Cohesive crack modelling of a simple concrete: Experimental and numerical results

Engineering Fracture Mechanics 76 (2009) 1398–1410

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Cohesive crack modelling of a simple concrete: Experimental and numerical results M. Elices a,*, C. Rocco a,b, C. Roselló a a b

Departamento de Ciencia de Materiales, E.T.S. de Ingenieros de Caminos, Universidad Politécnica de Madrid, c/ Profesor Aranguren s/n, 28040 Madrid, Spain Area Departamental Construcciones, Facultad de Ingeniería, Universidad Nacional de La Plata, Calle 48 y 115 s/n, 1900 La Plata, Argentina

a r t i c l e

i n f o

Article history: Received 6 July 2007 Received in revised form 11 March 2008 Accepted 18 April 2008 Available online 27 April 2008 Keywords: Concrete Fracture mechanics Softening curve Cohesive zone modelling

a b s t r a c t Fracture tests were performed on six types of simple concrete made with two types of mortar matrix w/c = 0.32 and w/c = 0.42, two types of spherical aggregates (strong aggregates that debonded during concrete fracture, and weak aggregates, able to break), and two kinds of matrix–aggregate interface (weak and strong). The tensile strength, fracture energy and elasticity modulus of the six types of concrete were measured. These results are intended to serve as an experimental benchmark for checking numerical models of concrete fracture and for providing certain hints to better understand the mechanical behaviour of concrete. A bilinear softening function was used to model the fracture of concrete. The influence of the type of matrix, aggregate, and interface strength on the parameters of the softening curve are discussed: particularly, the fracture energy, the cohesive strength and the critical crack opening. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction This paper offers experimental results that can be profitably used in concrete research and design as well as being useful for checking numerical models of fracture of concrete. Plain concrete is a heterogeneous material composed of cement paste and rock particles. From a mechanistic point of view at meso-level, plain concrete can be considered as a two-phase material made of a continuous matrix of cement paste and fine aggregates (the mortar phase), and a dispersion of particles of coarse aggregates (the aggregate phase) embedded into the matrix. This approach has proven useful for analysing the influence of the aggregate and of the mortar–aggregate interface on the fracture of concrete [1,2]. Experimental information on the load–displacement or load–CMOD, as reported in [1], is a significant input for constitutive models. It is well known that specimen geometry and boundary conditions limit the applicability of continuum-based constitutive models to the pre-peak regime and that, if preventive measures are not taken into account, structural response is measured rather than material behaviour. However, the cohesive crack model is accepted as a realistic simplification of the fracture of brittle or quasi-brittle materials such as concrete; see, for example, [4] and references therein. In this model, the softening curve is a material property that characterises the macroscopic fracture behaviour. Here, we provide experimental values of the softening curve for very simple concretes made with three types of spherical aggregates, and of two kinds of matrix–aggregate interfaces, which show the influence of the matrix–aggregate interface and of the aggregate strength on the softening curve.

* Corresponding author. Tel.: +34 915 43 39 74; fax: +34 915 43 78 45. E-mail addresses: [email protected] (M. Elices), [email protected] (C. Rocco). 0013-7944/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2008.04.010

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Nomenclature B1 B2 B3 E0 fst f(w) GF lch l1 M1 M2 PA PBA w1 wc  w wq rt rq

strong coarse aggregate (7 mm diameter) weak coarse aggregate (7 mm diameter) weak coarse aggregate (14 mm diameter) effective elasticity modulus splitting strength softening function fracture energy characteristic length initial characteristic length mortar matrix (w/c = 0.32) mortar matrix (w/c = 0.42) percentage of aggregates on the fracture surface percentage of broken aggregates on the fracture surface initial crack opening (defined in Fig. 2) critical crack opening abscissa of the centroid (defined in Fig. 2) abscissa of the kink point (defined in Fig. 2) cohesive strength kink point stress (defined in Fig. 2)

This paper is structured in the following way: A brief summary of the basic aspects of the cohesive crack model is given in Section 2. The materials, specimens, and tests performed are presented in Section 3. The experimental results, including the parameters of the softening curves and its graphic representation, appear in Section 4. The analysis of the experimental results is the subject of Section 5, and the paper ends with certain comments on the softening curves computed for this kind of simple concrete. 2. The cohesive crack model 2.1. Theoretical background The cohesive crack model is generally accepted as a realistic simplification of the fracture of brittle or quasibrittle materials. This model was proposed by Hillerborg, Modéer and Petersson in the late seventies [3] and initially called the fictitious crack model. In this model, the entire fracture process is lumped into a line, which makes it possible to treat the whole bulk of the body as elastic. A review of the main aspects of this model and relevant references appear in [4,5]. For a detailed exposition, see [6]. The simplest assumptions regarding the cohesive zone model (see Fig. 1) are: (a) A cohesive crack initiates at the point where the maximum principal stress rI first reaches a critical value called the cohesive strength rt. This cohesive crack forms normal to the direction of the major principal stress, and given that this stress is induced by the notch, the cohesive crack forms ahead of the notch itself. (b) After its formation, the cohesive crack opens while transferring stress from one face to another. Once the crack has been initiated, the stress transferred (the cohesive stress) is a function of the crack opening displacement history. For monotonic mode I opening, the stress transferred, r, is normal to the crack faces and is a unique function of the crack opening w r ¼ f ðwÞ

ð1Þ

The function f(w) is termed the softening function or softening curve. Two properties of the softening function are worth noting: the cohesive strength rt and the cohesive fracture energy GF. The cohesive strength is the stress at which the crack is created and starts to open, i.e. rt ¼ f ð0Þ

ð2Þ

The cohesive fracture energy GF is the external energy supply required to create and fully break a unit surface area of the cohesive crack, and is given by the area under the softening function, i.e. Z wc GF ¼ f ðwÞdw ð3Þ 0

where wc is the critical crack opening, after which the cohesive stress becomes zero.

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Fig. 1. The cohesive zone model; notation used and sketches of two softening functions.

A further parameter, important in the structural behaviour, is the characteristic length: lch ¼

E0 GF r2t

ð4Þ

where E0 = E/(1  m2) (for plain strain), and m is the Poisson ratio. This simple formulation of the cohesive crack model is able to capture the main aspects of the fracture of brittle materials, particularly of components with blunted notches that do not exhibit a precrack or singularity. This model can be generalised in different ways: (1) The material outside the process zone – considered initially as isotropic linear elastic – can behave in a more complex manner, (2) the softening function may depend on triaxiality or on a previous loading history, and (3) the uniaxial model formulation can be generalised to a mixed-mode one. Discussions of such possible extensions were published in [5,7] and updated in [4,8]. 2.2. The bilinear softening curve In concrete-like materials, the softening curve can be approximated by a bilinear function [9], as depicted in Fig. 1. This simple diagram captures the essential facts: Large-scale debonding, or fracture, of aggregates in the steepest part, and frictional pull-out of aggregates and crack face bridging in the shallow tail of the diagram. This function is completely characterised when the following four parameters are known, as shown in Fig. 2: The tensile strength rt, the specific fracture energy  and the initial slope, measured as the horizontal intercept w1 of the GF, the abscissa of the centroid of the softening area w, initial segment.

Fig. 2. Sample geometry and rigid-body kinematics at the end of the bending test. Simplified softening curve (four parameters) used in this study.

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The four parameters needed for the bilinear approximation can be easily computed following the fitting procedure developed in [10,11] which makes use of the results from stable three-point bending tests on notched beams and from Brazilian splitting tests. The splitting-tension test, ASTM C469, gives results close to the actual cohesive strength, as shown in [11]. Therefore, it can be assumed that rt ¼ fst ¼

2P u pBD

ð5Þ

where fst is the splitting stress, Pu is the maximum load, B is the length of cylinder or prism and D is the diameter of the cylinder or the side of the square cross-section of a prism. In a cubic sample B = D. The specific fracture energy GF was measured according to the RILEM proposal [12]. A discussion on the stability of the notched beam test can be found in [13]. A detailed procedure of how to proceed was given in [2]. It suffices to recall that WF BðD  a0 Þ

GF ¼

ð6Þ

where WF is the work supplied to fracture statically the notched specimen, B is the beam thickness, and D  a0 is the ligament length.  of the centroid of the area under the softening curve can be obtained from knowledge of the load–displaceThe abscissa w ment (P–d) tail recorded during experimentation. In essence, the reasoning is as follows [10]: In cohesive materials and beams where the self-weight is compensated, the last phase of a stable three-point bend test can be modelled by rigid-body kinematics. If f(w) is the softening function, the bending moment per unit thickness at the central section, M, may be approximated by Z wc Z zc 1 1  f f ½wðzÞz dz  2 f ðwÞwdw ¼ 2 wG ð7Þ M¼ h h 0 0 where z and h are shown in Fig. 2, and zc is the point where the softening is complete, i.e. w(zc) = wc; the second integral follows by setting h  z = w, the rigid-body kinematics approximation; the last expression states that the second integral in Eq. (7) is the first-order moment of f(w) and can be expressed as the area enclosed between the positive axes and the soft of the centroid of that area. ening curve, GF, multiplied by the abscissa, w, The centroid of the area may be written always in the form,  ¼ w

aGF rt

ð8Þ

where a is a dimensionless parameter depending on the shape of the softening function. For rectangular softening a = 1/2, for linear softening a = 2/3, for exponential softening a = 1, also a = 0.987 for the bilinear softening proposed by Petersson [14], and between 1 and 1.3 for the functions proposed by Reinhardt et al. [15]. The horizontal intercept w1 (see Fig. 2) can be computed by the procedure suggested in [10] or, alternatively, the initial characteristic length l1 can be first computed according to Ref. [11] and then w1 computed as w1 ¼

2rt l1 E0

ð9Þ

where E0 is the effective modulus, already defined. The elastic modulus can be measured from standard compression tests or from the initial compliance of the load–CMOD curve. 3. Experimental programme 3.1. Materials The ‘‘model concretes” were cast using two kinds of spherical aggregates, two types of mortar matrix and two kinds of aggregate–matrix interface. 3.1.1. Aggregates Commercial spheres of mullite were used as aggregate with three types of spheres (denoted as B1, B2 and B3) with two different heat treatments. High temperature treatment was used to obtain the strong aggregate B1, and a medium temperature treatment to produce weak aggregates B2 and B3. Details of chemical composition and heat treatments were given in [1]. The average diameter of the spheres was: B1 and B2 7 mm, and B3 14 mm. 3.1.2. Matrix Two types of mortar matrix, identified as M1 and M2, were used. The water/cement ratio was 0.32 in mortar M1 and 0.42 in mortar M2. Both mortars were made with a type III Portland cement (in accordance of ASTM standards) and a natural fine

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siliceous aggregate of a size between 0.2 and 0.4 mm. Silica fume was added to increase the strength of the mortar, and superplasticiser (Sikament 300Ò) to improve its workability. 3.1.3. Matrix–aggregate interface Two kinds of matrix–aggregate interface – weak and strong – were used. Weak interfaces were achieved by daubing the spheres with a release agent, whereas strong interfaces were naturally obtained from the matrix and aggregate interaction. Here, strong is used in the sense that the adherence between matrix and aggregates is stronger than when aggregates are smeared with the release agent. We refer to weak and strong interfaces as treated and untreated types. 3.1.4. Concrete The composition of the six concretes is shown in Table 1. In all cases the volume of spherical aggregates was 25.8%. To make the concrete, the matrix was prepared first using a mechanical mixer and then the aggregate was added and stirred by hand to avoid any damage. Details of this process and of the aggregate interface treatment appear in [1]. 3.2. Specimens Notched beams (see Fig. 2) of various sizes were manufactured with the model concrete: Prisms of 75 mm depth were cast with concrete made with aggregates B2 and matrix M1. For concrete made with aggregates B3 and matrix M2, prisms of 100 mm depth were cast. Finally, prisms of 40 mm depth were made with concrete using aggregate B1 and matrix M1. Table 2 shows specimen sizes, together with notch depths. For each type of concrete, six beams were tested. In all cases the specimens were cast in metallic moulds and compacted by external vibration. For the first 24 h the specimens were stored in a water saturated atmosphere to avoid shrinkage and cracking, and then, they were demoulded and immersed in lime-saturated water. The notch was cut in a humid environment, before testing, using a diamond steel disc, as detailed in [1]. The width of all notches was 1.5 mm and the different depths are shown in Table 2. 3.3. Testing procedures and experimental results Two kinds of test were carried out: Stable three-point bend tests with prismatic notched specimens, as recommended by RILEM TC 50 [12], and Brazilian splitting tests with cubic specimens. In the bending test, possible sources of experimental errors, as detailed by Planas, Elices and Guinea [16–18] were taken into account. Bending tests were performed in a 1MN servo-hydraulic testing machine INSTRON 1275, run in crack mouth opening displacement (CMOD) control mode, recording the crack opening, the load, and the displacement of the loading point. Load was measured with a 5 kN load cell of 5 N resolution and 0.25% accuracy. An extensometer, of ±2.5 mm and ±0.2% error at full scale displacement, was used to measure the CMOD. The load-point displacement was measured by two transducers of ±2.0 and ±0.2% error at full scale.

Table 1 Composition of the six types of model concrete Matrix type

Aggregate type

Interface type

cement (kg/m3)

water (kg/m3)

Sand (kg/m3)

Silica f. (kg/m3)

plasticiser (kg/m3)

aggregate (kg/m3)

M1 w/c = 0.32

B1 (strong)

Untreated Treated Untreated Treated

482 482 482 482

157 157 157 157

938 938 938 938

42 42 42 42

32 32 32 32

544 544 544 544

Untreated Treated

470 470

200 200

908 908

30 30

15 15

500 500

B2 (weak) M2 w/c = 0.42

B3 (weak)

Table 2 Dimensions of the prismatic specimens Matrix type

Aggregate type

Interface type

Depth (mm)

Width (mm)

Span (mm)

Notch (mm)

M1 w/c = 0.32

B1 (strong)

Untreated Treated Untreated Treated

40 40 75 75

40 40 50 50

160 160 300 300

10 10 25 25

Untreated Treated

100 100

100 100

400 400

30 30

B2 (weak) M2 w/c = 0.42

B3 (weak)

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Cubic specimens cut from the tested beams were used for splitting-tension tests which were carried out under displacement mode control following the recommendations of ASTM C 496, except for the specimen size and width of the load-bearing strip which was 4 mm. Previous work [19] had shown that the standard width of the bearing strip (16% of the specimen depth) is usually over dimensioned and can lead to an erroneous estimation of ft. The relative width of the bearing strip was maintained between 4% and 10%. 3.4. Fracture topography After testing, the proportion of debonded and broken aggregates in the fracture surfaces was measured by image analysis. The projected surfaces of both, debonded and broken aggregates, on the fracture surface were also measured. From these data the following indexes, which give an idea of the type of fracture, were drawn up: projected surface of aggregates projected broken surface projected surface of broken aggregates PBA ¼ projected surface of aggregates PA ¼

When the percentage of aggregates, PA, is below the average value of a concrete with homogeneously-distributed aggregates, being 25.8% in our model concrete the resultant explanation is that the crack has bypassed the aggregates and found its way mainly through the matrix. This indicates an intergranular fracture mode, while PA above 25.8% indicates transgranular fracture or extensive debonding. Another index, the percentage of broken aggregates, PBA, helps in classifying fracture modes. When the PBA is above 50%, the dominant type of fracture is transgranular. Values below 50% are indicative that less than half of the expected aggregates are broken and, qualitatively, the dominant type of fracture may be considered intergranular or/and due to debonding. For concrete made with strong aggregates (type B1) no aggregate failure was detected; all fractures were fully intergranular (PBA = 0%). The values of the PA index were below the theoretical value 25.8% and varied between 15% and 20%. For concrete made with weak aggregates (type B2 and B3) and weak interface (treated interface) the dominant type of fracture was intergranular with PBA values between 0% and 25%. For concrete made with weak aggregates (type B2 and B3) and strong interface (untreated interface) the PBA index varied between 40% and 100%. 4. Experimental results and softening curves 4.1. Mechanical and fracture properties of components and composites Average values of the modulus of elasticity, the tensile strength, and the fracture energy of mortar matrix (M1: w/c = 0.32 and M2: w/c = 0.42) and aggregates (B1; strong and B2, B3; weak) are shown in Table 3. Estimated values of debonding energies of interfaces are shown in Table 4. Details of the experimental procedure to determine these properties are given in [2]. Average values of the tensile strength, fracture energy and modulus of elasticity for the six simple concretes are summarised in Table 5.

Table 3 Average properties of matrix and aggregates Splitting tensile strength (MPa)

Modulus of elasticity (GPa)

Specific fracture energy (J/m2)

Matrix

M2 (w/c = 0.42) M1 (w/c = 0.32)

3.3 ± 0.1 4.7 ± 0.2

27 ± 1 31 ± 2

34 ± 2 53 ± 6

Aggregate

B1 strong (7 mm) B2 weak (7 mm) B3 weak (7 mm)

16 ± 2 1.7 ± 0.3 2.6 ± 0.3

19 ± 2 2.1 ± 0.5 6.1 ± 1.4

NM NM 60 ± 6

NM: not measured.

Table 4 Average properties of the matrix–aggregate interfaces Interface type

Estimated debonding energy (J/m2) Aggregate B1

Aggregate B2 and B3

Untreated Treated

30 ± 10 0

26 ± 10 0

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Table 5 Mechanical properties of concretes Matrix type

Aggregate type

Interface treatment

fst (MPa)

GF (J/m2)

E (GPa)

M1 w/c = 0.32

B1 (strong)

Untreated Treated Untreated Treated

4.31 4.79 3.30 2.36

66 48 40 40

34.0 36.3 24.0 20.0

Untreated Treated

3.08 1.86

48 42

22.9 20.9

B2 (weak) M2 w/c = 0.42

B3 (weak)

4.2. Softening curves  and w1, as explained in The four parameters of the bilinear softening curve can be computed from knowledge of fst, GF, w [10,11]. The critical crack opening wc is computed by solving the quadratic equation [9], w2c  wc

 F =rt Þ  2w1 ðGF =rt Þ 6ww  1 ðGF =rt Þ  4w1 ðGF =rt Þ2 6wðG þ ¼0 2ðGF =rt Þ  w1 2ðGF =rt Þ  w1

ð10Þ

and the coordinates of the kink point, (rq, wq) are given by [9] wq ¼ w1

wc  2ðGF =rt Þ 2ðGF =rt Þ  w1 rq ¼ ft wc  w1 wc  w1

ð11Þ

The basic parameters of the bilinear softening curves: cohesive strength, rt, critical crack opening, wc, and the coordinates of the kink point in the bilinear curve, wq and rq are reported in Tables 6a–6d. The mean value, as well as the maximum and minimum values, are shown. The bilinear softening curves were drawn from these parameters and are sketched in Figs. 3a–c, in which the continuous line represents the average behaviour of the concrete and the broken lines extreme values. 5. Comments on the softening curves The influence of the type of matrix, aggregate and interface strength on the parameters of the softening curve is discussed from the experimental results given in Section 4. We concentrate on the cohesive strength, the critical crack opening and on the shape of the softening curves. The specific fracture energy was considered in a previous paper by Roselló [2].

Table 6a Parameters of the bilinear softening curves: cohesive strength, rt Matrix

M1 w/c = 0.32

M2 w/c = 0.42

Specimen depth (mm)

Aggregate type

40

B1 strong

75

B2 weak

100

B3 weak

Interface treatment

Untreated Treated Untreated Treated Untreated Treated

Cohesive strength, rt(MPa) Mean value

Extreme values Maximum

Minimum

4.31 4.79 3.30 2.36 3.08 1.86

4.67 5.62 3.72 2.81 3.20 2.02

3.54 4.02 2.90 1.95 2.92 1.67

Table 6b Parameters of the bilinear softening curves: critical crack opening, wc Matrix

Specimen depth, mm

Aggregate type

Interface treatment

Critical crack opening, wc (lm) Mean value

M1 w/c = 0.32

M2 w/c = 0.42

40

B1 strong

75

B2 weak

100

B3 weak

Untreated Treated Untreated Treated Untreated Treated

123 113 152 176 357 483

Extreme values Maximum

Minimum

155 148 170 227 403 619

113 97 132 130 307 315

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Specimen depth (mm)

Aggregate type

Interface treatment

Break point crack opening, wq (lm) Mean value

M1 w/c = 0.32

M2 w/c = 0.42

40

B1 strong

75

B2 weak

100

B3 weak

Untreated Treated Untreated Treated Untreated Treated

4.6 6.5 10.2 19.2 22.0 40.1

Extreme values Maximum

Minimum

6.9 7.7 16.2 26.6 23.1 46.4

4.1 4.8 7.2 15.5 21.2 37.3

Table 6d Parameters of the bilinear softening curves: break point stress, rq Matrix

Specimen depth, mm

Aggregate type

Interface treatment

Break point stress, rq (MPa) Mean value

M1 w/c = 0.32

M2 w/c = 0.42

40

B1 strong

75

B2 weak

100

B3 weak

Untreated Treated Untreated Treated Untreated Treated

0.88 0.82 0.58 0.40 0.08 0.08

Extreme values Maximum

Minimum

1.02 0.92 0.65 0.49 0.09 0.10

0.68 0.73 0.51 0.34 0.05 0.06

5.1. Cohesive strength, rt Fig. 4a shows the variation of the cohesive strength as a function of the percentage of aggregate on the fracture surface, PA. The results are for concrete made with strong aggregates, B1, and a matrix of type M1 (w/c = 0.32). Due to the relatively high strength of the aggregate B1, no broken particles were found, and the concretes exhibited an intergranular fracture mode. In the figure the dotted lines show the experimental tendency and the horizontal line the mean splitting strength of the matrix. The tensile strength of the concrete is seen to decrease as the amount of aggregates on the fracture surface increases, in all the types of interface. The behaviour of concrete with strong and weak interfaces was similar but, with the same percentage of aggregate on the fracture surface, the concrete with the weak interface showed higher cohesive strength than that with a strong interface. A plausible explanation could be that the actual fracture surface of the crack path is larger in weak interfaces than in strong ones. Fig. 4b shows the variation of the cohesive strength as a function of the percentage of broken aggregates, PBA, on the fracture surface of concrete beams of 75 mm depth made with weak aggregates B2 and matrix type M1 (w/c = 0.32). This concrete showed a mixed transgranular and intergranular mode of fracture surface. In concrete with weak interfaces the dominant fracture mode was intergranular whereas for strong interfaces the dominant mode was transgranular. It appears that the maximum strength is reached when a full transgranular fracture mode occurs. From the ideally intergranular to the transgranular fracture mode, the cohesive strength increases from around 2.0 to 3.5 MPa. It should be noted that the weak aggregate strength means that the maximum cohesive strength of the concrete is lower than that of the matrix. Fig. 4c shows the variation of the cohesive strength with the percentage of the broken aggregates in concrete beams of 100 mm depth, made with weak aggregates, B3, and a matrix type M2 (w/c = 0.42). The dotted line shows the experimental tendency. This concrete displayed two extreme fracture modes: Fully intergranular (0% of broken aggregates) at weak interfaces, and fully transgranular (100% broken aggregate) at strong interfaces. The cohesive strength increases from 1.9 to 3.0 MPa as the fracture mode changes from intergranular to transgranular, a trend similar to that observed in Fig. 4b. 5.2. Critical crack opening, wc Fig. 5a shows the variation of the critical crack opening as a function of the percentage of aggregates on the fracture surface (PA index) for concrete made with strong aggregates B1 and a matrix M1. The dotted line represents the value of the critical crack opening in the mortar matrix. It should be remembered that in this type of concrete the fracture was fully intergranular, with all the aggregates on the fracture surface being debonded. It appears that the critical crack opening wc, remained almost constant, irrespective of the proportion of aggregates on the fracture surface. It should be noted that the values of wc of the concrete are close to those of the mortar. Fig. 5b shows the variation of the critical crack opening, wc, with the percentage of broken aggregates on the fracture surface (PBA index) in concrete with weak aggregates (types B2 and B3) and of both matrix types, M1 and M2. The dotted lines

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Fig. 3. Softening curves of the six types of concrete. Continuous line represents the average behaviour and broken lines extreme values. (a) Strong aggregates and matrix M1 (w/c = 0.32). (b) Soft aggregates and matrix M1 (w/c = 0.32). (c) Soft aggregates and matrix M2 (w/c = 0.42). (Notice the different scale in abcissas in c.)

show the values of the critical crack opening in the mortar matrix. The results show that the critical crack opening, wc, in both treated and untreated interface concrete is similar, and remains almost constant irrespective of the percentage of broken aggregates on the fracture surface. Such results also show that the critical crack opening for concrete made with weak aggregates is higher than the corresponding value for the matrix. 5.3. Shape of the softening curve Fig. 6a plots the average softening curves for the concrete made with strong aggregates B1 and matrix type M1. As mentioned above, this concrete shows intergranular fracture mode (all aggregates on the fracture surface were debonded) irrespective of the interface type. Fig. 6a also shows the softening curve of the matrix type M1. The softening curves of the concrete are close to those of the matrix: The initial part and the tail of the softening curves of concrete and mortar are similar. Nevertheless, individual analyses of the experimental results show that as the proportion of debonded aggregates on the fracture surface increases, the softening curve of the concrete deviates from that of the matrix. Fig. 6b and c shows the softening curves of concrete made with weak aggregates B2 and B3, together with those of the matrix, M1 or M2. As compared with the softening curve of the matrix, the concrete shows lower tensile strength and a higher critical crack opening. This is enhanced in concrete with weak interfaces, where the dominant mode of fracture is intergranular.

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Fig. 4. Cohesive strength values of the six types of concrete considered.

6. Summary and conclusions These experimental results have a dual purpose; To provide some hints to better understand the fracture and mechanical behaviour of concrete, and to serve as a benchmark for numerical models of composite materials, in particular for those based on the cohesive crack approach [20, and references therein], and specifically when considering bilinear softening [9,10]. Most of the results of this research can be gathered into two groups: Those pertaining to concrete made with strong aggregates (B1) where no broken aggregates were detected after tests, and those from concrete made with weak aggregates, irrespective of the mortar matrix (M1 or M2), where some or all broke during testing. Clearly, other combinations of aggregate, matrix and interface would expose different behaviour. When the experimental results are analysed under the frame of cohesive models and, particularly, when a bilinear softening function is assumed, the following conclusions emerge. The area under the softening curve (the specific fracture energy GF) was discussed in [2]. The highest GF values (66 J/m2) were obtained with strong aggregates well-bonded to the matrix. The reason is that the crack path avoids the aggregates and wanders through the matrix, increasing the fracture area and hence the work of fracture. When this work is divided

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Fig. 5. Critical crack opening in the six types of concrete considered.

by the actual area (and not the projected area) the matrix value is recovered. When weak aggregates are used, the highest GF values (59 J/m2) are recorded when the matrix–aggregate interface is strong enough that the crack path minimises the work of fracture by crossing the aggregates instead of wandering around them, and all the aggregates appear broken across the crack path. The recorded lowest values, for weak interfaces, were 41 J/m2. The cohesive strength of all the concretes is lower than that of the matrix (with the exception of a limited number of results from strong aggregates and lower PA values, already mentioned, whose behaviour is not well understood). This is a general result for the kind of aggregates, matrix and matrix–aggregate interfaces considered in this research. Analyses of concrete made with weak aggregates (B2 and B3) showed that the cohesive strength increases as the percentage of broken aggregates increases, reaching in some cases the matrix tensile strength when PBA = 100% (a fact reflected in Fig. 7) where the results of Figs. 4b and c are replotted and normalised. The critical crack opening appears to be almost insensitive to the type of interface and matrix of concrete made with strong aggregates, whereas this opening was always larger than the corresponding matrix when weak aggregates were used. It seems that the critical crack opening increases as the percentage of broken aggregates (PBA) decreases, as suggested by Fig. 5b. It is worth noting that when plotting the relative critical crack opening as a function of the relative concrete strength (Fig. 8), all the values of the same matrix mortar are located along a straight line (irrespective of the aggregate strength and type of matrix–aggregate interface). With regard to the shape of the softening function, no differences appear between the concrete made with strong aggregates (B1) and the corresponding mortar (M1). Concrete with weak aggregates does differ from the mortar matrix, as already

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Fig. 6. Shape of the softening curves of the six types of concrete considered.

Fig. 7. The concrete tensile strength increases as the percentage of broken aggregates increases, reaching the tensile strength of the mortar matrix when PBA = 100.

mentioned: Cohesive strength is always lower and the critical crack opening is always larger. The matrix–aggregate interface also plays a role; weak interfaces provide larger initial slopes w1 (see Fig. 2), larger critical openings wc and smaller cohesive strength values rt, than those of strong interfaces. Such conduct, obviously related to the debonding, fracture and pull-out of the aggregates during the bending test, can provide a certain amount of guidance in modelling the mechanical performance of concrete.

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Fig. 8. Concrete relative strength as a function of relative critical crack opening. Open symbols refer to weak matrix–aggregate interfaces and full symbols to strong matrix–aggregate interfaces.

Acknowledgements The authors gratefully acknowledge useful discussions with Profs. Jaime Planas and Gustavo V. Guinea and José Miguel Martínez for help with drawing the figures. Support for this research was provided by the Spanish Ministerio de Ciencia y Tecnología under Grants MAT2000-1355 and MAT2001-3863-C03-01 and the Programa Ramón y Cajal. This work was conducted within the framework provided by the projects DUMEINPA (S-0505/MAT-0155) sponsored by the Regional Government of Madrid, Spain, and SEDUREC, integrated in the Spanish national research program CONSOLIDER-INGENIO 2010. References [1] Roselló C, Elices M. Fracture of model concrete: 1. Types of fracture and crack path. Cement Concr Res 2004;34:1441–50. [2] Roselló C, Elices M, Guinea GV. Fracture of model concrete: 2. Fracture energy and characteristic length. Cement Concr Res 2006;36:1345–53. [3] Hillerborg A, Modéer M, Petersson P-E. Analysis of crack formation and crack growth by means of fracture mechanics and finite elements. Cement Concr Res 1976;1:773–82. [4] Elices M, Guinea GV, Gómez FJ, Planas J. The cohesive zone model: advantages, limitations and challenges. Engng Fract Mech 2002;69:137–63. [5] Cornec A, Scheider I, Schwalbe KH. On the practical application of the cohesive model. Engng Fract Mech 2003;70:1963–87. [6] Bazant ZP, Planas J. Fracture and size effect in concrete and other quasibrittle materials. Boca Raton, Florida: CRC Press; 1998. [chapter 7]. [7] Elices M, Planas J. Material models. In: Elfgren L, editor. Fracture mechanics of concrete structures. London: Chapman and Hall; 1989. p. 16–66. [8] Planas J, Elices M, Guinea GV, Gómez FJ, Cendón DA, Arbilla I. Generalizations and specializations of cohesive crack models. Engng Fract Mech 2003;70:1759–76. [9] Roester J, Paulino GH, Park K, Gaedicke C. Concrete fracture prediction using bilinear softening. Cement Concr Compos 2007;29:300–12. [10] Guinea GV, Planas J, Elices M. A general bilinear fitting for the softening curve of concrete. Mater Struct 1994;27:99–105. [11] Planas J, Guinea GV, Elices M. Size effect and inverse analysis in concrete fracture. Int J Fract 1999;95:367–78. [12] RILEM 50-FMC Recommendation, Determination of fracture energy of mortar and concrete by means of three-point bend test on notched beams. Mater. Struct. 1985;18:285–90. [13] Petersson PE. Fracture energy of concrete: practical performance and experimental results. Cement Concr Res 1980;10:91–101. [14] Petersson PE. Crack growth and development of fracture zone in plain concrete and similar materials. Report TVBM-1006, Division of Building Materials, Lund Institute of Technology; 1981. [15] Reinhardt HW, Cornelissen HAW, Hordijk DA. Tensile tests and failure analysis of concrete. J Struct Engng ASCE 1986;112(11):2462–77. [16] Guinea GV, Planas J, Elices M. Measurement of the fracture energy using three-point bend tests: I. Influence of experimental procedures. Mater Struct 1992;25:212–8. [17] Planas J, Elices M, Guinea GV. Measurements of the fracture energy using three-point bend tests: 2. Influence of bulk energy dissipation. Mater Struct 1992;25:305–12. [18] Elices M, Guinea GV, Planas J. Measurement of the fracture energy using three-point bend tests. 3. Influence of cutting the P–d tail. Mater Struct 1992;25:327–34. [19] Rocco C, Guinea GV, Planas J, Elices M. Size effect and boundary conditions in the Brazilian test: experimental verification. Mater Struct 1999;32:210–7. [20] Stang H, Olesen JF, Poulsen PN, Nielsen LD. On the application of cohesive crack modelling in cementitious materials. Mater Struct 2007;40:365–74.