CEMENT and CONCRETE RESEARCH. Vol. 17, pp. 249-262, 1987. Printed in the USA. 0008-8846/87 $3.00+00. Copyright (c) 1987 Pergamon Journals, Ltd.

A FRACTURE MECHANICS MODEL TO PREDICT THE RATE SENSITIVITY

OF MODE I FRACTURE OF CONCRETE

Reji John and Surendra P. Shah Department of C iv i l Engineering, Northwestern Un ivers i ty

Evanston, IL 60201 U.S.A.

and

Yeou-Shang Jenq Department of Civil Engineering, The Ohio State University

Columbus, OH 43210 U.S.A.

(Communicated by F.H. Wittmann) (Received Nov. 12, 1986)

ABSTRACT

The mechanical propert ies of cementit ious composites have been observed to be sens i t ive to the rate of loading and th is rate s e n s i t i v i t y has been a t t r i bu ted to the s t ra in rate ef fects on cracking. A nonl inear f racture mechanics model is proposed to predict the s t ra in rate ef fect on mode I f rac ture of concrete. This model r~quires three material propert ies (C r i t i ca l Stress In tens i t y Factor, KI~ , C r i t i ca l Crack Tip Opening Displacement, CTOD cand Young's Modulus, ~) which can be deter- mined from s ta t i c tes ts . The ana ly t ica l procedure is based on the ob- servation that the pre-peak non l inear i t y is due to the pre-peak (or stable or p r e - c r i t i c a l ) crack growth and that th i s p re -c~ i t i ca l crack growth decreases with increase in rate of loading. K~ and E are assumed to be rate independent while CTOD c is assumed to ~ecrease ex- ponent ia l l y with the logarithm of the re la t i ve s t ra in rate. The model predicted values corre lated well with the experimental ly observed trends in the s t ra in rate ef fects on mode I f racture of concrete.

INTRODUCTION

T r a d i t i o n a l l y , the tens i l e strength of concrete is neglected in s t ruc- tu ra l design. Modern computer-aided analysis and use of concrete for special s t ructures such as reactor containment vessels and miss i le storage s i l o s , has led to a growing in te res t in the cracking behavior of concrete ( i , 2). Such concrete s t ructures are also l i k e l y to be subjected to short durat ion impul- sive and impact loads in addi t ion to s ta t i c loads. Experimental resul ts i nd i - cate that the tens i l e strength increases with the rate of loading ( i , 2). This implies that neglect ing the ' rate e f fec t ' in s t ruc tura l design might lead

249

250 Vol. 17, No. 2 R. John, et al.

to underestimation of the strength and hence uneconomical designs.

The rate of loading effects on the mechanical properties of concrete have been reviewed in detail by Suaris and Shah (3), Mindess (4) and Reinhardt (5). Many have conducted experiments to study the rate effect in tension (6- 10), flexure (11-17) and compression (18-22). The available results indicate that the rate sens i t iv i ty of tensi le strength is higher than that of compressive strength and that the rate sensi t iv i ty of flexural strength is in between that of tensi le strength and compressive strength, Fig. 1 (23, 24). This implies that the rate sens i t iv i ty of ultimate strength is primarily related to the strain-rate effect on cracking.

THERMALLY ACTIVATED CRACK GROWTH MODELS

The rate-effect on crack propagation can be explained using thermally ac- tivated flaw growth models. Such models have been used by many to predict the strain rate effects on the fracture strength of materials such as concrete, ceramics and rocks. For example, Evans (25) assumed Eqn. 1 and derived Eqn. 2 which was or ig ina l ly proposed by Charles (26).

V = K~ (1)

1 ~f = (~) N+I (2)

where K I = stre}s intensity factor in mode I, V : crack velocity, of = frac- ture strength, o = rate of stress application and N is a constant. Mihashi and Wittmann (27) used this approach to predict the loading rate influence on strength of concrete (see equation in Fig. 2). Their predicted values are compared in Fig. 2 with the test results of Gopalaratnam, Shah and John (11), who used an instrumented impact testing system to obtain varying strain rates on Charpy-type of beam specimens. The value of the constant m in the Mihashi - Wittmann equation was adjusted so as to best f i t the data as shown in Fig. 2. I t can be seen that the thermo-dynamically activated rate type of model cannot predict the increase in fracture strength observed at the impact loading rates. A similar conclusion was also reached by Reinhardt (5) who compared various analytical models with several sets of experimental data. I t should be pointed out that for slower rates of loading (log (B/~o) < 5) the rate type describes the observed behavior well.

DYNAMIC CRACK GROWTH MODELS

The thermally activated crack models were developed to predict crack growth under a constant load or a slowly increasing monotonically applied load. Under such loads crack growth is slow (crack velocity less than 1 mm/s). Under impact loading the crack may grow at a much faster rate. To de- termine the dynamic stress distr ibut ion around a fast-moving crack-tip, one can use equations of motion including the inert ia term. Freund (28) obtained dynamic, elastic solutions for crack growth due to general loading. He con- cluded that the dynamic stress intensity factor decreases with increasing crack velocity and is given by:

KID = k(V) .KIs (3)

in which KID = dynamic stress i n t e n s i t y f ac to r , KIS = s t a t i c st ress i n t e n s i t y fac tor for ~he same loading and k(V) = v e l o c i t y cor rec t ion fac tor s im i l a r to the resu l t s of Broberg (29). The v a r i a t i o n of k(V) wi th crack ve l oc i t y (V) is shown in Fig. 3.

Vol. 17, No. 2 251 FRACTURE MECHANICS, MODEL, RATE SENSITIVITY

2.2

• 2.0

c~ v 1.8

1.6

1.4

b 1.2

1.0 10 -7

Experimental Results

Tension X /

Flexure . \ / / / /

Compression ~ , , / /

~ . ~ - /

, ,

i0 -5 i0-3 i0 -I

log(strain rate) (I/sec.)

FIG. 1

Effect of rate of loading on the fracture strength of concrete

(23,24)

0.30 k • Gopalaratnam, Shah and John • - Mihashi and Wittmann • •

°'15 r ~o LooJ ,--- "" _ ~ " " " - - ~o = 9.66 N/mm 0o= 0.03 N/mm2°s

O . O 0 ~ - - - , L , m ,

0 1 2 3 4 5 6

log( O'/ (7o)

FIG. 2

Comparison of data (11) with the Mihashi-Wittmann equation (27)

FIG. 3

Variation of k(V) with crack velocity (V) (28)

1.00

0.75

0.50 .=~

0.25

0.00 0.00 0.25 0.50 0.75

V / C R

~ ocity

CR= Rayleigh . n wave speed

& I I % 1.00

252 Vol. 17, No. 2 R. John, et a l .

Several researchers have observed crack ve loc i ty in concrete during im- pact loading (16, 30, 31). The resul ts observed by John and Shah (16) are shown in Fig. 4. I t can be seen that the maximum crack ve loc i ty for the s t ra in rate of about 1.0 per second is about I00 m/s. This value is less than

103

I01 E

lO -I

10 -3

concrete

V = 0.045 r

C R

/ /

/ /

t /

/ i CR= 2000 m/s

/ /

/ /

/ t

10 -4 l og (s t ra in rate)

! !

10 -2 i00 ( I / sec , )

FIG. 4

Rate s e n s i t i v i t y of crack ve loc i ty in concrete (16)

5% of the Rayleigh wave speed (CR) in concrete. For th is small value of V/CR, i t can be seen in Fig. 3 that ~he value of k(V) is close to 1.0 and hence KID ~ KIS in Eqn. 3.

From the preceding discussion i t seems that nei ther the thermal ly ac t i - vated crack growth model nor the dynamic crack propagation model can sa t is fac- t o r i l y explain the s t ra in rate sens i t i v t y of concrete subjected to impact loading up to a s t ra in rate of about i per second. A model based on the con- cept of nonlinear f racture mechanics is proposed in th is paper. This model is described in detai l fo l lowing presentation of some pert inent experimental ob- servat ions.

EXPERIMENTAL OBSERVATIONS ON CRACK GROWTH

To study crack propagation under impact loading, notched beam specimens of concrete were subjected to various loading rates using an instrumented im- pact system (see photograph of the tes t set-up at the end of th is manu- s c r i p t ) . The rate of crack growth was monitored using a res i s t i ve type of th in fo i l gage (Krak Gage-TTl D iv is ion , Hartrun Corp., FL, USA) which was glued on the surface of the specimen at the t i p of the notch (Fig. 5). A change in crack length is recorded as a change in potent ia l while a constant current is being supplied to the fo i l gage. The deta i ls of the test tech- niques are described in Ref. 16. In th is paper only the observations which are pert inent to the proposed model are reported.

A typical set of resul ts obtained during an impact event is shown in Fig. 6. The measured values of load (P), notch- t ip s t ra in as recorded by a s t ra in gage (c) and the crack extension (Aa) as recorded by the Krak Gage are given. The slope of the Aa VS. time curve gives the ve loc i ty of crack propagation. The maximum observed ve loc i ty at various loading rates are given in Fig. 4. From Fig. 6 i t can be seen that a stable crack growth occurs p r io r

Vol. 17, No. 2 253 FRACTURE MECHANICS, MODEL, RATE SENSITIVITY

1.0

0.8

v

0.6 , q

• 0.4 c=

v

0.2

0.0

10 -7

. ~ - K~c : 1.43 MNm -3/2

Analytical ~ C T O D c s = _ _ 0.0127 mm i • Data (16) ~ 2 " ~ E = 31450 N/mm

~a~(stat.):5 mm (data) ~ A = 0.00075

Krak Gage- - -~p ~ - - 3.65

! (dimensions in~Im) I I I I

10 .5 10 .3 10-1 101 log(strain rate) ( I /sec.)

FIG. 5

Effect of rate of loading on pre-critical crack growth

3.0

2.0

Ol.0

0.0 0 4

concrete

= O.l sec -l

load ~ / ~-notch-tip / / sray ~ ]'~crack

/ ~ / / extension

1 2 3 Time (ms)

30

E E

v

o

20 "F~ o-

R L~

lO ~

1500

lOOO

e -

"~, 500

C~

FIG. 6

Typical crack growth data. Impact velocity = 0.70 m/sec (16)

254 Vol. 17, No. 2 R. John, et al.

to the peak load. It was observed that this pre-critical (pre-peak) crack growth can be substantial for the specimens tested at static rate of loading (16, 32) and it decreases with increasing strain rate as shown in Fig. 5 (16).

A NONLINEAR FRACTURE MECHANICS MODEL

For conditions where LEFM (Linear Elastic Fracture Mechanics) is appli- cable, one can calculate the crit ical stress intensity factor K. from the notched beam tests using observed values of the maximum load of1~max , and in i - t ia l notch length a o. For cement based composites, since the pre-critical crack extension can be substantial, this stable crack growth (also often called fracture process zone or f ict i t ious crack) should be included in determining the fracture toughness value. It has been recognized that the exclusion of this pre-critical crack growth will yield values of K I . which will depend on specimen size and geometry (32-38). The extent of ~e pre- cr i t ical crack growth cannot be accurately determined from the surface crack growth measurement or from the conventional compliance technique since the crack front is tortuous and discontinuous (16, 39, 40).

w !

Effective Griffith Crack

~ X

0 . . . 1

pre-critical crack growth

'- ~ Critical Point

f / ' K I :

/ ~ CTOD : CTODcs

/ / / / MOD ~ K I = K~c

CMODe= elast, ic C

CMOD CMOD e

Typical Plot of Load vs. Crack Mouth Opening Displacement

FIG. 7

Definition of the two parameter fracture model (32,33)

Vol. 17, No. 2 255 FRACTURE MECHANICS, MODEL, RATE SENSITIVITY

To overcome this d i f f i cu l t y , Jenq and Shah (32) have proposed an effec- t ive elastic crack length approach to obtain a valid fracture toughness value. The effective crack length 'a' was defined such that the measured (elastic) crack-tip opening displacement (CTODr) was the same as that calcu- lated using LEFM (Fig. 7). They observed that K I and the corresponding CTOD

• , C

values determined using the effective crack length were essentially Indepen- dent of the size and geometry of the beams they tested (32,33). The effective pre-cr i t ica l stable crack growth as defined by them is dependent upon the value of a o, geometry of the specimen and the (LEFM calculated) value of CTOD c. Using this two-parameter fracture toughness model Jenq and Shah (33) were- able to explain many observed size-effect related p~enomena for cement composites. Note that the Kic defined by them is termed Kic .

I t is possible to calculate the maxiumum load #f a structure or a speci- men of a given geometry using the two parameters K~r and CTOD c. For example one can calculate the tensi le strength of a concret~ specimen from the know- ledge of the two parameters which can be determined from a notched beam test. Assuming that an unnotched specimen subjected to uniaxial tensi le stress fa i ls by cracks i n i t i a t i ng at the outer edges and progressing towards the center of the specimen, Jenq and Shah (33) have shown that

(K~c)2 s )2 (4) I

f t = 1.4705 E.CTOD for b ) 0.27 (E.CTOD c / Kic c

I

where 2.b = width of the specimen and f t = uniaxial tensi le strength.

This two parameter fracture model is used to predict the strain rate effects on the mode I fracture of concrete as discussed below.

PROPOSED MODEL

Based on the above discussion, a rate sensitive two parameter fracture model is proposed to predict the effect of rate of loading on mode I fracture of cementitious composites. The main features of the model are: ( I ) K s Ic is assumed to be rate independent Note that K s • Ic is different from

KIc-LEFM and includes the pre-cr i t ical crack growth.

(2) cTOpc is assumed to be rate dependent• I ts value decreases with an in- crease in strain rate (Fig. 8) and is given by the following equations•

CTODcd -A.( ~ )B CTOD = e (5)

cs

: log (~dy_n_) (6)

Cstat

where ~stat

~dyn = dynamic strain rate ,

CTODcs : CTOD c at ~stat '

CTODcd = CTOD c at ~dyn '

-1 = stat ic strain rate (= 10 -7 sec ),

~dyn ) ~stat'

and A and B are empirical constants.

256 Vol. 17, No. 2 R. John, et a l .

.

0.8

v U

0.6 0

L . )

= 0.4 "13 v

U

o 0.2 I---

0.0

i0-7

Proposed

I Data

A=0.00075 ~ * estimated

B=3"65 ~ ~ ; m s ~ -

gage data

S t r a i n CTOD c i ~ (16)

Rate

\ o.?o o , ,

10 -5 i0 -3 lO-I i01 103

log(s t ra in rate) ( I / sec . )

FIG. 8

Rate s e n s i t i v i t y of c r i t i c a l crack t i p opening d is -

placement, CTOD c

(3) Young's modulus of e l a s t i c i t y , E, is assumed to be rate independent (as has been observed in Refs. I0 and 17).

In general, LEFM y ie lds the fol lowing equations:

K I = ~ . ~ - ~ .FI(~ )

CMOD = ~ . V I ( ~ )

(7)

(8)

CTOD = CMOD.Z(~,B) (9)

where ~ = applied st ress, D = depth of the specimen, CMOD = crack mouth open- ing displacement, CTOD = crack t i p opening displacement, ~ = a/D = notch-depth ra t i o , a : a o + Aap , a o : i n i t i a l notch length, Aap = pre-peak crack

extension, B = ao/a and FI, V I and Z are geometry correct ion factors .

Combining Eqn. 7, Eqn. 8 and Eqn. 9 we obtain

KI _ CTOD.E.J-~ . y(~) ( i0) 4. D/~-

where

FI(~) (11) Y (~) : V - I - ~ " Z- ~ , ~

Knowing K~ and CTOD~, the peak stress ~ can be evaluated for any structure using Eqn. 7 and Eqn. i0. Note that for an unnotched specimen CTOD = CMOD

VoI. 17, No. 2 257 FRACTURE MECHANICS, MODEL, RATE SENSITIVITY

FIG. 9

Variation of the geometry correction factors with respect to the notch-

depth ratio

lO

8

>~6

~-4 LL

~2

0 0.0

LEFM p i

• ~I aO D // l

L ,,<o. -_+ , , , / ~'~"'Y (ao=O) Vl- ///// ' / " l l - - - F l

\ , ' J

, " - - - T ' - - - - - , " - - - - 0.2 0.4 0.6 0.8 1.0

notch-depth ratio,

(since a n : O, Z(~, B) = 1). The var ia t ion of the funct ions FI(~ ) , V l (~ ) and Y(~) wit f i respect to ~ is shown in Fig. 9.

Consider an unnotched specimen, s I t is proposed that CTOD c decreases with increase in rate of loading whi le Klc remains constant . An increase in rate of loading resu l ts in the fo l low ing :

( i ) a decrease of CTOD c (as proposed) (Eqn. 5 and Eqn. 6). i ) to maintain K{~ constant, Y(~) should increase (Eqn. i 0 ) .

l l i i ) increase of ~f'(a) impl ies decrease of a and consequently decrease of El(q ) (Fig. 9) .

( iv ) hence the s t rength , a, increases so as to maintain constant K s Ic ' as given by Eqn. 7.

A decrease of ~ means that the pre-crit ical crack growth (Aa~) is also de- creased. Thus this model is able to explain the general trend ~f the rate ef- fects, i .e. increase in strength and decrease of pre-peak nonlinearity with an increase in rate of loading.

Calibration of Proposed Model

To predict the strain rat~ effect using the proposed model one needs to know the fracture parameters Kit and CTODcs , the modulus of elast ic i ty E and the ~ariation of CTOD~d with ~espect to the rate of loading. The values of K~ , CTODcs and E ~an be obtained from a static test (that is, slowly ap- plie~Cmonot~nlcally increasing load at a strain rate approximately correspond- ing to 10- per second) using, for example, single edge notched (SEN) beam type of specimens (Fig. 9).

I t is d i f f i cu l t to accurately measure the crack t ip opening displacement during impact loading. As a result sufficient data ~re not available to pre- cisely establish CTOD_ variation with strain rate, c . An exponential func- tion is proposed base'on the observation that the higher the strain rate, the lower the extent of pre-crit ical crack growth and that beyond a particular high strain rate, the pre-crit ical crack growt~ wil l be negligible (16). At that rate, CTOD_ will also be negligible and K. = KT_ (LEFM). At very high rates, as the crack velocity approaches the Raleigh wave speed, dynamic con- siderations become important (Eqn. 3) and the proposed model may not be appli- cable.

258 Vol. 17, No. 2 R. John, et al.

To obtain the value of the constants A and B of Eqn. 5, the data obtained by other investigators (6-9) on rate sensitivity of tensile strength of con- crete were used. I t is l ikely that the constants A and B depend on the material compositions. However, only one set of values for the constants A and B were determined from the data. To determine the constants A and B from the tensile strength, Eqn. 4 was used. The values of E, KT~ and CTOD~: were taken from the notched beam test reported by Jenq and Sh~ (33, 41)~ ~ They

6

5 v

- . - ~ 4

3

2 v

l

0

lO -7

' 2 ** / f t (N/mm) o /

o Mellinger and Birkimer 3.40 / • Takeda and Tachikawa - - / • Cowell 3.55 /

Kormeling et al . 2.62 - -Ana l y t i ca l 3.65 /

K~c_ = 0.99 MPa~ , E = 31034 N/mm2 ~ ~ A = n _ • A=O.OOO75nnnT~

CTODcs oo127 mm

** static tensile strength i I i I i I I I

lO -5 lO -3 lO -I lO l log(strain rate) (I/sec.)

FIG. 10

Strain rate ef fect on tensi le strength of concrete

K S evaluated - and CTODcs for different mix proportions. The values used here correspond ~ an uniaxlal tensile strength comparable to those for the data in Fig. 10.

From Fig. 8 and Fig. 10 i t can be seen that the proposed variation of CTOD_ with strain rate seems plausible. To further check the validity of the

L

proposed model, the predicted rate effects on flexural strength were compared with the test data.

PREDICTION OF FLEXURAL RESPONSE

To demonstrate the appropriateness of the proposed model, the strain rate effect on flexural strength (~odulus of rupture, MOR) was predicted using the same parameters (E, CTOD c, KTr , A and B) as those used for predicting the rate sensitivity of tensiTe strength. The predicted response, which is shown as a solid line in Fig. 11, was obtained for beams with a depth of 76 mm (3 in.) and for LEFM solutions of K I and CTOD (similar to Eqn. 7, Eqn. 8 and Eqn. 9) based on span to depth ratio-of 4. From Fig. 11, i t can be seen that the theoretical curve correlates wel| with the experimental data obtained from several investigators.

I t should be noted that the data shown in Fig. 11 was obtained from beam specimens which had somewhat comparable static tensile strength values similar to that of the data in Fig. 10. In addition, the specimens tested did not al- ways have a span to depth ratio of 4. Perhaps an even better prediction can

Vol. 17, No. 2 259 FP~ACTURE MECHANICS, MODEL, RATE SENSITIVITY

2.6

2.2

~I ,8 0

c -

~1.4

l.O

oGopalaratnam, Shah and John • Butler and Keating

Suaris and Shah • Zech and Wittmann o Mindess and Nadeau

Analyt ical

K s = 0.99 MPa~ Ic

CTOD = 0.0127 mm cs

E = 31034 N/mm 2

A=0.00075, B=3.65 0

~7 ~7

' 2 ** ft(N/mm )

3.54 3.77 3.73 3.69

3.65

0

0

0

K7

** estimated s ta t ic tensi le strength

10 -7 10-5 10 -3 i0 -I i01 log(stra in rate) ( I /sec. )

FIG. i i

Strain rate effect on f lexural strength of concrete

S be obtained i f more appropriate LEFM solutions and material parameters (KTr , CTODcR and E) corresponding to the actual dimensions and loading configuratTon (3 p6Tnt or 4 point bending), and mix proportions respectively, were used.

DISCUSSION

I t has been reported that the strain rate sens i t i v i t y of concrete in uni- axial tension is higher than that in flexure (3, 23, 24). This can be pre- dicted using the proposed model. The theoret ica l ly predicted values (for the set of parameters indicated in the f igure) for uniaxial tension and for beams with two di f ferent depths are plotted in Fig. 12. I t can be seen that at a given strain rate, the MOR value approaches the uniaxial tensi le strength as the depth of the beam increases. Such a size effect on MOR values has been reported ear l ie r (33). The proposed model also predicts that the rate sensi- t i v i t y is less for f lexure (for commonly used beam depths) than that for uni- axial tension. This theoret ica l ly predicted trend is due to the interact ion of the size of the beam, p re-c r i t i ca l crack growth and the strain rate.

To further check the va l id i t y of the proposed model, the predicted pre- c r i t i ca l crack growth was compared with that measured by John and Shah (16) using the b r i t t l e Krak Gage, Fig. 5. Knowing K s , CTOD c and E, the theoret i - cal p re -c r i t i ca l crack growth was calculated ~ ing the LEFM based equations (s imi |ar to Eqn. 7 and Eqn° i0). A sat isfactory comparison is seen in Fig. 5.

260 Vol. 17, No. 2 R. John, et alo

2.2

l 8

"-" 1 .4 G

~- l.O

10 -7 i01

- -Tens ion / / • Flexure : D = 3810 mm /

- - Flexure : D = 76 mm - - / / /

s 0.99 MPa~ / / / Kic = / I

CTODcs = 0.0127 mm j / / 1 2 i

E = 31034 N/mm ~ 1 1 . . . . . _ i - A = 0 .00075

- B = 3.65

i I I l I I I

lO -5 10-3 lO-1

log(strain rate) ( I /sec.)

FIG. 12

Model predicted size effect on rate sens i t iv i ty of flexural strength

CONCLUSIONS

I . The effect of rate of loading on the fracture strength of concrete

subjected to impact loading cannot be explained sat is factor i ly by either

the thermally activated crack growth model or the dynamic crack propagation

model.

2. A two parameter fracture model is proposed to explain the strain rate ef-

fects on mode I fracture of concrete• This model is based on the observa-

tions that to evaluate a size independent fracture parameter(s) for con-

crete, one must include pre-cr i t ical (pre-peak) stable crack growth, and

that the pre-peak crack growth decreases with increasing strain rate.

3. To predict fracture strength using the proposed model one should know the

modulus of e las t ic i ty (E), c r i t i ca l stress intensity factor at the t ip of

an effective crack (K~c),_ and the cr i t i ca l crack t ip opening displacement

(CTODc). Only CTOD c is assumed to be rate sensitive•

4. The model predicted values correlated well with the experimentally observed

trends in tensile strength, flexural strength, size effect and pre-cr i t ical

crack growth with respect to the rate of loading.

5. The present model appears to be valid in the intermediate strain rate range

of 10 -7 to 10 sec - I . At higher rates of loading (~>10 sec -1) a dynamic

crack propagation analysis might be more appropriate and at lower rates of

loading (~<lO-7sec -1) a stress corrosion/creep based analysis would be

better suited to predict the effect of time, temperature and humidity•

Vol. 17, No. 2 261 FRACTURE MECHANICS, MODEL, RATE SENSITIVITY

ACKNOWLEDGEMENT

The research reported here is being supported by a grant (DAAG29-82-K- 0171, Program Manager: Dr. G. Mayer) from the U.S. Army Research Office to Northwestern University. The authors are also indebted to Dorothy Carlson for her flawless typing of the manuscript.

REFERENCES

1. Conc. Str. Under Impact and Impulsive Loading, Proceedings, RILEM-CEB- IABSE-IASS Inter. Symp., Berlin (West) (1982).

2. Cement Based Composites: Strain Rate Effect on Fracture (eds. S. Mindess and S. P. Shah) MRS Symp. Proceedings, 6_4_4 (1986).

3. W. Suaris and S. P. Shah, in Conc. Str. Under Impact and Impulsive

~ , Proceedings, RILEM-CEB-IABSE-IASS Inter. Symp., p. 33, Berlin (1982).

4. (e _S'dMindess' in Application of Fract. Mech. to Cementitious Composites, S. P. Shah), p. 617, Martinus Nijhoff Publ., The Netherlands,

(1985). 5. H. W. Reinhardt, in Cement Based Composites: Strain Rate Effects on

Fracture, (eds. S. Mindess and S. P. Shah) MRS Symp. Proceedings, 64 p. I , (1986).

6. F. M. Mel|inger and D. L. Birkimer, Tech. Rep. 4-46, Dept. of Army, Ohio River Div. Lab., (1966).

7. J. Takeda and H. Tachikawa, Proceedings, Int. Conf. on Mech. Beh. of Ma- ter ia ls, IV_, Kyoto, Japan, p. 267, (1971).

8. W. Cowel], Tech. Rep. R447, U.S. Naval Eng. Lab, Port Hueneme, CA (1966).

9. H. A. Kormeling, A. J. Zielinski and H. W. Reinhardt, Rep. 5-80-3, Stevin Lab., Delft Univ. of Tech., (1980).

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Modified Instrumented Charpy Impact Testing System (11, 16, 17)