Experiments on anisotropic and rate sensitive strain ratio and modulus of columnar-grained ice

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Publisher’s version / Version de l'éditeur:

Journal of Offshore Mechanics and Arctic Engineering, 111, 4, pp. 354-360, 1989

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Experiments on anisotropic and rate sensitive strain ratio and modulus

of columnar-grained ice

Sinha, N. K.

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EXPERIMENTS ON ANISOTROPIC AND RATE SENSITIVE STRAIN RATIO AND MODULUS OF COLUMNAR-GRAINED ICE

N. K. Sinha

National Research Council Canada Ottawa, Ontario. Canada

ABSTRACT

A n i s o t r o p i c and r a t e s e n s i t i v e c h a r a c t e r i s t i c s o f t h e r a t i o o f 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 d d i t i o n t o t h e r a t e s e n s i t i v e e f f e c t i v e modulus f o r columnar-grained f r e s h w a t e r i c e and sea i c e from t h e A r c t i c , have been i n v e s t i g a t e d . Tests were c a r r i e d o u t a t -20°C, f o r c o n d i t i o n s o f no m i c r o c r a c k i n g under u n i a x i a l l o a d s (normal t o t h e l e n g t h o f t h e g r a i n s ) i n t h e s t r e s s r a t e r a n e o f 1 x M N ~ - ~ s - ~ t o 1 x

l o 2

M N . ~ - ~ s-P o r an e q u i v a l e n t s t r a i n r a t e range o f 1 x

lo-'

s - l t o 1 x s - l . With i n c r e a s e i n s t r e s s r a t e , t h e r a t i o i n c r e a s e d f r o m about 0.2 t o 0.3 i n t h e p l a n e p a r a l l e l t o t h e columns whereas i t decreased from about 0.65 t o 0.3 i n t h e p l a n e normal t o t h e c o l umns. A c l o s e d - l o o p c o n t r o l l e d t e s t system i n a s s o c i a t i o n w i t h f o i l gauges, mounted d i r e c t l y on t h e specimen, was used. A c l ip o n displacement gauge was used t o e v a l u a t e t h e accuracy o f t h e f o i l gauges.

INTRODUCTION

A measure o f t h r e e dimensional deformation i n a m a t e r i a l i s g i v e n by t h e P o i s s o n ' s r a t i o . T h i s i s d e f i n e d as t h e r a t i o o f t h e l a t e r a l s t r a i n t o t h e l o n g i t u d i n a l s t r a i n i n a homogeneous m a t e r i a l f o r a u n i a x i a l e l a s t i c l o a d i n g c o n d i t i o n . It i s a m a n i f e s t a t i o n o f t h e f a c t o r s t h a t p l a y a c e n t r a l r o l e i n t h r e e dimensional c o n s t i t u t i v e f o r m u l a t i o n s f o r m a t e r i a l s i n c l u d i n g i c e . P o i s s o n ' s r a t i o i n t h e range 0.31 t o 0.37 f o r f r e s h w a t e r i c e was t a b u l a t e d some y e a r s ago by Gold ( 1 ) . L i n ' K o v ' s ( 2 ) i n - s i t u s e i s m i c d e t e r m i n a t i o n s f o r i c e ranged f r o m 0.36 t o 0.39. Both Peschansky ( 3 ) and Langleben and Pounder ( 4 ) r e p o r t e d dynamic values o f 0.29 f o r sea i c e . There i s l i t t l e d i s c r e p a n c y i n t h e d y n a m i c a l l y o b t a i n e d data, i r r e s p e c t i v e o f t h e i c e t y p e . 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 i n t h e r e s u l t s r e p o r t e d by d i f f e r e n t i n v e s t i g a t e r s . For t r a n s v e r s e l y i s o t r o p i c , c o l umnar-grained, f r e s h w a t e r i c e 1 oaded p e r p e n d i c u l a r t o t h e c o l umns, G o l d ' s s t a t i c experiments i n d i c a t e d values i n t h e Presented at the Seventh International Conference on

Offshore Mechanics and Arctic Engineering Houston, Texas - February 7-12, 1988

range o f 0.31 t o 0.54 f o r t h e r a t i o o f t r a n s v e r s e t o l o n g i t u d l n a l s t r a l n s (1). Corresponding t o t h i s ,

Wang ( 5 ) r e p o r t e d a range o f 0.8 t o 1.2 f o r sea i c e ,

however, he o b t a i n e d values i n t h e range of 5 t o 0.2 when t h e l o a d was a p p l i e d f 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 o f t h e columns. Saeki e t a1 (6) a l s o s t u d i e d d e f o r m a t i o n a l o n g t h e l e n g t h o f t h e columns i n sea i c e b u t r e p o r t e d a r a t h e r 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 f r o m 0.01 t o 0.5 M N * ~ - ~ s - l . I n t h e same s t r e s s r a t e range and f o r t h e d i r e c t i o n normal t o t h e columns a t -5OC, Murat and L a i n e y ( 7 ) r e p o r t e d t h e r a t i o d e c r e a s i n g from 0.48 t o 0.38. Moreover Saeki e t a1 ( 6 ) r e p o r t e d t h a t t h e r a t i o i n c r e a s e d w i t h decrease i n t e m p e r a t u r e whereas an o p p o s i t e temperature e f f e c t was observed by Murat and L a i n e y ( 7 ) . While examining t h e r a t e

s e n s i t i v i t y o f t h e compressive s t r e n g t h o f congealed f r a z i l sea i c e , Sinha ( 8 ) observed t h a t t h e s t r a i n r a t i o depends n o t o n l y on r a t e o f l o a d i n g b u t 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 s t a t e o f damage ( m i c r o c r a c k i n g ) o f t h e m a t e r i a l .

The v a r i a b i l i t y observed i n t h e s t a t i c a l l y determined values o f t h e r a t i o between t h e l a t e r a l s t r a i n and t h e a x i a l s t r a i n i n d i c a t e s t h e complex mechanical response o f i c e . The c o m p l e x i t i e s a r i s e p r i m a r i l y because working temperatures w i t h i c e a r e v e r y c l o s e t o i t s m e l t i n g p o i n t , T

,

and a r e u s u a l l y g r e a t e r t h a n 0.9 Tm where The t e m p e r a t u r e i s g i v e n i n K e l v i n scale. At h i g h temperatures,

>

0.4 T

,

p o l y c r y s t a l l i n e d e f o r m a t i o n i s g r e a t l y affecTed by i n t e r g r a n u l a r s l i d i n g mechanisms and g r a i n boundary e m b r i t t l e m e n t processes i n a d d i t i o n t o t h e i n t e r g r a n u l a r d e f o r m a t i o n due t o t h e e l a s t i c d i s t o r t i o n o f t h e l a t t i c e and m o b i l i t y o f l a t t i c e d i s l o c a t i o n s . C o n s t i t u t i v e e q u a t i o n s f o r i c e t h e r f o r e must g i v e c o n s i d e r a t i o n t o t h e s e

mechanisms. A micromechani c a l l y based r h e o l o g i c a l model was proposed (9) t o show t h a t t h e 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 . T h i s model a l s o p o i n t e d o u t t h e d i r e need f o r r e l i a b l e e x p e r i m e n t a l d a t a on t h r e e dimensional d e f o r m a t i o n o f i c e w i t h known f a b r i c , t e x t u r e , g r a i n s i z e and damage s t a t e , and o b t a i n e d

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under w e l l d e s c r i b e d experimental c o n d i t i o n s such as temperature, s t r e s s and s t r e s s r a t e , s t r a i n and s t r a i n h i s t o r y . The p r e s e n t paper d e s c r i b e s an e f f o r t towards t h i s goal and p r e s e n t s some r e s u l t s on s t r a i n r a t i o and e f f e c t i v e modulus o f

columnar-grained f r e s h - w a t e r and sea i c e over a wide ( o v e r f i v e o r d e r s o f magnitude) range of l o a d i n g r a t e s f o r c o n d i t i o n s w i t h no d e t e c t a b l e m i c r o c r a c k i n g a c t i v i t i e s . The maximum s t r e s s wat k e p t below t h a t r e q u i r e d t o form m i c r o c r a c k s .

ICE CHARACTERISTICS AND SPECIMEN PREPARATION I n v e s t i g a t i o n s were p r i m a r i l y c a r r i e d out on 1 a b o r a t o r y made t r a n s v e r s e 1 y i s o t r o p i c

c o l umnar-grained f r e s h w a t e r , S-2 t y p e i c e , w i t h l o a d a p p l i e d i n t h e p l a n e o f i s o t r o p y , t h a t i s , a t r i g h t angles t o t h e l o n g a x i s o f t h e columns. The i c e was t r a n s p a r e n t w i t h no a i r bubbles and had a d e n s i t y o f 917.8 ~ g - m - ~ , a t - l O ° C , which was t h e same as t h a t o f s i n g l e c r y s t a l d e n s i t y . The c r o s s s e c t i o n a l g r a i n s i z e s v a r i e d i n t h e range o f 2 t o 3 mm. G r a i n s t r u c t u r e i n t h e mid-plane o f one o f t h e specimen (L83) t e s t e d i s shown i n F i g u r e 1. Note t h e

u n i f o r m i t y i n t h e c r o s s s e c t i o n a l g r a i n s i z e s t h r o u g h t h e e n t i r e 250 mn l o n g specimen. D e t a i l s o f g r a i n s a r e g i v e n i n F i g u r e 2. I n t h i s t h i n s e c t i o n , 1498 g r a i n s were counted i n an area o f 90 mm x 100 mm, g i v i n g an average g r a i n area, a, o f 6.01 mm2 which

g i v e s an e q u i v a l e n t g r a i n diameter, d ( d = 2

G),

of 2.77 mm. The c o r r e s p o n d i n g average g r a i n s i z e

o b t a i n e d from t h e i n t e r c e p t method, u s i n g t e n random1 y s e l e c t e d 1 i nes over t h e same o b s e r v a t i o n area, was 2.33 mm. This i s 16% lower than t h a t g i v e n by t h e f i r s t method. I f a l l t h e g r a i n s a r e assumed t o be square i n cross s e c t i o n , t h e n 6.01 mn area i s e q u i v a l e n t t o a square w i t h s i d e s o f 2.4 mm which agrees w e l l w i t h t h e s i z e g i v e n by t h e i n t e r c e p t method.

Experiments were performed i n s i d e a c o l d room k e p t a t -20°C

+

O.l°C on specimens w i t h f i n a l

dimensions o f 50 mm x 100 mm x 250 mm, h a v i n g t h e 100 mm x 250

mn

faces a t r i g h t angles t o t h e l e n g t h o f t h e c o l umnar-grai ns .' They were machined w i t h a m i l l i n g machine and a l a t h e i n s i d e an a d j a c e n t c o l d

room a t -15OC. C u t t e r marks on t h e s u r f a c e s were removed w i t h f i n e sandpaper on a f l a t g r a n i t e b l o c k . F i n a l f i n i s h was g i v e n by w i p i n g w i t h a t i s s u e s l i g h t l y moistened w i t h a1 coho1

.

The samples were t h e n s t o r e d at-20°C w i t h i n t h e experimental c o l d room

i n i n d i v i d u a l bags u n t i l t h e y were t e s t e d w i t h i n a few days. It should be p o i n t e d o u t h e r e t h a t f o r t h e chosen o r i e n t a t i o n o f t h e g r a i n s , r e c t a n g u l a r

geometry o f t h e specimen i s more c o m p a t i b l e t h a n t h e c y l i n d r i c a l specimen. Though obvious, t h i s f a c t i s over1 ooked by most i n v e s t i g a t o r s .

A few t e s t s were a l s o c a r r i e d out a t -20°C on a specimen o f f i r s t - y e a r columnar-grained sea i c e w i t h p r e f e r r e d c - a x i s o r i e n t a t i o n i n t h e h o r i z o n t a l p l a n e (S-3 t y p e ) . The specimen had t h e f i n a l dimensions o f 100 mm x 100 mm x 250 mn w i t h t h e

l o n g dimension para1 1 e l t o t h e p r e f e r r e d c - a x i s o r i e n t a t i o n i n t h e p l a n e normal t o t h e columns. T h i s sample was made from an i c e b l o c k sampled i n May 1986 from A d m i r a l t y I n l e t near B a f f i n I s l a n d . The s a l i n i t y ( ~ O / O O ) and m i c r o s t r u c t u r e o f t h i s i c e was s i m i l a r t o t h a t r e p o r t e d i n Nakawo and Sinha (12) f o r sea i c e from E c l i p s e Sound, B a f f i n I s l a n d .

FIGURE 1 . H o r i z o n t a l ( a ) and v e r t i c a l ( b ) t h i n s e c t i o n s o f t e s t specimen L83, made a f t e r c o m p l e t i n g t h e t e s t s e r i ~ q

FIGURE 2. Mid-plane t h i n s e c t i o n o f t e s t specimen L83 e x h i b i t i n g no grain-boundary

d i s t o r t i o n s , small a n g l e boundaries and c r a c k s .

TEST METHODS

A commercial c l osed-1 oop, servohydraul i c t e s t machine, w i t h a d e s i g n l o a d c a p a c i t y o f 1 MN f o r t h e frame (MTS) and s i t u a t e d i n s i d e t h e c o l d room near an o b s e r v a t i o n window, was used. The c o n t r o l s , pumps, and r e c o r d i n g systems were k e p t o u t s i d e t h e c o l d room. The l o a d i n g t r a i n c o n s i s t e d o f t w o 152 mn diameter compression p l a t e n s (one a t t h e t o p and one a t t h e bottom o f t h e specimen), a s p h e r i c a l seat, a 250 kN c a p a c i t y l o a d c e l l and a 250 kN c a p a c i t y a c t u a t o r . The lower p l a t e n was a t t a c h e d t o t h e a c t u a t o r and t h e upper p l a t e n r e a c t e d a g a f n s t t h e l o a d frame t h r o u g h t h e

s p h e r i c a l seat and t h e l o a d c e l l . The lower p l a t e n

was s p e c i a l l y designed w i t h a l i q u i d c i r c u l a t i o n

system t o m a i n t a i n i t a t t h e c o l d roan a i r

t e m p e r a t u r e and thereby p r e v e n t jt from any wanning e f f e c t f r a n t h e o i l i n t h e a c t u a t o r . C m p r e s s i v e

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l o a d s were a p p l i e d p a r a l l e l t o t h e l o n g a x i s o f t h e specimens and hence normal t o t h e l e n g t h o f t h e

columnar g r a i n (See F i g u r e 3 ) . A x i a l l o a d r a t e and

hence s t r e s s r a t e was m a i n t a i n e d c o n s t a n t d u r i n g a

t e s t by t h e feedback system. Tests were conducted

over a wide range o f s t r e s s r a t e s f r o m 1

l o m 3

M N * ~ - ~ s - I t o 1.25 x

l o 2

M N * ~ - ~ S-l.

E a r l i e r experiments ( l o ) , c a r r i e d out on c o l umnar-grai ned i c e , showed t h a t m i c r o c r a c k s , about t h e s i z e o f t h e g r a i n f a c e t s , form when t h e

s t r e s s exceeds about 1.2 M N * ~ - ~ f o r s t r e s s r a t e s

g r e a t e r t h a n 1 x M N - ~ - ~ s - l . It was decided

t o keep t h e peak s t r e s s , a w e l l below c r a c k i n g

l e v e l t o a v o i d any e f f e c t s l i u e t o t h e cracks. The

system was programmed t o a p p l y t h e l o a d up t o an e q u i v a l e n t s t r e s s l e v e l o f about 0.75 ~ N - r n - ~ and

t h e n t o unload r a p i d l y t o zero s t r e s s . A l o n g

i n c u b a t i o n t i m e was g i v e n a f t e r each t e s t s d u r i n g which t h e specimen s t r a i n s were m o n i t o r e d i n o r d e r t o be c e r t a i n t h a t t h e specimen was a t an

e q u i l i b r i u m s t a t e b e f o r e l o a d i n g again. S t r e s s r a t e s were chosen randomly t o a v o i d any e f f e c t due t o m i c r o s t r u c t u r a l change t h a t c o u l d occur d u r i n g t h e t e s t s and s y s t e m a t i c a l l y i n f l u e n c e t h e r e s u l t s .

To a v o i d specimen t o specimen v a r i a t i o n , i t was

planned t o do as many t e s t s as p o s s i b l e on t h e same

specimen. However, an a c c i d e n t occured d u r i n g

h a n d l i n g and t h e f i r s t t e s t specimen ( d e s i g n a t e d as L81), a f t e r several t e s t s , broke i n t o small p i e c e s

when i t s l i p p e d and dropped on t h e f l o o r . A second

specimen ( ~ 8 3 ) was then used. Tests were t e r m i n a t e d

on t h i s specimen a f t e r s e v e r a l days o f t e s t i n g vlhen two f o i l gauges came o f f t h e specimen d u r i n g s t o r a g e

o v e r a weekend. T h i s specimen showed no c r a c k s

a f t e r a1 1 t h e t e s t s . M i c r o s t r u c t u r a l examination o f

t h i s specimen was t h e n made and a r e shown i n F i g u r e s

1 and 2. Tests on t h e sea i c e specimen (L82) were

t e r m i n a t e d when i t was damaged d u r i n g handl i n g . For t h e chosen maximum s t r e s s l e v e l o f 0.75

M N * ~ - ~ , t h e t o t a l l o a d i n g t i m e i s 0.0075 seconds a t

t h e i n t e n d e d maximum s t r e s s r a t e o f 1 x

l o 2

M N . ~ - ~

s . I n o r d e r t o r e c o r d a reasonably c o n t i n u o u s

h i s t o r y of l o a d i n g , t h e r e should a t l e a s t be 100

d a t a p o i n t s d u r i n g t h e l o a d i n g phase. This means

t h a t d a t a should be c o l l e c t e d a t a r a t e o f 1.3 x

l o 4

s - l

.

A v a i l a b i l i t y o f a s i x channel t r a n s i e n t r e c o r d e r , capable of r e c o r d i n g a1 1 t h e channels s i m u l t a n e o u s l y up t o 2 x

l o 6

s - l proved i d e a l . I t was i m p e r a t i v e t h a t t h e a x i a l l o a d and t h e deformation of e s s e n t i a l l y t h e e n t i r e specimen, g i v e n by t h e 200 mm l o n g MTS displacement gauge ( F i g u r e 3) mounted d i r e c t l y on t h e specimen, be

recorded. This procedure a1 1 owed on1 y f o u r

channels t o be used f o r o t h e r measurements. A l l

t h e e l e c t r o n i c equipment was kept o u t s i d e t h e c o l d room.

Four commercially a v a i l a b l e constantan a l l o y g r i d f o i l s t r a i n gauges, w i t h a c t i v e gauge l e n g t h of 12 mm, were mounted on t h e specimen a t t h e t e s t temperature o f -20°C as shown i n F i g u r e 3 f o l l o w i n g

t h e method d e s c r i b e d l a t e r . Longer gauges, though

p r e f e r a b l e , were not used because o f d i f f i c u l t i e s o f

rnounti ng them u n i form1 y

.

These gauges p r o v i d e d

independent measurements o f b o t h a x i a l and l a t e r a l s t r a i n s on two surfaces, one p a r a l l e l t o t h e l e n g t h o f t h e columnar g r a i n s and t h e o t h e r normal t o i t . The gauges were e x c i t e d by a commercial s i g n a l c o n d i t i o n e r and t h e gauge o u t p u t s were a m p l i f i e d b e f o r e r e c o r d i n g . An e x c i t a t i o n v o l t a g e o f 7 v o l t s , f o r t h e 350-ohm s t r a i n gauges, was found s a t i s f a c t o r y w i t h o u t showing any adverse e f f e c t due t o h e a t i n g .

Experiments were conducted w i t h two t y p e s o f

f o i l gauges. No d i f f e r e n c e s were noted i n t h e i r

1 . 2 , 3 . 4 - FOIL GAUGES

5 - DISPLACEMENT GAUGE

O

- ANCHOR I C E

FIGURE 3

EXPERIMENTAL ARRANGEMENT FOR THE GAUGES ON A SPECIMEh

response. Both were i n t e g r a l l e a d t y p e w i t h

a constantan a l l o y g r i d and a c t i v e gauge l e n g t h o f

12 mm. The gauge l e n g t h i s l a r g e enough t o cover

s e v e r a l g r a i n s , y e t small enough t o f r e e z e

u n i f o r m l y on t h e specimen surface. Two t y p e s o f

" o f f t h e s h e l f " M i c r o Measurement I n c . gauges were used: (model number CEA-06-500UW-350) w h i c h were c o m p l e t e l y encapsulated i n a f l e x i b l e p o l y i m i d e

c o a t i n g , and (model number EA-06-500-BL350W), which

were open faced w i t h o n l y a polymide backing. I n

b o t h cases t h e l e a d w i r e s were cleaned and

c a r e f u l l y s o l d e r e d o n t o t h e t a b s . The a c t i v e gauge p o r t i o n o f t h e open faced gauges was t h e n

waterproofed by c o a t i n g f i r s t w i t h a t h i n l a y e r o f Dow Corning 3140 RTV f o l l o w e d by a t h i n l a y e r o f M i c r o Measurements n i t r i l e rubber c o a t i n g , model M. The open ends o f t h e l e a d w i r e s and t h e t a b s were a l s o w a t e r p r o o f e d f o r b o t h t y p e s of gauges by t h e

two l a y e r technique. The encapsulated gauges were

p r e f e r r e d and used i n t h e main p a r t of t h i s study. Several methods, i n c l u d i n g t h a t d e s c r i b e d by Murat ( l l ) , were t r i e d i n s i d e t h e experimental c o l d room f o r mounting t h e s t r a i n gauges on t h e specimen

surfaces. The s i m p l e s t and y e t s a t i s f a c t o r y method

may be d e s c r i b e d as f o l l o w s . A f t e r s e l e c t i n g t h e d e s i r e d l o c a t i o n and p o s i t i o n i n g t h e gauges, about 1 0 mm o f t h e l e a d wires, a t a d i s t a n c e o f about 50 mm from t h e a c t i v e p o r t i o n , was f r o z e n t o t h e specimen u s i n g an eye dropper and two o r t h r e e drops o f water

from a c o n t a i n e r m a i n t a i n e d a t about 0°C. These

d r o p s a c t e d as t h e anchors and i s o l a t e d t h e gauges and t h e j o i n t s from any p u l l i n g a c t i o n o f t h e l e a d

w i r e s d u r i n g f u r t h e r handl i ng

.

Moreover, t h e y a1 so

p r o t e c t e d t h e gauges from any v i b r a t i o n a l s t r a i n induced by t h e l e a d w i r e s t h a t were o f t e n seen t o v i b r a t e d u r i n g t h e experiment when t h e l o a d was a p p l i e d suddenly f o r t h e r a p i d t e s t s o r due t o t h e a i r c i r c u l a t i o n i n s i d e t h e c o l d r o a n d u r i n g t h e s l o w

(7)

anchor, t h e gauge was l i f t e d s l i g h t l y , and two d r o p s o f c o l d water were a p p l i e d t o t h e area where t h e

gauge was t o be mounted. The gauge was then q u i c k l y

and a c c u r a t e l y pressed i n t o place, u s i n g a f l a t rubber pad, and h e l d u n t i l f r e e z i n g o f t h e water was

complete. The pad was t h e n removed. A f t e r

i n s p e c t i o n , t o ensure o v e r a l l f r e e z i n g and f l a t n e s s , a t h i n l a y e r o f water was placed over t h e gauge and a small p o r t i o n o f t h e l e a d w i r e s , f o r p r o t e c t i o n a g a i n s t s u b l i m a t i o n . Needless t o say, a g r e a t deal o f p r a c t i c e was r e q u i r e d b e f o r e t h e gauges c o u l d be

p r o p e r l y mounted. The common problem was t h e p a r t i a l f r e e z i n g and w r i n k l i n g o f t h e gauges. It should be mentioned here t h a t t h e use o f water a t about

f r e e z i n g p o i n t was found necessary t o a v o i d m i c r o c r a c k s on t h e specimen s u r f a c e due t o thermal

s t r e s s . Examination under a microscope r e v e a l e d

t h a t even one drop o f water a t 20°C ( u s u a l warm room t e m p e r a t u r e ) damages t h e i c e s u r f a c e a t -20°C.

RESULTS AND ANALYSIS

Two q u e s t i o n s were examined d u r i n g t h e t e s t s ( a ) how w e l l t h e machine performed and whether s t r e s s r a t e s were m a i n t a i n e d c o n s t a n t and ( b ) how we1 1 t h e d e f o r m a t i o n s were measured.

For a x i a l s t r e s s r a t e s f r o m 1 x M N * ~ - ~ s - l t o about 2 x

l o 1

M N * ~ - ~ s - l , t h e system was found capable o f 1 oading t h e specimen a t t h e i n t e n d e d c o n s t a n t s t r e s s r a t e s and o f u n l o a d i n g i t

a

6 N E

2

4 2 b- 2 0 10 LO 8 6 w- 4 2 0 "7 4

0

, 3 w 2 $ 1 0 0 4 8 12 16 20 24 T I M E , t , s F I G U R E 4 S T R E S S A N D S T R A I N H I S T O R I E S F O R A S L O W T E S T r a p i d l y a f t e r r e a c h i n g t h e peak l o a d bl as may be seen i n F i g u r e s (4) and ( 5 ) . Although !he s t r e s s r a t e was c o n s t a n t d u r i n g t h e major p a r t o f t h e l o a d i n g c y c l e , t h e l a c k o f constancy i n t h e l o a d r a t e d u r i n g t h e b e g i n n i n g and t h e end o f t h e c y c l e , was e v i d e n t a t r a t e s ,

b1

,

.20 M N * ~ - ~ s - l ( F i g u r e 6 ) . These were r a t h e r r a p i d t e s t s and i n v o l v e d l o a d r i s e t i m e , t

,

l e s s t h a n 40 m i l l i s e c o n d s o r frequency, f, ( = T/2 t r ) . g r e a t e r t h a n about 10 Hz. An average s t r e s s r a t e , a

-

a /t was t h o u g h t t o be a p p r o p r i a t e f o r d e s c r i b i n g tRe yoading r a t e s a t t h e s e h i g h r a t e s . F i g u r e 4 shows t h a t t h e a x i a l s t r a i n s measured by t h e f o i l gauges, 1 and 3, agree w e l l w i t h t h a t g i v e n by t h e displacement gauge 5. C o n s i d e r a t i o n i s g i v e n t o t h e f a c t t h a t t h e s t r a i n l e v e l s were e x t r e m e l y small and t h a t even a s l i g h t r o u n d i n g of t h e edges o f t h e 1 oad b e a r i n g end s u r f a c e s (50 mm x

100 mm) would s i g n i f i c a n t l y a f f e c t t h e s u r f a c e s t r a i n o f t h e s i d e s u r f a c e s . It should be n o t e d t h a t b o t h a x i a l and l a t e r a l s t r a i n s i n c r e a s e d l i n e a r l y w i t h t i m e g i v i n g i n d i c a t i o n s o f a l i n e a r

dependence o f s t r a i n on s t r e s s . Time dependence of

t h e s t r a i n r e c o v e r y and t o t a l r e c o v e r y a f t e r u n l o a d i n g were i n d i c a t e d by b o t h t h e f o i l gauges

and t h e displacement gauge ; permanent s t r a i n s , i f

any, were t h e r e f o r e beyond t h e accuracy o f t h e measurement. T h i s t y p e o f t o t a l r e c o v e r y was 0 . 8 I I I T ₁ ₁ ₁ ₁

-

0 . 6

-

T E S T : L 8 3 . 1 7

-

I - 8

-

1

-

Y) I

-

I

-

_I LO

'

4 - I 2 I I 10

-

I w I I- 2

-

N w I

-

d. 0 I 1 I 0 0 . 1 0.2 0.3 0.4 0.5 T I M E , t , s F I G U R E 5 S T R E S S A N D S T R A I N H I S T O R I E S F O R A M E D I U M R A T E T E S T observed i n a l l t h e l o a d i n g c y c l e s . T o t a l accumulated permanent s t r a i n i n a specimen a t t h e end of a l l t h e t e s t s was l e s s t h a n about 3 x 10-5

and t h e r e f o r e n e g l i g i b l e . Evidence o f 1 i t t l e permanent change i n t h e specimen L83, a f t e r 23 t e s t s , i s p r o v i d e d i n F i g u r e 2. Note t h a t t h e 58

(8)

, 4 0

-

I I I ,"

_-

0

_-

2 0

-

A t , = t, if0111 - t, ( d ~ s p l + . J C A * A A - a TEST : L83.20 - 2 0 ' ~ . 5 - 2 ICE 1.0 + 0. f l ' - a "? 2 0 6 - - w

.

-

- 0.4 , -A G D 2 - 0. 0 . i - - c - - o - r - u 0.-t

,-.

.

-

m

--

-

-.-

-

₆_-_._- _,

-

.

..

A. .*-A.,. A . . ; A

-

\A

*',

-

- 2 0 ° c

-

A - L E I A

-

8 -L82 \ -L83

f

-

t

I I I I 10-3 10-2 1 0 - 1 1 0 0 1 0 1 1 0 2 1 0 3 S T R E S S R A T E . 6,

.

~ ~ - r n - ~ r - l FIGURE 7

STRESS RATE DEPENDENCE OF THE RESPONSE OF THE FOIL GAUGES

a p p l i e d a l o n g x

,

t h e l a t e r a l compliances o f concern a r e

land

S31y and t h e major s t r a i n r a t i o s a r e g i v e n by P21 =

-

S 2 i

/

Sll =

-

E2/e1 and ( 2 ) 0 1 2 3 4 5 6 p 3 1 =

-

S31

/

SI1 =

-

c3/E1 TIME. 1 .

s

FIGURE 6

STRESS AND STRAIN HISTORIES FOR A R A P I D T E S T The h i s t o r i e s o f E2 and E3 f o r t h e t e s t c o r r e s p o n d i n g t o F i g u r e 4 a r e presented i n F i g u r e 8 which shows t h a t t h e l a t e r a l s t r a i n s i n t h e p l a n e g r a i n boundaries have no i r r e g u l a r i t i e s and no o f i s o t r o p y and normal t o i t d i f f e r e d s i g n i f i c a n t l y small a n g l e boundaries can be seen i n any g r a i n s . f r a n each o t h e r b u t b o t h m a i n t a i n e d

a

l i n e a r

A new phenomenon i n t h e response o f t h e f o i l r e l a t i o n s h i p w i t h t h e a x i a l s t r a i n . S i m i l a r gauges was observed as t h e r a t e o f l o a d i n g o b s e r v a t i o n s were made i n a l l t h e t e s t s i n c l u d i n g i n c r e a s e d . The gauges recorded n o t o n l y l o w e r t h o s e on S-3 sea i c e . Thus t h e s t r a i n r a t i o s do a x i a l s t r a i n s , w i t h i n c r e a s e i n s t r e s s r a t e , t h a n

t h a t g i v e n by t h e displacement gauge b u t a l s o

showed a s i g n i f i c a n t l y damped response ( F i g u r e s 5 5 , , , 1 1 , 1 , 1 , , 1

and 6 ) . The o u t p u t from t h e displacement gauge was ? o TEST : L83.7

-

n o t e d t o be i n phase w i t h t h e s t r e s s a t a l l s t r e s s 4

-

-20°c, s - 2 IC[,~-,

r a t e s b u t t h e c o r r e s p o n d i n g peak response i n t h e 10

-

4 - &,=0.05 MN. m

-

f o i l gauges o c c u r r e d a f t e r a s i g n i f i c a n t l a p s e of w HORIZONTAL

t i m e . R e s u l t s summarized i n F i g u r e 7 f o r t h r e e

-

s e t s o f measurements show t h a t (a) phase l a g i n t h e

-

f o i l gauges occurs a t about

b

>

3 M N * ~ - * s - I and

( b ) f o i l gauges c o u l d g i v e a c c e p t a b l e s t r a i n d a t a

-

f o r

b

<

0.1 M N * ~ - ~ s - l . C

'/l 2 -

-

The general i z e d Hooke1 s l a w r e l a t i n g s t r a i n ,

E ~ , t o s t r e s s u j i s g i v e n by

-

( 1 lJ J y i , j = 1 , 2 . . . 6

€ i = S . . u . '

-

where Sij denotes t h e compliances. F o r 1 1 1 1 1 1 1 1 I I I I ,

columnar-grained i c e w i t h t h e assumption of xl and 0 2 4 6 8 10 12 14

x 2 axes i n t h e h o r i z o n t a l p l a n e o r t h e p l a n e o f t h e _{AXIAL STRAIN. 6,.} i c e cover ( a l s o t h e p l a n e o f i s o t r o p y f o r S-2 i c e )

and x along t h e l o n g i t u d i n a l a x i s and hence t h e

g r o w t 2 d i r e c t i o n o f t h e columnar g r a i n s (see Fig. FIGURE 8

3). t h e usual concern i s w i t h t h e compliances SYll DEPENDENCE OF LATERAL STRAIN ON AXIAL SZ2 and S ; th e p r i n c i p a l s t r e s s e s a r e u s u a l l y i n STRAIN RECORDED B Y THE FOIL GAUGES x1 and x233i r e c t i o n s f o r most engineering

s i t u a t i o n s . I n case o f u n i a x i a l s t r e s s , al,

(9)

n o t depend, a t l e a s t n o t s i g n i f i c a n t l y , on t h e s t r e s s o r s t r a i n l e v e l w i t h i n t h e e x p e r i m e n t a l ranges i n v e s t i g a t e d . Dependence o f p 2 ~ and p 3 on s t r e s s r a t e s , u s i n g t h e peak s t r a i n s recorded k y t h e two p a i r s o f f o i l gauges a r e p r e s e n t e d i n F i g u r e 9. A c o n t i n u i t y i n t h e 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 s i s e v i d e n t here. Note t h a t t h i s c o n t i n u i t y extends from l o w r a t e s a t which f o i l gauges a r e r e l i a b l e t o h i g h r a t e s where t h e i r response i s q u e s t i o n a b l e . The two r a t i o s approaching a common dynamic v a l u e a t h i g h r a t e s a r e a1 so p h y s i c a l l y understandable. T h i s o b s e r v a t i o n i n d i c a t e s t h a t t h e damping e f f e c t on t h e f o i l gauges a f f e c t e d t h e t o t a l o u t p u t w i t h o u t i n f l u e n c i n g t h e i r response r e l a t i v e t o each o t h e r . Since t h e a x i a l and l a t e r a l s t r a i n s on t h e two o r t h o g o n a l s u r f a c e s were measured s e p a r a t e l y u s i n g p a i r s o f f o i l gauges, t h e r e 1 i a b i l i t y i n t h e measurement o f s t r a i n r a t i o s w i t h t h e s e gauges c o u l d perhaps be extended s i g n i f i c a n t l y t o h i g h e r s t r e s s r a t e s t h a n t h e l i m i t i n g r a t e of about 0.1 MN* m-2 s - I discussed e a r l i e r . Independent t e s t s a r e r e q u i r e d t o v e r i f y t h i s p o i n t . F i g u r e s 4-6 show t h a t t h e s t r a i n r a t e s were m w e - o r l e s s c o n s t a n t d u r i n g l o a d i n g i n a l l t h e t e s t s . The t e s t r e s u l t s t h e r e f o r e c o u l d a l s o be p r e s e n t e d i n t e n s o f c o n s t a n t s t r a i n r a t e s such as shown i n F i g u r e 10 i n which t h e s t r a i n r a t e s r e p r e s e n t t h o s e measured by t h e 200 mm gauge l e n g t h displacement gauge (No. 5). A s y s t e m a t i c v a r i a t i o n i n p 2 and ~ 3 1 i s again noted.

{ t r e s s r a t e dependence o f t h e e f f e c t i v e modulus, E, = ol

/

clp, where

E i s measured

gy

t h e displacement gauge, i s

? % r e n t e d i n F i g u r e 11. A complementary i l l u s t r a t i o n would be F i g u r e 12 e x h i b i t i n g t h e s t r a i n r a t e dependence o f t h e e f f e c t i v e modulus.

DISCUSSION

Complete s t r a i n r e c o v e r y a f t e r u n l o a d i n g demonstrates c l e a r l y t h e e l a s t i c n a t u r e o f t h e d e f o r m a t i o n processes i n v o l v e d . The t i m e dependent r e c o v e r y i n d i c a t e s t h a t a t l e a s t two processes a r e i n v o l v e d f o r t h e l o a d i n g c o n d i t i o n s used. T h i s I t y p e o f response was termed as " e l a s t o - d e l a y e d

e l a s t i c " (ede) by t h i s a u t h o r ( 9 ) . The "regime" of "ede" response i s e s s e n t i a l l y l i m i t e d t o t r a n s i e n t stage o f p o l y c r y s t a l l i n e behaviour when t h e s t r a i n s a r e small and t h e d e f o r m a t i o n i s homogeneous. I n t h i s regime, grain-boundary shear i n f r e s h w a t e r i c e o r i n t e r p l a t e l e t s l i d i n g i n sea i c e , c o n t r i b u t e s s i g n i f i c a n t l y t o t h e 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 e i n t r a g r a n u l a r e l a s t i c s t r a i n . These processes i n f l u e n c e b o t h e f f e c t i v e e l a s t i c modulus and s t r a i n r a t i o u n l e s s v e r y h i g h frequency l o a d i n g s a r e i n v o l v e d . I n t r a g r a n u l a r e l a s t i c s t r a i n , due t o t h e d i s t o r t i o n o f t h e l a t t i c e , i n v o l v e s v o l u m e t r i c change and a measure o f t h i s change i s g i v e n by " P o i s s o n ' s " r a t i o . I n t e r g r a n u l a r grain-boundary s l i d i n g o r delayed e l a s t i c process, i n t h e absence o f any c r a c k i n g a c t i v i t y , would be i s o t r o p i c and would i n v o l v e no a p p r e c i a b l e v o l umet r i c change ( 9 ) , f o r a l l p r a c t i c a l purposes. Thus i n t h e ede regime, s t r a i n r a t i o depends s i g n i f i c a n t l y on t h e degree o f c o n t r i b u t i o n o f delayed e l a s t i c s t r a i n t o t h e t o t a l s t r a i n . A t h e o r y has been proposed f o r i s o t r o p i c p o l y c r y s t a l l i n e m a t e r i a l and discussed i n ( 9 ) . T h i s t h e o r y has been extended t o p r e d i c t t h e a n i s o t r o p i c behaviour o f c o l umnar-grained i c e b u t w i l l be presented elsewhere because o f t h e space l i m i t a t i o n s here. However, t h e s o l i d l i n e s i n P21 F31 A - A - L 8 1 > ~ - 2 I C E 0 - 0 - L83 - L 8 2 - 5 - 3 I C E

.

-.

1

0 . 4 THEORY 0 . 2 O 0.0 I O - ~ IO-~ l o 3 - 2 - 1 S T R E S S R A T E , 6,. M N . m . S FIGURE 9

STRESS RATE DEPENDENCE OF STRAIN RATIO

0.4

THEORY

A A A A A

S T R A I N R A T E . il , S - l

FIGURE 10

STRAIN RATE DEPENDENCE OF STRAIN RATIO F i g u r e s 9 and

10,

o b t a i n e d u s i n g t h i s t h e o r y f o r c o l umnar-grai ned i c e o f average c r o s s - s e c t i o n a l g r a i n diameter o f 2.5 mm, g r a i n l e n g t h t o g r a i n diameter r a t i o (aspect r a t i o ) o f 20 and m a t e r i a l c o n s t a n t s g i v e n i n (9, 1 3 ) g i v e i n d i c a t i o n s t h a t m a t e r i a l response, however complex, i s r e a s o n a b l y

p r e d i c t a b l e . The t h e o r y a l s o c l a r i f i e s some o f t h e v a r i a b i l i t i e s i n s t a t i c a l l y determined s t r a i n r a t i o s p r e s e n t e d e a r l i e r i n t h e i n t r o d u c t i o n o f t h i s paper. Wang ( 5 ) r e p o r t e d values o f 0.8 t o 1.2 f o r p 2 ~ and Saeki e t a l l s ( 6 ) o b s e r v a t i o n s on ~ 3 1 , in c r e a s i n g f r o m 0.02 t o 0.48 w i t h s t r e s s r a t e , were c e r t a i n l y due t o t h e e f f e c t o f m i c r o c r a c k i n g a c t i v i t i e s as t h e s t r e s s l e v e l s i n v o l v e d i n t h e s e experiments were h i g h (up t o 6 M N . ~ - * ) . D i r e c t e x p e r i m e n t a l o b s e r v a t i o n s on t h e dependence o f t h e e f f e c t i v e modulus on s t r a i n r a t e , f o r c o n d i t i o n s o f no c r a c k i n g , a r e a v a i l a b l e up t o a s t r a i n r a t e o f 4 x s-1 as can be seen i n T r a e t t e b e r g e t a l , 1975 (14). The p r e s e n t r e s u l t s

(10)

* . .

_-

-

THEORY, d=2.5rnrn - - 2 0 ° c . 5 - 2 I C E

-

a;

= 0. 75 MN. m-'

-

d = 2 . 8 rnm lo-3 lo-2 10-l l o 3 S T R E S S R A T E , M N . ~ - ' s - ' FIGURE 11

STRESS RATE DEPENDENCE OF EFFECTIVE MODULUS

-

- 2 0 ° c , S - 2 I C E 5 = 0 . 7 5 M N . r n - ' - 4 - L 8 1

-

* - L 8 3 I I L I I 10" 1 0 . ~ 10-I S T R A I N RATE.;,

.

s-' FIGURE 12

STRAIN RATE DEPENDENCE OF EFFECTIVE MODULUS ( F i g u r e s 11, 1 2 ) , t h e r e f o r e , extend t h e a v a i l a b i l i t y o f experimental d a t a t o a s i g n i f i c a n t l y h i g h e r l o a d i n g r a t e . The s o l i d l i n e s i n F i g u r e 11 show t h e r e s u l t s c a l c u l a t e d f o r e f f e c t i v e modulus, E = 01,

/

~ ~ ~ , u s l n g ) t h e t h e o r y and m a t e r i a l c o n s t a n f s des r i b e i n 13 f o r i c e

w i t h g r a i n diameter o f 2.5 mm. The same r e s u l t s

a r e presented i n F i g u r e 1 2 as a f u n c t i o n of t h e

c a l c u l a t e d average s t r a i n r a t e . Again, it can b e

seen t h a t t h e e x p e r i m e n t a l r e s u l t s a r e reasonably p r e d i c t a b l e . CONCLUSIONS For s t r e s s r a t e s i n t h e range o f 1 x

l o m 3

t o 1.2 x

l o 2

M N * ~ - ~ s - l o r s t r a i n r a t e s i n t h e range o f 1 x

lo-'

t o 1 x s - l , f o r s t r e s s l e v e l s up t o 0.75 M N * ~ - ~ o r s t r a i n l e v e l s up t o 1.5 x a t a temperature o f -20°C and f o r l o a d s a p p l i e d i n t h e p l a n e normal t o t h e l e n g t h o f t h e columns, t h e l a t e r a l s t r a i n s i n c o l umnar-grained i c e m a i n t a i n a 1 in e a r r e l a t i o n s h i p w i t h l o n g i t u d i n a l o r a x i a l s t r a i n s d u r i n g l o a d i n g i n t h e p l a n e p a r a l l e l t o t h e l e n g t h o f t h e columns as w e l l as t h e o t h e r normal

t o it. The l a t e r a l s t r a i n , normal t o t h e l e n g t h o f

t h e columnar g r a i n s , i s about 3 t i m e s l a r g e r t h a n t h a t a r a l l e l t o t h e columns a t a s t r e s s r a t e o f 1

x

lo-!

M N * ~ - ~ s-1 o r s t r a i n r a t e o f 1 x s - I .

The s t r a i n r a t i o , , p 2

,

i n t h e l o a d i n g plane, decreases monoton~ca!ly from about 0.7 a t a s t r a i n

r a t e o f 1 x s - l t o about 0.3 a t 1 x s - l .

However, t h e s t r a i n r a t i o , b 3 ~ , c o r r e s p o n d i n g t o t h e p l a n e p a r a l l e l t o t h e l e n g t h o f t h e columns, i n c r e a s e s from about 0.2 t o about 0.3 f o r t h e same

range o f l o a d i n g r a t e s . Both f r e s h w a t e r i c e and

sea i c e shows s i m i 1 a r a n i s o t r o p i c response.

Since t h e response o f t h e f o i l gauges i s poor a t t h e h i g h end o f t h e l o a d i n g r a t e s i n v e s t i g a t e d , t h e d a t a o b t a i n e d on t h e s t r a i n r a t i o s i n t h e dynamic l o a d i n g ranges, though a p p a r e n t l y

reasonable, a r e o f q u e s t i o n a b l e value. Independent

t e s t s a r e r e q u i r e d t o v e r i f y t h e s e measurements. The s t r e s s / s t r a i n r a t i o o r t h e e f f e c t i v e modulus i n c r e a s e s m o n o t o n i c a l l y from 4 G N * ~ - ~ a t a s t r a i n r a t e o f 1 x s - l t o about 9.5 G N * ~ - ~ a t 1 x s - l

.

These r e s u l t s f i l l t h e gap i n t h e d a t a a v a i l a b l e i n t h e l i t e r a t u r e , i n t h e l o a d frequency range o f 3 t o 100 Hz and a r e i n agreement w i t h t h e t h e o r e t i c a l p r e d i c t i o n s made e a r l i e r .

ACKNOWLEDGEMENTS

The a u t h o r i s i n d e b t e d t o R. Jerome and R. S t a h l f o r t h e i r a s s i s t a n c e w i t h o u t which t h i s study

would n o t have been p o s s i b l e . T h i s i s a

c o n t r i b u t i o n from t h e I n s t i t u t e f o r Research i n C o n s t r u c t i o n , N a t i o n a l Research Council o f Canada. REFERENCES

1. Gold, L.W., "Some Observations on t h e

Dependence o f S t r a i n on S t r e s s f o r Ice," Canadian Journal o f Physics, Vol. 36, No. 10, 1958, PP. 1265-1276.

2. LinlKov, E.M., "Study o f t h e E l a s t i c P r o p e r t i e s

o f an I c e Cover i n t h e A r c t i c " ( i n Russian), Vestni k, Leningradskogo Univ. 13, 1958, pp. 17-22.

3. Peschansky, I.S., Problemy A r k t i k i , 2, 1957,

p. 161.

4. Langleben, M.P. and Pounder, E.R., " E l a s t i c

Parameters o f Sea Ice," I c e and Snow, e d i t e d by

W.D. Kingery, M.1 .T. Press, Cambridge, Mass.,

1963, pp. 69-78.

5. Wanq. Y.S.. " U n i a x i a l Compression T e s t i n q o f

~ r c t i c Sea

Ice,"

~ r o c e e d i n ~ s o f t h e 6 t h

-

I n t e r n a t i o n a l conTerence on P o r t and Ocean

E n g i n e e r i n g under A r c t i c C o n d i t j o n s , Laval

U n i v e r s i t y , Quebec, Canada, Yo1

.

1, 1981,

pp. 346-355.

6. Saeki, H., Ozaki, A. and Kubo, Y.,

"Experimental Study on F l e x u r a l S t r e n g t h and E l a s t i c Modulus o f Sea Ice," Proceedings o f t h e 6 t h I n t e r n a t i o n a l Conference on P o r t and Ocean E n g i n e e r i n g under A r c t i c C o n d i t i o n s , Laval

U n i v e r s i t v . Ouebec. Canada. Vol

.

1. 1981.

pp. 536-54;. '

7. Murat, J.R., and Lainey, L.M., "Some

Exoerimental Observations on t h e Poi ssonls

~ a t i o o f Sea Ice," C o l d Regions Science and

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8. sin ha,^

N.X.,

"Young A r c t i c F r a z i l Sea I c e : F i e l d and L a b o r a t o r y S t r e n g t h Tests

,"

Journal o f M a t e r i a l s Science, Vol

.

21, No. 5,

1

9

8

6 ,

pp. 1533-1546.

9. Sinha. N.K.. " E f f e c t i v e P o i s s o n ' s R a t i o o f

10. Sinha, N.K., " A c o u s t i c

mission

and M i c r o c r a c k i n g i n Ice," Proceedings, 1982 S o c i e t y o f ~xperimental- Japan

S o c i e t y o f Mech. Enaineers. Honolul u/Maui

.

~awaii", May 1982, p a r t 2,

pi.

767-772:

11. Murat, J.-B., "Small Scale Surface S t r a i n Measurement on Sea Ice," Proceedings o f t h e Workshop on Sea I c e F i e l d Measurement, St. John's, Centre f o r Cold Ocean Resources Engineering, k m o r i a1 U n i v e r s i t y o f Newfoundland, Pub1 i c a t i o n s No. 80-21, 1980, pp. 55-74.

reprinted f r o m

published b y

12. Nakawo, M., and Sinha, N.K., "A Note on B r i n e Layer Spacing o f F i r s t - Y e a r Sea I c e " ,

Atmosphere-Ocean, Vol

.

22,

No.

2, 1984, pp. 193-206.

13. Sinha, N.K., "Creep Model o f Ice,, f o r

M o n o t o n i c a l l y I n c r e a s i n g Stress, Cold Re I o n s Science and Techno1 ogy

,

Vo1

.

8,

*

o

N

14. Traetteberg, A., Gold, L.W. and Frederking, R.M.W., "The S t r a i n Rate and Temperature Deoendence o f Younq's Modulus o f Ice," ~ r b c e e d i n g s , 3 r d 1 i t . Symp. on I c e Problems, I n t . Assn. o f H y d r a u l i c Research, Hanover,

pp. 479-486.

Seventh International Conference o n Offshore Mechanics and

Arctic Engineerng - Volume IV

Editors: D.S. Sodlii, C.H. Luk, and N.K. Sinha (Book No. 10250D)

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