A tubular transducer for in situ measurement of stresses and strains in ice

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A tubular transducer for in situ measurement of stresses and strains in

ice

NATIONAL RESEARCH COUNCIL O F CANADA DIVISION O F BUILDING RESEARCH

A TUBULAR TRANSDUCER F O R IN SITU

MEASUREMENT O F STRESSES AND STRAINS IN ICE

R. F r e d e r k i n g I n t e r n a l R e p o r t No. 408 of t h e Division of Building R e s e a r c h Ottawa O c t o b e r 1973

P R E F A C E

Because of a need f o r t r a n s d u c e r s suitable f o r making in situ s t r e s s and s t r a i n m e a s u r e m e n t s in i c e , a thin-walled tube with c i r

-

cumferentially orientated s t r a i n g a g e s bonded to the inside h a s been designed. The e l a s t i c theory which r e l a t e s the s t r e s s and s t r a i n field in the ice t o the s t r a i n s in the tube has been developed in detail. Both l a b o r a t o r y and field experience show that although the t r a n s d u c e r successfully d e t e r m i n e s principal s t r e s s d i r e c t i o n s , i t s u s e f o r determining the magnitude of the principal s t r e s s e s i s limited by the inability t o e s t a b l i s h t r u e values of the e l a s t i c modulus of i c e .

Ottawa,

October 1973

N.B. Hutcheon D i r e c t o r

A TUBULAR TRANSDUCER F O R IN SITU

MEASUREMENT O F STRESSES AND STRAINS IN ICE by

R. F r e d e r k i n g

Introduction

T o d e t e r m i n e the loads floating i c e c o v e r s a r e capable of e x e r t i n g on s t a t i o n a r y s t r u c t u r e s , a m a t h e m a t i c a l m o d e l of t h e i n t e r a c t i o n between the i c e and the s t r u c t u r e c a n be u s e d . Such m o d e l s c a n be c o n f i r m e d by in s i t u m e a s u r e m e n t of t h e s t r e s s o r s t r a i n in t h e i c e a d j a c e n t t o the s t r u c t u r e and c o m p a r i n g t h e m e a s u r e d v a l u e s with t h o s e p r e d i c t e d by t h e m o d e l . Suitable t r a n s d u c e r s a r e r e q u i r e d f o r t h e s e m e a s u r e m e n t s . C o m m e r c i a l t r a n s d u c e r s f o r in s i t u m e a s u r e m e n t s i n r o c k , s o i l and c o n c r e t e a r e a v a i l a b l e , but none a r e s p e c i f i c a l l y d e s i g n e d f o r u s e i n i c e . It i s d e s i r a b l e f o r in s i t u t r a n s d u c e r s t o p r o d u c e m i n i m u m d i s t u r b a n c e t o t h e s t r e s s and s t r a i n p a t t e r n i n the h o s t m a t e r i a l , p a r t i c u l a r l y in a v i s c o e l a s t i c m a t e r i a l s u c h a s i c e . It i s a l s o d e s i r a b l e t h a t t h e t r a n s d u c e r have a f r e q u e n c y r e s p o n s e u p t o 10Hz and t h a t i t be a b l e t o m e a s u r e t h e p r i n c i p a l s t r e s s e s o r s t r a i n s and r e s o l v e t h e i r d i r e c t i o n . A t r a n s d u c e r m e e t i n g t h e above r e q u i r e m e n t s i s m o s t e a s i l y r e a l i s e d if u s e d i n c i r c u m s t a n c e s f o r which the h o s t m a t e r i a l p e r f o r m s e l a s t i c a l l y .

A suitable t r a n s d u c e r would include s u c h f e a t u r e s a s : s i m p l e c o n s t r u c t i o n

e a s y field deployment

r e l i a b i l i t y a t low t e m p e r a t u r e s s i m p l e d a t a r e c o r d i n g

s t r a i g h t f o r w a r d d a t a a n a l y s i s

T o m e e t t h e above r e q u i r e m e n t s a thin-walled tube with c i r c u m - f e r e n t i a l l y o r i e n t a t e d s t r a i n g a g e s bonded t o the i n s i d e w a s designed. T h e t r a n s d u c e r i s s i m p l e t o f a b r i c a t e , lends i t s e l f t o flexibility by t h r e a d e d s e c t i o n s which c a n be joined t o g e t h e r , can be deployed by f r e e z i n g into a v e r t i c a l hole in t h e i c e c o v e r , i s rugged and p r o v i d e s adequate p r o t e c t i o n f o r t h e s t r a i n g a g e s and c a n be u s e d with r e c o r d i n g

T h e o r y

T h e following t h e o r y a s s u m e s t h a t t h e i c e i s e l a s t i c , a r e a s o n a b l e provided t h e s t r a i n r a t e i s g r e a t e r than 1 0 - " s e c - l . The t r a n s d u c e r o p e r a t i o n i s modeled by t h e e l a s t i c a n a l y s i s of a thin-walled tube i m p l a n t e d in a n e l a s t i c host m a t e r i a l u n d e r biaxial loading.

T h e b a s i s f o r the t h e o r y u s e d i n designing t h e t r a n s d u c e r i s a r e p o r t by H i r a m a t s u , Niwa and 0 k a " ) . F i g . 1 i l l u s t r a t e s t h e g e o m e t r y of t h e p r o b l e m . T h e i n s i d e and outside r a d i i a r e given by b and a

r e s p e c t i v e l y , the biaxial s t r e s s f i e l d by p and q , the biaxial s t r a i n f i e l d by c p and B and t h e e l a s t i c c o n s t a n t s by E l and v' f o r t h e h o s t

9'

m a t e r i a l and E and

v

f o r t h e tube m a t e r i a l .

The A i r y s t r e s s function f o r the tube, which f o r m s the a c t i v e p a r t of t h e t r a n s d u c e r , i s d e s c r i b e d by

@

= A , r 2 + B o A n r + ( A 2 r 4

+

B 2 r - " +

c 2 r 2 +

D 2 ) c o s 28 _{( 1 )} f o r the d o m a i n a>r>b. The s t r e s s equations d e r i v e d f r o m the s t r e s s function a r e

=

2A0

+

3

-

(3

+

2C,

+

% '

c o s 28

r

r 2 r 2

I

S i m i l a r l y , t h e A i r y s t r e s s function f o r the h o s t m a t e r i a l i s

(8'

=

Abr2

+

Bb Anr

+

(Bgr-"

+

c&r2

+

Dg) c o s 28 ( 5 ) f o r t h e d o m a i n r>a. The s t r e s s equations d e r i v e d f r o m t h e s t r e s s function a r e

6q

t 2Ca

-9,

s i n 2 8

=

i-

r

The s t r e s s equations developed above m a y , in g e n e r a l , be applied t o two d i m e n s i o n a l p r o b l e m s i n p o l a r c o o r d i n a t e s . In the a b s e n c e of

body f o r c e s ( 2 )

,

t h e plane s t r e s s and plane s t r a i n A i r y s t r e s s functions a r c t h e s a m e . T o d e r i v e t h e s t r a i n s and d e f o r m a t i o n s , however, e i t h e r a plane s t r a i n o r plane s t r e s s condition m u s t be s e l e c t e d . The derivation by H i r a m a t s u , Niwa and 0ka(') a s s u m e s the plane s t r e s s condition.

In t h i s development both conditions will be c o n s i d e r e d .

i ) plane s t r e s s :

The e l a s t i c i t y equations f o r the plane s t r e s s condition a r e

and apply t o both the tube and host m a t e r i a l . Using t h e s e r e l a t i o n s s t r a i n equations c a n be developed, and i n t u r n the s t r a i n equations c a n be

i n t e g r a t e d t o obtain d i s p l a c e m e n t s . The c o n s t a n t s of i n t e g r a t i o n a r e eliminated by a s s u m i n g the s i m p l e s t solution consistent with the above s t r a i n equations. The r e s u l t i n g equations f o r s t r a i n and d i s p l a c e m e n t a r e : f o r the tube

- 4

-

B

=

1

{2(v

+

3 ) ~ , r ~ t 2(1 t v)

3

+

2(1

+

V) C , r E r - 2(1 - v) D

$

1 s i n 28 f o r t h e h o s t m a t e r i a l 1 Bb -

-

{2(1

-

v l ) Ah

-

(1

+

v t )

-p

€ \

-

E l

a'

1

+

(6(l

+

v')

f 2(1 f v ' ) C $ + 4Vr2 )COS 20, r - - 2(1

+

v t ) Y'r 0 E'

{63

-

2 ~ 4

+

9)

s i n 2 8

+

(2(1

+

v t )

3

-

2(1

+

vr) CQr

+

~ ; C O S20) r r (18) u) = 1

_-

_{2(1

+

_v')

J Ba

+

2(1

+

v')

C J r

-

2(1

-

v')

s i n 28 8 E t r (19) T h e following b o u n d a r y c o n d i t i o n s a r e a s s u m e d i n e v a l u a t i n g t h e c o n s t a n t s A , , A t , .

.

.

,D2,

DJ.

0' =

( p +

q ) +

(p

-

q ) c o s 28 a s r + r 2 01 = ( p + q)

-

( p

-

q ) c o s 28 8 2 ( p

-

q ) s i n 28 7 k e =

-

2 at i n t e r f a c e r

=

a

u

=

u:, 7 = 7' r re r 0 u = u:, u e = u' r 8

a t i n n e r s u r f a c e r = b o ₌ 7 = 0 r r 0 F o r t h e s c conditions, the c o n s t a n t s a r e 1

A,

=

- -

ZC,

+ %

3b2 \ b2

ii) plane s t r a i n :

The e l a s t i c i t y equations f o r t h e plane s t r a i n conditions a r e

l t v

1

c =

-

{ ( l

-

v) o @

-

vo r

8 E r J

S t r a i n equations can be developed and in t u r n can be i n t e g r a t e d to yield d i s - p l a c e m e n t s . The c o n s t a n t s of integration a r e again e l i m i n a t e d by a s s u m i n g t h e s i m p l e s t solution c o n s i s t e n t with t h e s t r a i n equations. T o f a c i l i t a t e t h e

solution the c o n s t a n t s D, and D i a r e t a k e n t o be z e r o . The equations f o r s t r a i n and d i s p l a c e m e n t d e r i v e d a r e f o r the tube - E 0 - +

v,

{2(1

-

Zv) A,

-

1 - V ) A 2 r 2 t

+

2C2 jcos 20 E r r 2(1

+

v) 6B r 8' E { 6 A 2 r 2

-

r t 2C2 jsin 28

- - 2B U 0

'

E l 2 ( 3

1

-

2 ~ ) A, r" t

-3

r t 2CBr} sin 2 0 f o r the host m a t e r i a l ( 1 t vl) E l = E ' {(l

-

2v1) 2A!, B b '6BB t 2 ~ ~ j c o s 20) 0

- ? + ( T C

U I = ( 1 t v l ) B I r 2B' t 2 c A r ) c o s 2 0 ) {2(1 - 2 v 1 ) A&

_-

(-

₇

r E 1

Again t h e c o n s t a n t s A,, A,',

.

.

.

,

C 2 , C' a r e evaluated using t h e following boundary conditions: a s r A r n 01 = ( p t q ) t ( p - q ) c o s 28 r 2 1 3 ( p t q )

-

( p

-

q ) c o s 28 8 2 7' -

-

re-

-

2 s i n 20 a t i n t e r f a c e r = a ; 0 = 0' ; r r a t i n n e r s u r f a c e r

=

b;

Using t h e s e conditions, constants a r e evaluated a s follows:

A s mentioned i t i s f u r t h e r r e q u i r e d that the t r a n s d u c e r produce the

minimum disturbance of the s t r e s s and s t r a i n p a t t e r n in the host m a t e r i a l , i. e . the d i s p l a c e m e n t s of the outer d i a m e t e r of the tube should approximate those of a solid cylinder of the s a m e s i z e of the host m a t e r i a l .

The A i r y s t r e s s function, s t r e s s e s and d i s p l a c e m e n t s f o r a solid cylinder (domain r ~ a ) , a s s u m i n g plane s t r e s s condition a r e

@

=

+

+

C a r 2 ) c o s 28

o _{= 2A0}

-

_{2C2 c o s 28}

r

cr

= 2Ao

+

( 1 2 ~ , r "

+

2 C 2 ) cos 28 8

- 9

-

.r = ( 6 A z r 2

+

2Cz) s i n 20

r 0

The c o n s t a n t s A,, Ah,

. . .

,

C:, D& evaluated by m e a n s of t h e boundary conditions:

a t i n t e r f a c e r = a 0 1 = ( p + q) - ( p - q) c o s 2 0 0 2 7' -

-

( p

-

q) sin 28

r e

2 a r e found t o be a s follows:

Using t h e s a m e e l a s t i c p r o p e r t i e s f o r the solid cylinder a s w e r e u s e d f o r the host m a t e r i a l , the equations f o r r a d i a l and c i r c u m f e r e n t i a l d e f o r m a t i o n s of the solid cylinder a t t h e i n t e r f a c e r = a a r e

r - E r ( P + 9) + + 2 v') ( p

-

q ) c o s 2 0 )

c a ( 1

+

v l ) u =

- -

E' 2 ( p

-

q) s i n 28 8

The r a d i a l and c i r c u m f e r e n t i a l d e f o r m a t i o n s f o r t h e tube a t r = a , a s d e r i v e d f r o m equations ( 1 3) and ( 1 4 ) , a r e

+

2(1

+

V)C,

-

3

j

cos 2 0 ) B

ut

=

5

I 2 ( v

+

3) A,aZ

+

2(1

+

v)

-$

+

(1

+

v)C2

-

( 1

-

v)

3)

sin 20

The d e s i g n p r o c e d u r e f o r invisibility c o n s i s t s of varying e l a s t i c p r o p e r t i e s and tube d i a m e t e r s t o produce a m i n i m u m difference between cylinder and tube d i s p l a c e m e n t s a t r = a .

Design

A n u m b e r of m a t e r i a l s and g e o m e t r i e s f o r the tube could be c o n s i d e r e d . ~t w a s d e c i d e d , however, t o r e s t r i c t t h e choice t o c o m m e r c i a l l y available

in s t a n d a r d s i z e s . A m a t e r i a l with t h e r m a l expansion c h a r a c t e r i s t i c s s i m i l a r t o t h o s e of i c e i s a l s o d e s i r a b l e and on t h i s b a s i s a n aluminum alloy was chosen. Aluminum h a s a high coefficient of t h e r m a l expansion f o r a m e t a l ,

2 3 x lo"/ " C c o m p a r e d t o 51 x 1 o - ~ / c " t h e coefficient of t h e r m a l expansion f o r i c e . A n u m b e r of aluminum a l l o v t u b e s w e r e evaluated i n o r d e r t o d e t e r m i n e one which would produce the m i n i m u m d i s t u r b a n c e t o the s t r e s s and s t r a i n field in the i c e . The i c e p r o p e r t i e s a s s u m e d w e r e Young's m o d u l u s 6900 M N / ~ ~ and P o i s s o n ' s r a t i o 0. 5. The m a t e r i a l which proved m o s t s a t i s f a c t o r y w a s

65ST6 a l u m i n u m of 2 in. (50 m m ) outside d i a m e t e r and 0. 120 i n ( 3 m m ) w a l l t h i c k n e s s . T h i s alloy h a s Young's m o d u l u s 6 9 , 0 0 0 M N / ~ ~ and P o i s s o n ' s r a t i o 0. 3 16. A s an indication of the invisibility of t h i s d e s i g n the r a d i a l and c i r c u m f e r e n t i a l d i s p l a c e m e n t s of the solid ice cylinder in m m a r e

C u = 4 4 x q ) t 131 x 10-'(p

-

q ) c o s 20 r C u = - 131 x 10-'(p

-

q ) s i n 20

8

t v e r s u s u = 4 2 ~ 1 0 - ~ ( p t q ) t 1 6 3 ~ 1 0 - ~ ( ~

-

q) cos 20 r t u =

-

131 x - q) s i n 20 A

f o r the t u b e , w h e r e s t r e s s e s p and q have units kN/m2,. The r a d i a l d i s -

p l a c e m e n t s of the tube a r e f r o m 17 t o 35 p e r cent g r e a t e r t h a n t h o s e of the solid cylinder while the c i r c u m f e r e n t i a l d i s p l a c e m e n t s a r e identical.

The o p e r a t i o n of the t r a n s d u c e r i s based on the change i n shape of the tube due t o the s t r e s s o r s t r a i n field in the i c e . T h i s change in shape p r o - d u c e s c i r c u m f e r e n t i a l s t r a i n s on t h e i n n e r wall of the tube. The c i r c u m - f e r e n t i a l s t r a i n s a s d e r i v e d f o r the plane s t r e s s and plane s t r a i n condition ( s e e equations 11 and 2 5 r e s p e c t i v e l y ) c a n be e x p r e s s e d in t e r m s of t h e s t r e s s field p and q o r the s t r a i n field E and E a s

t: = k l ( p t q ) t k , ( p

-

q) c o s 28

8

where k1, k2

,

k i and ka a r e constants dependent on the g e o m e t r y and e l a s t i c p r o p e r t i e s of the tube and host m a t e r i a l .

The constants kl and k, which r e l a t e the s t r e s s field p and q t o the t r a n s d u c e r s t r a i n € 0 a r e plotted in F i g . 2. It can be s e e n that variation in P o i s s o n ' s r a t i o h a s v e r y little influence on the values of the constants.

Comparing the plane s t r e s s and plane s t r a i n r e s u l t s shows that the difference between the two conditions i s not l a r g e provided Young's modulus i s g r e a t e r than 5000 M N / ~ ' . T h i s i s in g e n e r a l a g r e e m e n t with ~ i m o s h e n k o ( 3 , who

s t a t e s that if the d i a m e t e r of the hole i s m u c h s m a l l e r than the plate

t h i c k n e s s , the two conditions a r e s i m i l a r . Where the plate thickness i s about the s a m e a s the d i a m e t e r of the hole, the problem becomes t h r e e dimensional and beyond the scope of t h i s Report.

In F i g . 3 the constants k: and k& which r e l a t e t r a n s d u c e r s t r a i n c _{t o}

the s t r a i n field c _and c _{a r e plotted t o c o m p a r e the plane s t r e s s and p!ane}

P 9

s t r a i n conditions and t o show the influence of the i c e e l a s t i c p r o p e r t i e s . The plane s t r e s s and plane s t r a i n condition give r e s u l t s which differ

significantly. It i s c l e a r that the relation between the t r a n s d u c e r s t r a i n and the s t r a i n field i s v e r y dependent on the e l a s t i c p r o p e r t i e s of the i c e o v e r

the range i l l u s t r a t e d h e r e . M o s t importantly t h i s figure shows the t r a n s d u c c ~ r s t r a i n s and the s t r a i n s i n the i c e a r e of the s a m e o r d e r of magnitude

( 0 . 2 5 < ki < 2,

-/

< k; < 0.25).

The final design c o n s i s t s of a 2-in. (50 m m ) outside d i a m e t e r aluminum tube 100 m m long, the 100-mm dimension being chosen t o facilitate the

placement of the s t r a i n gages. T h r e e s t r a i n gages a r e bonded on the i n n e r wall a t the c e n t r a l c i r c u m f e r e n c e . T h e y a r e aligned c i r c u m f e r e n t i a l l y and spaced a t 120'. T h r e e aluminum t a b s a r e t a c k welded a t one end of the tube t o c a r r y t e m p e r a t u r e - c o m p e n s a t i n g g a g e s . E a c h t r a n s d u c e r contains t h r e e half

bridges, the half bridge comprising one active gage and one t e m p e r a t u r e compensating gage. The 100-mm s t r a i n gaged sections a r e threaded a t e i t h e r end t o allow t h e m t o be attached t o o t h e r tubular sections. In t h i s m a n n e r a s t r i n g can be built up and the s t r a i n gaged section placed a t any d e s i r e d level. The t r a n s d u c e r i s i l l u s t r a t e d in Fig. 4.

Calibration

On the basis of a s s u m e d p r o p e r t i e s of i c e i t i s possible t o analytically d e t e r m i n e the relation between the i n n e r wall c i r c u m f e r e n t i a l s t r a i n and the

aPP

lied s t r e s s field by m e a n s of equation ( 4 9 ) . F o r 120" spacing of the

strain g a g e s , s t r a i n s a t e a c h of the t h r e e locations would be

A

' e

= k, ( p t q) t k, (p

-

q) c o s 28 B F:8

=

k, ( p t q)

-

k2

( y )

( c o s 28

-

sin 28)

cC

_{= kl ( p t q)}

-

_{k2 (=) ( C O S 2 0 t}

0

_{sin 28)} 8 2

where the s u p e r s c r i p t s A , B,

C

r e f e r to s t r a i n gages A , B, C of F i g . 1. The above t h r e e equations can be solved f o r the principal s t r e s s e s p and q and the orientation 8 of the principal s t r e s s e s t o an a r b i t r a r y a x i s . T h e s e equations

B

C

A - k , ( p + q ) 8 P - q = _{k, c o s 28}

a r e used t o d e s c r i b e the operation of the t r a n s d u c e r . Note that the principal s t r e s s d i r e c t i o n i s independent of the m a t e r i a l p r o p e r t i e s ; i. e . p a r a m e t e r s k1 and k,

.

L a b o r a t o r y calibrations w e r e p e r f o r m e d by freezing the t r a n s d u c e r into the c e n t r e of a r e c t a n g u l a r block of columnar grained ice

200 x 200 x 100 m m . The a x e s of the columnar g r a i n s and the t r a n s d u c e r w e r e perpendicular t o the 200 x 200 mm f a c e . Load was applied in a

direction n o r m a l t o the 100 x 200 mm f a c e a t a s t r a i n r a t e of 4 x l o - " sec-l. In a l l thc calibration t e s t s the applied loads w e r e uniaxial. The t r a n s d u c e r s W e r e c a l i b r a t e d in two orientations obtained by rotating the block 90" about the t r a n s d u c e r a x i s . In Fig. 5a, where the t r a n s d u c e r i s orientated such that

~ r i ~ n t a t i o n of s t r a i n gage A ) , typical applied s t r e s s

-

m e a s u r e d t r a n s d u c e r s t r a i n r e s u l t s a r e shown. F i g . 5b shows c o r r e s p o n d i n g r e s u l t s w e r e

0 = 0 " . E r r o r b a r s show t h e e x p e r i m e n t a l v a r i a t i o n in the r e s u l t s . The t h e o r e t i c a l c u r v e s of F i g . 5 w e r e obtained using equations (50) and

taking 0= 90" and 0 " r e s p e c t i v e l y , plane s t r e s s condition, Young's m o d u l u s of i c e

=

7500 MN/m2 and P o i s s o n ' s r a t i o of i c e

=

0 . 4 . The e l a s t i c p r o p e r t i e s of the i c e c o r r e s p o n d t o t h o s e a t the t e s t t e m p e r a t u r e of

-10°C. T h e r e i s good a g r e e m e n t between the t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s . Note t h a t t h e sign convention t a k e s c o m p r e s s i v e s t r e s s e s and s t r a i n s t o be positive.

A l t e r n a t i v e l y , the s t r a i n r e s u l t s f r o m F i g . 5 could be substituted into equations ( 5 1 ) t o calculate the p r i n c i p a l s t r e s s e s p and q and t h e i r o r i e n t a t i o n w h e r e the e l a s t i c p r o p e r t i e s of t h e i c e a r e t h o s e a s s u m e d i n t h e p r e v i o u s p a r a g r a p h . T h i s w a s done by using t h e s t r a i n r e s u l t s f o r a n applied s t r e s s of 690 k N / m 2 . The r e s u l t s w e r e : f o r c a s e ( a )

f o r c a s e ( b )

It c a n be s e e n t h a t i n both c a s e s the calculated p r i n c i p a l s t r e s s d i r e c t i o n i s i n good a g r e e m e n t with t h e applied p r i n c i p a l s t r e s s d i r e c t i o n (within 2 " ) . F o r c a s c (a) t h c calculated s t r e s s , p, w a s 17 p e r cent g r e a t e r than t h e applicd s t r e s s and the calculated s t r e s s , q , w a s negligible c o m p a r e d t o the applied s t r e s s . F o r c a s e ( b ) the c a l c u l a t e d s t r e s s , p , w a s 16 p e r c e n t g r e a t e r t h a n t h e applied s t r e s s ; however, t h e calculated s t r e s s , q , w a s o v e r one - t h i r d of t h e applied s t r e s s .

Another a p p r o a c h t o t h i s p r o b l e m i s t o a s s u m e that t h e t r a n s d u c e r w o r k s and t o c a l c u l a t e t h e e l a s t i c p r o p e r t i e s of t h e i c e r e q u i r e d t o m a k e the t h e o r e t i c a l r e s u l t s m a t c h t h e o b s e r v e d r e s u l t s . F o r t h e c a s e 0

=

90"

Young's m o d u l u s w a s calculated t o be 5000 MN/m2 and v = 0.5. F i e l d E x p e r i e n c e

The t r a n s d u c e r h a s been u s e d in the field on s e v e r a l o c c a s i o n s o v e r t h e p a s t t h r e e w i n t e r s . T h e r e have been m o r e difficulties with t h e r e c o r d i n g i n e t r u m e n t a t i o n than with t h e t r a n s d u c e r .

The technique f o r implacing t h e t r a n s d u c e r i s t o d r i l l a hole slightly l a r g e r than the outside d i a m e t e r of t h e t r a n s d u c e r , put s o m e w a t e r i n t h e hole and i n s e r t the t r a n s d u c e r which d i s p l a c e s the w a t e r u p w a r d s t o f i l l

the annular s p a c e . Since the amount of w a t e r in this space i s s m a l l , f r e e z i n g o c c u r s rapidly. The t r a n s d u c e r i s e a s i l y removed by cutting with a chain saw.

With e a c h s u c c e s s i v e field u s e the instrumentation h a s been improved and m o r e complete r e s u l t s obtained. A field portable oscillograph r e c o r d e r with a p p r o p r i a t e s t r a i n a m p l i f i e r s h a s been obtained and found to work

s a t i s f a c t o r i l y .

The m o s t r e c e n t field u s e of the t r a n s d u c e r w a s in conjunction with in s i t u i c e s t r e n g t h t e s t s . The t r a n s d u c e r was placed in the i c e and the s t r a i n s w e r e r e c o r d e d continuously while the i c e cover w a s being loaded. The principal s t r e s s e s in the i c e m e a s u r e d with the t r a n s d u c e r w e r e about

half t h e i r value calculated f r o m the known load condition. The e l a s t i c p r o p e r t i e s a s s u m e d f o r the ice w e r e Young's modulus

=

6900 M N / ~ ~ and

Poisson's r a t i o = 0. 5. No information is available, unfortunately, on the value of the actual e l a s t i c p r o p e r t i e s of the i c e . If a l a r g e r Young's modulus w e r e a s s u m e d , s a y 1 3 , 8 0 0 M N / ~ ~ , the m e a s u r e d s t r e s s e s would only be about 20 p e r cent s m a l l e r than the calculated s t r e s s e s . The m e a s u r e d p r i n c i p a l s t r e s s d i r e c t i o n s w e r e within 5" of the calculated d i r e c t i o n s . Conclusions

L a b o r a t o r y c a l i b r a t i o n s and field experience have shown that the t r a n s d u c e r c a n successfully d e t e r m i n e the p r i n c i p a l s t r e s s direction.

T h i s s u p p o r t s the a n a l y s i s which shows that the calculation of the principal s t r e s s d i r e c t i o n i s independent of the e l a s t i c p r o p e r t i e s of the ice when the s t r a i n g a g e s in the t r a n s d u c e r a r e spaced a t 120". In the calibration t e s t s , w h e r e the e l a s t i c p r o p e r t i e s of the ice a r e known, the calculated principal s t r e s s e s w e r e within 20 p e r cent of the applied s t r e s s e s . In the field however, the a g r e e m e n t between the calculated and applied s t r e s s field h a s not been s a t i s f a c t o r y . This i s due p r i m a r i l y t o uncertainty i n the e l a s t i c p r o p e r t i e s of the i c e .

In the p r e s e n t stage of development, the m o s t important r e s u l t obtained f r o m the t r a n s d u c e r i s the principal s t r e s s direction in a biaxial s t r e s s field. By modifying the design t o produce a m o r e compliant t r a n s d u c e r , i. e. thinner tube wall o r lower modulus m a t e r i a l , the t r a n s d u c e r would produce s t r e s s r e s u l t s which a r e l e s s dependent on the e l a s t i c p r o p e r t i e s of the i c e .

R e f e r e n c e s

( 1 ) Y. H i r a m a t s u , Y . Niwa and Y . Oka. M e a s u r e m e n t of S t r e s s in Field by Application of P h o t o - E l a s t i c i t y , Report No. 37, Technical R e p o r t s of the Engineering R e s e a r c h Institute Kyoto University, Vol. 7 No. 3 , M a r c h , 1957.

( 2 ) S. T i m o s h c n k o . T h e o r y of E l a s t i c i t y , 2nd edition, 1951, M c G r a w - H i l l , p 25.

( 3 ) S. T i m o s h e n k o . T h e o r y of E l a s t i c i t y , 2nd edition, 1951, M c G r a w - H i l l , p 85.

F I G U R E 1

G E O M E T R Y O F C Y L I N D R I C A L I N C L U S I O N I N B l A X l A L L Y S T R E S S E D H O S T

--

--

PLANE STRAIN PLANE STRESS z 0 Z 5 0 0 0 1 0 , 0 0 0 1 5 , 0 0 0 ', hl Y O U N G I S M O D U L U S O F I C E , M N / ~ ~ E ( P O I S S O N ' S R A T I O 0 . 4 )

--- -

_ _

_ _ _

_ _ _ _ _

___-

---

-

Y O U N G ' S M O D U L U S O F I C E , M ~ / r n 2 ( Y O U N G ' S M O D U L U S 7 5 0 0 M ~ / m 2 ) z

z

1 V) P O I S S O N ' S R A T I O O F I C E I- z

z

or

Ln Z

I

( Y O U N G ' S M O D U L U S 7 5 0 0 M ~ / r n ~ ) F I G U R E 2 0 u - 1 k2 D E P E N D E N C E O F C O N S T A N T S k l A N D k 2 , W H I C H RELATE T R A N S D U C E R S T R A I N T O THE STRESS F I E L D , O N E L A S T I C PROPERTIES

O F I C E on 6 9 6 - 2

_ _ _ _ _ _ _

_-

- - - -

--

- - - -

- - - _

0 . 3 0 . 4 0 . 5

----

_{PLANE STRAIN}

-

PLANE STRESS ( P O I S S O N ' S RATIO 0 . 4 ) / 0 / / ,' / 0

-

/ 0 0 / 0 / / / / Y O U N G ' S MODULUS O F ICE, M ~ / r n ~ ( P O I S S O N ' S RATIO 0 . 4 ) Y O U N G ' S MODULUS O F ICE, M ~ / r n ~ ., 0 . 3 0 . 4 0 . 5 P O I S S O N ' S RATIO O F ICE O r ( Y O U N G ' S MODULUS 7500 M ~ / r n ? P O I S S O N ' S RATIO O F ICE FIGURE 3 DEPENDENCE O F CONSTANTS k ; A N D k;, WHICH RELATE TRANSDUCER STRAIN TO THE STRAIN FIELD, O N ELASTIC PROPERTIES O F

C i r c u m f e r e n t i a l S t r a i n G a g e s

Assembled T r a n s d u c e r

Exploded View

FIGURE 4

L

EXPERIMENTAL THEORETICAL - 5 0 2 0 0 4 0 0 6 0 0 8 0 0 A P P L I E D STRESS p , k N / m 2 - 1 5 0 2 0 0 4 0 0 6 0 0 8 0 0 A P P L I E D STRESS p , k ~ / r n ~ F I G U R E 5 C A L I B R A T I O N R E S U L T S O F A T Y P I C A L T R A N S D U C E R O* 5o.r.y