An energy-based analysis for aggregate size effect on mechanical strength of cement-based materials

Engineering Fracture Mechanics 102 (2013) 207–217

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An energy-based analysis for aggregate size effect on mechanical strength of cement-based materials C.F. Jin a, Q.C. He b, J.F. Shao a,⇑ a b

Laboratoire de Mécanique de Lille, UMR CNRS 8107, Université Lille 1, 59650 Villeneuve d’Ascq, France Laboratoire Modélisation et Simulation Multi Echelle, UMR 8208 CNRS, Université Paris-Est, 77454 Marne-la-Valle, France

a r t i c l e

i n f o

Article history: Received 1 June 2012 Received in revised form 25 January 2013 Accepted 2 February 2013

Keywords: Damage Fracture Inclusion size effect Debonding Cement-based materials Concrete

a b s t r a c t In this paper, the mechanical strength of cement-based materials is investigated. The emphasis is put on the aggregate size effect on the mechanical strength in uniaxial and triaxial compression tests. An analytical study based on Griffith’s surface energy criterion is performed. The surface energy for the creation of crack surfaces is related to the variation of elastic energy between the intact and cracked materials. The effective elastic properties of cement-based materials are estimated respectively by Mori–Tanaka scheme for the intact material and Reuss lower bound for the cracked material. The aggregate size effect is taken into account by the fact that the total energy needed for crack surface creation depends on the size of aggregates which are subjected to progressive debonding during the failure process. Further, a series of parametric studies are realized to study the effect of confining stress on the mechanical strength of cement-based materials. Comparisons between analytical results and experimental data are presented. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The mechanical behavior of cement-based materials is governed by nucleation and propagation of microcracks. Extensive experimental investigations have been conducted on the characterization of microcrack-induced damage at both macroscopic and microscopic scales. Without giving an exhaustive list of such previous works, most experimental results revealed some common characteristics on the micromechanical properties of cement-based materials. Under tensile stresses, open microcracks are generated and mainly propagate in the direction perpendicular to the tensile major stress. The kinetics of crack propagation is controlled by tensile strains. Under compressive stresses, most microcracks are closed and propagate in complex mixed modes. The crack propagation is inherently coupled with friction sliding along crack surface. At macroscopic scale, the frictional sliding along closed cracks can be the origin of irreversible strains and hysteretic cycles. On the other hand, the micromechanical properties of cement-based materials are strongly influenced by their microstructure and in particular by the presence of aggregates. It is known that the size of embedded aggregates dominates the density and type of microcracks which initiate in the transition zone around interfaces between aggregates and cement paste. For example, according to X-ray micro-tomographic observations on glass beads embedded concrete subjected to uniaxial and triaxial compression [1], microcracks first initiate at matrix-aggregate interfaces and progressively propagate into cement matrix. The density and extent of cracked zone are higher in the materials with large size glass beads. On the other hand, concrete structures are also subjected to desiccation process in many engineering applications. The characterization of desiccation-induced damage is an essential issue for long term durability analysis of such structures. During the last

⇑ Corresponding author. Tel.: +33 320434626. E-mail address: [email protected] (J.F. Shao). 0013-7944/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2013.02.013

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Nomenclature D H

R1 Rc R1 Wext Wel

U Wkin e(x) r(x)

R Shom X E(1)

m(1) E(2)

m(2) ci d f

diameter of concrete specimen height of concrete specimen axial stress in triaxial compression test lateral confining stress in triaxial compression test axial failure stress in triaxial compression test external work elastic energy total surface energy used in the creation of cracks macroscopic kinetic energy local strain tensor local stress tensor macroscopic stress tensor effective elastic compliance tensor of the homogenized medium volume of RVE elastic modulus of intact material Poisson ratio of intact material elastic modulus of cracked material Poisson ratio of cracked material surface energy density diameter of aggregate volumetric fraction of aggregate

decades, various laboratory investigations have been performed for the characterization of desiccation damage in different concrete materials [2–4] just to mention some reference works. The desiccation damage may lead to significant degradation of both mechanical and transport properties of concrete. Some previous works have been devoted to experimental study of shrinkage deformation effects on mechanical properties of concrete under uniaxial and triaxial compression [5–7]. Different kinds of constitutive models including desiccation damage and related numerical methods for structure analysis under coupled hydric mechanical conditions have also been developed [8–12]. Two competitive mechanisms are generally identified in desiccation damage evolution: non-uniform hydric gradient effect and material heterogeneity effect. In the first case, the desiccation damage is related to non-uniform saturation distribution leading to a constrained shrinkage strain field which is the origin of local tensile stress. In the second situation, the desiccation damage is directly related to the microstructural heterogeneity of concrete leading to differential shrinkage strain fields between cement paste and aggregates. A tensile stress zone may occur around aggregates. Therefore, the evolution of desiccation damage is not only influenced by loading conditions but also by the heterogeneous microstructure of materials. Like in mechanical damage, the size of aggregates plays also an important role in the evolution of desiccation damage. Some experimental investigations have been specifically devoted to this issue, for instance [1,13,14] just to mention a few. Bisschop and van Mier [13] used an artificial concrete, composed of spherical glass beads of different size embedded in a cement paste matrix, to explore the influence of aggregate size. Their experimental results evidenced the significant influence of inclusion size on the initiation and repartition of microcracks induced by the restrained shrinkage of cement matrix. The desiccation damage is clearly intensified by the presence of large size inclusions. This work was completed by Szczesniak et al. [1] by using the same material. In particular, X-ray micro-tomographic analyses have been performed on samples subjected to drying process and composed with different size inclusions. According to these microscopic analyses, in the concrete sample with small inclusion, no significant microstructural change was detected. However, in the samples with larger inclusions, the microstructure is significantly modified with the existence of a crack network. The cracks are localized essentially between the surface of the specimen and an inclusion, between two inclusions and finally around inclusions. These cracks are due to drying shrinkage of the cement matrix and mainly occur near rigid inclusions, which prevent free shrinkage. The presence of cracks at the interface matrix-inclusion seems to indicate the existence of an interfacial zone with weak mechanical properties. Further, from their observations, it is highlighted that the cracking pattern is strongly dependant on the size of inclusions. Indeed, the diameter of inclusions affected the number, mean length and opening of cracks. The change in the cracking pattern will have an influence not only on the macroscopic mechanical behavior of the composite but also on transport properties such as permeability and diffusivity, etc. These results are in accordance with those obtained by Grassl et al. [14], who found an increase of average crack width and permeability with increasing aggregates diameter. Similar results are reported by Cho et al. on polymeric composites with micro- and nano- particles [15]. In this paper, we propose an energy-based theoretical analysis for the aggregate size effect on mechanical strength of cement-based materials in saturated conditions. The problem of inclusion size effect on the effective mechanical properties of composites has been studied by a number of authors. Benveniste [16] studied the effective mechanical behavior of composites with imperfect contacts between particles and matrix and found a strong dependence of the effective properties of composites on the inclusion size. In Refs. [17,18], it is

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found that the inclusion/matrix interface plays an important role in the mechanical performance of composites. Most studies have shown that the tensile strength, as well as the compressive one, increases as the size of inclusions decreases. Using homogenization techniques, Duan et al. [19] derived two scaling laws to capture the size-dependence for multiphase composite with interface effects. Based on the energy release rate initially defined by Griffith [20] and Bhattacharya et al. [21] proposed an energy-based model for compressive splitting of polycrystals, showing that the failure strength decreases while the grain size increases. They considered that the material failure was driven by a progressive reduction of the total energy of solid system, defined as the sum of bulk strain energy and surface energy. In addition, Cho et al. [15] pointed out that the interface crack propagation for smaller particles required higher applied stresses. Quesada et al. [22] studied the compressive strength of a composite with a rigid inclusion embedded in a soft matrix. They proposed a mixed criterion involving both energy and stress conditions to predict the relation between the inclusion size and mechanical strength. In this study, the energy-based model proposed by Bhattacharya et al. [21] for brittle polycrystals will be extended to cement-based materials under uniaxial and triaxial compression. The analytical relation between the mechanical strength, aggregate size and confining stress will be derived by using the theory of Griffith [20] and the macroscopic analysis of strain energy of concrete composed of cement matrix and rigid aggregates. Particular attention will be put on the effect of confining stress on the mechanical strength of material. Comparisons between theoretical predictions and experimental data will be presented.

2. Energy analysis of failure stress in triaxial compression test Consider a cylindrical specimen with diameter D and height H as a representative volume element (RVE) of concrete as shown in Fig. 1. A number of rigid spherical inclusions, for instance glass beads, are embedded in the cement paste matrix. The specimen is subjected to a triaxial compression state defined by the following stress tensor:

0

R1

B R¼@ 0 0

0

0

1

Rc

C 0 A

0

Rc

ð1Þ

In the triaxial compression test, the axial stress R1 is progressively increased until the failure of specimen while the lateral confining stress Rc is hold at a constant value. Denote now R1 the value of axial stress at the failure state, here identified by the peak point on stress strain curves. Assume that under applied stresses, the RVE of concrete exhibits elastic deformation and crack nucleation and propagation. Based on the basic theory of Griffith [20], the energy balance equation can be written as follows:

W ext ¼ W el þ U þ W kin

ð2Þ

Wext is the external work, Wel the elastic energy. U denotes the total surface energy used in the creation of cracks. Wkin is the macroscopic kinetic energy which can be neglected for quasi static loading. Compare two different states of the RVE as shown in Fig. 1: an intact state and a cracked state at the failure condition. According to the balance Eq. (2) and consider a quasi static loading, the difference of surface energy between the two states is given by:

DU ¼ Uð2Þ  Uð1Þ ¼ DW ext  DW el

ð3Þ

The quantity DU represents the total surface energy related to the creation of new cracks between the two states of the material. DWext and DWel are respectively the difference of external work and elastic strain energy between the two states.

Σ1

Σc

*

Σ1

Σ1

Σc

Σ1 Cylinder specimen

Σc

Σc

Σ1 Intact state

Σc

Σc

*

Σ1

Cracked state

Fig. 1. Illustration of concrete specimen, intact and cracked states.

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Note that there is no crack occurrence in the RVE at the intact state (U(1) = 0). Further assume that for the same values of applied stress, the external works are the same for the two states of RVE, i.e. DWext = 0. Therefore, the total surface energy related to crack creation can be associated with the difference of elastic strain energy [21]:

  DU ¼ Uð2Þ ¼  W elð2Þ  W elð1Þ

ð4Þ

By using the energy equivalence principle of upscaling for heterogeneous materials [23,24], the total elastic energy of the RVE at a given strain state can be expressed as:

W el ¼

Z

1 1 rðxÞ : eðxÞdv ¼ XR : Shom : R 2 2

X

ð5Þ

In this relation, e(x) and r(x) are respectively the non-uniform local strain and stress fields inside the RVE. The tensor R is the uniform macroscopic stress applied to the boundary of RVE. The fourth order tensor Shom denotes the effective elastic compliance of the homogenized medium, and X is the volume of RVE. Assume now an isotropic linear elastic behavior for the cement-based material, characterized by the effective elastic modulus Ehom and Poisson’s ratio mhom, and consider the specific loading condition of triaxial compression tests, the elastic strain energy can be expressed by:

W el ¼

X h hom

2E

ðR1 Þ2 þ 2ðRc Þ2 ð1  mhom Þ  4mhom R1 Rc

i

ð6Þ

This relation is now applied to two comparative states of the material. Let E(1) and m(1) be the elastic properties of the intact state and E(2) and m(2) those of the cracked state. The elastic energy of the two states corresponding to the failure state is respectively given by:

X h

ð1Þ

W el ¼

ð1Þ

2E

X h

ð2Þ

W el ¼

2Eð2Þ

ðR1 Þ2 þ 2ðRc Þ2 ð1  mð1Þ Þ  4mð1Þ R1 Rc

ðR1 Þ2 þ 2ðRc Þ2 ð1  mð2Þ Þ  4mð2Þ R1 Rc

i

ð7Þ

i

ð8Þ

In order to evaluate the total surface energy related to crack creation in the cracked state U(2) and for the sake of simplicity, we assume that all cracks are entirely localized on the interfaces between the cement paste matrix and spherical aggregates. Therefore, the number of interfaces can be determined from the volume fraction of aggregates f as 6fX/pd3, with d being the diameter of aggregate. Further, let ci be the debonding energy for the unit surface, the total surface energy writes:

Uð2Þ ¼

6ci f X d

ð9Þ

Substituting (7)–(9) into the energy balance Eq. (4), one obtains:



R1

2



1

E

ð1Þ



1 ð2Þ

E



þ 2 ð Rc Þ 2

 1  mð1Þ E

ð1Þ



1  mð2Þ ð2Þ

E



 4R1 Rc



mð1Þ ð1Þ

E



mð2Þ E



ð2Þ

¼

12ci f d

ð10Þ

By solving this second order equation, the failure stress in a triaxial compression test R1 can be determined as follows:

1 R ¼ D1  1

(

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) 12D1 ci f 2D2 Rc þ 2ðRc Þ2 ð2D2  D1 ÞðD2 þ D1 Þ  d

D1 ¼

1 E

ð1Þ



1 ð2Þ

E

;

D2 ¼

mð1Þ E

ð1Þ



mð2Þ Eð2Þ

ð11Þ

From this result, we can note that the axial failure stress R1 depends on the confining pressure, the surface energy density for crack creation, the diameter of inclusion, as well as the contrast of elastic properties between the intact and cracked materials. In the particular case of uniaxial compression test, i.e. Rc = 0, the uniaxial compression strength is given by:

R1 ¼

1 D1

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12D1 ci f  ; d

D1 ¼

1 Eð1Þ



1 Eð2Þ

ð12Þ

In this case, the uniaxial strength depends on the difference of elastic modulus between the intact and cracked materials. It is worth noticing that the relations (11) and (12) are based on the assumption that all cracks are located at interfaces between cement paste matrix and aggregates, the proposed energy analysis can be extended to the case where cracks may propagate into the cement matrix. To do this, based on the information on the propagation mode of cracks inside the cement matrix, the total surface energy given by the relation (9) should be recalculated by taking into account the energy needed for the creation of crack surfaces inside the cement matrix. In the following sections, numerical studies will be conducted to evaluate the influences of various factors on the failure stress of concrete. The emphasis will be put on the influences of aggregate size and confining stress.

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211

2.1. Reference values of elastic properties According to the theoretical model given in (11), in order to evaluate the mechanical strength as a function of inclusion size, it is first needed to estimate the elastic properties of both intact and cracked materials. In this section, we will determine a set of reference values of elastic properties for both intact and cracked materials. Note that in the present work, the theoretical model will be applied to the experimental data obtained by Szczesniak et al. [1] on an artificial concrete composed of spherical glass beads embedded in cement paste. Due to this specific morphology of material, it seems pertinent to use Mori–Tanaka scheme [25] to estimate the elastic properties of the intact material. To do this, the elastic properties of constituents are determined from data in [26]. The elastic properties of glass beads are Ei ¼ 73 GPa and mi = 0.17, while those of the cement paste matrix are Em ¼ 16 GPa and mm = 0.27. In addition, the volume fraction of glass beads is f = 0.35. The use of Mori–Tanaka schema leads to the following estimates for the intact material: MT

k

¼ km þ

lMT ¼ lm þ

f ðki  km Þð3km þ 4lm Þ 3fkm þ 3ð1  f Þki þ 4lm 5f lm ðli  lm Þð3km þ 4lm Þ

ð13Þ

6ð1  f Þli ðkm þ 2lm Þ þ ð9 þ 6f Þkm lm þ ð8 þ 12f Þðlm Þ2

where ki and km are the bulk modulus of inclusion and matrix respectively, li and lm the shear modulus. For the concrete studied in the present work, the following elastic properties are obtained for the intact material: Eð1Þ ¼ EMT ¼ 25:3 GPa and m(1) = mMT = 0.243. For the cracked material, we assume that there is a full debonding of all interfaces and that the local stress field is nearly uniform. Thus, for the sake of simplicity, the Reuss lower bound is employed to estimate the effective properties of the cracked material [27]: 1 kR 1

lR

¼ ð1  f Þ k1 þ f i

1 km

ð14Þ

1

¼ ð1  f Þ l þ f l1 i m

Accordingly, the estimated elastic properties of the cracked material are Eð2Þ ¼ ER ¼ 22 GPa and m(2) = mR = 0.259. Now by using the experimental data from Refs. [1,26], the reference values of elastic properties are determined and summarized in Table 1. 2.2. Numerical application and experimental validation The theoretical model together with the reference values of elastic properties presented above is now applied to evaluate the mechanical strength of concrete investigated by Szczesniak et al. [1]. In their work, glass beads with different diameters were used as artificial aggregates, ranging from 1 mm to 6 mm. The glass beads were mixed with the cement paste as homogenously as possible. Uniaxial and triaxial compression tests, with different confining stresses, say 0, 5 and 15 MPa, were performed on each group of specimens. The failure stress was identified in each test from the peak point of stress–strain curves. According to (11), the surface energy density ci should be determined for the evaluation of failure stress. As the direct measurement of this parameter is not available, the following methodology is adopted. The value of surface energy is determined from (11) by using the experiment value of failure stress obtained from the uniaxial compression test on the specimen with 1 mm glass beads, that is:

ci ¼

ðR1ex Þ2 D1 d 12f

ð15Þ

Then the obtained value of ci is taken as constant and used to estimate the failure stress of other specimens. With this method, the surface energy density is estimated as ci ¼ 2:539 Pa m. Using the parameters presented above, the failure stress in different cases can be easily determined from the relation (11). In Figs. 2–4, the calculated values of failure stress are presented as functions of inclusion size for different confining stresses. We can see that there is a very good agreement between the theoretical values and experimental data for the case of uniaxial compression. The inclusion size effect on the mechanical strength is well described. This illustrates the suitability of the theoretical model in spite of its simplicity. However, for the case of triaxial compression tests, although the model predicts a good tendency of evolution of mechanical strength with inclusion size, there is a large difference between the theoretical predictions and experimental data. The Table 1 Reference values of elastic properties based on experimental data from Refs. [1,26]. Inclusion Cement paste matrix Effective properties of intact material

EI ¼ 73 GPa, mI = 0.17 Em ¼ 16 GPa, mm = 0.27

Effective properties of cracked material

ER ¼ 22 GPa, mR = 0.259

EMT ¼ 25:32 GPa, mMT = 0.243

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45

uniaxial compression failure strength (MPa)

40 35

MT&Reuss

30

test 25 20 15 10

d (mm) 0

1

2

3

4

5

6

7

Fig. 2. Failure strength versus aggregate size in uniaxial compression test: comparison between theoretical prediction and experimental data.

80

triaxial compression 5MPa

failure strength (MPa)

70

MT&Reuss 60

test

50 40 30

d (mm)

20 10

0

1

2

3

4

5

6

7

Fig. 3. Failure strength versus aggregate size in triaxial compression test with 5 MPa confining stress: comparison between theoretical prediction and experimental data.

triaxial compression 15MPa

failure strength (MPa)

90 80

MT&Reuss

70

test

60 50 40 30 20

d (mm)

10 0

1

2

3

4

5

6

7

Fig. 4. Failure strength versus aggregate size in triaxial compression test with 15 MPa confining stress: comparison between theoretical prediction and experimental data.

theoretical prediction largely underestimates the failure stress in triaxial compression tests and the effect of confining stress is not correctly taken into account. 3. Parametric studies In this section, parametric studies are performed in order to better capture the effect of confining stress on the mechanical strength of material. According to (11), the mechanical strength is dependent on the confining stress Rc through the terms D1 and D2. These two terms represent the contrast of elastic property between the intact and cracked materials. As a first approximation for the reference case presented in the previous section, the elastic properties are determined respectively

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C.F. Jin et al. / Engineering Fracture Mechanics 102 (2013) 207–217

by MT estimate and the Reuss lower bound. The values obtained from these estimations could be scattered from unknown real values of material. Therefore it is convenient to perform parametric studies in order to capture the sensibility of macroscopic failure stress to estimated elastic properties. 3.1. Influence of effective moduli of cracked material The influence of effective elastic moduli of cracked material is first studied. Instead to use the Reuss lower bound, we propose to determine the effective elastic moduli of cracked material by a linear combination of values respectively calculated from the Reuss lower bound and MT estimate. Introduce a partition factor a e [0, 1], and define the elastic moduli by: ð2Þ

k

¼ ð1  aÞk

MT

R

þ ak ;

lð2Þ ¼ ð1  aÞlMT þ alR

ð16Þ

We assume first that the surface energy density ci is not influenced by the confining stress. The effective moduli of the intact material remain unchanged and determined by MT estimate. Using Eq. (11), the values of failure stress can be calculated for different cases. In Table 2, the values of failure stress for the case of triaxial compression with 5 MPa confining stress are given for different values of a. We can see that the effective elastic properties of the cracked material are not significantly affected by the homogenization scheme and do not vary significantly from the lower bound and MT estimate. As a consequence, the failure stress remains largely underestimated. 3.2. Influence of elastic properties of intact material According to the experimental results by Szczesniak et al. [1], the slopes of the initial linear parts of stress–strain curves in a triaxial compression test are functions of confining stress. Determine now the elastic properties of intact material from these slopes. Generally, it is found that the elastic modulus increases and Poison ratio decreases with confining stress. For the concrete studied in this work, the following values are obtained:

Rc ¼ 0;

E ¼ 23; 000 MPa;

m ¼ 0:2

Rc ¼ 5 MPa; E ¼ 25; 000 MPa; m ¼ 0:15 Rc ¼ 15 MPa; E ¼ 33; 000 MPa; m ¼ 0:05 It is interesting to note that the values of elastic modulus identified from experiment are very close to that estimated by the MT scheme. However, the experimentally determined values of Poisson ratio are notably different and lower than those calculated from the MT scheme. As the elastic properties of intact material are changed for the uniaxial compression test, the surface energy density should be evaluated according to (15) and we obtained ci = 0.831 Pa m. We can see that the value of surface energy evaluated in this way is quite sensitive to the elastic modulus of intact material due to the term of D1 in equation (15). Applying now the experimentally determined elastic properties to evaluate the failure stress in triaxial compression tests using (11). The evolutions of failure stress with inclusion size are shown in Figs. 5 and 6 for two values of confining stress. Compared with the results shown in Figs. 3 and 4, the effect of confining stress is better described by using the experimentally determined elastic properties of intact material. According to the relation (11), when the values of surface energy and elastic properties of cracked materials are given, the failure stress is only dependent on the elastic properties of intact materials. The results obtained above show that the mechanical strength of material is affected by both elastic modulus and Poisson ratio of intact material. However, comparing the experimentally determined elastic properties with those from MT estimation, the difference of Poisson’s ratio is larger than that of elastic modulus. The improvement of numerical results is largely attributed to the change of Poisson’s ratio. Moreover, it is worth noting that the evolution of initial elastic properties with confining stress is a common phenomenon for most geomaterials, mainly related to progressive closure of initial microcracks under confining stress. Finally, although improved, the predicted failure stress is still lower than the experimental value in triaxial compression tests.

Table 2 Influence of effective elastic properties of cracked material on failure stress in a triaxial compression test with a confining stress of Rc = 5 MPa.

a

ER (MPa)

mR

0.2 0.4 0.6 0.8 1

24666.0 24006.1 23344.7 22681.7 22,017

0.246 0.249 0.253 0.256 0.259

R1ex (MPa)

Failure stress R1 (MPa) 1 mm

2 mm

4 mm

6 mm

45.94 45.87 45.25 45.86 45.84 67.8

33.43 33.36 33.35 33.36 33.33 50.97

24.53 24.45 24.44 24.45 24.42 48.2

20.54 20.46 20.45 20.46 20.43 47.64

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80

triaxial compression 5MPa

failure strength (MPa)

70

test&Reuss test MT&Reuss

60 50 40 30 20 10

d (mm) 0

1

2

3

4

5

6

7

Fig. 5. Failure strength versus aggregate size in triaxial compression test with 5 MPa confining stress: comparison between theoretical predictions (blue line, elastic properties of intact material given by MT estimate; green line, elastic properties of intact material identified from experimental curves) and experimental data. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

triaxial compression 15MPa

90

test&Reuss test MT&Reuss

failure strength (MPa)

80 70 60 50 40 30 20

d (mm)

10 0

1

2

3

4

5

6

7

Fig. 6. Failure strength versus aggregate size in triaxial compression test with 15 MPa confining stress: comparison between theoretical predictions (blue line, elastic properties of intact material given by MT estimate; green line, elastic properties of intact material identified from experimental curves) and experimental data. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3.3. Influence of surface energy density The analysis above is based on the assumption that the value of surface energy density is independent of the confining stress and calculated from the experimental data in uniaxial compression test with the inclusion size of 1 mm. Assume now that the surface energy density ci may vary with the confining stress. It needs to estimate the value of ci for each confining stress. The methodology used here is as follows: for each value of confining stress, the value of ci is first evaluated from the test performed on the specimen with the inclusion size of 1 mm. The value obtained is then used to calculate the failure stress of triaxial compression test in specimens with other sizes of inclusions. The values of ci calculated by inverting the relation (11) for each confining stress are given in Table 3. One can see that the surface energy density increases with the confining stress. Using these values of surface energy density, the values of failure strength are calculated using (11) and compared with experimental data in Figs. 7 and 8. Note that in this calculation, the effective elastic properties of intact material are determined by the MT scheme and those of cracked material by the Reuss bound. The comparisons with the experimental data show that although the theoretical prediction is improved with respect to the reference case with constant surface energy density, the effect of inclusion size on mechanical strength is still not correctly described for triaxial compression tests.

Table 3 Values of surface energy density for each confinement.

ci (Pa m) Rc ¼ 5 MPa Rc ¼ 15 MPa

5.837 7.461

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C.F. Jin et al. / Engineering Fracture Mechanics 102 (2013) 207–217

80

triaxial compression 5MPa

failure strength (MPa)

70

MT&Reuss- γi 60

test

50 40 30 20

d (mm)

10 0

1

2

3

4

5

6

7

Fig. 7. Failure strength versus aggregate size in triaxial compression test with 5 MPa confining stress: comparison between theoretical prediction with variable surface energy density and experimental data.

triaxial compression 15MPa

90

MT&Reuss- γi test

failure strength (MPa)

80 70 60 50 40 30 20 10

d (mm) 0

1

2

3

4

5

6

7

Fig. 8. Failure strength versus aggregate size in triaxial compression test with 15 MPa confining stress: comparison between theoretical prediction with variable surface energy density and experimental data.

90

confinement effect

failure strength (MPa)

80 70 60 50

1mm 40

2mm 30

4mm

20

6mm

confinement (MPa) 10

0

5

10

15

Fig. 9. Experimental values of failure strength as functions of confining stress for four sizes of aggregates.

3.4. Further analysis of confining stress effect In order to further investigate the effect of confining pressure on the mechanical strength of concrete. The derivation of the failure stress with respect to confining stress is calculated from (11) and expressed as: @ R1 @ Rc

2Rc ð2D2 D1 ÞðD2 þD1 Þ ffi ¼ 2DD12 þ D11 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðRc Þ

2

12D c f ð2D2 D1 ÞðD2 þD1 Þ d1 i

1 D1 ¼ Eð1Þ  E1ð2Þ ;

ð1Þ

ð2Þ

D2 ¼ mEð1Þ  Emð2Þ

ð17Þ

216

C.F. Jin et al. / Engineering Fracture Mechanics 102 (2013) 207–217 Table 4 Values of elastic properties of intact material identified with the condition 2D2/D1 > 1.

Rc ¼ 5 MPa Rc ¼ 15 MPa

80

Eð1Þ ¼ 23; 020 MPa; Eð1Þ ¼ 24; 390 MPa;

mð1Þ ¼ 0:18 mð1Þ ¼ 0:18

triaxial compression 5MPa

failure strength (MPa)

70

k&Reuss test

60 50 40 30 20

d (mm) 0

1

2

3

4

5

6

7

Fig. 10. Failure strength versus aggregate size in triaxial compression test with 5 MPa confining stress: comparison between theoretical prediction using elastic properties verifying the condition 2D2/D1 > 1 and experimental data.

90

triaxial compression 15MPa k&Reuss

failure strength (MPa)

80

test 70 60 50 40

d (mm) 30

0

1

2

3

4

5

6

7

Fig. 11. Failure strength versus aggregate size in triaxial compression test with 15 MPa confining stress: comparison between theoretical prediction using elastic properties verifying the condition 2D2/D1 > 1 and experimental data.

According to this relation, if the ratio 2D2/D1 is close to unit, the dependency of failure stress on confining stress is essentially controlled by this ratio. In Fig. 9, we present the experimental data from Szczesniak et al. [1], concerning the evolution of failure stress with confining pressure. We can see that the average slope of experimental curves is higher than 1 for all confining stresses. Therefore, in order to correctly describe the effect of confining stress on the failures stress related to aggregate surface debonding, the elastic properties of intact and cracked materials have to verify the following condition:

2D2 2 ¼ D1



mð1Þ

ð2Þ

m

Eð1Þ Eð2Þ 1 1  Eð2Þ Eð1Þ

 >1

ð18Þ

Let the elastic properties of cracked material be determined by the Reuss lower bound with the reference values of Eð2Þ ¼ ER ¼ 22 GPa and m(2) = mR = 0.259, as given in the reference case. However, the effective elastic properties of intact material are evaluated in order to meet the condition given in (18). To do this, we have used an average value of Poisson’s ratio commonly used for this class of concrete and then determined the value of elastic modulus from the failure stress obtained from the sample with 1 mm aggregate for each confining stress. The new values of elastic properties of intact material are given in Table 4 for two values of confining stress. Using these values, the failure stress is determined from (11) for different cases. The theoretical values are compared with experimental data in Figs. 10 and 11. We can see that the theoretical prediction is in good agreement with the experimental data. The effects of both inclusion size and confining stress are correctly described by the theoretical model. The mechanical strength of concrete increases with confining stress but it decreases with the size of aggregates.

C.F. Jin et al. / Engineering Fracture Mechanics 102 (2013) 207–217

217

4. Conclusion In this paper, an energy-based theoretical model is proposed for the determination of failure stress in concrete material by taking into account the size effect of aggregates. The theoretical model is based on the comparison of elastic energy between the intact and cracked materials. The cracking mode is limited to interface debonding between aggregates and cement paste. It is explicitly demonstrated that the mechanical strength decreases when the aggregates size increase. This dependence is controlled by the difference of elastic properties between the intact and cracked material. Further, the influence of confining stress on mechanical strength is also considered in the theoretical model. According to parametric studies, the effect of confining stress on mechanical strength is strongly influenced by the difference of Poisson ratio between the intact and cracked materials. Finally, the cracking mode is largely simplified in the present work. 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