THE DYNAMIC BEHAVIOR OF CONCRETE MATERIALS

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THE DYNAMIC BEHAVIOR OF CONCRETE

MATERIALS

R. Sierakowski, H. Adeli

To cite this version:

R. Sierakowski, H. Adeli. THE DYNAMIC BEHAVIOR OF CONCRETE MATERIALS. Journal de

Physique Colloques, 1985, 46 (C5), pp.C5-81-C5-89. �10.1051/jphyscol:1985511�. �jpa-00224741�

JOURNAL

DE

PHYSIQUE

Colloque C5, supplCment a u n08, Tome 46, aoQt 1985 page C5-81

THE DYNAMIC BEHAVIOR OF CONCRETE MATERIALS R.L. Sierakowski and H. Adeli

Department of C i v i l Engineering, The Ohio S t a t e U n i v e r s i t y , CoZwnbus, Ohio 43210, U.S.A.

Resume

-

Nous presentons l ' e t a t des connaissances s u r l e comportement des betons sous chargement dynamique (chocs ou charges i m p u l s i v e s ) . Nous passons en r e v u e e t nous d i s c u t o n s l e s t r o i s aspects q u i i n t e r a g i s s e n t e t q u i s o n t l a d e t e r m i n a t i o n e x p e r i m e n t a l e des p r o p r i & t & s dynamiques, l e developpement des e q u a t i o n s c o n s t i t u t i v e s e t l a p r o p a g a t i o n des ondes. Les recherches r G a l i s 6 e s dans ces domaines s o n t r6sum@es, d i s c u t e e s e t c o r r e l e e s en i n s i s - t a n t p a r t i c u l i Prement s u r l e s e q u a t i o n s c o n s t i t u t i ves e t 1

'

i n f l u e n c e de l a v i t e s s e de d 6 f o r m a t i o n s u r l e comportement dynamique.

A b s t r a c t

-

This paper i s a s t a t e - o f - t h e - a r t r e p o r t on t h e b e h a v i o r o f c o n c r e t e m a t e r i a l s under dynamic 1 oading, t h a t i s impact o r i m p u l s i v e loads. Three i n t e r a c t i n g events a r e discussed and reviewed. They a r e d e t e r m i n a t i o n o f dynamic p r o p e r t i e s t h r o u g h experiments, development o f c o n s t i t u t i v e e q u a t i o n s and p r o p a g a t i o n o f waves. Research done i n t h e s e areas i s summarized, reviewed and c o r r e l a t e d w i t h p a r t i c u l a r a t t e n t i o n t o c o n s t i t u t i v e r e l a t i o n s and t h e i n f l u e n c e o f m a t e r i a l s t r a i n r a t e on t h e dynamic behavior.

INTRODUCTION

The s u b j e c t o f p r e d i c t i n g and e v a l u a t i n g t h e i n f l u e n c e o f dynamic e f f e c t s i n m a t e r i a l s appears t o be a s u b j e c t o f i n c r e a s i n g awareness and importance t o t h e s c i e n t i f i c community. I n c i v i l e n g i n e e r i n g t e c h n o l o g y t h i s problem i s c o n s i d e r e d as a most i m p o r t a n t and t i m e l y t o p i c p a r t i c u l a r l y as i t r e l a t e s t o t h e e s t a b l i s h m e n t o f a p p r o p r i a t e d e s i g n methodology a s s o c i a t e d w i t h masonry and c o n c r e t e b u i l d i n g s t r u c t u r e s o f b o t h t h e r e i n f o r c e d and n o n - r e i n f o r c e d types. Dynamic e f f e c t s a r e produced by i n t r o d u c i n g r a p i d l y a p p l i e d l o a d s o f s h o r t d u r a t i o n i n t o s t r u c t u r a l elements. Therefore, by impact o r i m p u l s i v e 1 oading i s imp1 i e d t h e i n t e r a c t i o n s o c c u r r i n g between i m p a c t i n g bodies and s t r u c t u r e s , and t h e c o r r e s p o n d i n g f o r c e s a c t i n g a t t h e c o n t a c t i n g s u r f a c e s o f t h e s e s t r u c t u r e s . The l o a d i n g s o f s t r u c t u r e s s u b j e c t e d t o t h e s e t y p e s o f events a r e t h u s g e n e r a l l y c l a s s i f i e d as i m p a c t / i m p u l s i v e Loads. Of p r i n c i p a l i n t e r e s t t o engineers i s t h e mechanical b e h a v i o r of t h e s t r u c t u r a l m a t e r i a l s , as exposed t o t h e s e r a p i d l y a p p l i e d loads. A q u a n t i f i c a t i o n o f t h e s e events can g e n e r a l l y be c a t e g o r i z e d e i t h e r by t h e t i m e response o f t h e imposed l o a d r e l a t i v e t o t h e n a t u r a l p e r i o d o f t h e s t r u c t u r e , o r a l t e r n a t i v e l y by an. e s t a b l i s h e d s t r a i n r a t e . Summary t a b l e s i d e n t i f y i n g t h e s e c h a r a c t e r i s t i c s have been n o t e d i n Tables 1 and 2. One o f t h e more s i g n i f i c a n t c h a r a c t e r i s t i c s o f impact o r i m p u l s i v e t y p e l o a d s i n t h e regime where m a t e r i a l c o n s t i t u t i v e laws a r e a p p l i c a b l e i s t h e c o m p l i c a t e d m a t e r i a1 responses observed i n c l u d i n g a1 so modes o f d e f o r m a t i o n and f r a c t u r e mechanisms. G e n e r a l l y , t h e s e responses can be b r o a d l y c l a s s i f i e d as near f i e l d and f a r f i e l d e f f e c t s . The near f i e l d response can be c h a r a c t e r i z e d by t h e geometry, r i g i d i t y , and mass o f t h e t a r g e t media as w e l l as t h e i n t e n s i t y , mass, and v e l o c i t y o f t h e d e f i n e d i m p a c t / i m p u l s i v e load. The f a r f i e l d response i s c o n t r o l l e d p r i m a r i l y by t h e s t r e s s wave propagation, bond s t r e n g t h between a g g r e g a t e / r e i n f o r c e m e n t , m a t e r i a l c o n s t i t u t i v e e q u a t i o n s and f a i 1 u r e c r i t e r i a o f t h e c o n s t i t u e n t m a t e r i a l s .

In

a d d i t i o n , t h e v e l o c i t y / i n t e n s i t y o f t h e l o a d i n g i n f l u e n c e s t h e s t r a i n r a t e which has an e f f e c t on t h e s t r e s s wave p r o p a g a t i o n and t h e c o r r e s p o n d i n g f a i l u r e mechanisms o c c u r r i n g i n t h e s t r u c t u r e .

JOURNAL DE PHYSIQUE

(Const~tutive Equation)

F i g u r e 1

A key f e a t u r e associated w i t h dynamic loadings i s thus t h e i n t e r a c t i v e n a t u r e o f t h e events which occur. This i s g r a p h i c a l l y represented i n Figure 1. This f i g u r e d i s p l a y s t h e i n t e r a c t i v e r o l e of t h e important events necessary t o understand and model t h e behavior of m a t e r i a l s subjected t o impact/impulsive loads. That i s , i n o r d e r t o determine dynamic p r o p e r t i e s necessary as modeling parameters f o r dynamic events one must have knowledge o f wave propagation e f f e c t s as w e l l as knowledge o f t h e m a t e r i a l c o n s t i t u t i v e equations. However, t o understand wave propagation events t h e very same dynamic p r o p e r t i e s which are being sought must be known a p r i o r i as i n p u t t o t h e c o n s t i t u t i v e equations which are t o be determined. Thus i n p u t from a l l t h r e e events must be synthesized i n order t o understand and q u a n t i f y dynamic e f f e c t s . These t o p i c s a r e s e l e c t i v e l y high1 i g h t e d i n t h e accompanying paragraphs i n order t o review t h e c u r r e n t ongoing work i n these areas as w e l l as f o r p o t e n t i a l l y r e c o g n i z i n g where f u r t h e r work and d e f i c i e n c i e s i n our e x i s t i n g data base i s necessary.

Table 1 LON CLASSIFICATION

*

Load C l a s s i f i c a t i o n T I T Type o f Load

Q u a s i - S t a t i c

>4

Conventional t e s t i n g o f concrete

Quasi -Impact

Impul s i ve and Impact

-1 Transient 1 oading on s t r u c t u r e s

(0.25 K i n e t i c Energy, b l a s t loads

Shock Loads High energy expl o s i ves

*

T I T i s t h e r a t i o o f l o a d d u r a t i o n ( 7 ) t o c h a r a c t e r i s t i c response t i m e (T).

Table 2 STRAIN RATE REGIMES

Load C l a s s i f i c a t i o n S t r a i n Rate M a t e r i a l C h a r a c t e r i z a t i o n Ouasi-Static

l o 4

-

1 o 1 / e c C o n s t i t u t i v e Equation

-- - -- - - -

Intermediate 1 0

-

lol/sec C o n s t i t u t i v e Equation High

l o 2

-

104/sec C o n s t i t u t i v e Equation

DYNAMIC

PROPERTIES

While i t i s important t o recognize t h a t a number o f dynamic p r o p e r t i e s c o u l d be examined t h e f o l l o w i n g remarks w i l l be l i m i t e d t o t h e i n f l u e n c e o f m a t e r i a l s t r a i n r a t e as i t e f f e c t s dynamic p r o p e r t i e s . To develop an understanding o f s t r a i n r a t e e f f e c t s i n m a t e r i a l s recourse i s g e n e r a l l y made t o observations using a p p r o p r i a t e l y designed experimental devices. E a r l y r a t e e f f e c t s t u d i e s on t h e behavior i n concrete were i n v e s t i g a t e d using c o n t r o l l e d r a t e o f l o a d i n g t e s t s i n what we now c a l l conventional t e s t i n g machines. Only impact types t e s t s , however, achieve r a t e s comparable t o those i n t h e impact/impulsive l o a d i n g regime. Table 2 d i s p l a y s a c l a s s i f i c a t i o n schedule f o r i d e n t i f y i n g t h e various and important s t r a i n r a t e regimes.

At t h e h i g h end o f t h e q u a s i - s t a t i c l o a d spectrum and f o r measuring dynamic events, a dynamometer placed i n tandem w i t h t h e specimen i n t e s t i n g machines provides an i n d i c a t o r o f t h e impact force. At s t r a i n r a t e s above approximately 10 sec'l wave propagation e f f e c t s i n specimen and dynamometer complicate t h e s t r e s s determination and must be taken i n t o account. This has l e d t o t h e development o f s p e c i f i c t e s t i n g devices f o r various h i g h r a t e regimes, For i n t e r m e d i a t e s t r a i n rates, drop t e s t s u s i n g instrumented s t r i k e r weights have been used t o o b t a i n s t r a i n r a t e data. Pol important element i n v o l v e d w i t h a n a l y s i s o f data using t h i s t y p e o f

t e s t equipment i s t h e requirement f o r a c c o u n t a b i l i t y o f i n e r t i a e f f e c t s . For h i g h e r s t r a i n rates, a u s e f u l experimental apparatus i s t h e s p l i t Hopkinson pressure bar system which h ~ s been used e x t e n s i v e l y f o r dynam$c t e s t s o f metals. The l o n g t r a n s m i t t e r bar used w i t h t h i s system serves as a dynamometer measuring t h e force, and hence t h e average s t r e s s over t h e cross section, a t t h e specimen face i n contact w i t h t h e pressure

Mr.

For very h i g h s t r a i n rates, t h a t I s above

l o 3

sec", f l y e r p l a t e impact tests//'have been used as an experimental procedure. Gas guns a r e g e n e r a l l y used as ' t h e p r o p e l l i n g d e v i ces f o r t h e f l y e r p l a t e s w i t h s t r e s s e s generated ranging up t o t h e v i c i n i t y o f

l o 3

kbars.

Atchley and Furr ( 1 ) used

a

drop t e s t e r w i t h drop h e i g h t s up t o 20 ft and found dynamic s t r e n g t h s up t o more than

1.6

times t h e s t a t i c strengths. Seabold ( 2 ) s t u d i e d r a t e e f f e c t s . r e l a t e d t o t h e unconfined compressive s t r e n g t h i n concrete up t o s t r a i n r a t e s o f about 5 sec-I and i n t r o d u c e d e m p i r i c a l formula c o n t a i n i n g both a l i n e a r term and a l o g a r i t h m i c term i n t h e s t r a i n r a t e . By u s i n g , s i m p l e bar-wave t h e o r y and a drop t e s t e r , Hughes and Gregory ( 3 ) were able t o estimate t h e t r a n s i e n t stresses o c c u r r i n g and concluded t h a t impact s t r e n g t h s averaged about 1.92 times t h e s t a t i c compressive s t r e n g t h o f t h e m a t e r i a l . They found i n some o f t h e i r s t r o n g concretes values g r e a t e r than 37,000 psi. Hughes and Watson ( 4 ) used a s i m i l a r t e s t technique and observed t h a t t h e i n c r e a s e i n compressive s t r e n g t h i n dynamic t e s t s over t h e s t a t i c s t r e n g t h was g r e a t e r f o r low-strength concrete than f o r t h e corresponding h i g h - s t r e n g t h concrete.

Dynamic e l a s t i c p r o p e r t i e s o f concrete b a r specimens have a1 so been s t u d i e d u s i n g wave propagation t e s t s by Goldsmith et. al. (5, 6, 7, 8). Analyzing one dimensional wave propagation induced by impacting a l o n g bar w ~ t h a s t e e l sphere, t h e wave speeds, f r a c t u r e s t r a i n s , and energy r e q u i r e d t o f r a c t u r e g e o l o g i c a l and cementious m a t e r i a l s have been studied. The dynamic t e n s i l e s t r e n g t h and s t r a i n energy r e q u i r e d t o f a i l concrete specimens have been measured i n s i m i l a r l o n g i t u d i n a l impacts using various scaled p r o j e c t i l e s by Bi rkimer and Lindemann ( 9 ) , by G r i n e r ( 9 ) and Sierakowski et. a l .

(11).

Results obtained from these t e s t s showed t h a t dynamic compressive s t r e n g t h s around 26,400 p s i c o u l d be reached u s i n g 3/4-inch-diameter specimens. This was found t o be 1.93 times t h e s t a t i c compressive s t r e n g t h o f 13,690 psi.

h o n g t h e very few o t h e r a p p l i c a t i o n s o f s p l i t Hopkinson bar technology t o concrete i s t h a t made by Korrneling et. al. (12), f o r t h e case o f dynamic t e n s i l e t e s t s . I n these t e s t s , t h e r e p o r t e d dynamic t e n s i l e s t r e n g t h s were more than twice.

C5-84 JOURNAL DE PHYSIQUE t h e s t a t i c v a l u e a t s t r a i n r a t e s o f a p p r o x i m a t e l y 0.75 sec-l. Noteworthy . i s t h e f a c t t h a t t h e s e i n v e s t i g a t o r s as w e l l as o t h e r s have f o u n d s i g n i f i c a n t i n c r e a s e s i n t h e impact t e n s i l e s t r e n g t h i n f i b e r - r e i n f o r c e d c o n c r e t e o v e r t h a t o f p l a i n c o n c r e t e ( 1 3 ) . WAVE PROPAGATION I n o r d e r t o i n v e s t i g a t e dynamic e f f e c t s i n c o n c r e t e m a t e r i a l s o v e r i m p a c t / i m p u l s i ve 1 o a d i ng ranges where c o n s t i t u t i v e e q u a t i o n s model 1 i ng r e t a i n s i m p o r t a n c e s t u d i e s a s s o c i a t e d w i t h wave p r o p a g a t i o n events a r e u s e f u l . h o n g t h e more i m p o r t a n t parameters f o r s t u d y a r e p u l s e speed, a t t e n u a t i o n , g e o m e t r i c / m a t e r i a l d i s p e r s i o n e f f e c t s , as we1 1 as f r a c t u r e c h a r a c t e r i z a t i o n . Comprehensive s t u d i e s c o n c e r n i n g wave p r o p a g a t i o n i n c e r t a i n c l a s s e s o f rock and c o n c r e t e were performed by G o l d s m i t h e t . a l . (5, 6). R e s u l t s o f t h e s e t e s t s i n d i c a t e d t h a t t h e p u l s e propagates w i t h o u t d i s p e r s i o n and w i t h an a t t e n u a t i o n p r o p o r t i o n a l t o some power o f t h e r a t i o o f t h e d i s t a n c e t r a v e r s e d t o b a r d i a m e t e r f o r i n i t i a l wave passage. The f a c t o r s which most i n f l u e n c e d t h e dynamic b e h a v i o r o f t h e igneous r o c k t y p e s were found t o be g r a i n s i z e , g r a i n bond s t r e n g t h , and t h e e f f e c t o f r e p e a t e d shocks.

F o l l owing t h e s e s t u d i e s on r o c k specimens, Go1 dsmi t h and co-workers ( 7 ) conducted a s i m i l a r s e r i e s o f t e s t s on l a b o r a t o r y prepared specimens o f concrete. The specimens were a g a i n 0.75 i n c h e s i n d i a m e t e r and 24 i n c h e s i n l e n g t h . The wave p r o p a g a t i o n speed was found t o be s i m i l a r t o t h a t o f t h e coarse g r a i n e d igneous r o c k s and was a p p r o x i m a t e l y 144,000 i n c h e s p e r second.

F u r t h e r s t u d i e s on c o n c r e t e b a r s were performed by Goldsmith e t . a l . (8). Concrete composed o f a l i g h t w e i g h t aggregate was a l s o s t u d i e d and compared w i t h t h e dynami c p r o p e r t i e s o f c o n v e n t i o n a l c o n c r e t e . For t h e c o n c r e t e composed o f 1 i g h t w e i g h t aggregate t h e r e s u l t i n g d a t a i n d i c a t e d t h a t t h e wave speed was 123,000 i n c h e s p e r second and t h e r a t i o o f t h e dynamic Young's modulus t o t h e s t a t i c v a l u e was 1.2. For c o n v e n t i o n a l c o n c r e t e t h e wave speed n o t e d was found t o be 144,000 i n c h e s p e r second. For t h e s e m a t e r i a l s p u l s e a t t e n u a t i o n was found t o i n c r e a s e w i t h i n c r e a s e i n t h e s t r a i n l e v e l . It was f u r t h e r concluded t h a t f o r some r o c k s a t h r e s h o l d s t r a i n l e v e l was reached whereupon any f u r t h e r i n c r e a s e i n t h e impact generated by t h e s t r i k e r r e s u l t e d o n l y i n g r e a t e r comminution o f t h e impact end. Placement o f a m e t a l l i c cap on t h e impact end reduced t h i s e f f e c t and r e s u l t e d i n a 100 p e r c e n t i n c r e a s e i n t h e s t r a i n l e v e l reached w i t h i n t h e bar. Also, s i g n i f i c a n t d i f f e r e n c e s i n d a t a were n o t e d f o r s t r a i n gages p l a c e d on p i e c e s o f aggregate as d i s t i n c t f r o m r e s u l t s o b t a i n e d w i t h gages l o c a t e d on t h e cement proper.

More r e c e n t s t u d i e s conducted by G r i n e r e t . a l . ( 1 4 ) and Sierakowski e t . a l . ( 1 1 ) examined wave p r o p a g a t i o n d a t a a t v e l o c i t i e s i n t h e v i c i n i t y o f t h e t h r e s h o l d l e v e l f o r p r o d u c i n g f i r s t t e n s i l e f r a c t u r e i n c o n c r e t e b a r specimens o f 314" and 1- 1/2" d i a m e t e r by 36" long. The e f f e c t s o f i m p a c t o r l e n g t h on wave shape, a t t e n u a t i o n , and p u l s e speed f o r s e v e r a l c o n t r o l l e d cement aggregate specimens have been considered. Also e v a l u a t e d was t h e i n f l u e n c e o f end p r o t e c t i v e caps and epoxy r e b o n d i n g as e f f e c t i n g t h e p u l s e passage a l o n g t h e bars. Some o f t h e e f f e c t s n o t e d i n t h e s e s t u d i e s were t h a t compressive p u l s e speeds were changed by up t o 10% i n s h i f t i n g f r o m medium t o c o a r s e g r a i n aggregates and t h a t c u r e t i m e was s i m i l a r l y i n f l u e n c e d . I n a d d i t i o n , compressive p u l s e speeds were found t o i n c r e a s e w h i l e t e n s i 1 e p u l s e speeds decreased.

CONSTITUTIVE EQUATIONS

A p p r o p r i a t e m a t e r i a l m o d e l i n g as d e l i n e a t e d by c o n s t i t u t i v e e q u a t i o n s p l a y s an i m p o r t a n t r o l e o v e r t h e s t r a i n r a t e range e x t e n d i n g f r o m sec" up t o

l o 4

s e c - l . Beyond t h i s l a t t e r range, t h a t i s i n t h e shock wave regime, t h e m a t e r i a l can be c o n s i d e r e d as a f l u i d medium f o r response purposes w i t h s t r e n g t h c o n s i d e r e d as a p r i n c i p a l d e f i n i n g parameter. I n t h e f o l l o w i n g paragraphs a r e v i e w o f s e l e c t i v e areas i n t h e c o n s t i t u t i v e e q u a t i o n m o d e l l i n g o f c o n c r e t e i s presented.

Linear/Nonl i n e a r E l a s t i c Models

The observed n o n l i n e a r behavior o f c e r t a i n r e i n f o r c e d concrete s t r u c t u r a l elements, f o r which t h e s t r e s s s t a t e can be b a s i c a l l y i d e a l i z e d a.s of t h e compression-tension type, i s mainly due t o t e n s i l e c r a c k i n g o f t h e concrete and t h e r e s u l t i n g r e d u c t i o n i n concrete s t i f f n e s s . To model t h i s observed behavior, a number o f researchers have developed and employed l i n e a r e l a s t i c f r a c t u r e models as f o r example Ngo and Scordelis (15) and P h i l l i p s and Zienkiewicz (16). I n these models, t h e s t r e s s - s t r a i n r e l a t i o n s f o r uncracked concrete have been developed based upon a l i n e a r theory o f e l a s t i c i t y and defined completely by young's modulus E, Poisson's r a t i o v

, o r b u l k modulus

K = E/3(1-2v) and t h e corresponding shear modulus G = E / 2 ( 1 + ~ ) . Expressions i n v o l v i n g t h e v o l u m e t r i c s t r a i n , E ~ mean ~ , normal s t r e s s p = ukk/3, d e v i a t o r i c s t r e s s Si = oij

-

akkaij/3, and shear deformation d e f i n e d by eij = cij

-

E~~ 6. T j

/3

have been developed. The same type o f 1 in e a r e l a s t i c c o n s t i t u t i v e r e l a t i o n s has been used f o r cracked concrete. However, t h e instantaneous tangent modulus has been m o d i f i e d i n such a manner t h a t t h e stresses i n t h e d i r e c t i o n perpendicular t o t h e crack are zero. This model does not t a k e i n t o account t h e sudden Stress release due t o propagation o f t h e crack which may induce f u r t h e r cracking. Attempts have been made t o modify t h i s model by i n c o r p o r a t i n g t h e sudden release o f f r a c t u r e stresses i n a global manner o r by i n c l u d i n g p r o v i s i o n s f o r a d d i t i o n a l crack generation by t h e opening o r c l o s i n g o f e x i s t i n g cracks. _{These c r i t e r i a have been introduced by Chen and Suzuki (17) and by} Chen and Ting (18).

Nonlinear e l a s t i c c o n s t i t u t i v e r e l a t i o n s have a1 so been developed i n terms o f t h e s t r e s s r a t e and s t r a i n rate. That i s ,

I n these equations, Kt = Kt (aii) represents t h e tangent bulk modulus which i s a f u n c t i o n o f the f 4 r s t s t r e s s i-n-variant w h i l e Gt = Gt(Sij bii/2) ) i s t h e shear modulus which i s a f u n c t i o n o f t h e second s t r e s s invariant.-P Using t h e c u r r e n t values o f t h e tangent moduli, s i m i l a r m o d i f i c a t i o n s have been used f o r t h e i n c l u s i o n o f t h e cracks i n t h e model as i n t h e l i n e a r e l a s t i c f r a c t u r e model.

P l a s t i c Models

Such models take i n t o account t h e l i m i t e d p l a s t i c f l o w observed i n t h e s t r e s s - s t r a i n behavior o f concrete before crushing. The simp1 e s t model c o n s t r u c t e d i s based upon t h e p e r f e c t l y p l a s t i c model. The r e s u l t i n g c o n s t i t u t i v e r e l a t i o n i s d i v i d e d i n t o t h r e e p a r t s , t h e f i r s t o f these regimes being before p l a s t i c f l o w occurs, t h e second d u r i n g p l a s t i c flow, and t h e t h i r d a f t e r f r a c t u r e . Included i n such r e l a t i o n s are i n h e r e n t y i e l d and f r a c t u r e c r i t e r i a (e.g., Von Mises c r i t e r i o n , Drucker/Prager c r i t e r i o n , o r Coulomb c r i t e r i o n ) . For such models, t h e incremental p l a s t i c f l o w law which expresses t h e n o r m a l i t y o f the p l a s t i c deformation r a t e v e c t o r t o t h e y i e l d s u r f a c e can be w r i t t e n as,

where d~:.~ i s the incremental p l a s t i c s t r a i n tensor, oii i s t h e s t r e s s tensor, g i s t h e y i e i d f u n c t i o n , and A i s a p o s i t i v e p r o p o r t i o n a l i ~ y fa c t o r . This model has been argued as not meeting c e r t a i n requirements o f continuum mechanics when appl t ed t o b r i t t l e concrete m a t e r i a l s , as discussed f o r example by Chen and Ting (18).

To modify t h e p e r f e c t l y p l a s t i c model and remove some o f these d e f i c i e n c i e s , an e l a s t i c s t r a i n hardening p l a s t i c model has been introduced by Chen and Chen (19).

C5-86 JOURNAL DE PHYSIQUE

Using t h e n o r m a l i t y c o n d i t i o n o f t h e incremental t h e o r y o f p l a s t i c i t y , i t i s p o s s i b l e t o o b t a i n t h e f o l l o w i n g f l o w r u l e :

Here t h e f u n c t i o n f ( ) ~= 9(6..), h = d f / d ~

P 1

J

P' and E P

=/dEPJ

.

M a t r i x

c o n s t i t u t i v e equations based on t h i s model have been presented by Chen and Chen (19)

To describe t h e dynamic behavior o f concrete, as w e l l as t h e n o n - l i n e a r e f f e c t s o c c u r r i n g i n e l a s t i c - v i s c o p l a s t i c f r a c t u r e models a n a l y t i c a l methods have heen developed by N i l s s o n (20). I n t h i s model a r a t e hardening parameter, Hr, i s introduced i n order t o t a k e i n t o account t h e e f f e c t o f s t r a i n r a t e on t h e d u c t i l e y i e l d and b r i t t l e f r a c t u r e surfaces.

Where Eef i s t h e e f f e c t i v e s t r a i n r a t e expressed i n terms o f the octahedral normal and shear s t r a i n s , go and :o, r e s p e c t i v e l y .

Here t h e q u a n t i t i e s wl and w2 a r e weight c o e f f i c i e n t s i n t r o d u c e d t o t a k e i n t o account d i f f e r e n t m i c r o f a i l u r e mechanisms. The parameters cl, c2, and c3 a r e determined by f i t t i n g o f t h e data from dynamic compression t e s t s o f concrete.

Using a m o d i f i c a t i o n o f Perzyna's theory o f e l a s t i c - v i s c o p l a s t i c i t y (21), Bicanic and Zienkiewicz (22) have developed a r a t e and h i s t o r y dependent model f o r p l a i n concrete. The m a t e r i a l behavior i n t h i s model i s described using two surfaces i n t h e p r i n c i p a l s t r e s s space, these being, t h e d i s c o n t i n u i t y surface d e f i n i n g t h e departure from e l a s t i c i t y , and t h e s t r e n g t h l i m i t surface which monitors t h e i n i t i a t i o n o f t h e s o f t e n i n g regime.

Axiomatic Model s

I n t h e c l a s s i c a l incremental f l o w theory o f p l a s t i c i t y , a y i e l d c r i t e r i o n i s p o s t u l a t e d and a hardening r u l e i s s p e c i f i e d f o r t h e subsequent y i e l d surfaces. Any

c o n s t i t u t i v e law based upon such e l a s t o p l a s t i c behavior i s considered a d i s c o n t i nuous model c o n s i s t i n g o f separate stages f o r loading, unloading, and reloading. Experimental evidence i n d i c a t e s t h a t separation o f t h e e l a s t i c deformation from p l a s t i c deformation i s a mathematical i d e a l i z a t i o n o f t h e complicated a c t u a l phenomena. I n order t o circumvent some o f t h e d i f f i c u l t i e s associated w i t h t h e c l a s s i c a l t h e o r i e s o f p l a s t i c i t y , t h a t i s , p o s t u l a t i o n o f a y i e l d surface and i t s motion i n s t r e s s space, hardening r u l e s , and unloading c r i t e r i a , several i n v e s t i g a t o r s have attempted t o present continuous models. Among these axiomatic models, t h e endochronic t h e o r y presented by Valanis (23) has found considerable appl i c a t i o n s .

I n i t s basic format endochronic theory can be considered as a g e n e r a l i z a t i o n o f t h e c l a s s i c a l theory o f v i s c o p l a s t i c i t y which i s both h i s t o r y dependent and s t r a i n

r a t e dependent. Using i r r e v e r s i b l e thermodynamics, V a l a n i s ( 2 3 ) i n t r o d u c e d t h e n o t i o n o f a pseudo-time scale, t h e s o - c a l l e d i n t r i n s i c t i m e , and p r e s e n t e d a c o n s t i t u t i v e re1 a t i o n i n i n t e g r a l / d i f f e r e n t i a l f o r m and i n terms o f an i n t r i n s i c tinie. Such a c o n s t i t u t i v e model can d e s c r i b e d i f f e r e n t c h a r a c t e r i s t i c s o f m a t e r i a l b e h a v i o r i n c l u d i n g s t r a i n hardening, unloading, re1 oadi ng, c r o s s hardening, and c o n t i n u e d c y c l i c s t r a i n i n g . Bazant (24,25) extended and a p p l i e d t h e endochronic t h e o r y t o d i f f e r e n t t y p e s o f m a t e r i a l s i n c l u d i n g sand, rock, p l a i n c o n c r e t e , and r e i n f o r c e d concrete.

The s i m p l e endochronic t h e o r y r e s u l t s i n an u n l o a d i n g response which i s n o t e l a s t i c a t t h e b e g i n n i n g o f unloading. Consequently, i n f i n i t e s i m a l h y s t e r e s i s 1 oops i n t h e f i r s t quadrant o f t h e s t r e s s - s t r a i n space a r e n o t c l o s e d which i s i n c o n t r a s t t o e x p e r i m e n t a l evidence a v a i l a b l e on t h e b e h a v i o r o f metals. I n a r e c e n t paper, V a l a n i s and Lee (26) p r e s e n t e d a m o d i f i e d endochronic model based on a new i n t r i n s i c t i m e s c a l e which i s a measure o f l e n g t h i n t h e p l a s t i c s t r a i n space. T h i s model l e a d s t o t h e c l o s u r e o f t h e s e i n f i n i t e s i m a l h y s t e r e s i s loops.

Empi r i c a l Models ( E q u i v a l e n t u n i a x i a l Model )

The m a j o r i t y o f c u r r e n t a n a l y s i s t e c h n i q u e s uses an e q u i v a l e n t u n i a x i a l c o n s t i t u t i v e model f o r a n a l y z i n g two-dimensional problems such as beams, p l a t e s , and t h i n s h e l l s . I n t h i s model, t h e b i a x i a l s t r e s s - s t r a i n b e h a v i o r o f c o n c r e t e i s t r e a t e d by a s i n g l e e q u i v a l e n t s t r e s s - s t r a i n r e l a t i o n s h i p . Many e x p r e s s i o n s have been proposed i n t h e l i t e r a t u r e . The f o l l o w i n g e x p r e s s i o n o f a model o f t h i s t y p e can be c i t e d , as f o r example, t h e model p r e s e n t e d by Popovics ( 2 7 ) and Buyukozturk and Tseng (28) where,

Here a i s t h e p r i n c i p a l s t r e s s , a i s t h e maximum p r i n c i p a l s t r e s s , E i s t h e

P P

c o r r e s p o n d i n g e q u i v a l e n t s t r a i n , E i s t h e e q u i v a l e n t s t r a i n , and n i s t h e shape e

f a c t o r which i s g i v e n by

The q u a n t i t y Eo i s t h e i n i t i a l s l o p e o f t h e o-E curve. By d i f f e r e n t i a t i n g Eq. e ( I ) , t h e s l o p e a t a p o i n t w i t h e q u i v a l e n t s t r a i n ee i s found t o be 1

-

( )

]

n ( n - 1 ) E = [n-1

+($)

n ] Due t o s i g n i f i c a n t i n f l u e n c e o f h y d r o s t a t i c p r e s s u r e on t h e b e h a v i o r o f c o n c r e t e under t r i a x i a l states;this model cannot be a c c u r a t e l y used i n t h r e e dimensional analyses.

Rheol o g i c a l Models

These models have been developed based upon s t u d i e s o f t h e p r o p a g a t i o n o f one dimensional s t r e s s p u l s e s generated i n impact t e s t s o f l o n g b a r specimens. Several models can be c i t e d i n t h e l i t e r a t u r e . The s o l i d f r i c t i o n c o n s t i t u t i v e model

C5-88 JOURNAL

DE

PHYSIQUE

developed by Goldsmith e t a1 ( 6 ) and Pozzo ( 2 9 ) seems t o d e s c r i b e t h e d i s s i p a t i o n mechanism i n c o n c r e t e b e t t e r t h a n o t h e r models. The s t r e s s - s t r a i n r e l a t i o n f o r a s o l i d f r i c t i o n model can be r e p r e s e n t e d i n t h e f o l l o w i n g f o r m :

Here IA i s t h e n a t u r a l frequency and $ i s a damping c o n s t a n t .

CONCLUDING REMARKS AND RECOMMENDATIONS

The p r o p a g a t i o n o f s t r e s s waves r e p r e s e n t s a key f e a t u r e i n p r o d u c i n g t h e e f f e c t s which r e s u l t i n and c o n t r o l t h e f a i l u r e / f r a c t u r e mechanisms as w e l l as i n d e v e l o p i n g e x p e r i m e n t a l procedures f o r d e t e r m i n i n g dynamic p r o p e r t i e s and c o n s t i t u t i v e equations. Some o f t h e i m p o r t a n t parameters t h a t need t o be determined f rom wave s t u d i e s f o r c o n c r e t e , u s i n g a n a l y t i c a l o r e x p e r i m e n t a l procedures, i n c l u d e d a t a on t e n s i o n / c o m p r e s s i o n a l wave v e l o c i t i e s , e f f e c t s o f m i c r o v o i d s and m i c r o c r a c k s on wave p r o g r e s s i o n , e f f e c t o f aggregate s i z e and r e i n f o r c e m e n t on p u l s e a t t e n u a t i o n , g e o m e t r i c and m a t e r i a l p u l s e d i s p e r s i o n , damping, e f f e c t o f m a t e r i a l aging, s i z e e f f e c t s , and s c a l i n g parameters.

It would be i d e a l i f i n d e v e l o p i n g models f o r c o n s t i t u t i v e e q u a t i o n s s u b j e c t e d t o i m p a c t / i m p u l s i v e l o a d s a u n i v e r s a l l y a p p l i c a b l e methodology c o u l d be developed t o s y n t h e s i z e a l l l o a d i n g modes. For t h e p r e s e n t , when c o n s i d e r i n g t h e development o f c o n s t i t u t i v e e q u a t i o n s f o r use i n dynamic l o a d i n g o f c o n c r e t e , a number o f i m p o r t a n t i s s u e s a r e i n need o f f u r t h e r study. P r i n c i p a l among t h e s e i s t h e t e s t i n g o f a number o f t h e models c u r r e n t l y a v a i l a b l e beyond t h e one dimensional s t r e s s s t a t e t o m u l t i a x i a l s t r e s s s t a t e s . The i n c o r p o r a t i o n o f p r e v i o u s s t r a i n h i s t o r y , t h a t i s , t h e number o f p r i o r i m p a c t / i m p u l s i v e l o a d i n g s s h o u l d be known i n o r d e r t o expand p r e d i c t i v e model l i n g c a p a b i l i t y . I n a d d i t i o n , t h e r e c o g n i z e d r a t e s e n s i t i v i t y as i t r e l a t e s t o d i f f e r e n t l o a d i n g modes s h o u l d be i n c o r p o r a t e d i n any model development. Also i m p o r t a n t a r e t h e s i z e o f t h e components b e i n g t e s t e d . That i s , t h e response o f c o n c r e t e and s i m i l a r s t r u c t u r a l m a t e r i a l s i s dependent on t h e s i z e o f t h e geometric element as w e l l as t h e d i s t r i b u t i o n o f d e f e c t s such as developed m i c r o c r a c k s and i n h e r e n t m i c r o v o i d s w i t h i n t h e m a t e r i a l s . A d d i t i o n a l r e s e a r c h t h r u s t s a r e a l s o necessary t o develop a more complete u n d e r s t a n d i n g o f e m p i r i c a l and semi-empi r i c a l c o n s t i t u t i v e laws. C o n s i d e r a b l e p h y s i c a l e v i d e n c e f o r model 1 in g purposes may be gleaned by examining t h e m i c r o s t r u c t u r a l response regime o f c o n c r e t e as w e l l as t h e m a c r o s t r u c t u r a l regimes.

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