mixed mode fracture of concrete under proportional and nonproportional loading

International Journal of Fracture94: 267–284, 1998.© 1998Kluwer Academic Publishers. Printed in the Netherlands.

Mixed mode fracture of concrete under proportional andnonproportional loading

J.C. GÁLVEZ, M. ELICES, G.V. GUINEA and J. PLANASDepartamento de Ciencia de Materiales, Universidad Politécnica de Madrid, ETS de Ingenieros de Caminos,Ciudad Universitaria, 28040-Madrid, Spain. e-mail: [email protected]

Received 13 July 1998; accepted in revised form 17 December 1998

Abstract. A novel testing procedure for mixed mode crack propagation in concrete is presented: four pointbend of notched beams under the action oftwo independentforce actuators. In contrast to classical procedures,this method allowsnonproportional loadingand crack trajectory modifications by changing the action of oneactuator. Different experimental crack trajectories, under mixed mode and nonproportional loading, are presentedtogether with the corresponding curves of load-CMOD and load-displacement. The tests were performed for threehomotetic specimen sizes and two mixed mode loading conditions. The results are useful for checking the accuracyof mixed mode fracture analytical and numerical models. The models should predict the crack trajectory and acomplete group of experimental records of load and displacements on several control points in the specimen.

Keywords: Concrete, fracture, mixed mode, numerical analysis.

Abbreviations: ASTM – American Society for Testing and Materials; CMOD – crack mouth opening displace-ment; LEFM – linear elastic fracture mechanics; MTS – maximum tangential stress; PMMA – polymethil-methacrylate; RILEM – Réunion Internationale des Laboratoires d’Essais et de Recherches sur les Materiauxet les Constructions

1. Introduction

Mixed mode fracture of concrete is a complex problem, even in two dimensions. Duringthe last two-decades much work has been done to develop analytical and numerical toolsto describe the initiation and propagation of the cracks in mixed mode I/II in concrete struc-tures. A relatively large number of experimental results of crack initiation and propagation inmixed mode on notched beams are based on the Iosipescu geometry (Iosipescu, 1967; Arreaand Ingraffea, 1982; Bažant and Pfeiffer, 1986; Biolzi, 1990; Bocca et al., 1991; Schlangen,1993a,b; Swartz and Taha, 1990) and the results developed in the RILEM 89-FMT Com-mittee (Ballatore et al., 1990; Bocca et al., 1990), among others. Other sets of experimentalresults are based on the three point bend of notched beams with eccentric notch: (Guo etal., 1994a,b; Jenq and Shah, 1988; Swartz et al. 1988a), among others. Moreover, there areexperimental results developed on notched cylinders (Barr et al., 1989) and double notchedprismatic specimens (Bažant and Pfeiffer, 1986; Schlangen, 1993a; Barr and Deradj, 1990;Davies, 1989). Other important sets of tests have been developed by (Nooru-Mohamed, 1992;Nooru-Mohamed and Van Mier, 1992; Hassanzadeh, 1992) on notched specimens partiallycracked in tension. The advance has been important, but some difficulties still remain. TheIosipescu geometry and the nonsymmetric three point bend tests give trajectories of the crackthat are easily guessed from kinematic considerations and predicted within the wide scatterof the results by various models. This means that the performance of the models cannot be

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268 J.C. Galvez et al.

Table 1. Concrete prismatic specimens in each batch.

Sample Depth Length Thickness Number of Objective

mm mm mm specimens

D1 75 340 50 8 MP and MM

D2 150 675 50 4 MM

D3 300 1350 50 2 MM

MP: Mechanical properties; MM: Mixed mode tests.

assessed from crack path predictions for these geometries. The tests of the Delft group (Nooru-Mohamed, 1992; Nooru-Mohamed and Van Mier, 1992) led to overlapping several cracks.Then, there are confusing aspects: for example, (Swartz et al. 1988b) have concluded fromtheir tests that mode II fracture energy is eight to ten times larger than mode I fracture energy,and (Bažant and Pfeiffer, 1985) thirty times larger.

The numerical aspects of the cohesive crack model, developed by Hillerborg and co-workers (Hillerborg et al., 1976) for mode I fracture of concrete, have been included in finiteelement codes (Cervenka, 1994; Xie and Gerstle, 1996; Reich et al., 1993; Valente, 1995)as well as in boundary element codes (Saleh and Aliabadi, 1995; Saleh and Aliabadi, 1996).In these works it is assumed that the crack grows in the direction normal to the maximumprincipal stress, the MTS criterion (Erdogan and Sih, 1963). This hypothesis has been verifiedfor materials of almost linear elastic behaviour (Gálvez et al., 1996), but it needs to be verifiedfor mortar and concrete with more involved trajectories of the cracks.

The purpose of this paper is to develop a novel testing procedure for mixed mode crackpropagation on concrete, undernonproportionalloading. This procedure can provide a wholerange of crack trajectories, useful information as abenchmarkfor numerical codes devoted toprediction of crack trajectories and fracture loads.

Such procedures were developed for materials with an elastic linear behaviour (Gálvez etal., 1996). The purpose of this contribution is to extend this testing procedure to quasibrittlematerials, more specifically to concrete and mortar, and to provide additional experimentalinformation on mixed mode fracture of concrete: crack trajectories, load-displacement andload-CMOD curves that may help research in this field. The set of tests has been developed forgeometrically similar specimens of three sizes. Moreover it has been found that even thoughfracture is actually nonlinear, the crack path can be approximated by that predicted by LEFM.

2. Experimental program

2.1. NEW TYPE OF TEST ON MIXED MODE FRACTURE OFCONCRETE AND MORTAR

Figure 1a shows the geometry of the test specimens and the loading of the testing procedure.ForcesP1andP2are supplied byindependentforce actuators, which permitsnonproportionalloading of the specimen when properly controlled. Different combinations ofP1 and P2provide different trajectories of the crack. In this work, stable tests were achieved by applyingP1 through a servocontrolled testing machine running under CMOD control andP2 was setto a spring boundary condition (Figure 1b), using a servocontrolled actuator.

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Mixed mode fracture of concrete under proportional and nonproportional loading269

Figure 1. Geometry and loading diagram of testing procedure under nonproportional mixed mode. (a) Idealized.(b) Experimental device.

Table 2. Mechanical properties of themixtures.

Mixture fc ft GF E

MPa MPa Nm GPa

1 54 3.0 69 38

2 56 3.2 70 38

3 56 2.8 61 38

4 61 3.0 75 39

5 57 3.0 69 39

2.2. MATERIALS AND SPECIMENS

Five batches of a single microconcrete mixture were used to cast the specimens. The mi-croconcrete was composed of Portland cement, siliceous sand as the fine aggregates, andsiliceous crushed coarse aggregates of 5 mm maximum size. The cement was supplied inbulk to guarantee the homogeneity of the specimens. The water/cement ratio was 0.45 and theslump was 100 mm, measured according to ASTM C143. A vertical axis concrete mixer, with100 l. of capacity, was used.

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Figure 2. Geometry, forces and boundary conditions in mixed mode tests. (a) Type 1 (K = 0). (b) Type 2(K = ∞).

From each batch, three geometrically similar sizes of specimens were cast. The dimensionsand number of specimens are detailed in Table 1. The fracture energy, compressive strength,elastic modulus and tensile strength were determined in accordance with RILEM 50-FMC,ASTM C39, ASTM C469 and ASTM C496, respectively. Mechanical properties, measured atthe testing age, are detailed in Table 2.

The specimens were cast horizontally in one layer in ground steel moulds, which werevibrated 12 seconds on a vibrating table. The specimens were left in the moulds 72 hours,covered with saturated sacking at room temperature. They were then left in a curing room at20◦C and 99 percent relative humidity for 28 days; from then until testing they were immersedin lime saturated water at room temperature.

The notches were machined with a low speed diamond cutting disk. The notch was 2 mmwide with right-angled tip. The notch depth was 0.5D (see Figure 2), except for the smallspecimens of the type 2 test, where it was 0.6D. The support surfaces were ground to avoidspurious displacements due to crushing of the irregularities of the surface.

The specimen nomenclature is the following: Mixture. Size. Number of specimen. Forexample, the denomination M2.D1.2 corresponds to mixture 2, size D1 (depthD = 75 mm)and specimen number 2 of these characteristics.

2.3. EXPERIMENTAL PROCEDURE

To obtain different crack trajectories, clearly different even for large experimental scatter,the extreme values of the spring stiffness(K) were adopted: (a) In type 1 tests,K was 0(Figure 2a), (b) in type 2 tests,K was∞ (Figure 2b). Type 1 tests were performed without

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Table 3. Values ofα andβcoefficients.

Sample α β

D1 1.133 1

D2 1 1

D3 1 0.89

displacement control at pointB. In the type 2 tests the boundary condition at pointB wasimposed by a servocontrolled actuator that prevented vertical displacement throughout thetest. During the tests the following parameters were recorded: the CMOD, the loadP, theload-point displacement of forceP, the displacement of the pointB (test type 1) and thereaction force at pointB (test type 2).

The tests were performed in CMOD control, at a rate of 0.004 mm/min until 40 per-cent of the peak load in the descending branch and 0.08 mm/min until the end of the test.Table 4 shows the mixed mode tests under proportional and nonproportional loading. Table 5summarizes the number of valid and stable tests.

2.4. TESTING EQUIPMENT

LoadP was applied by a servocontrolled testing machine and the reaction force at pointB bya servocontrolled actuator. The loadP and the reaction force at pointB were measured with5, 25 and 100 kN load cells with±0.1 percent error of cell-rated output.

An extensometer, with±2.5 mm travel and±0.2 percent error at full scale displacement,was used to measure the CMOD. An inductive transducer, with±20 mm travel and±0.2percent error at full scale displacement, was used to measure the load-point displacement offorceP. The displacement of pointB was controlled with LVDT devices, with±1.0 mm and±5.0 mm travel, and±0.18 percent error full scale displacement, in type 2 and type 1 tests,respectively.

3. Experimental results

3.1. TRAJECTORIES OF THE CRACKS

For each specimen size, two different families of crack trajectories appeared in type 1 andtype 2 tests. Figures 3a, 3b and 3c show the experimental envelopes of the crack trajectoriesand the mean crack trajectory for the three sizes. The experimental scatter decreases as thespecimen size increases. To understand the trend in the behaviour of the scatter, the twofollowing remarks may help.

(1) For this concrete, made with strong siliceous aggregates, the crack grows throughthe mortar matrix, surrounding the aggregates and leading to a sinuous path of the crack.Therefore, since the aggregate size is always the same, the sinuosity of the crack relative tothe beam size is larger the smaller the beam size, which justifies a larger relative experimentalscatter for the smallest specimens.

(2) Several specimens of the smallest size have shown a little warping of the crack path,due to the existence of heterogeneities inside the mass of concrete. As these related to the

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Table 4. Mixed mode tests under proportional and nonproportional loading.

Sample Batch Depth Type Output Sample Batch Depth Type Output

mm test

M1.D1.1 1 75 1 Stable M3.D2.1 3 150 2 Non valid

M1.D1.2 1 75 1 Stable M3.D2.2 3 150 2 Stable





M1.D2.3 1 150 1 Stable


M4.D2.2 4 150 1 Stable

M1.D3.1 1 300 2 Unstable M4.D2.3 4 150 2 Stable

M1.D3.2 1 300 2 Stable M4.D2.4 4 150 2 Unstable



M2.D1.3 2 75 1 Stable





M2.D2.4 2 150 1 Stable

M5.D2.1 5 150 2 Unstable

M2.D3.1 2 300 2 Stable M5.D2.2 5 150 2 Unstable


M5.D2.4 5 150 2 Stable

M5.D3.1 5 300 1 Stable

M5.D3.2 5 300 1 Stable

Table 5. Number of valid andstable tests.

Sample Type 1 Type 2

D1 6 4

D2 6 6

D3 4 5

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maximum size of the aggregates, their effect is relatively more important in the smallestspecimens. Moreover, the warping leads to different crack paths in the front and the backfaces of the specimens, and again all the specimens are of equal thickness this phenomenon ismore marked in the smallest specimens.

3.2. LOAD-CMOD AND LOAD -DISPLACEMENT CURVES

Figures 4, 5 and 6 show respectively, for type 1 tests, the experimental records loadP vs.CMOD, loadP vs.load-point displacement, and loadP vs.vertical displacement of the pointB, for the three sizes. It is worth noting that the tests are stable and repeatable and the scatterband is narrow for all the specimen sizes.

Figures 7, 8 and 9 show respectively, for type 2 tests, the experimental records loadP vs.CMOD, loadP vs.load-point displacement, and the reaction force at pointB vs.CMOD, forthe three sizes. Again the tests are stable and repeatable and the scatter band, although largerthan for type 1 tests, is narrow enough for practical purposes, for all the specimen sizes.

4. Numerical prediction of the crack trajectory

For brittle materials such as PMMA, LEFM has proven its worth to predict the crack path,even for complex trajectories (Gálvez et al., 1996). To check LEFM crack path predictions forconcrete, where a significant nonlinear fracture zone is present, numerical computations wereconducted using the MTS criterion.

This criterion was proposed by Erdogan and Sih (Erdogan and Sih, 1963) and is stated asfollows: Crack extension starts at the crack tip in radial direction. This extension occurs in thedirection perpendicular to the direction of the greatest tension. Crack extension begins whenthis tension reaches a certain critical value at a given distance from the crack tip.Accordingto reference (Broek, 1986), for example, the stresses near the tip of the crack are given, inpolar coordinates, by

σθθ = 1√2πr

cos12θ[KI cos2 1

2θ − 32KII sinθ] (1)

τrθ = 1

2√

2πrcos1

2θ[KI sinθ +KII (3 cosθ − 1)], (2)

whereKI andKII are the stress intensity factors of the main crack prior to kinking. Themaximum ofσθθ happens whenτrθ is zero, and consequently the initiation angle,θm is givenby

θm = 2 tan−1

1

4

KI

KII± 1

4

√(KI

KII

)2

+ 8

. (3)

If it is postulated that the crack extension takes place whenσθθ (θ = θm) has the same valueas an equivalent I case, whereK = KIc andθ = 0, one has

KIc = KI cos3 12θm − 3KII cos2 1

2θm sin 12θm. (4)

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Figure 3. Experimental envelope of crack trajectories and mean crack trajectory. (a) Small size specimens(D = 75 mm). (b) Medium size specimens (D = 150 mm). (c) Large size specimens (D = 300 mm). (d)Axes of reference.

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Figure 4. Experimental curves for small size specimens (D = 75 mm) undertype 1conditions test. (a) Curves ofloadP vs.CMOD. (b) Curves of loadP vs.load-point displacement. (c) Curves of loadP vs.vertical displacementof point B.

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Figure 5. Experimental curves for medium size specimens (D = 150 mm) undertype 1conditions test. (a)Curves of loadP vs.CMOD. (b) Curves of loadP vs.load-point displacement. (c) Curves of loadP vs.verticaldisplacement of pointB.

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Figure 6. Experimental curves for large size specimens (D = 300 mm) undertype 1conditions test. (a) Curves ofloadP vs.CMOD. (b) Curves of loadP vs.load-point displacement. (c) Curves of loadP vs.vertical displacementof point B.

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Figure 7. Experimental curves for small size specimens (D = 75 mm) undertype 2conditions test. (a) Curvesof loadP vs.CMOD. (b) Curves of loadP vs.load-point displacement. (c) Curves of reaction force at pointB vs.CMOD.

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Figure 8. Experimental curves for medium size specimens (D = 150 mm) undertype 2conditions test. (a) Curvesof loadP vs.CMOD. (b) Curves of loadP vs.load-point displacement. (c) Curves of reaction force at pointB vs.CMOD.

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Figure 9. Experimental curves for large size specimens (D = 300 mm) undertype 2conditions test. (a) Curvesof loadP vs.CMOD. (b) Curves of loadP vs.load-point displacement. (c) Curves of reaction force at pointB vs.CMOD.

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It is worth observing that the MTS criterion, to first order, implies the condition of localsymmetry and that the crack initiation is locally under mode I conditions. For linear elasticbehaviour, this is equivalent to assuming that the crack initiation takes place in the direction ofmaximunk1, wherek1 andk2 are the stress intensity factors at the tip of an infinitesimal crackat an angleα to the main crack. Furthermore,k2 along the direction of crack propagation canbe shown to be zero. Sometimes this criterion -based onk1 andk2- is preferred for numericalcomputations.

Figures 10a, 10b and 10c compare the numerical prediction of the crack path with theexperimental envelope of the crack trajectories for the three sizes and the two types of tests.The numerical prediction of the crack path is based on the MTS criterion, and the simulationof the crack propagation was done step by step, supported on the finite element code FRANC(FRANC2D, 1994). The accuracy of the prediction lends further support to the hypothesis thatthe elastic fracture crack path is a good approximation for concrete structures, even thoughfracture is clearly nonlinear. This novel result is important because, to the authors’ knowledge,no verification of this hypothesis based on such wide experimental conditions as presentedhere was available.

5. Final remarks

A novel testing procedure, for mixed mode fracture undernonproportionalloading, was de-veloped: the four point bend on a notched beam under the action oftwo independentforceactuators. In contrast to classical procedures, based on the three point bend and four point bendon notched beams under the action of one force actuator, the new testing procedure allows themodification of the crack trajectory, and the curves load-CMOD and load-displacement, underthe action ofnonproportionalloading supplied bytwo independentforce actuators.

Two dispositions of the testing procedure were used (type 1 and type 2 tests) leading totwo different families of crack trajectories. Three sizes of homotetic beams were tested. Theexperimental records loadP-CMOD, loadP-displacement of the application point of the loadP, load P-displacement of the point B (type 1 tests) and reaction force on point B-CMOD(type 2 tests) are shown. The tests were stable and repeatable and the experimental curvesshowed a narrow scatter band.

The experimental results presented in this paper may serve as to provide a strong validationof analytical and numerical fracture models. The model should predict the crack trajectory anda very exigent set of experimental records of loads and displacements at several control pointsof the specimen, all for three sizes of specimen and two mixed mode loading conditions.

For the tested beams, the elastic fracture crack path is a good approximation of the ex-perimental crack path for concrete structures, even though the fracture behaviour is clearlynonlinear.

Acknowledgements

The authors wish to thank Prof. A. Ingraffea (Cornell University) for providing the computerprogram FRANC and Portland Valderribas Inc. for supplying the cement. The authors alsogratefully acknowledge financial support for this research by the Ministerio de Educacióny Cultura of Spain and the Universidad Politécnica de Madrid, under grants PB97-0579,MAT97-1022 and I+D 14989.

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Figure 10. Numerical prediction of crack trajectories. (a) Small size specimens (D = 75 mm). (b) Medium sizespecimens (D = 150 mm). (c) Large size specimens (D = 300 mm). (d) Axes of reference.

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mixed mode fracture of concrete under proportional and nonproportional loading

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