Construction and Building Materials 26 (2012) 357–361

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Construction and Building Materials

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Size independent fracture energy of concrete

Vítezslav Vydra ⇑, Karel Trtík, František VodákFaculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7, 16629 Praha 6, Czech Republic

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 October 2009Received in revised form 31 May 2011Accepted 18 June 2011Available online 29 July 2011

Keywords:Local fracture energySize effectEffective crack length

0950-0618/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.conbuildmat.2011.06.034

⇑ Corresponding author.E-mail address: vydra@fsv.cvut.cz (V. Vydra).

It has been confirmed by many authors that the specific fracture energy of concrete determined by lab-oratory experiments depends on the shape and size of the specimen because the local energy in the frac-ture process zone is influenced by the free surface of the specimen. As the crack propagates towards theback face of the specimen the shape of the fracture process zone changes having a direct impact on thelocal fracture energy. In this paper, the authors present an original local fracture energy concept within aframework of effective crack model. Within this model the crack is modeled as an ‘‘elastically equivalentnotch’’ with a depth equal to the effective crack length ae. The local fracture energy can be – within thismodel – determined using three-point-bend tests as a derivative of energy release rate with respect tothe effective crack length ae. As long as the local fracture energy exhibits a plateau, size-independent frac-ture energy can be determined. The main advantage of suggested method is that it does not require (con-trary to other methods) testing of many samples of different shapes, sizes or notch to depth ratios, thusgreatly simplifying the determination of the size independent fracture energy.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The specific fracture energy GF determined by the three-point-bend tests is commonly used as a suitable parameter in the frac-ture analysis of concrete structures. It is well known, however, thatGF varies with sizes and shapes of test specimens. To avoid this var-iability in the specific fracture energy determination the Rilemcommittee issued guidelines [1] for the specimen sizes and testarrangement. The main drawback of these recommendations, how-ever, are quite enormous sizes of specimens. A number of research-ers have tried to identify reasons for variability of GF. Hu andWittman [2] have addressed the possibility that the specific frac-ture energy may vary along the crack path as it propagatesthroughout a relatively thin sample. This local specific fractureenergy being a function of crack length a is denoted as gf(a) inorder to distinguish it from the averaged GF (see Eq. (2)). The modelof Duan et al. [3] which addresses this problem is based on anassumption that energy required for the crack propagation de-pends also on the width w of the fracture-process-zone (FPZ).The width of FPZ is affected by the free boundary of the specimenand therefore the local fracture energy gf is a function of position ofthe crack tip in the specimen. Hu and Wittman [4] suggested bilin-ear distribution of the local fracture energy along the crack tip path(see Fig. 2). Two similar methods have been suggested for gf(a)determination [5,6]. Both methods are based on measurementsof GF on samples with several (at least two) notch depth. These

ll rights reserved.

methods assume that a notch with a depth a0 is equivalent to acrack with the same length. However, there is a difference becausethere is no fracture process zone around the notch contrary to thecrack. Model of a crack equivalent elastically to a notch with depthae has already been published by Nallathambi and Karihaloo [7] asthe well known effective crack model. In this paper, the concept oflocal fracture energy in viewpoint of the effective crack model isfurther developed. The local fracture energy is determined by theauthors of this paper as a derivative of energy release rate with re-spect to the effective crack length ae. The derivative is calculatedusing recorded load–displacement curves of three-point-bendingtests of concrete beams. The size-independent fracture energy (de-noted as G1) can be determined by this method as long as the localfracture energy exhibits a plateau.

2. Bilinear model

The fracture energy GF according to the Rilem recommendation[1] is the average specific energy obtained as a ratio of total work offracture and the fracture area (i.e. cross-section of initially un-cracked ligament). For a specimen of depth W, thickness B and ini-tial notch depth a0 (see Fig. 1), the fracture energy is given by

GF ¼R dmax

0 Pddþ 12 mgdmax

W � a0ð ÞB ; ð1Þ

where P is applied load, d is displacement of the load point (seeFig. 4 for illustration of the load–displacement curve), dmax isdisplacement of the load point at which applied load reaches zero,

Fig. 1. Three-point-bending test arrangement.

358 V. Vydra et al. / Construction and Building Materials 26 (2012) 357–361

m is mass of the specimen and g is acceleration due to gravity. Onthe other hand the fracture energy GF can be also determined usinga local energy concept as [6,3]

GF ¼1

ðW � a0Þ

Z W�a0

0gf ðxÞdx: ð2Þ

Hu and Wittman [4] suggested to use a bilinear model (Fig. 2)for gf(a):

gfðaÞ ¼G1 a < W � a0 � al

G1 1� a�ðW�a0�alÞal

� �a P W � a0 � al

(; ð3Þ

where G1 is the size independent fracture energy and al is the lig-ament transition length. Substituting Eq. (3) into Eq. (2) they got

GFa0

W

� �¼

G1 1� al2WðW�a0Þ

� �W � a0 > al

G1 1� ðW�a0Þ2al

� �W � a0 6 al

8><>: : ð4Þ

The size independent fracture energy G1 and also the transitionligament length al can be determined by the method of Abdalla andKarihaloo as soon as measurements of GF(a0/W) for at least twodifferent a0/W ratios are available [5,8].

3. Specimen size effect on fracture energy in viewpointof effective crack

The Rilem fracture energy concept was originally developed byHilleborg [9] being based on the Hilleborg’s fictious crack modelthat ignores the fracture process zone width. The local fracture en-ergy concept [2] expands this model as it accepts that the fractureprocess zone width w can vary as the crack propagates through thespecimen (as it is demonstrated schematically by authors of thispaper in Fig. 3) and consequently accepts that experimentallydetermined value of GF may change with the specimen size ornotch depth.

Fig. 2. Bilinear model of gf(a) [4].

In the first approximation it is possible to suppose that the en-ergy G required for creating of a unit volume of the process zone isconstant. The local energy gf can be then expressed as a function ofthe fracture process zone volume VFPZ as

gf ¼GB

dVFPZ

dae:

Although the effective crack length ae was originally suggested to beevaluated at peak load Pu only [7], the same calculation procedurecan be easily applied at any point of the load–displacementcurve – see Fig. 4. For this purpose the authors of this paperdeveloped a new straightforward method of effective crack lengthdetermination that is described in Section 5. A derivative of the frac-ture process zone volume dVFPZ

daecan be in the first approximation

(according the Fig. 3) expressed as:

dVFPZ

dae¼

B �wLðaeÞ wL < wFPZ ðaÞB �wFPZ wL > wFPZ and l < W ðbÞB � wFPZ �wUðaeÞð Þ l > W ðcÞ

8><>: ; ð5Þ

where Eqs. (5a–c) are related to sub-figures (a)–(c) in Fig. 3 respec-tively, wL and wU is the fracture process zone width on the lower(front-face) and the upper (back-face) surfaces of the beam, wFPZ

is the width of fully developed fracture process zone. Using linearapproximation of the functions wL(ae) = c1ae + c2 and wU(ae) = -c3ae + c4 in Eq. (5) (c1–c4 are positive constants) one can express gf

as:

gf ¼G � ðc1ae þ c2Þ wL < wFPZ ðaÞG �wFPZ wL > wFPZ and l < W ðbÞG � ðwFPZ � ðc3ae þ c4ÞÞ l > W ðcÞ

8><>: ; ð6Þ

The linear approximation was used also by Hu and Wittman [4] inthe bilinear model (Eq. (3)) corresponding to Eqs. (6b and c). Thepresence of linearly increasing part (a) of the curve expresed byEq. (6) is justified by the expected approximately triangular shapeof the fracture process zone at the beginning of the bending(Fig. 3a), and also experimentally, as seen in the experimental partof the article, namely Fig. 8.

Introducing another symbols, whose meaning is obvious fromFig. 5, the local specific fracture energy gf can be expressed, finally,as a trilinear function of the effective crack length in the form sim-ilar to the bilinear function (Eq. (3)) of Hu and Witman [4]:

gf ðaeÞ ¼

G1�G0atðae � a0Þ þ G0 ae < at þ a0

G1 a1 þ a0 < ae < W � al

G1 � ðae �W þ alÞ G1al

ae > W � al

8><>: : ð7Þ

4. Experimental determination of gf

Energy release rate of the structure can be determined as aderivative of work of external and elastic forces

G ¼ �1B@W tot

@ae¼ 1

B�dUe

daeþ dWP

dae

� �: ð8Þ

WP ¼R d

0 PðxÞdx is work of external forces and Ue ¼ P2ðdÞ2E is poten-

tial energy of elastic forces, E ¼ PðdÞ=d is stiffness of the beam andWtot = WP � Ue. As long as the process can be supposed to beadiabatic, the local fracture energy gf is equal to the energy releaserate. Therefore (see Fig. 6):

gfðaeÞ ¼1B

dR d

0 PðxÞdx� P2ðdÞ2E

� �dae

: ð9Þ

Fig. 3. The fracture process zone shape changes as the fracture propagates throughout the sample and therefore energy required to its propagation changes too(schematically). (a) The fracture process zone is triangular at the beginning. (b) As soon as the fracture process zone is fully developed its width wFPZ is constant. (c) Thefracture process zone reaches the specimen backface, finally.

Fig. 4. Effective crack length ae for arbitrary value of a load point displacement d.Effective crack length determined at peak load is labeled aeu.

Fig. 5. Distribution of the fracture energy gf along the notched beam of depth W andnotch depth a0 (trilinear approximation Eq. (7)).

V. Vydra et al. / Construction and Building Materials 26 (2012) 357–361 359

Eq. (9) can be solved using two different methods:

Method No. 1: Integral in Eq. (9) is evaluated numerically andthen derivation with respect to ae is performed (numerically,too). As soon as an effective crack reaches the specimen back-face (Dae ? W � a0), the method fails as seen in Fig. 8. The rea-son for this is obvious: accuracy of determination of effectivecrack length is much worse in this region. Fortunately this noisy‘‘tail’’ does not influence determination of the size independentfracture energy G1.Method No. 2: Integral in Eq. (9) is performed numerically andfitted by any suitable mathematical function of ae using the

least square method. The fitted curve is derived analyticallywith respect to ae. This method should be preferred as it allowsto avoid cumbersome numerical derivation. As a suitableexpression for fitting the integrated Eq. (7) may be chosen, i.e.W totðaeÞ ¼ B

R ae0 gfðxÞdx:

W totðaeÞ¼BG1�G0

2atðae�a0Þ2þBG0aeþ c1 ae < atþa0

BG1aeþ c2 atþa0 < ae <W�al

BG1ae�Bðae�WþalÞ2 G12alþ c3 ae >W�al

8><>: ;

ð10Þ

where constants c1, c2 and c3 assure that the curves are continuous.The example of fitting by Eq. (10) and the example of fitting by thepolynomial function is presented in Fig. 7.

Fig. 6. Work of external and elastic forces Wtot = WP � Ue as a function ofdisplacement of the load point d. Normalized to ligament cross-section B � (W � a0).

Fig. 7. Work of external and elastic forces Wtot fitted by polynomial and trilinearfunction as a function of Dae = ae � a0.

360 V. Vydra et al. / Construction and Building Materials 26 (2012) 357–361

The results of both above mentioned methods are presented inFig. 8. The local specific energy gf calculated by the first methodshows a plateau between two linear segments, which can be inter-preted as an experimental proof of the assumptions expressed inEq. (7). If there is no observable plateau between the rising andthe falling segments of the experimental curve, the sample thick-ness is not sufficient to determine size-independent fracture en-ergy G1.

g f [

J/m

2]

Fig. 8. Size independent fracture energy gf determined using Eq. (9). Curve labeledgf was determined using method 1 (Section 4), curve labeled ‘‘trilinear best fit’’wasdetermined by method 2 (Section 4).

5. Effective crack length determination

Several different methods of effective crack length determina-tion have been published [10,11]. The implicit equation of Stiborwas proved by FEM studies as the most accurate:

E¼1B

P4d

SW

� �3

1�0:387WSþ12:13

WS

� �2;5" #

þ92

Pd

SW

� �2

F1 aeð Þ( )

;

ð11Þ

where ae = ae/W is a relative crack length, S is a span of the beam, Eis a modulus of elasticity, F1ðaeÞ ¼

R ae

0 xY2ðxÞdx; and Y is a functionof geometry according to Pastor (as cited by Stibor in [11]), whichis suitable for any sample geometry:

YðaÞ ¼ Y1ðaÞ þ 4WSðY4ðaÞ � Y1ðaÞÞ;

where

Y4ðaÞ ¼1:9�a½�0:089þ0:603ð1�aÞ�0:441ð1�aÞ2þ1:223ð1�aÞ3�

ð1þ2aÞð1�aÞ3=2

and

Y1ðaÞ ¼1:989� að1� aÞ½0:448� 0:458ð1� aÞ þ 1:226ð1� aÞ2�

ð1þ 2aÞð1� aÞ3=2 :

The relative crack length ae is normally evaluated numerically as aroot of implicit Eq. (11), but the author succeded to express itapproximately in an explicit way as

aeðF1Þ ¼12þ

arctan b1 þ b2 ln F1 þ b3ðln F1Þ2 þ b4ðln F1Þ3� �

p;

where

b1 ¼ 0:2872þ 0:2091WS;

b2 ¼ 0:5268� 0:0067WS; ð12Þ

b3 ¼ 0:0381þ 0:017WS;

b4 ¼ 0:0284� 0:002WS;

where expression

F1 ¼2B9

WS

� �2 EdP� 1

4BS

W

� �3

1� 0:387WSþ 12:13

WS

� �2;5 ! !

;

has been determined solving Eq. (11).This equation is valid for the width to span ratio W/S within an

interval (0.1,1) with accuracy better than 0.7%. Modulus of elastic-ity E is obtained from the three point bending tests by means of Eq.(11) using an initial slope of l–d curves for P/d ratio and an initialnotch depth to depth-of-specimen ratio (a0/W) for a.

6. Experimental example and discussion

A set of ordinary concrete samples made of Portland cementwas investigated by the suggested method. The samples were for2 h exposed to elevated temperatures up to 1000 �C in age of180 days. Three point bend testing was carried out using theclose-loop testing machine. Three samples were used for everytemperature. All samples were of uniform sizes, i.e. B = 100 mm,W = 100 mm, beam span S = 300 mm, length of samplesL = 400 mm and initial notch depth a0 = 25 mm (see Fig. 1).

Experimentally determined size-independent fracture energyG1 is presented in Fig. 9 and compared with the the fracture

,GF [

J/m

2 ]G

,GF[-

]G

Fig. 9. Size independent fracture energy G1, GF determined according to Rilem (Eq.(1)) and its relation.

V. Vydra et al. / Construction and Building Materials 26 (2012) 357–361 361

energy GF calculated according to Eq. (1) recommended by Rilem.The relative values of G1/GF behave as expected, i.e. they remainin a reasonably narrow range of values being always higher thanone. The best agreement between G1 and GF can be observed onthe samples treated at 300 �C probably due to acceleration ofhydration processes in the samples made of relatively young con-crete and consecutive strengthening of the concrete at this temper-ature [12]. Width of the fracture process zone (wFPZ) in the wellhydrated concrete is lower and therefore Eq. (1) is more accurate.On the other hand a ratio G1/GF grows with temperature of treat-ment above 400 �C. This growing discrepancy can be attributed tothe increasing width of the fracture process zone caused by deteri-oration of the material at high temperatures [12], and thusdecreasing accuracy of the fracture energy determination accord-ing to Rilem Eq. (1).

7. Conclusions

The fracture energy GF according to the Rilem methodology isessentially a standard parameter to compare the fracture proper-ties where experiments are performed on samples of the same size,but it cannot be considered a material parameter, unless it is mea-sured on sufficiently large samples. The fracture energy GF dependson the size of the sample and the sample size required for a properdetermination is quite high especially when the high width of theprocess zone is expected. The new methodology for determining

the fracture energy G1, independent of sample size and size ofthe fracture process zone was designed in this work. This methodshould give similar results as the method of Abdalla and Karihaloo[5,8] which is yet to be experimentally confirmed. Our method has,however, some advantages over the method of Abdalla and Kariha-loo. First, there is no need to carry out experiments on a set of sam-ples of different sizes and with various deep notches, second it isnot necessary to perform load–displacement experiments withprecision down to the end of the curve, which is always experi-mentally challenging.

Acknowledgement

This work was supported by the Ministry of Education, Youthand Sports of the Czech Republic, Contract Number 6840770020.

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