm; hence i t i s of very l i m i t e d use. The a u t h o r h a s n o t , however, s e e n Lin'Kov's ( 4 ) o r i g i n a l p a p e r and d o e s n o t know e x a c t l y what t y p e of i c e was used d u r i n g t h a t i n v e s t i g a t i o n . It should be recognized t h a t t h e p r e d i c t e d v a l u e s (Fig. 3) f o r s t r a i n r a t i o a r e s i g n i f i c a n t l y h i g h e r t h a n t h e assumed P o i s s o n ' s r a t i o ( p e l of 0.3. What, t h e n , should be t h e r a t e of l o a d i n g s u i t a b l e f o r measuring t h e t r u e P o i s s o n ' s r a t i o of i c e ? C a l c u l a t e d s t r a i n r a t i o s o v e r a wide range of l o a d i n g r a t e s , g r a i n s i z e s and tempera- t u r e s a r e p r e s e n t e d i n F i g s . 4 and 5. Frequency, f , of l o a d i n g was e s t i m a t e d from

where t = all: i s t h e r i s e time. Maximum l o a d l e v e l of b.5 MN-m-2 i s chosen f o r convenience of p r e s e n t a t i o n and t o a v o i d microcracking. S t r e s s e s lower t h a n 0.5 t h a t miaht be encountered i n

-

s e i s m i c experiment would n o t n o t i c e a b l y a f f e c t t h e c a l c u l a t e d v a l u e s o v e r t h e e n t i r e range shown. ISOTROPIC ICE

o.,

t

F i g u r e 4. Frequency, s t r e s s r a t e and g r a i n - s i z e dependence of s t r a i n r a t i o i n i s o t r o p i c i c e ISOTROPIC ICE U = 0.50 MPo 1 r =d./2u 1 1 GRAIN SIZE, d = 1 mm a 0.35

-

tx C

-

A few o b s e r v a t i o n s can be made from Figs. 4 and 5. P o i s s o n ' s r a t i o of i c e may n o t be determined p r o p e r l y f o r f i n e - g r a i n e d i c e u n l e s s t h e frequency of l o a d i n g i s i n t h e MHz range. The behaviour of coarse-grained i c e , of c o u r s e , could be e v a l u a t e d a t much lower

f r e q u e n c i e s . F i e l d s t u d i e s c a r r i e d o u t i n w i n t e r o r i n summer could show d i f f e r e n c e s , even i n s e i s m i c methods, i f low f r e q u e n c i e s and f i n e - g r a i n e d i c e were t o be involved i n t h e measurements. T h i s could e x p l a i n why Langleben and Pounder ( 6 ) , who u s e d f r e q u e n c i e s i n t h e range of 8-80 kHz, d i d n o t d e t e c t any e f f e c t of s a l i n i t y , temperature, o r i c e t y p e and recommended a low v a l u e of 0.295, whereas Lin'Kov ( 4 ) o b t a i n e d

F i g u r e 5. Frequency, s t r e s s r a t e and temperature dependence of s t r a i n r a t i o i n i s o t r o p i c i c e

s i g n i f i c a n t l y h i g h e r v a l u e s and n o t i c e a b l e t e m p e r a t u r e dependence.

CONCLUSION

The r h e o l o g i c a l model proposed by Sinha ( 2 ) , i n c o n j u n c t i o n w i t h t h e assumption t h a t d e l a y e d - e l a s t i c and v i s c o u s deformation i n v o l v e s no v o l u m e t r i c change, h a s been used t o show t h a t s t r a i n r a t i o depends n o t only on i c e t y p e b u t a l s o on l o a d i n g h i s t o r y .

Homogeneous and i s o t r o p i c m a t e r i a l s have been d i s c u s s e d f o r l o a d i n g c o n d i t i o n s i n v o l v i n g no c r a c k i n g . It i s f a i r l y c l e a r t h a t s t r a i n r a t i o d e c r e a s e s w i t h i n c r e a s e i n g r a i n s i z e and r a t e of l o a d i n g and w i t h d e c r e a s e i n temperature. The r a t e and t e m p e r a t u r e e f f e c t i s more

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noticeable in fine-grained ice than in coarse-grained ice, complex behaviour due primarily to the

delayed-elastic effect. Mechanisms leading to viscous or plastic deformation play practically no role for loading conditions relevant to most engineering problems. Determination of Poisson's ratio or the strain ratio due to pure elastic effect requires loading rates in the range of a few MHz for ice with a grain size of 0.5 mm. A frequency range of a few kHz is sufficient, however, for coarse-grained ice of 5 mm. ACKNOWLEDGEMENT

The author is indebted to R. Jerome and R. Stahl for their assistance in the preparation of this paper, which is a contribution from the Institute for Research in Construction, National Research Council of Canada. REFERENCES

1. Sinha, N.K., 'Rheology of Columnar-grained Ice, "

Experimental Mechanics, Vol. 18, No. 12, 1978,

pp. 464-470.

2. Sinha, N.K., 'Grain-boundary Sliding in

Polycrystalline Materials

,'-

~hilosophical Magazine A. Vol. 40. No. 6. 1979. DD. 825-842.

3. Gold, L.w., '~ome.0bser;ations on the Dependence of Strain on Stress for Ice,' Canadian Journal of Physics, Vol. 36, No. 10, 1958, pp. 1265-1276.

4. _{Lin'Kov, E.M.,} _{'Study of the Elastic Properties of}

an Ice Cover in the Arctic' (in Russian), Vestnik, Leningradskogo Univ. 13, 1958, pp. 17-22.

5. Peschansky, I.S., Problemy Arktiki, 2, 1957, p. 161.

6. Langleben, M.P. and Pounder, E.R., 'Elastic Parameters of Sea Ice,"

_Ice

_{edited by} W.D. Kingery, M.I.T. Press, Cambridge, Mass.,

1963, pp. 69-78.

Wang, Y.S.. 'Uniaxial Comvression Testing of Arc& Sea.Ice,' ~ r o c e e d i ~ g s of the 6th International Conference on Port and Ocean Engineering under Arctic ~onditione, Lava1 University, Quebec, Canada. Vol. 1. 1981. pp. 346-355.

Saeki, H., Ozaki, A. and Kubo, Y., 'Experimental Study on Flexural Strength and Elastic Modulus of Sea Ice,' Proceedings of the 6th International Murat, J.R., and Lainey, L.M., 'Some Experimental Observations on the Poisson's Ratio of Sea Ice,' Cold Regions Science and Technology, Vol. 6, 1982, pp. 105-113.

Sinha. N.K.. 'Young Arctic Frazil Sea Ice: Field and ~ i b o r a t b r ~ strength Tests,' Journal of Materials Science, Vol. 21, No. 5, 1986,

UP. 1533-1546.

- -

Sinha, N.K., 'Discussion: Uniaxial Compression Testing of Arctic Sea Ice,' by Y.S. Wang,

Proceedings of the 6th International Conference on

Port and Ocean Engineering under Arctic Conditions. Lava1 University, Quebec, Canada, Vol. 3, 1981, pp. 1465-1466.

Sinha, N.K., 'Creep Model of Ice for Monotonically Increasing Stress," Cold Regions Science and Tecbology Vol. 8, No. 1, 1983, pp. 25-33.

Flitcher,~.~.,

The Chemical Physics of Ice, Cambridge University Press, U.K., Chapter 8, 1970, pp. 165-173.

Weeks, W. and Assur, A., 'The Mechanical Properties of Sea Ice," U.S. Army, Cold Regions Research and Engineering Laboratory,

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reprinted from

Proceedings of the Sixth (1987) International Offshore Mechanics and Arctic Engineering Symposium

-

Volume I V

Editors: V. J. Lunardini, N. K. Sinha, Y. S. Wang, and R. D. Goff (Book No. 10217D)

published by

THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 East 47th Street, New York, N.Y. 10017