Basis of method for predicting thermal stresses and deformations in frozen soils

(1)

Publisher’s version / Version de l'éditeur:

Vous avez des questions? Nous pouvons vous aider. Pour communiquer directement avec un auteur, consultez la première page de la revue dans laquelle son article a été publié afin de trouver ses coordonnées. Si vous n’arrivez pas à les repérer, communiquez avec nous à PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca.

Questions? Contact the NRC Publications Archive team at

PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca. If you wish to email the authors directly, please see the first page of the publication for their contact information.

https://publications-cnrc.canada.ca/fra/droits

L’accès à ce site Web et l’utilisation de son contenu sont assujettis aux conditions présentées dans le site LISEZ CES CONDITIONS ATTENTIVEMENT AVANT D’UTILISER CE SITE WEB.

Technical Translation (National Research Council of Canada), 1976

READ THESE TERMS AND CONDITIONS CAREFULLY BEFORE USING THIS WEBSITE.

https://nrc-publications.canada.ca/eng/copyright

NRC Publications Archive Record / Notice des Archives des publications du CNRC :

https://nrc-publications.canada.ca/eng/view/object/?id=a261d401-48ba-43a3-9906-aa59715c06ea https://publications-cnrc.canada.ca/fra/voir/objet/?id=a261d401-48ba-43a3-9906-aa59715c06ea

NRC Publications Archive

Archives des publications du CNRC

For the publisher’s version, please access the DOI link below./ Pour consulter la version de l’éditeur, utilisez le lien DOI ci-dessous.

https://doi.org/10.4224/20359113

Access and use of this website and the material on it are subject to the Terms and Conditions set forth at

Basis of method for predicting thermal stresses and deformations in

frozen soils

Grechishchev, S. E.; National Research Council of Canada. Division of

Building Research

(2)

CANADA INSTITUTE

FOR SCIENTIFIC AND TECHNICAL

INFORMATION

I S S N 0 0 7 7

-

5 6 0 6

I NSTITUT CANADl EN

DE L'INFORMATION SClENTlFlQLlE

ET TECHNIQUE

TECHNICAL TRANSLATION TRADUCTION TECHNIQUE

S.E. GRECHISHCHEV

B A S I S OF METHOD FOR P R E D I C T I N G THERMAL STRESSES

AND DEFORMATIONS I N FROZEN S O I L S

MINISTERSTVO G E O L O G I I SSSR.

VSESOYUZNYI

NAUCHNO-ISSLEDOVATEL ' S K I

I INSTITLIT GIDROGEOLOGI I I

INZHENERNOI GEOLOGI I

(VSEGINGEO)

.

MOSCOW, 1 9 7 0 .

53 PP.

TRANSLATED BY ITRADUCTION DE

V. POPPE

I S I S THE TWO HUNDRED AND TWENTY-EIGHTH I N THE S E R I E S OF TRANSLATIONS

PREPARED FOR THE D I V I S I O N

OF B U I L D I N G RESEARCH

4 , , ,

TRADUCTION NLIMERO

228 DE L A S E R I E PRkPAREE POUR

L A D I V I S I O N DES RECHERCHES EN BATIMENT

OTTAWA

National Research

Conseil national

(3)

NATIONAL RESEARCH COUNCIL OF CANADA CONSEIL NATIONAL DE RECHERCHES DU CANADA

TECHNICAL TRANSLATION 1886 TRADUCTION TECHNIQUE B a s i s of method f o r p r e d i c t i n g t h e r m a l s t r e s s e s and d e f o r m a t i o n s i n f r o z e n s o i l s (K osnovam m e t o d i k i prognoza t e m p e r a t u r n y k h n a p r y a z h e n i i i d e f o r m a t s i i v m e r z l y k h g r u n t a k h ) A u t h o r /Aut e u r : S .E. G r e c h i s h c h e v R e f e r e n c e / R g f Grence: M i n i s t e r s t v o G e o l o g i i SSSR. V s e s o y u z n y i Nauchno- I s s l e d o v a t e l ' s k i i I n s t i t u t G i d r o g e o l o g i i i I n z h e n e r n v i G e o l o g i i (vSEGINGEO). Moscow, 1970. 53pp. T r a n s l a t o r / T r a d u c t e u r : V. Poppe, T r a n s l a t i o n S e r v i c e s / S e r v i c e d e t r a d u c t i o n Canada I n s t i t u t e f o r I n s t i t u t c a n a d i e n d e S c i e n t i f i c and T e c h n i c a l 1

'

i n £ ormat i o n s c i e n t i f i q u e e t I n £ orma t i o n t e c h n i q u e O t t a w a , Canada

KlA

OS2

(4)

PREFACE

The i n f l u e n c e o f t e m p e r a t u r e on t h e s t r e n g t h and deformation b e h a v i o u r of f r o z e n ground i s o f p a r t i c u l a r i n t e r e s t t o t h e D i v i s i o n of B u i l d i n g Research i n i t s s t u d i e s of p e r m a f r o s t . T h i s p a p e r d e a l s w i t h some of t h e fundamental f a c t o r s t o b e considered i n a s s e s s i n g and p r e d i c t i n g c r e e p t h e r m a l s t r e s s e s and d e f o r m a t i o n s of f r o z e n s o i l s .

The D i v i s i o n wishes t o r e c o r d i t s t h a n k s t o M r . V. Poppe of t h e T r a n s l a t i o n S e r v i c e s , CISTI, who t r a n s l a t e d t h i s p a p e r and t o M r .

G.H. J o h n s t o n of DBR, who checked t h e t r a n s l a t i o n f o r t e c h n i c a l a c c u r a c y . O t t awa November 1976 C

.

B

.

Crawf o r d

,

D i r e c t o r , D i v i s i o n of B u i l d i n g Research

(5)

T a b l e of C o n t e n t s

Page

Chapter I. Some Data on t h e Mechanical P r o p e r t i e s of

F r o z e n s o i l s . . . 4

1. Creep

. . .

.

. .

.

. . .

.

. .

.

. . .

4

2. R e l a x a t i o n

. .

. . .

.

. . .

.

. .

8

3. S t r e n g t h

. . .

11

Chapter 11. Thermal Deformation of F r o z e n S o i l s

. . . .

.

. . .

1 6 1. General

.

. . . .

.

. . .

.

. .

.

. .

1 6 2. Some E x p e r i m e n t a l Data on Thermal Deformation of Frozen S o i l s

.

. . .

.

. . .

. .

.

. .

18

3. Main P a t t e r n s of Thermal Deformation of F r o z e n S o i l s

. .

.

. .

.

. .

.

. . .

.

1 9 C h a p t e r 111. Thermal S t r e s s e s i n U n d i s t u r b e d Frozen Ground

. . .

28

1. E q u a t i o n of One-dimensional Thermo-rheological S t a t e of F r o z e n s o i l .

. .

.

. . .

.

. . .

.

28 2. Thermal S t r e s s e s During F r e e z i n g of t h e A c t i v e Layer

. . .

.

. . . .

.

. . . .

. .

32 3 . Thermal S t r e s s e s i n a F r o z e n A c t i v e Layer

. .

.

38 R e f e r e n c e s

.

. .

. . .

.

. .

. . .

.

50

(6)

Chapter I

SOME DATA ON THE MECIIANICAL PROPERTIES OF FROZEN SOILS

1. Creep

A t s t r e s s e s exceeding t h e u l t i m a t e long-term s t r e n g t h , t h e t o t a l d e f o r m a t i o n of f r o z e n s o i l s may b e r e p r e s e n t e d a s a sum of t h e i n i t i a l e l a s t i c d e f o r m a t i o n

EH, t h e e l a s t i c d e f o r m a t i o n E and t h e v i s c o u s - p l a s t i c f l o w & (Vyalov, 1959) :

3 n A t s t r e s s e s below t h e u l t i m a t e long-term s t r e n g t h , E i s a b s e n t . n The f i r s t two t e r m s i n e q u a t i o n (1.1) r e p r e s e n t a r e v e r s i b l e d e f o r m a t i o n , and t h e r e f o r e i n t h e t r e a t m e n t t h a t f o l l o w s we s h a l l u s e t h e f o l l o w i n g e x p r e s s i o n i n s t e a d of e q u a t i o n (1.1) : where tzYnP i s t h e r e v e r s i b l e ( c o n d i t i o n a l l y e l a s t i c ) d e f o r m a t i o n .

The r e l a t i o n s h i p between t h e r e v e r s i b l e d e f o r m a t i o n , s t r e s s and t i m e i s

d e s c r i b e d by t h e f o l l o w i n g n o n l i n e a r e q u a t i o n (Vyalov e t a l . , 1962):

where

a

i s t h e c o m p r e s s i v e o r t e n s i l e s t r e s s ,

A ( t ) i s t h e time-dependent modulus of n o n l i n e a r d e f o r m a t i o n , and

m i s t h e c o e f f i c i e n t of n o n l i n e a r i t y .

S i n c e i t i s d i f f i c u l t t o u s e t h e n o n l i n e a r e q u a t i o n ( 1 . 3 ) i n c a l c u l a t i o n s , we may u s e , i n t h e f i r s t a p p r o x i m a t i o n , a l i n e a r e x p r e s s i o n on t h e a s s u m p t i o n t h a t m

-

1, i . e . :

(7)

where E ( t ) i s t h e time-dependent modulus of l i n e a r d e f o r m a t i o n which v a r i e s between i t s i n s t a n t a n e o u s v a l u e E _Mr' a t t = 0 t o i t s long-term v a l u e Em a t t +

Em

and E Wr d i f f e r from each o t h e r and Em may exceed E

m

by a f a c t o r of 1 0 o r more.

The d e f o r m a t i o n modulus d u r i n g compression i s q u i t e d i f f e r e n t from t h a t d u r i n g t e n s i o n . During compression t h e i n s t a n t a n e o u s v a l u e of t h e modulus i s

l e s s t h a n d u r i n g t e n s i o n , w h i l e t h e long-term v a l u e i s g r e a t e r d u r i n g compression t h a n d u r i n g t e n s i o n . For example, f o r f r o z e n sand a t - 3 " ~ , t h e i n s t a n t a n e o u s d e f o r m a t i o n moduli a r e : 2 x

l o 3

kg/cm2 d u r i n g t h e compression and 5 x

l o 3

kg/cm2 d u r i n g t e n s i o n . The long-term v a l u e s a r e : 0.6 x

l o 3

kg/cm2 d u r i n g compression and 0.4 x

l o 3

kg/cm2 d u r i n g t e n s i o n . S i m i l a r d a t a were p r e s e n t e d g r a p h i c a l l y by S . E . G o r o d e t s k i i (1969).

Changes i n t h e d e f o r m a t i o n modulus i n r e l a t i o n t o t e m p e r a t u r e may be r e p r e s e n t e d a s f o l l o w s :

where

a l ,

61

and K 1 a r e e m p i r i c a l c o e f f i c i e n t s , and

8

i s t h e a b s o l u t e n e g a t i v e t e m p e r a t u r e i n O'C. C o n s i d e r i n g t h a t t h e f i r s t term i n e q u a t i o n (1.5) i s s m a l l compared t o t h e second t e r m , we s h a l l u s e t h e f o l l o w i n g s i m p l i f i e d e x p r e s s i o n i n t h e t r e a t m e n t t h a t f o l l o w s : E x p r e s s i o n (1.6) h o l d s f o r b o t h i n s t a n t a n e o u s and long-term v a l u e s of t h e d e f o r m a t i o n modulus, a l t h o u g h t h e e m p i r i c a l c o e f f i c i e n t s i n i t w i l l d i f f e r

.

According t o N. A. T s y t o v i c h (1937), f o r t h e i n s t a n t a n e o u s v a l u e s we may t a k e K

-

1 . 0 , i . e . , t h e dependence i s l i n e a r . However, a c c o r d i n g t o o t h e r a u t h o r s ,

e . g . , A. D. F r o l o v and A . A . Smirnov ( 1 9 6 1 ) , who measured t h e speed of u l t r a - sound i n f r o z e n s o i l , K 1

<<

1.

(8)

A t s t r e s s e s exceeding t h e u l t i m a t e long-term s t r e n g t h , f r o z e n s o i l b e g i n s t o c r e e p a t a c o n s t a n t r a t e which i s p r o p o r t i o n a l t o t h e d i f f e r e n c e between t h e a p p l i e d s t r e s s and t h e u l t i m a t e long-term s t r e n g t h . The r e l a t i o n s h i p between t h e c r e e p r a t e and t h e s t r e s s i s u s u a l l y n o n l i n e a r and i s d e s c r i b e d by a n n e m p i r i c a l e x p r e s s i o n suggested by S. S. Vyalov (1959) : 4 m

L,,=q-(6-64~)

,

( 1 - 7 ) where q i s t h e c o e f f i c i e n t of p l a s t i c v i s c o s i t y ,

a

i s t h e a p p l i e d s t r e s s , 0 i s t h e u l t i m a t e long-term s t r e n g t h , and Wr m i s t h e c o e f f i c i e n t of n o n l i n e a r i t y .

Expression (1.7) h o l d s f o r any u n i a x i a l t y p e s of s t r e s s : u n i a x i a l compres- s i o n , t e n s i o n , and pure s h e a r , b u t t h e e m p i r i c a l f a c t o r s rl and U

_m

w i l l be

d i f f e r e n t i n each i n d i v i d u a l c a s e .

There a r e c o n s i d e r a b l e d i f f e r e n c e s between t h e c o e f f i c i e n t s of p l a s t i c v i s c o s i t y d u r i n g compression q and d u r i n g t e n s i o n rl :

P rlc i s 2 t o

4

t i m e s C

g r e a t e r t h a n q

.

According t o our own r e s u l t s (1963), f o r f r o z e n sand a t - 3 ' ~ : P

QC = 0.4 x

l o *

kg*days/cm2 and 17 = 0.1 x

l o *

kg*days/cm2; f o r

P

*

f r o z e n s u g l i n o k a t t h e same t e m p e r a t u r e :

rlC = 1.2 x

l o 4

kg*days/cm2, 17 _P= 0.67 x

l o 4

kg*days/cm2.

The c o e f f i c i e n t s of p l a s t i c v i s c o s i t y a r e s t r o n g l y dependent on t e m p e r a t u r e . According t o S. E. G o r o d e t s k i i (1969), t h i s dependence may b e r e p r e s e n t e d a s

f o l l o w s :

where q o , B 2 and K 2 a r e e m p i r i c a l c o e f f i c i e n t s , and 8 i s t h e a b s o l u t e n e g a t i v e

t e m p e r a t u r e , O C .

*

"Suglinok"

-

c l a y e y s i l t w i t h some sand, c l a y e y s i l t y loam; c o n t a i n s 1 0 t o 30% c l a y by weight w i t h c l a y p a r t i c l e s l e s s t h a n 0.005 mm i n s i z e .

(9)

The f i r s t term i n e x p r e s s i o n (1.8) i s u s u a l l y c o n s i d e r a b l y s m a l l e r t h a n t h e second term. T h e r e f o r e , i n t h e t r e a t m e n t t h a t f o l l o w s , we s h a l l u s e t h e f o l l o w i n g s i m p l i f i e d e x p r e s s i o n i n s t e a d of (1.8) :

For c o a r s e - g r a i n e d and sandy s o i l s , t h e n o n l i n e a r i t y coef f i c i e n t m i n e x p r e s s i o n (1.7) i s a p p r o x i m a t e l y e q u a l t o u n i t y . For c l a y e y s o i l s , m > 1 and i n c r e a s e s w i t h t h e i n c r e a s e i n t h e c l a y f r a c t i o n . The r a t e of v i s c o u s - p l a s t i c f l o w d u r i n g s h e a r i n t h e p r e s e n c e of normal s t r e s s e s depends on t h e l a t t e r , a s d e s c r i b e d by t h e f o l l o w i n g e x p r e s s i o n which r e p r e s e n t s t h e b a s i c law of t h e v i s c o u s - p l a s t i c f l o w of f r o z e n s o i l s : where i s t h e r a t e of t h e v i s c o u s - p l a s t i c s h e a r d e f o r m a t i o n , T i s t h e a c t i n g s h e a r s t r e s s , 0 i s t h e normal s t r e s s , @ i s t h e u l t i m a t e long-term v a l u e of t h e a n g l e of i n t e r n a l f r i c t i o n , C i s t h e long-term v a l u e of t h e c o e f f i c i e n t of c o h e s i o n , WI i s t h e c o e f f i c i e n t of p l a s t i c v i s c o s i t y , and m i s t h e n o n l i n e a r i t y f a c t o r . The c o e f f i c i e n t s @,

Cm'

m and

rl

depend on t h e p h y s i c a l p r o p e r t i e s of t h e s o i l and a r e determined e x p e r i m e n t a l l y .

Let u s n o t e t h a t C i s s t r o n g l y dependent on t h e t e m p e r a t u r e and i n c r e a s e s ,

Jm

a s t h e t e m p e r a t u r e d r o p s , a p p r o x i m a t e l y a s f o l l o w s ( T s y t o v i c h , 1958):

The f i r s t term i n (1.11) may b e n e g l e c t e d , and we may u s e t h e f o l l o w i n g approximate e x p r e s s i o n :

(10)

According t o Vyalov (1959), K g

"

0.5. According t o Grechishchev, f o r sand B 3 = 1

-

2 kg/cm2 ( d e g l K 3 and f o r s u g l i n o k 0g = 0.5

-

1 . 5 kg/cm2 ( d e g ) K 3 . The e m p i r i c a l c o e f f i c i e n t s rl and m i n e x p r e s s i o n (1.10) depend on t h e p h y s i c a l

p r o p e r t i e s of t h e s o i l , a s was t h e c a s e w i t h t h e same c o e f f i c i e n t s i n e x p r e s s i o n ( 1 . 7 ) .

C o m p r e s s i b i l i t y of f r o z e n s o i l s i s low. According t o Brodskaya (1962),

*

f r o z e n sands cannot be compressed below -0. ~ O C , f r o z e n s u p e s e s below -2. O'C,

and f r o z e n s u g l i n o k s and c l a y s below - 4 . 0 ' ~ .

2. R e l a x a t i o n

Many a u t h o r s c a r r i e d o u t experiments t o d e r i v e e q u a t i o n s d e s c r i b i n g t h e r h e o l o g i c a l s t a t e of v a r i o u s m a t e r i a l s . S i m i l a r s t u d i e s were done on f r o z e n

s o i l s a s w e l l (Vyalov, 1959; V o i t k o v s k i i , 1961; Grechishchev, 1961, 1963; Vyalov, 1962)

.

Equations d e r i v e d i n t h e s e i n v e s t i g a t i o n s were based on

e x p e r i m e n t a l c r e e p d a t a . S t r e s s r e l a x a t i o n was s t u d i e d t h e o r e t i c a l l y and i n t h e c a s e of f r o z e n s o i l s was checked e x p e r i m e n t a l l y o n l y r e c e n t l y (Vyalov, Ermakov, 1966; G o r o d e t s k i i , 1969). S t u d i e s of c r e e p showed t h a t s e v e r a l e q u a t i o n s of s t a t e c o u l d be s u g g e s t e d . For example, t h e r e s u l t s of e x p e r i m e n t a l i n v e s t i g a t i o n s of c r e e p i n f r o z e n s o i l s c a n b e d e s c r i b e d e q u a l l y w e l l by e q u a t i o n s of n o n l i n e a r i n h e r i t e d c r e e p (Vyalov, 1959) and e q u a t i o n s of n o n l i n e a r f l o w (Grechishchev, 1961; V o i t k o v s k i i , 1961). However, i f t h e same e q u a t i o n s a r e s o l v e d w i t h r e s p e c t t o s t r e s s e s a t c o n s t a n t d e f o r m a t i o n , i . e . , on proceeding from c r e e p t o r e l a x a t i o n , t h e r e s u l t s w i l l b e v a s t l y d i f f e r e n t , and t h i s i s c o m p l e t e l y u n a c c e p t a b l e i n c a l c u l a t i o n s of t h e s t a t e of t h e r m a l s t r e s s of f r o z e n s o i l , where s t r e s s r e l a x a t i o n i s of primary importance

.

By way of i l l u s t r a t i o n , we s h a l l c i t e t h e i n t e r e s t i n g a t t e m p t by Lachen- bruch (1962) t o c a l c u l a t e t h e f i e l d of t e m p e r a t u r e s t r e s s e s i n f r o z e n s o i l . S i n c e Lachenbruch p o s t u l a t e d t h i s a s a problem of t e m p e r a t u r e s t r e s s e s i n a c o n t i n u o u s semispace, t h e s t r e s s f i e l d d i f f e r e d from z e r o , w h i l e d i s p l a c e m e n t s

*

_"Supes"

_-

_{s i l t y sand w i t h c l a y , sandy s i l t y loam; c o n t a i n s}₃ _{t o 10% c l a y by}

(11)

a l o n g t h e c o o r d i n a t e a x e s (except v e r t i c a l d i s p l a c e m e n t ) were z e r o , i . e . , it was a c a s e of r e l a x a t i o n . The p r o p e r t i e s of f r o z e n s o i l were i n t e r p r e t e d by Lachenbruch f i r s t w i t h t h e h e l p of t h e Maxwell e q u a t i o n ( c a s e 1 ) and t h e n u s i n g t h e model of n o n l i n e a r f l o w ( c a s e 2 ) . C a l c u l a t i o n s showed t h a t a f t e r a b r u p t changes i n t h e s o i l t e m p e r a t u r e , t h e s t r e s s was reduced by 90% i n t h e c o u r s e of o n e day i n c a s e 1, and i n t h e c o u r s e of one week i n c a s e 2, i. e.

,

t h e r e s u l t s of c a l c u l a t i o n d i f f e r e d by a f a c t o r of

7 .

T h e r e f o r e , i n c a s e s where t h e s t r e s s f i e l d v a r i e s w i t h t i m e , t h e s e l e c t e d e q u a t i o n of s t a t e must b e i n good agreement w i t h t h e r e s u l t s of t h e r e l a x a t i o n t e s t s .

The e q u a t i o n s of s t a t e were v e r i f i e d e x p e r i m e n t a l l y i n r e l a x a t i o n t e s t s on f r o z e n s o i l s i n v o l v i n g b o t h t e n s i o n and u n i a x i a l compression. The r e s u l t s of some of t h e s e t e s t s a r e summarized i n T a b l e I.

It may b e ' s e e n from T a b l e I t h a t a t i n i t i a l s t r e s s e s g r e a t e r t h a n t h e u l t i m a t e long-term t e n s i l e s t r e n g t h , s t r e s s r e l a x a t i o n w i l l occur up t o t h e u l t i m a t e long-term s t r e n g t h i r r e s p e c t i v e of t h e i n i t i a l s t r e s s . However, i f

i n i t i a l s t r e s s e s a r e l e s s t h a n t h e u l t i m a t e long-term s t r e n g t h , r e l a x a t i o n i s i n s i g n i f i c a n t

.

S i m i l a r c o n c l u s i o n s were reached by Vyalov and Ermakov (1966) who noted t h a t t h e r e i s a c e r t a i n s t r e s s l i m i t below which r e l a x a t i o n and hence c r e e p a r e p r a c t i c a l l y a b s e n t .

Experimental v e r i f i c a t i o n of t h e e q u a t i o n s of s t a t e showed t h a t i f t h e

i n i t i a l stresses exceed t h e u l t i m a t e long-term s t r e n g t h , 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 d e s c r i b e d b e s t by t h e e q u a t i o n of s t a t e f o r a n e l a s t i c - v i s c o u s body s u g g e s t e d by I s h l i n s k i i and R z h a n i t s y n and modified f o r f r o z e n s o i l s by Vyalov (1962). However, i t i s v e r y d i f f i c u l t t o u s e t h i s e q u a t i o n i n p r a c t i c e s i n c e , a s a f i r s t

s t e p , i t must b e g e n e r a l i z e d f o r t h e c a s e of complex s t r e s s e d s t a t e f o r which f u r t h e r i n v e s t i g a t i o n s a r e r e q u i r e d .

Theref o r e , w i t h a l l o w a n c e s f o r t h e f a c t t h a t r e l a x a t i o n i s n e g l i g i b l e a t i n i t i a l s t r e s s e s l e s s t h a n t h e u l t i m a t e long-term s t r e n g t h , we c a n u s e , a s a n a p p r o x i m a t i o n , a s i m p l i f i e d e q u a t i o n of s t a t e of t h e Bingam-Shvedov t y p e :

(12)

I t s s o l u t i o n f o r t h e c a s e of r e l a x a t i o n

( t

= 0) a t OH > 0 _is:

m

6 = 6,. + (6.- ~..)-CXP(-

v)

,

(1.14) and a t 6'" B 6 4 4

6'

= C O ~ S ~ .

.

(1.15) The l a s t e q u a t i o n shows t h a t i n a c c o r d a n c e w i t h e q u a t i o n (1.13) t h e r e i s no r e l a x a t i o n a t CJ < CJ A s was noted e a r l i e r , t h i s i s q u i t e f e a s i b l e . A t H An' > 0 e q u a t i o n (1.14) y i e l d s a r e l a x a t i o n c u r v e which i s less s t e e p t h a n OH

m y

t h e e x p e r i m e n t a l c u r v e . The d i f f e r e n c e , however, i s n o t v e r y g r e a t (10

-

1 5 % ) . Furthermore, i n t h e c a l c u l a t i o n of t h e s t a t e of t h e r m a l s t r e s s , t h i s w i l l y i e l d e x a g g e r a t e d v a l u e s of t h e s t r e s s , which w i l l improve t h e s a f e t y f a c t o r . E q u a t i o n (1.13) was d e r i v e d f o r t h e s t a t e of u n i a x i a l s t r e s s ( t e n s i o n o r compression). It c a n be g e n e r a l i z e d f o r t h e s t a t e of complex s t r e s s a s was s u g g e s t e d e a r l i e r (Grechishchev, 1963). T h i s a u t h o r f e e l s t h a t t h e f o l l o w i n g e q u a t i o n should b e used i n t h e f i n a l c a l c u l a t i o n of t h e s t a t e of t h e r m a l s t r e s s of f r o z e n s o i l : where rl i s t h e c o e f f i c i e n t of p l a s t i c v i s c o s i t y , t g 0 i s t h e c o e f f i c i e n t of i n t e r n a l f r i c t i o n , T

m

i s t h e u l t i m a t e long-term s t r e n g t h i n t h e c a s e of p u r e s h e a r , GH i s t h e s h e a r modulus, S i s t h e o c t a h e d r a l s h e a r s t r e s s , H i s t h e r a t e of s h e a r . 0

.

T

...

a r e t h e s t r e s s components, X x Y b

...,

T x y . . .

a r e t h e components.,of s t r a i n r a t e s , X

(13)

3 . S t r e n g t h

It i s e v i d e n t t h a t f o r t h e same s o i l t h e r e may b e any number of r e l a x a t i o n c u r v e s d e f i n e d by e q u a t i o n ( 1 . 1 4 ) , depending on t h e i n i t i a l s t r e s s OH. I t i s assumed t h a t t h e u l t i m a t e r e l a x a t i o n c u r v e i s t h e c u r v e of long-term s t r e n g t h ( a t OH = 0 where Om i s t h e i n s t a n t a n e o u s s t r e n g t h ) . T h e r e f o r e , i n t h e o r y , M r

'

t h e e q u a t i o n f o r t h e c u r v e of long-term s t r e n g t h i s o b t a i n e d from e q u a t i o n (1.14) i n t h e f o l l o w i n g form:

where t i s t h e t i m e p r i o r t o f a i l u r e under t h e l o a d 0

,

and T = l /i s ~t h e ~

P nP

r e l a x a t i o n time.

By t p i s meant t h e moment of t r a n s i t i o n from t h e d e f o r m a t i o n s t a g e t o t h e a c c e l e r a t i n g s t a g e of c r e e p .

For f r o z e n s o i l s , t h e i n s t a n t a n e o u s and long-term compression s t r e n g t h i s

2 t o 4 t i m e s h i g h e r t h a n t h e i n s t a n t a n e o u s and long-term t e n s i l e s t r e n g t h . The r e l a x a t i o n t i m e i n t h e c a s e of compression may b e 2 t o 1 0 t i m e s l o n g e r t h a n i n t h e c a s e of t e n s i o n .

F i g u r e 1 shows t h e e x p e r i m e n t a l c u r v e s of long-term s t r e n g t h f o r f r o z e n sand and s u g l i n o k . The r e l a x a t i o n t i m e f o r sand i n t h e c a s e of compression i s T = 1 0 d a y s and i n t h e c a s e of t e n s i o n T =

3.3

days. The c o r r e s p o n d i n g

C P

r e l a x a t i o n t i m e s f o r s u g l i n o k a r e T = 4.5 d a y s and T = 2 . 2 d a y s , r e s p e c t i v e l y .

C P

The i n s t a n t a n e o u s and long-term s t r e n g t h of f r o z e n s o i l s depends on t h e t e m p e r a t u r e i n t h e c a s e of b o t h compression and t e n s i o n . According t o T s y t o v i c h

(1958)

,

t h i s dependence may b e d e s c r i b e d by t h e f o l l o w i n g e m p i r i c a l e x p r e s s i o n :

where 0 i s t h e a b s o l u t e n e g a t i v e t e m p e r a t u r e , OC.

For f ine-grained s o i l s , t h e f i r s t term i n e x p r e s s i o n (1.18) i s u s u a l l y v e r y s m a l l compared t o t h e second term, and may b e o m i t t e d :

(14)

For some f r o z e n s o i l s we may t a k e n

'

0.5 ( ~ y a l o v , 1959).

The s t r e n g t h of f r o z e n s o i l s i s g r e a t l y a f f e c t e d by t h e i r w a t e r c o n t e n t ( i c e c o n t e n t ) . The s t r e n g t h i s a t a maximum a t a c e r t a i n water c o n t e n t c h a r a c t e r i s t i c of a g i v e n s o i l ( F i g u r e 2 ) .

I n problems concerning t h e s t a t e of t h e r m a l s t r e s s i n f r o z e n s o i l , t h e s t r e n g t h of s o i l and c o n d i t i o n s l e a d i n g t o i t s f a i l u r e under v a r i a b l e s t a t i c l o a d s a r e e s p e c i a l l y i m p o r t a n t . T h i s was i n v e s t i g a t e d t h e o r e t i c a l l y by Kachanov

(1961, 1967), who p o s t u l a t e d a g e n e r a l t h e o r y of f a i l u r e under c o n d i t i o n s of c r e e p f o r any m a t e r i a l s , and by Vyalov (1968) who c a r r i e d out a t h e o r e t i c a l s t u d y of t h e long-term s t r e n g t h of f r o z e n s o i l s under c o n d i t i o n s of v a r i a b l e l o a d s and temperature. E v i d e n t l y t h i s problem was n o t s t u d i e d e x p e r i m e n t a l l y .

According t o Vyalov (1968), t h e s t r e n g t h of f r o z e n s o i l s s u b j e c t e d t o v a r i a b l e s t a t i c l o a d s i s expressed a s f o l l o w s :

where

B

and B a r e c o n s t a n t s , t i s t h e time p r i o r t o f a i l u r e , and ~ ( t ) i s t h e P

s t r e s s which v a r i e s w i t h time.

I f t h e change i n ~ ( t ) i s known, t h e n t c a n b e determined from e q u a t i o n P

(1.20). I n p a r t i c u l a r , i f t h e change

i n

~ ( t ) i s g i v e n a s

i t f o l l o w s from e q u a t i o n (1.20) t h a t t i s d e s c r i b e d by t h e f o l l o w i n g e x p r e s s i o n : P

where

t,=Be

i s t h e t i m e p r i o r t o f a i l u r e under a c o n s t a n t l o a d T I .

To d e t e r m i n e t h e a p p l i c a b i l i t y of e q u a t i o n (1.20), t h i s a u t h o r c a r r i e d o u t a s e r i e s of experiments. The e x p e r i m e n t a l m a t e r i a l w a s k a o l i n c l a y c o n t a i n i n g

30% water a t - 3 " ~ . I t s t e n s i l e s t r e n g t h was: i n s t a n t a n e o u s 8.5 kg/cm2, long- term 1 . 2 kg/cm2. The specimens were s u b j e c t e d t o a t e n s i l e s t r e s s of 2.2 kg/cm2, which was k e p t c o n s t a n t f o r a c e r t a i n p e r i o d of t i m e and t h e n r a p i d l y i n c r e a s e d

(15)

( b u t w i t h o u t impact) u n t i l f a i l u r e o c c u r r e d . I n o t h e r words, we determined t h e temporary t e n s i l e s t r e n g t h a f t e r keeping t h e samples under a l o a d of 2.2 kg/cm2. The r e s u l t s of t h e s e experiments a r e shown i n T a b l e 11.

A s may b e s e e n from T a b l e 11, t h e temporary s t r e n g t h d e c r e a s e s v e r y l i t t l e i f a specimen i s preloaded f o r a s h o r t p e r i o d of t i m e , b u t l a t e r , a s t h e pre- l o a d i n g t i m e i n c r e a s e s and e v e n t u a l l y becomes e q u a l t o t h e t i m e t o f a i l u r e a t

0 = 2.2 kg/cm2, t h e temporary s t r e n g t h b e g i n s t o d e c r e a s e e x p o n e n t i a l l y .

S i n c e t h e l o a d was a p p l i e d Fn accordance w i t h e q u a t i o n ( 1 . 2 1 ) , t h e d e c r e a s e i n t h e temporary s t r e n g t h ( t h e s t r e s s T2) should have been d e s c r i b e d by e q u a t i o n (1.22) p r o v i d i n g t l = t (because i n t h e experiments t h e l o a d T 2 l e d t o f a i l u r e

P

a s soon a s t h e t i m e t 1 , i. e . , t h e time of a p p l y i n g t h e l o a d T i = 2.2 kg/cm2, was up). However, a s i n d i c a t e d by e q u a t i o n ( 1 . 2 2 ) , i t i s i m p o s s i b l e t o d e t e r m i n e

T 2 under such c o n d i t i o n s . Moreover, i n accordance w i t h e x p r e s s i o n ( 1 . 2 2 ) , no m a t t e r how g r e a t t h e second l o a d i n g s t a g e (K -+ a ) , t h e time t o f a i l u r e w i l l always b e l o n g e r t h a n t l . Consequently, t h e b a s i c e q u a t i o n (1.20) c o n t a i n s some

d i s c r e p a n c i e s , t h e e x t e n t of which must b e determined e x p e r i m e n t a l l y .

To d e t e r m i n e t h e t i m e t o f a i l u r e , Kachanov (1961) i n t r o d u c e d t h e "damage" f u n c t i o n

Ji

which i s e q u a l t o 1 when m a t e r i a l i s undamaged, and t o 0 when i t h a s f a i l e d . He suggested t h e f o l l o w i n g e q u a t i o n r e l a t i n g $ t o t h e s t r e s s : where A and n a r e c o n s t a n t s c h a r a c t e r i s t i c of a g i v e n m a t e r i a l . I n t h e c a s e of t o t a l f a i l u r e , )I = 0 and t h e t i m e t o f a i l u r e i s determined from t h e f o l l o w i n g e x p r e s s i o n : I f t h e l o a d s v a r y i n accordance w i t h ( 1 . 2 2 ) , i t f o l l o w s from (1.24) t h a t It f o l l o w s from t h e l a s t e x p r e s s i o n t h a t a t t l = t

,

K -+ a , i . e . , t h e P

(16)

t e m p o r a r y s t r e n g t h d o e s n o t d e c r e a s e , and r e m a i n s i n d e f i n i t e l y l a r g e i r r e s p e c t i v e of t h e p r e l o a d i n g t i m e . A s may b e s e e n from T a b l e 11, t h i s t o o c o n t r a d i c t s t h e e x p e r i m e n t a l d a t a . We s h o u l d n o t e t h a t t h e problem of t h e s t r e n g t h o f f r o z e n s o i l s s u b j e c t e d t o v a r i a b l e s t a t i c l o a d s e v i d e n t l y r e m a i n s u n s o l v e d . I t i s a v e r y i m p o r t a n t problem, e s p e c i a l l y i n t h e c a s e of t h e r m a l s t r e s s e s ( w h i c h , i n c i d e n t l y , may a l s o c h a n g e s i g n s ) . Hence, f u r t h e r i n v e s t i g a t i o n s a r e e s s e n t i a l . T a b l e I S t r e s s r e l a x a t i o n d u r i n g t e n s i o n (sand c o n t a i n i n g 20% w a t e r , t e m p e r a t u r e -3°C) Note: 0 ₌ 1.8 _{kg/cm2 d e t e r m i n e d i n c r e e p t e s t s .}

-

_Wr I n i t i a l s t r e s s S t r e s s 0 , kg/cm2

a,,

kg/cm2

t

A f t e r A f t e r ( ~ f t e r A f t e r No. o f t e s t s

(17)

T a b l e I1 T e n s i l e t e s t s on f r o z e n k a o l i n c l a y c o n t a i n i n g 30% w a t e r a t - 3 ' ~

-

- - .- - - - .- . - .. P r e l o a d i n g t i m e ,

a

= 2 . 2 kg/cm2, h r Temporary s t r e n g t h a £ t e r p r e l o a d i n g

a

= 2 . 2 kg/cm2 No. of t e s t s F i g . 1 Long-term s t r e n g t h of f r o z e n s u g l i n o k ( a ) , a n d f r o z e n s a n d ( b ) : 1

-

u n d e r c o m p r e s s i o n , 2

-

u n d e r t e n s i o n F i g . 2 C o m p r e s s i v e s t r e n g t h of f r o z e n s o i l a s a f u n c t i o n of i t s t o t a l w a t e r c o n t e n t ( a f t e r T s y t o v i c h ) 1

-

s a n d ; 2

-

s u p e s ; 3

-

c l a y ; 4

-

s i l t y c l a y

(18)

Chapter I1

THERMAL DEFORMATION OF FROZEN SOILS

1. General

There have been v e r y few q u a n t i t a t i v e s t u d i e s of t h e r m a l d e f o r m a t i o n s and s t r e s s e s i n f r o z e n s o i l s . There a r e two r e a s o n s f o r t h i s . F i r s t l y , i t i s d i f f i c u l t t o c a r r y out e x p e r i m e n t a l i n v e s t i g a t i o n s of t h e s e phenomena under n a t u r a l c o n d i t i o n s . Secondly, t h e r m a l d e f o r m a t i o n s and s t r e s s e s i n f r o z e n s o i l s a r e o f t e n r e g a r d e d i n our c o n s t r u c t i o n p r a c t i c e s i n t h e North and N o r t h e a s t a s phenomena of secondary importance, a l t h o u g h t h e r e i s no j u s t i f i c a t i o n f o r t h i s . I t i s known from e x p e r i e n c e t h a t t h e r m a l d e f o r m a t i o n , and e s p e c i a l l y development of f r o s t f i s s u r e s , may r e s u l t i n t o t a l d e s t r u c t i o n of e n g i n e e r i n g s t r u c t u r e s

(Grechishchev, Zhigul' s k i i , 1969).

Experimental s t u d i e s of t h e r m a l d e f o r m a t i o n of f r o z e n s o i l s were i n i t i a t e d by P.V. Shvetsov and I . N . Votyakov (1963, 1966) a t t h e P e r m a f r c s t I n s t i t u t e i n Yakutsk. These s t u d i e s showed t h a t f r o z e n s o i l s have an abnormally l a r g e

c o e f f i c i e n t of t h e r m a l expansion between O O C and

-loOc,

i . e . , w i t h i n a t e m p e r a t u r e

r a n g e which i s important i n p r a c t i c e . For example, t h e c o e f f i c i e n t of thermal expansion of some c l a y s o i l s (Votyakov, 1966) i s about 300 x I l d e g between -1. O"C and - 0 . 4 " ~ , and a b o u t 100 x I / d e g between - 8 . 0 ' ~ and - 3 . 6 ' ~ . L e t u s r e c a l l t h a t t h e l a r g e s t c o e f f i c i e n t e v e r recorded i s 100 x I l d e g

(capron*)

.

Normal t h e r m a l expansion c o e f f i c i e n t s of b u i l d i n g m a t e r i a l s a r e

1 4 x I l d e g ( c o n c r e t e ) and 1 2 x 1/deg ( s t e e l ) . According t h e Votyakov, t h e c o e f f i c i e n t of i c e , which i s one of t h e components of f r o z e n s o i l , i s

35 t o 50 x I / d e g .

Furthermore, t h e c o e f f i c i e n t of t h e r m a l expansion of f r o z e n s o i l i s v e r y s t r o n g l y dependent on t h e t e m p e r a t u r e ; i t d e c r e a s e s almost e x p o n e n t i a l l y w i t h a n i n c r e a s e i n t h e a b s o l u t e temperature. Consequently, t h e dependence of t h e r m a l

(19)

d e f o r m a t i o n on t h e i n i t i a l and f i n a l s o i l t e m p e r a t u r e i s d i s t i n c t l y n o n l i n e a r .

A l l t h e s e f a c t s i n d i c a t e t h a t t h e mechanism of t h e r m a l d e f o r m a t i o n and s t r e s s e s i n f r o z e n s o i l s i s more complex t h a n i n o t h e r m a t e r i a l s . T h i s may b e e x p l a i n e d by t h e multicomponent s t r u c t u r e of f r o z e n s o i l s and t h e p h y s i c a l p r o p e r t i e s of p h a s e components: t h e c o l l o i d a l p r o p e r t i e s of m i n e r a l p a r t i c l e s and t h e s p e c i f i c c h a r a c t e r i s t i c s of u n f r o z e n w a t e r f i l m s .

A t t e m p t s t o p o s t u l a t e a g e n e r a l t h e o r y of t h e r m a l e x p a n s i o n of two-

component systems were made by V.M. Levin ( 1 9 6 7 ) , S.D. Volkov (1968) and o t h e r s . However, e q u a t i o n s d e r i v e d by t h e s e a u t h o r s c a n n o t b e a p p l i e d t o f r o z e n f i n e - g r a i n e d c l a y s o i l s , s i n c e t h e s e f o r m u l a e do n o t a c c o u n t f o r t h e r h e o l o g i c a l p r o p e r t i e s , t h e i n t e r n a l phase t r a n s i t i o n s of u n f r o z e n w a t e r t o i c e , and t h e s t r u c t u r a l c h a r a c t e r i s t i c s of f r o z e n s o i l s . N e v e r t h e l e s s , t h e y may b e u s e f u l i n t h e c a s e of low-temperature f r o z e n c o a r s e - g r a i n e d and sandy s o i l s , which i n t h e f i r s t a p p r o x i m a t i o n may b e r e g a r d e d a s two-component systems ( m i n e r a l

p a r t i c l e s and ice-cement). However, t h e r e i s s t i l l no e x p e r i m e n t a l c o n f i r m a t i o n f o r a l l t h i s .

The c o m p l e x i t y of p r o c e s s e s t a k i n g p l a c e w i t h i n f r o z e n f i n e - g r a i n e d s o i l s d u r i n g t h e r m a l e x p a n s i o n c a n o n l y b e g u e s s e d a t . For example, Votyakov (1966) s u g g e s t s t h a t , t o g e t h e r w i t h t h e phase t r a n s i t i o n of w a t e r f i l m s , t h e physico- c h e m i c a l c o a g u l a t i o n - p e p t i z a t i o n t r a n s i t i o n and i n t e r n a l s t r u c t u r a l t r a n s f o r - m a t i o n s a r e a l s o e x t r e m e l y i m p o r t a n t . Votyakov came t o t h i s c o n c l u s i o n

b e c a u s e s o i l s t e s t e d by him a t "high" t e m p e r a t u r e s (about - 1 ' ~ ) r e t a i n e d a p o s i t i v e c o e f f i c i e n t of t h e r m a l e x p a n s i o n , i . e . , t h e t e m p e r a t u r e i n c r e a s e was accompanied by s o i l e x p a n s i o n r a t h e r t h a n by t h e e x p e c t e d c o n t r a c t i o n due t o a p a r t i a l t r a n s i t i o n of i c e t o l i q u i d w a t e r .

On t h e b a s i s of contemporary i d e a s on t h e p h y s i c s of f r o z e n s o i l s , we may assume t h a t a change i n t h e volume of f r o z e n s o i l s d u r i n g t e m p e r a t u r e changes i s accompanied by s e v e r a l d i f f e r e n t i n t e r n a l p r o c e s s e s proceeding i n d i f f e r e n t d i r e c t i o n s . These a r e : 1) c o n v e n t i o n a l expansion o r c o n t r a c t i o n of s o i l components; 2 ) c o n t i n u o u s e q u i l i b r i u m t r a n s i t i o n s a t t h e i n t e r f a c e " i c e

-

u n f r o z e n w a t e r " , which have s i g n s o p p o s i t e t o t h o s e of e x p a n s i o n o r c o n t r a c t i o n ; and 3 ) t h e r m a l d e f o r m a t i o n of i n t e r n a l m i c r o s t r u c t u r e s , which i s r e l a t e d t o t h e a f o r e m e n t i o n e d p r o c e s s e s b u t t h e s i g n of which i s d i f f i c u l t t o d e t e r m i n e , s i n c e

(20)

i t i s t o some e x t e n t a r e s u l t a n t of "normal" t h e r m a l expansion and i n t e r n a l phase t r a n s i t i o n s . E v i d e n t l y i t was Shvetsov (1958) who f i r s t p a i d a t t e n t i o n t o t h e p e c u l i a r p h y s i c a l a s p e c t s of t h e r m a l expansion of f r o z e n s o i l s .

C o n s i d e r i n g a l s o t h a t a l l components of f r o z e n s o i l a r e i n a m e c h a n i c a l i n t e r a c t i o n w i t h each o t h e r ( e . g . , t h e i n t e r n a l p h a s e t r a n s i t i o n s should r e s u l t

i n a c o n s i d e r a b l e h y d r o s t a t i c p r e s s u r e t r a n s m i t t e d by w a t e r f i l m s t o m i n e r a l p a r t i c l e s and i c e ) , we must a g a i n c o n c l u d e t h a t t h e mechanism of t h e r m a l d e f o r m a t i o n i s v e r y complex and t h a t i t i s d i f f i c u l t t o p o s t u l a t e even a n

a p p r o x i m a t e t h e o r y . I n t h e meantime, we c a n m e r e l y p r o v i d e a g e n e r a l d e s c r i p t i o n of t h e p r o c e s s based on a v a i l a b l e e x p e r i m e n t a l d a t a .

2 . Some E x p e r i m e n t a l Data on Thermal Deformation of Frozen S o i l s

We s t c d i e d changes i n t h e r m a l d e f o r m a t i o n of f r o z e n s o i l samples under c o n d i t i o n s of f r e e e l o n g a t i o n , i . e . , i n t h e a b s e n c e of e x t e r n a l l o a d s . The e x p e r i m e n t s were c a r r i e d o u t by Yu.B. S h e s h i n under t h e s u p e r v i s i o n of t h e a u t h o r i n c o l d chambers a t t e m p e r a t u r e s below O O C . Deformation was r e c o r d e d

by c l o c k - t y p e ICh-3 i n d i c a t o r s l o c a t e d a t b o t h ends of t h e specimen. The specimens were p r e p a r e d i n a s t e e l frame and had t h e s h a p e of a p a r a l l e l e p i p e d

( c r o s s s e c t i o n 30 x 30 mm2, l e n g t h 380 mm). The w a l l s of t h e specimens were i n s u l a t e d w i t h a p o l y e t h y l e n e l a y e r c o a t e d w i t h g r e a s e f o r r e l i a b l e p r o t e c t i o n a g a i n s t w a t e r . The t e m p e r a t u r e was measured by a thermocouple l o c a t e d i n t h e c e n t r e of t h e specimen.

A t t h e s t a r t of each e x p e r i m e n t , t h e t e m p e r a t u r e of specimens was above

O'C and no d a t a were r e c o r d e d u n t i l a l l p h a s e t r a n s i t i o n s i n t h e specimens were complete (which c o u l d b e e a s i l y f o l l o w e d from t h e r e a d i n g s of t h e thermo- c o u p l e ) . The f i r s t r e a d i n g of t h e i n d i c a t o r was t h e n t a k e n . T h i s p r o c e d u r e was adopted t o s e p a r a t e (however r o u g h l y ) heaving of t h e s o i l (which i s

e s p e c i a l l y pronounced d u r i n g p h a s e t r a n s i t i o n s ) from t r u e t h e r m a l d e f o r m a t i o n . Readings were t h e n t a k e n e v e r y hour f o r f i v e h o u r s . Once t h e t e m p e r a t u r e of a specimen r e a c h e d t h a t i n t h e c o l d chamber, r e a d i n g s were t a k e n o n c e a day. The t o t a l d u r a t i o n of each experiment a t c o n s t a n t t e m p e r a t u r e was 1 5 t o 20 d a y s . The e x p e r i m e n t s were d i s c o n t i n u e d when d e f o r m a t i o n s of specimens were s t a b i l i z e d

(21)

We t e s t e d t h r e e t y p e s of s o i l : k a o l i n c l a y c o n t a i n i n g 222, 29% and 37% w a t e r ; medium-grained s a n d c o n t a i n i n g 6 . 5 2 , 1 0 % and 1 7 % w a t e r ; and a m i x t u r e of 70% c l a y and 30% sand c o n t a i n i n g 9%, 14% and 26% w a t e r . T y p i c a l d e f o r m a t i o n c u r v e s a r e shown i n F i g u r e 3. The s h a p e of t h e c u r v e s i n d i c a t e s t h a t s t a b i l i z a t i o n of t h e r m a l d e f o r m a t i o n s i n f r o z e n s o i l s o c c u r s r a t h e r s l o w l y f o r s e v e r a l d a y s a f t e r t h e t e m p e r a t u r e of s p e c i m e n s becomes c o n s t a n t . F i n a l d e f o r m a t i o n ( i n d i c a t e d by E~ i n F i g u r e 3) i s c o n s i d e r a b l y g r e a t e r t h a n t h a t d e v e l o p e d i n t h e s p e c i m e n p r i o r t o t h e s t a b i l i - z a t i o n of i t s t e m p e r a t u r e . F i g u r e

4

shows E~ a s a f u n c t i o n of t h e f i n a l t e m p e r a t u r e a n d a l s o t h e d a t a o n t h e r m a l e x p a n s i o n of f r o z e n s u p e s t a k e n f r o m a p a p e r by Votyakov and G r e c h i s h c h e v ( 1 9 6 9 ) . The c u r v e s i n F i g u r e

4

w e r e u s e d t o i n v e s t i g a t e t h e r m a l e x p a n s i o n a s a f u n c t i o n of t h e w a t e r c o n t e n t a n d t h e g r a i n - s i z e c o m p o s i t i o n ( F i g u r e s 5 a n d 6 ) . I n c o n t r a s t t o F i g u r e 4 , w h e r e E~ s t a n d s f o r t h e r m a l d e f o r m a t i o n , t h e l a t t e r i v F i g u r e s 5 a n d 6 i s c h a r a c t e r i z e d by t h e c o e f f i c i e n t of l i n e a r e x p a n s i o n s urn, which was c a l c u l a t e d a s a d e r i v a t i v e of t h e c u r v e s i n F i g u r e 4.

3 . Main P a t t e r n s of Thermal D e f o r m a t i o n of F r o z e n S o i l s T u r n i n g t o t h e a n a l y s i s of t h e e x p e r i m e n t a l d a t a d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n , we s h o u l d f i r s t of a l l n o t e o n e v e r y i m p o r t a n t phenomenon, i . e . , t h e f a c t t h a t t h e r m a l d e f o r m a t i o n c o n t i n u e s t o d e v e l o p f o r a c o n s i d e r a b l e t i m e a f t e r t h e t e m p e r a t u r e of a s p e c i m e n becomes c o n s t a n t . T h i s phenomenon, f i r s t d i s c o v e r e d by Votyakov, i s e v i d e n t l y r e l a t e d t o t h e r h e o l o g i c a l p r o p e r t i e s of f r o z e n s o i l s , a s w e l l a s s l o w t r a n s f o r m a t i o n of t h e i r t e x t u r e which l a g s b e h i n d t h e more r a p i d t e m p e r a t u r e c h a n g e s . A s a r e s u l t of t h i s , t h e s t a b i l i z a t i o n of t h e b u l k of f r o z e n s o i l e x t e n d s o v e r a p e r i o d o f time. T h i s phenomenon was c a l l e d t h e

thermal aftereffect

(Votyakov, G r e c h i s h c h e v , 1969) by a n a l o g y w i t h t h e a f t e r e f f e c t which m a n i f e s t s i t s e l f i n t h e c r e e p of f r o z e n s o i l s a n d i s t h e r e s u l t of t h e i r r h e o l o g i c a l p r o p e r t i e s .

E v i d e n t l y t h i s t h e r m a l a f t e r e f f e c t c a n m a n i f e s t i t s e l f n o t o n l y i n f r o z e n s o i l s . An a b s t r a c t model of t h e r m a l e x p a n s i o n of a c o m p o s i t e m a t e r i a l c o n s i s t i n g

(22)

of v i s c o - e l a s t i c m a t r i x (cementing m a t e r i a l ) w i t h i d e a l l y r i g i d i n c l u s i o n s c a n s e r v e a s a t h e o r e t i c a l model of t h e a f t e r e f f e c t . Thermal d e f o r m a t i o n i n a n a b s t r a c t two-component model was c a l c u l a t e d by Levin (1967).

I n a c c o r d a n c e w i t h t h e e q u a t i o n d e r i v e d by L e v i n , t h e r m a l d e f o r m a t i o n of such a model i n c r e a s e s w i t h t i m e , and t h e n a t t e n u a t e s e x p o n e n t i a l l y . T o t a l d e f o r m a t i o n a f t e r s t a b i l i z a t i o n ( E ~ ) i s e x p r e s s e d a s f o l l o w s :

where

a2

_and_{C 2}_{a r e t h e c o e f f i c i e n t s of expansion and c o n c e n t r a t i o n of cementing}

m a t e r i a l r e s p e c t i v e l y . T h i s i s t h e t o t a l l i n e a r d e f o r m a t i o n of t h e m a t r i x m a t e r i a l c o n t a i n e d i n u n i t volume of a two-component model.

Frozen sand and p e b b l e s a r e two t y p e s of f r o z e n s o i l which a r e d e s c r i b e d b e s t w i t h t h e h e l p of a two-component model. However, i t i s i n t e r e s t i n g t h a t i n our e x p e r i m e n t s w i t h f r o z e n sand t h e t h e r m a l a f t e r e f f e c t was t r i v i a l . S i n c e i n L e v i n ' s model t h e development of t h e r m a l d e f o r m a t i o n s w i t h t i m e i s r e l a t e d t o t h e r e l a x a t i o n of m i c r o s t r e s s e s o n l y , t h i s i n d i c a t e s t h a t i n t e r n a l s t r u c t u r a l r e a r r a n g e m e n t s and p h a s e t r a n s i t i o n s a t t h e "unfrozen w a t e r

-

i c e " i n t e r f a c e p r e v a i l o v e r t h e i n t e r n a l c r e e p which i s r e l a t e d t o m i c r o s t r e s s e s .

F i g u r e

4

shows a l a r g e v a r i e t y of " s t a b i l i z e d d e f o r m a t i o n

-

t emperatur el' c u r v e s . F i r s t l y , t h e y a r e d i s t i n c t l y n o n l i n e a r . F a i r l y s t e e p c u r v e s a r e c h a r a c t e r i s t i c of t e m p e r a t u r e s c l o s e t o O O C . The maximums o c c u r w i t h i n t h e

f i r s t few d e g r e e s below O'C, and t h e c u r v e s l e v e l o u t a s t h e t e m p e r a t u r e f a l l s . Secondly, t h e s i g n o f t h e r m a l d e f o r m a t i o n ( c o n t r a c t i o n o r expansion) depends on t h e w a t e r c o n t e n t and t h e g r a i n - s i z e c o m p o s i t i o n of t h e s o i l . For example, k a o l i n c l a y c o n t a i n i n g 22% and 29% w a t e r undergoes c o n s i d e r a b l e c o n t r a c t i o n i n volume a s t h e t e m p e r a t u r e d r o p s from 0' t o - 2 ' ~ ( p o s i t i v e t h e r m a l d e f o r m a t i o n ) . The same c l a y c o n t a i n i n g 37% w a t e r expands w i t h i n t h e same t e m p e r a t u r e r a n g e . Supes expands w i t h i n any t e m p e r a t u r e r a n g e ( i f t h e i n i t i a l t e m p e r a t u r e i s oOC), w h i l e a m i x t u r e of c l a y and sand c o n t r a c t s , e t c . T h i r d l y , t h e s i g n of t h e r m a l d e f o r m a t i o n f o r t h e same s o i l b u t a t d i f f e r e n t f i n a l t e m p e r a t u r e s may d i f f e r . For example, sand c o n t a i n i n g 6.5% w a t e r expands up t o - 3 ' ~ and c o n t r a c t s a t lower t e m p e r a t u r e s .

(23)

a ~ ~ / 3 0 )

o n t h e w a t e r c o n t e n t i s e q u a l l y n o t e w o r t h y . As may b e s e e n from F i g u r e 5 , t h e r m a l d e f o r m a t i o n s h a v e a maximum a t d e f i n i t e w a t e r c o n t e n t s : f o r c l a y a b o u t 302, and f o r s a n d a b o u t 1 0 % . A t w a t e r c o n t e n t s g r e a t e r t h a n t h e n o r m a l v a l u e , d e f o r m a t i o n s d e c r e a s e and may e v e n a c q u i r e a n e g a t i v e s i g n . T h i s i n d i c a t e s t h a t a t low w a t e r c o n t e n t s s o i l w i l l c o n t r a c t w i t h f a l l i n g t e m p e r a t u r e , b u t a t h i g h w a t e r c o n t e n t s t h e same s o i l w i l l expand ( l e t u s r e c a l l t h a t , i n o u r m e t h o d , h e a v i n g d u r i n g p h a s e t r a n s i t i o n s was n o t c o n s i d e r e d ) . A l s o , s i n c e w a t e r c o n t e n t a n d c o n s o l i d a t i o n o f c l a y s o i l s a r e i n t e r r e l a t e d , we may c o n c l u d e t h a t w e a k l y c o n s o l i d a t e d s o i l s must c o n t r a c t w i t h f a l l i n g t e m p e r a t u r e , w h i l e c o n s o l i d a t e d s o i l s must expand. T h i s c o n c l u s i o n i s i m p o r t a n t f o r f o r e c a s t s of t h e r m a l s t r e s s e s a n d d e f o r m a t i o n s i n embankments and dams b u i l t of l o c a l m a t e r i a l s . However, i t must b e c h e c k e d e x p e r i m e n t a l l y .

A s may b e s e e n from F i g u r e 6 , which i s b a s e d o n t h e e x p e r i m e n t a l d a t a

o b t a i n e d by t h i s a u t h o r , t h e r m a l d e f o r m a t i o n s i n c l a y s o i l s a r e q u i t e c o n s i d e r a b l e a n d d e c r e a s e e x p o n e n t i a l l - y w i t h a n i n c r e a s e i n t h e s a n d f r a c t i o n . S i m p l e c a l c u l a t i o n s (Votyakov, G r e c h i s h c h e v , 1969) showed t h a t t h e d e p e n d e n c e of t h e r m a l d e f o r m a t i o n on t h e t e m p e r a t u r e w i t h i n t h e i m p o r t a n t r a n g e O O C t o -28' t o - 3 0 ' ~ c a n b e e x p r e s s e d i n t h e f o l l o w i n g n o n l i n e a r form: where : W i s t h e w a t e r c o n t e n t ,

~,d,,8.,8~

a r e e m p i r i c a l c o e f f i c i e n t s , t i s t h e c o e f f i c i e n t of e x p a n s i o n d u r i n g f r e e z i n g of w a t e r , P a l P r n a r e t h e d e n s i t i e s of w a t e r a n d t h e s o i l s k e l e t o n , r e s p e c t i v e l y , a n d O K ,

8

_{a r e t h e f i n a l and t h e i n i t i a l t e m p e r a t u r e s , r e s p e c t i v e l y .} H We s h o u l d n o t e t h a t i n a l l e q u a t i o n s g i v e n a b o v e , E i s t a k e n a s p o s i t i v e d u r i n g e x p a n s i o n , and n e g a t i v e d u r i n g c o n t r a c t i o n . I n a s e r i e s of e x p e r i m e n t s Votyakov and G r e c h i s h c h e v (1969) a n a l y z e d t h e d e p e n d e n c e of 8 0 , 81, Bo and B I on t i m e and t h e w a t e r c o n t e n t . They c o n c l u d e d t h a t t h e f o l l o w i n g e m p i r i c a l e q u a t i o n s d e s c r i b e t h e e x p e r i m e n t a l d a t a b e s t :

(24)

where Born, B l m , a r e t h e v a l u e s of c o e f f i c i e n t s a f t e r s t a b i l i z a t i o n of d e f o r m a t i o n s w i t h t i m e ( a s a p p r o x i m a t i o n s a t t + a ) ; T o , T I , V o , V l a r e e m p i r i c a l c o e f f i c i e n t s which have t h e d i m e n s i o n a l i t y of t i m e , and X o , X I ,

B o ,

61 a r e e m p i r i c a l c o e f f i c i e n t s .

The same a u t h o r s noted t h a t B o and B1 o n l y a r e dependent on t h e w a t e r c o n t e n t . A s t h e w a t e r c o n t e n t i n c r e a s e s , B o d e c r e a s e s w h i l e B1 i n c r e a s e s , and Bo depends on t h e w a t e r c o n t e n t more s t r o n g l y t h a n B 1 . A l l t h i s f o l l o w s from e q u a t i o n (2.4)

.

I t i s v e r y d i f f i c u l t t o u s e t h e p r e c i s e e q u a t i o n ( 2 . 2 ) , i f t h e c o e f f i c i e n t s i n c o r p o r a t e d i n i t a r e time-dependent i n a c c o r d a n c e w i t h e q u a t i o n ( 2 . 5 ) .

T h e r e f o r e , t h i s a u t h o r h a s s i m p l i f i e d e q u a t i o n (2.2) c o n s i d e r i n g t h a t d e p e n d e n c i e s 80 (t) and 8 1 ( t ) i n e q u a t i o n (2.5) a r e n o t s t r o n g ( e x c e p t t h e r e g i o n t = 01,

s i n c e

60

and

B1

a r e p r a c t i c a l l y g r e a t e r t h a n u n i t y . F u r t h e r m o r e , we may assume t h a t i n e q u a t i o n ( 2 . 5 ) , T O

'

~1 and

X o

-

X 1

-

1. With a l l o w a n c e s f o r a l l t h i s , we c a n r e w r i t e e q u a t i o n ( 2 . 2 ) i n t h e f o l l o w i n g s i m p l i f i e d form, which n e v e r t h e l e s s r e t a i n s a l l e s s e n t i a l p o i n t s of t h e b a s i c e q u a t i o n :

t ~r(t)-O(8r(,8n),

where

O(@~,B.)=B,[(~&

-

A

f g

)-(e-h

-

ne-k!];

~ ( q

=

1-

e-!a

;

and Bm, 0 0 , 8 1 ,

X

and t o a r e f r o z e n s o i l c o n s t a n t s .

On t h e b a s i s of e q u a t i o n (2.4) and t h e a f o r e m e n t i o n e d s i m p l i f i c a t i o n s , t h e dependence of Brn and

X

on t h e w a t e r c o n t e n t c a n now b e r e p r e s e n t e d a s f o l l o w s :

(25)

I f t h e i n i t i a l t e m p p r a t u r e of a sample i s O ' C , t h e f u n c t i o n @(6

_OH),

K

'

which r e f l e c t s t h e dependent.e of km on t h e t e m p e r a t u r e , w i l l assume t h e f o l l o w i n g form:

Q u a l i t a t i v e e x a c , l n d t i o r c q r i l ~ i : i ( 2 . 1 0 ) shnws t h a t i n p r i n c i p l e i t c a n

r e f l e c t t h e e n t i r e spectrunl :f Irvcis

.

L e t u s s i n g l e o u t t h e most t y p i c a l c u r v e s ( F i g u r e 7 )

.

i; i m p l e ni f i t a t i o r ~ ot e q u a t i o n (2.10) makes i t p o s s i b l e t o s u g g e s t t h e *,:L >wir,g c l a s i i f i c a t i ~ i r i o f t h e s e c u r v e s :

1.

a ) @ ~ . , i

, the

.:...

as r x.::c7 iLm i n <:pansion r e g i o n a t

fl&=&,h

A 0. 9

w h i l e a t 6,

-&$

, t r er i h c c . b v l t r 3 ~ t i n : :gion.

b )

q c

**I * ,**

t 1 7 l ~ i i m 4 cl e n t i r e l y w i t h i n t h e

r e g i o n of e x p a n s i o n .

2 , ' > l

a )

e>i

,

t h c

.

L L ~

.I

.JI) .i i n ~ l n i . I m '*t

0,=&-b#~

and l i e s e n t i r e l y

w i t h i n t h e r e g i o n of c o n t t ac t i o : i .

b ) 1,

,

t h e .;me has no maximum and l i e s e n t i r e l y w i t h i n t h e r e g i o n of c o n t r a c t i o ~ ,

For p r a c t i c a l purp--,t.,-. t 2 ir;,pcr t a n t t o g e n e r a l i z e e q u a t i o n ( 2 . 6 ) f o r t h e c a s e of temperatu:-,a , . I

I

..rl,

.

L.. t h tine. I n t h e a b s e n c e of e x p e r i m e n t a l

d a t a , t h i s c a n be done on t h e b , ~ s i s sf l c g i c a l on, i d e r a t i o n s a p p l y i n g t h e

p r i n c i p l e of summ'it i . I o r i . ! i v :,iuai el i r c t s , c h ~ a r y w i t h t i m e . I n a c c o r d a n c e w i t h t h i s p r i n c i p l e , w t i i , h w l d e l ) u::?d In c r e e p r h e o r y , t h e f o l l o w i n g more g e n e r a l e x p r e s s i o n c a n b e , I : t a i l ~d i n s t e a d oL , q t . , , r Lan ( 2 . 6 ) :

where K(t

-

'I) i s t h e lur?et 11 l r t . . ' t l i c : m a i , ! f f e c t s ,

(26)

Function K(t

-

T) c a n b e e a s i l y o b t a i n e d by s u b s t i t u t i n g

O K

= c o n s t i n t o e q u a t i o n (2.11) and comparing i t t o e x p r e s s i o n ( 2 . 6 ) . Then:

F i n a l l y , l e t u s a p p l y e q u a t i o n (2.121 t o t h e c a s e of r e l a x a t i o n of thermal s t r e s s e s ( f o r e x p e r i m e n t a l d a t a s e e F i g u r e 8 ) . The t h e o r e t i c a l s o l u t i o n , w i t h allowances f o r e q u a t i o n (2.11) and i f t h e s o i l i s simulated by a Bingham-

Shvedov v i s c o - e l a s t i c - p l a s t i c medium, i s a s f o l l o w s ( l e a v i n g o u t t h e d e t a i l s of t h i s s i m p l e s o l u t i o n :

and a t

6J

>

6;n

( t

at,)

(2.14)

7

where

dUI

and

6"'

a r e t h e s t r e s s e s i n t h e e l a s t i c and t h e v i s c o p l a s t i c s t a g e s , E i s t h e d e f o r m a t i o n modulus,

T i s t h e r e l a x a t i o n time, P

0 i s t h e u l t i m a t e long-term s t r e n g t h ,

and t I i s t h e t i m e of t r a n s i t i o n from t h e e l a s t i c t o t h e v i s c o p l a s t i c s t a g e (found from

oYnP

=

am).

It f o l l o w s from e q u a t i o n (2.14) t h a t s t r e s s e s a t t h e moment of t r a n ~ i t f u n from t h e e l a s t i c t o t h e p l a s t i c s t a g e a t t = t l a r e e q u a l t o U

_m'

They )lave a

T &

maximum a t

tmt,-hk*

and t h e n r e l a x a t t - + Y D t o 0 i . e . , t h e c u r v e i s i n

m y

good agreement w i t h experimental d a t a i n F i g u r e 8. The d c t t e d c u r v e i n F i g u r e

8 i s t h e e x p e r i m e n t a l c u r v e of nornril r e l a x a t i o n of t h e same s o i l a t t h e same t e m p e r a t u r e but a t a n i n i t i a l s t r e s s c l o s e t o t h e maximum on t h e t h e r m a l r e l a x a t i o n c u r v e . Comparison of t h e two c u r v e s i n d i c a t e s t h a t t h e t h e r m a l a f t e r e f f e c t p l a y s a n i m p o r t a n t r o l e i n t h e development of s t r e s s e s w i t h time. L e t u s n o t e i n c o n c l u s i o n t h a t t h e g e n e r a l i z e d e q u a t i o n (2.11) should b e used i n p r a c t i c a l c a l c u l a t i o n s , s i n c e i t r e f l e c t s t h e main c h a r a c t e r i s t i c s of t h e r m a l d e f o r m a t i o n of f r o z e n s o i l more f u l l y t h a n o t h e r e q u a t i o n s .

(27)

30% medium- g r a i n e d s a n d ) J to ' * d a y s Temp;rature i n t h e s a m p l e t. d a y s -5' F i g . 3 E x p e r i m e n t a l c u r v e s of c h a n g e s i n t h e r m a l d e f o r m a t i o n w i t h t i m e f.-' 10'

a

_{K a o l i n}_{c l a y} Medium-grained s a n d ~ t & f t u r $ (702 k a o l ' n c l me ium-gralneA s a d 7

'

S u p e s ( I . N. Votyakov) 1.0 0 - t o F i g . 4 S t a b i l i z e d t h e r m a l d e f o r m a t i o n ( E ~ ) a s a f u n c t i o n of t e m p e r a t u r e ( i n i t i a l t e m p e r a t u r e O'C)

(28)

Mixture (70% k a o l i n c l a 30% medium-grained sand7 20 w, % IMedium-grained sand Fig. 5 S t a b i l i z e d c o e f f i c i e n t of t h e r m a l expansion (arn) a s a f u n c t i o n of water c o n t e n t 0 50 I00 Sand c o n t e n t , % F i g , 6

(29)

C o n t r a c t i o n

+ t

.

. Expans i o n F i g . 7 Main t y p e s of t h e

@(OK,

0 _H) c u r v e s d e s c r i b e d by e q u a t i o n (2.10) Fig. 8 E x p e r i ~ n e n t a l c u r v e s showing t h e r e l a x a t i o n of compressive s t r e s s e s i n f r o z e n s u p e s 1

-

r e l a x a t i o n of t h e r m a l s t r e s s e s d u r i n g a t e m e r a t u r e r i s e from -38OC t o -3Oc;

2

-

s t r e s s r e l a x a t i o n a t c o n s t a n t t e m p e r a t u r e ( - 3 ' ~ ) and i n i t i a l s t r e s s of 8 kg/cm2;

(30)

Chapter I11

THERMAL STRESSES I N UNDISTURBED FROZEN GROUND

1. Equation of One-dimensional Thermo-rheological S t a t e of Frozen S o i l

A s o u t l i n e d i n t h e f i r s t two c h a p t e r s , t h e thermo-rheological p r o c e s s e s i n f r o z e n s o i l s a r e d e s c r i b e d b e s t i f t h e s o i l i s r e g a r d e d a s a v i s c o - e l a s t i c model of t h e Bingham-Shvedov t y p e . The d e f o r m a t i o n s c o n s i s t of two p a r t s :

e l a s t i c and v i s c o - p l a s t i c .

To d e r i v e t h e b a s i c d i f f e r e n t i a l e q u a t i o n , i t i s e s s e n t i a l t o e x p r e s s t h e r e l a t i o n s h i p between d e f o r m a t i o n and s t r e s s . The r e l a t i o n between t h e e l a s t i c p a r t of d e f o r m a t i o n and s t r e s s c a n b e expressed i n t h e c o n v e n t i o n a l l i n e a r form a s used i n t h e t h e o r y of e l a s t i c i t y :

where

cynp,

E ~

cynp

~ ~ynp ,

-

a r e t h e components of t h e e l a s t i c p a r t of

X Y z yxy

d e f o r m a t i o n ;

a

-

a r e t h e components of s t r e s s , x' y ' z

and E and G

-

a r e t h e moduli of e l a s t i c i t y and s h e a r , r e s p e c t i v e l y

.

To f o r m u l a t e t h e r e l a t i o n s h i p between t h e v i s c o - p l a s t i c p a r t of d e f o r m a t i o n and s t r e s s , we c a n u s e t h e well-known Genki-Mizes i d e n t i t i e s which hold f o r any c o n t i n u o u s media a t low d e f o r m a t i o n s :

.

Nl

where

_&

_,

_&

_,

i n

_,

-

a r e t h e components of t h e r a t e of v i s c o - p l a s t i c deformat i o n ;

H

-

i s t h e r a t e of t h e v i s c o - p l a s t i c s h e a r deformat i o n ;