# A Trans-Scale Young’ Modulus Calculation Model of ITZ Based on Void Shape Randomness and Calcium Hydroxide Enrichment

Authors: Junqing Liu; Fan Zuo; Chao Liu
Journal: Advances in Materials Science and Engineering
 (2020)

---

## Abstract

The randomness of void shape and enrichment of calcium hydroxide are significant in interfacial transition zone (ITZ) of concrete; however, current theoretical models of ITZ do not include these features. In this article, ITZ was regarded as a three-phase composite material, and the pore morphological parameters were defined according to the characteristics of microscopic pores, and the corresponding random distribution function was constructed. The calcium hydroxide enrichment factor was introduced, and a cross-scale ITZ Young’s modulus calculation model was established in combination with the Mori–Tanaka method. The reliability of the proposed model in this paper was verified through comparison to experimental results in a reference.


---

## Body

## 1. Introduction

As the weak area between aggregate and bulk paste in concrete, the interfacial transition zone (ITZ) influences mechanical properties of concrete, including Young’s modulus, strength, and cracking behavior significantly [1–7]. Concrete was treated as a two-phase composite composed of aggregate and bulk paste in early studies, but study on limestone aggregate concrete conducted by Nilsen and Monterio [8] revealed that the tested Young’s modulus values were lower than Hashin–Shtrikman lower limit under two-phase assumption. Simeonov and Ahamd [5] observed increasing deviation between test results and Hashin–Shtrikman limit under two-phase assumption with the increasing of aggregate volume fraction.ITZ was began to be taken into consideration [3] to solve the problem of large difference between the model-calculated results and experimental results. Analysis of Chun’an et al. [9] showed that the properties of transition zone between inclusions and the matrix influence the stress field inside the composite, which then influences damage location and failure mode consequently. Studies of Liu et al. [10, 11] revealed the impact of ITZ on long-term deformation and bond behavior of recycled concrete. Three-phase models considering ITZ were established by Lee and Park [1], Zhao and Chen [12], and Nadeau [13] et al., and reliable results were obtained. The failure mechanism of concrete under high strain rate with a three-phase model was studied by Zhou and Hao [7], and this study showed that it influences the failure mode, failure time, and tensile strength remarkably. Berger [14] held that the wall effect led to higher porosity of cement paste on aggregate surface, which created a suitable circumstance for migration of water and ions and resulted in enrichment of calcium hydroxide and ettringite on aggregate surface. Simeonov and Ahamd [5] et al. pointed out the unsteadiness of ITZ properties along direction perpendicular to aggregate surface.Experimental and theoretical methods were used to study many aspects of ITZ. The ITZ nanoindentation tests of gravel concrete and limestone concrete by Mondal [15] revealed that Young’s modulus of ITZ has a gradient of change in the direction perpendicular to the surface of the aggregate, and the test results at different measuring points have large dispersion. Other experimental results on ITZ in self-compacting concrete [16], rock-filled concrete [17], nanosilica modified concrete [18], and sugarcane industrial ash modified concrete [19] also showed similar characteristics. X-ray diffraction (XRD) was used by Hagis [20] to study the microstructure of cement hydration products, and the result showed that the microstructure and pore characteristics of calcium hydroxide, ettringite, and hydrated calcium silicate gels are significantly different. Mercury intrusion test was used by Liu et al. [21] to study the influence of porosity on water and ion migration in ITZ. Scanning electron microscopy (SEM) was used by Barnes et al. [22] to study the evolution of the microstructure of the glass-cement mortar interface transition zone. The result showed that calcium hydroxide enrichment contributes greatly to the formation of ITZ high porosity. Sun et al. investigated the effects of aggregate surface by coating slag and silica fume on the ITZ microstructure [23] and studied the transport behavior of sulfate ions in concrete-containing ITZ [24]. Their studies showed that the ettringite formation inside the concrete pores influences the elastic deformation behavior of concrete significantly [25].The most common theoretical method for calculating ITZ Young’s modulus is the discount coefficient method: a discount coefficient is imposed on Young’s modulus of cement paste, and the discounted result is used as the corresponding parameter of ITZ [7, 12, 13]. Luts and Zimmerman [26] believed that ITZ Young’s modulus is related to aggregate size, distance from aggregate surface, and modulus of bulk paste. They assumed that ITZ Young’s modulus can be calculated by a power-law function. Nadeau [27] proposed a theoretical calculation method of ITZ Young’s modulus, which comprehensively considered the influence of water-cement ratio, aggregate volume fraction, cement specific gravity, ITZ thickness, and other factors. However, so far, there is still no ITZ Young’s modulus calculation model that can essentially consider the void shape randomness and calcium hydroxide enrichment caused by the characteristics of ITZ components and reflect the variation along direction perpendicular to aggregate surface and the discreteness among results of different measured points.A trans-scale calculation model of ITZ Young’s modulus is proposed in this paper. In this model, ITZ is adopted to three-phase composite, void shape parameter and calcium hydroxide enrichment parameter are defined, and then ITZ Young’s modulus could be calculated with the Mori–Tanaka method. The calculation model comprehensively considers the randomness of void shape and the degree of calcium hydroxide enrichment caused by the characteristics of ITZ components. It has the advantages of covering the upper and lower limits of Young’s modulus, reflecting the gradient of Young’s modulus change and small calculation. By comparison with the test results in a reference, the validity of the calculation model is verified. This model could calculate ITZ Young’s modulus reliably and quickly and provides a theoretical basis for the calculation of concrete mechanical properties.


## 2. Three-Phase Composite Model of ITZ

ITZ is a thin layer caused by the wall effect occurred between the aggregate and the bulk paste [14]. The resistance of aggregate on water permeation and cement accumulation lead to higher water-cement ratio and porosity compared with cement paste; meanwhile, the enrichment of calcium hydroxide crystal is rather notable. Thus, ITZ could be considered as a layer of near-field cement paste with high water-cement ratio, high porosity, and enriched calcium hydroxide crystal, and the water-cement ratio, porosity, and enrichment of calcium hydroxide reduce with the distance to the aggregate increasing.The major components of ITZ remain consistent with the cement paste [15], which include hydrated calcium silicate gel (CSH, which contains low-density CSH and high-density CSH, contracts to LD-CSH and HD-CSH, respectively), calcium hydroxide (CH), and ettringite. However, the volume fractions of components in ITZ are different with those in cement paste, considering the enrichment of CH. The microcrystal shape and pore features of ITZ components differ largely through electron microscopy [20, 28, 29], as is shown in Figure 1.Figure 1
Micromorphology of components in ITZ [20, 28, 29]. (a) Micromorphology of ITZ [28]. (b) Micromorphology of CSH [20]. (c) Micromorphology of CH [29]. (d) Micromorphology of ettringite [20].
(a)(b)(c)(d)The macromechanical properties of cement paste in far or near-field are closely related to micromechanical properties and volume fraction of components, while those of components are closely related to corresponding microstructures and pore features. The micro- and macrofeatures of components should be taken into consideration in the calculation of ITZ mechanical properties, and a trans-scale model could be established.Cement paste was treated as a four-phase composite in previous studies [30]: micropores, LD-CSH, HD-CSH, and ettringite. However, it is necessary to take the differences between ITZ and cement paste into account: pores stem from the special crystal structures of CH and ettringite rather than air holes; the volume fraction of micropores is not equal to the absolute porosity of bulk paste, which leads to the difference on mechanical properties and volume fractions between ITZ components and bulk paste components.In addition, phases of ITZ grow and stagger throughout the hydration of cement. Influence of local pores leads to local randomness of property in each phase, following the local randomness of phases leads to general randomness of composite. More attention on local randomness led by pores is necessary in trans-scale modeling.This paper presents a trans-scale ITZ model used for Young’s modulus calculation, which takes the randomness of pores and phases into account. This model adopted a three-phase model, which consists of LD-CSH, HD-CSH, and CH, voids with shape randomness distributed in three phases. The assumption of continuously uniform media was applied into cement paste, and the relative porosity of ITZ to cement paste was set as the porosity of phases; methods of composite materials were used in mechanical property calculation of phases.The following 3 steps were needed in trans-scale modeling:(1)
In the microscale level1st step: it was assumed that all the pores distribute in the phases continuously. Each phase was regarded as an accumulation of representative volume element (RVE) with an independent void in the center, and the porosity of each RVE was consistent with others. The shape of voids corresponded with some distribution, the shape parameter was defined, and corresponding distribution functions were constructed so that the features of void shape and distribution were described. The method of composite material was used to gain the mechanical properties of each RVE.2nd step: each RVE was treated as a part of a certain phase, and thus, each phase could be regarded as a combination of n-RVEs, that is, an n-phase composite. The mechanical properties of each phase were obtained through the composite material method.(2)
In the macroscale level3rd step: ITZ was considered as a 3-phase composite consisted of LD-CSH, HD-CSH, and CH, and the macro Young’s modulus of ITZ was obtained.The modeling of the trans-scale model is shown in Figure2.Figure 2
Trans-scale modeling. (a) Trans-scale analyzing procedure. (b) Trans-scale modeling procedure.
(a)(b)


## 3. Trans-Scale Model of Young’s Modulus in ITZ

### 3.1. Equivalence of Microvoid Shape and the Shape Parameter

An accurate description on void features of phases in ITZ is necessary for calculation. Voids could be treated as special inclusions with zero modulus. The inclusion with a shape of rotating ellipsoid, in which 2 in 3 axes are of equal length, is a kind of inclusions with excellent universal adaptability. Various kinds of inclusions, such as spherical, ellipsoidal, penny-shaped, and cylindrical inclusions [31], are the generalization of rotating ellipsoidal inclusion. Considering the complexity of void shape in ITZ, this paper assumed that all the voids in ITZ phases are in the shape of rotating ellipsoid.The phases in cement paste appear significantly different according to the SEM images: the crystal polymer of CSH is dense and uniform, and that of CH appears like a peony, while that of ettringite is needle-like, which is easy to form an extremely loose structure. Based on the shape and shape randomness of voids in phases, it is a reasonable assumption that the RVEs corresponding to CSH contain mainly spherical voids, CH contains mainly rotating ellipsoidal voids, and ettringite are treated as a pure void without any matrix. Considering the staggering of all the phases, it is acceptable to put the “ettringite voids” into other RVEs as part of porosity instead of an independent phase, as is shown in Figure3.Figure 3
Equivalence of pores in phases of cement paste.A typical RVE was set to be a cube with 2L in side length, containing a rotating ellipsoidal void witha1,a2,anda3 in 3 axis length, respectively, where a2=a3. The void shape parameter α was defined as(1)α=a1a3.The rotating ellipsoidal void could be classified to 3 types according to the value ofα: flat rotating ellipsoidal void, spherical void, and prolate rotating ellipsoidal void, as is shown in Figure 4. Eshelby tensors [31] of the voided RVE corresponding to different α are shown in Appendix A.Figure 4
Voided RVE.Based on the fact that0<a1,a2,a3≤L and a2=a3, α=a1/a3, it follows that(2)f0=4/3πa1a2a32L3=π6αa3L3=π6α−2a1L3,and by reforming equation (2), we can get(3)α=6f0πa3/L3=6f0πa1/L3−1/2,0<a1<L,0<a3<L.The domain ofα could be concluded as(4)π6f0−1<α<π6f01/2.Denote(5)αcr=π6f01/2,and then the domain of α was reformed to(6a)αcr−2<α<αcr.For the purpose of a more tractable parameter, denoteγ=αcr1/2α, and then equation (6a) could be reformed further as(6b)αcr−3/2<γ<αcr3/2.The void is spherical whenα=1,γ=αcr1/2, flat rotating ellipsoidal when α>1,γ>αcr1/2, and prolate rotating ellipsoidal when α<1,γ<αcr1/2. γ and α are equivalent in the following analysis.


### 3.2. Distribution Functions of Shape Parameter

The differences among microstructures of each phase in cement paste led to the differences among corresponding void shapes. For the voids in the RVE corresponding to CSH, the most probable values of shape parameters wereMoα=1,Moγ=αcr1/2, while those to CH were Moα>1,Moγ>αcr1/2, or Moα<1,Moγ<αcr1/2. In accordance with the most probable values, the possible probability density functions of α and γ are shown in Figure 5. Five kinds of distribution functions were constructed consequently to describe the distribution of α and γ.Figure 5
Possible probability density function curves ofα.


#### 3.2.1. Distribution of Shape Parameter forMoα=1

In this case,Moα=1, Pα>1=Pα<1=1/2; that is, Moγ=αcr1/2, Pγ>αcr1/2=Pγ<αcr1/2=1/2. A reasonable assumption is that γ follows the lognormal distribution [32]:(7)γ∼LNlnαcr1/2,σ,and the distribution function is(8)Fγ=Φlnγ−lnαcr1/2σ,αcr−3/2<γ<αcr3/2.


#### 3.2.2. Distribution of Shape Parameter forMoα<1

In this case,αcr−2≤Moα<1,αcr−3/2≤Moγ<αcr1/2. The lognormal-Pareto combined distribution was constructed [33] to describe the distribution of shape parameters, denoted as γ∼LNPβ,θ,σ, and the distribution function is(9)Fγ=1−ωFLNγFLNθ,αcr−3/2≤γ<θ,1−ω1−FPγ1−FPθ,θ≤γ<αcr3/2,where(10a)FLNγ=Φlnγ−μσ,−∞<γ<+∞,(10b)FPγ=1−θβγ−β,0<θ≤γ,(11a)μ=lnθ−βσ2,(11b)ω=11+2πβσΦβσexp0.5βσ2.In the extreme case,Moα=αcr−2,Moγ=αcr−3/2, and ω=1. γ follows Pareto distribution, denoted as γ∼Pβ,θ. The distribution function is(12)Fγ=1−θβγ−β,θ=αcr−3/2,αcr−3/2≤γ≤αcr3/2.


#### 3.2.3. Distribution of Shape Parameter forMoα>1

In this case,1<Moα≤αcr,αcr1/2<Moγ≤αcr3/2. The domain of γ is completely conjugated with that in Section 3.2.2. Therefore, transforming γ to a conjugated parameter u and then the conjugated distribution function could be reconstructed. The method is as follows.Denoteu as the conjugated parameter of γ, GPu,GLNu as the conjugated distribution function of FPu,FLNu. Obviously,(13)u=tγ=1γ,αcr−3/2≤u≤αcr3/2.The monotonicity ofu and γ is in contrary, then FPu and FLNu should be monotone decreasing consequently, and numerical range of which should be 0,1. Guaranteeing the monotonicity and numerical range, GPu,GLNu could be constructed as(14a)GPu=1−FPu=θβuβ,(14b)GLNu=1−FLNu=Φlnu+μσ.The reconstructed distribution is named conjugated Pareto-lognormal combined distribution, denoted asγ∼con_LNPβ,θ,σ. The distribution function is(15)Gγ=ωGPγGPθ−1,αcr−3/2<γ<θ−1,1−1−ω1−GLNγ1−GLNθ−1,θ−1≤γ<αcr3/2.The calculation ofω and μ is the same with equation (11).In the extreme case,Moα=αcr,Moγ=αcr3/2and ω=1. γ follows the conjugated Pareto distribution, denoted as γ∼con_Pβ,θ, and the distribution function is(16)Gγ=GPγGPθ−1=θβγβ,αcr−3/2<γ<θ−1,θ−1=αcr3/2.As the distribution functions are obtained,αcr could be obtained with a known f0, and other parameters in distribution functions could be calculated with a corresponding quantile.


### 3.3. Equivalent Modulus of RVEs Corresponding to Phases

The porosity of theith phase was denoted as ci, the Poisson ratio as νi0, and Young’s modulus as Ei0. This phase is composed of n-RVEs, each RVE contains a void with a shape parameter of α, and the porosities of all the RVEs are the same ci. Massive α values could be gain through the Monte Carlo sampling method, with a certain distribution function correspondingly.The average mechanical properties of RVEs are worthy to be adopted in consideration of the randomness of void axis directions in the RVE accumulation. The average volume modulusκ and the average shear modulus μ of an RVE with an α valued shape parameter and random axis direction could be obtained by Mori–Tanaka method [34]:(17)κα=κi01+cip,μα=μi01+ciq,respectively, where p and q are decided by the Eshelby tensor [34].The macroporosity, Poisson ratio, and Young’s modulus of each phase would be determined in the next part then the macro Young’s modulus of ITZ, consequently.


### 3.4. Related Parameters of Cement Paste

As the distance between the cement paste and the aggregate surface increases, the influence of the side wall effect gradually weakens, the local water-cement ratio decreases, the porosity decreases, the degree of calcium hydroxide enrichment weakens, and the porosity and volume fraction of the constituent phases decrease. The porosity and volume fraction of the constituent phases also change.ITZ is a layer of high-porosity cement paste. Taking the porosity of the cement paste and the compositional phase volume fraction as a reference, the local porosity of ITZ was determined based on the local water-cement ratio, and the relative porosity of ITZ was obtained. The degree of calcium hydroxide enrichment of ITZ was determined, based on which the volume fraction of the cement mortar composition phase was adjusted, and the volume fractions of the ITZ composition phases were obtained. After ITZ’s relative porosity and composition phase volume fraction were obtained, the ITZ modulus calculation can be performed.The volume fractions of components in cement paste were denoted ascHD−CSH0 and cLD−CSH0 and CH as cCH0. The relative volume fractions among LD-CSH, HD-CSH, and CH are almost unchanged under the water-cement ratio of 0.5 and 0.65, respectively [15]. Without test result under higher water-cement ratio, it is reasonable to assume that, for the cement paste with a water-cement ratio w0≥0.5, the relative volume fractions of the 3 phases keep as the average value of that under 0.5 and 0.65 water-cement ratio, as is shown in Table 1.Table 1
Relative volume fraction of phases in cement paste whenw0≥0.5 [15].
cLD−CSH0cHD−CSH0cCH068.5%23.4%8.1%Young’s modulus of phases in cement paste is shown in Table2, in which that of micropores(MC) was set to be zero in this paper. The Poisson ratios of all the phases were set to be 0.2.Table 2
Young’s modulus of phases in cement paste [15].
PhasesMCLD-CSHHD-CSHCHYoung’s modulus (GPa)016.52533The absolute porosity of cement paste,φbulkabs, was set to be 30.7% when w0=0.5 [35].The absolute porosity of cement paste,φ bulkabs, was set to be 30.7% when w0 = 0.5 [35].It is worth noting that Young’s modulus of phases in cement paste provided by [15] is affected by the micropores in the phases themselves, while the micropores are contained in the absolute porosity of cement paste. There would be a higher estimation on actual porosity if the test results in [13] are adopted directly. To avoid this problem, it is supposed to treat the volume fraction of micropores, cmc0, as the relative porosity of cement paste, φbulkre. Then, the porosity of phases could be calculated as follows:(18)φphaseabs=φbulkabs−φbulkre1−φbulkre⋅100%,in which φbulkre=10% and φbulkabs=30.7% when w0=0.5 [35], and thus, φphaseabs=23%.


### 3.5. Macro Young’s Modulus of ITZ considering the Enrichment of CH

#### 3.5.1. Local Water-Cement Ratio and Relative Porosity of ITZ

As is discussed in 3.4, in accordance with the influence of bulk paste porosity on tested modulus of cement paste phases, it is necessary to adopt the relative porosity of ITZ to bulk paste in calculation of ITZ modulus, to avoid an over estimation on porosity.The local water-cement ratio,w, could be calculated as [27](19)wd=1−αcdGcαcd,where(20a)αcd=α¯c1+acd−δδ2,0<d≤δ,α¯c,d≥δ,(20b)α¯c=101−ca1+w0accaδ/raδ/ra2+5δ/ra+10+101−ca,in which Gc=3.15,ac=−0.4959,δ is the thickness of ITZ, ra is the radius of aggregate, ca is the volume fraction of aggregate, and w0 is the water-cement ratio of bulk paste, as is shown in Figure 6.Figure 6
Absolute porosity.The relation between the distance from aggregate surface and porosity is [35](21)φabsd=φITZabs−φITZabs−φbulkabsdδβ,0<d≤δ,φbulkabs,δ≤d,where φITZabs is the absolute porosity when d=0. φITZabs=78.9%,β=0.49, and φbulkabs=30.7% [35] when w0=0.5.With a determinedφabsd, the relative porosity, φred, could be obtained as follows:(22)φred=φabsd−φphaseabs1−φphaseabs.


#### 3.5.2. Enrichment Parameter of CH

The enrichment degree of CH is proportional to the local relative porosity. An enrichment parameter,η, was introduced to estimate the enrichment degree of CH as follows:(23)η=ηφre.The form of equation (23) depends on test results, which were determined in the 4th part of this paper. In accordance with an assumption of a constant volume fraction ratio between LD-CSH and HD-CSH, the volume fraction ratio of ITZ with consideration of CH enrichment could be obtained as follows:(24a)cCH=ηcCH0,(24b)cLD−CSH=1−ηcCH0cLD−CSH0+cHD−CSH0cLD−CSH0,(24c)cHD−CSH=1−ηcCH0cLD−CSH0+cHD−CSH0cHD−CSH0.


#### 3.5.3. Macro Young’s Modulus of ITZ

The volume and shear modulus of isotropic multiphase composite could be obtained by the Mori–Tanaka method [36]:(25)κκ0=1+a1−α0a,μμ0=1+b1−β0b,where(26a)α0=3κ03κ0+4μ0,β0=65κ0+2μ03κ0+4μ0,(26b)a=∑rcrκr−κ0α0κr−κ0+κ0,b=∑rcrμr−μ0β0μr−μ0+μ0.The subscript 0 represents the bulk matrix,r is the rth phase, and cr is the volume fraction ratio of rth phase.


## 3.1. Equivalence of Microvoid Shape and the Shape Parameter

An accurate description on void features of phases in ITZ is necessary for calculation. Voids could be treated as special inclusions with zero modulus. The inclusion with a shape of rotating ellipsoid, in which 2 in 3 axes are of equal length, is a kind of inclusions with excellent universal adaptability. Various kinds of inclusions, such as spherical, ellipsoidal, penny-shaped, and cylindrical inclusions [31], are the generalization of rotating ellipsoidal inclusion. Considering the complexity of void shape in ITZ, this paper assumed that all the voids in ITZ phases are in the shape of rotating ellipsoid.The phases in cement paste appear significantly different according to the SEM images: the crystal polymer of CSH is dense and uniform, and that of CH appears like a peony, while that of ettringite is needle-like, which is easy to form an extremely loose structure. Based on the shape and shape randomness of voids in phases, it is a reasonable assumption that the RVEs corresponding to CSH contain mainly spherical voids, CH contains mainly rotating ellipsoidal voids, and ettringite are treated as a pure void without any matrix. Considering the staggering of all the phases, it is acceptable to put the “ettringite voids” into other RVEs as part of porosity instead of an independent phase, as is shown in Figure3.Figure 3
Equivalence of pores in phases of cement paste.A typical RVE was set to be a cube with 2L in side length, containing a rotating ellipsoidal void witha1,a2,anda3 in 3 axis length, respectively, where a2=a3. The void shape parameter α was defined as(1)α=a1a3.The rotating ellipsoidal void could be classified to 3 types according to the value ofα: flat rotating ellipsoidal void, spherical void, and prolate rotating ellipsoidal void, as is shown in Figure 4. Eshelby tensors [31] of the voided RVE corresponding to different α are shown in Appendix A.Figure 4
Voided RVE.Based on the fact that0<a1,a2,a3≤L and a2=a3, α=a1/a3, it follows that(2)f0=4/3πa1a2a32L3=π6αa3L3=π6α−2a1L3,and by reforming equation (2), we can get(3)α=6f0πa3/L3=6f0πa1/L3−1/2,0<a1<L,0<a3<L.The domain ofα could be concluded as(4)π6f0−1<α<π6f01/2.Denote(5)αcr=π6f01/2,and then the domain of α was reformed to(6a)αcr−2<α<αcr.For the purpose of a more tractable parameter, denoteγ=αcr1/2α, and then equation (6a) could be reformed further as(6b)αcr−3/2<γ<αcr3/2.The void is spherical whenα=1,γ=αcr1/2, flat rotating ellipsoidal when α>1,γ>αcr1/2, and prolate rotating ellipsoidal when α<1,γ<αcr1/2. γ and α are equivalent in the following analysis.


## 3.2. Distribution Functions of Shape Parameter

The differences among microstructures of each phase in cement paste led to the differences among corresponding void shapes. For the voids in the RVE corresponding to CSH, the most probable values of shape parameters wereMoα=1,Moγ=αcr1/2, while those to CH were Moα>1,Moγ>αcr1/2, or Moα<1,Moγ<αcr1/2. In accordance with the most probable values, the possible probability density functions of α and γ are shown in Figure 5. Five kinds of distribution functions were constructed consequently to describe the distribution of α and γ.Figure 5
Possible probability density function curves ofα.


### 3.2.1. Distribution of Shape Parameter forMoα=1

In this case,Moα=1, Pα>1=Pα<1=1/2; that is, Moγ=αcr1/2, Pγ>αcr1/2=Pγ<αcr1/2=1/2. A reasonable assumption is that γ follows the lognormal distribution [32]:(7)γ∼LNlnαcr1/2,σ,and the distribution function is(8)Fγ=Φlnγ−lnαcr1/2σ,αcr−3/2<γ<αcr3/2.


### 3.2.2. Distribution of Shape Parameter forMoα<1

In this case,αcr−2≤Moα<1,αcr−3/2≤Moγ<αcr1/2. The lognormal-Pareto combined distribution was constructed [33] to describe the distribution of shape parameters, denoted as γ∼LNPβ,θ,σ, and the distribution function is(9)Fγ=1−ωFLNγFLNθ,αcr−3/2≤γ<θ,1−ω1−FPγ1−FPθ,θ≤γ<αcr3/2,where(10a)FLNγ=Φlnγ−μσ,−∞<γ<+∞,(10b)FPγ=1−θβγ−β,0<θ≤γ,(11a)μ=lnθ−βσ2,(11b)ω=11+2πβσΦβσexp0.5βσ2.In the extreme case,Moα=αcr−2,Moγ=αcr−3/2, and ω=1. γ follows Pareto distribution, denoted as γ∼Pβ,θ. The distribution function is(12)Fγ=1−θβγ−β,θ=αcr−3/2,αcr−3/2≤γ≤αcr3/2.


### 3.2.3. Distribution of Shape Parameter forMoα>1

In this case,1<Moα≤αcr,αcr1/2<Moγ≤αcr3/2. The domain of γ is completely conjugated with that in Section 3.2.2. Therefore, transforming γ to a conjugated parameter u and then the conjugated distribution function could be reconstructed. The method is as follows.Denoteu as the conjugated parameter of γ, GPu,GLNu as the conjugated distribution function of FPu,FLNu. Obviously,(13)u=tγ=1γ,αcr−3/2≤u≤αcr3/2.The monotonicity ofu and γ is in contrary, then FPu and FLNu should be monotone decreasing consequently, and numerical range of which should be 0,1. Guaranteeing the monotonicity and numerical range, GPu,GLNu could be constructed as(14a)GPu=1−FPu=θβuβ,(14b)GLNu=1−FLNu=Φlnu+μσ.The reconstructed distribution is named conjugated Pareto-lognormal combined distribution, denoted asγ∼con_LNPβ,θ,σ. The distribution function is(15)Gγ=ωGPγGPθ−1,αcr−3/2<γ<θ−1,1−1−ω1−GLNγ1−GLNθ−1,θ−1≤γ<αcr3/2.The calculation ofω and μ is the same with equation (11).In the extreme case,Moα=αcr,Moγ=αcr3/2and ω=1. γ follows the conjugated Pareto distribution, denoted as γ∼con_Pβ,θ, and the distribution function is(16)Gγ=GPγGPθ−1=θβγβ,αcr−3/2<γ<θ−1,θ−1=αcr3/2.As the distribution functions are obtained,αcr could be obtained with a known f0, and other parameters in distribution functions could be calculated with a corresponding quantile.


## 3.2.1. Distribution of Shape Parameter forMoα=1

In this case,Moα=1, Pα>1=Pα<1=1/2; that is, Moγ=αcr1/2, Pγ>αcr1/2=Pγ<αcr1/2=1/2. A reasonable assumption is that γ follows the lognormal distribution [32]:(7)γ∼LNlnαcr1/2,σ,and the distribution function is(8)Fγ=Φlnγ−lnαcr1/2σ,αcr−3/2<γ<αcr3/2.


## 3.2.2. Distribution of Shape Parameter forMoα<1

In this case,αcr−2≤Moα<1,αcr−3/2≤Moγ<αcr1/2. The lognormal-Pareto combined distribution was constructed [33] to describe the distribution of shape parameters, denoted as γ∼LNPβ,θ,σ, and the distribution function is(9)Fγ=1−ωFLNγFLNθ,αcr−3/2≤γ<θ,1−ω1−FPγ1−FPθ,θ≤γ<αcr3/2,where(10a)FLNγ=Φlnγ−μσ,−∞<γ<+∞,(10b)FPγ=1−θβγ−β,0<θ≤γ,(11a)μ=lnθ−βσ2,(11b)ω=11+2πβσΦβσexp0.5βσ2.In the extreme case,Moα=αcr−2,Moγ=αcr−3/2, and ω=1. γ follows Pareto distribution, denoted as γ∼Pβ,θ. The distribution function is(12)Fγ=1−θβγ−β,θ=αcr−3/2,αcr−3/2≤γ≤αcr3/2.


## 3.2.3. Distribution of Shape Parameter forMoα>1

In this case,1<Moα≤αcr,αcr1/2<Moγ≤αcr3/2. The domain of γ is completely conjugated with that in Section 3.2.2. Therefore, transforming γ to a conjugated parameter u and then the conjugated distribution function could be reconstructed. The method is as follows.Denoteu as the conjugated parameter of γ, GPu,GLNu as the conjugated distribution function of FPu,FLNu. Obviously,(13)u=tγ=1γ,αcr−3/2≤u≤αcr3/2.The monotonicity ofu and γ is in contrary, then FPu and FLNu should be monotone decreasing consequently, and numerical range of which should be 0,1. Guaranteeing the monotonicity and numerical range, GPu,GLNu could be constructed as(14a)GPu=1−FPu=θβuβ,(14b)GLNu=1−FLNu=Φlnu+μσ.The reconstructed distribution is named conjugated Pareto-lognormal combined distribution, denoted asγ∼con_LNPβ,θ,σ. The distribution function is(15)Gγ=ωGPγGPθ−1,αcr−3/2<γ<θ−1,1−1−ω1−GLNγ1−GLNθ−1,θ−1≤γ<αcr3/2.The calculation ofω and μ is the same with equation (11).In the extreme case,Moα=αcr,Moγ=αcr3/2and ω=1. γ follows the conjugated Pareto distribution, denoted as γ∼con_Pβ,θ, and the distribution function is(16)Gγ=GPγGPθ−1=θβγβ,αcr−3/2<γ<θ−1,θ−1=αcr3/2.As the distribution functions are obtained,αcr could be obtained with a known f0, and other parameters in distribution functions could be calculated with a corresponding quantile.


## 3.3. Equivalent Modulus of RVEs Corresponding to Phases

The porosity of theith phase was denoted as ci, the Poisson ratio as νi0, and Young’s modulus as Ei0. This phase is composed of n-RVEs, each RVE contains a void with a shape parameter of α, and the porosities of all the RVEs are the same ci. Massive α values could be gain through the Monte Carlo sampling method, with a certain distribution function correspondingly.The average mechanical properties of RVEs are worthy to be adopted in consideration of the randomness of void axis directions in the RVE accumulation. The average volume modulusκ and the average shear modulus μ of an RVE with an α valued shape parameter and random axis direction could be obtained by Mori–Tanaka method [34]:(17)κα=κi01+cip,μα=μi01+ciq,respectively, where p and q are decided by the Eshelby tensor [34].The macroporosity, Poisson ratio, and Young’s modulus of each phase would be determined in the next part then the macro Young’s modulus of ITZ, consequently.


## 3.4. Related Parameters of Cement Paste

As the distance between the cement paste and the aggregate surface increases, the influence of the side wall effect gradually weakens, the local water-cement ratio decreases, the porosity decreases, the degree of calcium hydroxide enrichment weakens, and the porosity and volume fraction of the constituent phases decrease. The porosity and volume fraction of the constituent phases also change.ITZ is a layer of high-porosity cement paste. Taking the porosity of the cement paste and the compositional phase volume fraction as a reference, the local porosity of ITZ was determined based on the local water-cement ratio, and the relative porosity of ITZ was obtained. The degree of calcium hydroxide enrichment of ITZ was determined, based on which the volume fraction of the cement mortar composition phase was adjusted, and the volume fractions of the ITZ composition phases were obtained. After ITZ’s relative porosity and composition phase volume fraction were obtained, the ITZ modulus calculation can be performed.The volume fractions of components in cement paste were denoted ascHD−CSH0 and cLD−CSH0 and CH as cCH0. The relative volume fractions among LD-CSH, HD-CSH, and CH are almost unchanged under the water-cement ratio of 0.5 and 0.65, respectively [15]. Without test result under higher water-cement ratio, it is reasonable to assume that, for the cement paste with a water-cement ratio w0≥0.5, the relative volume fractions of the 3 phases keep as the average value of that under 0.5 and 0.65 water-cement ratio, as is shown in Table 1.Table 1
Relative volume fraction of phases in cement paste whenw0≥0.5 [15].
cLD−CSH0cHD−CSH0cCH068.5%23.4%8.1%Young’s modulus of phases in cement paste is shown in Table2, in which that of micropores(MC) was set to be zero in this paper. The Poisson ratios of all the phases were set to be 0.2.Table 2
Young’s modulus of phases in cement paste [15].
PhasesMCLD-CSHHD-CSHCHYoung’s modulus (GPa)016.52533The absolute porosity of cement paste,φbulkabs, was set to be 30.7% when w0=0.5 [35].The absolute porosity of cement paste,φ bulkabs, was set to be 30.7% when w0 = 0.5 [35].It is worth noting that Young’s modulus of phases in cement paste provided by [15] is affected by the micropores in the phases themselves, while the micropores are contained in the absolute porosity of cement paste. There would be a higher estimation on actual porosity if the test results in [13] are adopted directly. To avoid this problem, it is supposed to treat the volume fraction of micropores, cmc0, as the relative porosity of cement paste, φbulkre. Then, the porosity of phases could be calculated as follows:(18)φphaseabs=φbulkabs−φbulkre1−φbulkre⋅100%,in which φbulkre=10% and φbulkabs=30.7% when w0=0.5 [35], and thus, φphaseabs=23%.


## 3.5. Macro Young’s Modulus of ITZ considering the Enrichment of CH

### 3.5.1. Local Water-Cement Ratio and Relative Porosity of ITZ

As is discussed in 3.4, in accordance with the influence of bulk paste porosity on tested modulus of cement paste phases, it is necessary to adopt the relative porosity of ITZ to bulk paste in calculation of ITZ modulus, to avoid an over estimation on porosity.The local water-cement ratio,w, could be calculated as [27](19)wd=1−αcdGcαcd,where(20a)αcd=α¯c1+acd−δδ2,0<d≤δ,α¯c,d≥δ,(20b)α¯c=101−ca1+w0accaδ/raδ/ra2+5δ/ra+10+101−ca,in which Gc=3.15,ac=−0.4959,δ is the thickness of ITZ, ra is the radius of aggregate, ca is the volume fraction of aggregate, and w0 is the water-cement ratio of bulk paste, as is shown in Figure 6.Figure 6
Absolute porosity.The relation between the distance from aggregate surface and porosity is [35](21)φabsd=φITZabs−φITZabs−φbulkabsdδβ,0<d≤δ,φbulkabs,δ≤d,where φITZabs is the absolute porosity when d=0. φITZabs=78.9%,β=0.49, and φbulkabs=30.7% [35] when w0=0.5.With a determinedφabsd, the relative porosity, φred, could be obtained as follows:(22)φred=φabsd−φphaseabs1−φphaseabs.


### 3.5.2. Enrichment Parameter of CH

The enrichment degree of CH is proportional to the local relative porosity. An enrichment parameter,η, was introduced to estimate the enrichment degree of CH as follows:(23)η=ηφre.The form of equation (23) depends on test results, which were determined in the 4th part of this paper. In accordance with an assumption of a constant volume fraction ratio between LD-CSH and HD-CSH, the volume fraction ratio of ITZ with consideration of CH enrichment could be obtained as follows:(24a)cCH=ηcCH0,(24b)cLD−CSH=1−ηcCH0cLD−CSH0+cHD−CSH0cLD−CSH0,(24c)cHD−CSH=1−ηcCH0cLD−CSH0+cHD−CSH0cHD−CSH0.


### 3.5.3. Macro Young’s Modulus of ITZ

The volume and shear modulus of isotropic multiphase composite could be obtained by the Mori–Tanaka method [36]:(25)κκ0=1+a1−α0a,μμ0=1+b1−β0b,where(26a)α0=3κ03κ0+4μ0,β0=65κ0+2μ03κ0+4μ0,(26b)a=∑rcrκr−κ0α0κr−κ0+κ0,b=∑rcrμr−μ0β0μr−μ0+μ0.The subscript 0 represents the bulk matrix,r is the rth phase, and cr is the volume fraction ratio of rth phase.


## 3.5.1. Local Water-Cement Ratio and Relative Porosity of ITZ

As is discussed in 3.4, in accordance with the influence of bulk paste porosity on tested modulus of cement paste phases, it is necessary to adopt the relative porosity of ITZ to bulk paste in calculation of ITZ modulus, to avoid an over estimation on porosity.The local water-cement ratio,w, could be calculated as [27](19)wd=1−αcdGcαcd,where(20a)αcd=α¯c1+acd−δδ2,0<d≤δ,α¯c,d≥δ,(20b)α¯c=101−ca1+w0accaδ/raδ/ra2+5δ/ra+10+101−ca,in which Gc=3.15,ac=−0.4959,δ is the thickness of ITZ, ra is the radius of aggregate, ca is the volume fraction of aggregate, and w0 is the water-cement ratio of bulk paste, as is shown in Figure 6.Figure 6
Absolute porosity.The relation between the distance from aggregate surface and porosity is [35](21)φabsd=φITZabs−φITZabs−φbulkabsdδβ,0<d≤δ,φbulkabs,δ≤d,where φITZabs is the absolute porosity when d=0. φITZabs=78.9%,β=0.49, and φbulkabs=30.7% [35] when w0=0.5.With a determinedφabsd, the relative porosity, φred, could be obtained as follows:(22)φred=φabsd−φphaseabs1−φphaseabs.


## 3.5.2. Enrichment Parameter of CH

The enrichment degree of CH is proportional to the local relative porosity. An enrichment parameter,η, was introduced to estimate the enrichment degree of CH as follows:(23)η=ηφre.The form of equation (23) depends on test results, which were determined in the 4th part of this paper. In accordance with an assumption of a constant volume fraction ratio between LD-CSH and HD-CSH, the volume fraction ratio of ITZ with consideration of CH enrichment could be obtained as follows:(24a)cCH=ηcCH0,(24b)cLD−CSH=1−ηcCH0cLD−CSH0+cHD−CSH0cLD−CSH0,(24c)cHD−CSH=1−ηcCH0cLD−CSH0+cHD−CSH0cHD−CSH0.


## 3.5.3. Macro Young’s Modulus of ITZ

The volume and shear modulus of isotropic multiphase composite could be obtained by the Mori–Tanaka method [36]:(25)κκ0=1+a1−α0a,μμ0=1+b1−β0b,where(26a)α0=3κ03κ0+4μ0,β0=65κ0+2μ03κ0+4μ0,(26b)a=∑rcrκr−κ0α0κr−κ0+κ0,b=∑rcrμr−μ0β0μr−μ0+μ0.The subscript 0 represents the bulk matrix,r is the rth phase, and cr is the volume fraction ratio of rth phase.


## 4. Result and Verification

Nanoindentation tests on gravel-cement ITZ [13] were chosen for verification. Samples were made using Type I Portland cement, river sand, and gravel. Water-cement ratio was fixed at 0.5, river sand in the size of 1.18 mm to 2.36 mm, and gravels with an approximate size of 5 mm were used. All the samples were designed to be 1 inch × 1 inch × 0.5 inch, with a single gravel in the center, and cured under water at 25°C.Seven groups of test results located at 2.5μm to 27.5 μm from the aggregate surface were provided, 2 of which were used for parameter fitting, and the thickness of ITZ, δ, was determined to be 20 μm. With these data, according to least square principle, the form of enrichment parameter η was determined as follows:(27)ηφre=P1⋅expP2⋅φre+Q1⋅expQ2⋅φre,and fitted parameters in which are shown in Table 3.Table 3
Parameter values in equation (27).
P1P2Q1Q28.02.66.4−391.3Considering the randomness of void shape, the calculation results were not equal to test results directly, so it is reasonable to adopt all the test results for verification. The calculated local water-cement ratiow and local porosity φabs and φre are shown in Table 4, and the parameters in distribution are shown in Appendix B.Table 4
Local water-cement ratio and local porosity of ITZ.
dμmwφabs%φre%2.51.0061.5050.007.50.7049.0933.8912.50.5640.6222.8817.50.5133.7513.9622.50.5030.7010.0027.50.5030.7010.00Assumed the void shape of CH follows LN, P, con_P, PLN, and con_PLN distribution, respectively, 5 curves of ITZ Young’s modulus along the distance from aggregate surface were obtained, and comparisons with test results were executed, as is shown in Figures7 and 8.Figure 7
Comparison among all test and calculated results.
(a)(b)Figure 8
Comparison between test average results and calculated results assuming LN distribution of CH voids.Among the tested points within ITZ (0μm < d < 20 μm), the nearer located to the surface of aggregate, the more significant discreteness of the test results, and the lager difference among all the calculated results under different distributions. Based on the analysis in Part 3, the local porosity increases with the distance from the surface of aggregate decreases. The calculated results showed that high porosity has a greater influence on Young’s modulus of ITZ than low porosity, which was proved by the discreteness of test results.Among the tested points within ITZ (0μm < d < 20 μm), the further located to the surface of aggregate, the lower porosity of ITZ, and the smaller difference among all the calculated results under different distributions. This phenomenon showed that the low porosity has a lesser influence on Young’s modulus of ITZ, and which keeps consistent with the discreteness of test results.One possible reason for the large discreteness of test results within bulk paste (d > 20 μm) is the thickness of ITZ was set 20 μm, but the actual thickness of ITZ among the tested points varied. For the 2nd and 4th group of test results, the peak value of Young’s modulus appeared at d = 22.5 μm, which indicated a lager thickness of ITZ than 22.5 μm; similarly, the 6th group of test results indicated 27.5 μm. Another possible reason is the randomness of Young’s modulus in bulk paste, which results in the difference of corresponding ITZ Young’s modulus. The deviation between test and calculated results would significantly decrease when ignoring the 2nd, 4th, and 6th group.The calculated results were suitable for the estimation on upper limit of Young’s modulus when the void in CH was assumed to follow P or PLN distribution, the lower limit when con_P or con_PLN distribution, and the average value when LN distribution. The estimation on upper and lower limit is conservative relatively, while that on average value showed small deviation. Considering the discreteness of test results themselves, there is no sense to analyse the discreteness of calculated results. Deviations about all the estimation are shown in Tables5–7.Table 5
Deviation of average ITZ Young’s modulus.
ITZBulk pastedμm2.57.512.517.522.527.5Average deviation (LN)−0.0880.016−0.009−0.068−0.098−0.087Table 6
Deviation of upper limit of ITZ Young’s modulus.
ITZBulk pastedμm2.57.512.517.522.527.5Average deviation (con_P)0.102−0.152−0.325−0.201−0.372−0.366Average deviation (con_PLN)0.142−0.091−0.286−0.157−0.340−0.342Table 7
Deviation of lower limit of ITZ Young’s modulus.
ITZBulk pastedμm2.57.512.517.522.527.5Average deviation (con_P)0.0520.7120.0810.0520.1010.248Average deviation (con_PLN)0.0160.7740.0860.0520.1050.256


## 5. Conclusion

A macro Young’s modulus calculation model of ITZ was proposed in this paper. With the introduction of void shape parameter and calcium hydroxide enrichment parameter, and the combination of local water-cement ratio, aggregate volume fraction, and local porosity, Young’s modulus of ITZ could be obtained.This model, with an advantage of small computation, could cover the upper and lower limit and average value of Young’s modulus of ITZ and reflect the gradient variation along direction perpendicular to aggregate surface. The availability of the model whenw = 0.5 was verified through comparison with test results in a reference paper.For the sake of limited research scope of gravel-cement paste transition zone and limited test data for verification, the proposed model was mainly adopted to ITZ in natural aggregate concrete at present. In addition, this suggested model could provide both theoretical support for meticulous modeling and failure analysis on concrete, and theoretical tool for improvement of ITZ properties in concrete strength research.The suggested formula of calcium hydroxide enrichment parameter was empirical, while the coupling effect of water enrichment and ion migration was the basic cause of calcium hydroxide enrichment. This coupling effect would be the theme of further study.



---

*Source: 9430875-2020-08-26.xml*