Mixed-mode fracture of concrete

$Page 1: Mixed-mode fracture of concrete$
International Journal of Fracture 38:123-142 (1988) © Kluwer Academic Publishers, Dordrecht Printed in the Netherlands 123

Mixed-mode fracture of concrete

Y.S. JENQ ~ and S.P. SHAH 2 1Department of Civil Engineering, The Ohio State University, Columbus, OH 43210, USA "Department of Civil Engineering, Center for Concrete and Geomaterials, Northwestern University, Evanston, IL 60201, USA

Received 15 December 1987; accepted in revised form 22 March 1988

Abstract. Mixed-mode fracture of concrete is encountered frequently in design of concrete structures. To accurately analyze and design a concrete structure, it is essential to understand the failure mechanisms of concrete under mixed-mode types of loading. A model, which was originally proposed for mode I failure of concrete, is extended to characterize mixed-mode crack propagation. Crack initiation angles, final failure path, and crack instability of concrete specimens/structures are discussed. A series of mixed-mode notched beam tests were performed to verify the proposed model. The theoretical predictions were found to be in good agreement with the experimental results.

1. Introduction

Crack propagation in concrete is complicated by different mechanisms (crack initiation, crack curving, crack propagation, and crack arrest) involved in the failure process. These failure mechanisms can be illustrated through the load-CMOD (crack mouth opening displacement) response curve of a notched specimen, Fig. 1. It can be seen that in the initial stage the load-CMOD curve is linear since there is no (or negligible) crack growth. In this stage, the whole system behaves linearly. As the applied load increases, the specimen starts to respond nonlinearly. This nonlinear effect is mainly due to the generation and coalescence of microcracks (also termed stable crack growth) and possible material nonlinearity (e.g., nonlinear creep effects which are not considered here). Objective determination of the onset of nonlinearity at this stage is very difficult. As the microcracks coalesce into a macrocrack, further propagation of this macrocrack may be arrested by aggregates and air voids unless the applied load is increased. As a result of this arresting mechanism, the crack may deviate from its original propagation plane and additional load and energy are required for the crack to propagate further. An increasing K R (Stress Intensity Factor-Resistance) curve can be observed at this stage due to the toughening mechanisms resulting from the arrest and deviation of crack propagation [31]. When the applied load reaches the peak load, the crack will begin to propagate in an unstable manner under load-control testing environment (i.e., the incremental rate of change of load is kept constant). However, if CMOD control (i.e., the incremental rate of change of CMOD is kept constant) is used, then a stable post-peak behavior can be observed as shown in Fig. 1.

The aforementioned behavior is a general description for all modes of loading. Since mode I tests can be conducted easily in the laboratories compared to other modes, studies of fracture mechanics of concrete have focused on mode I failure. Several different models [4, 11, 17, 18] have been proposed to characterize mode I crack propagation in

124 Y.S. Jenq and S.P. Shah

8O0 Z

8_

600 O 0

CMOD (lO-2mm) 0 5, 10~ 15, 201200

I I : l inear range II " stable crack propagat ion of nonlinear range

III • post peak stage 1"= 25.4 mm - 1 0 0 0 mor ta r mix." C" S" A" W

200- - I 1.0:2.6:0.0:0.65 ~/P j=:I.125,,

1 5 o - . Io.8 n

clip gage / ' ~ x o 100- I "hL contro Is ~ - ~

CMOD 400

0 1 2 3 4 5 6 7 8 crack mouth opening displacement, CMOD (10 -3 in.)

Fig. 1. Typical experimental results of load vs. CMOD curve.

concrete. The Two Parameter Fracture Model (TPFM) was proposed by the authors [15, 16-18] to characterize mode I failure of concrete. The principles and some results of this model, which are relevant to the present mixed-mode fracture study, are summarized next.

2. Mode I fracture

The extent of stable precritical crack growth (stage II in Fig. 1) can be substantial for concrete. It has been observed that when the critical stress intensity factor, Ku, is calculated ignoring this precritical crack growth (i.e., initial notch length, a0, is used in evaluation of K~c ), the value of fracture toughness (K~c) depends on the geometry and dimensions of specimens [16, 17].

An effective Griffith crack model has been proposed by the authors to incorporate the precritical crack growth [15-18]. It is assumed that a crack of initial length a0 will critically propagate when the stress intensity factor at the tip of the effective crack a reaches the critical value K{, (the superscript s denotes consideration of stable precritical crack growth). The effective crack a is defined such that the crack tip opening displacement at the tip of the original crack a0 reaches a critical value CTOD, (Fig. 2). One can determine these parameters from pertinent linear elastic fracture mechanics (LEFM) equations and from the measured values of initial compliance (Ci in Fig. 2) and unloading compliance (C, in Fig. 2) at or near the maximum load as described next.

Mixed-mode ,fracture of concrete 125

i C T O D ¢

_ a o _

L ' a ! -

%

8 _o

i

I CMODe

..pre-critical crack growth crit ical point

K I = K~c " ~ T O D = CTODc

~ / p o s t critical

~ KI: Ks~

CMODe = elastic CMOD

CMOD

Fig. 2. Typical plot of load vs. crack mouth opening displacement according to two parameter fracture model for concrete proposed by Jenq and Shah (see [17]).

In general, L E F M yields the following equations:

K l = G ~ - a F , ( ~ ) (1)

C M O D = 4aa/E" V~(~) (2)

CTOD = C M O D . Z(~, fl) (3)

where a = applied far field stress, E = Young's modulus, D = depth of the specimen, C M O D = crack mouth opening displacement, CTOD = crack tip opening displacement,

= a/D = notch-depth ratio, a = a 0 + Aap, a 0 = initial notch length, Aap = pre-peak crack extension, fi = ao/a and Fj, V~ and Z are geometry correction factors. Knowing the initial compliance ((7,. in Fig. 2) and using (2) and a = a0, the value of Young's modulus E can be determined. Then substituting this value of E in (2), an effective crack length a is calculated such that the compliance obtained using (2) is equal to the measured elastic compliance at the peak load (C, in Fig. 2). K~. and CTOD c are then calculated at peak load using (1)-(3) and the effective crack length q. Thus, the two fracture parameters, Ki~,. and CTOD~ (as well as Young's modulus E ) can be determined from a single notched beam test [26].

I f the fracture parameters K~. and CTOD~ and Young's modulus E are known, the peak stress and hence peak load of any structure can be calculated using (1)-(3). Combining


(1)-(3) leads to:

K~ = CTOD. E" ~ . /71(¢z) 4w/-a V 1 (~z) • Z(~, ,B) " (4)

The value of effective _a at the critical load is determined from (4) for a given set of values of K~ c and CTOD c. Then using (1) the corresponding stress value at the critical load can be determined.

Based on a large number of notched beam tests, it has been shown that the two parameters, K{c and CTOD c are independent of size and geometry of beams [15-18]. Using this two parameter fracture model (TPFM), it is possible to explain the experimentally observed size-effect on modulus of rupture and on conventional fracture values [15-18], strain rate effects [19] and notch-sensitivity behavior observed in cement based composites [16, 17, 20].

3. Mixed-mode fracture of concrete

In reality, mixed-mode (i.e., combination of tensile and shear modes) crack formation is often observed in concrete structures. Attempts to apply fracture mechanics concepts to study mixed mode failure of concrete have been made by Arrea and Ingraffea [1], Rots et al. [27], Wium et al. [33], Kobayashi et al. [21] and Ingraffea and Gerstle [13]. It was reported by Ingraffea and Gerstle [13] that the final failure load is very sensitive to the material constants (i.e., tensile strength, fracture energy and shape of the stress-separation curve) used in the Fictitious Crack Model proposed by Hillerborg et al. [11]. Furthermore, it is very difficult to accurately determine these material properties from laboratory tests. Therefore, a trial and error method (or numerical experiment) has to be used to determine the appropriate material properties for the particular test result. Rots et al. [27] reported that the crack trajectories of mixed-mode tests predicted using the Crack Band Theory [4] (or smeared crack approach) is dependent on the finite element meshes and is not very objective. In addition, the post peak behavior could not be reproduced satisfactorily using the smeared crack approach.

For mixed-mode failure of concrete, the determination of the final failure path and the criteria for crack instability are more complicated than that of pure mode I failure. In mode I failure, the crack always propagates along its own plane, which is not the case in mixed- mode crack propagation. Therefore, criteria for determining the crack propagation trajec- tory and crack instability are necessary for mixed-mode fracture as discussed in the following sections.

Crack initiation theories

Several different criteria have been proposed for determining the direction of crack growth under general mixed-mode loading conditions [2, 5, 7, 8, 12, 29, 32]. The most commonly used criteria are the maximum tangential stress criterion [7], minimum strain energy density criterion [29], and maximum energy release rate criterion [12]. Only the maximum energy release rate criterion is reviewed next for reasons discussed later.

Mixed-mode .fracture o f concrete 127

Maximum energy release rate criterion, (G0)ma x

The energy release rate (G) can be expressed in terms of stress intensity factors as:

K:, - ~ + E'' (5) G m

where

E / =

E ' =

E =

K, =

Kll =

E for plane stress case

El(1 - v 2) for plane strain case

Young's modulus

Poisson's ratio

mode 1 stress intensity factor

mode II stress intensity factor.

In the derivation of (5), the crack is assumed to propagate along its own plane (self- similar crack growth) which is valid only for the pure mode I case. To relate the energy release rate (G) and stress intensity factors (K l and Kn) under mixed-mode loading conditions, Hussain et al. [12] proposed that the energy release rate for an arbitrary direction of crack propagat ion is necessary in order to apply the Griffith-Irwin energy release rate criterion. The stress intensity factors for the propagation branch in the limit as the branch goes to zero can be expressed as [12]:

( 4 )(i K~(O) = 3 + cos 20 + O/rc/ (K~cosO + } K n s i n 0 ) (6)

( 4 )(, K.(O) = 3 + cos 20 1 + 0--~J ( - 0 . 5 K l s i n 0 + K. cos0) , (7)

where 0 = the branch angle, K~(O)= mode I stress intensity factor at angle 0 and K.(O) = mode II stress intensity factor at angle 0. The energy release rate, G(O), is then expressed as:

1 a(0) = ~ [K?(0) + K?,(0)] (8)

o r

4 3 + cos 20 + O/n/ {(1 + 3cos:O)K21 G(O) = E---;

+ 8 cos 0 sin 0 KIK n + (9 - 5 cos 20)K~I }. (9)


The maximum energy release rate criterion postulates that a crack will initiate along the direction in which G(O) is maximum at a critical value of G(O). Based on this criterion, the fracture initiation angle can be determined by satisfying the following equations:

L~G c32G 00 -- 0, ~ < 0; (10)

and

K~(O) > O. (11)

The first set of equations are used to determine the local maxima. Since there are more than one maximum, (11) is introduced [30] so that crack propagation is confined along the direction in which a tensile stress exists. Note that (9) can be reduced to (5) by setting 0 = 0.

Nuismer [24] derived the closed form solution of energy release rate based on the stress intensity factors for vanishing branch crack at initiation. He proposed that this form should be used in the maximum energy release rate criterion. The crack initiation angle calculated using Nuismer's approach is exactly the same as that derived from maximum tangential stress criterion [7]. Other than the aforementioned criteria, Nemat-Nesser and Horii [23] and Ballarini et al. [2] have proposed that a crack will initiate along the direction of the maximum value of K~ in a new field and start propagating when K~ reaches a critical value.

Discussions of crack initiation theories

The theoretically predicted crack initiation angles based on different crack initiation theories are plotted in Fig. 3 for different values of K~I/Kj ratios. The crack initiation angle reaches

8 0

4O

• 2oL! I I , energy releos e rote ~ ~ strain energy density

of 0 2 4 6 8 10

Kz]/Kz

Fig. 3. Crack initiation angles predicted by different crack initiation theories.

Mixed-mode fracture of concrete 129

a constant plateau value when the K./K~ ratio is larger than 10. The initiation angles predicted using different theories were almost identical as compared to the scatter observed in experimental results of concrete specimens, which will be discussed later. All the theories predict that a crack will propagate along its original plane (0 = 0) for pure mode I and will deviate from its own plane (at an angle of about 0 = 73 degrees) for pure mode II.

Any of the above fracture theories can possibly be used as fracture theories to govern crack propagation and crack instability for concrete materials if proper consideration of stable crack growth is taken into account. In the present paper, only the maximum energy release rate criterion which is consistent with the original Griffith energy concept [9, 10] and Two Parameter Fracture Model [17] was used to predict the mixed-mode fracture behavior of concrete.

4. Proposed model

Mixed-mode crack stability

For plain concrete under mixed-mode (mode I and mode II) loading, a mixed-mode stress intensity factor (K) was defined as

K = [K2(0) + K(~(O)] '/2. (12)

Similar to the mode I model, the critical stress intensity factor (K,.), which is determined at the effective mixed-mode crack tip, is used as the fracture parameter to govern the mixed- mode crack propagation. The crack tip displacement (CTD) for mixed-mode crack was expressed as the vector sum of crack tip opening displacement (CTOD) and crack tip sliding displacement (CTSD) as shown in Fig. 4. Similarly, the crack mouth displacement (CMD) is the vector sum of crack mouth opening displacement (CMOD) and crack mouth sliding displacement (CMSD) (see Fig. 4). It should be noted that the fracture parameters K, and

Y

= X

MSD

CMOD

Fig. 4. Composition of crack tip displacement and crack mouth displacement.


CTD c (critical crack tip displacement) can be determined directly from mode I failure which is a special case of mixed-mode failure, i.e.,

Kc = KI,, (13)

and

CTD,, = CTOD~. (14)

The maximum load resistance of any section under mixed-mode loading was calculated following the same principles as that of mode I fracture by replacing K~ and CTOD by K and CTD, respectively, and using the predicted crack path. The section that offers the least resistance to the applied load will be the final dominant failure section. Therefore, depending on the loading setup, the least resistance may not be at the location of the imposed notch. An example of this situation will be demonstrated in the following experimental program.

5. Experimental verification

Three point bend notched beam tests with notches at different off-set ratios (y = 2x/s) (Fig. 5) were performed to verify the proposed model. It can be seen that when the off-set ratio (y) equals to zero, the tests will be reduced to pure mode I three-point bend tests. Thus, pure mode I and mixed-mode tests can be performed using similar specimens and the same testing setup. Taking advantage of the unique design of this testing setup, the material parameters (K~c and CTODc or K c and CTDc) can be determined directly from the mode I tests. Using the predetermined values of K{,. and CTOD, and the proposed model, one can then predict the peak loads for mixed-mode tests. When the off-set ratio is not equal to zero, both KI and Ktl will exist and mixed-mode type of failure is expected to occur.

Three different mix-proportions and two different specimen sizes were used in the mixed- mode experiments. The mix proportions are listed in Table 1. The dimensions (span x depth x thickness) of the specimens were 609.6mm x 152.4mm x 57.15mm (24" x 6" x 2 .25")and 304.8mm x 70.2mm x 28.6mm (12" x 3" x 1.125")for large and small beams respectively. The off-set ratio, which varied from 0 to 2/3 by an increment of 1/6 with a notch-depth ratio equal to 1/3, was used to study the shear effect. Another set of specimens with fixed off-set ratio (0.5) and different notch-depth ratios (ranging from 0.5 to 2/3) was used to study the notch-depth effects on mixed-mode failure. All beams were tested in a closed-loop testing machine with CMD or load-line deflection as feedback signal

P

xi---~° I ' i i I ~ 0 . 5 S ~ ~ 0 . 5 S

~ 1 - - - - I

Fig. 5. Dimensions of mixed-mode tests.


Fig. 6. Load-line deflection, crack mouth displacement, and crack mouth sliding displacement measurement arrangements for mixed-mode tests.

Table l. Mix-proportion used for mixed-mode tests

Max Coarse aggregate

Series Cement Sand aggregate Water size, mm (inches)

C 1.0 2.6 2.6 0.55 9.52(0.375) M 1.0 2.6 0 0.45 4.76(0.1875) P 1.0 0.5 0 0.35 4.76(0.1875

to maintain a "stable" failure. All the notches were saw-cut and all beams were kept in a curing room with 90 percent average humidity and 27°C temperature. Some specimens of the P (paste) series experienced different curing conditions due to the malfunction of the humidifier. Thus, the material properties among the specimens of the P series were different. However, the crack paths of the P series were included in the analysis since the crack paths are probably not sensitive to the curing conditions. The applied load, crack mouth displacement, load-line displacement and crack sliding displacement were monitored and stored using a Nicolet analog-digital oscilloscope. CMD and load-line deflection were measured using an MTS clip gage and an MTS extensometer, respectively. The crack sliding displacement was monitored using LVDT (Linear Voltage Displacement Transducer) (Fig. 6). Sixty beams were tested in the present investigation.

Calculation of stress intensity factors and crack displacement

Analytical solutions of the stress intensity factors and the associated displacement fields are not available for the present testing configuration when off-set ratios (7 = 2x/s, Fig. 5) are not equal to zero. A numerical approach was used to obtain these values for the present study. There are several methods that can be used to calculate the stress intensity factors for two dimensional elasticity problems. The quarter-point singular element approach is used in


(a)

(b) Fig. 7. Finite element meshes (a) Before crack initiation and (b) After crack initiation.

the present study. It has been shown by Barsoum [3] that by moving the mid-node of an eight-node or six-node isoparametric element to quarter point around the crack tip, the stress field will be square-root singular, which is the same order of singularity as the asymptotic near tip stress field of an elastic solid. Using this characteristic property of singular element, the stress intensity factors can be calculated from the displacement components of singular elements with a reasonable number of elements [14].

The numerical performance of quarter-point singular elements has been studied by Saouma and Schwemmer [28]. A 2 x 2 Gaussian numerical integration scheme with eight triangular (6-node) singular elements surrounding the crack tip was recommended to give consistent numerical results. This recommendation was adopted in the present numerical approach. The finite element mesh given in Fig. 7a was used to calculate the stress intensity factors for the specimens used in this investigation. A beam subjected to center loading and a pair of unit shear forces were used to test the accuracy of the developed singular finite element code. The accuracy of the numerical results of K~, K H and CMOD was judged to be satisfactory [18]. Using the same FEM program and rearranging the meshes, the stress intensity factors (K~, K,) of initial crack with different off-set ratios were calculated for the testing configurations used in the present study. When the crack propagated away from its own plane under mixed-mode loading conditions, the stress intensity factors were calculated in a similar manner by properly shifting its notch tip (Fig. 7b). For the present mixed-mode testing configuration, numerical results of K~ and KH values were obtained using different branch lengths along different angles. It was found that the values of KH were close to zero and /<1 was close to its maximum value along the direction of theoretical initiation angle regardless of the length of the branch. It is suggested that for the present testing configuration,


the crack can be reasonably assumed to propagate in the direction predicted at crack initiation in a straight line. This implies that after the crack initiates, crack propagation is mainly dominated by mode I for the present testing configuration.

6. Discussions of experimental results

Crack initiation angle

For the sake of brevity, only the principal test results are reported here. The maximum loads, final failure angles, conventional critical mixed-mode stress intensity factor (/(,.) and other related information for the C (concrete) series are listed in Table 2. Figure 8 shows some of the results of Load-CMD curves of the C series with different off-set ratios. The crack trajectories for the C series tested are plotted in Fig. 9. It can be seen that the crack initiation angles are difficult to measure due to the tortuosity of crack paths. However, as discussed earlier, the crack can be reasonably assumed to initiate and propagate in a straight line along the predicted initiation angle. The initiation angle was then assumed as the final failure angle measured from the notch tip to the final failure point. Figure 10 gives the experimental measurements of the final failure angles and the theoretical predictions (using maximum energy release rate criterion) of initiation angles for small and large specimens, respectively. The theoretical predictions of the crack initiation angles always underestimated the final crack angles for small specimens (Fig. 10a). This may be due to the aggregate arresting mechanism which forces the crack to deviate from the original initiation angle. To accurately predict the actual crack path, effects of aggregates may have to be considered in the analysis.

Table 2. Experimental results of peak load, conventional mixed-mode stress intensity factor and final failure angle - C-series

Off-set Final ratio /~, failure

Series ao/b ()') Pm~ N(lbs) MNm-32(psi ,~ . ) angle (°)

SI 1/3 0 840 (189.6) 0.708 (644.6) 1 $2 1/3 0 950 (213.6) 0.798 (726.2) 10 $3 1/3 1/6 1050 (236.0) 0.809 (736.7) 16 $4 1/3 1/6 1100 (246.4) 0.845 (769.2) 25 $5 1/3 2/6 1190 (267.2) 0.747 (679.6) 33 $6 1/3 2/6 1480 (332.0) 0.928 (844.4) 15 $7 1/3 3/6 1850 (416.0 0.883 (803.6) 20 $8 1/3 3,/6 1830 (410.4) 0.871 (792.8) 16 $9 1/6 1/2 2300 (517.6) Failed at center - S10 1/2 1/'2 1090 (244.0) 0.805 (732.5) 30 Sl l 2/3 1/2 430 (97.6) 0.570 (519.2) 19 S 12 1/3 4/6 2150 (482.4) Failed at center SI 3 t/3 4/6 2270 (509.6) Failed at center - S14 0 0 2200 (493.6) .... S15 - - - L1 1/3 0 3160 (710.4) 0.938 (854) 4 L2 1/3 1/6 3280 (738.0) 0.895 (814.5) 7 L3 1/3 2/6 3870 (870.0) 0.860 (782.4) 16 L4 1/3 3/6 5270 (1184.4) 0.889 (808.9) 12 L5 1/3 4/6 7880 (1771.2) 0.851 (774.4) 26


0 5 1,0 15 20 2,5 :~0 I I / 1 . t I 2ooL CMD (xlO-2mm) specimen: $1 /

1=°° l O O ~ / / / / / / / ' ~ 4 o o

o 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12

CMD (x 10-3 in.)

0 5 10 15 20 25 30 ! I ~ , , ~ 1 . . I I 2 I . I I

t - / / / / °% : "° 7=°° 200[- / - # F ~ ' 7 CMD (xlO- mm) specimen : S3 /

ool/ i i °° __j oo o Or I l l y t o ' - , o ~ f ~ ~ J 4 - - - ~ I .I i i IO

0 1 2 3 4 5 6 7 8 9 10 11 12 CMD (x lO -3in.)

3,~,0 5 10 15 20 25 30 u k . ~ l I I I I ~ I

"~ h ~ CMD (xlO-2mm) specimen :S~ - - - - - - ao A I/. I/.

: ' ° ° F / / 2 - ' : " > ' -

- O r I v v a , , . ' - ~ ~ I I 0 0 1 2 3 4 5 6 7 8 9 10 11 12

CMD (x lO -3 in.)

0 5 10 15 20 25 30 2000 ._.400 t / / ~ CMD (xlO-Zmm) ' a°/b='/3, V:'/2specirhen: $7 '

_~ ~°° f

I 1001 400

- - O / 0 0 1 2 3 4 5 6 7 8 9 10 11 12

CMD (x 10 -3 in.)

Fig. 8. Typical experimental results of load-CMD curves of C-series.

Z v

n

o 0

Z

o o

1200 Z

800 n

400 u 0

1600

Z 1200 v a_

800 ~ 0 0

Interestingly, it was observed that the experimentally measured final failure angles are scattered about the theoretical predictions for large specimens (Fig. lOb). This may be because the uncracked ligaments of large specimens are long enough for the crack to correct its crack path and go back to the original initiation path. Moreover, since the specimens are relatively large compared to the grain size, these specimens exhibited more homogenous behavior. Despite the large scatter of the experimental results, the theoretical predictions of crack initiation angles seem to be reasonable compared to the accuracy of the experimental measurements.

°%: . .7, .0 Y6


Fig. 9a. Crack trajectories of small specimens - C-series.

135

a%: 1,,3 ,-2,: 0

½

i e'

[ ½ ½

Fig. 9b. Crack trajectories for large specimens - C-series.

Critical fracture parameters

The conventional critical mixed-mode stress intensity factors (/£,), which were calculated from the measured peak loads, the original notch lengths and the theoretical crack initiation angle, are shown in Fig. 11 for the two different sizes of specimens of the C series. Though the size effects on the conventional total stress intensity factor are not clearly revealed due to the relatively small range of specimen sizes used, Fig. 11 does imply that the /C<. is dependent on the sizes of the specimens. Figure 12 shows the observed size effect on the conventional mixed-mode stress intensity factor of different notch-depth ratios for the C series. Table 3 lists the average values of critical stress intensity factors (K~I:), critical crack tip opening displacement (CTOD<) and other basic material properties for the corresponding series determined from mode I tests. In the proposed model, the values of K~<. and CTOD< are assumed identical to those of critical mixed-mode stress intensity factor (K<) and critical crack tip displacement (CTD<.) in the mixed-mode failure case. Thus, based on the Kit< and CTOD< values determined from mode I tests, the peak loads of different mixed-mode testing configurations were calculated. Depending upon the values of off-set ratios, material properties


40 L_ ~ 3 5 "o "-" 3 0 (I)

25 D

2 0

lO I

0

+ [] ¢ e x p e r i m e n t 0 - t h e o r y u} +

B

9 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 K[b/Kz

Fig. lOa. Comparison of theoretical predictions and experimental measurements of final failure angles - small specimens.

3 5

30 ̧

25

o

15 "1

21o

O

+ o 0 e x p e r i m e n t t h e o r y +

- ; +

_ 0

-

+ 0 G

I i I I I I I I 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

KIVK z

Fig. lOb. Comparison of theoretical predictions and experimental measurements of final failure angles - large specimens.

Table 3. Critical stress intensity factors and critical crack tip opening displacement for mixed-mode tests

Compressive Young's strength modulus K;c CTODc

Series MPa (psi) 104 MPa (106 psi) MNm 3/2 (psi ,~-ff.) 10 3 mm (10 3 in.)

C 34.3 (4979) 3.28 (4.76) 1.054 (959) 10.0 (0.394) M 36.2 (5245) 2.62 (3.80) 0.791 (720) 7.49 (0.295)


g 900

8501

'~ 800

+a

u 750 2

700 c 0 *~ 65C c

6 0 0 - L.

.+J u3 o 5 5 0 - "o 0

E 50C -o 0 0 ×

E

pec imens

-

I t t i_ s _m

7: 2z/s, ao/b: ,/~

I I I I I I 0.1 0.2 0.3 0.4 0.5 0.6

o f f - s e t r a t i o , ~'

9.9

'E 9.8 z

3.7

0.6

Fig. 11. Size effect o f convent iona l m i xed -mode critical stress intensi ty factor for different off-set rat ios - C-series.

• ~_- 900

800 u

c 700 0

600

5 0 0 C

~ 4 o o r . -

300- ,t_

2 0 0 - "o 0 E 1 0 0 -

o o 0 × 0

E

t i _

&

, oo1:} 4b _1

T = 0 . 5

I I I I I I 0.1 0.2 0.3 0.4 0.5 0.6

n o t c h - d e p t h r a t i o , ao/b

9.8

0.6

g-

o.4'z E

v

u

ix. "

0.2

Fig. 12. Size effect o f convent iona l m i xed -mode critical stress intensi ty factor for different no tch -dep th ratios.


2.0

1.5

0 1.0

0.~

O.

I • t h e o r y f a i l ed at - - - - I - [] e x p e r i m e n t no tch i

I

_ i ,Too,i} / 4b q d / ,

large s p ~

v

sma I I s p e c i m e n s m ~ _ . ~ ~ i

~ I v

I I I I I i I 0 0.2 0.4 0.6

o f f - set r a t i o, ")/

10 -,--fai led at center

---8

~2

I 0 0.8

Z

6 ~ "o o O

4 u

Fig. 13. Comparison of experimental peak load and theoretical predictions using two parameter fracture model for different off-set ratios - C-series.

and specimen size, the final failure may not occur at the location of notch as discussed earlier. For the present testing configuration, location of notch (section at which the strength was reduced) and the mid-span (which is subjected to the highest bending moment) are the two possible paths that offer the least resistance to external load. The resistance to the external load offered by these two crack paths can be calculated using the proposed model by assuming that there is no interaction between these two cracks. The lower resistance was determined as the critical failure load. For beams close to the bifurcation configuration, the resistance to applied external load offered by the two critical sections are about the same. Small differences in material properties or distribution of imperfections may significantly influence the final location of crack at the critical configuration. Therefore, at configurations around the bifurcation point, the final failure could be expected to occur randomly at the two critical sections and the peak load is not so sensitive to the location of the final crack. The experimental results and theoretical predictions of the peak loads for C- series are given in Figs. 13, 14. It can be seen that the peak loads are correctly predicted by the proposed model. In addition, when the off-set ratio approaches unity (Fig. 13) or the notch-depth ratio is close to zero (Fig. 14), the final failure occurs at the center of the span (Fig. 15) which is correctly predicted by the proposed model. It should be noted that the failure always occurs at the location of notch and the predicted failure load will approach infinity when the off-set ratio is close to unity or the notch-depth ratio approaches zero if the conventional energy release rate criterion is used. This shows that the conventional single fracture parameter approach cannot be used to characterize the general mixed-mode fracture behavior of concrete. The experimental results of M-series (mortar) were also found to be in good agreement with the theoretical predictions. Figure 16 shows the comparison of theoretical

Mixed-mode .fracture o/" concrete 139

8 0 0

600 ..Q

"13

0 - - 40C c~

cl.

2 0 0 -

C 0

,oi,eO ot._:_.,o,,oO ot I 4 } cen te r I notch I-

i t ~ t

i • B T = ° 5

,

Z

2 v

O

o .

+ theo ry [] exper i ment

I I I I I 0.2 0.4 0.6

n o t c h - d e p t h ra t io , ao/b

Fig. 14. Comparison of experimental peak load and theoretical predictions using two parameter fracture model for different notch-depth ratios - C-series.

prediction (using the values of K{,, CTOD c, and E listed in Table 3) and experimental measurements of CMD. It can be observed that the theoretical prediction is always underestimated compared to the experimental results. This is because the inelastic deformation was not considered in the analysis. It can be noted that the inelastic deformation is about 26 percent of the total deformation at peak load which appears reasonable when compared with the experimental results shown in Fig. 8. Attempts were made to evaluate the CTD c and K,, directly from the unloading compliance of mixed-mode tests as was done in determination of K~,, and CTOD, from mode I test results (Fig. 2). However, due to the irreversible shear deformation during unloading, the compliance measured from the loading- unloading curve could not fully represent the elastic response. Hence, values of K, and CTD, cannot be calculated using the measured unloading compliance under mixed-mode conditions.

Effects of aggregate interlock

In the above discussion, Ku/K~ ratios of the specimens tested are relatively small. Thus, the assumption of a traction free crack is probably valid [19]. However, for higher ratios of K,/K~, the tractions generated through the frictional force due to the so-called aggregate interlock effect [6, 22] may not be negligible. Under this situation, the contribution of the tractions should be considered in the determination of ultimate load and released energy [16, 25]. The tractions acting along the crack interface are indirectly involved in the calculation of stress intensity factors, global force balance, and global energy balance. Based on the fact that the near tip stress field is dominated by the net stress intensity factors, the crack propagation direction can be reasonably assumed to be governed by the net stress intensity factors. Thus, the crack propagation direction can be determined from the ratios of net


0

o

,fi

o

o

.6

E

e.,

,.o

o

o E

.=.


CMD (measured) (mm) 0.02 0.04 0.06

I u I /4'/10.06

141

0.05

- - - • Z ] 'o d .74 ~. - ~ 1 E v ~ 1 - 0 . 0 4 E u 0 .0015 - • • v

. ,u •

u ~ .2 , , -- 0.03

a 0 .0010 - • -~

U • - 0.02 r~ • U

0.0005

O i l I I I I 0 0.0005 0.0010 0.0015 0.0020

CMD (measured) (in.)

0.01

0 0.0025

Fig. 16. Comparison of experimental values of CMD and theoretical predictions of elastic CMD.

K/K. and the crack initiation theories discussed earlier. Further results of experimental determination of the magnitude of aggregate interlock and numerical implementation of the proposed approach for higher values of K./K~ ratios are in progress.

7. Conclusion

1. Different crack initiation theories have been reviewed in the present paper for the applicability to concrete materials. It is concluded that all the different theories give essen- tially similar results compared to the experimental scatter of the measured initiation angles of concrete. Therefore, the choice of the crack initiation theories may be dependent on ease of computational implementation and theoretical consistence of fracture parameters used.

2. The proposed model has been shown to correctly predict the final failure path and ultimate load carrying capacity of the mixed-mode concrete beams tested in the present study.

3. The two fracture parameters (Ki~c and CTOD,.), which are originally proposed to characterize mode I failure of concrete can be used to characterize the mixed-mode fracture resistance of concrete.

4. For high ratio of K./KI values, it is speculated that the tractions results from aggregate interlock may not be negligible.


Acknowledgments

The research reported here is supported by the US Air Force Office of Scientific Research through Northwestern University (Grant: AFOSR-85-0261, Program Manager: Dr. Spencer T. Wu), Additional support to the first author by Department of Civil Engineering (Chair- man: Dr. R.L. Sierakowski) at The Ohio State University is also appreciated.

References

1. M. Arrea and A.R. Ingraffea, "Mixed-mode Crack Propagation in Mortar and Concrete," Report No. 81-13, Department of Structural Engineering, Cornell University, Ithaca, New York.

2. R. Ballarini, S.P. Shah and L.M. Keer, Proceedings of the Royal Society of London, A404 (1986) 35-54. 3. R.S. Barsoum, International Journal for Numerical Methods" in Engineering 10 (1976) 25-37. 4. Z.P. Bazant and B.H. Oh, Materiaux et Constructions 16, No. 93 (1983) 155 177. 5. B. Cotterell and J.R. Rice, International Journal of Fracture 16 (1980) 155-169. 6. M.P. Divakar, A. Fafitis and S.P. Shah, Journal of Structural Engineering (1987) accepted for publication. 7. F. Erdogan and G.C. Sih, Journal of Basic Engineering, ASME 85 (1963) 519-527. 8. I. Finnie and A. Saith, International Journal of Fracture 9 (1973) 484-486. 9. A.A. Griffith, Philosophical Transactions, Royal Society of London, A 221 (1920) 163-197.

10. A.A. Griffith, Proceedings, 1st International Congress of Applied Mechanics (1924) 55-63. 11. A. Hillerborg, M. Modeer and P.E. Petersson, Cement and Concrete Research 6, No. 6 (1976) 773-782. 12. M.A. Hussain, S.L. Pu and J.M. Underwood, in Fracture Analysis, ASTM STP 560 (1974) 2-28. 13. A.R. Ingraffea and W.H. Gerstle, in Application of Fracture Mechanics to Cementitious Composites, S.P. Shah

(ed.), Kluwer Academic Publishers, Dordrecht, The Netherland (1985). 14. A.R. Ingraffea and C. Manu, International Journal for Numerical Methods' in Engineering 15 (1980) 1427 1445. 15. Y.S. Jenq and S.P. Shah, in Application of Fracture Mechanics to Cementitious Composites, S.P. Shah (ed.),

Kluwer Academic Publishers, Dordrecht (1986) 319-360. 16. Y.S. Jenq and S.P. Shah, Engineering Fracture Mechanics 21, No. 5 (1985) 1055-1069. 17. Y.S. Jenq and S.P. Shah, Journal of Engineering Mechanics, ASCE 111, No. 4 (1985) 1227-1241. 18. Y.S. Jenq, "'Fracture Mechanics Studies of Cementitious Composites," Ph.D. dissertation, Northwestern

University (1987). 19. R. John, S.P. Shah and Y.S. Jenq, Cement and Concrete Research 17, No. 2 (1987) 249-262. 20. R. John and S.P. Shah, in Proceedings, R1LEM-SEM International Conference on Fracture of Concrete and

Rock, June 17-19, 1987, Houston, Texas. 21. A.S. Kobayashi, N.M. Hawkins, D.B. Barker and B.M. Liaw, in Application of Fracture Mechanics to

Cementitious Composites, S.P. Shah (ed.), Kluwer Academic Publishers, Dordrecht, The Netherlands (1985). 22. A. Maji and S.P. Shah in Proceedings of Society of Experimental Mechanics, Spring Conference, New Orleans,

LA, June 8 13, 1986. 23. S. Nemet-Nasser and H. Horii, Journal of Geophysical Research 87, No. 68 (1982) 6805-6821. 24. R.J. Nuismer, International Journal of Fracture 11 (1975)203-222. 25. J.R. Rice, Journal of Applied Mechanics 325 (1968) 379-386. 26. RILEM recommendation: "Determination of Fracture Toughness (K~,, and CTOD,) of Plain Concrete Using

Three-point Bend Tests," revised version (1987). 27. J.G. Rots, P. Nauta, G.M. Kustera and J. Blaauwendraad, HERON 30 (1985) No. 1. 28. V.E. Saouma and D. Schwemmer, International Journal for Numerical Methods in Engineering 20 (1984)

1629-1641. 29. G.C. Sih, International Journal of Fracture Mechanics 10, No. 3 (1974) 305-321. 30. J.L. Swedlow, in Crack and Fracture, ASTM, 506-521. 31. M. Wecharatana and S.P. Shah, Journal of Engineering Mechanics Division, ASCE (1982) 1100-1113. 32. J.G. Williams and P.D. Ewing, International Journal of Fracture 8 (1972) 441-446. 33. D.J.W. Wium, O. Buyukozturk and V.C. Li, Journal of Engineering Mechanics, ASCE (1985).

R~sum6. Dans la construction en b6ton, on rencontre fr6quemmcnt des ruptures de mode mixte. Pour analyser et conccvoir de mani6re s fir une structure en b&on, il est essentiel de comprendre les m6chanismes de ruine du beton sous des types de sollicitation de mode mixte. On proc6de fi l 'extension d 'un module propos6

I'origine pour les ruptures de mode I dans le b6ton, pour caract6riser une propagat ion de fissure scion un mode mixte. On discute les angles d 'amor~age de la fissure, le parcours final de la rupture et l 'instabilit6 d 'une fissure dans des 6prouvettes et dans des structures en b6ton. On a proc6d6 ~ une s~rie d'essai dc sollicitation scion un mode mixte de poutres entaillees en vue de v6rifier le mod61e propose. On trouve que les pr6dictions th6oriques sont en bon accord avcc lcs r6sultats exp6rimentaux.

Documents

Mixed-mode fracture of concrete