Size Effects in Concrete Fracture

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International Journal of Fracture 102: 2542, 2000.

2000 Kluwer Academic Publishers. Printed in the Netherlands.

Size effects in concrete fracture Part II: Analysis of test results

MOHSEN A. ISSA, MAHMOUD A. ISSA, MOHAMMAD S. ISLAM and ALEXANDERCHUDNOVSKYDepartment of Civil and Materials Engineering, University of Illinois at Chicago, Chicago, IL 60607, U.S.A.

Received 16 July 1998; accepted in revised form 25 June 1999

Abstract. This paper presents an analysis of the extensive experimental program aimed at assessing the influence

of maximum aggregate size and specimen size on the fracture properties of concrete. Concrete specimens used

were prepared with varying aggregate sizes of 4.75, 9.5, 19, 38, and 76 mm. Approximately 250 specimens varying

in dimension and maximum aggregate size were tested to accomplish the objectives of the study. Every specimen

was subjected to the quasi-static cyclic loading at a rate of 0.125 mm/min (0.005 in./min) leading to a controlled

crack growth. The test results were presented in the form of load-crack mouth opening displacement curves,

compliance data, surface measured crack length and crack trajectories as well as calculated crack length, critical

energy release rate, and fracture toughness (G1). There is a well pronounced general trend observed: G1 increases

with crack length (R-curve behavior). For geometrically similar specimens, where the shape and all dimensionless

parameters are the same, the R-curve for the larger specimens is noticeably higher than that for the smaller ones.

For a fixed specimen size, G1 increases with an increase in the aggregate size (fracture surface roughness). For

the same maximum aggregate size specimens, the apparent toughness increases with specimen size. It was clear

that the rate of increase in G1, with respect to an increase of the dimensionless crack length (the crack length

normalized by the specimen width), increases with both specimen size and maximum aggregate size increase.

The crack trajectory deviates from the rectilinear path more in the specimens with larger aggregate sizes. Fracture

surfaces in concrete with larger aggregate size exhibit higher roughness than that for smaller aggregate sizes.

For completely similar specimens, the crack tortuosity is greater for the larger size specimens. The crack path is

random, i.e., there are no two identical specimens that exhibit the same fracture path, however, there are distinct

and well reproducible statistical features of crack trajectories in similar specimens. Bridging and other forms

of crack face interactions that are the most probable causes of high toughness, were more pronounced in the

specimens with larger maximum size aggregates.

Key words: Maximum aggregate size, specimen size, crack length, split tensile test, R-curve, stress intensity

factor, strain energy release rate, CMOD, crack trajectory, fracture toughness, roughness.

1. Introduction

In part I of this paper Issa et al. (2000, this issue) reported the experimental setup and obser-

vations for determining the effects of aggregate and specimen size on the fracture behavior

of concrete, in the presence of compressive forces parallel to the fracture path. The existence

of a trend in dimensionless critical load-CMOD envelopes despite the apparent geometricaland physical similarity of the test conditions was found to be a direct indication of a scale

effect. The presence of hidden parameters not accounted for in the simple dimensional analysis

was also suggested. Application of Fracture Mechanics formalism and further analysis of the

experimental data are reported in this paper.

Generally, structural designs are based on ultimate loading conditions. Slow crack growth

starts long before the concrete reaches its ultimate state. To account for this phenomena in


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26 M. A. Issa et al.

design, parameters such as K1c, G1c or GF provide an insight into how the materia behaves

under normal service loads. The nations infrastructure system is constantly deteriorating due

to cracking, corrosion, etc. Research on these problems is essential in order to determine the

cause behind the damage of the structural materials. New materials such as ceramics and

high performance cementitious materials are being developed to meet the demands of modern

technology. These materials are remarkably strong and prohibitively brittle. As a result, it is

necessary to develop dependable reliability models that provide the criteria for small scale

testing of brittle materials. This research incorporates a special experimental design that is

utilized in an effort to study the reliability of brittle materials as well as a development of a

methodology for reliability assessment.

The consistent scale effect analysis of concrete fracture in the presence of compressive

forces parallel to the fracture plane is presented below. It is based on the three forms of

comparison of test results. The first corresponds to specimens of various sizes cast with the

same maximum aggregate size. The second form of comparison deals with geometrically

identical specimens cast with various maximum aggregate sizes. The third and final form of

comparison entails equally scaled both specimen dimension and maximum aggregate size, i.e.,

having identical t/dmax ratio. Figures 1, 2 and 3 display typical halves of fractured specimens

employed in the three forms of comparison described above.

Figure 1. Specimens prepared by equally

scaling geometry.

Figure 2. Specimens prepared by equally

scaling max. aggregate size.


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Concrete fracture Part II: Analysis of test results 27

Figure 3. Specimens prepared by equally scaling both geometry and maximum aggregate size.

2. CMOD crack length dependence

When the rising load-CMOD curve of a particular loading cycle crosses the unloading branch

of the load-CMOD curve of the preceding cycle, a significant increase in intensity of acousticevents was recorded. Consequently, within a very short period of time, load reduction was

observed, indicating a crack reinitiation. At this stage, the active loading cycle was stopped,

switched to an unloading and the crack length was measured. When the specimen was un-

loaded to nearly zero load, residual CMOD was observed and recorded. This type of residual

CMOD appears to be typical in concrete and may be attributed to the fracture surface rough-

ness, which creates an obstacle to crack closure. The size of the residual crack opening

depends on several factors, such as aggregate size, specimen geometry and type of the test. An

increase in the residual CMOD, r with increasing crack length is apparent in the Figures 7

to 9 of (Issa et al., 2000). The total CMOD, total at the onset of crack extension and residual

CMOD dependence on the crack length is reconstructed from the data reported in Figures 7 to

9 (Issa et al., 2000) and presented in the Figures 4 to 6. For same size specimens, an increase

in aggregate size resulted in an increase in CMOD as shown in Figure 7.

3. Estimation of crack length

At the end of each cycle, when the loading was switched to unloading, the crack was arrested.

At this point the crack length was measured (defined as measured surface crack length). From


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Figure 4. Total and residual CMOD dependence on crack length for typical S1A3 specimen.



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the linear portion of the unloading compliance of a particular load cycle, the crack length was

evaluated (defined as an estimated crack length).

The crack length estimation is performed in a conventional way on the basis of variation

of apparent compliance with crack length via linear elastic solution. Elastic CMOD, e due

to a pair of forces P applied via rollers (Issa et al., 1999) is proportional to the force P

e = C(a,H d) . P , e = total rk (a ,d ).total, (1)

where, k(a,d) is determined from data on Figures 4 to 6.

Here the effective compliance, C depends on elastic modulus, crack length, a and spec-

imen shape and dimension. It is evaluated following the numerical solution reported in Tada

(1973).

The results of the computation in dimensionless form are shown in Figure 8. The process

of estimation is depicted by arrows on the Figure 8 that translates the measured slope of load

displacement curve into corresponding crack length.

The average of the two trajectories obtained from both faces of the specimen was con-

sidered as the final crack trajectory of the plane strain problem. This trajectory was then

used in the computation of various fracture parameters. The data pertaining to the crack-tip location and maximum load at each cycle were also used as the input data for the finite

element analysis.

Surface measured crack lengths versus estimated crack lengths for different specimen sizes

made with 9.5 mm maximum size aggregates are plotted in Figure 9 for illustration. Although

scatter is significant, a general trend is noticed. For all the specimens, the magnitude of cal-

culated crack length for the initial few cycles are slightly higher than the measured length.


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Figure 7. Typical CMOD versus crack length curves for S3 specimens.

Figure 8. Estimation of crack length.


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Figure 9. Surface measured vs estimated crack lengths for different specimen sizes made with 9.5 mm maximum

size aggregates.

During the last few cycles, the measured crack lengths were higher than the calculated ones.

Although the cracks on both sides of some specimens were found to be somewhat different,

in most of the specimens, the magnitudes of the crack lengths were approximately the same.

4. Observation of crack trajectories

There are two obvious mechanisms of crack path formation in concrete. First, the crack de-

flects around the aggregates, and second, the crack penetrates through the aggregates. The

possibility of either or both patterns occurrence depends on the strength of the cement paste,

strength of aggregate and aggregate location with respect to the crack front. Since the cement

paste was designed deliberately to be weaker than the aggregates, the crack pattern has a

meandering path and tends to travel around the aggregates instead of penetrating through

them (Issa and Hammad, 1993; Hammad and Issa, 1994a, b). This type of process tends to

raise the energy consumption and affects the apparent toughness.

The observed crack path is random, i.e., no two macroscopically identical specimens ex-

hibit the same fracture path. Concrete fracture surfaces with the larger aggregate sizes appearto have a higher roughness than those with smaller aggregate sizes. The crack path is less

tortuous for geometrically identical specimens with smaller size aggregates. Similarly, the

crack path deviates from the centerline of the specimen to a lesser degree for the smaller size

aggregates than that for the larger ones as illustrated in Figure 10. For geometrically identical

specimens made with larger maximum aggregate sizes, the degree of tortuosity is significantly

higher. As a result, bridging is clearly noticed in specimens with large aggregate sizes (see


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Figure 10. Crack trajectories for geometrically identical S1 specimens.

Figure 11). For different size specimens made with the same maximum aggregate size, crack

tortuosity followed the same trend, i.e., crack tortuosity increases with specimen size. In the

third form of comparison of similar specimens with respect to maximum aggregate as well as

specimen size, crack tortuosity is greater for the larger size specimens within the group.

5. Evaluation of the critical values of the energy release rate,G1c

Two-dimensional finite element modeling of the above described test configuration was per-

formed using the FRANC2D (Cornell Fracture Group, 1995). A standard, direct stiffness

FEM approach, with quadratic order isoparametric elements was used. Eight-noded quadrilat-

eral and six-noded triangular isoparametric elements were used. An appropriate singularity in

these elements is achieved by placing the mid-side node near the crack-tip at the quarter point.

Linear elastic material is assumed in the computation. An incremental solution is performed

with the geometry of the specimen and crack trajectory, modified at each step to reflect the

evolving configuration of the crack. Fracture mechanics parameters, especially the stress in-tensity factors (KI and KII), and the energy release rate (G1) are computed at each step of the

analysis.

The specimen configuration employed in the computation replicates that used in the exper-

imental program. The experimental maximum load and crack-tip location at the end of each

loading cycle resulting from the quasi-static splitting tensile test (wedge splitting test) were

used as the input data to run the step by step analysis for each crack.


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Figure 11. Bridging in large aggregate size specimen.

When a crack deviates from the specimen centerline, a mixed mode loading condition

exists, where both KI and KII are nonzero. Material nonlinearity at the crack-tip may limit theapplicability of linear elastic solution and the concept of SIFs. However, the energy release

rate, G1 of LFM may still be applicable, since it is much less sensitive to nonlinear effects

than SIF (Slepjan, 1990). Thus, the critical value of ERR G1c associated with the crack-tip

extension is a natural candidate for the fracture parameter in the characterization of brittle

fracture.

Figure 12 shows the plot of the critical energy release rate G1c as a function of the crack

tip coordinate for a typical specimen. Comparison of the results for G1c as a function of crack

length for various specimen and aggregate size are presented in the next section. The minimal

values ofK1c and G1c observed at crack initiation are summarized in Table 1.

As can be seen in Table 1 for the same specimen size, the value of critical stress intensity

factor K1c, increases as the maximum aggregate size increases. The critical stress intensity

factor, K1c also increases as the specimen size increases. However, the concept of critical SIF

values K1c as a measure of toughness have obvious limitations due to

(a) mixed mode fracture resulting from tortuous crack path and

(b) possible material nonlinearity in the vicinity of the crack tip.

Therefore, the authors concentrate on the analysis of G1c behavior since,


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Figure 12. Critical values of the energy release rate G1c vs crack projection length for a typical specimen and

FEM mesh.

(a) it naturally accounts for mixed mode fracture and

(b) is insensitive to nonlinearity (Slepjan, 1990).

G1c values of crack initiation display the same trend as K1c.

6. Critical energy release rate (G1c) analysis

Critical energy release rate G1c values at the crack initiation from the notch tip for different

size specimens with different aggregate sizes are presented in Table 1. From Table 1, it is

observed that for the same specimen size, G1c increases as the maximum aggregate size

increases.

The energy release rate for geometrically identical specimens increases as the maximum

aggregate size increases as shown in Figure 13. Fracture behavior has an apparent correlation

with the different texture, angularity and size of the coarse aggregate in the concrete. The

increase in fracture toughness with increasing maximum aggregate size is associated with

the increasing resistance encountered by the propagating crack. Similarly, for different size

specimens made with the same maximum aggregate size, G1c increases as the specimen sizeincreases as shown in Figure 14. The propagating crack encounters resistance due to the

presence of more coarse aggregate particles.

For geometrically similar specimens (i.e., the shape and all dimensionless parameters are

the same), the R-curve for larger specimens is noticeably higher than that for the smaller ones.

G1c values for S1A38 specimens are higher than those for S3A316 specimens in group I of the

similarity chart. For group II, G1c values for S1A34 specimens are the highest for the observed


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Figure 13. Comparison ofG1c for geometrically identical S3 specimens.

Figure 14. Comparison ofG1c for 3/2 maximum aggregate size specimens.


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Table 1. Average values ofK1c and G1c at crack initiation.

Specimen Maximum aggregate Average K1c Average G1c

size size

mm in. MNm3/2 psi in.1/2 J/m2 lb-in./in.2

9.5 38

1.628 1481.4 76.007 0.434

S1 19.0 34

1.456 1324.6 63.573 0.363

(1680 1680 200 mm) 38.0 1 12

1.781 1620.3 99.475 0.568

76.0 3 1.734 1577.9 95.972 0.548

9.5 38

1.481 1347.6 63.047 0.360

S2 19.0 34

1.568 1426.7 73.730 0.421

(1050 1050 125 mm) 38.0 1 12

1.595 1451.4 80.035 0.457

76.0 3

4.75 316

1.263 1148.9 47.285 0.270

9.5 38

1.519 1382.6 66.200 0.378S3

(840 840 100 mm) 19.0 34 1.546 1407.1 71.629 0.40938.0 1 12 1.583 1440.1 78.634 0.449

76.0 3 1.619 1473.2 83.888 0.479

4.75 316

1.124 1023.1 38.004 0.217

S4 9.5 38

1.443 1312.7 59.545 0.340

(420 420 50 mm) 19.0 34 1.465 1333.1 64.799 0.370

38.0 1 12

1.314 1195.6 54.466 0.311

4.75 316

1.103 1004.1 36.252 0.207

9.5 38

1.239 1127.3 43.958 0.251S5

(210 210 25 mm) 19.0 34

1.356 1233.8 55.342 0.316

S6 4.75 316 1.247 1134.8 46.585 0.266

(105 105 12.5 mm) 9.5 38

1.295 1178 48.862 0.279

crack length to width ratios of 0.4 and greater. Values corresponding to S4A316 are the lowest

for any ratio. For group III, the order of specimens in terms of G1c values from the highest

to the lowest is S1A32, S3A34, S4A38, and S5A316 (Figure 15). For group IV, the order of

specimens in terms of G1c values from the highest to the lowest is S1A3, S3A32, S4A34,

S5A38, and S6A316 (Figure 16). For group V, the order of specimens in terms of G1c values

from the highest to the lowest is S3A3, S4A32, S5A34, S6A38. The standard deviations at a

crack length to width ratio of 0.5 for each specimen type are also shown in Figures 15 and

16. It is clearly noticed in Figures 15 to 16, that within the groups the rate of increase in G1cas an increase in a/w, increases as both the specimen size and the maximum aggregate size

increase for the same t/dmax ratio. The fracture toughness (G1c) exhibited scatter.


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Figure 15. Comparison ofG1c for group III.

7. Size effect inG1c

The set of data presented in Figures 1316 indicate a strong size effect in values of apparent

toughness G1c. Indeed, G1c increases with the crack length a/w, with the specimen size w,

and with the maximum aggregate size dmax.

To quantify the size dependence of G1c we consider G1c values of various specimen sizes

for fixed relative crack length a/w from Figures 1316. The computed values of G1c forspecimen S3 with varying maximum aggregate size dmax = 76, 38, 19, 9.5 and 4.75 mm at

fixed values of a/w = 0.2; 0.3; 0.4; 0.5 and 0.6 are shown in log-log scales in Figure 17.

The specimens made ofdmax = 4.75 mm display a significantly lower value ofG1c. This may

be attributed to the fact that the maximum coarse aggregate size used in these specimens is

similar to the maximum grain size of sand, i.e., the specimens are made of a homogeneous

mortar. The linear relationship in log-log scales suggests the size effect in form of power

relation for maximum aggregate size larger or equal 9.5 mm

G1c

aW

, dmax

= G1c

aW

, dmax

|dmax9.5 mm

. d(a/W)max . (2)

The computed values ofG1c for specimens S4, S3, and S1 with fixed maximum aggregate

size dmax = 38 mm at fixed values of a/w=0.2; 0.3; 0.4, 0.5 and 0.6 are shown in log-log

scales in Figure 18. The linear relationship in log-log scales suggests the size effect in form

of power relation for specimens larger or equal to S4 (W = 420mm)

G1c

aW

, W= G1c

aW

|W420 mm

. W (3)


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Figure 16. Comparison ofG1c for group IV.

with a = 0.64.

It should be noticed that the smaller specimens S5 and S6 do not follow the trend expressed

by (3). The fractional power in Equations 2 and 3 and the fact that the size effect becomes

pronounced for relatively large size specimens and maximum aggregate size call for a rational

explanation. Comparing size effects expressed in (2) and (3) one notices the power in (2),

i.e., the effect of aggregate size, varies with crack length whereas is practically constant.

8. Summary of the observations

Toughness of six groups of similar concrete specimens with various dimensions and aggre-

gate sizes were evaluated and compared. Similarity of the specimens was strictly maintained

by scaling the specimen dimensions from one group to another by a factor of two start-

ing from a specimen size of (width total depth thickness) 105 105 12.5mm to

1680 1680 200 mm. These specimens were prepared with varying aggregate sizes of

4.75, 9.5, 19, 38, and 76 mm, and tested in the quasistatic loading and unloading at a rate of

0.125 mm/min (0.005 in./min). Approximately 250 specimens varying in size and maximum

aggregate dimension were tested to accomplish the objectives of the study.

Apparent toughness is found to be in direct relation with the texture, angularity and size of

the coarse aggregate in the concrete, for maximum aggregate size larger or equal to 9.5 mm;

and with the specimen size beginning with 420 mm.

The experimental observations combined with the finite element analysis revealed the

following trend in apparent fracture toughness G1c and crack trajectories:


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Figure 17. Aggregate size dependency ofG1c

There is a consistent trend in all groups of specimens tested: an apparent toughness (G1c)

increases with crack length (R-curve behavior, see Figures 1316).

For same size specimen, the apparent toughness increases with an increase in the maxi-

mum aggregate size (see Figure 13).

For the same maximum aggregate size specimens, the apparent toughness increases with

specimen size (see Figure 14).

For geometrically similar specimens, i.e., the shape and all dimensionless parameters

including normalized maximum aggregate size are the same, the apparent toughness of

the larger specimens is noticeably higher than that of the smaller ones (see Figures 15

16).

In macroscopically identical specimens with smaller size aggregates, the crack path is

less tortuous. The crack trajectory deviates more from the rectilinear path in the speci-

mens with larger aggregate size.

For completely similar specimens, the crack tortuosity is greater for the larger size spec-

imens.

9. Discussion

The burst of acoustic emission that correlates very well with the onset of nonlinearity in the

loading part of Load-CMOD curves suggests that the nonlinearity is associated with the frac-

ture process in the vicinity of crack front. Thus, the onset of nonlinearity can be considered


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Figure 18. Specimen size dependency of G1c.

to be the critical point of fracture initiation on the first loading cycle and reinitiation in the

consecutive ones. Moreover, practically linear load-CMOD relationship prior to this point

justifies an application of linear elastic analysis.

Our monitoring of acoustic emission combined with the optical observations support widely

discussed in literature proposition: the observed increase of apparent fracture toughness (G1c)

with crack length (R-curve behavior) in concrete is most likely associated with an interaction

between crack faces (bridging, interlocking, friction). Thus, the observed cracks in concrete

significantly depart from an ideal traction free crack model of conventional fracture mechan-

ics, that has been employed in computation ofG1c in this work. Therefore, the observed trends

in G1c reflect the fracture surface traction characteristics rather than true material toughness.

The surface traction, unlike invariant material parameters, are naturally dependent on particu-

lar test conditions and specimen size. Indeed, since surface traction acting over cross section

t w, at least partially depend on compressive forces, e.g., weight, that varies as t w2, one

may expect an increase in surface traction with the specimen size. However, different scaleeffect may be expected in simple tension (no compression) or bending etc. It may partially

explain the controversial reports on scale effect in concrete and cementitious materials (Issa

et al., 1999).

The surface tractions naturally account for all the trends in tortuosity of fracture paths and

G1c variations with the specimen and maximum aggregate sizes reported above, including the

fractional power in the Equations 2 and 3. The challenge is to construct a predictive model


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of such traction, i.e., an apparent fracture toughness, that can be employed in the engineering

design of large scale concrete structures. The equations 2 and 3 is a step in that direction.

For the specimens made with larger maximum size aggregates, bridging was clearly no-

ticed. This is evident since the aggregates are generally an order of magnitude stronger than

the paste. When a growing crack reaches an aggregate, it either goes around or penetrates

through it. Since the fracture energy of the interface is smaller than the fracture energy for the

aggregate, a crack more often travels around the aggregate. Furthermore, bigger aggregates

have a larger surface area and the overall bond strength between the aggregates and the

mix is higher. Bridging and other forms of crack faces interactions, i.e., the most probable

causes of higher toughness, are more pronounced in the specimens with larger maximum size

aggregates.

The apparent toughness G1c dependence on the maximum aggregate size has been found in

a power form similar to specimen size dependence, but with a different power (see Figures 17

18). The similarity in specimen size and maximum aggregate size dependence of G1c suggests

a likelihood of the same common origin for both, i.e., the fracture surface interaction associ-

ated with its tortuosity. The observed difference in the power ( < ) may be an indication

that the specimen size (weight) or in a more general case, the presence of compressive forces,

has an effect on the apparent toughness, not accounted for by the maximum aggregate sizedependence only.

The randomness of fracture path, i.e., there are no two identical specimens that exhibit

the same fracture path, however there are distinct and well reproducible statistical features

of crack trajectories in similar specimens, significant scatter and fractional size effect in G1ccall for statistical modeling concrete fracture. Statistical fracture mechanics developed in the

last two decades provides an adequate framework for modeling of the observed phenomena

(Chudnovsky and Kunin, 1987; Chudnovsky and Kunin, 1992 and Chudnovsky et al., 1997).

10. Conclusions

Connections between the resistance to crack growth in concrete in the presence of compressive

forces, i.e., an apparent fracture roughness G1c and the interactions of fracture surfaces is

documented by means of direct optical observations and acoustic imaging of crack growth.

Surface traction along the crack faces in concretes are called to account for:

R-curve behavior, i.e., G1c dependence on crack length,

G1c dependence on maximum aggregate size,

G1c dependence on specimen size (size effect),

fractional power form of size effect.

The surface traction, reflecting the fracture surface interaction, apparently depends on

crack path tortuosity, the fracture mechanisms, e.g., trans-aggregate vs inter-aggregate fractureand the stress state.

The randomness of crack trajectories, a significant scatter of the apparent toughness and

fractional power form ofG1c dependence on maximum aggregate and specimen sizes point to

the statistical nature of the observed phenomena. A probabilistic model, based on statistical

fracture mechanics, constitute the third part of the present work. It is presently being prepared

for publication.


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Acknowledgements

This study was funded by a contract awarded to the University of Illinois at Chicago by the

National Science Foundation (Grant No. CMS 9522306). Their financial support is gratefully

acknowledged.

Thanks are also due to Ahmad Hammad, Alfred Yousif and graduate students, Wassim

Kmeid, and Mark Bendok.

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Size Effects in Concrete Fracture

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