DIRECTION OF PLASTIC FLOW - Application of Plasticity Theory to Soil Behavior: A New Sand Mo

CHAPTER V CHAPTER V

5.3 DIRECTION OF PLASTIC FLOW

Contrary t o g e n e r a l p r a c t i c e i n e l a s t i c - p l a s t i c models, t h e e l a s t i c c o n t r i b u t i o n may be coupled w i t h t h e p l a s t i c response. The e l a s t i c parameters, such a s t h e b u l k and s h e a r modulus, may depend n o t only upon t h e s t r e s s but upon t h e i n t e r n a l v a r i a b l e s . When no p l a s t i c s t r a i n s a r e c r e a t e d , i.e., when t h e i n t e r n a l v a r i a b l e s a r e unchanged, t h e e l a s t i c model s t i l l p r e d i c t s no energy d i s s i p a t i o n o r c r e a t i o n . When p l a s t i c flow occurs, t h e e l a s t i c moduli become dependent on t h e i n t e r n a l v a r i - a b l e s , and t h e e l a s t i c model may c r e a t e o r d i s s i p a t e energy, which any- way becomes meaningless s i n c e energy i s d i s s i p a t e d a t t h e same time by p l a s t i c s t r a i n .

Such an e l a s t i c - p l a s t i c coupling, i f used i n t h e f u t u r e , w i l l allow a b e t t e r d e s c r i p t i o n of d e n s i f i c a t i o n . For i n s t a n c e i n o r d e r t o r e p r e s e n t t h e d e n s i f i c a t i o n of Sacramento River sand, t h e m a t e r i a l c o n s t a n t s B ₀ and n may be s e l e c t e d a s f u n c t i o n s ( i n a d d i t i o n t o s t r e s s ) of i n t e r n a l v a r i a b l e s , such a s t h e p l a s t i c v o l u m e t r i c s t r a i n .

much a s p o s s i b l e , well-known and observed a s p e c t s of t h e behavior of g r a n u l a r m a t e r i a l , i n p a r t i c u l a r , t h e t h r e e main concepts known a s t h e c r i t i c a l s t a t e , t h e c h a r a c t e r i s t i c s t a t e and t h e s t r e s s - d i l a t a n c y r e l a - t i o n s . Each of t h e s e p o i n t s , which a r e f i r s t reviewed and t h e n v e r i f i e d f o r t h e dense Sacramento River sand, enforce c o n s t r a i n t s on t h e r e l a t i o n between p l a s t i c flow d i r e c t i o n and s t r e s s s t a t e , and consequently on t h e p o s s i b l e shape of t h e bounding s u r f a c e . Ultimately, a bounding s u r f a c e i s s e l e c t e d ; i t s p r e d i c t i o n of t h e p l a s t i c flow d i r e c t i o n i s compared w i t h r e a l v a l u e s o b t a i n e d f o r t h e Sacramento River sand. Since t h e bounding s u r f a c e e q u a t i o n i s e s t a b l i s h e d from g e n e r a l o b s e r v a t i o n s on g r a n u l a r m a t e r i a l , i t may be a p p l i e d t o any sand.

5.3.1 C r i t i c a l S t a t e

A s defined by Schof i e l d and Wroth, 15.223 t h e c r i t i c a l s t a t e i n s o i l mechanics i s a n asymptotic s t a t e , e v e n t u a l l y reached during loading, c h a r a c t e r i z e d by no volume change, no s t r e s s change and i n f i n i t e d e v i a t o r i c s t r a i n : "It i s a s i f t h e m a t e r i a l h a s melted under s t r e s s . "

For axisymmetric s t a t e s of s t r e s s and s t r a i n , t h i s d e f i n i t i o n i s t r a n s l a t e d i n t o t h e f o l l o w i n g mathematical terms:

According t o Schof i e l d and Wroth, t h e c r i t i c a l s t a t e h a s a l o c a t i o n i n t h e p-q-e space given by t h e following r e l a t i o n s :

It i s d e f i n e d o n l y by four m a t e r i a l c o n s t a n t s :

I"

value of c r i t i c a l void r a t i o a t u n i t mean p r e s s u r e .

A s l o p e of c r i t i c a l s t a t e l i n e p r o j e c t i o n i n Xn(p)-e plane.

Mc,Ilfe s l o p e s of p r o j e c t i o n o f c r i t i c a l s t a t e l i n e s i n p-q plane corresponding t o p o s i t i v e and n e g a t i v e d e v i a t o r i c s t r e s s .

For a c l a y , normally o r s l i g h t l y overconsol ida t e d , t h e c r i t i c a l s t a t e l i n e i s unique, and i t s p r o j e c t i o n i n t h e In(p)-e plane i s p a r a l l e l t o t h e v i r g i n c o n s o l i d a t i o n l i n e (response of a normally c o n s o l i d a t e d sample t o a n h y d r o s t a t i c compression). It does n o t depend on t h e previous l o a d i n g h i s t o r y of t h e m a t e r i a l .

For most sands, t h e c r i t i c a l s t a t e i s d i f f i c u l t t o e x h i b i t experimentally. A s observed by Lade C5.111 and o t h e r e x p e r i m e n t a l i s t s , t h e

s t r a i n s w i t h i n a sand sample become o f t e n non-uniform, f o r i n s t a n c e , due t o a s t r a i n - l o c a l i z a t i o n such a s a shear band. The volume measurement, g e n e r a l l y performed w i t h i n t e r s t i t i a l w a t e r s a t u r a t i n g t h e sample, u n d e r e s t i m a t e s t h e v o l u m e t r i c s t r a i n i n t h e shear band, a c t i v e p a r t of t h e sample. The d i s c o n t i n u i t i e s , even more a c c e n t u a t e d i n t h e presence

of l a r g e deformation (e.g., 20% of a x i a l s t r a i n i n t r i a x i a l t e s t ) , render t h e a s w ~ t o t i c c r i t i c a l s t a t e more d i f f i c u l t t o determine experimentally i n sands.

However, t h e c r i t i c a l s t a t e remains a v a l i d assumption i n t h e case of t h e Sacramento r i v e r sand. I n Fig. 5.3, t h e experimental p o i n t s of v o i d r a t i o v e r s u s mean p r e s s u r e a t c r i t i c a l s t a t e obey a r e l a t i o n

s i m i l a r t o ( 5 . 1 1 ~ ) f o r d r a i n e d and undrained t e s t s a t d i f f e r e n t confining p r e s s u r e s . The c r i t i c a l s t a t e f o r sands depends on t h e i n i t i a l sand d e n s i t y , and i s n o t p a r a l l e l t o t h e i s o t r o p i c c o n s o l i d a t i o n response curve.

By applying a l i n e a r r e g r e s s i o n a n a l y s i s t o t h e d a t a of Tables 5.3, corresponding t o t h e e q u a t i o n (5.131, t h e following v a l u e s were found (Fig. 5.4):

dense sand : Mc = 1.38

, I(

= 0.88

,

5 = 0.088 and

loose sand : M = 1.35

,

⁼1.0

,

5 ⁼0.084

Within t h e c o n t e x t of an e l a s t i c - p l a s t i c t h e o r y , t h e c r i t i c a l s t a t e r e q u i r e s t h e p l a s t i c s t r a i n increment t o be p a r a l l e l t o t h e q a x i s , which means t h a t t h e unit v e c t o r d e f i n i n g t h e p l a s t i c flow d i r e c t i o n h a s t h e following components:

TABLE 5.3 a CRITICAL STATE DATA FROM DRAINED AND UNDRAINED TESTS ON DENSE S A C W N T O RIVER SAND (AFTER SEED [5.12,5.231)

TABLE 5.3b CRITICAL STATE DATA FROM DRAINED AND UNDRAINED TESTS ON

LOOSE SACRAEifENTO RIVER SAND (AFTER SEED [ 5.1 2,5.23

1

)

I n i t i a1 Void

l R a t i o

M 1.40 1.37 1.43 1.3 9 1 . 3 1 t e s t s exc 1 ude d ( c a v i t a t i o n )

1.42 1 . 4 1 1 . 3 0 P

kg/cm 2 2.64 7.53 16.3 27.09 46.4

5.92 34.7 47.5 43.6 43.2 41.2

R a t i o I n i t i a l

Void R a t i o e

0 0.638 0.629 0.61 0.592 0.576 0.628 0.606 0.587 0.587 0.571 0.564

I n i t i a l C o n f i n i n g P r e s s u r e I n i t i a l C o n f i n i n g

P r e s s u r e (kg/cm 2

1 3 6 10.5 2 0

1 1 0 . 5 1 5 . 1 20.2 29.9 40.1

C r i t i c a l Void R a t i o 0.826 0.719 0.691 0.617 0.576 0.628 0.606 0.587 0.587 0.571 0.564

C r i t i c a l Void p kg/cm2

1.88 5.51 11.43 19.55 35.47 3.82 23.0 31.8 30.63 30.7 31.8

The c r i t i c a l s t a t e i m p l i e s a l s o t h a t t h e amplitude of t h e p l a s t i c s t r a i n increment i s i n f i n i t e , i. e., t h e p l a s t i c modulus H becomes zero,

5 .3 .2 CHARACJXRISTIC STATE

F i r s t observed e x p e r i m e n t a l l y by Shibata and Karube 15.241

,

t h e

" c h a r a c t e r i s t i c s t a t e " was d e f i n e d Luong 15.151 a s t h e s t r e s s s t a t e where t h e r a t e of volumetric s t r a i n becomes equal t o zero. D i f f e r e n t from t h e c r i t i c a l s t a t e , which i s always obtained f o r l a r g e s t r a i n s , t h e c h a r a c t e r i s t i c s t a t e corresponds t o small deformations. But l i k e t h e c r i t i c a l s t a t e , t h e c h a r a c t e r i s t i c s t a t e i s independent of t h e i n i t i a l d e n s i t y . The c h a r a c t e r i s t i c s t a t e s e p a r a t e s t h e c o n t r a c t i n g and d i l a t i n g b e h a v i o r s (Fig. 5-41. The c o n t r a c t i o n o c c a r s i n t h e s u b c h a r a c t e r i s t i c r e g i o n , which i s bounded i n t h e p-q plane by t h e c h a r a c t e r i s t i c s t a t e l i n e s , and t h e d i l a t i o n t a k e s place i n t h e s u p e r c h a r a c t e r i s t i c region.

According t o Luong

'

s experiments on Fontainebleaa sand, t h e c h a r a c t e r i s t i c s t a t e l i n e s a r e s i m i l a r t o t h e c r i t i c a l s t a t e l i n e s of r e l a t i o n (5.11) :

F i g . 5 . 4 . C h a r a c t e r i s t i c s t a t e l i n e from d r a i n e d t e s t s a t d i f f e r e n t c o n f i n i n g p r e s s u r e s and c r i t i c a l s t a t e l i n e from d r a i n e d and undrained t e s t s a t d i f f e r e n t c o n f i n i n g p r e s s u r e s ; a l l t e s t s a r e performed o n t h e l o o s e o r dense Sacramento R i v e r sand.

where M* ,M*e a r e two m a t e r i a l c o n s t a n t s , g e n e r a l l y s l i g h t l y d i f f e r e n t

from Mc and Me. The dense Sacramento River sand obeys a l s o r e l a t i o n (5.13a), a s shown i n Fig. 5.4.

However Luong's c h a r a c t e r i s t i c s t a t e i s n o t s a t i s f a c t o r y . T e n s i l e t r i a x i a l t e s t s a t c o n s t a n t c o n f i n i n g p r e s s u r e performed by Robine t 15.191 h 9 shown t h a t some dense sands d i l a t e continuously from t h e beginning of d e v i a t o r i c loading. This absence of c o n t r a c t i o n i m p l i e s t h a t t h e s u b c h a r a c t e r i s t i c domain does n o t e x i s t and consequently v i o l a t e s Lnong 's c h a r a c t e r i s t i c s t a t e . This d e f i c i e n c y may be avoided i f t h e c h a r a c t e r i s t i c s t a t e i s r e d e f i n e d a s t h e s t r e s s s t a t e where t h e r a t e of p l a s t i c ( i n s t e a d of t o t a l ) volumetric s t r a i n becomes equal t o zero. This new s t a t e i s s t i l l d e f i n e d by t h e r e l a t i o n s (5.13). But t h e continuous d i l a t i o n observed by Robinet C5.191 i n t e n s i l e t e s t i s now j u s t i f ied: t h e e l a s t i c v o l u m e t r i c s t r a i n may be n e g a t i v e ( d i l a t i n g ) and l a r g e r i n a b s o l u t e v a l u e t h a n t h e p o s i t i v e p l a s t i c s t r a i n ( c o n t r a c t i n g )

,

s o t h a t t h e t o t a l volumetric s t r a i n i s n e g a t i v e ( d i l a t i n g ) . Within t h e context of p l a s t i c i t y , t h e c h a r a c t e r i s t i c s t a t e , l i k e t h e c r i t i c a l s t a t e , r e q u i r e s t h e u n i t v e c t o r , c o l l i n e a r t o t h e p l a s t i c flow, t o s a t i s f y r e l a t i o n s (5.12), while t h e s t r e s s s t a t e s a t i s f i e s (5.13). How- e v e r , no c o n d i t i o n i s imposed on t h e p l a s t i c modulus H.

The " c h a r a c t e r i s t i c s t a t e " i s a n a t t r a c t i v e f e a t u r e t o r e p r e s e n t t h e c y c l i c behavior of sand, which may be i l l u s t r a t e d by t h e complex

c y c l i c t e s t performed by Luong E5.141 and shown i n Fig. 5.5. During t h i s drained t e s t a t c o n s t a n t confining p r e s s u r e , e i g h t successive s e r i e s of twenty c y c l e s of d e v i a t o r i c s t r e s s q were applied t o Fontainebleaa sand. These c y c l e s had a constant amplitude (0.1 MPa), a r e centered f o r d i f f e r e n t v a l u e s of t h e r a t i o q/p, and were d i s t r i b u t e d on both s i d e s of t h e c h a r a c t e r i s t i c s t a t e l i n e (Fig. 5.5a). The r e s u l t - ing volumetric s t r a i n s , which a r e shown v e r s u s q i n Fig. 5.5b, change i n agreement w i t h t h e c h a r a c t e r i s t i c s t a t e . I n t h e s u b c h a r a c t e r i s t i c domain, when t h e r a t i o q/p i s between -0.75 and 1.26, densif i c a t i o n i s observed, whereas, i n t h e s u p e r c h a r a c t e r i s t i c domain, d i l a t i o n i s recorded. The c h a r a c t e r i s t i c l i n e remains f i x e d even during such a complex cycl i c 1 oading h i s t o r y

.

5.3.3 Stress-Dilatancy Theories

A l l s t r e s s d i l a t a n c y t h e o r i e s have one common goal: t o e x p l a i n how granular m a t e r i a l d i l a t e s while i t i s subjected t o shear s t r e s s e s .

Since they give q u a l i t a t i v e and q u a n t i t a t i v e information on t h e d i r e c - t i o n of t h e s t r a i n increment, they, t o g e t h e r w i t h t h e c h a r a c t e r i s t i c and c r i t i c a l s t a t e s , a r e u s e f u l f o r c h a r a c t e r i z i n g t h e p l a s t i c flow d i r e c t i o n . The f i r s t of these t h e o r i e s was developed i n 1962 by Rowe

[5.21] and placed on a mathematical b a s i s i n 1965 by Horne 15.91. Since then, o t h e r t h e o r i e s have appeared: Tatsuoka 15.271

,

Nwa [5.171, Momen and Ghabonssi 15.161.

F i g . 5 . 5 a . Drained c y c l i c l o a d i n g a t c o n s t a n t c o n f i n i n g p r e s s u r e on F o n t a i n e b l e a u sand ( a f t e r Luong 15 .I51 )

.

F i g . 5 . 5 b . Drained c y c l i c l o a d i n g a t c o n s t a n t c o n f i n i n g p r e s s u r e on F o n t a i n e b l e a u sand ( a f t e r Luong [5.151).

A s a n i l l u s t r a t i o n , only Rowets and N w a t s t h e o r i e s a r e presented.

5.3.3.a Rowe's S t r e s s D i l a t a n c y Theory (1962)

Rowe C5.211 c o n s i d e r s , experimentally and t h e o r e t i c a l l y , t h e behavior of a s s e m b l i e s of c o h e s i o n l e s s , s p h e r i c a l p a r t i c l e s of uniform s i z e , arranged i n i t i a l l y i n r e g u l a r a r r a y s . These assemblies a r e s u b j e c t e d t o a n axisymmetric s t a t e of s t r e s s a s defined i n r e l a t i o n (2.51, with t h e a x i s of symmetry c o i n c i d i n g w i t h t h e a x i s of symmetry of t h e i n i t i a l packing. The a s s o c i a t e d s t r a i n s s a t i s f y t h e r e l a t i o n (2.3).

I n t r o d u c i n g t h e a n g l e of s o l i d f r i c t i o n between p a r t i c l e s , denoted by

d

and assumed t o be uniform and independent of p r e s s u r e , Rowe obtained e x p e r i m e n t a l l y a r e l a t i o n between s t r e s s and s t r a i n increments i n t h e f 01 lowing way :

where a l l s t r a i n s and s t r e s s e s a r e d e f i n e d i n s e c t i o n s 2.2 and 2.3.

The v a l i d i t y of Rowets t h e o r y i s t e s t e d i n Fig. 5.6 by p l o t t i n g

Q v e r s u s 1

-

^dsv/dsl i n t h e s p e c i a l case of t h e l o o s e and dense Sacramento River sand s u b j e c t e d t o d r a i n e d t e s t s a t d i f f e r e n t c o n f i n i n g p r e s s u r e . I n s p i t e of a n o t i c e a b l e s c a t t e r i n g , p a r t i a l l y due t o f i n i t e increments of s t r a i n , dsl and ds,, t h e experimental p o i n t s t e n d t o l i e on a s t r a i g h t l i n e , a s i n d i c a t e d i n r e l a t i o n (5.14). Fran Fig. 5.6 and r e l a t i o n (5.14) t h e average a n g l e of f r i c t i o n

d

i s found eq-1 t o 32'

f o r t h i s s o i l .

constant fT3 fkglcln 2 1

a 1

-

⁰ ³

0 6

A 10.5

bm v ²⁰

4 , -

6

1 + d € v l d € l

0 I A

F i g . 5 . 6 . Rowe's s t r e s s - d i l a t a n c y t h e o r y a p p l i e d t o d r a i n e d t e s t s w i t h c o n s t a n t c o n f i n i n g p r e s s u r e performed o n t h e Sacramento River sand.

a ) l o o s e sand b) d e n s e sand.

Within t h e context of p l a s t i c i t y theory, i f t h e e l a s t i c s t r a i n increment i s n e g l i g i b l e w . r . t . t h e p l a s t i c one, t h e r e l a t i o n (5.14) i n d i c a t e s t h a t t h e p l a s t i c flow d i r e c t i o n , represented by del/daV, depends upon t h e o b l i q u i t y of t h e s t r e s s s t a t e , c h a r a c t e r i z e d by al/a3.

For t h e loose and dense Sacramento River sand, c o n t r a c t i o n occurs i f t h e s t r e s s r a t i o a i s l e s s t h a n 3, followed by d i l a t i o n i f a1/a3 i s g r e a t e r t h a n 3 (Fig. 5.6). This r e s u l t a g r e e s w i t h Luong's c h a r a c t e r i s - t i c s t a t e .

5.3.3 .b Nova's Theory (1982)

Retaining t h e n o t i o n of c h a r a c t e r i s t i c s t a t e and t h e dependence of p l a s t i c flow d i r e c t i o n on t h e s t r e s s o b l i q u i t y , Nova (5.17) proposed a d i f f e r e n t s t r e s s - d i l a t a n c y theory based on t h e experimental work by Strond [5.261, and defined a s follows:

This r e l a t i o n i s n o t s a t i s f a c t o r y f o r t h e i s o t r o p i c s t a t e (q=O) s i n c e i t p r e d i c t s a p l a s t i c d e v i a t o r i c s t r a i n . I n order t o c o r r e c t t h i s d e f i c i e n c y , Nova a s s m e s t h a t , f o r low values of t h e r a t i o

3

^{t h e}

s t r e s s - d i l a t a n c y i s governed by another equation:

where t h e c o n s t a n t a i s found by assuming a smooth t r a n s i t i o n between r e l a t i o n s (5.15) and (5.161,

T h i s t r a n s i t i o n i s reached when t h e r a t i o 9 i s equal t o M

1.

Nova's s t r e s s d i l a t a n c y theory i s checked by p l o t t i n g i n Fig. 5.7 t h e r a t i o det/ds: v e r s u s ^q= q/p f o r t h e dense Sacramento River sand. A l l r e s u l t s a r e o b t a i n e d by a s p e c i a l computer code E5.21, which u s e s t h e f o l l o w i n g technique. A f t e r s e l e c t i n g t h e e l a s t i c model

i n s e c t i o n 5.2.2, t h e incremental e l a s t i c response dae,dee i s c a l c u l a t e d v q

f o r t h e f i n i t e s t r e s s increment dp, dq which connects two s u c c e s s i v e experimental s t r e s s s t a t e s . Then t h e p l a s t i c s t r a i n increments d~!r d s i a r e obtained by s u b t r a c t i o n of t h e c a l c u l a t e d e l a s t i c s t r a i n increment from t h e t o t a l s t r a i n increment drvD daq which r e l a t e s two s u c c e s s i v e recorded s t r a i n s t a t e s ^E,,, a T h e r e f o r e t h e r a t i o dav/daq i s a v a i l a b l e

P' a s a f u n c t i o n of t h e r a t i o ^q= q/p.

The i m p o r t a n t s c a t t e r i n g i n Fig. 5.7 i s p a r t i a l l y due t o t h e s p a r s e r e c o r d i n g s of t h e s t r a i n and s t r e s s s t a t e s p, q, and ^Ev * a ^q*which produce t o o l a r g e s t r a i n o r s t r e s s increments between two s u c c e s s i v e s t a t e s . It i s a l s o p a r t i a l l y caused by t h e e l a s t i c model s e l e c t e d i n s e c t i o n 5.2.2. However, b e a r i n g i n mind t h e s e sources of e r r o r s , Fig. 5.7 shows t h a t da;/dsP depends upon t h e r a t i o ^qand t h i s i s n o t i c e -

a b l e f o r any type of t e s t s , d r a i n e d o r =drained, a t any c o n f i n i n g pres- sure. I n Fig. 5.7bD during t h e undrained t e s t s ( c o n s t a n t volume), t h e

F i g . 5 . 7 . D i r e c t i o n o f p l a s t i c f l o w , r e p r e s e n t e d by deP/dsP v e r s u s t h e r a t i o q during d i f f e r e n t t e s t s on t h e dense ~ a c r i e n t o River sand.

a ) d r a i n e d t e s t s a t c o n s t a n t c o n f i n i n g p r e s s u r e

b) undrained t e s t s a t c o n s t a n t t o t a l c o n f i n i n g p r e s s u r e .

q u a n t i t y dsP/deP i s always p o s i t i v e and smaller t h a n during drained

v q

t e s t s (Fig. 5.7a). This discrepancy r e s u l t s from t h e e l a s t i c model;

during a n undrained t e s t , i n o r d e r t o keep t h e volume constant, t h e p l a s t i c volumetric s t r a i n i s equal but opposite t o t h e e l a s t i c volumetric s t r a i n . Fran Fig. 5.7, f o r low values of q, da:/daP i s

4 p o s i t i v e ( t h e sand c o n t r a c t s ) . When q exceeds some f i x e d value c l o s e t o 1.4. de:/dsP becomes negative ( t h e sand d i l a t e s ) . After reaching a

minimum negative value, dsf/deP i n c r e a s e s back t o , and s t o p s f i n a l l y a t 4

z e r o ( c r i t i c a l s t a t e ) . From Fig. 5.7a, t h e branch, along which deP/deP

v q

decreases i s d i f f e r e n t from t h e branch where i t i n c r e a s e s back t o zero.

This n o w r e v e r s i b l e phenomenon i s c e r t a i n l y due t o l o s s of measurement accuracy f o r l a r g e deformation.

I n summary, r e l a t i o n s (5.15) and (5.16) d e s c r i b e qua1 i t a t i v e l y t h e d i r e c t i o n of t h e p l a s t i c flow shown i n Fig. 5.7; however, t h e s c a t t e r i n g of t h e experimental r e s u l t s p r e v e n t s us from c a l c u l a t i n g t h e v a l u e s f o r p and M of r e l a t i o n (5.15).

5.3.4 D e f i n i t i o n of t h e New Bounding Surface

So f a r , from t h e review of t h e c r i t i c a l s t a t e , c h a r a c t e r i s t i c s t a t e and t h e s t r e s s d i l a t a n c y t h e o r i e s , some c o n s t r a i n t s have been imposed on t h e p l a s t i c flow d i r e c t i o n . These r e s t r i c t i o n s a r e smn- marized i n Table 5.4 by using t h e u n i t v e c t o r , with components n and

P n c o l l i n e a r t o t h e p l a s t i c flow. Also from t h e experimental r e s u l t s

on t h e Sacramento River sand (Fig. 5.41, no d i f f e r e n c e appears between t h e c r i t i c a l and t h e c h a r a c t e r i s t i c s t a t e s . For convenience, and

-

¹¹⁵

-

TABLE 5.4 SUMMARY OF CONTWAINTS ON THE DIRECTION OF PLASTIC FLOW T r a n s l a t i o n i n E l a s t i c - P l a s t i c Terms i f 9 = I * ~ o r M * ~ ,

t h e n n = 0 and n = 1

P 9

no c o n d i t i o n on H

i f M * ~

< <

^{M * ~ ,}

P t h e n n

>

⁰

i f 9

<

M * ~ o r 9

>

M * ~ ,

P P

t h e n n

<

⁰

i f = M o r Dlo.

P C

t h e n n = 0 and n = 1

P q

-

H = O

n and n a r e f u n c t i o n s of 9

P 9 P

Nature of C o n s t r a i n t A t t h e c h a r a c t e r i s t i c s t a t e ( r e l a t i o n 5.13) t h e p l a s t i c v o l u m e t r i c s t r a i n deP e q u a l s z e r o .

I n t h e s u b c h a r a c t e r i s t i c domain, de P i s p o s i t i v e

( c o n t r a c t i n g ) . v

-

I n t h e s u p e r c h a r a c t e r i s - t i c domain, deP i s n e g a t i v e ( d i l a x i n g )

.

A t t h e a s s y m p t o t i c c r i t - i c a l s t a t e ( r e l a t i o n 5 . I l l , t h e p l a s t i c v o l u m e t r i c s t r a i n de P v e q u a l s z e r o , and

t h e p l a s t i c d e v i a t o r i c s t r a i n deP i s i n f i n i t e

The p l a s t i c flow d i r e c - t i o n depends on t h e s t r e s s o b l i q u i t y , i. e., 9.

P -,

O r i g i n of C o n s t r a i n t

S t a t e

C r i t i c a l S t a t e

S t r e s s - d i l a t a n c y

invoking t h e d i f f i c u l t y of d e f i n i n g a c c u r a t e l y both s t a t e s , t h e c r i t i c a l and t h e c h a r a c t e r i s t i c s t a t e s a r e assumed t o c o i n c i d e i n t h e p-q space, i . e . , r e l a t i o n s (5.13) and (5.11) a r e i d e n t i c a l .

I n conclusion, from t h e experimental o b s e r v a t i o n s on t h e Sacramento River sand and from t h e l i t e r a t u r e review, t h e u n i t v e c t o r normal t o t h e bounding s u r f a c e obeys t h e f o l l o w i n g r u l e : when q i s p o s i t i v e , t h e component n i s p o s i t i v e , zero, o r n e g a t i v e depending on whether t h e

r a t i o 9 i s r e s p e c t i v e l y s m a l l e r t h a n Mc, equal t o Mc, o r l a r g e r t h a n Mc;

when q i s n e g a t i v e , t h e same r e s u l t holds f o r n by s u b s t i t u t i n g Me f o r P

Since t h e r a t i o i s o n l y a f u n c t i o n of t h e r a t i o t h e following n 9

e q u a t i o n i s o b t a i n e d

which r e s u l t s from e q u a t i o n (3.72). The b a r added t o dp and dq r e f e r s t o t h e image p o i n t l y i n g on t h e bounding s u r f a c e . I n o r d e r t o s o l v e t h e e q u a t i o n (5.18), a r e l a t i o n between t h e image and s t r e s s s t a t e must be s p e c i f i e d , One such a r u l e i s r a d i a l mapping, which i s s p e c i f i e d i n r e l a t i o n ( 4 . 6 ) . and becomes h e r e

where x i s s c a l a r . When s e l e c t i n g such a mapping, t h e e q u a t i o n (5.18) becomes hmogeneous, and may be solved t o give t h e bounding s u r f a c e equation. However, according t o t h e experimental s c a t t e r i n g observed i n Fig. 5.7, t h e f u n c t i o n

k]

i s d i f f i c u l t t o d e f i n e . The following

a l t e r n a t i v e approach i s t h e r e f o r e p r e f e r r e d .

Keeping t h e r a d i a l mapping, a s enunciated i n r e l a t i o n (5.191, a simple s u r f a c e , w i t h a s u i t a b l e normal, i s chosen. Composed of p o r t i o n s of e l l i p s e s , i t i s d e s c r i b e d by t h e following e q u a t i o n s : i f M i s

( c o n t r a c t i n g domain),

and i f

141 >

M;, ( d i l a t i n g domain),

where )I i s equal t o M o r Me, depending on whether

<

^{i s} p o s i t i v e o r

negative. P l o t t e d i n Fig. 5.8 t h i s s u r f a c e h a s a general e q u a t i o n

From t h e r e l a t i o n s (5.19) and ( 5 . 2 0 ~ ) t h e s t r e s s s t a t e and t h e image p o i n t s a r e r e l a t e d such t h a t :

F i g . 5 . 8 . The bounding s u r f a c e i n t h e p-q s p a c e .

where y i s found by s o l v i n g t h e following equation:

f(yA* yA ,A,M.a,p) = 0

The e q u a t i o n (5.22) y i e l d s t h e f o l l o w i n g r e s u l t s

and i f Iql

>

^Mp,

where

Y = 2 ( a-z) a+( a-2) z 2

The d i s t a n c e 6 , connecting t h e s t r e s s s t a t e and t h e image p o i n t , i s given a s f o l l o w s

8 = [(p-;12 + (q-<12

y

and, by using t h e r e l a t i o n s (5.211, i t becomes

The u n i t v e c t o r normal t o t h e bounding s u r f a c e a t t h e image p o i n t i s c a l c u l a t e d from (3.571, (5.20) and (5.231, i n t h e d i l a t i n g domain t o have components

where

and i n t h e c o n t r a c t i n g domain

where

The Parameter M e , l c d e f i n e t h e c o n t r a c t i n g and d i l a t i n g domain. The d i r e c t i o n of p l a s t i c flow i s governed by two parameters: one i n t h e c o n t r a c t i n g domain, p , and a n o t h e r i n t h e d i l a t i n g domain, a. The parameter A, which c o n t r o l s t h e s i z e of t h e bounding s u r f a o e , does n o t a f f e c t t h e p l a s t i c flow d i r e c t i o n ,

The q u a n t i t i e s n /n c a l c u l a t e d from r e l a t i o n s (5.251, (5.261, a r e P Q'

p l o t t e d v e r s u s t h e r a t i o 9 f o r d i f f e r e n t v a l u e s of t h e c o n s t a n t s p and a P

i n Fig. 5.9. Although no p e r f e c t agreement between t h e o r e t i c a l and experimental r e s u l t s i s p o s s i b l e , due t o t h e s c a t t e r i n g of experimental p o i n t s , t h e p l a s t i c flow d i r e c t i o n p r e d i c t e d by t h e bounding s u r f a c e a g r e e s w i t h t h e observed v a l u e s .

Dalam dokumen Application of Plasticity Theory to Soil Behavior: A New Sand Model (Halaman 105-129)