Application of Plasticity Theory to Soil Behavior: A New Sand Model

APPeICATIOM OF PLASTICITY THEOKY TO SOIL BERAVIORr A NEW SAHD MODEL

In P a r t i a l Falf i l l m a n t sf t h e Reqeirement s

f o r the Degree of Doctor of P h i l o s o p h y

C a l i f o r n i a I n s t i t u t e sf Technology Pasadena, C a l i f o r n i a

(Submitted September 2 7 , 1983)

A m e s p a r e n t s (To my p a r e n t s )

-iii-

ABSTRACT

The r e p r e s e n t a t i o n of r h e o l o g i c a l s o i l b e h a v i o r by c o n s t i t u t i v e e q u a t i o n s i s a new branch of s o i l mechanics which h a s been expanding f o r 3 0 y e a r s . Based on continuum mechanics, numerical methods ( f i n i t e e l e - ments) and experimental t e c h n i q u e s , t h i s new d i s c i p l i n e a l l o w s p r a c t i c - ing e n g i n e e r s t o s o l v e complex g e o t e c h n i c a l problems. A1 though a l 1 s o i l s a r e c o n s t i t u t e d of d i s c r e t e mineral p a r t i c l e s , f o r c e s and d i s - placements w i t h i n them a r e r e p r e s e n t e d by continuous s t r e s s e s and s t r a i n s . Most s t r e s s - s t r a i n r e l a t i o n s h i p s , which d e s c r i b e t h e s o i l b e h a v i o r , a r e d e r i v e d from p l a s t i c i t y theory. O r i g i n a t e d f o r m e t a l s , t h e conventional p l a s t i c i t y i s p r e s e n t e d and i l l u s t r a t e d s i m u l t a n e o u s l y w i t h a metal and a s o i l model. Each p l a s t i c i t y concept may be c r i t i c i z e d when a p p l i e d t o s o i l . A r e c e n t t h e o r y , c a l l e d "bounding s u r f ace p l a s t i c i t y ,

"

g e n e r a l i z e s t h e conventional p l a s t i c i t y and d e s c r i b e s more a c c u r a t e l y t h e c y c l i c r e s p o n s e s of m e t a l s and c l a y s . T h i s new t h e o r y i s f i r s t p r e s e n t e d and l i n k e d w i t h t h e c o n v e n t i o n a l p l a s t i c i t y , t h e n a p p l i e d t o a new m a t e r i a l , sand. Step by s t e p a new sand model i s c o n s t r u c t e d , mainly from d a t a a n a l y s i s w i t h an i n t e r a c t i v e computer code. I n i t s p r e s e n t development, only monotonic l o a d i n g s a r e i n v e s t i g a t e d . I n o r d e r t o v e r i f y t h e model a b i l i t y t o d e s c r i b e sand r e s p o n s e s , i s o t r o p i c , d r a i n e d and undrained t e s t s on t h e dense Sacramento R i v e r sand a r e s i m u l a t e d n u m e r i c a l l y and compared w i t h r e a l t e s t r e s u l t s and p r e d i c t i o n s w i t h a n o t h e r model. F i n a l l y t h e new c o n s t i t u t i v e e q u a t i o n , which was formulated i n t h e p-q space f o r

axisymmetric l o a d i n g s , i s g e n e r a l i z e d i n t h e six-dimensional s t r e s s s t a t e w i t h the assumption o f i s o t r o p y and a p a r t i c u l a r

ode's

a n g l e c o n t r i b u t i o n . T h i s new model i s ready t o be used i n f i n i t e element c o d e s t o r e p r e s e n t a sand behavior.

I would l i k e t o e x p r e s s my s i n c e r e a p p r e c i a t i o n t o my a d v i s o r , P r o f e s s o r R.F. S c o t t , f o r h i s p a t i e n c e , guidance and f i n a n c i a l s u p p o r t d u r i n g my s t u d i e s and r e s e a r c h e s a t t h e C a l i f o r n i a I n s t i t u t e of Tech- nology. I am p a r t i c u l a r l y indebted t o him f o r l e t t i n g me t e a c h a term of h i s g r a d u a t e s o i l mechanics course.

My g r a t i t u d e i s a l s o extended t o Mrs. Sharon Beckenbach and Mrs.

G l o r i a Jackson f o r t h e i r accomplished t y p i n g of t h e manuscript

-v i-

TABLE OF CONTENTS

PAGE

ABSTRACT

. . .

ⁱⁱⁱ

CHAPTER 11: THE MATERIAL SOIL AS A CONTINUUM

. . .

¹⁰

2.1. APPLICATION OF CONTINUUM MECHANICS TO ASSEMBLY OF DISCRETE PARTICLES

. . .

¹⁰

. . . .

2 2 DESCRIPTION OF KINEMATICS 17 CHAPTER 111: CLASSICAL PLASTICITY THEORY

. . .

²¹

. . . .

3.1

FUNDAMENTAL

PLASTICITY CONCEPTS F'ROM UNIAXIAL TESTS 2 1

. . .

3.2 GENERALIZATIONTOSIX-DIMENSIONAL STRESS.SPACE 26 3.2.1 Existence of Unrecoverable S t r a i n : Yield C r i t e r i o n

. . . . . .

²⁹

3.2.2 D i r e c t i o n of P l a s t i c Flow : Flow Rule 39

. . . .

3.2.3 Amplitude of P l a s t i c F l o w : HardeningRules 43 3.3 FORMULATION OF ELASTIC-PLASTIC RELATIONS FOR AXISYMMETBIC LOADING

. . .

⁵⁰

3.4 ISOTBOPIC ELASTIC-PLASTIC CONSTITUTIVE EQUATIONS

. . .

⁵³

3.4.1 I s o t r o p i c E l a s t i c - P l a s t i c Models

. . .

⁵⁴

3.4.2 I s o t r o p i c Models i n t h e ^p-q Space

. . .

⁵⁸

3

.

^{5 A CRITICAL} REVIEW OF CONVENTIONAL PLASTICITY THEORY

. . .

APPLICABLE TO SOILS 60

. . .

3.5.1 Existence of Yield Surface 61

. . .

3.5.2 Y i e l d s u r f a c e s h a p e 63

. . .

3.5.3 Existence of a P l a s t i c P o t e n t i a l Surface 65

. . .

3.5.4 S h a p e o f P l a s t i c P o t e n t i a l Surface 66

. . .

3.5.5 HardeningRules 68

. . .

3.5.6 Imp1 i c a t i o n s of Drucker' s P o s t u l a t e 69

. . .

CHAPTER IV: BOUNDING SURFACE PLASTICITY 7 1

. . .

4.1 BOUNDING SURFACE IDEAS FROM UNIAXIAL TESTS 7 2

4.2 GENERALIZATIONTOSIX-DIMENSIONAL STRESSSPACE

. . .

^{7 4}

4.3 ADVANTAGES OF BOUNDING SURFAC% PLASTICITY OVER

. . .

CONVENTIONAL PLASTICITY 7 8

4.4 FORMULATION OF BOUNDING SURFACE TBEORY I N p-q SPACE

. . .

CHAF!CER V : APPLICATION OF BOUNDING SURFACE PLASTICITY TO

SOILS: A NEX SAND MODEL

. . . . . .

5 . 1 INTRODUCTION

. . .

5.2 PLASTIC CONTRIBUTION

. . . .

5.2.1 P r e l i m i n a r y Remarks on E l a s t i c Sand Models

. . .

5.2.2 Choice of an E l a s t i c Model

. . .

5.2.3 C a l c u l a t i o n of E l a s t i o M a t e r i a l Constants 5.2.4 General Remarks and Suggestions f o r Future

. . .

E l a s t i o Models

. . .

5 . 3 . 1 C r i t i c a l S t a t e

5.3.2 ^5.3.3 C h a r a c t e r i s t i c S t a t e S t r e s r d i l a t a n c y T h e o r i e s

. . . . . .

5.3.3.a Rowets S t r e s s - d i l a t a n c y Theory

. . . . . .

5.3.3

.

b Nova's Theory

. . .

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

5.4 BOUNDING SURFACE MOTION

. . . . . .

5.4.1 Choice of I n t e r n a l V a r i a b l e s

. . .

5.4.2 Motion of t h e Bounding Surface

. . .

5.5 A M P L I ~ D E OF PLASTIC

n o w

5.5 ^5.5.2

.

¹ P l a s t i c Modulus R e l a t i o n Between l a s t i c Moduli

5

on Bounding Surf ace H and

# . . . . . .

a n d D i s t a n c e 6

. . .

5.6 MODEL CONSTANTS 136

. . .

5.7 THEORY VERSUS EXPERIMENT 1 3 9

CHAPrER I INTRODUcr I O N

S o i l Mechanics i s a n o l d d i s c i p l i n e i n A r t s and S c i e n c e s . From t h e e a r l i e s t t i m e s , when people s t a r t e d t o b u i l d , i t was n e c e s s a r y t o c o n s t r u c t a p p r o p r i a t e f o u n d a t i o n s . Designed w i t h e m p i r i c a l , b u t n e v e r t h e l e s s a d e q u a t e t e c h n i q u e s , some f o u n d a t i o n s s t i l l s u p p o r t t h e i r s t r u c t u r e s , a f t e r hundreds t o thousands of y e a r s . The E g y p t i a n s knew how t o make a wide h o r i z o n t a l p l a n e almost p e r f e c t t o s t a r t t h e i r pyramids, w i t h o u t t h e h e l p of o p t i c a l d e v i c e s : t h e y covered t h e founda- t i o n a r e a w i t h w a t e r and l e v e l e d o f f t h e emerging ground. The Romans c o n s t r u c t e d r o a d s w i t h pavement and a b a s e composed of d i f f e r e n t t y p e s of a g g r e g a t e s . As i n most s c i e n c e s , t h e knowledge t h a t Greeks and Romans accumulated was l o s t d u r i n g t h e Middle Ages and r e c o v e r e d d u r i n g t h e R e n a i s s a n c e . I n I t a l y , s u c c e s s f u l wooden p i l e f o u n d a t i o n s were a p p l i e d i n t h e Lagoon of Venice; l e s s s u c c e s s f u l were t h e f o u n d a t i o n s a t P i s a , b u t s t i l l t o our day a d e q u a t e . I n t h e e i g h t e e n t h c e n t u r y , m i l i t a r y e n g i n e e r s such a s Coulomb (1736-1806) a p p l i e d h i s f r i c t i o n t h e o r y t o o b t a i n t h e p r e s s u r e s a c t i n g on r e t a i n i n g w a l l s . The major development of f o u n d a t i o n t e c h n i q u e s came l a t e r w i t h t h e e x p a n s i o n of t h e r a i l w a y s owing much t o famous c o n t r i b u t o r s such a s Rankine (1820- 1872) and Winkler (1835-1888). Up t o t h e b e g i n n i n g of t h e t w e n t i e t h c e n t u r y , f o u n d a t i o n e n g i n e e r i n g was n o t d i s a s s o c i a t e d from s t r u c t u r a l e n g i n e e r i n g . T e r z a g h i (1883-1963) c a u s e d S o i l Mechanics t o g a i n i t s

independence and t o be recognized a s a p a r t i c u l a r l y important d i s c i p l i n e i n C i v i l Engineering. Since t h a t time, sub-discipl i n e s have emerged i n S o i l M e c h a n i ~ a .

From about 1955. a new branch i n s o i l mechanics h a s been s t e a d i l y i n c r e a s i n g : t h e c o n s t i t u t i v e modeling of s o i l behavior. By d e f i n i t i o n , i t d e s c r i b e s t h e s o i l behavior w i t h c o n s t i t u t i v e e q u a t i o n s using t h e framework of continuum mechanics. R e l a t i o n s between s t r a i n and s t r e s s a r e used e v e n t u a l l y i n f i n i t e element programs t o s o l v e e n g i n e e r i n g problems formulated a s boundary-value problems. The s o i l i s t r e a t e d a s a n e n g i n e e r i n g m a t e r i a l i n t h e same way a s s t e e l , c o n c r e t e o r polymers.

The development of t h i s new branch may be a p p r e c i a t e d i n Fig. 1 . l a by t h e number of p u b l i c a t i o n s i n two s o i l s mechanics j o u r n a l s : a B r i t i s h J o u r n a l , "Geotechnique," and a n American one, " t h e Geotechnical Engi- n e e r i n g D i v i s i o n J o u r n a l of t h e American S o c i e t y of C i v i l Engineers

."

These two j o u r n a l s of s t e a d y p o p u l a r i t y f o r many y e a r s i n s o i l mechanics c a p t u r e t h e i n c r e a s i n g i n t e r e s t i n t h i s new f i e l d . However, t h e y g r e a t l y u n d e r e s t i m a t e t h e t o t a l p r o d u c t i o n of p a p e r s s i n c e t h e r e c e n t c r e a t i o n of s p e c i a l i z e d magazines such a s t h e " I n t e r n a t i o n a l J o u r n a l f o r Numerical and A n a l y t i c a l Methods i n Geomechanics," "Mechanics of Materi- a l s , ' * "Numerical J o u r n a l f o r Gemechanics," e t c .

The d i f f e r e n t peaks observed i n Fig. l . l a correspond t o s i g n i f i c a n t e v e n t s of t h e new branch h i s t o r y . P r i o r t o 1940. t h e s o i l behavior was r e p r e s e n t e d by e l a s t i c i t y theory. The p l a s t i c i t y t h e o r y , i n i t i a t e d i n

30-

Finite elements

Offshore technology

20,

C r i t i c a l state m a

U L

<

..

t e , 0

a

1

10- 2

Soil plasticity

3

(Z

0 , ^I

Fig. 1.1. Number of papers on soil properties and soil constitutive equations published in two journals of steady popularity in soil mechanics: a British journal, "Geote~hniqae,'~ and an American one, "The Geotechnical Engineering Division Journal of the American Society of Civil Engineers,

"

a) accumulated publications of both journals b) separate publications of each journal.

1868 by Tresca and S a i n t Venant was o n l y a p p l i e d t o m e t a l . L a t e r , about 1945, s o i l p l a s t i c i t y a t i t s b e g i n n i n g was s t r i c t l y d e r i v e d from m e t a l p l a s t i c i t y . But s o i l , compared t o m e t a l , h a s a d i f f e r e n t r h e o l o g i c a l b e h a v i o r , which depends m a i n l y on mean p r e s s u r e and d e n s i t y . The f i r s t s o i l models d e r i v e d from metal p l a s t i c i t y c o u l d n o t d e s c r i b e t h e d i f f e r - ence and were n o t s u c c e s s f u l n s o i l mechanics. I n 1955, Drucker, Gibson and Henkel [1.111 compensated f o r t h i s d e f i c i e n c y and proposed t h e f i r s t o r i g i n a l s o i l model which d i s a s s o c i a t e d s o i l from metal p l a s t i c i t y . I n 1956 a n i m p o r t a n t s e r i e s of e x p e r i m e n t a l d a t a on c l a y s was produced by P a r r y E1.191 i n G r e a t B r i t a i n . A few y e a r s l a t e r , t h i s abundant c o l l e c t i o n of t e s t s , d r a i n e d and u n d r a i n e d a t d i f f e r e n t confin-

i n g p r e s s u r e s , were i n t e r p r e t e d by Roscoe, S c h o f i e l d and Wroth 11.241.

These r e s e a r c h e r s i n t r o d u c e d t h e concept of " c r i t i c a l s t a t e " i n t o t h e p l a s t i c i t y t h e o r y t o c r e a t e a c l a y model, s a t i s f a c t o r y o v e r a wide range of t e s t s and o v e r c o n s o l i d a t i o n r a t i o s .

I n t h e 1 9 5 0 1 s , t h e s o i l models were o n l y c o n s i d e r e d a s a u n i f y i n g approach t o e x p l a i n t h e s o i l r e s p o n s e d u r i n g v a r i o u s l a b o r a t o r y t e s t s and f o r d i f f e r e n t i n i t i a l d e n s i t y . I n p a r t i c u l a r Roscoe, Schof i e l d , and Wroth 11.241 c o r r e l a t e d n o r m a l l y and o v e r c o n s o l i d a t e d c l a y b e h a v i o r s d u r i n g d r a i n e d and u n d r a i n e d t e s t s . However, t h e s o i l models c o u l d n o t be used t o s o l v e p r a c t i c a l e n g i n e e r i n g problems. I f t h e s e problems were f o r m u l a t e d a s boundary v a l u e problems, t h e n o n l i n e a r s t r e s s - s t r a i n r e l a - t i o n s h i p s were l e a d i n g t o n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s , which c o u l d n o t be s o l v e d w i t h t h e e x i s t i n g t e c h n i q u e s , i . e . , w i t h

a n a l y t i c a l o r g r a p h i c a l methods. About 1965, t h e development of com- p u t e r f a c i l i t i e s and numerical methods was a t u r n i n g p o i n t i n s o i l model ing. Most p r a c t i c a l and complex g e o t e c h n i c a l problems, which had been approached u n t i l t h e n w i t h e m p i r i c a l methods, could be solved a s boundary value problems w i t h t h e h e l p of a c c u r a t e and r i g o r o u s numerical

t e c h n i q u e s . F i n i t e element and f i n i t e d i f f e r e n c e methods o f f e r e d t h e s o l u t i o n t o n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s w i t h complex boundary c o n d i t i o n s and geometries. Simple s o i l models s t a r t e d t o be used i n f i n i t e element codes, mostly d e r i v e d from n o n l i n e a r e l a s t i c i t y o r m e t a l - p l a s t i c i t y . They attempted t o d e s c r i b e t h e n o n l i n e a r and n o n r e v e r s i b l e s o i l b e h a v i o r a s simply a s p o s s i b l e w i t h a minimum number of parameters. Simultaneously, w i t h t h e t h e o r e t i c a l development of s o i l models, t h e t e s t i n g t e c h n i q u e evolved. The s t a n d a r d t r i a x i a l t e s t became more r e f i n e d . Although o r i g i n a t e d i n 1936 by Kjellman [1.141, t h e t r u e t r i a x i a l t e s t s were a p p l i e d t o c u b i c a l s o i l samples i n o r d e r t o i n v e s t i g a t e t h e i n f l u e n c e of t h e i n t e r m e d i a t e p r i n c i p a l s t r e s s (e.g., KO and S c o t t [1.151). I n 1975, s o i l modeling was reanimated, mainly due t o t h e growth of o f f s h o r e o i l p r o d u c t i o n technology. The main c h a l l e n g e i n t h i s a r e a i s t o d e s c r i b e i n e l a s t i c c y c l i c s o i l behavior o c c u r r i n g around f o u n d a t i o n s o r p i l e s d u r i n g p e r i o d i c wave o r earthquake l o a d i n g . The s o i l models a r e founded n o t only on p l a s t i c i t y t h e o r y , but a l s o on d i f f e r e n t c o n s t i t u t i v e t h e o r i e s such a s r a t e - t y p e e q u a t i o n s . T h e i r f o r m u l a t i o n becomes v e r y t h e o r e t i c a l and i n v o l v e s more and more m a t e r i a l c o n s t a n t s . The t e s t i n g t e c h n i q u e s a l s o become more s o p h i s t i c a t e d . For i l l u s t r a t i o n , i n o r d e r t o d e f i n e t h e s t r a i n f i e l d w i t h i n a sand sample,

X-ray p i c t u r e s a r e t a k e n t o measure t h e d i s p l a c e m e n t of l e a d s h o t s l o c a t e d w i t h i n t h e sample. The t e s t i n g e f f o r t i s now o r i e n t e d towards t h e e f f e c t of r o t a t i o n of p r i n c i p a l s t r e s s e s e i t h e r w i t h t h e hollow c y l i n d e r a p p a r a t u s (e.g. Symes e t a l . 11.2711 o r w i t h t r i a x i a l a p p a r a t u s (e.g. A r t h u r e t a l . 11.31).

R e c e n t l y , a s a r e s p o n s e t o t h i s i n c r e a s e i n t h e nmnber of models and t o demands from i n d u s t r y , i n t e r n a t i o n a l c o n f e r e n c e s have t a k e n p l a c e a l l over t h e world: Montreal (19801, D e l f t ( 1 9 8 1 ) , Z u r i c h (19821, Grenoble (19821, and Tucson (1983) a r e examples. I n t h e s e forums of m a t e r i a l modeling, s o many d i f f e r e n t and c o n t r a d i c t o r y o p i n i o n s have been f o r m u l a t e d c o n c e r n i n g t h e r e 1 i a b i l i t y and c a p a b i l i t y of e x i s t i n g models t h a t a n a t t e m p t a t a u n i f y i n g approach becomes n e c e s s a r y .

A s a supplement t o Fig. l . l a , Fig. l . l b i l l u s t r a t e s t h e d i f f e r e n c e of a c t i v i t y between Europe and t h e U n i t e d S t a t e s . The B r i t i s h p u b l i c a - t i o n s a r e s t e a d i l y i n c r e a s i n g w h i l e t h e American o n e s f o l l o w t h e r u l e of t h e f r e e market: demand and supply.

Although t h e s o i l models a r e t o o numerous t o be d e s c r i b e d i n d e t a i l , t h e b a s i c t h e o r i e s , on which t h e y a r e founded, a r e more l i m i t e d i n q u a n t i t y . To our knowledge, e i g h t main d i v i s i o n s may be d i s t i n g u i s h e d i n t h e r e p r e s e n t a t i o n of t h e r a t e - i n d e p e n d e n t b e h a v i o r of s o i l .

E l a s t i c i t y

P l a s t i c i t y ( s i n g l e y i e l d s u r f a c e ) P l a s t i c i t y ( m u l t i p l e y i e l d s u r f a c e s ) P l a s t i c i t y (bounding s u r f ace

Endochroni c Rate- type

Nonlinear incremental E m p i r i c a l model

T h e o r i e s such a s t h e above have been a p p l i e d t o a l l k i n d s of m a t e r i a l s . I n t h i s r e s p e c t , s o i l h a s become a n e n g i n e e r i n g m a t e r i a l such a s s t e e l o r c o n c r e t e .

For each c l a s s of t h e o r y , a few examples a r e given i n T a b l e 1.1.

No a t t e m p t a t a n e x h a u s t i v e p r e s e n t a t i o n h a s been made; o n l y models b e l i e v e d t o be mostly s i g n i f i c a n t a r e p r e s e n t . From t h i s t a b l e d i f f e r e n t i n f o r m a t i o n may be e x t r a c t e d :

m Age of t h e model

Type of s o i l i t i s b e s t q u a l i f i e d t o d e s c r i b e , e . g , , normally c o n s o l i d a t e d c l a y s , sand, e t c .

e Number of m a t e r i a l c o n s t a n t s which need t o be s p e c i f i e d by experiment t o f u l l y c h a r a c t e r i z e t h e s o i l behavior. I n t h e same way t h a t i s o t r o p i c l i n e a r e l a s t i c i t y needs two c o n s t a n t s , E Young's modulus and V P o i s s o n ' s r a t i o , a l l models r e q u i r e s i m i l a r c o n s t a n t s d i f f e r e n t f o r every m a t e r i a l . The number of c o n s t a n t s i s a s i g n of complexity, not only r e g a r d i n g t h e

TABLE 1.1 PRINCIPAL MODELS IN SOIL HECBAh'ICS

Fr*rework P l a s t i c i t y S i n g l e Y i e l d Surf ace

P l a s t i c i t y M u l t i p l e Y i e l d S u r f a c e s

Boundin8 S u r f a c e (B.S.) P 1 a a t i c i t y Endochronic Theory

Bate Type

N o r l i n e a r Incremental

Authors Data of M a t e r i a l Number of

h b l i- %PO P a r m e t e r s

I

Coament a

I c a t i o n 1 I

Schof i e l d 1968 Normally and o v e r 3

Wroth c o n s o l i d a t e d c l a y .

1 * ~ s o t r o p i o

1

One of t h e f i r s t models f o r clay.

Large e l a s t i c domain f o r overcon- s o l i d s t e d c l a y .

S i m i l a r t o Scbof ield-Wroth model but d i f f e r e n t shape f o r y i e l d s u r f a c e .

Roscoe 1968

Burland

B a l a d i Lade

I s o t r o p i c I I ^-

Sands Applied f o r high c o n f i n i n g p r e s s u r e

Normally and over-

c o n s o l i d a t e d c l a y . 4

1975 1977

I ¹ ( n u c l e a r b l a s t ) . ^{- -}

Pander

1

¹⁹⁷⁸

I

P r o v o s t 1978

I

-

Daf a 1 i a a 1 1980

Bead 1

Bazant 1 1976

Sands I 9

I o r

I

-

C l a y - K

I

⁷

w a r c o n s o l ida?ed Normally and o v e r consol ida t e d c l a y . KO c o n s o l i d a t e d

1 4 4

c o n s o l i d a t e d c l a y .

1

Sands 1 8

Sands 13

Sands

I

⁷

Clays Sand Normal 1s and over-

2 models f o r medium range p r e s s u r e . F i r s t model w i t & three-dimensional f a i l u r e

Depending on number on YS.

8

s u r f ace.

Attempt t o d e s c r i b e t h e a v e r c o n s o l i d a t e d c l a y b e h a v i o r .

2 models, i n f i n i t s number of y i e l d a n r f a c e s ( I N S ) , o r only 2 s n r f a c e s . 2 models w i t h n e s t e d y i e l d s u r f a c e s . c y l i n d e r s o r e l l i p s o i d r .

A d a p t a t i o n of Roscoe Burland i n t o B.S.

framework.

A simple B.S. model f o r sand.

A d a p t a t i o n of B.S. t h e o r y t o sand.

A p p l i c a t i o n t o sand of new i n t r i n s i c time measure ( c r i t i c a l s t a t e ) .

S e v e r a l models w i t h adapted endochronic

Sands I I theory.

Sanda 11 I Another a p p l i c a t i o n of new i n t r i n s i c t i m e

i I i 1 measure.

D a v i s I 1978 Clays 1 2 1 A p p l i c a t i o n of h y p o e l a s t i c i t y t o c l a y

Mu1 l e n g e r Gudehns Kolymbss

Darva Sands b c l a y s

-1

^{l 4}

A r a t e - t y p e model from Germany.

D i f f e r e n t from a11 o t h e r t h c o r i o s . T h i s model i s based on combination of t r i - a x i a l p a t h s t o s i m u l a t e t h e m a t e r i a l memory. A l l t h e t h e o r e t i c a l framework h a s been i n i t i a t e d f o r s o i l , but i t i s a p p l i e d a t p r e s e n t t o c o n c r e t e .

--- ---..----

t h e o r e t i c a l notions, but a1 so i n the experimental processes required t o e s t a b l i s h the constants f o r a particular theory or model.

CHAPTER I1

THE MATERIAL SOIL AS A CONTINUUM

2.1 APPLICATION OF CONTINUUM MECHANICS

TO

ASSEMJ3LY

OF

DISCRETE PARTICLES

A l l s o i l s ( s a n d s , s i l t s o r c l a y s ) c o n s i s t of an assemblage of many i n d i v i d u a l p a r t i c l e s w i t h a i r a n d l o r l i q u i d f i l l i n g t h e v o i d between p a r t i c l e s . A sand i s composed of approximately r o t u n d p a r t i c l e s w i t h s h a r p o r rounded c o r n e r s and s i z e v a r y i n g from 0.06 t o 2 m i l l i m e t e r s a s u s u a l l y d e f i n e d , S i l t s have p a r t i c l e s i z e s r a n g i n g from 0.06 t o 0.002 m i l l i m e t e r s . A v a r i e t y of m i n e r a l s such a s q u a r t z and f e l d s p a r s may be found i n b o t h m a t e r i a l s . The i n t e r a c t i o n r e l a t i o n between g r a i n s i s f r i c t i o n a l . C l a y s c o n s i s t of s m a l l p l a t e s w i t h s i z e from 0.002 t o m i l l i m e t e r s h e l d t o g e t h e r by short-range i n t e r a c t i o n f o r c e s . Clay m i n e r a l s , kaol i n i t e , i l l i t e , m o n t m o r i l l o n i t e , e t c . , a r e d i f f e r e n t from t h o s e i n sands and s i l t s . A b e t t e r d e s c r i p t i o n of s o i l s c o n s t i t u t i o n may be found i n most s o i l mechanics t e x t b o o k s , e.g., Lambe and Whitman l2.101. According t o t h e i r s i z e , t h e g r a i n s of sands and s i l t s may be observed w i t h t h e naked eye o r a magnifying l e n s ; c l a y p a r t i c l e s c a n o n l y be examined w i t h a powerful microscope o r electron-microscope.

S i n c e t h e i r p h y s i c a l c o n s t i t u t i o n i s more e a s i l y o b s e r v a b l e , our a t t e n - t i o n i s drawn p a r t i c u l a r l y t o t h e c o a r s e r g r a n u l a r media.

When e x t e r n a l l o a d i n g s , composed of p r e s c r i b e d d i s p l a c e m e n t a n d / o r f o r c e s , a r e a p p l i e d t o a g r a n u l a r mass a s shown i n Fig. 2 . l a , r e d i s t r i b u t i o n of f o r c e s and d i s p l a c e m e n t o c c u r s i n s i d e t h e m a t e r i a l volume. Looking c l o s e l y i n t h e neighborhood of a g i v e n p a r t i c l e , neighbor p a r t i c l e s form c o n t a c t s a s shown i n Fig. 2 . l b . Each c o n t a c t a c t i o n , which i s d i s t r i b u t e d o v e r a small a r e a of a g r a i n , may be r e p r e s e n t e d by a r e s u l t a n t f o r c e and a t o r q u e , l o c a t e d a t t h e c o n t a c t p o s i t i o n . I n most m i c r o s c o p i c s t u d i e s , t h i s t o r q u e i s n e g l e c t e d a c c o r d i n g t o t h e s m a l l s i z e of c o n t a c t a r e a . The p o s i t i o n s of c o n t a c t f o r c e s change d u r i n g a p p l i c a t i o n of l o a d i n g ; e v e n t u a l l y t h e c o n t a c t s may be d e l e t e d , o r r e l o c a t e d . Each g r a i n , which i s g e n e r a l l y r e p r e s e n t e d by a r i g i d body, h a s a d i s p l a c e m e n t which i s c h a r a c t e r i z e d by t h e t r a n s l a - t i o n and t h e r o t a t i o n of t h e c e n t r o i d . The g l o b a l mass d e f o r m a t i o n r e s u l t s from r e l a t i v e p a r t i c l e motions, s l i p p a g e of c o n t a c t s and r e o r g a n i z a t i o n of l o c a l s t r u c t u r e , i . e . , from m i c r o s c o p i c changes i n f o r c e s and d i s p l a c e m e n t s .

Although i t c o r r e s p o n d s t o r e a l i t y , t h i s d e s c r i p t i o n of t h e p h y s i c s o f a g r a n u l a r m a t e r i a l cannot be a p p l i e d t o s o l v e e n g i n e e r i n g problems p r a c t i c a l l y f o r v a r i o u s r e a s o n s : It i s i m p o s s i b l e t o d e f i n e e x a c t l y t h e geometry of each g r a i n , c o r n e r , edge, e t c . , f o r t h e volume of m a t e r i a l c o n s i d e r e d i n a g e o t e c h n i c a l problem. But even f o r w e l l - d e f i n e d g r a i n geometry such a s a r e g u l a r assembly of uniform s p h e r e s , t h e number of unknowns c o n s t i t u t e s a b a r r i e r t o any a n a l y s i s . For i l l u s t r a t i o n

9 9

p u r p o s e s , a c u b i c m e t e r c o n t a i n s a p p r o x i m a t e l y 2.10 t o 3.10 s p h e r e s of

Fig. 2.1. Transition from an assembly of discrete particles to a continuous medium.

(a) Volume of granular material subjected along its boundary to prescribed forces andlor displacements.

(b) Normal and tangential forces acting o n grain contacts at a microscopic level.

(c) Translation and rotation of each grain at a microscopic 1 eve1

.

(dl Approximation of the discrete system of contact forces in (b) by a continuous distribution of traction vector.

(e) Representation of displacement of each grain in (c) by a cont inuous displacement field.

1 mm i n r a d i u s , depending on whether t h e y a r e o r g a n i z e d i n uniform sim- p l e c u b i c o r c l o s e d hexagonal packing ( a f t e r Deresiewicz E2.81). Since each s p h e r e and each c o n t a c t ( f o r c e - p o s i t i o n ) p o s s e s s e s 6 d e g r e e s of freedom, t h e t o t a l number of unknown q u a n t i t i e s may r e a c h 4 . 1 0 ~ ~ which i s s t i l l a p r o h i b i t i v e number f o r a l l p r e s e n t a n a l y s e s , t h e o r e t i c a l , e x p e r i m e n t a l o r numerical.

A . a l t e r n a t i v e way t o s o l v e t h e problem of Fig. 2 . l a i s t o assume t h a t t h e d i s c r e t e d i s t r i b u t i o n of f o r c e s and d i s p l a c e m e n t i n t h e g r a n u l a r mass may be r e p r e s e n t e d by c o n t i n u o u s q u a n t i t i e s . The d i s - placement of p a r t i c l e s i n t h e neighborhood of a p a r t i c l e , a t i n i t i a l p o s i t i o n x t i l d e i n a r e f e r e n c e frame, i s smoothed by a c o n t i n u o u s d i s - placement f i e l d denoted u t i l d e ( x t i l d e , t ) depending on i n i t i a l p o s i t i o n x t i l d e and time t ( F i g . 2 . l e ) . I n a s i m i l a r way, t h e c o n t a c t f o r c e s a r e averaged by a c o n t i n u o u s d i s t r i b u t i o n of s t r e s s v e c t o r s , sometimes c a l l e d t r a c t i o n v e c t o r s ( F i g . 2 . l d ) . Each v e c t o r depends o n l y on t h e p o s i t i o n a t time t of a p a r t i c l e i n i t i a l l y a t p o s i t i o n 5 , and o n t h e u n i t v e c t o r g t o some u n i t a r e a s u r f a c e on which i t i s a c t i n g . The s t r e s s v e c t o r i s denoted t a u t i l d e . It i s assumed t o depend o n l y on t h e d i r e c t i o n of t h e c o n t a c t s u r f a c e where i t a c t s , n o t on i t s c u r v a t u r e .

These a s s u m p t i o n s of c o n t i n u i t y f o r t h e d i s p l a c e m e n t s and a c t i n g f o r c e s w i t h i n a g r a n u l a r mass r a i s e a few i n t e r e s t i n g q u e s t i o n s which seem t o be r e g a i n i n g f a v o r r e c e n t l y i n m a t e r i a l modeling. These h y p o t h e s e s may be examined w i t h t h r e e d i f f e r e n t approaches; e x p e r i m e n t a l t e c h n i q u e s i s one of them. I n such a n approach, d i s c r e t e q u a n t i t i e s

such a s t h e d i s p l a c e m e n t of i n d i v i d u a l p a r t i c l e s and f o r c e s a t t h e c o n t a c t s c a n be measured on some s i m p l e r p h y s i c a l m a t e r i a l model. Exam- p l e s a r e a s s e m b l i e s of o p t i c a l l y s e n s i t i v e d i s k s . F o r c e s a t c o n t a c t s a r e o b t a i n e d from p h o t o e l a s t i c i t y e f f e c t s ; d i s p l a c e m e n t s a r e found by comparison of photographs made a t v a r i o u s s t a g e s of t h e experiment.

T h i s method was used by Dantu C2.61, De J o s s e l i n de Jong and V e r r u i j t C2.71, and Van d e r Kogel C2.181. I n a r e c e n t e x p e r i m e n t , Luong 12.111 used i n f r a r e d thermography t o d e t e c t l o c a l i z a t i o n of energy d i s s i p a t i o n by f r i c t i o n i n r e a l sand samples.

Complementing t h e e x p e r i m e n t a l approach on simp1 i f i e d m a t e r i a l Cundall and S t r a c k C2.31 and S c o t t and C r a i g C2.161 propose t o use com- p u t e r s t o s i m u l a t e two-dimensional m a t e r i a l b e h a v i o r . Numerical and r e a l e x p e r i m e n t s o n i d e a l i z e d m a t e r i a l c o n s i s t i n g of two-dimensional assembly of d i s k s were compared by Cundall e t a l . 12.51.

The t h i r d approach i n v o l v e d g e n e r a l t h e o r e t i c a l work, s t a r t i n g from m i c r o s c o p i c b e h a v i o r t o o b t a i n macroscopic r e s p o n s e . Such a t tempts have g i v e n some u s e f u l r e s u l t s i n t e r m s of p h y s i c s of m a t e r i a l b e h a v i o r such a s t h e s t r e s s - d i l a t a n c y t h e o r y , which was o r i g i n a l l y f o r m u l a t e d i n 1962 by Rowe C2.151 and m a t h e m a t i c a l l y founded i n 1965 by Horne 12.91 f o r s p h e r e s . The s t r e s s - d i l a t a n c y t h e o r y was a b l e t o e x p l a i n q u a l i t a t i v e l y and q u a n t i t a t i v e l y how sands d i l a t e when s u b j e c t e d t o s h e a r i n g s t r e s s e s .

I n 1982, r e c e n t works, such a s Nemat-Nasser 12.13,2.21 and Oda and Konishi 12.141, p r w e t h a t t h e m i c r o s c o p i c i s r e g a i n i n g f a v o r i n

m a t e r i a l modeling, and i s e s p e c i a l l y j u s t i f i e d i n view of t h e i n c r e a s i n g complexity of c o n s t i t u t i v e e q u a t i o n s f o r s o i l s .

A 1 1 t h e above approaches, e x p e r i m e n t a l , numerical o r t h e o r e t i c a l , g i v e r e s u l t s on d i s c r e t e q u a n t i t i e s . Since t h e y a r e founded o n r e a l p h y s i c s of m a t e r i a l s , t h e i r outcome, may be i n t e r p r e t e d s i g n i f i c a n t l y i n terms of c o n t i n u o u s q u a n t i t i e s . A l l t r a n s i t i o n s from d i s c r e t e t o c o n t i n u o u s i n v o l v e a v e r a g i n g p r o c e s s e s , g e n e r a l l y performed on some f i n i t e m a t e r i a l volume. For i l l u s t r a t i o n , Cundall e t a l . t2.51 d e f i n e s t h e d e f o r m a t i o n g r a d i e n t (which c h a r a c t e r i z e s u l t i m a t e l y t h e s t r a i n s ) , i n a n assembly of d i s c r e t e d i s k s , i n t h e f o l l o w i n g way: F i r s t , t h e e q u i v a l e n t c o n t i n u o u s d i s p l a c e m e n t f i e l d , a t any p o i n t x t i l d e of t h e continuum, i s d e f i n e d e i t h e r by t h e r e a l d i s p l a c e m e n t of a d i s k p a r t i c l e o r by t h e imaginary c o n t i n u o u s d i s p l a c e m e n t which e x t r a p o l a t e s t h e m o t i o n of p a r t i c l e s s u r r o u n d i n g t h e i n s t e r s t i t i a l space; t h i s c h o i c e depends on whether o r n o t t h e p o i n t x t i l d e l i e s on a d i s k .

The a v e r a g e d i s p l a c e m e n t g r a d i e n t t e n s o r w i t h C a r t e s i a n component

-

u may be d e f i n e d a s i, j

where u i s t h e ith component of t h e e q u i v a l e n t continuous displacement i

aui

f i e l d ,

-

i s t h e g r a d i e n t of c o n t i n u o u s d i s p l a c e m e n t f i e l d , and V i s a ax;

m a t e r i a l volume s e l e c t e d l a r g e enough t o y i e l d s i g n i f i c a n t c o n t i n u o u s r e s u l t s .

It i s n o t c e r t a i n t h a t a l l d i s c r e t e q u a n t i t i e s may be averaged w i t h o u t l o s i n g some s i g n i f i c a n t p h y s i c s of t h e m a t e r i a l b e h a v i o r . For i n s t a n c e , t h e a v e r a g i n g p r o c e s s f o r t h e d e f o r m a t i o n g r a d i e n t may h i d e d i s c o n t i n u i t i e s i n t h e d i s p l a c e m e n t f i e l d , such a s l o c a l i z e d s h e a r band.

Such d i s c o n t i n u i t i e s , n o t always e a s y t o c h a r a c t e r i z e w i t h i n a d i s c r e t e system, make m e a n i n g l e s s t h e t r a n s i t i o n from d i s c r e t e t o c o n t i n u o u s . It w i l l be i n t e r e s t i n g t o e s t a b l i s h c r i t e r i a f o r such t r a n s i t i o n s ; t h e y c o u l d y i e l d i n s t r u c t i v e d a t a on how d i s c o n t i n u i t i e s emerge and s t r e s s - l o c a l i z a t i o n a p p e a r s . I n a continuum mechanics c o n t e x t , t h e s e r e s u l t s would be u s e f u l w i t h b i f u r c a t i o n methods a p p l i e d t o boundary v a l u e problems.

However, d i s r e g a r d i n g t h e m i c r o s c o p i c a s p e c t of m a t e r i a l b e h a v i o r , t h e g e n e r a l a t t i t u d e i n s o i l modeling i s more pragmatic. G e n e r a l l y urged by immediate c o n c e r n s , p r a c t i c i n g e n g i n e e r s do n o t q u e s t i o n t h e v a l i d i t y of t h e c o n t i n u i t y a s s u m p t i o n t o r e p r e s e n t s o i l b e h a v i o r . T h i s short-term a t t i t u d e i s q u i t e j u s t i f i e d by t h e combined power of continuum mechanics and of numerical t e c h n i q u e s coupled w i t h computers.

The m a t e r i a l b e h a v i o r d e s c r i b e d i n t e r m s of mathematical c o n s t i t u t i v e e q u a t i o n s i s c o m p a t i b l e w i t h numerical methods such a s f i n i t e e l e m e n t s . Most e n g i n e e r i n g problems w i t h complex g e o m e t r i e s and s o p h i s t i c a t e d m a t e r i a l s become s o l v a b l e a s boundary v a l u e problems.

Once t h e c o n t i n u i t y assumption h a s been s e l e c t e d t o d e s c r i b e a s o i l , a l l continuum mechanics r e s u l t s may be a p p l i e d . The f i r s t s t e p i s t o d e f i n e t h e k i n e m a t i c s of t h e m a t e r i a l , i . e . , t h e motion w i t h i n i t i n t e r m s of d i s p l a c e m e n t , v e l o c i t y and a c c e l e r a t i o n and t h e k i n e t i c s a c t i n g , i . e . , t h e f o r c e s developed c o m p a t i b l e w i t h t h e motion.

2.2 DESCRIPTION OF KINEMATICS

Only t h e n e c e s s a r y c o n c e p t s f o r t h i s p r e s e n t a t i o n a r e d e f i n e d . A more g e n e r a l d e s c r i p t i o n of t h e k i n e m a t i c s i n a continuum may be found i n v a r i o u s t e x t b o o k s on continuum mechanics ( e . g . , Malvern [2.121

,

T r u e s d e l l and Toupin E2.171).

I n a c o n v e n t i o n a l way, f o r s i m p l i c i t y , o n l y s m a l l d i s p l a c e m e n t s a r e c o n s i d e r e d . The d e f o r m a t i o n s a r e c h a r a c t e r i z e d by t h e i n f i n i t e s i m a l symmetric s t r a i n t e n s o r e p s i l o n t w i t h C a r t e s i a n components e i j such t h a t

where u i ( x t i l d e , t ) a r e t h e C a r t e s i a n components of t h e d i s p l a c e m e n t a u i f i e l d a t time t of a m a t e r i a l p o i n t i n i t i a l l y a t p o s i t i o n x t i l d e and

-

_ax

j a r e p a r t i a l d e r i v a t i v e s of t h e ith d i s p l a c e m e n t f i e l d c o o r d i n a t e w i t h r e s p e c t t o t h e j th p o s i t i o n c o o r d i n a t e .

From t h i s t o t a l s t r a i n , which d e s c r i b e s a s t a t e of d e f o r m a t i o n w i t h i n a m a t e r i a l , a n i n c r e m e n t a l q u a n t i t y may be d e f i n e d a s B i o t 12.11,

t o r e p r e s e n t changes about t h i s s t a t e . Between time t and i n f i n i t e l y c l o s e time t + d t i n c r e m e n t s of s t r a i n s a r e d e f i n e d by s u b s t i t u t i n g t h e i n c r e m e n t a l d i s p l a c e m e n t f i e l d between time t and time t + d t i n r e l a t i o n ( 2 . 2 ) . Another approach, commonly used i n s o l i d mechanics, i s t o employ r a t e i n s t e a d of increment. Within our p r e s e n t a t i o n b o t h d e s c r i p t i o n s a r e e q u i v a l e n t , s i n c e any increment ( d i s p l a c e m e n t o r o t h e r ) i s a r a t e m u l t i p l i e d by an increment of time d t . I n c r e m e n t s a r e used h e r e s i n c e we a r e o n l y concerned w i t h r a t e - i n d e p e n d e n t m a t e r i a l .

The p r i n c i p a l v a l u e s of t h e s t r a i n t e n s o r a r e denoted E ~ , ~ ~ , ~ ~ .

For t h e s p e c i a l axisymmetric c o n d i t i o n s , d e f i n e d by:

E i j = 0 when i P j ( i * j = 1 , 2 , 3 ) (2.3b)

two new c o o r d i n a t e s a r e i n t r o d u c e d s and aq; t h e y a r e r e l a t e d t o v

p r i n c i p a l components i n t h e f o l l o w i n g way:

These l a s t d e f i n i t i o n s have been commonly used i n s o i l mechanics f o r axisymmetric s t a t e s s i n c e t h e e a r l y s o i l models were developed i n 1960.

A s a common c o n v e n t i o n i n s o i l mechanics, a l l compressive s t r a i n s a r e

t a k e n p o s i t i v e , and t e n s i l e s t r a i n s n e g a t i v e , i . e . , t h e y have t h e o p p o s i t e s i g n t o t h e u s u a l s o l i d mechanics convention.

2.3 DESCRIPTION OF KImTICS

The f o r c e s i n s i d e a m a t e r i a l a r e r e p r e s e n t e d by t h e Cauchy s t r e s s t e n s o r sigmat w i t h C a r t e s i a n c o o r d i n a t e s o i j a t p o s i t i o n = t i l d e and time t . The increment of s t r e s s between t i m e t and t + d t i s d e s c r i b e d by t h e i n c r e m e n t a l s t r e s s t e n s o r dsigmat w i t h components doij

.

The p r i n c i p a l v a l u e s of t h e s t r e s s t e n s o r a r e denoted ol,a2,a3.

For a x i symme t r i c 1 oading

,

i. e.

,

o i j = 0 , when ifij ( i , j = 1 , 2 , 3 ) o n l y two s t r e s s components a r e n e c e s s a r y : p and q d e f i n e d a s :

As f o r s t r a i n s , a l l compressive s t r e s s e s a r e p o s i t i v e , t e n s i l e s t r e s s e s n e g a t i v e . Now, t h e c o e f f i c i e n t 2 e n t e r i n g t h e d e f i n i t i o n of E i n r e l a -

Q

t i o n (2.4b) may be j u s t i f i e d ; i t was s e l e c t e d i n o r d e r t o conserve a simple form f o r t h e i n c r e m e n t a l work dW d e f i n e d a s

dW = b i j d e i j (sum on i , j = 1 , 2 , 3 ) ( 2 . 7 a ) By c a l c u l a t i o n , u s i n g t h e axisymmetric c o n d i t i o n s p e c i f i e d i n r e l a - t i o n s (2.5) and ( 2 . 3 )

I t should be kept i n mind t h a t t h i s r e p r e s e n t a t i o n of s t r e s s e s and s t r a i n s w i t h p, q, e

v s E q i s r e s t r i c t e d t o axisymmetric l o a d i n g ; i t s convenience i n s t u d y i n g s o i l b e h a v i o r i n t h e s t a n d a r d t r i a x i a l a p p a r a t u s j u s t i f i e s i t s u s e .

CHAPTER I11

CLASSICAL PLASTICITY THEORY

The p l a s t i c i t y t h e o r y s t a r t e d i n 1868 when Tresca p r e s e n t e d two n o t e s d e a l i n g w i t h t h e flow of m e t a l s under g r e a t p r e s s u r e s t o t h e French Academy. Saint-Venant (1797-1886)

,

who had t o p r e p a r e a r e p o r t on t h a t work a s member of t h e Academy, became i n t e r e s t e d i n t h i s s u b j e c t . He was t h e f i r s t t o s e t up t h e fundamental e q u a t i o n s of p l a s t i c i t y and t o use them i n p r a c t i c a l problems. Around 1950, famous c o n t r i b u t o r s l i k e H i l l 13.41 completed t h e mathematical s t r u c t u r e of p l a s t i c i t y .

O r i g i n a l l y concerned w i t h y i e l d i n g of metal, t h i s t h e o r y was o n l y a p p l i e d t o s o i l around 1960. I t s fundamental b a s i s l i e s i n o b s e r v a t i o n s made from t h e s i m p l e s t t e s t i n g o n m a t e r i a l : t h e u n i a x i a l t e s t . A l l t h e p h y s i c a l o b s e r v a t i o n s , o b t a i n e d f o r t h e simple nnidimensional s t a t e , have been g e n e r a l i z e d t o t h e combined s t a t e of s t r e s s (six-dimensional) i n a t e n s o r i a l form w i t h t h e h e l p of geometrical c o n s i d e r a t i o n s .

3.1 FUNDAMENTAL PLASTICITY CONCEPTS FROM UNIAXIAL TESTS

A t y p i c a l s t r e s s - s t r a i n curve o b t a i n e d i n a u n i a x i a l t e s t of most m a t e r i a l ( m e t a l , s o i l , e t c . ) i s shown i n Fig. 3.la. Only t h e a x i a l component of s t r e s s and s t r a i n i s recorded; b o t h a r e assumed t o be uniformly d i s t r i b u t e d i n t h e sample. I n t h e p a r t i c u l a r example of Fig.

3 . l a , t h r e e c y c l e s of loading-reloading a r e performed. I n o r d e r t o be

F i g . 3 . l a . T y p i c a l s t r e s s - s t r a i n curve i n u n i a x i a l t e s t performed o n a r e a l m a t e r i a l .

F i g . 3 . l b . I d e a l i z e d g l o b a l and i n c r e m e n t a l m a t e r i a l r e s p o n s e s f o r development of p l a s t i c i t y t h e o r y .

d e s c r i b e d w i t h mathematical terms, t h i s complex n o n l i n e a r and i r r e v e r s i - b l e behavior must be approximated without l o s i n g s i g n i f i c a n t f ea t a r e s . Such a schematic m a t e r i a l response i s shown i n Fig. 3 . l b and may be de s c r i b e d by t h e f 01 lowing remarks :

1. The response i s r e v e r s i b l e around t h e o r i g i n ; it may be r e p r e s e n t e d by an i s o t r o p i c l i n e a r l y e l a s t i c behavior t o a f i r s t approximation.

2 . A t h r e s h o l d f o r r e v e r s i b i l i t y i s reached when t h e s t r e s s exceeds a " y i e l d shear" o r e l a s t i c l i m i t . Beyond t h i s l i m i t , denoted 6 , i r r e v e r s i b i l i t y i s c h a r a c t e r i z e d by permanent o r unrecoverable deformation a f t e r a s t r e s s removal ( P i g . 3 .lb).

3 . Following y i e l d i n g , more apparent n o n l i n e a r e f f e c t s m a n i f e s t themselves: t h e s t r e s s - s t r a i n curve bends towards t h e s t r a i n a x i s .

4. Daring unloading and r e l o a d i n g , t h e response i s almost p a r a l l e l t o t h e i n i t i a l response about t h e o r i g i n ( i f t h e h y s t e r e t i c phenomenon i s i g n o r e d ) .

5. A t t h e end of a r e l o a d i n g , t h e s t r a i n e i s t h e sum of an anre- coverable s t r a i n sP and a r e c o v e r a b l e s t r a i n ee (Fig, 3 , l b ) :

6. The response d u r i n g a r e l o a d i n g phase i s r e v e r s i b l e u n t i l t h e s t r e s s exceeds a new y i e l d s t r e s s : t h e p r e v i o u s y i e l d i n g s t r e s s h a s i n c r e a s e d i t s v a l u e t o t h e h i g h e s t v a l u e t a k e n by

t h e s t r e s s s t a t e d u r i n g t h e l o a d i n g h i s t o r y . The m a t e r i a l l o o k s h a r d e r t h a n i t was o r i g i n a l l y ; t h i s phenomenon i s known a s " s t r a i n hardening." T h i s remark a l s o a p p l i e s f o r s u c c e s s i v e unl oading-reloading proce s s e s.

7. The m a t e r i a l e x h i b i t s some memory of i t s p r e v i o u s l o a d i n g h i s - t o r y . T h i s remembrance a b i l i t y may be c a p t u r e d by t h e y i e l d s t r e s s v a l u e . Other v a r i a b l e s c a l l e d ' I i n t e r n a l v a r i a b l e s , " may be d e f i n e d t o c h a r a c t e r i z e t h i s memory. T h i s terminology comes from i r r e v e r s i b l e thermodynamics. I n o r d e r t o f u l l y c h a r a c t e r i z e a n i r r e v e r s i b l e thermodynamic s t a t e , t h e

"observed" o r " e x t e r n a l " v a r i a b l e s such a s t e m p e r a t u r e and s t r e s s a r e i n s u f f i c i e n t ; new v a r i a b l e s , c a l l e d "hiddent' o r

" i n t e r n a l " must a l s o be d e f i n e d .

8. The m a t e r i a l f a i l s when t h e s t r e s s r e a c h e s a f i n a l f a i l u r e s t r e s s . T h i s f a i l u r e s t r e s s , t h e l i m i t of t h e p o s s i b l e s t r e s s f o r a m a t e r i a l , d i f f e r s from t h e y i e l d s t r e s s which i s t h e t h r e s h o l d f o r i r r e v e r s i b l e s t r a i n .

I n a d i f f e r e n t system of c o o r d i n a t e s , s t r e s s - p l a s t i c s t r a i n , ( t h e r e v e r s i b l e s t r a i n i s removed from t h e t o t a l s t r a i n ) t h e m a t e r i a l response of Fig. 3 . l b i s shown i n Fig. 3 . l c .

A t a n i n c r e m e n t a l l e v e l , i . e . , f o r a n i n f i n i t e s i m a l increment of s t r e s s , d a , a s shown i n Fig. 3 .lb, t h e above remarks a l s o p e r t a i n .

The incremental s t r a i n de, corresponding t o a s t r e s s increment da, from a s t a t e a , i s equal t o :

a ) o n l y t h e e l a s t i c s t r a i n increment dee, i.e.:

i f t h e s t r e s s s t a t e a i s l e s s t h a n t h e y i e l d s t r e s s a* o r i f t h e s t r e s s increment d a i s n e g a t i v e .

b ) t h e sum of e l a s t i c and p l a s t i c incremental s t r a i n r e s p e c t i v e l y dee and deP, i.e.:

i f t h e s t r e s s s t a t e a i s equal t o t h e y i e l d s t r e s s a* and t h e s t r e s s increment d a i s p o s i t i v e .

I n c o n t r a s t t o most m a t e r i a l s , which strain-harden i n u n i a x i a l t e s t s a s i n d i c a t e d i n Fig. 3 . l a , s o i l s , under some circumstances, p r e s e n t a more complex behavior, known a s "strain-sof t e n i n g , " r e p r e s e n t e d i n Fig. 3 .ld.

Obtained w i t h s t r a i n - c o n t r o l l e d t e s t i n g d e v i c e s , t h e s t r e s s i s observed t o decay a s t h e s t r a i n exceeds some value. Although t h e s t r e s s a d e c r e a s e s , t h i s behavior i s d i f f e r e n t from unloading a s shown i n Fig.

3.ld. During unloading b o t h s t r a i n and s t r e s s increments a r e n e g a t i v e , while during s t r a i n - s o f t e n i n g t h e s e increments have o p p o s i t e signs.

These two d i f f e r e n t p r o c e s s e s a r e d i s t i n g u i s h e d by t h e s i g n

$

^{da, where}

S i s t h e slope of t h e s t r e s s - s t r a i n curve a t s t r e s s a. For unloading

d a i s n e g a t i v e , whereas f o r loading, with o r without s t r a i n - s o f t e n i n g ,

5

1 d a i s p o s i t i v e .

3.2 GENERALIZATION TO SIX-DIMENSIONAL STRESS SPACE

A l l fundamental concepts, introduced f o r t h e simple unidimensional s t a t e of s t r e s s and s t r a i n , need t o be g e n e r a l i z e d f o r every p o s s i b l e combined s t a t e i n t h e space of t h e symmetric Cauchy s t r e s s t e n s o r and of t h e i n f i n i t e s i m a l s t r a i n t e n s o r . A conventional s t r e s s f o r m u l a t i o n h a s been p r e f e r r e d t o a s t r a i n f o r m u l a t i o n (Naghdi and Trapp 13.71, Yoder 13 .I31 1. A l l g e n e r a l i z a t i o n s t o t h e six-dimensional s t r e s s space a r e performed i n C a r t e s i a n t e n s o r i a l form, with t h e h e l p of Euclidean geometrical c o n s i d e r a t i o n s . T w o p a r t i c u l a r s i m p l e m o d e l s , one a p p l i e d t o m e t a l , t h e o t h e r t o s o i l , i l l u s t r a t e s t e p b y - s t e p t h e p r e s e n t a t i o n of each new p l a s t i c i t y concept.

The s t r e s s s t a t e i s r e p r e s e n t e d by t h e Cauchy s t r e s s t e n s o r ^Q, with s i x independent C a r t e s i a n components a . .

,

( i, j = 1 , 2 , 3 )

.

The increment

1J

of s t r e s s i s d e s c r i b e d by t h e incremental s t r e s s t e n s o r dg with components d a . . ( i , j = 1,2,3). The s t r a i n s , corresponding t o e,e ,eP i n e

1J

t h e ~ i d i m e n s i o n a l case, a r e r e p r e s e n t e d by t h e following q u a n t i t i e s :

L ( t o t a l ) s t r a i n s t a t e w i t h components s i j

,

^{( i ,} j=1,2,3) Ee e l a s t i c s t r a i n s t a t e w i t h components s e _{i j} * ( i , j=1,2,3)

-

²⁷

-

gp p l a s t i c s t r a i n s t a t e w i t h components eP

i j * ^{( i ,} j=1,2,3) The increments of s t r a i n s , t o t a l , e l a s t i c o r p l a s t i c a r e denoted by:

d g increment of t o t a l s t r a i n with components de

,

( i s j=1,2,3) i j

dge increment of e l a s t i c s t r a i n w i t h components d e i j

,

( i , j=1,2,3) dgp increment of p l a s t i c s t r a i n w i t h components dep

.

( i , j = 1 , 2 , 3 )

i j

The d e s c r i p t i o n of t o t a l s t r a i n i n t o e l a s t i c and p l a s t i c s t r a i n i s s t i l l assumed t o be v a l i d i n t h e six-dimensional s t r a i n space

o r i n terms of components

For i n c r ements, t h e same a s s m p t i o n holds:

d e . . = dee

+

deP i j i j 1J

The increment d& i s f u l l y c h a r a c t e r i z e d by i t s Euclidean norm and i t s d i r e c t i o n . The norm, denoted

1

ldgl

1 ,

i s d e f i n e d such t h a t :

The f a c t o r 2 i n f r o n t of t h e shear s t r a i n , deij, where i i s d i f f e r e n t from j , t a k e s i n t o account t h e symmetry of t h e i n f i n i t e s i m a l s t r a i n t e r r s o r dg. T h i s l a s t d e f i n i t i o n may be c o n t r a c t e d using E i n s t e i n ' s implied snm n o t a t i o n :

The implied snm convention i s a n a t t r a c t i v e way t o condense e q u a t i o n s and i s used f r e q u e n t l y i n t h e r e s t of t h i s p r e s e n t a t i o n . The d i r e c t i o n of d!Z, i s d e f i n e d by t h e u n i t v e c t o r ar with t h e same d i r e c t i o n and t h e f 01 lowing components :

m . .

- -

^{d = i i}

1J l l d g l l

'

^{( i n j=1,2,3)}

The d e f i n i t i o n s (3.6) and (3.7) a p p l y t o any increments, i n p a r t i c - u l a r t o p l a s t i c and e l a s t i c s t r a i n increments dgp and dlZe. To d e f i n e d g r e s u l t i n g from an increment of s t r e s s dg a t a s t a t e a, dae and dgp need t o be determined. dae i s found by using a n e l a s t i c model, whereas t o f a l l y c h a r a c t e r i z e dgP, t h r e e p o i n t s remain t o be examined:

1) e x i s t e n c e , 2 ) d i r e c t i o n and 3) amplitude. These a s p e c t s r e f e r i n t a r n t o t h e y i e l d c r i t e r i o n , t h e flow r u l e and t h e hardening r ~ l e s ~ which a r e d e s c r i b e d a s follows.

3.2.1 E x i s t e n c e of Unrecoverable S t r a i n ( Y i e l d C r i t e r i o n )

The y i e l d p o i n t a* d e f i n e d i n t h e u n i a x i a l t e s t , i s g e n e r a l i z e d i n t o a h y p e r s u r f a c e i n t h e six-dimensional s t r e s s space, c a l l e d t h e

"Yield Surface" ( t h e p r e f i x "hyper" i s u s u a l l y o m i t t e d ) . By d e f i n i t i o n , t h e y i e l d s u r f a c e e n c l o s e s a domain i n s t r e s s space i n which any i n f i n i t e s i m a l s t r e s s change produces o n l y r e c o v e r a b l e s t r a i n . T h i s s u r f a c e i s r e p r e s e n t e d by a g e n e r a l equation:

o r expl i c i t l y

where E o r rl,r2,...,rn r e f e r t o t h e parameters c o n t r o l l i n g t h e s i z e , shape, e t c . , of t h e y i e l d s u r f a c e . Once t h e y i e l d s u r f a c e h a s been d e f i n e d , t h e presence o r absence of u n r e c o v e r a b l e s t r a i n , a t some s t r e s s s t a t e , i s c h a r a c t e r i z e d by t h e " y i e l d c r i t e r i o n " .

Y i e l d c r i t e r i o n :

a ) f ( g , x )

<

0, t h e s t r e s s s t a t e a l i e s i n s i d e t h e y i e l d s u r f a c e ; any increment of s t r e s s c r e a t e s o n l y incremental e l a s t i c s t r a i n dge, i.e., dgP = jZ.

b ) f ( g , g ) = 0 , t h e s t r e s s s t a t e ^g, l i e s

on

t h e y i e l d s u r f a c e . I f dg i s

o r i e n t e d outwards ( l o a d i n g ) , i r r e v e r s i b l e s t r a i n r e s u l t s , i.e.

d!Ap

#Q;

i f dg p o i n t s inwards (unloading) o r i s t a n g e n t t o t h e y i e l d s u r f a c e , only e l a s t i c s t r a i n develops, i.e. dgp = 6.

c ) The case f ( g , x )

>

⁰ i s impossible.

T h i s i m p o s s i b i l i t y i s j u s t i f ied by invoking t h e following argument:

i f a s t a t e of s t r e s s q, i s allowed t o e x i s t o u t s i d e t h e y i e l d s u r f a c e , t h e n f o r any increment of s t r e s s from t h i s s t a t e q,, i r r e v e r s i b l e d e f o r mation would occur. No unloading w i t h r e v e r s i b l e response would be p o s s i b l e , which i s c o n t r a d i c t o r y t o t h e p r e v i o u s u n i a x i a l concept d e f i n e d from u n i a x i a l t e s t s shown i n Fig. 3 , l a . T h e r e f o r e t h e y i e l d s u r f a c e must change i t s p o s i t i o n , o r deform i n o r d e r t o f o l l o w t h e s t r e s s s t a t e when p l a s t i c flow occurs, so t h a t t h e s t r e s s s t a t e q, l i e s o n o r w i t h i n t h e y i e l d s u r f a c e . T h i s c o n d i t i o n i s known a s t h e

"consistency condition."

A more p r e c i s e d e f i n i t i o n of l o a d i n g and unloading e n t e r i n g t h e y i e l d c r i t e r i o n may be g i v e n w i t h t h e h e l p of t h e u n i t v e c t o r normal and p o i n t i n g outwards from t h e y i e l d s u r f a c e a t ^Q ( s e e Fig. 3 . 2 ) . Denoted g , t h i s u n i t v e c t o r h a s i t s components n such t h a t :

i j

+

a f ^{( i ,} j = l J 2 , 3 )

n ^{' -} (sum on k , f s l J 2 , 3 ) ( 3 09)

a 0

i j - ]

F i g . 3 . 1 ~ . I d e a l i z e d m a t e r i a l r e s p o n s e a f t e r s u b t r a c t i n g t h e e l a s t i c s t r a i n .

F i g . 3 . l d . T y p i c a l s t r e s s - s t r a i n curve i n u n i a x i a l t e s t on a s t r a i n - s o f t e n i n g s o i l .

- ^-

^{9 -}

plastic potential

F i g . 3 . 2 . Y i e l d s u r f a c e and p l a s t i c p o t e n t i a l s u r f a c e i n s t r e s s s p a c e .

where

- ^{a f}

i s t h e p a r t i a l d e r i v a t i v e of f ( p , r ) with r e s p e c t t o t h e s t r e s s component a..

.

A f t e r c h a r a c t e r i z i n g t h e s c a l a r product between a

1J

and dg i n t h e following way:

t h e l o a d i n g and unloading, which i n d i c a t e i f t h e s t r e s s increment p o i n t s outwards o r inwards t o t h e y i e l d s u r f ace, may be r e d e f i n e d i n t h e following way:

1 oading ~ ' d g

>

0

n e u t r a l l o a d i n g a'dg = 0

an1 oading a d %

<

0

The s i m p l e s t e l a s t i c - p l a s t i c model, a p p l i e d t o metal, i s c e r t a i n l y t h e von Mises' model. The y i e l d suxface i s a c y l i n d e r i n t h e p r i n c i p a l s t r e s s space, t h e a x i s of which i s p a r a l l e l t o t h e h y d r o s t a t i c a x i s , a s r e p r e s e n t e d i n Fig. 3.3. The e q u a t i o n of such a s u r f a c e i s o b t a i n e d by s t a t i n g t h a t t h e d i s t a n c e t o any p o i n t on t h i s s u r f a c e t o i t s a x i s i s equal t o t h e r a d i u s denoted R, i.e.

Fig. 3.3. v o n Mises' yield surface i n principal stress space.

- -

^predicted

tension

Fig. 3.4. Typical experimental response curves of a real material and predicted response curves by v o n Mises' model during t w o uniaxial tests (compression and tension).