Notch sensitivity of concrete and size effect Part I: Effect of specimen size and crack length by 3-point bending

CEMENT and CONCRETE RESEARCH. Vol. 15, pp. 979-987, 1985. Printed in the USA. 0008-8846/85 $3.00+00. Copyright (c) 1985 Pergamon Press, Ltd.

PART

I

:

NOTCH SENSITIVITY OF CONCRETEAND SIZE EFFECT EFFECT OF SPECIMEN SIZE AND CRACK LENGTH BY 3-POINT BENDING Yu.V. Zaitsev All-Union Politechnical Institute (VZPI) P. Korchagina, 22, Moscow 129278, USSR K.L. Kovler Moscow Civil Engineering Institute (MISI) Shliuzovaia nab., 8, Moscow 113114, USSR

(Communicated by F.H. Wittmann) (Received July 15, 1985)

ABSTRACT The problem of notch s e n s i t i v i t y of concrete is discussed with the influence of size effect taken into account. The need to consider the size effect on the tensile strength of concrete was noted. It is shown that the c r i t i c a l stress intensity factor, KIC, is connected with the tensile strength of the material, which gives an opportunity to estimate the Kic-value on the results of standard strength tests of the unnotched specimens. In conclusion, a practical condition for quasi-brittle manner of failure has been found. INTRODUCTION The methods of linear and nonlinear Fracture Mechanics have often been used recently in the calculations for concrete and reinforced concrete structures /1/. However, the limitations of the application of LEFM are not clearly defined and often the use of LEFM is not well j u s t i f i e d . In particular, due to insufficient experimental data for the size effect on the c r i t i c a l stress intensity factor KIC, there is no distinct explanation of one of the fundamental terms of fracture mechanics : notch sensitivity. The latest research /1-5/ has established that Kic-value increases in accordance with the growth of specimen size h with decreasing speed and stabilizes at h = hmi n. At h > hmin one may consider that KIC = const and use the LEFM methods. At h < hmi n the failure is also defined by the crack growth, but i t can't be described with the aid of LEFM. One must use more complex models : Nonlinear Fracture Mechanics or the models of crack growth, based on the real structure of materials /6,7/. Unfortunately, in practice i t is rather d i f f i c u l t to obtain the value of hmi n from the direct fracture 979

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Vol. 15, No. 6 Yu. V. Zaitsev and K.L. Lovler

toughness test because with increases / 2 / .

increasing accuracy of the experiment, hmi n also

Size e f f e c t on KIC was studied in many works both experimentally and t h e o r e t i c a l l y / i - 5 , 8 , 9 / . Z. Bazant has had great success in the research of this problem / 9 / . Having considered the balance of energy at crack propagation in concrete, he has formulated a hypothesis : that the potential energy released at the f a i l u r e is equal to the sum of two components, proportional to the square of crack length and the crack zone area respectively. The width of the crack f r o n t is considered constant for every material and proportional to aggregate size. Using this hypothesis and dimensional analysis, Z. Bazant has come to a simple r e l a t i o n of size e f f e c t for geometrically similar specimens :

(i)

n : B Rt ( i + " h / d a ) - l / 2

where o n = stress on the net section at f a i l u r e ; Rt = t e n s i l e strength; h = c h a r a c t e r i s t i c size of the specimen; da = maximum aggregate size; B and = empirical constants. Size e f f e c t on the strength has not been studied f u l l y , but the s t a t i s t i cal theory proposed for its explanation agrees with the experimental data in general. According to this theory the p r o b a b i l i t y of the presence of the most dangerous defect determining the strength increases with increase in specimen size. The size e f f e c t of flexure beams of heavy concrete (d a = 2 cm) was estimated in p a r t i c u l a r by K.A. Malzov /10/, who proposed the formula for size factor Rt,b/Rt :

(2)

Rt,b/R t -- i + hl/h

where Rt = t e n s i l e strength of the specimens of any depth h in bending; Rt a x i ~ t e n s i l e strength of the standard specimens with the section 15 x 15 cm; hl = 13 cm. Thus, in p a r t i c u l a r Jim (Rt,b/Rt) = 1. h÷~ Though in the work of Bazant /9/ size e f f e c t on the strength was marked, q u a n t i t a t i v e estimates with the aid of the expression (1) supposed that Rt = const. In practice the value of Rt decreases analogically Rt, b with increasing specimen size h.

NOTCH SENSITIVITY Let us consider the test of the bent beam as one of the most well known schemes of the K i c - d e f i n i t i o n . According to / 8 / , °n

1

KIC

Rt,b

11/2(1-1/h)2y(l/h)

Rt,b

(3)

where Rt h = t e n s i l e strength in bending; h : specimen depth; l = crack length, ~q%al to Xh; on = stress on the net section, weakened by notch, at f a i l u r e (stress concentration excluding). Y(I/h) = Y(X) = correcting polynome in the form :

Y(X) = Ao+AIX+A2X2+A3X3+A4Xh+...

(4)

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981 NOTCH SENSITIVITY, FRACTUREMECHANICS, CONCRETE, SIZE EFFECT

where for 3-point bending : Ao = 1,93; AI = -3,07; A2 = 14,53; A3 = -25,11; A4 = 25,80; for 4-point bending : A0 = 1,99; AI = -2,47; A2 = 12,97; A3 = -23,17; A~ = 24,80. The r a t i o on/Rt b is often called notch s e n s i t i v i t y (8), but i t seems i t would be correc{ to call i t notch i n s e n s i t i v i t y because when %/Rt, b increases to I, real notch s e n s i t i v i t y decreases to O. Thus real notch senslt i v i t y is : S : 1 - %/Rt, b

(5)

I f we introduce the function Z(X) = xl/2(I-X)2Y(X)

(6)

then Eq. (3) can be rewritten in the form : KIC = Rt,b(1-S)hl/2z(L )

(7)

It is known that concrete at usual specimen sizes is almost insensitive to the notch, but with increasing homogeneity of the material (the decrease of the grain size d in comparison with the specimen size h), S increases. In addition, S depends on the r e l a t i v e crack length X, approaching the maximum S = S. at X = X. : 0,25 /2,4,8,11,12/. Thus, S = S(h/d, X); S. = S.(h/d). NOTCH SENSITIVITY : SIZE EFFECT Let us f i x some X, i . e . X . KIC and S with the growth of Z(L.) = 0,5, S = S. and

= 0,25, and let us consider the change of h at given d for 3-point bending. Because

(8)

KIC/Rt, b = (1-S.)hi/2/2 Considering

that

1.)

lim KIC = KIC; 2.) S. ÷ h÷m 0 at h ÷ O, and 3.) notch s e n s i t i v i t y S. has zero dimension, one may at the f i r s t approach (with one independent undimensional constant before h) imagine notch i n s e n s i t i v i t y (1-S.) in the form :

I - S . = [ l + ~ ( h / d ) n ] "1/2n

there

is

a finite

limit

(9)

where n > O. Let us consider 3 possible cases, Ol, and examine the character of the change on/Rt, b = I-S. depending on h (Fig. l ) . For these cases lim d(on/Rt,b)/dh is equal to ®, const and -0, respectively.The experimental results show that both notch s e n s i t i v i t y (or i n s e n s i t i v i t y ) and i t s rate of change d(~n/Rt,b)/dh vary monotonically with h. Then the case n>l (Fig. lc) does not apply. On the other hand, the experimental data show that the rate of change of notch i n s e n s i t i v i t y is f i n i t e for h÷O. Thus the case O
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Vol. 15, No. 6 Yu.V. Zaitsev and K.L. Lovler

(i0)

1-S, = on/Rt,b= (i+~ h/d) -1/2

Eq (10) coincides with Bazant's formula ( i ) and is a separate case of Eq (9) at n=l. Thus formula ( i ) , obtained by Z. Bazant with the aid of the energy approach, is confirmed by the a n a l y s i s of the notch s e n s i t i v i t y behaviour on the basis of the force approach. Analogous equations can be obtained also at other X-values. To define ~ we have from Eq (8), t a k i n g into account Eqs (2) and (10), i/2(1+~h/d)-1/2/2= 1/2/ KIC : ] i m Rt,bh Rt(d/c ) 2 h-~ which corresponds to the relation KIC = 4 Rtd

:

(11)

1/2

(12)

obtained by the analysis of the experimental data on KIC / 3 / . In Eq (12) d is the structure grain size, approximately equal to (2/3)d a /3/. By structure grain we mean some characteristic volume appearing during the fracture surface formation and in which all structure components are represented. From Eqs (11) and (12) i t follows that ~ : 0.016. From Eqs (8),

( i 0 ) and (11)

KIC/KIC = ( R t , b / R t )

-1/2 (l+d/~h)

(13)

Let us t r y to formulate a condition which j u s t i f i e s the use of LEFM, choos.ing as the main parameter S,. I t is known that the deviations of the strength d e f i n i t i o n usually do not exceed 15-20 % for concrete (13). Thus i t is logical to consider the achieving a Kic-value 80-85 % of KIC-Value as a practical c r i t e r i o n for the s t a b i l i z a t i o n of KIC proportional to Rt, b. From (13) taking into account Eqs (2) and (10) at KIC/KIc>O.85, we obtain S,>0.4.

.....................

. . . . . . . . . . . . . .- g

,

..... I

I_/

a

...................

b

J

:

N

L ..............

c

FIG. i : The character of the change of the notch i n s e n s i t i v i t y on/Rt depending on specimen size h at fixed r e l a t i v e crack length X : Ol (c).

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983 NOTCH SENSITIVITY, FRACTUREMECHANICS, CONCRETE, SIZE EFFECT

NOTCH SENSITIVITY : RELATIVE CRACK LENGTH EFFECT There are generalized experimental results of the an/Rt h-definition for specimens of concrete, mortar and hardened cement paste (hcp)'depending on according to S. Ziegeldorf /1/ (see Fig. 2a). We note that experimental curves pass through characteristic points (X=O; S=O) and (X=0.25; S=S,). Usua l l y S, = 0...0.2 for concrete and 0.25...0.4 for mortar. For hcp as a rule S,>0.4. Thus hcp can be considered a s u f f i c i e n t l y b r i t t l e material that one may use the LEFM methods. Let us imagine that i t follows :

Kic/Rt,bhl/2

= const.

Then from Eqs (7) and (8)

(I-S)Z(X) = (I-S,)Z(X,) : (I-S,)/2

(14)

I-S = (1-S,)/2Z(X)

(15)

In Fig. 2b the dependence (15) for different S,-values is graphed. At the large S,-values (more than 0.4) these graphs are close to experimental curves (see Fig. 2a) : they pass through the extremum (X=0.25; S=S,) and quite near the point (~=0; S=O). But with the decrease of S,-value the likeness of calculated curves to experimental ones disappears : the calculated curves become further removed f r o ~ e point (X=O, S=O). This means that the accepted hypothesis (KIc/Rt,bh~/~ = const) is not correct at small S,-values, namely S,<0.4. At S,>0.4 i t is convenient to approximate the function S = S(h/d, x) in the following form :

16)

S(h/d,k) = S,(h/d)f(X) At S,>0.4 we will suppose that formula (15) is correct.

............

dol

/i --S,~

C~

i

i

•

.

................

_

b

o7:

o~

~

96 a

_

J.

Og

.

.

.

.

~

.

J

IG

_

1.0

o

C~S

" S ; __I..................... J......................... Io 01 ~ a.4 06 08 ~ 1o b

FIG. 2 : Experimental (a) and calculated (b) dependences of notch insensitiv i t y an/Rt, b on relative crack length x.

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Vol. 15, No. 6 Yu. V. Z a i t s e v and K.L. L o v l e r

For border value S,=0.4 Eqs (15) and (16) must be realized simultaneously. Thus we find the form of the function f(X) : f(X) = 2.5 [1-0.3/Z(X)]

(17)

In conclusion we have from Eqs (7) and (16) : KIC/Rt, b = [ l - S , ( h / d ) f ( X ) ] h l / 2 z ( x )

(18)

where at S,<0.4 the function Z(L) and f(X) are accepted according to Eqs (6) and (17), and at S,>0.4 Z(X)=Z(X,)=0.5; f(X)=f(X,)=l.

SOME ILLUSTRATIVE EXAMPLES Dependences KIc/Rt h, calculated for concrete (d=1.6cm), mortar (d=O.16cm) and hcp Cd=~.'~16cm) according to Eq (18) are shown in Fig. 3. The value of hmin, which corresponds to q u a s i - b r i t t l e manner of f a i l u r e are equal to 180; 18 and 1.8 cm, accordingly. In other words, the specimen size must exceed the structure grain size approximately by 200 times in t h i s case. At usual specimen sizes (5-20 cm) dependences KIC/Rt, b from X have t h e i r maxima at X=X,=O.25, but going from concrete to mortar and hcp they transform into horizontal lines. This agrees with A. Carpinteri's experimental results / 4 / , which show the existence of the maximum Kic-value at x=0.25-0.30 for concrete and marble and shows no maximum for mortar.

"--= , c , ~~12

FIG. 3 : Dependences of KIC/Rt h on specimen size h and r e l a t i v e crack T~}. for concrete (d=Icm), mo~r't~ar (d=O.lcm) and hcp (d=O.Olcm).

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985 NOTCH SENSITIVITY, FRACTURE MECHANICS, CONCRETE, SIZE EFFECT

Let us consider the detailed size effect on KI C using P. Walsh's experimental data, the expression (10) and the transformation from tensile strength of concrete, determined for the beams with h=7.5cm, to Rt-value with the aid of the coefficient 0.58 (15). In Fig. 4 we w i l l see that size effect greatly influences the strength of notched specimens. Ignoring the size effect on the strength of such specimens can result in rough mistakes at smal] specimen depths. •

.~

•

..

i

•

c)

--

~5

~

FIG. 4 : Dependences of size factor notched beams accordingly P. Walsh /14/.

Rt,b/R t

IIkcm

on specimen depth

h for

Dependences Rt b/Rt, ~n/Rt b and KIC/KIC at X=0.25 for usual heavy concrete (da=~Cm), achieve~ by the calculations on the Eqs (2), (10) and (13) are shown. In Fig. 5. We can see that the increase of KIC begins only after h=2Ocm. This corresponds to the experimental results of S. Mindess /16/, who found no differences between Kic-values in tests with beams with depths of 10 and 20 cm but did find a greater Kic-value at h=40 cm. This phenomenon can be explained by the fact that up to h=2Ocm the size effect on the strength, resulting in the decrease of Rt, b and, accordingly, KIC, is compensated by the size effect on notch s e n s i t i v i t y , resulting in the increase of S, a n d , accordingly, KIC. Note that at t h i s aggregate size (da=2Cm) the value of hmin, which corresponds to q u a s i - b r i t t l e manner of f a i l u r e (S,>0.4), is equal to 160 cm.

CONCLUSIONS 1) The correctness of Bazant's formula of size effect (1), obtained on the basis of the energy approach, is confirmed by the analysis of the notch s e n s i t i v i t y behaviour on the base of the force approach. However i t is necessary to take into account the size effect on the tensi|e strength of the material, according to Eq (2). 2) As a practical condition of the q u a s i - b r i t t l e manner of f a i l u r e (the condition for the use of LEFM) one may consider the achievement of the notch

996

Vol. 15, No. 6 Yu. V. Zaitsev and K,L. Kovler 1,~',

!

,.~ ..

~

C

.

. . . . . . . . . . . . . . . .

FIG. 5 : Dependences of Rt h/Rf ~/Rt,bb size h for usual heavy concrete~'da ~ ~ c .

....

:.•

?2.

.j

and

KIC/KIc

on

specimen

s e n s i t i v i t y S, value of 0.4. In t h i s case the specimen size must exceed the s t r u c t u r e grain size by approximately 200 times. Thus Kic-values , obtained in the experiments with concrete beams of usual sizes, are e x t r e mely far from extreme Kic-values (KIc) and can be interpreted only as g i v i n g empirical c h a r a c t e r i s t i c s of the resistance of the crack propagation of concrete. 3) I t is shown that c r i t i c a l stress i n t e n s i t y f a c t o r KIC in bending depends on the t e n s i l e strength in bending Rt,b, r e l a t i v e crack length 4, specimen depth h and s t r u c t u r e grain size d : KIC = Rt, b L Z - S , ( h / d ) f ( X ) ] h l / 2 z ( z ) where the f u n c t i o n S,(h/d) denotes notch s e n s i t i v i t y at ~=~,= 0.25 and is equal to 1-(I+0.016 h / d ) - i / 2 ; for 3-point bending at S,<0.4 one hasZ(Z)=~l-X)2xl/2(l.93-3.07X+14.53X2-25.11X3+25.80~), f(X)=2.5[ I - 0 . 3 / Z ( x ) ] ; at S,>0.4 Z(~)=Z(X,)=O. 5, f(X)=f(X,)=1. The i n t e r r e l a t i o n between KIC and Rt h permits us to make an i n d i r e c t KlC-estimation from the standard '~ strength tests of the unnotched specimens. With increasing specimen size Kic-value grows to the maximum KIC = 4 Rt d l / ~ . REFERENCES /1/ /2/ /3/

Fracture Mechanics of Concrete, Ed. F.H. Wittmann, Amsterdam, E l s e v i e r , 1983, 680 p. HIGGINS D.D., BAILEY J.E., Fracture Measurements on Cement Paste, J. of Materials Science, i i , 1995-2003, (1976). ENTOV V.M., YAGUST~.I., Experimental I n v e s t i g a t i o n of Laws Governing Quasi-Static Developement of Macrocracks in Concrete, Mechanics of Solids

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987 NOTCH SENSITIVITY, FRACTUREMECHANICS, CONCRETE, SIZE EFFECT

/4/ /5/

/6/ /7/ /8/ /9/ /i0/

/11/ /12/ /13/ /14/ /15/ /16/

(Transl. from Russian), i0, No 4, 87-95, (1975). CARPINTERI A., Applicatio~ of Fracture Mechanics to Concrete Structures, Proc. of ASCE, 108, No ST4, 833-848, (1982). PAK A.P., TRAP-E~-NIKOV L.P., Experimental Investigation Based on The G r i f f i t h - l r w i n Theory processes of the Crack Developement in Concrete, in Advances in Fracture Research, V. 4, Proc. of the 5th Intern. Conf. on Fracture, Cannes, 1531-1539, (1981). ZAITSEV Yu.V., WITTMANN F.H., Simulation of Crack Propagation and Failure of Concrete, Mat~riaux et Constructions, 14, 357-365, (1981). ZAITSEV Yu.V., Inelastic Properties of Solids with Random Cracks, Prepr. W. Prager Symp. on Mechanics of Geomaterials : Rocks, Concretes, Soils, Evanston, I l l . , 75-148, (1983). ZIEGELDORF S., MULLER H.S., HILSDORF H.K., A Model Law for the Notch Sensitivity of B r i t t l e Materials, Cement and Concrete Research, 10, No 5, 589-599, (1980). BAZANT Z.P., Size Effect in Blunt Fracture : Concrete, Rock, Metal, Proc. of ASCE, 110, No EM4, 518-535, (1984). MALZOV K.A., Le sens physique de la r6sistance conventionnelle de rupture en traction du b~ton mesur~e par flexion, B~ton Arm~, 21, (1960). SHAH S.P., MC GARRY F.J., Griffith Fracture CriteriOn and Concrete, Proc. of ASCE, 97, No EM6, 85-97, (1971). GJORV O.E., SORENSEN S.I., ARNESEN A., Notch Sensitivity and Fracture Toughness of Concrete, Cement and Concrete Research, Z, No 3, 333-344, (1977). LJUDKOVSKIJ A.M., About the Influence of the Specimen Size on the Characteristics of Mortar, Concrete and Reinforced Concrete, No 10, 14-15, (1983) (in Russian). WALSH P.F., Fracture of Plain Concrete, Indian Concrete Journal, 4__66, No 11, 469-470, 476, (1972). LESCHJINSKIJ M.Yu., Concrete Testing, Moscow, Stroyizdat, 1980, 360 p. (in Russian). MINDESS S., The Effect of Specimen Size on the Fracture Energy of Concrete, Cement and Concrete Research, 1_44, No 3, 431-436, (1984).