Three-scale stochastic homogenization of elastic recycled aggregate concrete based on nano-indentation digital images

Abstract

In this paper, three-scale stochastic elastic finite element analyses are made for recycled aggregate concrete (RAC) based on nano-indentation digital images. The elastic property of RAC contains uncertainties across scales. It has both theoretical and practical values to model and predict its mechanical performance. Based on homogenization theory, effective stochastic elastic moduli of RAC at three different scales are obtained using moving window technique, nano-indentation digital images, and Monte-Carlo method. It involves the generation of a large number of random realizations of microstructure geometry based on different volume fraction of the inclusions and other parameters. The mean value, coefficient of variation and probability distribution of the effective elastic moduli are computed considering the material multiscale structure. The microscopic randomness is taken into account, and correlations of RAC among five phases are investigated. The effective elastic properties are used to obtain the global behavior of a composite structure. It is indicated that the response variability can be considerably affected by replacement percentage of recycled aggregates.

Keywords

RAC / nano-indentation digital image / multiscale / microscopic randomness / homogenization

Introduction

Recycled Aggregate Concrete (RAC) is a new-type of concrete composed of cement, admixture and aggregates. Concrete aggregate collected from demolition sites is put through a crushing machine. Crushing facilities accept only uncontaminated concrete, which must be free of trash, wood, paper and other such materials. Before being used again, these aggregates are broken, screened and cleaned and then replace natural gravel aggregates. Also, the use of the green material alleviates lack of natural resources and reduces pollution from dumped concrete. However, the mechanical properties of RAC differ from those of general concrete. It is more heterogenous than general concrete. Nowadays there has been increasing researches focused on randomness of basic mechanical properties of RAC using laboratory experimental methods [¹–⁴]. However, one of the drawbacks is that it might be very expensive to provide enough statistical data through laboratory experiments.

Recent research has suggested that numerical simulation might be an important tool to analyze microstructural randomness of concrete material properties. For example, Xiao et al. [⁵] simulated uniaxial compression on modeled RAC. Li et al. [⁶] investigated the characteristics of compressive stress distribution. Peng et al. [⁷] developed microscopic convex polygon aggregate model and carried out numerical simulation for uniaxial tension test using Base Force Element Method (BFEM). Unfortunately, there has been far less research on stochastic numerical analysis for material properties of RAC across different scales.

As shown in Fig.1, RAC is a five-phase structure including natural aggregate, old inter-facial transition zone (ITZ), old mortar, new ITZ and new mortar. The recycled aggregate is formed by the first three phases. The new mortar, the old mortar and the ITZs are composed of many different components, such as C-S-H gel (involves LD C-S-H and HD C-S-H), un-hydrated particles, calcium hydroxide (CH) and pores, etc. In general, ITZ might have more defects because of the wall effect on aggregate surface. Microstructure of the new mortar might be quite different from one of the old mortar because of low hydration in a short period of time. It is indicated that RAC is a multi-phase highly heterogeneous material. Uncertainties of its material properties might exist at different scales.

Obviously, global mechanical properties and durability of RAC might be greatly affected by randomness from microstructures. The material properties at nano-scale might be the starting point for researchers to find more clues. Due to limitation of experimental techniques at the nano scale, little is known about C-S-H gel, although it is an important and significant component of cement-based materials. Fortunately, a new-type of testing methodology, called nano-indentation technique, has been developed lately. In a traditional indentation test, a hard tip whose mechanical properties are known (frequently made of a very hard material like diamond) is pressed into a sample whose properties are unknown. The load placed on the indenter tip is increased as the tip penetrates further into the specimen and soon reaches a user-defined value. While indenting, various parameters such as load and depth of penetration can be measured. A record of these values can be plotted on a graph to create a load-displacement curve (such as the one shown in Fig. 2). These curves can be used to extract mechanical properties of the material.

The continuing improvements in nanoindentation technology have led to many new and fascinating studies in microstructures of RAC. For instance, Hughes and Trtik [⁸] studied micro-mechanical properties of cement paste measured by depth-sensing nanoindentation. It was a preliminary correlation of physical properties with phase type. Wu et al. [⁹] investigated determination of elastic-plastic and creep of Calcium-Silicate-Hydrate Gel. Li et al. [¹⁰] done experimental research to investigate mechanical properties of interracial transition zones in recycled aggregate concrete. Sorelli et al. [¹¹] explored the nano-mechanical signature of ultra high performance concrete by statistical nanoindentation techniques. Combined scanning electron microscope images with nano-indentation data, Němeček and coworkers [¹²–¹⁴] completed statistic analysis and studied the micro-structures of materials that involve in hardened mortar and aluminum alloy. Nevertheless, within the extensive literature on nanoindentation techniques, comparatively little research has focused on homogenization of RAC considering randomness from geometric distribution of different constituents.

In this paper, three-scale stochastic elastic finite element analyses are made for recycled aggregate concrete (RAC) based on nano-indentation digital images. Based on homogenization theory, effective stochastic elastic moduli of RAC at three different scales are obtained using moving window technique, nano-indentation digital images, and Monte-Carlo method. It involves the generation of a large number of random realizations of microstructure geometry based on different volume fraction of the inclusions and other parameters. The mean value, coefficient of variation and probability distribution of the effective elastic moduli and Poisson ratio are computed considering the material multiscale structure. The microscopic randomness is taken into account, and correlations of RAC among five phases are investigated. The effective elastic properties are used to obtain the global behavior of a composite structure.

Characteristics of microstructures in RAC

Microscopic elastic modulus

There are many methods to calculate material elastic modulus and hardness, in which Oliver-Pharr method [¹⁵] has been used widely. In the traditional hardness test, the projection area was calculated based on the indentation photo after removal of loading. However, the projected area of Nano-indentation was difficult to obtain. Also, it was not accurate to measure from the photo. Oliver and Pharr derived the relationship among contacting area, indentation depth and indenter shape. In Eqs. (1) and (2), the elastic modulus, E, is given as

(1)E=(\mathrm{1}-{\nu }^{\mathrm{2}}){\left[{\displaystyle \frac{\mathrm{1}}{E_{\mathrm{r}}}}-{\displaystyle \frac{(\mathrm{1}-{\nu }_{\mathrm{i}}^{\mathrm{2}})}{E_{\mathrm{i}}}}\right]}^{-\mathrm{1}},

(2)E_{\mathrm{r}}={\displaystyle \frac{S\sqrt{\mathrm{\pi }}}{\mathrm{2}\beta \sqrt{A_{\mathrm{c}}}}},

where v is the Poisson ratio in the cement paste of RAC, and set to be 0.22; E_r the reduced modulus; β the relevant parameter of Berkovich indenter 1.034; A_c the contact area; E_i and v_i are indenter material parameter and Poisson ratio, respectively.

Microstructure of mortar

Assisted by atomic force microscope, nano-indentation technology is capable to preciously focus the indentation on the new ITZ, the old ITZ, the new mortar and the old mortar areas in RAC. This study used Hysitron nano-indentation instrument with Berkovich diamond indenter of trihedral taper shape, 1200 nm of diameter and 142.3°of oblique angle. During the loading process, the indentation velocity was 240 µN/s at the first 5 s and then increased gradually. When the load reached the maximum value, 1200 µN/s, it was kept the same for 2 s, and then the creep effect between material surface and indenter was removed. Next, the unloading process began in the same velocity. In the ITZ area, indentation velocities in the horizontal and vertical directions were set to be 3 µm/s to eliminate superposition from double indentation. In the mortar area, it was set to be 10 µm/s. Figures 3(a-d) show the four indent areas of old mortar, new mortar, old ITZ, and new ITZ. Figures 4(a-d) present the binary digital images of elastic moduli of the four indent areas.

Verification

To verify the accuracy of these binary digital images of elastic moduli, a comparison between the constituent proportion of the present study and one from literature [¹¹] was made. In Table 1, ranges of elastic moduli for different constituents are given. Figure 5 show the comparison. It is indicated that proportion of HD C-S-H gel in the new mortar is relatively low. The reason might be hydration in the new mortar was not completely finished during its formation process. However, because the difference between results of this paper and the literature is approximately 10%, it might be indicated that the proposed method is accurate enough.

Homogenization theory

One of the major preoccupations of multiscale numerical analysis in the past decade has been investigating the asymptotic homogenization method [¹⁶]. In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients. It turns out that the study of these equations is also of great importance in physics and engineering, since equations of this type govern the physics of inhomogeneous or heterogeneous materials. Of course, all matter is inhomogeneous at some scale, but frequently it is convenient to treat it as homogeneous. Procedures of asymptotic homogenization method include various scales are introduced, the asymptotic expansion is substituted into the equilibrium equation of displacement field, and governing equations are derived based on assumption of periodicity.

The analogue of the differential element in the continuum concept is known as the Representative Volume Element (RVE) in homogenization and micromechanics. This element contains enough statistical information about the inhomogeneous medium in order to be representative of the material. In this paper, Y can be used to describe minimum size of the unit cell based on Y-periodicity assumption. Two coordinates at coarse and fine scales are expressed as x(x₁, x₂, x₃) and y(y₁, y₂, y₃), respectively, where y = x/β. Here β is a non-negative value which is far less than 1.

The response function at fine scale is given as

(3)F^{\beta }(x)=F(x/\beta )=F(x,y),

Partial differentiation with respect to the fine-scale coordinate x yields

(4){\displaystyle \frac{\partial F(x)}{\partial x_i}}={\displaystyle \frac{\partial F(x,y)}{\partial x_i}}+{\displaystyle \frac{\mathrm{1}}{\beta }}{\displaystyle \frac{\partial F(x,y)}{\partial y_i}},

In the solution domain Ω, it satisfies

(5)E_{ijkl}^{\beta }(x)=E_{ijkl}(x,y)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x,y\in \Omega ,

Equations of equilibrium, strain displacement relations, constitutive law and boundary conditions can be written as

(6){\displaystyle \frac{\partial {\sigma }_{ij}^{\beta }}{\partial x_j}}+f_i=\mathrm{0}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in \Omega ,

(7){\epsilon }_{kl}^{\beta }={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left({\displaystyle \frac{\partial U_k^{\beta }}{\partial x_l^{\beta }}}+{\displaystyle \frac{\partial U_l^{\beta }}{\partial x_k^{\beta }}}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in \Omega ,

(8){\sigma }_{ij}^{\beta }=E_{ijkl}^{\beta }{\epsilon }_{kl}^{\beta }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in \Omega ,

(9)U_i^{\beta }={\bar{U}}_i\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in {\Gamma }_u,

(10){\sigma }_{ij}^{\beta }{n}_j=t_i\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in {\Gamma }_t,

The asymptotic expansion for displacement field function, U^b(x), can be given as

(11)U^{\beta }(x)=U^{\mathrm{0}}(x,y)+\beta U^{\mathrm{1}}(x,y)+{\beta }^{\mathrm{2}}{U}^{\mathrm{2}}(x,y)+O({\beta }^{\mathrm{3}}),

where the superscripts, 0, 1, 2..., denote expanding orders of b. Substitution Eq. (11) into Eqs. (4) and (7) yields,

(12)\begin{array}{l}{\epsilon }_{kl}^{\beta }(x)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left({\displaystyle \frac{\partial U_k^{\beta }}{\partial x_l^{\beta }}}+{\displaystyle \frac{\partial U_l^{\beta }}{\partial x_k^{\beta }}}\right)={\displaystyle \frac{\mathrm{1}}{\beta }}{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left({\displaystyle \frac{\partial U_k^{\mathrm{0}}}{\partial y_l}}+{\displaystyle \frac{\partial U_l^{\mathrm{0}}}{\partial y_k}}\right)+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left({\displaystyle \frac{\partial U_k^{\mathrm{0}}}{\partial x_l}}+{\displaystyle \frac{\partial U_l^{\mathrm{0}}}{\partial x_k}}+{\displaystyle \frac{\partial U_k^{\mathrm{1}}}{\partial y_l}}+{\displaystyle \frac{\partial U_l^{\mathrm{1}}}{\partial y_k}}\right)\\ +\beta \left({\displaystyle \frac{\partial U_k^{\mathrm{1}}}{\partial x_l}}+{\displaystyle \frac{\partial U_l^{\mathrm{1}}}{\partial x_k}}+{\displaystyle \frac{\partial U_k^{\mathrm{2}}}{\partial y_l}}+{\displaystyle \frac{\partial U_l^{\mathrm{2}}}{\partial y_k}}\right)+{\beta }^{\mathrm{2}}{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left({\displaystyle \frac{\partial U_k^{\mathrm{2}}}{\partial x_l}}+{\displaystyle \frac{\partial U_l^{\mathrm{2}}}{\partial x_k}}+{\displaystyle \frac{\partial U_k^{\mathrm{3}}}{\partial y_l}}+{\displaystyle \frac{\partial U_l^{\mathrm{3}}}{\partial y_k}}\right)+O({\beta }^{\mathrm{3}})\\ ={\displaystyle \frac{\mathrm{1}}{\beta }}{\epsilon }_{kl}^{-\mathrm{1}}(x,y)+{\epsilon }_{kl}^{\mathrm{0}}(x,y)+\beta {\epsilon }_{kl}^{\mathrm{1}}(x,y)+{\beta }^{\mathrm{2}}{\epsilon }_{kl}^{\mathrm{2}}(x,y).\end{array}

Through substitution Eq. (12) into Eq. (8), the asymptotic expansion equation in the stress field is given as

(13){\sigma }_{kl}^{\beta }(x)={\displaystyle \frac{\mathrm{1}}{\epsilon }}{\sigma }_{kl}^{-\mathrm{1}}(x,y)+{\sigma }_{kl}^{\mathrm{0}}(x,y)+\beta {\sigma }_{kl}^{\mathrm{1}}(x,y)+{\beta }^{\mathrm{2}}{\sigma }_{kl}^{\mathrm{2}}(x,y)+O({\beta }^{\mathrm{3}}),

where

{\sigma }_{ij}^n(x,y)=E_{ijkl}^{\beta }(x){\epsilon }_{kl}^n(x,y).

Thus, substitution Eq. (13) into Eq. (6) yields

(14)\begin{array}{l}{\displaystyle \frac{\mathrm{1}}{\beta }}\left[{\displaystyle \frac{\partial {\sigma }_{ij}^{-\mathrm{1}}(x,y)}{\partial x_j}}+{\displaystyle \frac{\mathrm{1}}{\beta }}{\displaystyle \frac{\partial {\sigma }_{ij}^{-\mathrm{1}}(x,y)}{\partial y_j}}\right]+\left[{\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{0}}(x,y)}{\partial x_j}}+{\displaystyle \frac{\mathrm{1}}{\beta }}{\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{0}}(x,y)}{\partial y_j}}\right]+\\ \beta \left[{\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{1}}(x,y)}{\partial x_j}}+{\displaystyle \frac{\mathrm{1}}{\beta }}{\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{1}}(x,y)}{\partial y_j}}\right]+{\beta }^{\mathrm{2}}\left[{\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{2}}(x,y)}{\partial x_j}}+{\displaystyle \frac{\mathrm{1}}{\beta }}{\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{2}}(x,y)}{\partial y_j}}\right]+O({\beta }^{\mathrm{3}})+f_i=\mathrm{0},\end{array}

Then, by multiplying β on both sides, Eq. (12) can be written as

(15)\begin{array}{l}{\displaystyle \frac{\partial {\sigma }_{ij}^{-\mathrm{1}}(x,y)}{\partial y_j}}+\beta \left({\displaystyle \frac{\partial {\sigma }_{ij}^{-\mathrm{1}}(x,y)}{\partial x_j}}+{\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{0}}(x,y)}{\partial y_j}}\right)+{\beta }^{\mathrm{2}}\left({\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{0}}(x,y)}{\partial x_j}}+{\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{1}}(x,y)}{\partial y_j}}+f_i\right)\\ {\beta }^{\mathrm{3}}\left({\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{1}}(x,y)}{\partial x_j}}+{\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{2}}(x,y)}{\partial y_j}}\right)+{\beta }^{\mathrm{4}}\left({\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{2}}(x,y)}{\partial x_j}}+{\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{3}}(x,y)}{\partial y_j}}\right)+O({\beta }^{\mathrm{3}})=\mathrm{0},\end{array}

Finally, governing equations are given as

(16){\displaystyle \frac{\partial {\sigma }_{ij}^{-\mathrm{1}}(x,y)}{\partial y_j}}=\mathrm{0},

(17){\displaystyle \frac{\partial {\sigma }_{ij}^{-\mathrm{1}}(x,y)}{\partial x_j}}+{\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{0}}(x,y)}{\partial y_j}}=\mathrm{0},

(18){\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{0}}(x,y)}{\partial x_j}}+{\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{1}}(x,y)}{\partial y_j}}+f_i=\mathrm{0},

(19){\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{1}}(x,y)}{\partial x_j}}+{\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{2}}(x,y)}{\partial y_j}}=\mathrm{0},

Based on the periodicity assumption, U^{\mathrm{0}}only depends on x coordinate. Then, Eq. (16) can be written as

(20){\sigma }_{ij}^{-\mathrm{1}}=\mathrm{0},

Then, Eq. (17) can be written as

(21){\displaystyle \frac{\partial {\sigma }_{ij}^{\mathrm{0}}(x,y)}{\partial y_j}}=\mathrm{0},

That is,

(22){\displaystyle \frac{\partial }{\partial y_j}}\left[E_{ijkl}^{\beta }\left({\displaystyle \frac{\partial U_k^{\mathrm{0}}}{\partial x_l}}+{\displaystyle \frac{\partial U_k^{\mathrm{1}}}{\partial y_l}}\right)\right]=\mathrm{0},

Here U_p^{\mathrm{1}} is decomposed into oscillating and smooth parts as

(23)U_p^{\mathrm{1}}=\overline{U_p^{\mathrm{1}}}+\chi {(y)}_p^{kl}{\displaystyle \frac{\partial U_k^{\mathrm{0}}}{\partial x_l}},

where (y) is homogenization function that satisfied Y-periodicity. Substitution Eq. (23) into Eq. (21) yields

(24){\sigma }_{ij}^{\mathrm{0}}=E_{ijpm}^{\beta }\left({\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}({\delta }_{pk}{\delta }_{ml}+{\delta }_{pl}{\delta }_{mk})+{\displaystyle \frac{\partial {\chi }_p^{kl}}{\partial y_m}}\right){\displaystyle \frac{\partial U_k^{\mathrm{0}}}{\partial x_l}},

(25){\displaystyle \frac{\partial }{\partial y_j}}\left[E_{ijpm}^{\beta }\left({\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}({\delta }_{pk}{\delta }_{ml}+{\delta }_{pl}{\delta }_{mk})+{\displaystyle \frac{\partial {\chi }_p^{kl}}{\partial y_m}}\right){\displaystyle \frac{\partial U_k^{\mathrm{0}}}{\partial x_l}}\right]=\mathrm{0},

U_k^{\mathrm{0}} is independent ofycoordinate, so Eq. (25) can be written as

(26){\displaystyle \frac{\partial }{\partial y_j}}\left[E_{ijpm}^{\beta }\left({\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}({\delta }_{pk}{\delta }_{ml}+{\delta }_{pl}{\delta }_{mk})+{\displaystyle \frac{\partial {\chi }_p^{kl}}{\partial y_m}}\right)\right]=\mathrm{0},

Thus, Eq. (26) is the equilibrium equation at fine scale. By Integration at both sides yields

(27)\langle {\sigma }_{ij}^{\mathrm{0}}\rangle ={\displaystyle {\int }_Y{\sigma }_{ij}^{\mathrm{0}}}dY={\displaystyle \frac{\mathrm{1}}{Y}}{\displaystyle {\int }_Y{E}_{ijpm}^{\beta }\left({\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}({\delta }_{pk}{\delta }_{ml}+{\delta }_{pl}{\delta }_{mk})+{\displaystyle \frac{\partial {\chi }_p^{kl}}{\partial y_m}}\right){\displaystyle \frac{\partial U_k^{\mathrm{0}}}{\partial x_l}}\mathrm{d}Y=}{E}_{ijkl}^{eff}{\displaystyle \frac{\partial U_k^{\mathrm{0}}}{\partial x_l}},

Finally, the effective elastic tensor can be written as

(28)E_{ijkl}^{eff}={\displaystyle \frac{\mathrm{1}}{Y}}{\displaystyle {\int }_Y{E}_{ijpm}^{\beta }\left({\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}({\delta }_{pk}{\delta }_{ml}+{\delta }_{pl}{\delta }_{mk})+{\displaystyle \frac{\partial {\chi }_p^{kl}}{\partial y_m}}\right)\mathrm{d}Y},

At the fine scale, the problem can be written as

(29){\displaystyle \frac{\partial \langle {\sigma }_{ij}^{\mathrm{0}}\rangle }{\partial x_j}}+f_i=\mathrm{0}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in \Omega ,

(30){\sigma }_{ij}^{\mathrm{0}}=E_{ijkl}^{eff}{\displaystyle \frac{\partial U_k^{\mathrm{0}}}{\partial x_l}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in \Omega ,

(31)U_i^{\mathrm{0}}=\overline{U_i}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in {\Gamma }_u,

(32)\langle {\sigma }_{ij}^{\mathrm{0}}\rangle n_j=t_i\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in {\Gamma }_t.

Stochastic homogenization at microscale

Development of RVE using moving window technique

The size of each binary digital image of elastic modulus from nanoindentation is about 100 µm × 100 µm, approximately 500 × 500 pixels. In this study, the moving window technique is used to develop RVE at microscale. In this method, an initial window of area is set at a starting point of the image, and then by moving this window over the moving distance step is assumed to be the same along both direction. Figure 6 presents illustration of the moving window technique. According to literature [¹²], an appropriate size of the moving window is approximately 20 µm × 20 µm, about 100 × 100 pixels. Monte Carlo simulation was carried out for 1000 RVEs randomly picked from binary digital images of the new mortar, old mortar, new ITZ and old ITZ areas. Thus, mean values and standard deviations of effective elastic moduli of different constituents are obtained.

Effective moduli at microscale

Figure 7 shows histograms of effective elastic modulus (E_eff ) of different constituents. Figure 8 presents mean values and coefficient of variation (COV) of E_eff at (a) new mortar and old mortar; (b) new ITZ and old ITZ. It is indicated that the mean value of E_eff of the new mortar was about 10% lower than one of the old mortar. It might be resulted from insufficient hydration. But, coefficients of variation of the old ITZ was considerably higher than ones of other constituents. The reason might be that higher porosity and more damages existed in the old ITZ area.

In addition, Monte Carlo simulation was made for 500 samples for two different sizes of RVE. 20 µm × 20 µm and 1 mm × 1 mm. Table 2 presents mean, standard deviation and COV for RVEs of different sizes. It was found that as the size of RVE increased from 20 micr to 1 mm, the variation in the output is extremely decreased. It might indicate that global material property is more stable than one of microstructures.

Stochastic homogenization at mesoscale

RVE development

Lately, Povirk [¹⁷] and Gusev [¹⁸] have investigated microscopic properties of composite materials based on the same size of RVEs. It is suggested that the best volume proportion of RVE to the largest aggregate might be 3 to 4 [¹⁹]. As shown in Fig. 9, an ideal square RAC model with tens of randomly distributed circle inclusions is used at the mesoscale. The side of the RVE is of 100 mm in length, with the inclusions of minimum radius 5 mm and maximum 20 mm.

Random generation

In this study, Fuller expansion is used to investigate the effect of aggregate gradation on the homogenized material property. Based on transformation of the expansion into two-dimensional problem by Walraven [²⁰], probability of occurrence for aggregate of diameter D_i can be written as

(33)

where D_max denotes the maximum diameter among aggregate particles, V_a volume fraction of aggregate. Random generation is made for particles’ diameters ranged from D_max and D_min as well as their positions.

Procedures of the random generation are given as follows.

(1) The proportion of aggregates to the whole area is defined.

(2) Aggregate particle size distribution based on Fuller expansion is calculated. The volume of aggregates is determined as follows.

(34)

The equation is solved until the Fuller expansion curve is met.

(3) The aggregate numbers are arranged in descending order according to particle size, and stored in an array.

(4) According to aggregates’ numbers, their positions are randomly generated without overlapping successively.

(5) Particles numbers, sizes and coordinates are stored into the other array as the input data for the following finite element analysis.

Boundary conditions and FEM results

As shown in Fig. 10, boundary conditions of RVE at the macro-scale are given.

Figure 11 shown the horizontal displacement and the normal stress after finite element analysis. From the figure of displacement. It was found that all displacements satisfy boundary conditions. In addition, it was found that clear stress concentration near ITZ. It is indicated that the results might be reasonable and acceptable.

Verification

Based on homogenization theory investigated by Voigt [²¹] and Reuss [²²], the equation upper and lower boundaries for effective elastic modulus of a multi-phase materials are given as

(35)E_{\delta }^V(\theta )=f_{\mathrm{m}}(\theta )\cdot E_{\mathrm{m}}+f_{\mathrm{i}\mathrm{n}\mathrm{1}}(\theta )\cdot E_{\mathrm{i}\mathrm{n}\mathrm{1}}+f_{\mathrm{i}\mathrm{n}\mathrm{2}}(\theta )\cdot E_{\mathrm{i}\mathrm{n}\mathrm{2}}+O(\theta ),

(36)E_{\delta }^R(\theta )={\left[f_{\mathrm{m}}(\theta )\cdot {\displaystyle \frac{\mathrm{1}}{E_{\mathrm{m}}}}+f_{\mathrm{i}\mathrm{n}\mathrm{1}}(\theta )\cdot {\displaystyle \frac{\mathrm{1}}{E_{\mathrm{i}\mathrm{n}\mathrm{1}}}}+f_{\mathrm{i}\mathrm{n}\mathrm{2}}(\theta )\cdot {\displaystyle \frac{\mathrm{1}}{E_{\mathrm{i}\mathrm{n}\mathrm{2}}}}+O(\theta )\right]}^{-\mathrm{1}},

where f_m(q) is volume fraction of matrix, and f_in(q) volume fraction of inclusions.

Figure 12 presented the effective elastic modulus obtained under conditions with aggregate volume fraction of 20%, replacement percentage of 100%, and aggregate particle size of 5 mm. Here the elastic modulus of new mortar modulus is 30 GPa, old mortar 20 GPa, and aggregate 60 GPa. It is shown that the elastic moduli are within a reasonable range between the upper and lower boundaries. It is indicated that effective elastic modulus under boundary conditions of uniform traction might be a little bit lower than one of linear displacement.

Stochastic homogenization at macroscale

Three-scale stochastic homogenization

As shown in Fig. 13, illustration of the three-scale stochastic homogenization is given. The micro-structural concrete composite RVE of size 20 µm × 20 µm was used as a finite element at meso-scale. Its effective elastic modulus was from homogenization of nano indentation images. Then, the meso-structural concrete composite RVE of size 1 mm × 1 mm was used as a finite element at macro-scale. Its effective elastic modulus was from homogenization of meso-scale images. Boundary conditions at macro-scale were given in Fig. 10. Uncertainties across scales were considered through randomly generation of components for models at various scales.

Results and discussion

The effect replacement percentage of recycled aggregates

Monte Carlo simulations of replacement percentages of 0%, 25%, 50%, 75%, 100%, respectively, were made for more than 500 samples. It is assumed that the elastic modulus of natural aggregate is approximately 60 GPa. Figure 14 shown histograms and fitted PDFs of E_eff with different recycled aggregate replacement percentages. In this study, some probability density functions have been tested before the best one is chosen. For this issue, Vu-Bac et al. [²⁴] have made stochastic predictions of interfacial characteristic of polymeric nanocomposites. More details please refer to [²⁴].

Goodness of fit

The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing. Table 3 shown the goodness of fit adjusted R² to different distributions for effective elastic moduli of RAC. As shown in Table 3, it seems that lognormal and normal distributions fit better than Weibull distribution. However, because elastic modulus is non-negtive, lognormal distribution might be more reasonable to be chosen in this study to investigate goodness of fit. As shown in Fig. 14, all histograms are very good to fit lognormal distribution. In addition, Hamdia et al. [²⁵] have investigated uncertainty quantification of the fracture properties of polymeric nanocomposites based on phase field modeling. Vu-Bac et al. [²⁶] have proposed a unified framework for stochastic predictions of mechanical properties to investigate PCNs. Vu-Bac et al. [²⁷] have used Mori-Tanaka method to make homogenization for PMCs. They applied stochastic homogenization on different materials, but the basic concepets and approaches might be quite similar.

COV of RAC effective elastic moduli

Figure 15 presented the relation between recycled aggregate replacement percentages and mean effective moduli. Figure 16 shown the relation between recycled aggregate replacement percentages and standard deviation of effective moduli. Figure 17 illustrated the relation between recycled aggregate replacement percentages and COV of effective moduli. As the replacement percentage of recycled aggregates increased, mean effective elastic moduli decreased linearly. As the modulus decreased about 10% when the displacement percentage was approximately 25%, while the modulus decreased to about 30% when the displacement percentages was approximately 100%. Standard deviation and coefficient of variation (COV) decreased gradually as replacement percentage increased. The reason might be that as replacement percentage increases, proportion of natural aggregates of larger elastic modulus decreases. It might lead to decrease of material property randomness. The effect of microscopic uncertainty, however, would be getting more considerable when the replacement percentage increased. When microscopic uncertainty was taken into account, standard deviation and COV were higher than ones ignoring it. Also, it was found that probably impact of microscale randomness was less significant than one of mesoscale.

Conclusions

In the past data studies on RAC, few of them thought of micro uncertainty. Even if researchers analyzed homogenization based on experimental data of nano-indentation, proportion of compositions was the only factor taken into account and impact of space distribution and randomness were neglected. Effective parameters of elastic modulus in three different scales and their statistical probability were both obtained using stochastic homogenization methods, the moving window technique, and Monte Carlo simulation based on nano-indentation binary digital images. Correlations across three scales were investigated. The findings are summarized as follows.

(1) The accuracy of binary digital images of elastic moduli at microscale is verified through a comparison between the constituent proportion of the present study and one from literature.

(2) The elastic moduli at mesoscale are within a reasonable range between Voigt-Reuss upper and lower boundaries.

(3) At macroscale, as the modulus decreased about 10% when the displacement percentage was approximately 25%, while the modulus decreased to about 30% when the displacement percentages was approximately 100%.

(4) Standard deviation and coefficient of variation (COV) decreased gradually as replacement percentage increased.

(5) Impact of microscale randomness was less significant than one of mesoscale. It might be indicated that mesoscale uncertainty was a primary factor to affect variation of effective elastic moduli.

(6) When microscopic uncertainty was taken into account, standard deviation and COV were higher than ones ignoring it.

This manuscript regards to stochastic modeling of a multiscale model where the correlation of the output is considered. In the future, the research might be extended to implementation of sensitivity analysis [²⁸] to quantitatively assess the quality of the proposed model. It will be an interesting direction.

References

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