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Structural Integrity

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Procedia Structural Integrity 2 (2016) 2889-2895

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21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy

Mode I fracture; toughness of fibre-reinforced concrete by means of a modified version of the two-parameter model

Sabrina Vantadori*, Andrea Carpinteri, Giovanni Fortese, Camilla Rnnchei, Dpniela Scorza

Dept. of Civil-Environmental Engineeringand Architecture (DICATeA) — University of Parma Parco Area delle Scienze 181/A, 43124 Parma, Italy

Abstract

The present paper proposes a method to calculate Mode I plane-strain fracture toughness of concrete, by taking into account the possible crack deflection (kinked crack), even in the case of a far-field Mode I loading. As a matter of fact, during fracture extension, cracks may deflect as a result of microstructural ieelomogeneities inside tine material. Concrete is an inhomogeneous mixture due to aggregates embedded in the cementitious matrix, but additional inhomogeneities may be represenued by fibres. Firstly, n two-parametdr fracture model based on Mode I analytical expreesions of the linear elastic fracture machanics is amployed. Then in arder to take into acaount Hie possible crack deflection as a result of the abodm inhomogeneities, a modified version of such a model is hcre discussed. elnee-poinr bending tests on both plain concuete spocimens and concrete opecimens reinforced with micro-synthetic polypropylene fibrilMed fibres are experimentally perfoпнld, and SShs modified model is applied

Copyyright © 2016 Hie Authors. Published by Elsnvier B.V. This is an apen access article under the CC BY-NCAND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Stientifiu Committee ofECF21.

Keywords: Fsbrr-rrseПorcrn concretr; fracturr toughness; micro-synthetic polypropylene fibrillatrd fibres rrseПorcrn concretr; two-parameter fracturr modal

1. Introductiun

eheewo-Parameter Model (TPM) originally proposed "to deteпнsne the value of Mode I plane-strain fracture touehnrst of plain concrete (Jenq and Shah (19985), RILEM (1990), Kaiihaloo and Nallathambi (19991)) in herein modified in order to take into account the possible crack deflection (kinked craiO).

According to the TPM, the value of the fracture toughness is obtained from three-point bending tests on single edge-notched specimens. Firstly, the registration of the applied load against the crack mouth opening displacement

* Corresponding author. Tel.: +39 0521-905962; fax: +39 0521-905924. E-mail address: sabrina.'vantadori@unipr.it

Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer review under responsibility of the Scientific Committee of ECF21.

10.1016/j.prostr.2016.06.361

(CMOD) is needed. Then, the critical stress-intensity factor is computed through the expressions related to a crack loaded in Mode I (Tada et al. (2000)).

However, a crack in concrete may deflect during fracture extension, even in the case of a far-field Mode I loading, as a result of inhomogeneities embedded in the cementitious matrix. Inhomogeneities can be represented by aggregates for a plain concrete, and by aggregates and fibres for a fibre-reinforced concrete. Since the crack is loaded under both Mode I and Mode II in such a case, the TPM cannot be applied in its original formulation being proposed for crack under Mode I loading only. Therefore, to determine the critical stress-intensity factor (or Mode I plane-strain fracture toughness), a modified version of the TPM is here proposed by employing both the Castigliano theorem and the analytical solutions for the SIFs of a bent crack (Kitagawa et al. (1975), Cotterell and Rice (1980)).

Three-point bending tests on concrete specimens are performed in order to assess the proposed model, by considering the inhomogeneities represented by both only aggregates (for plain specimens) and aggregates and randomly-distributed micro-synthetic polypropylene fibrillated fibres with a fibre volume content equal to 2.5% (for fibre-reinforced specimens).

Nomenclature

a effective critical crack length

a0 notch length

A cracked area

B specimen thickness

C initial compliance

cu unloading compliance

E elastic modulus

F virtual load

G total energy rate

Mode I critical stress-intensity factor

Ky+jj )C Mixed Mode critical stress-intensity factor

P max peak load

S specimen loading span

UT total energy

W specimen depth

a0 relative notch length

a relative crack length

A F displacement along F - direction

2. Two-Parameter Model

According to the Two-Parameter Model (TPM) (Jenq and Shah (1985), RILEM (1990), Karihaloo and Nallathambi (1991)), the specimens have a prismatic shape and present a notch in the lower part of the middle cross section (Fig. 1(a)). The tests are performed under three-point bending loading and crack mouth opening displacement (CMOD) control.

Each specimen is monotonically loaded up to the peak load. When such a load is achieved, the post-peak stage follows and, when the force is equal to about 95% of the peak load, the specimen is fully unloaded. Then, the specimen is reloaded up to failure.

"■i ', *: 4 •

Fig. 1. Crack propagates under: (a) pure Mode I; (b) Mixed Mode.

The initial compliance, Ci, is used to calculate the elastic modulus, E (Tada et al. (2000)):

E = 6 Sao Vjao ) (j)

C W 2 B

where S, W and B areloading span, depth and thickness of the specimen, respectively, a0 is the notch length (Fig. 1(a)), Ci is the linear elastic compliance. Further, the parameter V(a0) is expressed as follows (Tada et al. (2000)):

V(a0) = 0.76 - 2.28a0 + 3.87a02 - 2.04a03 + 0'662 with a0 = (2)

1 - a02 W

Therefore, if the crack propagates under pure Mode I loading, the effective critical crack length, a, is determined through the following equation, by employing an iterative procedure (Tada et al. (2000)):

E = (3)

CUW 2 B

where Cu is the unloading compliance, and V(a) is obtained from Eq.(2) by replacing a0 with a . Since a stable three-point bend test cannot be performed in some cases, the value Cu can approximately be computed by assuming that the unloading path will return to the origin.

Finally, the Mode I critical stress-intensity factor, K-C, is computed by employing the measured value of the peak load, Pmax, as follows (Tada et al. (2000)):

3 p - _

Kfc = -^f-^af (a) (4)

where:

,, 1 1.99 - a(1 - a)(2.15 - 3.93a+ 2.70a2) a

f (a) =-—-----—- with a = -=- (5)

■U (1 + 2a)(1 - a)3'2 W

Note that the value of KIC , computed by assuming an unloading path to the origin, is about 10 to 25% higher than the corresponding one computed using the actual unloading compliance.

3. Modified Two-Parameter Model

Now a modified procedure is proposed when crack propagates under Mixed Mode loading (Mode I and Mode II). Specimens geometry and experimental test procedure are analogous to those discussed in the previous Section. Firstly, the elastic modulus is determined according to Eq.(1).

Under Mixed Mode loading, the effective critical crack length, a = a0 + a1 + a2 (Fig. 1(b)), is obtained from the following equation by employing an iterative procedure:

6 S \y( a0 . +

c,, w2 b a° I W 1 +

6 0 , -2 0 4 0 COS--h Sin — COS —

■ / COS

0)y\ ao + /iœs6l-a0 V¡^

i ■ 2n J I „ „w/ a0 + a COS0 + a2cosd\ ! ^ ( a0 + a cos0 + [cos 6 + Sin 0 COS 00 (a0 + a1 COS 0 + a2 COS 0) VI -0-1-w--I - (a0 + a1 COS 0) VI -0—W-

Equation (6) is deduced by employing the Castigliano theorem in the manner suggested by Paris (1957), being 0 the crack kinking angle (Fig. 1(b)) and a1 = 0.3 a0 . More precisely, the Castigliano theorem states that the displacement, AF, of any load F (in its own direction) may be computed as follows:

dUT dF

where UT iS the total energy expreSSed by:

UT = UNo Crack + J dA

being dA an increase in the cracked area. By assuming constant loading forces, the total energy rate G is equivalent to the rate of increase of the total strain energy UT , that is:

G =9Ut

the displacement, AF, of a virtual load F can be computed by replacing Eqs (8) and (9) in Eq.(7):

'dUT\ (dUr -

No Crack

+ JA \— l dA

F=0 J0 IdF )f=0

For the plane-stress problem herein examined, that is, a prismatic specimen tested under three-point bending (Fig. 2), the first term on the right-hand side of Eq.(10) is equal to zero, because it corresponds to the displacement

produced by F (in their own direction) in an uncracked beam, whereas the second term is a function of the SIF values for each Mode of loading (Kitagawa et al. (1975), Cotterell and Rice (1980), Tada et al. (2000)) due to both the loading force, P , and the virtual force, F.

F F 160

Fig. 2. Specimen geometry and the actual and the virtual forces in Castigliano theorem. Lengths are in mm.

Note that, as is shown in Figure 2, the kinked crack path consists of two segments, named a1 and a2 . If the value of a2 obtained from Eq.(6) is negative, it means that the effective crack length is a = a0 + a1 with a1 < 0.3 a0 . Such a length a1 is obtained from the following equation by employing an iterative procedure:

E = 6 S 2 X V\ C„W2 B I 0 I W

6 0 -2 0 4 0

cos — + sin — cos — 2 2 2

, „..„a» + a1cos0| ( a0 a + ai cos0) V| -I - a» VI wW

Finally, the critical stress-intensity factor, Ky+11 )C , is computed through Eqs (4) and (5) by considering a

straight crack having length equal to the projected length of the effective kinked crack (Kitagawa et al. (1975); Cotterell and Rice (1980)):

(I+II )C

2W 2 B

-\jn[a0 +(a1 + a2)cos0] f(a) with:

(a1 + a2 )cos0

when a1 = 0.3a0 (12)

( I+II )C

2W 2 B

+ a1cos0] f(a) with: a = °0 + aicos^ when a1 <0.3a0

4. Experimental and theoretical results

Specimens are tested under three-point bending (Figure 2). Testing is performed by means of an Instron 8862 testing machine under crack mouth opening displacement (CMOD) control, employing a clip gauge at an average speed equal to 0.1 mm h-1.

The specimen matrix is a cementitious matrix characterised by the following proportions: cement: water: aggregates (by weight) = 1: 0.7 : 3.6. This mixture presents a compressive strength of 30MPa at 28 days.

Two types of specimens are tested: plain concrete specimens (from P-1 to P-3 in Table 1) and concrete specimens reinforced by randomly-distributed micro-synthetic polypropylene fibrillated fibres (from R25-1 to R25-3 in Table 1). Such fibres are generally used for concrete secondary reinforcement and to control the plastic shrinkage of

concrete. The fibre aspect ratio and content are equal to 0.003 (fibre length equal to 18mm) and 2.5% by volume, respectively. The maximum aggregate size is 4mm.

Values of fracture toughness Ky+II)C are computed according to Eqs (12) e (13). Values of K(S)C are also

computed according to Eq. (4) and listed in the last column of Table (1).

Table 1. Elastic modulus, E, and critical SIF under Mixed, KSi + ¡¡)c , and Mode I, kC , loading.

Specimen Na E (MPa) K(y+II)C (MPa m1/2) KfC (MPa m1/2)

~~P-1 16028.85 0.460 0.547

P-2 16294.31 0.423 0.473

P-3 16242.12 0.488 0.503

R25-1 17068.80 0.622 0.639

R25-2 16862.08 0.664 0.696

R25-3 16838.15 0.600 0.660

For each specimen, crack growth under Mixed Mode (Figs 3 and 4) is observed. Higher values of crack kinked angle are generally found for plain concrete specimens.

In Table 1, it can be noted that fracture toughness values, Ky+II)C, for reinforced specimens (averaged value

equal to 0.629 MPa m1/2) are significantly different from those related to plain specimens (averaged value equal to 0.457 MPa m1/2).

Therefore, it can be concluded that randomly-distributed micro-synthetic polypropylene fibrillated fibres are able both to reduce the value of the kinked angle and to increase the concrete resistance to fracture. This is probably due to the fact that such fibres, tending to slip with respect to the matrix, reduce shear stress field with respect to that related to the case where only aggregates are embedded in the matrix. Since lower shear stresses produce a reduction of Mode II loading, both a decrease in kinked angle values and an increase in fracture toughness,

Ky+II )C, are expected to occur.

As is listed in Table 1, the fracture toughness value determined by employing Eq.(4) instead of Eqs (12) and (13) is overestimated up to 19% for the plain concrete specimens, and up to 10% for the fibre-reinforced concrete specimens here examined.

Fig. 3. Front and back side of fracture region of plain concrete specimens: (a)-(b) P-1; (c)-(d) P-2; (e)-(f) P-3.

5. Conclusions

In the present paper, a method to calculate Mode I plane-strain fracture toughness of concrete, by taking into account the possible crack deflection (kinked crack), has been proposed. Concrete and fibre-reinforced concrete are inhomogeneous mixtures due to aggregates and fibres embedded in the cementitious matrix. Due to such microstructural inhomogeneities, cracks may deflect during fracture extension. Therefore, to take into account such a possibility, a modified version of the TPM model has been proposed. Three-point bending tests on both plain concrete specimens and concrete specimens reinforced with micro-synthetic polypropylene fibrillated fibres have experimentally been performed in order to assess the proposed model. It can be concluded that, by applying the TPM according to its original formulation, Mode I plane-strain fracture toughness is overestimated up to 19% for plain concrete and up to 10% for fibre-reinforced concrete.

Acknowledgements

The authors gratefully acknowledge the financial support of the Italian Ministry of Education, University and Research (MIUR).

References

Cotterell, B., Rice J.R., 1980. Slightly Curved or Kinked Cracks. International Journal of Fractur 16, 155-169.

Jenq, Y., Shah, S., 1985. Two Parameter Fracture Model for Concrete. Journal of Engineering Mechanics 111, 1227-1241.

Karihaloo, B.L., Nallathambi, P., 1991. Notched Beam Test: Mode I Fracture Toughness, in "Fracture Mechanics Test Methods for Concrete.

Report of RILEM Technical Committee 89-FMT". In: Shah, S.P and Carpinteri, A. (Eds). Chapman & Hall, London, pp. 1-86. Kitagawa, H, Yuuki, R, Ohira, T., 1975. Crack-Morphological Aspects in Fracture Mechanics. Engineering Fracture Mechanics 7, 515-529. Paris, P.C., 1957. The Mechanics of Fracture Propagation and Solutions to Fracture Arrester Problems. Document D2-2195, The Boeing Company, Seattle.

RILEM Draft Recommendations (TC89-FMT), 1990. Determination of fracture parameters (KsIc and CTODc ) of plain concrete using three-point

bend tests. Materials and Structures 23, 457-460. Tada, H., Paris, P.C., Irwin, G.R., 2000. The Stress Analysis of Cracks Handbook. ASME Press, New York.