RHEOLOGY OF SOILS

by S.S. Vyalov

RHEOLOGICAL FUNDAMENTALS OF SOIL MECHANICS

Objec t ives of rheo log ica l s tud ie s . Rheology i n gene ra l may be de f ined , a f t e r Reiner [1,2] and academician Sedov [ 3 ] , a s t h e sc i ence dea l ing w i t h the deformed s t a t e of a medium and changes occurr ing i n i t w i t h t ime, a s w e l l a s w i t h t h e c h a r a c t e r i s t i c behaviour of t h a t medium under loads exceeding t h e limits p red ic t ed by c l a s s i c a l t h e o r i e s of e l a s t i c i t y and p l a s t i c i t y .

As is we l l known, t r a d i t i o n a l s o i l mechanics i s based on t h e fo l lowing fundamental concepts:

1) s o i l is sub jec t t o l i n e a r deformation behaviour; 2) compaction ( conso l ida t ion ) of s o i l i s dependent on t h e movement of water

through voids i n t h e s o i l ; 3) s o i l cons is tency is determined by t h e f o r c e s of i n t e r p a r t i c l e cohesion

and i n t e r n a l f r i c t i o n ; 4 ) t h e process of deformation of s o i l s is descr ibed by t h e e l a s t o - p l a s t i c

law: s o i l deforms l i n e a r l y up t o a c e r t a i n maximum s t r e s s va lue , a t which t h e s t a t e of l i m i t i n g stress occurs t h a t i s cha rac te r i zed by a continuous development of s t r a i n a t a cons tan t stress.

Theses p o s t u l a t e s have played an important p a r t i n t h e formation and development of modern s o i l mechanics. They have g r e a t l y con t r ibu ted t o so lv ing many geotechnica l problems and w i l l probably cont inue t o be used i n engineer ing p r a c t i c e f o r a long t ime t o come. These laws, however, a r e somewhat i d e a l i z e d and do not correspond t o t h e a c t u a l deformation behaviour of a s o i l under load. The d i sc repanc ie s between t h e i d e a l and r e a l p r o p e r t i e s of s o i l a r e n a t u r a l l y r e f l e c t e d i n t h e r e s u l t s of engineer ing c a l c u l a t i o n s , and t h e i r e f f e c t s become inc reas ing ly more p e r c e p t i b l e a s t h e problems i n p r a c t i c a l geotechnica l engineer ing grow p rogres s ive ly more complex: e.g., i n c r e a s i n g loads on foundat ions of bu i ld ings and o t h e r s t r u c t u r e s ; t h e use of cons t ruc t ion s i t e s on uns t ab le s o i l s , i nc lud ing l a n d s l i d e a reas ; cons t ruc t ion on weak s o i l s w i th h igh f l u i d i t y , i nc lud ing sea- f loor sediments; and s o on.

It i s t h e r e f o r e impera t ive t o review t h e t r a d i t i o n a l concepts of s o i l mechanics, t o in t roduce new deformation laws r e f l e c t i n g more a c c u r a t e l y t h e r e a l p r o p e r t i e s of e a r t h m a t e r i a l s , and t o develop new schemes and models t h a t would allow u s t o t ake i n t o account t h e d i s t i n c t i v e deformation p a t t e r n s of t h e s e ma te r i a l s . This i s p a r t i c u l a r l y important f o r f rozen s o i l s , t h e behaviour of which d i f f e r s even more markedly from t h e i d e a l behavour descr ibed by c l a s s i c a l laws.

Hence, t h e o b j e c t i v e s of f u r t h e r s t u d i e s a r e a s fol lows: 1. To i d e n t i f y t h e s p e c i a l a s p e c t s of deformation and f a i l u r e of s o i l s

( p a r t i c u l a r l y those of f rozen s o i l s ) compared t o c l a s s i c a l media.

2. To c o n s t r u c t mathematical models d e s c r i b i n g t h e p a t t e r n of deformation of s o i l s under l oad and t o work ou t fundamental equa t ions on t h e b a s i s of t h e s e models.

3. To t a k e i n t o cons ide ra t ion any s p e c i a l p a t t e r n s of s o i l deformation i n formula t ing engineer ing problems, and t o develop methods f o r s o l v i n g t h e s e problems, e s p e c i a l l y w i th t h e u s e of computers.

4 . To a s s e s s t h e r e l a t i v e importance of t h e s p e c i a l a s p e c t s of t h e p r o p e r t i e s of s o i l s , and t o i d e n t i f y t h e cond i t i ons under which each of t h e s e a s p e c t s must be taken i n t o account.

5. To develop f i e l d and l a b o r a t o r y methods f o r determining t h e parameters of t h e c o n s t i t u t i v e equa t ions used i n eng inee r ing c a l c u l a t i o n s , and t o accumulate exper imenta l d a t a on t h e va lues of t h e s e parameters.

6 . To compare t h e r e s u l t s of t h e o r e t i c a l s o l u t i o n s wi th t h e d a t a from f i e l d observa t ions . This should be t h e c r i t e r i o n determining t h e choice of t he model and f low c h a r t f o r s o l v i n g engineer ing problems.

Spec ia l a s p e c t s of deformation and f a i l u r e of s o i l s . The process of deformation of s o i l s is accompanied by changes i n t h e s o i l s t r u c t u r e , namely, rearrangement of s o i l p a r t i c l e s , breaking o r r e s t o r a t i o n of i n t e r p a r t i c l e bonds, development of s t r u c t u r a l d e f e c t s ( d i s r u p t i o n of i n t e r p a r t i c l e bonds of t h e s o i l s k e l e t o n , formation of mic ro f r ac tu re s , e t c . ) and t h e i r heal ing. Owing t o t h e v iscous n a t u r e of i n t e r p a r t i c l e bonds, t h e s e processes a r e time-dependent and account f o r t h e c r eep p r o p e r t i e s i n h e r e n t i n c l a y s o i l s and i n a l l t ypes of f rozen s o i l s , which owe t h e i r v i scous p r o p e r t i e s l a r g e l y t o t h e presence of i c e a s t h e bonding ma te r i a l . The process of deformation i s thus accompanied by s o f t e n i n g o r hardening of t h e s o i l s t r u c t u r e . Hardening of t h e s t r u c t u r e r e s u l t s i n damping of s t r a i n s , whi le s o f t e n i n g l e a d s t o t h e i r undamped development, ending i n f a i l u r e . Moreover, i t has been demonstrated by Maksimyak ( see Vyalov e t a l . [ 4 1 ) t h a t f a i l u r e occu r s when t h e d e n s i t y of t h e d e f e c t s reaches a c e r t a i n c r i t i c a l value. Since t h e r a t e of formation of d e f e c t s i s a func t ion of s t r e s s , t h e lower t h e s t r e s s , t h e more t i m e i t t a k e s f o r t h e d e n s i t y of d e f e c t s t o a t t a i n t h a t c r i t i c a l va lue , and t h e longer t h e t ime t o f a i l u r e . This is where t h e p a t t e r n of t h e long time s t r e n g t h of s o i l s mani fes t s i t s e l f .

Although from t h e s t r u c t u r a l p o i n t of view s o i l is a d i s c r e t e multicomponent medium, w e apply t h e concepts of continuum mechanics and t r e a t i t a s an e l a s t o - p l a s t i c v i scous body ( s i n c e s o i l is s u b j e c t t o a l l t h e s e types of deformation) . On t h e o t h e r hand t h e micro- and macro-structural c h a r a c t e r i s t i c s of s o i l s a l s o a f f e c t t h e i r deformation under load and account f o r t h e f a c t t h a t i t does n o t conform t o t h e c l a s s i c a l p a t t e r n . Never the less , even t h e s e s p e c i a l c h a r a c t e r i s t i c s could be descr ibed by equa t ions of continuum mechanics ( u n t i l a new branch of mechanics, i.e. mechanics of d i s c r e t e media, was developed). I n r e c e n t y e a r s t h i s problem has a t t r a c t e d a t t e n t i o n of many r e s e a r c h e r s both i n t h e U.S.S.R. and abroad. Noteworthy among t h e works by Sovie t au tho r s a r e monographs [5-201 and proceedings of conferences and symposia on rheology of e a r t h m a t e r i a l s , i nc lud ing f rozen s o i l s [21-251. Among t h e f o r e i g n works, i n a d d i t i o n t o t h e works mentioned e a r l i e r [ 1 , 2 ] , we should mention monographs [26-301, proceedings from t h e I n t e r n a t i o n a l Rheology and S o i l Mechanics Symposium i n Grenoble [31] , symposium on rheology of s o i l s [32] , etc. A g r e a t d e a l of a t t e n t i o n i s focussed on t h e problems p e r t a i n i n g t o rheology of s o i l s i n t h e review papers presented a t t h e 7 t h I n t e r n a t i o n a l Congress of S o i l Mechanics and Foundation Engineering (Mexico, 1969), and on

rheology of f rozen s o i l s , i n t h e review papers from t h e 2nd I n t e r n a t i o n a l Conference on Permafrost (Yakutsk, 1973) [361.

Summarizing t h e r e s u l t s of experimental s t u d i e s , w e can i d e n t i f y t h e fo l lowing c h a r a c t e r i s t i c a s p e c t s of t h e behaviour of s o i l s under load.

I r r e c o v e r a b l e volumetr ic s t r a i n s (due t o v a r i a t i o n s i n p o r o s i t y ) and s h e a r s t r a i n s i n t h e s o i l s , developing r i g h t from t h e very beginning of loading. It should be noted t h a t t h e s e p r o p e r t i e s of s o i l s have been t aken i n t o cons ide ra t ion from t h e e a r l i e s t days of t h e development of s o i l mechanics.

The non-linear c h a r a c t e r of t h e s t r e s s - s t r a i n r e l a t i o n s h i p wi th t h e f u n c t i o n f f o r s h e a r s t r a i n s d i f f e r i n g from t h a t f o r volume s t r a i n s f*:

where zi and yi a r e t h e s h e a r s t r e s s e s and t h e s h e a r s t r a i n s , r e s p e c t i v e l y , while am and E, a r e t h e average normal s t r e s s ( conf in ing p r e s s u r e ) and t h e average l i n e a r s t r a i n ( equa l t o 113 volumetr ic s t r a i n ) , r e spec t ive ly .

The most widely used forms of r e l a t i o n s h i p f o r s h e a r s t r a i n s a r e power and l i n e a r f r a c t i o n a l func t ions :

The former i s very s imple , but t h e l a t t e r i nc ludes both t h e s h e a r modulers (Go) and u l t i m a t e s t r e n g h t Ti(,).

S imi l a r r e l a t i o n s h i p s may be used f o r volumetr ic s t r a i n s , bu t t h e s t r e s s - s t r a i n curves descr ibed by them a r e concave towards t h e s t r e s s a x i s r a t h e r than towards t h e s t r a i n a x i s ( 2 ) , s i n c e wi th i n c r e a s i n g s t r e s s volume s t r a i n s tend towards a c e r t a i n l i m i t (maximum conso l ida t ion ) whi le s h e a r s t r a i n s develop continuously.

For engineer ing c a l c u l a t i o n s t h e non-l inear v a r i a t i o n of s t r a i n s w i t h stress i s a very important cons idera t ion . This enables u s t o c a l c u l a t e s t r a i n s over t h e whole range of stress up t o t h e l i m i t i n g s t r e s s , whereas i n t h e l i n e a r model t h e c a l c u l a t i o n s a r e r e s t r i c t e d on ly t o t h a t segment of t h e s t r e s s - s t r a i n curve f o r which a l i n e a r r e l a t i o n s h i p can be assumed. Mechanical p r o p e r t i e s of s o i l s can then be u t i l i z e d more f u l l y , and des ign loads on s o i l d s can be increased. Once t h e non-l inear r e l a t i o n s h i p of t h e d i s t r i b u t i o n of r e a c t i o n f o r c e s underneath t h e f o o t i n g of t h e foundat ion i s taken i n t o account , t h e v a r i a t i o n of s t r e s s w i t h depth of t h e f o o t i n g and t h e depth of t h e p l a s t i c zone around t h e f o o t i n g , approach values c l o s e r t o t h e r e a l s i t u a t i o n . A s a r e s u l t of t h a t t h e des ign va lues of s e t t l e m e n t s of t h e f o o t i n g a l s o approach c l o s e r t o t h e i r a c t u a l values.

Development of s t r a i n s wi th time i s a d i s t i n c t i v e proper ty of c l a y s o i l s and a l l t ypes of f rozen s o i l s . I n t h e c a s e of s h e a r s t r a i n s i t mani fes ts i t s e l f i n t h e form of c reep , which i s t r a n s i e n t a t smal l s t r e s s e s , b u t i s v iscous a t l a r g e s t r e s s e s and t e rmina te s i n v iscous o r b r i t t l e

failure. In the presence of volumetric strains the development of strains with time manifests itself in the form of a consolidation process induced both by drainage with redistribution of pore and effective pressures in the soil, and by creep of the soil skeleton volumetric strains are invariably damped.

The time-dependent equation of state can be written in one of the following forms, based on the accepted hypothesis regarding the relationship between strain or strain rate, stress and time:

where pi = dy/dt, and < = d~,/dt, are the strain rates. In equation (3) and ( 4 ) (the theory of aging and the theory of flow,

respectively) time is considered in the explicit form, and in equation (5) and (6) (the theory of hardening and the theory of inherited creep, respectively) it is considered in the implicit form. At a constant stress these equations give identical results, but under variable stress the results are different. Equation ( 3 ) is the simplest and most widely used relationship; it does not, however, take into account changes occurring in the load with time. Equation ( 6 ) takes into consideration the variability of the load and the loading history. The hereditary theory has therefore gained wide recognition, but it makes no allowances for various distinctive aspects of deformation of soils, which will be discussed later. Equations (4) and, particularly, (5) are mre promising from that viewpoint.

Time is one of the most important factors to be considered in problems pertaining to deformation of soils. This factor controls the settlements of foundations, consolidation of earth structures (embankments, earth-fill dams, etc.), landslide movements on slopes and scarps, redistribution of reaction pressures of the soil under the foundation and on the thin bearing walls, redistribution of the bending moment in the above structures, and SO on.

Variability of the lateral strain ratio. Since Poisson's ratio can be expressed through equations (3) as

where K = u,/f*(u,,t), G = ~~/f(~~,t), and since f* # f, the value of v is

not a cons tan t , but is a func t ion of t he r a t i o of shea r s t r a i n t o volume s t r a i n , and of t h e development of t h e s e s t r a i n s wi th time. It should be noted, however, t h a t i n the commonly used s i m p l i f i e d r e l a t i o n s h i p :

- om/K, where K = cons tan t , but yi = f ( z i , t ) , v i s considered cons tan t . 'm - However, t o accep t t h e condi t ion , v = cons tan t , i s t o assume t h a t volume and shea r s t r a i n s a r e descr ibed by i d e n t i c a l r e l a t i o n s h i p s . S t r i c l y speaking t h i s assumption is no t v a l i d t o r ep resen t t h e phys ica l processes of shea r and volume deformations, but i t s i g n i f i c a n t l y s i m p l i f i e s t h e s o l u t i o n of t h e problems i n s o i l mechanics, and o f t e n does not s e r i o u s l y d i s t o r t t h e r e s u l t s . For example, experimental d a t a and computer c a l c u l a t i o n s have shown t h a t t h e p a t t e r n of development of s e t t l e m e n t s under p l a t e loading is analogous t o t h e non-linear law of shea r s t r a i n s , a l though t h e r e l a t i o n s h i p of volumetr ic s t r a i n s is c l o s e t o l i n e a r .

The f a c t t h a t t h e r e s i s t a n c e t o deformation under compression is d i f f e r e n t from t h a t under ex tens ion i s due t o t h e i n t r i n s i c n a t u r e of i n t e r p a r t i c l e bonds i n s o i l . It accounts f o r t h e f a c t t h a t f o r c e s of i n t e r n a l f r i c t i o n mani fes t themselves n o t only i n t h e l i m i t * * s t a t e , bu t a l s o i n t h e pre-limit** s t a t e , hence shea r s t r a i n s a r e func t ions of conf in ing pressure:

Volumetric s t r a i n i t s e l f is a func t ion of shea r s t r e s s ( t h e d i l a t a n c y phenomenon):

Moreover,

where E: i s t h e volumetr ic s t r a i n induced by t h e s p h e r i c a l ( d i l a t a t i o n a l ? ) D s t r e s s t enso r (conf in ing p res su re am) , and em = Ayi i s t h e volumetr ic s t r a i n

induced by t h e a c t i o n of t h e s t r e s s dev ia to r ( shear s t r e s s yi) , and A is t h e d i l a t a n c y c o e f f i c i e n t .

The mutual e f f e c t of am and zi on shape deformation and volumetr ic s t r a i n p l ays a s i g n i f i c a n t r o l e i n t h e process of s o i l deformation, a s w e l l as i n t h e process of f a i l u r e . The e f f e c t of om is t o i n h i b i t t h e shea r ing process. The e f f e c t of zi on volumetr ic deformation may, however, l ead t o a d d i t i o n a l conso l ida t ion of t h e s o i l ( p o s i t i v e d i l a t a n c y ) o r t o i t s remolding (negat ive d i l a t ancy) . It i s p r e c i s e l y t h e l a t t e r t h a t causes s o i l f a i l u r e , s i n c e i n cohesive s o i l s ( inc lud ing permafros t ) , f a i l u r e occurs a s a r e s u l t of microf rac tur ing . Since mic ro f rac tu r ing i s caused by t h e a c t i o n of d e v i a t o r i c s t r e s s e s , t h e g r e a t e r i t s magnitude, t h e sooner w i l l f a i l u r e occur. I f conf in ing p res su re p r e v a i l s , t h e s o i l becomes merely compressed

*Note form E i o Actual va lue of v c a l c u l a t e d from e l a s t i c i t y comes o u t a s v = + ( K ~ G 5-5j 3

(K/G)+1/3 **The pre- l imi t and l i m i t s t a t e s appear t o correspond t o t h e e l a s t i c and

- -

plastic cond i t ions , r e spec t ive ly .

or consolidated with healing of microcracks, and no failure of the soil occurs.

Treatment of the deformation and failure processes becomes more reliable by taking into consideration the mutual effect of a,,, and T~ on the processes of shear and volume deformation, and the results of theoretical calculations are brought closer to the actual data.

The effect of the type of stress. For conventional materials the results of the tests represented in the form of the ~ ~ - y ~ (generalized stress-strain) diagram do not depend on the type of stress. In other words the experimental points obtained from testing that type of materials under any type of load (compression, extension, shear, or combined stress), will fall on the same general curve when plotted in the ~ ~ - y ~ diagram. For soils with different resistances to compression and extension each type of test will yield its own zi-yi curve. This means that the process of soil deformation depends on the state of the stress defined by a certain relationship between the principal stresses expressed by the Lode-Nadai parameter pa: (-1 < p < +I), or by a characteristic angle wa: (n/3 > oa > O), which fs an invariant of magnitude y.

To account for the effect of the state of stress, the parameter p (or w should be introduced into the rheological equations of state, f 8 ) and (81, i.e.

Experiemnts have shown that characteristics of soils (including the angle of friction) may differ substantially, depending on the value of po at which they are determined. Further studies are needed, however, to determine when that factor must be considered and when it can be ignored. If the parameter pa (or wa) is not introduced into the equation of state, the design specifications should be determined from tests of soil specimens under conditions close to natural. For example, to calculate the bearing capacity of footings and circular or square foundations, the soils should be tested under conditions of axi-symmetric loading (triaxial compression, p, = -1)) whereas for strip loading the tests should be carried out under conditions of plane shear.

The effect of the loading path. A particular state of stress may be caused to occur at a point M ( u ~ ,ai,a$) by changing the components al, a2 and 63, according to some program. The representation of that program in the form of a curve in the stress space (61-a2-a3) is known as the loading path. For conventional materials the strain at point M depends only on the ultimate value of the stresses o1 = = a' and a3 = a;, independent of

"1 f the loading path. But in the case of soi s wh ch possess distinctive structural characteristics, the loading path may affect the pattern of deformation. For example, if a specimen is first consolidated by a confining pressure and then subjected to a shearing force, the resultant deformation will be different from what it would be if the specimen had been first sheared and then subjected to confining pressure.

The effect of the loading path is observable if the differences in the loading conditions are sufficently sharp. This has been demonstrated by Zaretskii et al. [37], who filled a reservoir during, as well as after, the

e r e c t i o n of t h e dam. Data is a l s o a v a i l a b l e , however, showing t h a t f o r a foo t ing , t h e d i f f e r e n c e i n t h e s t a t e of s t r a i n obta ined by s imultaneous and consecutive a p p l i c a t i o n of v e r t i c a l and shea r ing loads i s small. It is t h e r e f o r e impera t ive t o c a r r y out s p e c i a l s t u d i e s t o determine t h e degree of i n f luence of t h e loading path and t h e cond i t ions under which t h a t i n f luence must be taken i n t o cons idera t ion .

The e f f e c t of t h e loading program. Rate of loading a l s o has a c e r t a i n e f f e c t on t h e deformed s t a t e of t h e s o i l . This e f f e c t may manifest i t s e l f a s progress ive accumulation of p l a s t i c s t r a i n s wi th i n c r e a s e i n t h e loading r a t e , o r a s t h e delayed d i s s i p a t i o n of t h e pore p res su re due t o such an i n c r e a s e , a s a r e s u l t of which t h e e f f e c t i v e s t r e s s , and hence t h e shea r ing r e s i s t a n c e of t h e s o i l , w i l l l a g behind t h e i n c r e a s e i n t h e load.

The absence of s i m i l i t u d e between t h e s t a t e s of stress and s t r a i n . The c l a s s i c a l theory of p l a s t i c i t y is based on t h e fo l lowing pos tu l a t e s :

a ) s i m i l a r i t y of t h e s t r e s s e d and s t r a i n e d s t a t e s of t h e medium, which corresponds t o t h e cond i t ion of e q u a l i t y of t h e Lode parameters f o r s t r e s s e s and s t r a i n s po = pE, and

b) c o a x i a l i t y of stress and s t r a i n v e c t o r s , which corresponds t o t h e e q u a l i t y of t h e angles of i n c l i n a t i o n of t h e axes of p r i n c i p a l s t r e s s and s t r a i n ( a b = a E ) . Experiments show, however, t h a t t h e s e cond i t ions a r e not always observed f o r s o i l s . The degree of depa r tu re from t h e condi t ions of s i m i l i t u d e and c o a x i a l i t y , and t h e case where t h e s e dev ia t ions a r e pe rcep t ib l e , should be s t u d i e d f u r t h e r . It can merely be s t a t e d t h a t v i o l a t i o n of t h e cond i t ions of s i m i l i t u d e and c o a x i a l i t y of t h e s t r e s s e d and s t r a i n e d s t a t e s s i g n i f i c a n t l y complicates s o l u t i o n of p r a c t i c a l problems and should be r e s o r t e d t o only i n extreme cases .

Geometrical non- l inear i ty . I n a d d i t i o n t o t h e phys ica l non- l inea r i ty of t h e laws of deformation d iscussed above, i t i s sometimes necessary t o t ake i n t o cons ide ra t ion t h e geometr ical non- l inea r i ty of t h e fundamental equat ions. Normally only i n f i n i t e s t i m a l s t r a i n s a r e considered, and t h e r e f o r e , non-linear terms i n t h e c o n s t i t u t i v e equat ions of s t r a i n s and displacements and i n t h e equat ions of equ i l ib r ium can be removed, thereby transforming them i n t o l i n e a r equat ions. In s o i l s , owing t o t h e i r g r e a t compress ib i l i t y , t h e s t r a i n s i n t h e r eg ions of s t r e s s concen t ra t ions may be q u i t e l a r g e , and s t r i c t l y speaking, should be regarded a s f i n i t e . This makes t h e equat ions geometr ica l ly non-linear, which i n t u r n l ead t o c e r t a i n changes i n t h e r e s u l t s of c a l c u l a t i o n s of t h e deformed s t a t e of t h e s o i l [39]. The degree of such changes and t h e need t o t a k e them i n t o account a r e sub jec t t o f u r t h e r s t u d i e s .

Long-term s t r eng th . A s was mentioned e a r l i e r , one of t h e manif-ogical p r o p e r t i e s i s t h e decreas ing s t r e n g t h of s o i l a s a r e s u l t of c reep , and t h e f a c t t h a t t h e r e s i s t a n c e of s o i l is a func t ion of t h e time of a c t i o n of t h e load [40,41]. Equations of long-term s t r e n g t h can be obta ined from t h e creep equat ion by in t roduc ing i n t o i t one of t h e fol lowing cond i t ions of f a i l u r e :

( a ) a c e r t a i n c r i t i c a l value of p l a s t i c s t r a i n given by: yf = yics) = cons tan t

t ( b ) c e r t a i n cons tan t s of magnitude: l P ~ ( t ) d t = cons tan t , where A = aij rpj

0 i s t h e u n i t work of v i s c o p l a s t i c s t r a i n , and t i s t h e t ime t o f a i l u r e , P ( c ) i n t h e gene ra l cFse ( i f t h e thermodynamic approach i s used) con tan t s of

t h e magnitude: j p S ( t ) d t = ASkp = cons tan t , where S i s entropy and AS 0 kp

i s t h e c r i t i c a l va lue of i ts increment.

Proceeding from t h e f i r s t of t h e above c o n d i t i o n s , t h e fo l lowing l o n g t i m e s t r e n g t h equat ion i s obta ined , i n which t h e v a r i a b l e load i s accounted f o r :

o r i n a s p e c i a l ca se [401

where zi(s ,O) i s t h e ins tan taneous s t r e n g t h .

The l i m i t s t a t e of s o i l w i th cons ide ra t ion of i t s rheo log ica l p r o p e r t i e s . The l i m i t s t a t e of t h e s o i l s possess ing c reep p r o p e r t i e s corresponds t o t h e development of p rogres s ive f low s t r a i n s te rminat ing i n b r i t t l e o r v iscous f a i l u r e ( o r i n t h e development of un l imi t ed s h e a r s t r a i n s ) . Since t h e r e s i s t a n c e t o f a i l u r e v a r i e s i n t ime, t h e equat ion of t h e l i m i t s t a t e should inc lude t h e t ime f a c t o r a s i n equa t ion (11). Accordingly t h e equat ion of t h e l i m i t s t a t e i s der ived from equat ion (10) by assuming t h a t t r a n s i t i o n t o t h a t t h e s t a t e i s achieved when t h e s t r a i n a t t a i n s t h e maximum value ye = yi(s) = constant . Hence

where t h e r e l a t i o n s h i p between ri and t i s determined by equa t ion (12).

The gene ra l form of t h e equat ion of s t a t e . The q u a n t i t i e s am, em, ri, po ( o r woj a r e def ined i n t e r m s of t h e r a t i o between t h e p r i n c i p a l normal s t r e s s e s and t h e p r i n c i p a l l i n e a r s t r a i n s , and a r e f u n c t i o n s of t h r e e i n v a r i a n t s of t h e s t r e s s and s t r a i n t e n s o r , o r more a c c u r a t e l y , of t h e i r d i l a t a t i o n a l and d e v i a t o r i c segments. The equa t ions of s t a t e can t h e r e f o r e be w r i t t e n i n a gene ra l form:

where J1,2 a r e t h e f i r s t and second i n v a r i a n t s of t h e s t r a i n t e n s o r , and '1,2,3 a r e t h e f i r s t , second and t h i r d i n v a r i a n t s of t h e stress tensor . A

similar notation can be made for the invariants of the strain rate tensor jl and j2.

The equation of the limit state (13) can be written in the following form:

Soil temperature 8, which is an extremely important factor for frozen soils, is included in these equations.

The possible forms of equations of the pre-limit and limit states of soil. The equations of state (14) and (15) or (8) and (13) proposed by -

different researchers, are only phenomenological and vary fairly widely in form [42]. To clarify this two equations are considered in which the effect of the three invariants of stresses and of the time factor is accounted for. One of these equations is derived from the exponential function of zi and yi and has the following form:

The equation of limit equilibrium is derived from (16) by assuming that yi = ys = constant and, accordingly, that ~ ~ ~ t ( ~ ) = T ,(,), where 7 the instantaneous (t = 0) ultimate strength in simple shear. Then s(0) is

When am = A = t = 0, equation (16) changes to an ordinary exponential function yT = zi/Ao, and equation (17) at A = t = 0 to the von Mises-Shleikher*-Botkin condition

'm T = i s (1 + -) = (Hs + a,,,) tgY . Hs

The equation that follows from the linear-fractional functions of zi and yi, has the form

-1 =i t a

~i = - [l + ] (18) Go

The equation of the limit state is derived from the condition yi + a, and is written in the form

where E . . = E; + EP . and a a r e components of s t r a i n and s t r e s s t e n s o r s i j ( i , j = t i 2 , 3 ) , 6 $4 t h e Kronecker d e l t a , and y and y* a r e t h e f u n c t i o n c h a r a c t e r i z i n g s i & a r and volume ( i n c l u d i n g d i l a t a n c y ) s t r a i n s and a r e e q u a l to :

I n gene ra l y # y*, consequently v f const. When t h e t r a n s i t i o n t o t h e l i m i t s t a t e occurs , equa t ion (20) becomes equa t ion (13) .

Equations of t h e theory of p l a s t i c flow a r e w r i t t e n i n t h e form of r e l a t i o n s h i p s between t h e increments of s t r a i n and s t r e s s e s . In doing s o i t i s assumed t h a t t h e increment of t o t a l s t r a i n i s composed of t h e increments of i t s e l a s t i c ( e ) and p l a s t i c ( p ) components:

The r e l a t i o n s h i p between a and ee is determined by equat ions of t h e i j theory of e l a s t i c i t y , and idat between a and cPj , by t h e equat ions : i j dcPj = ( a i j + dijam)/dh + 6ijamdc , where

The l i m i t s t a t e i s descr ibed by equa t ion (13 ) , which i s a l s o t r u e f o r t h e s t r a i n theo ry , bu t w i t h t h e a d d i t i o n of t h e cond i t ion of d i l a t a n c y . According t o t h a t cond i t ion , when l i m i t s t a t e * occur s increments i n shea r s t r a i n a r e a s s o c i a t e d w i t h i r r e v e r s i b l e changes i n volume

where A i s t h e d i l a t a n c y r a t e c o e f f i c i e n t . Assoc ia t ive and non-associat ive laws. The r e l a t i o n s h i p between p l a s t i c

s t r a i n and s t r e s s can be dep ic t ed by t h e s t r e s s f u n c t i o n F known a s t h e p l a s t i c p o t e n t i a l

D i s t i n c t i o n is made between t h e a s s o c i a t i v e law (according t o von Mises) and non-associat ive law. I n t h e a s s o c i a t i v e law t h e p l a s t i c

* P l a s t i c deformation

p o t e n t i a l co inc ides wi th t h e loading func t ion (13) , i.e. F = $. I n t h e non-associat ive law, t h e va lue of t h e f u n c t i o n F i s d i f f e r e n t from t h a t of 4.

The a s s o c i a t i v e law and t h e Drucker-Prager model based on t h a t law have a number of l i m i t a t i o n as given below: t h e v e c t o r s d ~ 4 . of t h e loading s u r f a c e should be orthogonal ; t h e cond i t ion A = t g I Jhould be met, where A and Y a r e parameters of equa t ion (23) and I 1 I '1; and t h e cond i t ion h < 0 should be m e t , which says t h a t d i l a t a n c y occurs only i n t h e form of remolding, whi le d i l a t a n c y due t o a d d i t i o n a l conso l ida t ion does n o t occur. Severa l assumptions a r e made t o remedy these l i m i t a t i o n s . For example, t o meet t h e cond i t ion A = t g Y, c o r r e c t i o n are proposed [44] f o r t h e va lues of Y ( o r 4, i f t h e MohrCoulomb cond i t ion and t h e cond i t ion ] A ) = s i n 4 t h a t folows from i t , a r e used). On t h e o t h e r hand Bugrov [381 has demonstrated t h a t from t h e viewpoint of t h e f i n a l r e s u l t s t h e use of t h e a s s o c i a t i v e law i s accep tab le even f o r cases where I A ) # s i n 4 ( o r 111 # t g I ) . A s f o r t h e f a c t t h a t t h e Drucker-Prager a s s o c i a t i v e model f a i l s t o account f o r a d d i t i o n a l d i l a t a n c y conso l ida t ion of t h e s o i l , according t o Gibson-Drucker-Henkel, i t can be remedied by assuming t h e cond i t ion of c losed y i e l d s u r f a c e ( l i m i t state). Z a r e t s k i i , Lombardo e t a l . , [37,45] have proposed a model wi th a s i n g u l a r y i e l d sur face . I n t h e well-known "Canrclay" model of Roscoe e t a l . [461, t h e loading s u r f a c e is assumed t o be i n t h e form of an e l l i p s o i d .

These and o t h e r adjustments make t h e a s s o c i a t i v e law q u i t e accep tab le f o r u s e i n engineer ing p r a c t i c e .

On t h e o t h e r t h e non-associat ive law of p l a s t i c f low i s f r e e from t h e l i m i t a t i o n s l i s t e d above, and i s more a c c u r a t e and more gene ra l , a l b e i t more complicated, s i n c e i t r e q u i r e s t h e inco rpora t ion of a d d i t i o n a l parameters , de terminat ion of which is f a r from simple.

I n apply ing t h e non-associat ive law e i t h e r a va lue d i f f e r e n t from t h e va lue of t h e load ing f u n c t i o n 4 i n (13) i s s p e c i f i e d f o r p o t e n t i a l F 1241, o r two mutual ly uncoupled ( 1 # t g I ) c o n d i t i o n s [ t h e cond i t ion of t h e l i m i t state (13) and t h e cond i t ion of d i l a t a n c y (23 ) ) a r e used i n a d d i t i o n t o equa t ions (21) and (22). The models cons t ruc ted wi th t h e u s e of t h e non- a s s o c i a t i v e law have been analysed by Nikolaevski i 144, 471. However, most r e s e a r c h e r s p r e f e r t h e a s s o c i a t i v e law (wi th co r rec t ions ) .

So lu t ion of non-l inear problems wi th t h e use of t h e s t r a i n theory. Experimental s t u d i e s on t h e b a s i s of which s o i l models were developed w i t h i n t h e framework of t h e s t r a i n theory , have been c a r r i e d ou t a t t h e NII* by Gorodetsk i i and o t h e r s , wi th t h e main focus on t h e t i m e f a c t o r [7 ,41] , and a t t h e MISI** by Lomise, Kryzhanovskii and o t h e r s ( s e e Refs. 48, 49) , who performed l a rge - sca le exper imenta l s t u d i e s under cond i t ions of s imple and composite loading.

The scope of t h e non-linear problems of s o i l mechanics t h a t a r e so lved by a n a l y t i c a l methods, i s very l i m i t e d even w i t h i n t h e framework of t h e s impler s t r a i n theory. For example, l i n e a r problems of t h e deformed s t a t e

* N I I = Research I n s t i t u t e of Foundations (Tr.) **MIS1 = t h e Moscow C i v i l Engineering I n s t i t u t e (Tr.)

of t h e f o o t i n g (p lane and axio-symmetric problems) can be so lved only f o r t h e exponent ia l s t r a i n l a w and only wi th c e r t a i n l i m i t a t i o n s [61, [50].

Only s p e c i a l problems were solved f o r t h e p a r t i a l l i n e a r func t ion , such a s t h e flow of s o i l down t h e s l o p e , t h e axi-symmetric problem of v iscous flow i n c y l i n d r i c a l p r o t e c t i v e i c e - s o i l b a r r i e r s , e t c . Straganov [51] used i n t h e s e s o l u t i o n s h i s own non-linear v iscous model r e p r e s e n t i n g a combination of t h e p a r t i a l l i n e a r law of shea r s t r a i n and Newtonian law of v iscous flow. The phenomenon of d i l a t a n c y was a l s o taken i n t o account.

A gene ra l s o l u t i o n t o non-l inear problems can be obta ined wi th t h e he lp of a computer according t o t h e method of f i n i t e d i f f e r e n c e s (MFD) o r t h e method of f i n i t e elements (MFE). Vinokurov [52] was one of t h e f i r s t r e s e a r c h e r s t o apply t h e method of f i n i t e d i f f e r e n c e s i n so lv ing non-linear problems of s o i l mechanics. H e worked out an i t e r a t i v e s o l u t i o n based on I l y u s h i n ' s method of e l a s t i c s o l u t i o n s us ing t h e s t r a i n theory of p l a s t i c i t y . By us ing a s i m i l a r method Shirokov e t a l . [53] and Malyshev e t a l . [54] obta ined a s o l u t i o n t o t h e problem of t h e s t r e s s - s t r a i n state of a f o o t i n g wi th p r o p e r t i e s descr ibed by equat ions (18) and (19) ; moreover, t h e s e equa t ions va r i ed i n form. So lu t ions of s i m i l a a r problems wi th t h e use of t h e method of f i n i t e d i f f e r e n c e s , but wi th fundamental equat ions of a somewhat d i f f e r e n t form, a r e given i n [55] , [56] , and i n o t h e r works, and those wi th t h e u s e of t h e method of f i n i t e elements , i n [42] , where s o l u t i o n s based on t h e s t r a i n theory are compared w i t h t h e s o l u t i o n s based on t h e theory of p l a s t i c flow.

So lu t ion of non-linear problems wi th t h e use of t h e theory of p l a s t i c flow. The b e s t known model based on t h e theory of p l a s t i c flow is t h e -

"Cam-clay" model developed a t Cambridge by Roscoe, Burland, Shof ie ld and o t h e r s ( s e e Refs. 46, 58). In t h a t model s o i l i s regarded a s a n e l a s t o - p l a s t i c hardening medium, t h e p l a s t i c hardening o r s o f t e n i n g of which i s induced by p o s i t i v e o r nega t ive d i l a t ancy . The s t r e s s - s t r a i n state of t h e s o i l i s f u l l y def ined by t h e t h r e e parameters am, T~ and e (where e i s t h e void r a t i o ) ; moreover, t h e increments of s t r a i n s a r e r e l a t e d t o t h e stress by t h e a s s o c i a t i v e l a w of p l a s t i c flow. The p a t t e r n of deformation i n t h e pre- l imi t state is descr ibed by t h e loga r i thmic equat ion of compression according t o Terzaghi e = eo - hIno. The l i m i t s t a t e i s de f ined a s t h e state a t t a i n e d when poros i ty ' e ' reaches t h e c r i t i c a l va lue ekr, a t which increment of p l a s t i c shea r s t r a i n occur a t cons t an t s t r e s s and volume. That s t a t e i s achieved when t h e r a t i o zi/am reaches a c e r t a i n c r i t i c a l value: z /am = M. The equat ion of conso l ida t ion then assumes t h e form: V = r - hf"cm, where V is the s p e c i f i c volume and r i s t h e s o i l cons t an t .

The "Cam-clay" model was subsequently modified.

Io se l ev ich e t a l . [59] developed t h e i r own v a r i a n t of a non-l inear model w i t h p l a s t i c hardening, which, similar t o t h e "Caurclay" model, i s a l s o based on t h e p o s t u l a t e of t h e e x i s t e n c e of a c losed loading s u r f a c e F (wi th both p o s i t i v e and nega t ive d i l a t a n c y being accounted f o r ) and i s dependent on t h e loading path. Proceeding from t h e a s s o c i a t i v e law, i t is assumed t h a t t h e loading s u r f a c e F co inc ides w i t h t h e p l a s t i c p o t e n t i a l (24).

I n t h e model w i th a s i n g u l a r loading s u r f a c e t h a t w a s mentioned e a r l i e r ( Z a r e t s k i i [45, 5011, t h e s o i l i s t r e a t e d a s a non-linear p l a s t i c hardening

medium wi th p l a s t i c shea r and volume s t r a i n s a c t i n g a s t h e hardening parameters. An unl imi ted accumulation of t h e s e s t r a i n s produces t h e l i m i t s t a t e . The l i m i t s u r f a c e i s descr ibed by a s i n g u l a r (piecewise smooth) curve making i t p o s s i b l e t o t a k e i n t o account t h e nega t ive and p o s i t i v e d i l a t a n c y and t h e in f luence of t h e loading path. Solu t ions t o a number of problems p e r t a i n i n g t o t h e s t a b i l i t y of l a r g e e a r t h dams and t h e i r foo t ings have been obta ined wi th t h e h e l p of t h a t mode, u s ing f i n i t e element method. Problems concerned wi th t h e process of conso l ida t ion of s o i l s have a l s o been solved.

Another noteworthy model is t h e e l a s t o - p l a s t i c model (Bugrov [38,611). I n t h a t model t h e s o i l i s regarded a s an e l a s t w p l a s t i c medium, e l a s t i c deformations of which fo l low Hooke's law, and p l a s t i c deformations fo l low t h e law of p l a s t i c f low ( i n t h e form of p l a s t i c i t y wi th hardening o r p e r f e c t p l a s t i c i t y ) . The r e l a t i o n s h i p between t h e s t r e s s e s and s t r a i n s was considered i n t h e incrementa l form, i.e. i n t h e form of s t r a i n increments. The cond i t ions of t h e l i m i t s t a t e accepted f o r t h e gene ra l ca se was i n t h e form given by Mises-Botkin, and f o r p lane problems i n t h e form of t h e Mohr-Coulomb c r i t e r i o n . Problems of t h e s t r e s s - s t r a i n s t a t e of t h e f o o t i n g wi th d i v e r s e i n i t i a l cond i t ions were so lved by t h e f i n i t e element method.

The i n i t i a l cond i t ions , des ign parameters , etc., were va r i ed i n a l l t h e aforementioned s o l u t i o n s obta ined w i t h t h e h e l p of t h e method of f i n i t e d i f f e r e n c e s o r f i n i t e elements. With t h i s d i f f e r e n t a s p e c t s of s o i l deformation could be taken i n t o account i n t h e models used, and t h e e f f e c t of s p e c i f i c parameters on t h e r e s u l t i n g s o l u t i o n s , a s w e l l a s t h e e x t e n t t o which t h e s e s o l u t i o n s d i f f e r e d from t h e t r a d i t i o n a l s o l u t i o n s , p a r t i c u l a r l y from t h e s o l u t i o n s us ing t h e e l a s t i c i t y theo ry , could be determined.

It should be noted t h a t i n r ecen t yea r s a g r e a t number of r e s e a r c h e r s have focussed on developing non-l inear models. Noteworthy works i n t h a t a r e a , i n a d d i t i o n t o those mentioned e a r l i e r , a r e as fol lows: a ) t h e model of hardening e l a s t i o - p l a s t i c s o i l proposed by Matsuoko e t a l . [62] , based on t h e hypothes is of s o i l f a i l u r e on a s p e c i a l s u r f a c e des ignated a s t h e " s p a t i a l mob i l i za t ion plane", ob ta in ing a new cond i t ion of f a i l u r e i n t h e form of a c o r r e l a t i o n of t h r e e i n v a r i a n t s of t h e stress t e n s o r (and d e v i a t o r ) , b) t h e model of Lode* e t a l . [631, who proposed t h e i r own cond i t ion of f a i l u r e i n t h e form of t h e maximum value of t h e r a t i o of t h e f i r s t and t h i r d i n v a r i a n t s ; c ) and t h e model of Huang Wen-Ki e t a l . [641, i n which t h e fundamental equat ion inc ludes t h e loading func t ion , hardening modulus and p l a s t i c func t ion ; and d ) t h e model of Akai and Nish i [651, who examined p l a s t i c flow and s o i l f a i l u r e wi th in t h e framework of t h e a s s o c i a t i v e law cons ider ing t i m e and o t h e r f a c t o r s .

A s may be seen from t h e above review, many d i f f e r e n t models have a l ready been developed f o r s o i l s , t ak ing i n t o account va r ious c h a r a c t e r i s t i c s of i t s behaviour: t h e phys ica l and geometr ic non- l inea r i ty , c r eep and conso l ida t ion , mutual e f f e c t of t h e t h r e e i n v a r i a n t s of t h e s t r e s s t enso r (which r e f l e c t s t h e e f f e c t of conf in ing p res su re on shea r s t r a i n and t h e e f f e c t of shea r stress on volume deformation ( d i l a t a n c y ) , and t h e e f f e c t of t h e loading p a t h and t h e depa r tu re from t h e cond i t ions of s i m i l i t u d e of t h e s t a t e s of stress and s t r a i n , and of c o a x i a l i t y of t h e stress and s t r a i n t enso r . It may be s t a e d t h a t a l l t h e c h a r a c t e r i s t i c s l i s t e d above can be taken i n t o cons ide ra t ion i n d i f f e r e n t s o i l models (perhaps even i n a u n i f i e d model), as w e l l a s i n t h e s o l u t i o n s of va r ious geo techn ica l problems, and

that all these characteristics affect the results of the solutions obtained to varying degrees, bringing them closer to the real situation. Further development of non-linear soil mechanics, however, may not be worthwhile, since increasing the number of factors considered in a model increases the number of initial parameters and hence the characteristics to be determined experimentally. Overall, the accuracy of solutions obtained with the large number of the factors considered may be lost due to the spread and inaccuracy in the determinations of the initial characteristics of soils, because of the heterogeneity of soils on the one hand and inadequacy of geotechnical methods of investigation on the other. The main objective should therefore be to identify the role of different factors on the occurrence and development of the stress-strain state in soils, and to evaluate the conditions under which these factors can be ignored, or else, must be taken into account. Lastly it is also imperative to keep improving the methods of laboratory and field determinations of the properties of soils, bearing in mind that the reliability of these methods is a function of the dependability of theoretical solutions.

BIBLIOGRAPHY

1. Reiner M. Deformatsiya i techenie (Deformation and Flow). Moscow, Gostoptekhizdat, 1963, 3 8 2 p.

2. Reiner M. Reologiya (~heology). Moscow, Nauka, 1965, 224 p . 3. Sedov L.I. Mekhanika sploshnoi sredy (Mechanics of Continuum).

Moscow, Nauka, 1970, V. 1, 492 p. Vyalov S.S., Pekarskaya N.K., Maksimyak R.V. 0 fizicheskoi sushchnosti deformirovaniya i razrusheniya gruntov (The physics of deformation and failure of soils). - Osnovaniya, fundamenty i mekhanika gruntov (Footings, foundations and soil mechanics), 1970, No. 1, pp. 7-9. Budin A.Ya., Demina G.A. Naberezhnye (Quays). Moscow, Stroiizdat, 1979, 264 p. Vyalov S.S. Reologicheskie osnovy mekhaniki gruntov (Rheological Fundamentals of Soil Mechanics). Moscow, Vyssh. shkola, 1978, 448 p. Vyalov S.S., Zaretskii Yu.K., Gorodetskii S.Z. Raschety na prochnost' i polzuchest' pri iskustvennom zamorazhivanii (Calculations of Strength and Creep during Artificial Freezing). Leningrad, Stroiizdat, 1981, 200 p. Gal'perin A.M., Shafarenko E.M. Reologicheskie raschety gornotekhnicheskikh sooruzhenii (Rheological Calculations of Mining Engineering Structures). Moscow, Nedra, 1977, 246 p. Gol'dshtein M.N. Mekhanicheskie svoistva gruntov (Mechanical Properties of Soils). Moscow, Stroiizdat, V. I., 1971, 368 p.; V. 11, 1973, 376 p.; V. 111, 1979, 304 p. Gorelik L.V. Raschety konsolidatsii osnovanii i plotin iz gruntovykh materialov (Consolidation Calculations for Earth Footings and Dams). Leningrad, Energiya, 1985, 154 p. Erzhanov Sh.S Teoriya polzuchesti gornykh porod i ee prilozhenie (Creep Theory of Earth Materials and its Application). Alma-Ata, Nauka, 1964, 174 p. Zaretskii Yu .K. Teoriya konsolidatsii gruntov (Theory of Soil Consolidation). Moscow, Nauka, 1967, 270 p. Maslov N.N. Dlitel'naya ustoichivost' i deformatsii smeshcheniya podpornykh sooruzhenii (Long-term Strength and Shear Deformations of Support Structures). Moscow, Energiya, 1968, 160 p. Maslov N.N. Mekhanika gruntov v praktike stroitel'stva (Soil Mechanicas in Construction Practice): Opolzni i bor'ba s nimi (Landslides and their Control). Moscow Stroiizdat, 1977, 320 p. Meschyan S.R. Polzuchest' glinistykh gruntov (Creep of Clay Soils). Erevan. Izd-vo ANB ArmSSR, 1967, 318 P. ~ e s c h ~ a n S.R. Nachal'naya i dlitel'naya prochnost' glinistykh gruntov (Initial and Long-term Strength of Clay Soils). Moscow, Nedra, 1978, 308 D. Ter-Stepanyan G.I. O dlitel'noi ustoichivosti sklonov (Long-term Stability of Slopes). Erevan, Izd-vo AN ArmSSR, 1961, 4 2 p. Tsytovich N.A. Mekhanika gruntov (Soil Mechanics). Vyssh. shkola, 1979, 240 p. Tsytovich N.A., Zaretskii Yu.K., Malyshev M.V. et al. Prognoz skorosti osadok osnovanii sooruzhenii (Prediction of the Rate of Settlements of Footings under Superstructures). Moscow, Stroiizdat, 1967, 240 p. Tsytovich N.A., Ter-Martirosyan Z.G. Osnovy priklandnoi geomekhaniki v stroitel'stve (Fundamentals of Applied Soil Mechanics in Construction). Moscow, Vyssh. shkola, 1981, 318 p.

Prochnost' i polzuchest' merzlykh gruntov (Strength and Creep of Frozen Soils). Moscow, Izd-vo AN SSSR, 1963, 218 p. Issledovaniya reologicheskikh svoistv gruntov (Studies of Rheological Properties of Soils). Tr. koordinats. soveshch. po gidrotekhnike. Leningrad, Energiya, 1968, Issue 38, 288 p. Mater~aly~~seso~u~nogo simposiuma po problem reologii gornykh porod i relaksatsii v tverdykh telakh (Proceedings of the All-Union Symposium on Problems of Rheology of Rocks and Relaxation in Solid Bodies). Kiev. Nauk. dumka. 1969. 162 P. ~rud; Vsesoyuznogo~simpo~iuma po reologii gruntov (Proceedings of the All-Union Symposium on Rheology of Soils). Erevan, Izd-vo EGU, 1973, 292 p.; 1978, 406 p.; 1980, 364 p. ~nzhenernoe merzlotovedenie (Permafrost Engineering Studies). Novosibirsk, Nauka, Sib. otd-nie, 1979. Reologiya: Teoriya i prilozhenie (Rheolom: Theory and Application). Moscow, Izd-vo inostr. lit., 1962, 824 p. Folgue J.B. Reologia de solos nao saturados Liboa. 1966, 209 p. Kisiel J., Lysik B; Zarys reologii gruntov (Outlines of ~heology of Soils). W-wa, 1966, 315p. -

Pusch R. Creep of soils. Bochum: Ruhr-Univ., 1979, 74 p. Suklje L. Rheological aspects of soil mechanics. L.; N.Y.: Wiley-Intersci., 1969, 572 p. IUTAM: Symposium Grenoble. Rheology and soil mechanics/Ed. J. Kravtchenko, P.M. Sirieys. B.; N.Y.: Springerverl., 1966, 502 p. Limit analysis and rheological approach in soil mechanics/Ed. W. Olszak, L. Suklje. Wien; New York: Springer-Verl., 1979, 125 p. Scott R.F., Hon-Yim KO. Stress-deformation and strength characteristics. - In: Proc. 7th Intern. Conf. Soil Mech. Found. Eng. State-Art-Volume, Mexico, 1969, p. 291-340. Mello V.F.B. Foundation of buildings in clay. - In: Proc. 7th Intern. Conf. Soil Mech. Found. Eng. State-Art-Volume, Mexico, 1969, p. 49-139. Skempton A.W., Hutchinson J. Stability of natural slopes and embankment. - In: Proc. 7th Intern. Conf. Soil Mech. Found. Eng. State-Art-Volume, Mexico, 1969, p. 291-340. Anderson M., Morgenstern W.R. Physics, chemistry and mechanics of frozen ground. - A review Permafrost second Intern. Conf. North. Amer. contribution. Wash.: Nat. Acad. Sci., 1973, p. 257-288. Zaretskii Yu.K., Lombardo V.N. et al. Ustoichivost' gruntovykh otkosov (Stability of earth slopes). -0snovaniya fundamenty i mekhanika gruntov, 1980, No. 1, pp. 23-27. Bugrov A.K. 0 vliyanii traektorrii nagruzheniya na napryazhenno- deformirovannoe sostoyanie osnavaniya (The effect of the loading path on the stress-strain state of the footing). -Osnovaniya, fundamenty i mekhanika gruntov, 1980, No. 2, pp. 24-26. Kopeikin V.S., Solomin V.I. Raschet peschanogo osnovaniya s pomoshch'yu fizicheski i geometricheski nelineinykh uravnenii (Calculation of sand footings with the help of physically and geometrically non-linear equations). -Osnovaniya, fundamenty i mekhanika gruntov, 1977, No. 1, pp. 30-32. Vyalov S.S. Reologicheskie svoistva i nesushchaya sposobnost' merzlykh gruntov (Rheological properties and bearing capacity of frozen soils). Moscow, Izd-von AN SSSR, 1959, 160 p.

Vyalov S.S., Gmoshinskii V.G., Gorodetskii S.E. et al. Prochnost' i polzuchest' merzlykh gruntov i raschety ledo-gruntovykh ograzhdenii (Strength and creep of frozen soils and calculations of ice-soil protective barriers). Moscow, izd-vo AN SSSR, 1962, 254 p. Vyalov S.S. Plastichnost' i polzuchest' svyaznoi sredy (Plasticity and creep of cohesive media). -In: Doklady k VI Mezhdunar. kongr. po mekhanike gruntov i fundamentostroeniyu (Papers from the VI International Congress of Soil Mechanics and Foundation Engineering). Moscow, Stroiizdat, 1965, pp. 45-54. Gorbunov-Posadov M.1 . Metod resheniya smeshannoi zadachi teorii uprugosti i teorii plastichnosti gruntov (A method of solving a composite problem of the theory of elasticity and of the theory of plasticity of soils). Osnovaniya, fundamenty i mekhanika gruntov, 1971, No. 2, pp. 2-6. Nikolaevskii V.N. Mekhanicheskie svoistva gruntov i teoriya lastichnosti (Mechanical Properties of Soils and the Theory of

glasticity). Moscow, VINITI, 1972, 86 p. Zaretskii Yu.K. Novaya kontseptsiya vyazkoplasticheskogo techeniya gruntov (A new concept of the visco--plastic flow of soils). -In: Tr.

-

3 3 0 Vsesoyuzn. simpoz. po reologii gruntov (Proceedings of the 3rd All-Union Symposium of Soil Rheology). Erevan, Izd-vo EGU, 1980, pp. 58-73. Roscoe K.H. The influence of strains in soil mechanics. - Geotechnique, 1970, vol. 20, No. 2, p. 129-170. Nikolaevskii V.N. Dilatansiya i zakony neobratimogo deformirovaniya gruntov (Dilatancy and the laws of irreversible deformation of soils). -Osnovaniya, fundamenty i mekhanika gruntov, 1979, No. 5, pp. 29-32. Voprosy prochnosti i deformiruemosti gruntov (Problems of Strength and Deformability of Soils). Baku, Azerb. gos. izd-vo, 1966, 110 p. Voprosy mekhaniki gruntov i stroitel'stva na lessovykh osnovaniyakh (Problems of Soil Mechanics and Construction on Loess Footings). Groznyi, Check. -1ng. kn. izd-vo, 1970, 138 p.

50. Zaretskii Yu.K. 0 radiallnom raspredelenii napryazhenii v poluploskosti pod deistviem sosredotochennoi sily (Radial distribution of stresses in a half-plane under the effect of a concentrated force). -In: Issledovaniya po mekhanike gornykh porod (Studies on Mechanics of ~ocks). Alma-Ata, Nauka, 1965, pp. 88-100. -

51. Stroganov A.S. Osnovnye uravneniya i nekotorye zadachi nelineinoi vyazkouprugosti gruntov (Fundamental equations and some problems of non- linear visco-elasticity of soils). -In: Eksperimental 'no-teore ticheskie issledovani ya nelineinykh zadach v oblasti osnovanii i fundamentov (Experimental and Theoretical Studies of Non-linear Problems Pertaining to Footings and Foundations). Novocherkassk, the Novocherkassk Polytechnical Institute, 1979,

knokurov E .F. Iteratsionnyi metod rascheta osnovanii i fundamentov s pomoshch'yu EVM (Interative Method of Calculating Footings and Foundations with the Help of a Computer). Minsk, Nauka i teknika, 1972, 246 p.

53. Shirokov V.N. Solomin V.I., Malyshev M.V., Zaretskii Yu.K. Napryazhennoe sostoyanie i peremeshchenie vesomogo nelineino-deformiruemogo gruntovogo poluprostranstva pod druglym zhestkim shtampom (The stressed state and displacement of a finite soil half-space undergoing non-linear deformation under a rigid spherical loading plate). -Osnovaniya, fundamenty i mekhanika gruntov, 1970, No. 1, pp. 2-5.

54. Malyshev, M.V., Zaretskii Yu.K., Shirokov V.N., Cheremnykh V.A. 0 sovmestnoi rabot zhestkikh fundamentov i nelineino-deformiruemogo poluprostranstva (Combined work of rigid foundations and non-linearly deformed half-space). -In: Tr. k VIII Mezhdunar. kongr. po mekhanika puntov i fundamentostroeniy (Proceedings of the VIII International Congress of Soil Mechanics and Foundation Engineering). Moscow, Stroiizdat, 1973, pp. 97-104.

55. Kryzhanovskii A.L., Chevikin A.S., Kulikov O.V. Effektivnost' rascheta osnovanii s uchetom nelineinykh deformatsionnykh svoistv gruntov (Efficiency of calculations of footings in which non-linear deformation properties of soils are taken into account). -Osnovaniya, fundamenty i mekhanika gruntov, 1975, No. 5, pp. 37-40.

56. Fedorovskii V.G., Kaganovskaya S.E. Zhestkii shtamp na nelineino-deformiruemom svyaznov osnovanii (A grid loading plate on a cohesive footing undergoing non-linear deformation). +snovaniya, fundamenty i mekhanika gruntov, 1975, No. 1, pp. 41-44.

57. Bugrov A.K., Grebnev K.K. Chislennoe reshenie fizicheski nelineinykh zadach dlya gruntovykh osnovanii (Numerical solutions of physically non-linear problems for earth footings). -Osnovaniya, fundamenty i mekhanika gruntov, 1977, No. 3, pp. 39-42.

58. Shofield A., Wroth P. Critical state soil mechanics, London: McGrow Hill, 1968, 306 p.

59. Ioselevich V.A., Didukh B.I. 0 primenenii teorii plasticheskogo uprochneniya k opisaniyu deformiruemosti gruntov (Application of the theory of plastic hardening to description of deformability of soils). -In: Voprosy mekhaniki gruntov i stroitel'stvo na lessovykh osnovaniyakh (Problems of Soil Mechanics and Construction on Loess Footing). Groznyi, Chech.-Ing. kn. izd-vo, 1970, pp. 125-133.

60. Zaretskii Yu.K., Lombardo, V.N., Groshev M.E. Plasticheskoe techenie gruntovykh massivov (Plastic flow of soil masses). -In: Eksperimental'noteoreticheskie issledovaniya nelineinykh zadach v oblasti osnovanii i fundamentov (Experimental and Theoretical Studies of Non-linear Problems Pertaining to Footings and Foundations). Novocherkassk, Novocherkasskii politekhn. in-t, 1979, pp. 16-26.

61. Bugrov A.K. 0 reshenii smeshannoi zadachi teorii uprugosti i teorii plastichnosti gruntov (Solution of a composite problem of the theory of elasticity and theory of plasticity of soils). -Osnovaniya, fundamenty i mekhanika gruntov, 1974, No. 6, pp. 20-23.

62. Matsuoko H., Nokoi T. Stress-strain relationship of soil based on the "SMPw. -In: Proc. 9th Intern. Conf. Soil Mech. Found. Eng. Special Section, Tokyo, 1977, p. 153-162.

63. Lode P.V. Three-dimensional stress-strain behavior and modelling of soils. Bochum: RuhrUniv., 1979, 179 p.

64. Huang Wen-Ki, Pu Yia-Liu, Chem Yu, Jiong nasdening Eul and yield function for soils. - In: Proc. 10th Intern. Conf. Soil Mech. Found. Eng. Stockholm, 1981, vol. 1, p. 631-634.

65. Akai T., Nishi K. Mechanical properties of soft rocks. -In: Proc. 9th Intern. Conf. Soil Mech. Found. Eng. Tokyo, 1977, vol. 1, p. 7-10.