Numerical Study on Drying Shrinkage of Concrete Affectedby Aggregate SizeIppei Maruyama, Ai Sugie

Journal of Advanced Concrete Technology, volume ( ), pp.12 2014 279-288

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Journal of Advanced Concrete Technology Vol. 12, 279-288, August 2014 / Copyright © 2014 Japan Concrete Institute 279

Scientific paper

Numerical Study on Drying Shrinkage of Concrete Affected by Aggregate Size Ippei Maruyama1* and Ai Sugie2

Received 1 May 2014, accepted 16 August 2014 doi:10.3151/jact.12.279

Abstract This paper describes new insights on the influence of aggregate size on drying shrinkage of concrete through numerical simulation. Using a two-dimensional finite element method, it is demonstrated that it is impossible to reproduce the impact of aggregate size on concrete drying shrinkage by simply considering crack opening in the mortar matrix. From this result, it is deduced that the effect of aggregate size on concrete shrinkage is influenced by aggregate surface area. The application of a constitutive law that considers the characteristics of ITZ to coarse aggregate elements successfully re-produced the damage and shrinkage strain of concretes with different aggregate sizes. A certain range of virtual ITZ pore thickness in the model was found to allow simulation of the real phenomenon. It is concluded that ITZ characteristics should be taken into account when considering aggregate behavior in drying concrete.

1. Introduction

A variety of factors affect drying shrinkage of concrete, but it has become generally accepted in recent years that the properties and amount of aggregate have a major influence within the normal range of the water-cement ratio of mix design in Japan (Architectural Institute of Japan 2006). A high degree of correlation has been found between the drying shrinkage and specific surface area of aggregate in fourteen types of concrete using different aggregates, leading researchers to point out the impor-tance of aggregate properties (Imamoto and Arai 2008). Zhang et al. (2013) investigated drying shrinkage strain of mortar and concrete using various types of aggregate and reported that the type of aggregate has a large in-fluence on drying shrinkage. These findings underline the contemporary relevance of the findings of Hansen & Nielsen (1965).

Furthermore, the drying shrinkage strain of concrete in which limestone aggregate with low drying shrinkage strain is used has been reported to be small in general (Architectural Institute of Japan 2009), and use of such aggregate is one of the control measures recommended in shrinkage crack control guidelines in Japan (Architec-tural Institute of Japan 2009). However, limestone ag-gregate availability is limited in some production areas, and therefore, countermeasures based on the use of general aggregate are also needed in reality.

On the other hand, the drying shrinkage strain of concrete has also been reported to vary as a function of aggregate size (Carlson 1939). It is generally known that

mortar shrinks to a great extent, and that the shrinkage varies with the use of different sizes of uniformly sized aggregate (e.g. Maruyama and Sasano 2014). This could be applied as one of the countermeasures for control of concrete shrinkage. However, none of these reports dis-cuss the mechanisms involved. Hansen and Nielsen (1965) noted that such variations in shrinkage are caused by differences in the amount of paste, but did not discuss this in depth.

Concrete is considered to be a two-phase composite material consisting of cement paste and aggregate. The rules of mixture, which calculate the drying shrinkage strain of concrete from the stiffness and shrinkage driv-ing force in each phase, are often employed (Hansen and Nielsen 1965; Hobbs 1974; Eguchi and Teranishi 2005), but evaluation of the stress generated in the paste at that time shows that it clearly exceeds tensile strength, and thus the rules of mixture cannot be considered to allow realistic evaluation. In fact, as many experiments have shown, the amount of shrinkage of paste exceeds the amount of shrinkage of aggregate by more than ten times, and under drying, cracks are observed around aggregate (Bisschop and Mier 2002; Grassl et al. 2010; Hubert et al., 2003; Lagier et al. 2011; Maruyama and Sasano 2014). Such crack openings might be considered to con-tribute to the inhibition of concrete shrinkage.

This study investigates the influence of coarse aggre-gate size on drying shrinkage strain using a two-dimensional finite element method that allows con-sideration of cracking that occurs in the mortar matrix. First, calculation based on a model that does not take into account the interfacial transition zone (ITZ) shows that drying shrinkage strain does not depend on aggregate size, even when cracking is taken into account. Next, a constitutive law that takes into account the ITZ is intro-duced, and the behavior of aggregate in concrete is evaluated by considering the influence of the thickness of the ITZ.

1Assoc. Prof., Graduate School of Environmental StudiesNagoya University, Nagoya, Japan. *Corresponding author, E-mail: ippei@dali.nuac.nagoya-u.ac.jp 2Graduate student, Graduate School of Environmental Studies, Nagoya University, Nagoya, Japan.

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2. Analytical method

Non-linear two-dimensional finite element analysis us-ing 4-point isoparametric elements was employed in this study. The coarse aggregate element was assumed to be an elastic element in which cracking does not occur. On the other hand, for the mortar element, a crack band model was employed to allow consideration of cracks, as comparison of the crack development path with past experiment results would provide an additional verifica-tion method. Further, in the crack band model, a single crack was assumed to occur within the element, and the model was normalized so as to convert the energy con-sumed at the fracture surface into energy consumed per element volume.

With regard to the tensile stress softening characteris-tic of the mortar element, the 1/4 model was adopted (Rokugo et al. 2011). In addition, the shear stiffness after cracking was taken into account by multiplying the healthy shear stiffness by a reduction factor. The fol-lowing equation using the strain after cracking as a variable was used to obtain the shear stiffness reduction factor (Walraven and Keuser 1987).

1/1 4447cr

cr

G Gβε

= =+

(1)

where β = shear stiffness reduction factor (-), G = shear stiffness coefficient (MPa), crG = shear stiffness coefficient after cracking (MPa), and crε = strain per-pendicular to the crack (-).

Table 1 lists the material parameters used in the analysis. The experimental values of limestone aggregate were adopted for Young's modulus and Poisson's ratio of the coarse aggregate element (Horiguchi et al. 2011a). The experimental results of Young's modulus, tensile strength (Maruyama et al. 2014) and fracture energy (Miura and Sato 2010) of cement paste specimen by the three point bending test are used for the parameters of mortar element. For Poisson's ratio, the cement paste value obtained through ultrasonic testing was used (Horiguchi et al. 2011b). The reason for using the ex-perimental values of cement paste as the mortar element parameters was that the finite elements under considera-tion were assumed to consist of very small elements with sides measuring between 0.5 mm and 1.0 mm, and it would not be appropriate for the analysis to use the fracture energy and tensile strength of mortar in which multiple cracks are actually distributed through the me-diation of aggregate at the fracture. In addition, the

bending strength of paste, whose equilibrium at 90 to 80% RH in room temperature (Maruyama et al. 2014) was almost the same value as that of mortar (Miura and Sato 2010). Therefore, the used strength in the calcula-tion is considered to be realistic. Young’s modulus and Poisson’s ratio are not sensitive parameters in this cal-culation because the shrinkage input of mortar is much larger than that of aggregate, and as a result cracks form under very small shrinkage in mortar and after that, crack openings expand continuously.

The creep behavior of mortar was not considered in the present study. In addition, almost all the elements in the calculation show the tension side, barring out ob-servation of the compressive side.

Drying shrinkage strain was introduced for all mortar elements as equivalent nodal force. From the literature (Maruyama and Sasano 2014), the drying shrinkage strain input into the mortar element was assumed to be the amount of drying shrinkage strain (-1300 μ) when the mortar specimens approximate the constant mass state under drying at 20C° 60%RH. Shrinkage was not con-sidered for the coarse aggregate element because in pre-liminary studies crushed limestone, which is the coarse aggregate used in the concrete of the literature (Maru-yama and Sasano 2014), exhibited drying shrinkage strain of approximately -30 μ during drying from the water saturated state to 20C° 60%RH, a sufficiently small value in relation to the drying shrinkage strain of the mortar specimens.

A definition of the term crack is called for here. In general, cracks in concrete are defined as stress-induced discontinuity areas in a concrete body. However, damage in concrete is not localized in a part and there should be plural and numerous scales of cracks. In the following chapters, some numerical analysis results are compared with existing experimental data, for which shrink-age-induced cracking in concrete was confirmed by the fluorescent epoxy impregnation method (Maruyama and Sasano 2014). With regards to experimental data, dam-age in concrete due to shrinkage of cement paste was confirmed, and the minimum damage was 0.03 x 0.03 mm2 in concrete section. This is defined as a crack caused by the cement paste matrix. On the other hand, FE analysis always deals with a continuous body, and nor-mally, elements whose Gaussian integration point is in the tension softening zone are considered to be damaged elements and composing a part of the crack path. Therefore, for evaluating the FE analysis results, the localized strain due to tension softening is considered as a crack.

3. Single particle size model analysis not considering ITZ

3.1. Analysis object model Figure 1 shows single aggregate particle size models. In each circle in this figure, the color black represents the coarse aggregate element, and the color white the mortar

Table 1 Physical properties used in FEM. Young’ modulus (GPa) 71.3 Aggregate

Poisson’s ratio (-) 0.30 Young’s modulus (GPa) 12.1

Poisson’s ratio (-) 0.26 Tensile strength (MPa) 10.2

Mortar

Fracture energy (N/m) 15.0

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element. From the left, the coarse aggregate particle size is S (5 mm coarse aggregate particle size), M (10 mm coarse aggregate particle size), and L (20 mm coarse aggregate particle size). Coarse aggregate that is circular in shape and of uniform particle size is simulated, and the total area (aggregate ratio) of the aggregate element to the concrete cross section area is 52.9% for all three models. The element division is 100 elements in the diametrical direction (1 element = 1 × 1 mm) and the total number of elements is 7860.

The assessment of the drying shrinkage strain of con-crete was done by averaging the values obtained by di-viding the X-direction displacement difference of X1 and X2 and the Y-direction displacement difference of Y1 and Y2 by the concrete diameter (100 mm).

3.2. Analysis results Figure 2 compares the drying shrinkage strains obtained from the analysis of concretes using different aggregate particle sizes and shows the results of analysis done by greatly varying the fracture energy (Gf) (Gf = 1.0 N/m). To begin with, there is almost no difference in drying shrinkage strain based on particle size. This result par-allels that of analysis done by greatly varying the fracture energy, showing that fracture energy has only a small influence on drying shrinkage strain.

To verify the dependence (on model) of the distance ratio of coarse aggregate in the cross section on the po-sitional relationship of aggregate, analysis was con-ducted using a similarity model where the element size of Model L in Fig. 1 multiplied by 0.5 is coarse aggregate particle size M, and that multiplied by 0.25 is coarse aggregate particle size S. The results of this analysis showed little difference in drying shrinkage strain, similarly to the results in Fig. 2.

Figure 3 shows the relation between the damage index and drying shrinkage strain. The damage index is the degree of damage arising in the cross section of the concrete averaged for the total cross section, as ex-pressed by Eq. (2).

. 0i i ii

ii

AD

Aε

ε ε ≥= ∑

∑ (2)

where Dε = damage index (μ), iε = maximum princi-pal strain (μ), and iA = effective area of measurement point i (mm2).

In this analysis, the damage index was calculated by integrating the values indicating strain of 0 μ or more among the maximum principal strain at all the Gaussian integration points, and then dividing the result by the total number of strain output points (total number of Gaussian integration points). Figure 3 shows that the coarse aggregate of large particle size has a damage index more than twice that of coarse aggregate of small particle size. This is attributed to the fact that the larger the particle size of coarse aggregate, the more the ag-gregate restrains shrinkage of the mortar, causing greater

crack width in the mortar. This is to be expected, con-sidering the similarity of shapes, including those of cracks, which means that the crack width of each crack increases, resulting in a greater degree of damage. While coarse aggregate particle size was observed to influence the degree of damage that occurs within the concrete cross section, similarly to the results of previous ex-periments, no influence on drying shrinkage strain was

Fig. 1 Schematics of models with different aggregate size. (Black spots and white spots represent aggregate element and mortar element, respectively.)

Fig. 2 Calculated drying shrinkage of concrete with dif-ferent aggregate size. The effect of ITZ is not considered in FE analysis.(S = 5 mm, M = 10 mm, and L = 20 mm in aggregate size.)

Fig. 3 Relationship between drying shrinkage of aggre-gate and damage index of concrete with different ag-gregate size. The effect of ITZ is not considered in FE analysis. (S = 5 mm, M = 10 mm, and L = 20 mm in ag-gregate size.)

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confirmed. This result indicates that the influence of coarse aggregate particle size on the drying shrinkage strain of concrete cannot be properly evaluated without considering the cracks that occur in the cross section.

The results of this numerical analysis lead to a dif-ferent interpretation with regard to the relation between the damage index and the amount of shrinkage previ-ously confirmed in the literature (Maruyama and Sasano 2014). In other words, the effect of cracks caused by drying shrinkage in terms of reducing the amount of shrinkage of concrete is considered to be extremely lim-ited. Consequently, the effect of aggregate size on con-crete shrinkage is produced by some other factor, most likely the surface area, which is a function of the volume and size of the aggregate. This constitutes the most im-portant conclusion in the present study.

As the necessity of examining the phenomena that depend on the particle size of coarse aggregate, which differ from those that depend on cracks, in order to re-produce the influence of coarse aggregate particle size on drying shrinkage strain, has been established, the authors decided to investigate ITZ as a new influencing factor.

4. Model analysis considering ITZ

The ITZ is the region between the aggregate and the cement paste matrix. Compared with cement paste, it is porous and of low stiffness and strength (Diamond 1992a, 1992b; Oliver et al. 1995; Sonebi 2006; Erdem et al. 2012). In general use, the term ITZ includes both the pores on the aggregate surface and the coarser paste or mortar matrix around the aggregate rather than the matrix located far from the surface of aggregate. ITZ is consid-ered to be created by the wall effect of the packing process of unhydrated cement particles and the water membrane at the surface of the aggregate. Even though many researchers have pointed out the existence of ITZ, there is no broadly agreed upon definition of ITZ. In addition, research on ITZ with regards to concrete shrinkage is quite scarce. Given the presence of pores at the surface of aggregate (Scrivner and Nemati 1996), there is a dramatic increase in porosity in the vicinity of the aggregate surface (Scrivner and Gartner 1988), and the local cement paste is of higher W/C ratio (Meso 1980). It is consequently deduced that the locally higher W/C cement may shrink much under the drying process, and that the pores at the surface of the aggregate will not restrain this shrinkage.

The objective is to prove that it is possible to simulate the concrete shrinkage strain and cracking pattern in the section of concrete at the same time when the proper assumption is used for ITZ modeling, rather than pro-posing a numerical constitutive law for the physical properties of ITZ. When the numerical modeling of the behavior of the interfacial zone is focused on, many kinds of treatment or modelling approaches are applica-ble (Grondin and Matallah 2014), and due to the scanti-ness of experimental data on the physical properties of

ITZ and related phenomena, such modelling warrants further studies. Even though it is quite natural that con-crete shrinkage increases when the volume of ITZ, whose shrinkage is greater than the other matrix, is in-creased in the numerical calculation, the following dis-cussion to the effect that there is a certain applicable scope for ITZ modelling may contribute to the under-standing of ITZ behavior.

4.1 Coarse aggregate element constitutive law considering ITZ Based on the experimental facts mentioned above, it is assumed that as the mortar around aggregate shrinks and the pores at the surface of aggregate are flattened and the mortar deformation reaches a certain distance, stress transmission between mortar and aggregate is produced and the restraining effect of the aggregate comes into being.

It was decided to incorporate a constitutive law that takes into account the influence of the ITZ on the coarse aggregate element in this study.

As the ITZ is more porous the closer it is to the ag-gregate, Young's modulus in the ITZ is distributed along a gradient that depends on the distance of the ITZ from the aggregate (e.g. Erden et al. 2012). Figure 4 shows this distribution schematically. In this study, it was de-cided to simplify this distribution and simulate the be-havior around the ITZ by using a constitutive law com-

Fig. 4 Schematic of Young’s modulus distribution from aggregate surface and applied model in FE analysis.

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bining void areas where volume can freely change and solid areas that exhibit the stiffness of aggregate (Fig. 4). In other words, the porous parts in ITZ rather than the mortar matrix are gathered into the pores, which are named “virtual ITZ pore”, at the surface of aggregate, and these pores are modeled as part of the aggregate with a given thickness of virtual ITZ pores, which is named the “virtual ITZ pore thickness”. A conceptual diagram of that constitutive rule is shown in Figure 5. The ag-gregate element having an ITZ pore (Element for Ag-gregate with interfacial transition zone EA-ITZ) has two aspects, namely when mortar and aggregate are in con-tact, and when they are not in contact. The strain of the EA-ITZ element when the ITZ pores are absent and the aggregate and mortar are in contact with each other is termed threshold strain εcr. εcr can be expressed by the following equation, which uses the ITZ pore thickness and coarse aggregate particle size.

3×10ITZcr

aggregate

hd

ε −= (3)

where crε = strain when mortar and coarse aggregate are in contact (μ)， ITZh = virtual ITZ pore thickness (representative pore which is equivalent to the weakness of ITZ) (μm), and aggregated = coarse aggregate particle size (mm)

Assuming that when the element strain exceeds this crε strain (Fig. 5 Zone B), the influence of the ITZ

becomes pronounced and the stiffness of the EA-ITZ is low, 1/1000 was used as the value of Young's modulus of coarse aggregate. Conversely, for the case when the element strain is smaller than the crε strain (Fig. 5 Zone A), the same constitutive rule as that of the coarse ag-gregate element was used for the coarse aggregate ele-ment considering ITZ.

The strength of ITZ is not considered in the present study. Even though it is not considered, the stress in mortar next to the coarse aggregate becomes quite large and shortly the mortar shows cracking. Therefore, the crack along the ITZ is reproduced due to localization of

the shrinkage-induced stress in the mortar.

4.2. Investigation of virtual ITZ pore thickness in coarse aggregate constitutive law Figure 6 shows the results of the analysis of the damage index and drying shrinkage strain based on the single particle size (S, M, L) model when the virtual ITZ pore thickness is changed. This figure shows that for all par-ticle sizes, the greater the virtual ITZ pore thickness is, the smaller the damage index and the greater the drying shrinkage strain become. This can be said to be a natural result as the greater the virtual ITZ pore thickness, the larger the amount of the shrinking strain aggre-gate-bound mortar. Furthermore, cracks occur within the concrete cross section as a result, which reduces the damage index.

Moreover, depending on the ITZ pore thickness, cracking does not occur in the concrete section and the damage index value is 0 μ, as seen in the case of the virtual ITZ pore thicknesses of 4 μm and 5 μm for coarse aggregate of particle size S. Given that this differs from the actual phenomena, these virtual ITZ pore thickness values are unrealistic.

For the combinations of aggregate and mortar ele-ments in this study, the virtual ITZ pore thickness that satisfies simultaneously the macroscopic drying shrink-age strain, the crack initiation state in the concrete, and the influences of virtual ITZ pore thickness on drying shrinkage strain of concrete is considered to be in the range of 1 μm to 3 μm.

5. Cross-sectional analysis of the concrete specimens

To confirm the validity of the present analysis, a com-parison of the analytical values and experimental values was carried out, using specimens subjected to drying shrinkage strain measurement and fluorescent epoxy

Fig. 5 Constitutive law of element containing ITZ.

Fig. 6 Relationship between damage index and drying shrinkage of concrete with different characteristic ITZ thickness. The effect of ITZ is considered in FE analysis. (S = 5 mm, M = 10 mm, and L = 20 mm in aggregate size.)

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impregnation tests as the analysis objects reported in the literature (Maruyama and Sasano 2014). 5.1. Analytical model The analysis object was the cross section of specimens fabricated by casting concrete in φ100 × 200 molds, allowing it to become fully hydrated, and then cutting it into specimens measuring φ100 × 9 mm. For the coarse aggregate, limestone aggregate of particle sizes of 5 to 10 mm (specimen name S-ex), 10 to 15 mm (specimen name M-ex), and 15 to 20 mm (specimen name L-ex) was used. Photos of the respective aggregates are shown in Fig. 7. The material parameters listed in Table 1 were used for the coarse aggregate element and the mortar element. Figure 8 shows the particle size distribution of the limestone coarse aggregate in the specimens used in the experiment. As the coarse aggregate particle size used for the constitutive law of coarse aggregate, the value of the coarse aggregate particle size whose cumulative per-centage is 50% in Fig. 8 was used as the average particle size of the respective specimens (S: 8.7 mm, M: 14.3 mm, L: 19.7 mm).

Moreover, since viscosity agent was added to the concrete for analysis to minimize segregation as much as possible, the influence of coarse aggregate particle size on the ITZ was considered to be minimal, and the virtual ITZ pore thickness was set to the constant of 3 μm re-gardless of the coarse aggregate particle size.

The number of element divisions was set as 200 ele-ments in the specimen diameter direction (one element

measuring 0.5 mm × 0.5 mm), and the total number of elements as 31420 elements. Figure 9 shows an outline of the element division. The coarse aggregate positions were determined from a photo of the section taken before the start of drying (Fig. 9) and material properties were assigned as coarse aggregate elements and mortar ele-ments. At that time, if the ratio of coarse aggregate oc-cupied by one element was under 50%, that element was not classified as a coarse aggregate element and instead it was regarded as a mortar element. Then, the aggregate amount was adjusted to achieve the same aggregate proportion for all the specimen models, using 41.4% as the volume fraction of aggregate.

5.2. Analysis results Figure 10 shows the drying shrinkage strain affected by aggregate particle size, and Fig. 11 the relationship be-tween the damage index and drying shrinkage in speci-men analysis. As a result, the extent of the influence of coarse aggregate particle size on drying shrinkage strain was reproduced. In addition, the damage index and dry-ing shrinkage strain results show similar tendencies to those of previous studies.

Figure 12 presents crack patterns obtained from the fluorescent epoxy impregnation test and crack patterns obtained by analysis for each various coarse aggregate particle size. Regardless of the particle size, cracks around the aggregate and cracks connecting aggregates to each other are nicely reproduced in the crack patterns, and there are no major discrepancies between the ex-perimental and numerical analysis crack patterns. In

Aggregate element

Mortar element

Fig. 9 Meshing process and material properties assign-ment process.

Dry

ing

shrin

kage

(µ)

Analysis

S-ex M-ex L-ex

-1000

-800

-600

-400

-200

0

Exp.

Fig. 10 Comparison of calculated drying shrinkage of S-ex, M-ex, and L-ex with those of experimental results.

Fig. 7 Sections of samples of concrete with different ag-gregate size.

Fig. 8 Particle size distribution of used aggregate in S-ex, M-ex, and L-ex (Maruyama and Sasano 2014).

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addition, in Fig. 13, which shows a close up of L-ex pattern in Fig. 12, the positions of the cracks are in good agreement. This numerical analysis was able to repro-duce the shape of cracks caused by the physical interac-tion of aggregate and mortar, as well as the influence of coarse aggregate particle size on shrinkage strain asso-ciated with the drying of concrete.

6. Discussion

Based on the analysis results, it is inferred that in terms of the influence of aggregate size on concrete shrinkage, the shrinkage that occurs in the ITZ at the aggregate-mortar interface is the dominant factor. Although the larger the aggregate, the greater the crack opening tends to be, because localization of the cracks in the mortar caused by the restraining effect of the aggregate is determined by aggregate particle size, the accumulation of crack open-ings in the concrete does not much to compensate for the macroscopic concrete shrinkage amount.

This may be only natural as the mortar's shrinkage driving force works as the macroscopic concrete shrinkage driving force when no cracks connecting grains of aggregate occur, but after cracking occurs, only crack expansion occurs, and this does not contribute to the macroscopic shrinkage behavior of concrete.

Hansen and Nielsen (1965) pointed out that although cracks occur around the aggregate (Carlson 1939) when paste shrinks greatly, which departs from the rules of mixture, the fractional volume ratio of the paste, the aggregate amount, and shrinkage strain remain the major factors that influence the drying shrinkage of concrete. Further, Hansen and Nielsen indicated also that the rea-son for the large shrinkage of mortar is that the volume fraction of paste in mortar is greater than that of paste in concrete. This may be why the issue of aggregate size dependence failed to be taken up for such a long time.

However, as evidenced by the experiments of Carlson and the authors (Maruyama and Sasano 2014), differ-ences in shrinkage amount when the amount of aggregate is held constant are clearly apparent.

In considering rational mix design taking into account

the association between drying shrinkage strain of con-crete and mix proportions, the issue of aggregate size dependence cannot be ignored.

The FEM without ITZ model showed that aggregate size did not affect the shrinkage strain of concrete. This makes sense because the meshes for FEM have similarity in geometric shape. On the other hand, in reality, ag-gregate size has an impact. Therefore, the interface of the aggregate becomes an issue.

The numerical analysis result that shrinkage of con-crete is increased as the number of shrinking aggregate interfaces is increased is quite obvious. However, the virtual voids that influence shrinkage at this interface, though they are extremely small in amount, perform

Fig. 11 Relationship between damage index and drying shrinkage in FE analysis.

Exp. Calc.

L-ex

M-ex

S-ex

Fig. 12 Comparison of crack patterns of L-ex, M-ex, and S-ex evaluated by FEM with experimental results.

Fig. 13 Close-up of L-ex crack pattern (left: experiment, right: FEM).

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significant work, and the fact that an appropriate region exists in the thickness of these voids has been revealed through correspondence with crack initiation.

To consider this effect in a previous study, the authors attempted to verify the ITZ in concrete in a wet state through the fluorescent epoxy impregnation method, but were unable to inject fluorescent epoxy resin in the ITZ (Maruyama and Sasano 2014). At that time, the areas that can be considered as voids or in a very coarse state were shown to be 12 μm or less in size based on the relation between the surface tension of the epoxy resin and the negative pressure of the vacuum created for impregna-tion purposes.

A virtual ITZ pore thickness of 3 μm represents a strain value of as much as 1500 μ for 20 mm aggregate. Given that the average particle size of cement particles is on the order of 10 μm, gaps on that order are considered to easily occur, even taking into account the wall effect. The 3 μm thickness of the virtual ITZ under discussion here varies depending on the physical property values applied to the FE analysis, such as Young's modulus of the mortar part and the shrinkage driving force as equivalent nodal forces. Young's modulus of the mortar here is assumed to be the same as Young's modulus of the cement paste itself. However, even though Young's modulus of the paste and that of the mortar might differ, that difference does not result in a 200% change in the thickness of the virtual ITZ, and the variation in thickness is thus ex-pected to fall within the range of 50% to 200%. Based on the above, a virtual void of 3 μm may be adopted as realistic for the purpose of this discussion.

In addition to the above, the distribution of ITZ from the aggregate surface is said to be about 10 to 70 μm25)~29), and it has been pointed out that Young's modulus of the ITZ is between 10% and 50% smaller than that of bulk material. Considering the Young's modulus reduction to be 20% in ITZ, which is assumed to be within 50 μm from the aggregate surface owing to the presence of voids, virtual ITZ pore thickness is calculated as 10 μm. In this light, the proposed value is plausible.

Next, the Young’s modulus of concrete is affected by cracking in concrete due to drying. Recently, it was proved that there is a relationship between the damage index and Young’s modulus reduction, and concrete with larger aggregate size shows a dramatic reduction of Young’s modulus due to drying (Maruyama et al. 2014). These experimental evidences also support the validation of the present analysis.

Finally, let us discuss the mix proportions of concrete. The results of this study suggest that changing the parti-cle distribution so as to reduce the specific surface of aggregate is effective for reducing the drying shrinkage strain of concrete. This was partially confirmed in the Mix and L specimens in the experiments of Maruyama and Sasano (2014).

On the other hand, changing the particle size distribu-tion of aggregate for the purpose of reducing its specific area may cause a decrease in the actual solid content of

aggregate and lead to a reduction in flow properties. Under constant aggregate volume, the flowability and shrinkage of concrete may be in a trade-off relationship. This hypothesis is consistent with the previously ad-vanced hypothesis that the amount of cement paste that contributes to concrete shrinkage is the excess paste (Maruyama and Sasano 2014).

7. Concluding remarks

This study examined the influence of coarse aggregate size on drying shrinkage strain using a two-dimensional finite element method that allows consideration of cracks that occur in the mortar matrix. Furthermore, a constitu-tive law that considers the ITZ was introduced to discuss its influence. The findings from this study are listed below. (1) No influence of coarse aggregate particle size on the

drying shrinkage strain of concrete could be con-firmed by analysis considering cracks using a two-phase model of the coarse aggregate element and mortar element, for concrete sections using rounded coarse aggregate of S (5 mm), M (10 mm), and L (20 mm) particle sizes and holding the area of the aggregate element constant in relation to the cross-sectional area of the concrete. Comparing the damage indices, the extent of damage was found to be L > M > S in order of increasing damage. It be-came apparent that the influence of coarse aggregate particle size on drying shrinkage strain cannot be reproduced if only cracks are taken into considera-tion.

(2) Single particle size model analysis using a constitu-tive law that takes into account the ITZ revealed the existence of a specific parameter that allows the re-production of both the extent of damage dependent on aggregate particle size and the drying shrinkage strain. That parameter is based also on the hypothesis that there are voids that shrink without resistance in the ITZ.

(3) The specimens fabricated in the experiment were modeled and analysis incorporating a constitutive law taking into account the ITZ was conducted, yielding the same occurrence tendencies of crack patterns as those observed in fluorescent epoxy im-pregnation tests. The authors were also able to suc-cessfully reproduce the tendency of influence of coarse aggregate particle size on drying shrinkage strain of concrete.

(4) This analytical study indicates that the shrinkage of concrete is influenced by the ITZ, and therefore, that it may be possible to reduce the amount of shrinkage by adjusting the particle size distribution so as to reduce the amount of ITZ.

Acknowledgments Some parts of this study were conducted as part of the framework of the “Japan Ageing Management Program

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