Materials and Structures/Matdriaux et Constructions, Vol. 33, June2000, pp 309-316

The effect of specimen size on the fracture energy and softening function of concrete X. H. Guo and R. L Gilbert School of Civil and Environmental Engineering, The University of New South Wales, Sydney, Australia, 2052

Paper received:June 16, I999; Paperaccepted:January25, 2000

AaSTRACT

RrSt0ME Cet article examine l'influence de la dimension des &hantillons sur l'&ergie de fissuration du b~ton, GF, mesur{e h l'aide du test deflexion en trois points sur des poutres pr~ssur&s selon les prescriptions du R I L E M TC-50 [1]. Le concept d'dnergie partielle de fissuration est introduit et utilis~ pour expliquer les effets de dimensions observ&. L ecartement, wr, de I echantdlon enfin d essa, au hau de la fi'ssure d~pend de la dimension de l' e'chantillon, qui a son tour affecte l'&ergie de rupture du b~ton. Th&riquement, quand l'~chantillon est sufiTsamment grand pour permettre le d&eloppement complet de la zone de rupture, wf; atteindra sa valeur crin'que, We, et l'influence de la dimensl-'on de l'e~chantillon sur G z disparaftra. Les r&ultats expMmentaux inclus dans ce papier montrent qu'en rdalit~ la dimension de I'&hantillon qffecte ta mesure de GF, mdme si sa dimension est telle que la zone defissuration a pu compl~tement se de;velopper. Ceci s'explique par une d~formation plastique locale dans la zone du point de chargement, dSformation qui est particuli&ement importante pour de grands &hantillons. En outre, une proce'dure est propos& pour d~terminer pour des b~tons la fonction unique d'adoucissement bas& sur l'~nergie de fissuration mesure'e dans les essais R I L E M . Dans cette procMure, tes ~chantillons sont suffisamment petits pour que l'&ergie mesur& soit re'ellement due a une rupture et non pas une d~formation plastique.

This paper examines the effect of specimen size on the fracture energy of concrete, G F, as measured using the three-point bending test on pre-notched b.eams prescribed by RILEM TC-50 [1]. The concept of partial fracture energy is introduced and used to explain the observed size effect. The opening displacement at the top of the notch in the test specimen at the end of the test, wf, is affected by the size of the specimen, which in turn affects the measured value of the concrete fracture energy. In theory, when the specimen is large enough to allow the fracture process zone to develop fully, reach its critical value, w~, and the effect of specimen size on Gv will be eliminated. The experimental results included here show that in reality the size of the specimen does affect the measurement of G> even when the size is such that the fracture process zone develops fully. This may be due to local plastic deformation in the area around the loading point, which is particularly significant in larger specimens. It may also be due to differences in the influence of the boundary conditions of the test for different specimen sizes. In addition, a procedure is outlined for the determination of the softening function for concrete based on the fracture energy measured in RILEM tests, in which the specimens are small enough to ensure that the energy measured is actually due to fracture and not plastic deformation.

Wfwill

1. INTRODUCTION The fracture energy Gp is an important parameter for describing and modelling the fracture behaviour of cohesive materials like concrete, mortar or other cementitious materials. It is defined as the amount of energy necessary to create a unit area of a crack (or fracture surface). There are difficulties in measuring G~ in a direct, uniaxial tensile test. It is often difficult to perform a stable direct tensile test. In addition, secondary flexure 1359-5997/00 9 RILEM

309

may cause additional crack growth and the boundary conditions in the test may significantly influence the measured fracture energy. ILILEM [1] has proposed a simple procedure based on a stable three-point bending test on a notched beam. The RILEM test is frequently used for the experimental determination of the fracture energy of concrete. A typical test specimen and test setup is shown in Fig. 1. Measured values of G E obtained using the RILEM procedure depend on the specimen size and, in general,

Materials and Structures/Mat~riaux et Constructions, Vol.33, June2000

when the tensile stress drops to zero (see Fig. 2a). According to Wittmann et al., w c depends on the dimensions of the fracture process zone (FPZ), which in turn depends directly on the size and shape of the specimen. In other words, the tensile softening curve of concrete is not a unique material property, but depends on specimen size, in addition to the concrete composition and environmental factors. Wittmann et al. also concluded that, when the size of a specimen is large enough for the FPZ to fully develop, a constant or asymptotic value of G v is measured. In this paper, the FPZ is assumed to be fully developed when the opening displacement, ug; at the tip of the initial notch in the test specimen at theend of the test reaches or exceeds w,.. The three-point bending test is assumed to end when instability occurs due to the self-weight of the specimen. It is further assumed that wcis a material parameter. This paper offers another explanation of the measured dependence of Gf on the specimen size by introducing the concept of partial fraaure enegy, Gt: This is similar in some respects to Wittmann's explanation, but differs in that it maintains the assertion that for a given tensile strength,f, the tensile softening function of concrete is unique, consistent with the classic fictitious crack model. This actually requires size independent values of G > f and us, and leads to simple procedure for determining the softening curve. Of course, the presumption of a unique softening curve for concrete in tension is contentious. It is well known that the tensile strength of concrete is dependent on the size and shape of the specimen, as well as the boundary conditions of the test, so that even the ascending part of the curve is not unique.

p

Fig. 1 - RILEM three-point bending test set-up.

(y

f,

OlW )

W

Wf

(a)

Wc

11 P

2. RILEM FRACTURE ENERGY GFR AND ITS PHYSICAL MEANING The fracture energy G F has already been defined as the energy required to create a unit area of fracture surface. Actually, the area under the softening curve (Fig. 2a) represents the fracture energy and can be expressed as:

W2

"l

/

W1 GF =

15)

cy d w

(1)

The critical crack opening displacement w c is defined in Fig. 2a and is dependent on a number of factors, including the maximum size of the aggregate dmax and has been approximated by [16]:

Fig. 2 - Softening curve for concrete and load-deflection plot from R I L E M test.

increase with increasing specimen size [2-6]. Several interpretations oftbis size effect have been proposed, but most are phenomenological and do not attempt to explain the mechanics of the problem [7-13]. An interpretation, which is based on mechanics, was proposed by Wittmann et al. [14-15]. They suggested that the size dependence of the fracture energy may be caused in part by a change of the critical crack opening displacement, wc, with specimen size. The critical crack width wc corresponds to the point on the softening curve in tension

w c = 0 . 0 6 dmax

(2)

However, for normal concretes where dmax may be 10 to 20 nun, any tension being carried across the crack at widths approaching that given by equation (2) is likely to be due to the mechanical interlocking of aggregates rather than tension softening. The partial fracture energy, Gf, is here defined as:

310

@ = Ior O dw

(3)

Guo, Gilbert

and represents the area under the softening curve for concrete in tension from w = 0 to w = w: (as illustrated in Fig. 2a). The partial fracture energy is fiseful in explaining the effect of specimen size on the fracture energy measured in the RILEM test. For the sake of convenience here, the fracture energy measured using the RILEM procedure is called the P,.ILEM energy and is denoted by GFR. Fig. 2b shows a typical load-deflection plot obtained from a RILEM test (in solid lines). The dashed lines represent a hypothetical complete loaddeflection curve when the weight of the beam and the testing equipment (not included in the measured load P) are taken into account. Referring to Fig. 2b , the RILEM energy is calculated using GFR -

W

_ Wo +

+ W2 + W3

(a)

I]"

w: "'J

(4)

Alig Alig where dti_ is the fracture area assumed to be equal to b(D - ao); ~ and D are, respectively, the width and depth of the beam; ao is the depth of the initial notched in the test specimen; W o is the work done by imposed load P and is given by:

Wo

(5)

P

W 1 is the work done by the weight of the beam itself (mlg) and the weight of any testing equipment not fixed

Fig. 3 - (a) Fully-developed fracture process zone when wf>_w: and (b) Fully-developed quasifracture process zone w h e n wf
to the testing machine (m2g) from the start of the test to the end of the test (when the central deflection is ~f). That is: (6) Both W 2 and W:3 are also components of work done by m~ and m2g. The work W 3 can be neglected since it is small. The work W 2 can not be obtained directly from the knowledge of P-8 curve (unlike W 0 and WI) because the curve in this region is not measured (ie. the dashed curve in Fig. 2b when 8 > 8f). One can reasonably estimate W 2 by assuming the shap~e of the curve. Petersson [16] proposed that it was reasonable to assume that W 2 = W 1, and in that case, GpR can be rewritten as GFR - W _ Wo + WI + W2 _ Wo + 2W1 Alig

f~Jo: P dS+ 2QS:

(7)

It is essential to understand and interpret the physical meaning of each term in equation (7) and its part in the fracture mechanism. A very important point on the P-8 curve for this purpose is the point at which the test stops. This point corresponds to the last stage of the threepoint beam test (when P has dropped to zero). Petersson [16] modelled the FPZ at the end of the test using rigidbody kinematics, as shown in Fig. 3. In this approach, it is assumed that the beam is divided into two rigid rectangular pieces which are connected only through a fracture process zone, with the compression zone assumed to be reduced to a point at the top of the beam. Using this model, at the end of the test when the cen-

tral displacement at midspan is 50 if the opening displacement of the crack at the top of theeinitial notch wfis ~eater than or equal to w~, the sum of W o and W I represents the work dissipated in the fracture process zone 011'2'2 0 shown in Fig. 3a. W 2 is the work done in the potential sub-fracture process zones 013 and 024 shown in Fig. 3a. The total work in the fully-developed fracture process zone 1'342' (in Fig. 3a)is therefore W = W o + W 1 + W 2 and the RILEM energy GFR obtained from equation (7) is the fracture energy Gi~. However, when wfis less than w~, Wis the total work done in the quasifra&ure process zone 1342 in Fig. 3b. The sum of W o and W 1 is now the work done in the fracture process zone 102 (Fig. 3b). W 2 is the work done in the potential sub-fracture process zones 103 and 204 (Fig. 3b) and the RILEM energy Gi~R obtained from equation (7) is the partial fracture energy Gf(wt-), as illus" see the- - Appen ~ d " ix. trated in Fig. 2a. For further details,

3. ANALYSIS OF THE EFFECTOF SPECIMEN SIZE ON GrR Experimental results now available show that there is always a size effect in the measured values of GpR, with GFR tending to increase as the specimen size increases. If the sizes of the test specimen used for the measurement of G~R are not large enough to allow Wf tO reach w~, the observed size effect is explained in terms of the partial

311

Materials and Structures/Materiaux et Constructions, Vol. 33, June 2000

ffacturc energy. As thc specimcn size is increased, thc value of u,: increascs and so too does the partial fracture energy Gl-.(wl) (= G/..R). If the specimen size and test procedure Were such that w:> w c (which would be possible if a weight-compensatedtesting set-up was employed, as suggested by Elices et al. [10], because a long tail in the P-8 curve would be obtained), G~R would then represent the fracture energy G>. and the effect of specimen size on GFR would disappear. However, test results obtained using the P,,ILEM test procedure indicate that the energy G~R continues to increase with specimen size, even when wf > w c.

0

f,

I WI

(a)

4. DETERMINATION OF THE SOFTENING FUNCTION

I

~

W2

W Wi

We

I

G:(w)

The effect of specimen size on GFR may be used to obtain information on the softening function for concrete in tension. As pointed out above, GvR actually represents the partial fracture energy Gf and from equation (3):

o(w)=dGf

r....... / /I

i J

/ (8)

dw From a series of RILEM tests in which the fracture energy GFR (= GD is measured for specimens of different size (with differefit values of _wD, a plot of G/versus w can be constructed, as shown by the solid line in Fig. 4b. The tail of the softening curve can then be obtained from equation (8) (and is illustrated in Fig. 4a). From the tests, the critical crack width wc corresponding to ~ = 0 can also be estimated and the corresponding value of G F can be determined. In addition, the steep initial portion of the softening curve can be identified by using the known point (o = ft, w = 0) and the estimated value of G F to achieve an appropriate fit (Fig. 4a). The argument proposed here in support of a unique softening curve for concrete in tension is opposed to the view of Wittmann and his coworkers. They argued that a different softening function is obtained for each specimen size because of the different critical opening displacements, Wci, which result from variations in the dimensions of the FPZ (as illustrated in Fig. 4c).

W II-

(b)

14/

(c)

Wcl We2

I

!

Wci

Wen

Fig. 4 - (a) Softening function; (b) @versus w diagram; and (c) Effect of specimen size on softening function due to Wittmann [14].

5. ANALYSIS OF EXPERIMENTAL RESULTS In order to study the effect of specimen size on the fracture energy of concrete, comprehensive three-point beam tests were carried out by Xu and Zhao [6]. Fourteen beam specimens with different lengths, depths, widths and volumes were tested. Table 1 provides details of the overall dimensions of each specimen, together with the experimental results. All of the beams were constructed using the same concrete mix. The maximum size of the coarse aggregates was 20 mm. The cube compressive strength, the tensile strength, the modulus of elasticity and the Poisson ratio were, respectively, 39.1 MPa, 2.79 MPa, 32.4 GPa and 0.197.

I

Pmax

In Table 1, is the maximum load measured during the test; 8: is the measured deflection at the loading point at the efid of the test; and V is the measured opening displacement at the mouth of the notch (i.e. at the soffit of the specimen) at the end of the test. The terms wj and u/f are the final calculated opening crack displacements at the bottom of the crack (i.e. at the top of the notch) obtained using the assumption of rigid body kinematics, as used by Petersson [16]. wf is calculated from the measured beam deflection 8c (andis given by wf = 4(D - ao)6j/1, see Fig. 7). w/f is &lculated from th~ measured displacement at the mouth of the notch V (and is given by (D - ao) V~ D).

312

~r

Guo, Gilbert

Table 1 - Experimental test result

Series Specimen Specimen number Volume Sedes

V1

3

V2

2

V3

3

V4

3

Length

L1

3

Series

L2

3

Width

T1

3

Sedes

T2 13

3 3

T4

3

H1

3

Depth Series

H2

2

H3

3

H4

3

Pmax

Specimen size Span(S)•215

(mm)

2400x600x600 1600x400x400 1200x300x300 800x200x200 1500x150x150 1200x150x150 1200x300x300 1200x200x300 1200x150x300 1200x100x300 1200x150x300 1200x150x250 1200x150x200 1200x150x150

/. ./

0.3.~__....s

0.2-

/ ~" '~ i '"

/

/

/ depth o width • length +

v~

M

G eR -, "0.148wf 2 +0.2431wy +0.06381

I

ws(mm)

,

!

0i 0

0.5

1

1.5

ao/D

(N)

V

6f

w/f

(mm)

(mm)

(mm)

wf

G~

( m m ) (N/ram)

45122

0.4

1.031

2.520

0.619

1.512

.4426

23625

0.4

1.720

1.832

1.032

1.099

.2846

15125 7051

0.4 0.4

1.200 0.956

1.460 1.195

0.720 0.574

0.876 0.717

.2012 .1526

1094

0.4

0.475

1.208

0.285

0.290

.1204

2074

0.4

0.645

1.275

0.378

0.383

.1361

15125

0.4

1.200

1.456

0.720

0.874

.2030

10960

0.4

1.182

1.632

0.709

0.979

.2035

7883

0.4

1.209

1.216

0.725

0.730

.1652

5170

0.4

1.239

1.269

0.743

0.761

.1512

7883

0.4

1.209

1.216

0.725

0.730

.1652

5060

0.4

0.987

1.312

0.592

0.656

.1552

3397

0.4

1.060

1.067

0.636

0.427

.1428

2074

0.4

0.645

1.275

0.387

0.383

.1361

be neglected for the specimens with large cross-sections (invalidating the assumption that the compressive zone reduces to a point at the top fibre). A comparison between w:with u/r in Table 1 shows that wris significantly greater than iJf for all the volume serie~specimens and for specimens Tland T 2 of the width series, with wf being about 2.44 times greater than u/f for the specimen with the largest cross-sectional dimensions, V 1. Therefore, there appears to be an upper limit to the specimen size for reliable measurements of fracture energy using the RILEM three-point beam test. In order to determine the softening function for the concrete in these tests a bilinear unloading model has been adopted. A second order polynomial was used to fit the experimental results of GFR and wf, for both the length series and the depth series tests, a~shown in Fig. 5. This relationship is:

0.5 q ~" 0.4

[6]

2

Fig. 5 - Test results, RILEM fracture energy GFR versus final crack width wf.

Fig. 5 shows the relationship between experimental GFR and wf. It is expected that GFR should increase with wf and dG}tffdwf should decrease with wf, because G~R a&ually represents the partial fracture energy (see Fig. 4b). This is the case for the length series and depth series specimens, where wf = u/f. In other words, the assumption of rigid-body kinematics can be used on the specimens in these series, which typically had relatively small cross-sectional dimensions. However, the slope of the relationship in Fig. 5, dGi:v,/dwf, shows no tendency to decrease with wf for the volume series and width series specimens, although GpR continues to increase with w/-. For the specimens tested in the volume series (V1, V 2, V 3 and V4) and the width series (T 1 and T2), the cross-sectional dimensions were significantly greater than for the length and depth series specimens. For these larger specimens, wfand u/fare significantly different indicating that the assumptioh of rigid-body kinematics does not apply. A possible explanation is that local non-linear &formation in compression around the loading point can not 313

GFR =-0.1480w~ + 0.2431wf + 0.06381

(9)

and from equation (8),

dGFR. = G(w)= -0.296wf

+ 0.2431

(10)

dw When G(w) = 0, equation (10) indicates that wf = wc = 0.821 mm. Substituting this value into equation (9) gives GvR = G v -- 0.1636 N/mm, which equals the total area under the softening curve in Fig.6. Equation (10) represents the linear part of the softening curve in Fig. 6 between w1 and wc. Knowing that G =ft' = 2.79 MPa whenw = 0, the remaining two unknowns in Fig. 6, % and wl, can be calculated from equations (11) and (12). Equation (11) equates the area under the softening curve (from w = 0 to w = w~) to G F = 0.1636 N/ram and equation (12) is obtained from equation (10). 0.5ft'w1 + 0.Swc% = 1.395w 1 + 0.411% = 0.1636 (11) % = - 0.2960w I + 0.2431

(12)

Materials and Structures/Mat~riaux et Constructions, Vol. 33, June 2000

A

(Y

obtained using the RILEM procedure, is due to the procedure itself. 5. The hypothesis and results presented here support the fictitious crack model in that the fracture energy, the critical opening displacement and the softening function are unique. From measurements of the size-dependent RILEM energy G~,R, the size-independent fracture energy G F, the critical opening displacement w c and the softening function can be estimated by appropriate curve fitting and analysis.

w , = 0.050 (mm) cr1 = 0.228 (MPa)

it

w e = 0.821 (ram) f t = 2.790 (MPa) G F =0.1636 (N/mm) W

ACKNOWLEDGEMENTS

I

WI

Wc

This research was supported by the Australian Research Council as well as the National Natural Science funds of China.

Fig. 6 - Calculated bilinear softening relationship.

Solving equations (11) and (12) gives o 1 = 0.228 MPa and wI = 0.050 mm. The softening function so determined is shown in Fig. 6. The calculated value ofw c (0.822 ram) is reasonably consistent with the approximation given by equation (2) (1.2 mm).

6. CONCLUSIONS 1. The effect of specimen size on the fracture energy (GFR) of concrete as measured using the RILEM threepoint bending test has been explained in terms of the concept of partial fracture energy, Gf, introduced in this study and in the previous work [17]. Gf is directly related to the width w of the FPZ. Gf increases with w and dGffdw decreases with w. 2. RILEM fracture energy GFR actually represents Gf and the corresponding opening displacement of the prenotched crack tip wf represents the width w of the FPZ. The variation of G/;R with wf is the same as the variation ofGfwith w. The effect ofs]3ecimen size on the value of GFRis ascribed to the size sensitivity of the opening displacement wf of the pre-notched crack tip at the end of the test. 3. In theory, when the specimen is large enough t O allow the fracture process zone to develop fully, wj[is greater than or equal to the limiting value wc and the effect of specimen size on Gt: is eliminated. However, if the specimen size is too large, the measured energy includes energy associated with non-linear deformation in the compression zone immediately under the loading point and becomes an unreliable estimate of fracture energy. The experimental results presented herein show that, for the measurement of GFn, the size of the specimen must be large enough to allow the fracture process zone to develop fully, but not too large to introduce significant non-linear (plastic) deformation in the compressive zone under the loading point. 4. Gv appears to be a unique material property. Its apparent size-dependence, as observed in test results

REFERENCES [1] RILEM Technical Committee 50 (Draft Recommendation), 'Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams', Mater. Struct. 18 (1985) 287-290. [2] Hillerborg, A., 'Results of three comparative test series for determining the fracture energy G F of concrete', Mater. Struct. 18 (107) (1985) 407-413. [3] Wittmann, F. H. (Ed.), 'Fracture toughness and fracture energy of concrete', (Elsevier, Amsterdam, 1986). [4] Mihashi, H., Takahashi, H. and Wittmann, F. H. (Eds.), 'Fracture toughness and fracture energy: test methods for concrete and rock', (Balkema, Rotterdam,1989). [5] Swartz, S. E. and Refai, T. M. E., 'Influence of size effects on opening mode fracture parameters for precracked concrete beams in bending', Proceedings of the SEM/RILEM International Conference on Fracture of Concrete and Rock, (Shah and Swartz eds., Houston, 1987). [6] Xu, S. and Zhao G., 'Fracture energy of concrete and its variational trend in size effect studied by using three point bending beams', Journal of Dalian Universityof Technology 31 (1) (1991) 79-86. [7] Planas,J. and Elices, M., 'Conceptual and experimental problems in the determination of the fracture energy of concrete', in 'Fracture Toughness and fracture energy: Test Methods for Concrete and Rock' (Balkema, Rotterdam, 1989) 165-181. [8] Planas, J., Maturana, P., Guinea G. V. and Elices, M., 'Fracture energy of water saturated and partially dry concrete at room and at cryogenic tempratures', in 'Advances in Fracture Research' (Pergmon, Oxford, 1989) 1809-1817. [9], Maturana, P., Planas, J. and Elices, M., 'Evolution of fracture behaviour of saturated concrete in low temperature range', Engineering FractureMechanics 35 (1990) 827-834. [10] Elices, M., Guinea G. V. and Planas, J., 'Measurement of the fracture energy using three point bend tests: Part 3-influence of cutting the P-d tail', Mater. Struct. 25 (1992) 327-334. [11] Qian J. and Luo H., 'Size effect on fracture energy of concrete determined by three point bending', Cement and Concrete Research 27 (1997) 1031-1036. [12] Guinea G. V., Planas, J. and Elices, M., 'Measurement of the fracture energy using three point bend tests: Part 1-influence of experimental procedures', Mater. Struct. 25 (1992) 212-218. [13] Planas, J., Elices, M. and Guinea G. V., 'Measurement of the fracture energy using three point bend tests: part 2-influence of bulk energy dissipation', Mater. Struct. 25 (1992) 305-312 [14] Wittmann, F. H., Mihashi, H. and Nomura, N., 'Size effect on fracture energy of concrete', Engineering Fracture Mechanics 35 (1990) 107-115. [15] Hu, X. Z. and Wittmann, F. H., 'Fracture energy and fracture

314

Guo,Gilbert process zone', Mater. Strua. 25 (1992) 319-326. [16] Petersson, P. E., 'Crack growth and development of fracture zones in plain concrete and similar materials', Report TVBM1006 (Division of Building Materials, University of Lund, Sweden, 1981). [17] Guo, X. H., Tin-Loi, F. and Li, H., 'Determination ofquasibritfle fracture law for cohesive crack models', Cementand Concrete Research29 (1999) 1055-1059.

APPENDIX CONCEPT AND CALCULATION OF QUASIFRACTURE ENERGY

U1 = W0 + W 1

Crucial in the analysis of the effect of specimen size on G~R is the calculation of the partial fracture energy Gf, shown shaded in Fig. 2a. Thi's quantity, referred to h~nceforth as "quasifracture" energy, is defined mathematically by: Gf =

(A1)

o dw

Physically, this energy is the specific fracture energy or work needed to displace the two faces of a unit surface of crack9 apart by .a distance ofw~ Ifw~5 = w,c then Gf J" = G~. In this paper, it has been shown ttiat the fracture energy measured in the RILEM three-point bending test G~R (equation (7)) represents the partial fracture energy Gf(wf), when the opening displacement of the prenotched crack tip wc_ wc at the end of the test. The corresponding deflectibn of the loading point is 8f (Fig. 7). The quantity G,-(wt-)Ali can be calculated'as the sum of the unrecoverable specl~c work U 1 (the energy &ssxpated in the sub-fracture process zone 102, shown in Fig. 3b, as the beam deflects to 8r) and the specific work U 2 (the energy dissipated in both the sub-fracture process zones 103 and 204, shown in Fig. 3b, as the .

jr

.

crack propagates to a width of wf at the pre-notched crack tip). Because a condition is being considered where the imposed load tends to zero, corresponding to the last phase of the three-point bending test, the crack can be modelled using rigid-body kinematics as proposed by Petersson [16]. This idealization allows the calculation of the aforementioned specific work required to propagate a crack of constant width u~ throughout the depth of the beam until the beam fails . First consider the unrecoverable work U 1 required to produce a central deflection of fir. If both the corresponding recoverable elastic deforination energy of the beam and the energy dissipation outside the fracture process zone are neglected, U~ is given by:

.

.

(A2)

Also, the central bending moment M f corresponding to 8f (when P -- 0) is given (Fig. 2) by: l a4i =30

(A3)

where Q is the equivalent central load corresponding to the weights of the specimen and the test equipment not fixed to the testing machine and is defined in equation (6). The specific work U2 is the work required to propagate a crack of constant width wf throughout the depth (D - ao) of the beam until the be[am "fails". For a cohesive law o = <5(w), the central bending moment M x resisted by the sub-fracture process zone of length x and corresponding maximum crack width w = 2xsin0 (see Fig. 7) is: Mx = So o ( w ) b x c o s O d x = 4sin 2bc~

J0 rw~

'

'

(A4)

where b is the beam thickness. Hence, when 0 = Of and the maximum crack width at x = D - a o is wf, the total moment Mfsupplied by the entire process zone is: Mf

bcos0f 4sin2 0f Gf~

(A5)

Noting, from Fig. 2a, that Io' o ( w ) w d w = G f ~ , where ~ is the distance from the origin to the centroid of the area representing Gd. Combining equations (A3) and (A5) gives: Qlsin 2 0[ Gf~

bcos0f

(A6)

and, hence, the total moment, from equations (A4) and (A6), can be written, after some simplifications, as a function of 0 as: M= Fig. 7 - Rigid body kinematics at the end of the RILEM test (P = 0 and 8 = 8f).

Q/sin 2 of cos0 4eos0f sin e 0

(A7)

Finally, the energy dissipation U 2 in the sub-fracture process zones 103 and 204 (Fig. 3b), when a crack of

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Materials and Structures/Matariaux et Constructions, Vol. 33, June2000

width u,f propagatcs throughout the ligament (D - a0) of the beam (corresponding to an increase in rotation from 0 = 0fto 0 = rt/2) can be written as: U2 = 2 I d M d 0 2s = or ~

cos0d0 sin20

(A8)

Since the partial fracture energy Gf has been defined as (U I + U2)[ Alis, and from equation (7), o i _ w0 +2wl

(A10)

Alig

equation (AI0) can be rewritten as a function ofwfas:

after substitution of equation (A7). When Of is small tan0f =- sin0f = Of, and 0y can be neglected. In this case, U2 = Q ~f

(A9)

since 8/= l Oj / 2.

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It is noted that wf --

4(D-ao)6f"