The effect of micro-structural uncertainties of recycled aggregate concrete on its global stochastic properties via finite pixel-element Monte Carlo simulation

Abstract

In this paper, the effect of micro-structural uncertainties of recycled aggregate concrete (RAC) on its global stochastic elastic properties is investigated via finite pixel-element Monte Carlo simulation. Representative RAC models are randomly generated with various distribution of aggregates. Based on homogenization theory, effects of recycled aggregate replacement rate, aggregate volume fraction, the unevenness of old mortar, proportion of old mortar, aggregate size and elastic modulus of aggregates on overall variability of equivalent elastic properties are investigated. Results are in a good agreement with experimental data in literature and finite pixel-element method saves the computation cost. It is indicated that the effect of mesoscopic randomness on global variability of elastic properties is considerable.

Keywords

RAC / Monte Carlo analysis / stochastic / finite pixel-element method / homogenization / coefficient of variation

Introduction

In recent years, there was a dramatic proliferation of research concerned with recycled aggregate concrete (RAC) in literature. Many scholars used experimental methods to study RAC. For instance, Li et al. [¹] investigated interfacial transition zones in recycled aggregate concrete with different mixing approaches. Xiao et al. [²] studied seismic performance of frame structures with recycled aggregate concrete. Xiao et al. [³] did research on compressive strength of recycled aggregate concrete. Xiao et al. [⁴] explored mechanical properties of recycled aggregate concrete under uniaxial loading. Xiao et al. [⁵] studied shear transfer across a crack in recycled aggregate concrete. On the other hand, some other scholars tried to use numerical methods to analyze material properties of recycled aggregate concrete. For example, Xiao et al. [⁶] made FEM simulation of chloride diffusion in modeled recycled aggregate concrete. Xiao et al. [⁷] studied the stress distribution in modeled recycled aggregate concrete under uniaxial compression. Xiao et al. [⁸] carried out mesoscopic numerical simulation of uniaxial compressive behavior of model recycled aggregate concrete. Zhou et al. [⁹] investigated effect of coarse aggregate on elastic modulus and compressive strength of high performance concrete. Stock et al. [¹⁰] explored the effect of aggregate concentration upon the strength and modulus of elasticity of concrete. There has been relatively little progress in the highly heterogenous material considering micro-structural uncertainties.

The last two decades have seen growing importance placed on research in stochastic finite element method. Stefanou [¹¹] reviewed the stochastic finite method: past, present and future. Wall and Deodatis investigated variability response functions of stochastic plane stress/strain problems. Argyris et al. [¹³] proposed stochastic finite element analysis of shells. Craham and Deodatis studied response and eigenvalue analysis of stochastic finite element systems with multiple correlated material and geometric properties. Noh [¹⁵] derived a formulation for stochastic finite element analysis of plate structures with uncertain Poisson’s ratio. Stefanou and Papadrakakis [¹⁶] made stochastic finite element analysis of shells with combined random material and geometric properties. Kamiński and Świta [¹⁷] used perturbation stochastic finite element method to analyze the stability and reliability of the underground steel tanks. The results shown that the geometrical uncertainty played a more important role in the critical pressure uncertainty than the material uncertainty. Xia et al. [¹⁸] developed perturbation stochastic finite element method to analyze static responses of stochastic structures. Fink and Nackenhorst [¹⁹] compared the spectral representation method coupled with Monte Carlo simulation with classical Latin hypercube sampling in simulating the uncertainty of inelastic material behavior. Gunzburger et al. [²⁰] investigated different approaches of stochastic finite element method (SFEM) for partial differential equations with random input data. SFEM can also be applied to the analysis of composite structures [²¹].

Nonetheless, in recent years the study of SFEM apparently moved to both stochastic and multiscale considerations. To solve scale-coupling stochastic elliptic problems, Xu et al. [²²–²⁴] investigated the feasibility of the multiscale stochastic finite element method (MsSFEM) and got a quite satisfactory result. Kamiński [²⁵–²⁷] used a stochastic homogenization method to analyze the modelled composite structures with stochastic interface defects. The results shown the great efficiency of this method in modelling stochastic interface defects in composite materials in any geometrical scale. The stochastic homogenization method proposed by Sakata et al. [²⁸–³⁰] had a lower computational cost compared to the Monte-Carlo simulation. Ma and Zabaras [³¹–³²] developed an adaptive hierarchical sparse grid collocation method as an alternative to spectral stochastic finite element method to analyze the effects of random inputs in the solution of partial differential equations. This method was also more efficient than the traditional Monte Carlo and multi-element based random domain decomposition method. Upon all the researches, the research did by Hou and Liu [³³] was a quite perfect approach to elliptic partial differential equations. A recent surge of research on MsSFEM has given us new opportunities and challenges. However, there has thus far been relatively little application into material property research on RAC considering micro-structural uncertainties.

In this paper, a novel finite pixel-element method, in which pixels of digital images are used as elements, is proposed to simulate elastic properties of recycled aggregate concrete. Thousands of two-dimensional meso-scale RAC samples with random distribution of recycled aggregates are generated to make Monte Carlo simulation. After accuracy and efficiency of the proposed numerical method are validated, it is used to investigate the effects of replacement ratio, volume fraction, size, material property of recycled aggregates, position of natural aggregate, and proportion of old mortar, on macroscopic variability of elastic properties of RAC.

Random generation for RAC samples

The three-phase RAC model

RAC is usually considered to be five-phase composite material composed of natural aggregate, the old interface transition zone (ITZ), the old mortar, the new interface transition zone, the new mortar, as shown in Fig.1. Since the thickness of the new and old transition zone is generally 20‒40 µm at the microscale, the coarse aggregate size is usually above 5 mm at the mesoscale. In the finite element method, the specimen usually needs to be meshed at the corresponding scale, so the presence of ITZ area greatly increased the amount of computation. In this study, the influences of the new and old ITZ on the global elastic properties of recycled concrete are neglected. The recycled concrete model is simplified as three-phase material composed of natural aggregate, old mortar and new mortar. Results of verification test are in a good agreement with experimental data which means the effect of ITZ can be neglected when calculating the equivalent elastic modulus.

Random generation program

The random generation program considering arbitrarily distribution of recycled aggregate contains two steps, such as aggregate formation and random distribution of aggregates.

Aggregate formation

Shapes of the recycled aggregates may be very different. They can be simulated using various convex polygons. The random generation of convex polygon is mainly based on triangulation. It is assumed that V is a finite set of points on a real number field, and E is a closed line formed by the points in a set of points. Triangulation of the point set V, T= (V, E), on a plane graph G satisfies the following condition. First of all, except for the endpoints, the edges in the plane graph do not contain any points in the point set without intersecting edges. Secondly, all faces of the plane graph are triangular, and these triangles are assembled as a point set V. Finally, the corresponding points of V are connected as a convex polygon. The number of points can be arbitrarily defined. As shown in Figs. 2(a‒c), these convex polygons are generated form 10, 20, and 100 points, respectively. It can be seen that as the number of random points increases, the polygon becomes more plump and rounded. In this study, it is assumed that recycled aggregate is completely encapsulated by old mortar. One of recycled aggregate samples generated from the previous algorithm is shown in Fig. 2(d).

Random distribution of aggregates

After aggregate formation, recycled aggregate can be randomly distributed in the research domain. It has to meet the conditions of non-intersection, non-coincidence and non-inclusion among aggregates. Specific volume fractions of recycled aggregate are set to make the following analysis. All aggregates are required to be inside the specimen. In other words, the case in which there is any aggregate partly inside partly outside the specimen are eliminated. The number of aggregates in a specimen is computed form Walraven function given as follows

(1)

where P_k denotes the volume fraction of aggregates, D₀ the aggregate size, and D_max the maximum aggregate size. Figure 3 shows an example of 100 mm × 100 mm single-graded recycled aggregate concrete model with volume fraction of 40%. The number of points used to produce the previous recycled aggregate sample is 100.

Homogenization method

In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients. Of course, all matter is inhomogeneous at some scale, but frequently it is convenient to treat it as homogeneous. There are lots of homogenization methods, researchers have done lots of work on it [³⁴]. In this study, the method proposed in [³⁵] is used to obtain the effective elastic moduli of RAC samples generated previously. According to the Hill’s energy principle,

(2)\overline{\mathit{\sigma }}:\overline{\mathit{\epsilon }}={\displaystyle \frac{\mathrm{1}}{\left|V\right|}}{\displaystyle {\int }_{\mathrm{\Omega }}\sigma :\epsilon \mathrm{d}Y},

where \bar{\sigma } and \bar{\epsilon } represent equivalent stress and strain at coarse scale, \mathit{\sigma } and \mathit{\epsilon } stress and strain at fine scale. V denotes the volume of representative volume element, and Y the spatial coordinate of microstructure.

Boundary conditions of the rectangular RAC specimen are shown in Fig. 4. When the free edge has uniform strain of 1, the strain at coarse scale can be expressed as \overline{\mathit{\epsilon }}=\left[{\overline{\epsilon }}_{\mathrm{11}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{\epsilon }}_{\mathrm{22}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{\epsilon }}_{\mathrm{33}}\right]=[\mathrm{1}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{0}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{0}]. In elastic mechanics, the constitutive equation is given as

(3)\left[\begin{array}{l}{\bar{\sigma }}_{\mathrm{11}}\\ {\bar{\sigma }}_{\mathrm{22}}\\ {\bar{\sigma }}_{\mathrm{12}}\end{array}\right]=\left[\begin{array}{lll}{D}_{\mathrm{11}}& D_{\mathrm{12}}& \mathrm{0}\\ D_{\mathrm{12}}& D_{\mathrm{11}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& D_{\mathrm{33}}\end{array}\right]\left[\begin{array}{l}{\bar{\epsilon }}_{\mathrm{11}}\\ {\bar{\epsilon }}_{\mathrm{22}}\\ {\bar{\epsilon }}_{\mathrm{12}}\end{array}\right],

where

(4)D_{\mathrm{11}}=\{\begin{array}{ll}{\displaystyle \frac{E_{\mathrm{eff}}}{\mathrm{1}-v_{\mathrm{eff}}^{\mathrm{2}}}}& \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{\hspace{0.17em}}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\\ {\displaystyle \frac{(\mathrm{1}-v_{\mathrm{eff}})E_{\mathrm{eff}}}{(\mathrm{1}+v_{\mathrm{eff}})(\mathrm{1}-\mathrm{2}{v}_{\mathrm{eff}})}}& \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{\hspace{0.17em}}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\end{array},

(5)D_{\mathrm{12}}=\{\begin{array}{ll}{\displaystyle \frac{v_{\mathrm{eff}}{E}_{\mathrm{eff}}}{\mathrm{1}-v_{\mathrm{eff}}^{\mathrm{2}}}}& \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{\hspace{0.17em}}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\\ {\displaystyle \frac{v_{\mathrm{eff}}{E}_{\mathrm{eff}}}{(\mathrm{1}+v_{\mathrm{eff}})(\mathrm{1}-\mathrm{2}{v}_{\mathrm{eff}})}}& \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{\hspace{0.17em}}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\end{array},

(6)D_{\mathrm{33}}={\displaystyle \frac{E_{\mathrm{eff}}}{\mathrm{2}(\mathrm{1}+v_{\mathrm{eff}})}}\mathrm{.}

Stress at coarse scale can be calculated by

(7)\overline{\mathit{\sigma }}={\displaystyle \frac{\mathrm{1}}{\left|V\right|}}{\displaystyle \sum _{n=\mathrm{1}}^N{D}_{\mathrm{q}}{f}_{\mathrm{q}}},

where f_{\mathrm{q}} is the nodal reaction forces on the boundary of the RAC specimen, and D_{\mathrm{q}} is a geometric matrix that depends on the coordinates of the nodal point q which lies on the boundary of the model,

(8)D_{\mathrm{q}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left[\begin{array}{ll}\mathrm{2}{x}_{\mathrm{1}}& \mathrm{0}\\ \mathrm{0}& \mathrm{2}{x}_{\mathrm{2}}\\ x_{\mathrm{2}}& x_{\mathrm{1}}\end{array}\right],(x_{\mathrm{1}},x_{\mathrm{2}})\in \mathit{Y}.

The equivalent elastic modulus and the equivalent Poisson’s ratio can be obtained through Eq. (3) when the stress at coarse scale is got as D_{\mathrm{11}}={\bar{\sigma }}_{\mathrm{11}}/{\bar{\epsilon }}_{\mathrm{11}}={\bar{\sigma }}_{\mathrm{11}}, D_{\mathrm{12}}={\bar{\sigma }}_{\mathrm{22}}/{\bar{\epsilon }}_{\mathrm{11}}={\bar{\sigma }}_{\mathrm{22}}. Strain boundary condition can be transformed into displacement boundary condition through

(9)u={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{D}_{\mathrm{q}}^{\mathrm{T}}\left[\begin{array}{l}{\overline{\epsilon }}_{\mathrm{11}}\\ {\overline{\epsilon }}_{\mathrm{22}}\\ {\overline{\epsilon }}_{\mathrm{12}}\end{array}\right].

See [³⁵] for more details.

The finite pixel-element method

Random distribution of recycled aggregates brings a great challenge for free mesh generation of finite element modelling. When the distance between two aggregates is short, it is very difficult to generate elements using current commercial finite element software with low computational cost. In this paper, the pixel element method in which pixels of digital images are used as elements, is implemented in MATLAB. In the pixel element method, the research domain is meshed based on rectangular elements. Then, the generated RAC model is pixelated and projected into the research domain. The question is how fine the grid needs to be to achieve sufficient accuracy. To answer this question, a control experiment can be done. For example, a single-graded RAC specimen with an aggregate volume fraction of 40% can be meshed into from 30 ´ 30 to 100 ´ 100 or even more or less elements like Fig. 5. The representative aggregate size is 12.5 mm and the thickness of the old mortar is 2.5 mm. For elastic properties, the equivalent elastic moduli and Poisson’s ratio can be chosen as the coefficient to check the accuracy. For each kind of mesh, 1000 samples are generated to calculate the mean value and the coefficient of variation (COV) of the elastic moduli and Poisson’s ratio. The results are shown in Fig. 6, and the material parameters used here are shown in Table 1.

It can be seen from Fig. 6 that the mean value of the elastic moduli decreases as the grid becomes thinner and Poisson’s ratio shows a completely opposite trend. But the decreasing or the rising of the mean value caused by the difference of the number of elements is really small. However, in order to study the effect of microscopic randomness on macroscopic properties, the value of COV is much more important. In Fig. 6, the COV of both the elastic moduli and Poisson’s ratio decreases as the grid becomes thinner. When the number of mesh elements is less than 70 ´ 70, the decrease is obvious. And when the number of mesh elements is more than 70 ´ 70, the COV of both equivalent coefficients becomes steady. So a 70 ´ 70 pixel mesh is accurate enough in this case to get a stable COV so as to investigate the variability. The time cost of each control group is shown in Table 2. It can be found that the calculation time can be saved by using the appropriate element size. As the size of the RAC specimen is 100 mm ´ 100 mm, the appropriate element size is about 100 mm/70 ≈ 1.4 mm. However, as the thickness of the old mortar is 2.5 mm, the appropriate element size is smaller than it. So the case of which the thickness of the old mortar is less than 1.4 mm needs to be checked to see if this appropriate size still works. Another control experiment is done with the thickness of the old mortar of 1 mm and the results shown in Fig. 7 show that the previous conclusion is correct. It can also be seen from Figs. 6 and 7 that the mean value of the equivalent elastic parameters is of low mesh dependency but the COV is on the contrary.

Numerical experiments

Verification of finite element modeling

In this section, a comparison is made among results of in the present study and ones in literature [⁹,¹⁰] to verify the accuracy of the proposed finite pixel-element method. As shown in Table 3, the average error is approximately 6%, where the error is defined as

err={\displaystyle \frac{\mathrm{difference}\hspace{0.17em}\mathrm{between}\hspace{0.17em}\mathrm{experimental}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\mathrm{numerical}\hspace{0.17em}\mathrm{results}}{\mathrm{experimental}\hspace{0.17em}\mathrm{results}}},

where err denotes the error. It is indicated that results of the present study are in a good agreement with ones in literature [⁹,¹⁰]. And it is also indicated that in calculating the equivalent elastic modulus or Poisson’s ratio of the RAC specimen, a three-phase model which neglects the influence of the new and the old ITZ is workable. Besides the equivalent elastic modulus and Poisson’s ratio calculated by the RAC model considering the random distribution of aggregates is always between the results calculated by the series model and the parallel model which defines the upper and lower bounds in modeling the elastic properties. So the results get by the RAC model can be used to investigate the effect of microscopic parameters on macroscopic elastic properties.

Monte Carlo simulation

Monte Carlo method provides a convenient way to get the mean value and coefficient of variation of the elastic moduli and Poisson’s ratio. In this section, a series of Monte Carlo simulations are made to explore the effects of recycled aggregate replacement rate, aggregate volume fraction, the unevenness of old mortar, proportion of old mortar, aggregate size and elastic modulus of aggregates on the variability of global behavior of concrete. The number of samples in each Monte Carlo simulation is 500. First, a control experiment needs to be done to check if the number of 500 samples is large enough to get a stable mean value and COV, and of course the statistical distribution of elastic properties is also considerable. In this control experiment, four control groups with different material and geometric parameters are tested. The parameters used here are shown in Table 4.

The convergence results

The convergence of the equivalent elastic moduli and Poisson’s ratio of 500 samples is shown in Fig. 8. It can be seen from Fig. 8 that both the mean value and the coefficient of variation of equivalent elastic moduli and Poisson’s ratio of four control groups change sharply when the number of samples is below 100. And when the number of samples is larger than 300, the results tend to be stable which means a 500-sample Monte Carlo simulation for equivalent elastic properties of RAC is enough to get a credible stable mean value and COV. As four control groups have different material and geometric parameters which covering the parameters that will be studied in the subsequent numerical experiments, a 500-sample Monte Carlo simulation is workable for all these cases.

The probability distribution of the results

The probability distribution of the equivalent elastic moduli and Poisson’s ratio is also worth studying. In statistics, there are many ways to visually give a rough estimate of the probability distribution. The most commonly used method is to plot the probability distribution of frequency (PDF) histogram. When the number of segments is chosen to be appropriate, the histogram can roughly reflect the shape of the probability distribution. According to a large number of statistical experience, the number of segments can be chosen by the following equation

(9)num=\mathrm{1}+{\log }_{\mathrm{2}}N,

where N is the number of sample. In this study, num = 1+ log₂500= 10.

The histogram of the mean value and COV of the equivalent elastic moduli and Poisson’s ratio of the four control groups is shown in Fig. 9. From Fig. 9 it can be found that the histogram shows single peak characteristics. So it can be fitted by this king of distribution such as normal distribution, logarithmic normal distribution, Weibull distribution and so on [³⁶]. Here the normal distribution is used to fit due to its convenience as if the mean value and the standard deviation is known, the probability distribution is determined. Though in Fig. 9 the normal distribution seems to fit well, it still needs to be checked. In statistics, the Kolmogorov-Smirnov test is an effective method to test fitting effect. Let the results of the sample be arranged from small to large. What is going to be tested is F_n(x) = F(x) where F_n(x) is the sample cumulative frequency and F(x) is the supposed cumulative frequency. If the following formula is tenable, then the distribution fitting is passed.

(10)

where D_nα is the critical value when the significance level is α. By using OriginPro, a convenient data processing software, the normality test is very easy to be taken out. The results show that at the 0.05 level, the data of all the four control groups is significantly drawn from a normally distributed population. Table 5 lists the data obtained from the test. Besides the Kolmogorov-Smirnov test, the goodness of fit can also show the fitting effect. The goodness of fit (R²) is defined as a ratio of two statistical data which are the total sum of squares (TSS) and explained sum of squares (ESS). Here

(11)\mathrm{TSS=}\mathrm{\Sigma }{y}_i^{\mathrm{2}}=\mathrm{\Sigma }{(Y_i-\overline{Y})}^{\mathrm{2}},

(12)\mathrm{ESS=}\mathrm{\Sigma }{\widehat{y}}_i^{\mathrm{2}}=\mathrm{\Sigma }{({\widehat{Y}}_i-\overline{Y})}^{\mathrm{2}}.

The closer R² is to 1, indicating that the actual observed points are closer to the fit distribution and the goodness of fit is better. Figure 6 lists out the mean value, standard deviation, COV and the goodness of fit obtained by the Monte Carlo simulation. From the results we can see that R² of all the four groups are larger than 0.94 which again proves that the equivalent elastic moduli and Poisson’s ratio obtained by Monte Carlo simulation can be fitted by the Gauss normal distribution.

Results and discussion

The effect of recycled aggregate replacement rate

Since the properties of RAC are usually not as good as the natural aggregate concrete, the natural aggregate concrete is usually partially replaced by RAC. In this section, replacement rates of 0%, 25%, 50%, 75%, 100% are set in the following Monte Carlo simulations. In addition, the diameter of recycled aggregate and original natural aggregate is set as 25 mm, the size of natural aggregate inside recycled aggregate 20 mm, and the volume fraction 40%. The other material coefficients are the same as shown in Table 1. The mean value and COV of the equivalent elastic moduli and Poisson’s ratio obtained by 500-sample Monte Carlo simulation of each control group is shown in Fig. 10. It is indicated that as the replacement rate of recycled aggregates increases, the mean value of the equivalent elastic moduli decreases, but the mean value of Poisson’s ratio increases. The mean value of equivalent elastic moduli of replacement rate 0% is about 13% higher than one of replacement rate 100%. However, COV of both parameters decreases as the replacement rate increases. It might because the properties of old mortar is worse than the natural aggregate, but more like new mortar. As the replacement rate increases, difference between the various phase decreases. Here the effect of the stoma and microcracks in old mortar is neglected as it belongs to the micro scale but in this paper the meso scale influence is what we concerned.

The effect of volume fraction of recycled aggregate

Volume fraction of 10%, 15%, 20%, 25%, 30%, 35%, 40%, 45% and 50% are set in the following Monte Carlo simulations. The replacement rate of recycled aggregate is set as 100%, the diameter of recycled aggregate 25 mm, and the diameter of initial natural aggregate 20 mm. The other material parameters are the same as shown Table 1. The effect of volume fraction of recycled aggregate on the mean value and COV of the elastic properties is shown in Fig. 11. It is indicated that the mean value of the equivalent elastic moduli increases as volume fraction of recycled aggregate increases but the mean value of Poisson’s ratio shows an opposite rule. For the equivalent elastic moduli, 40% volume fraction increasing causes the mean value growing up about 20%. The COV of both the equivalent elastic moduli and Poisson’s ratio increases as the volume fraction of recycled aggregate increases. When the volume fraction of RA increases from 10% to 30%, increase of COV is considerable. But COV tends to stabilize after the volume fraction is higher than 30%. One of reasons might be that as the volume fraction of aggregates increases, uncertainty caused by position of aggregates has a peak of growth. And in this Monte Carlo simulation, 30% is the value of the peak. In the actual civil engineering, the volume fraction of aggregates in concrete is always higher than 50%, so the variability cause by the volume fraction can be described by a constant.

The effect of the unevenness of old mortar

Since the current recycled aggregate processing level is limited, the old mortar of the RA is not evenly covering natural aggregate. In some special cases, the natural aggregate may be exposed to the new mortar such as Fig. 12. This kind of unevenness will also have impact on macro variability. In this section, a Monte Carlo simulation comparison is made between recycled aggregates with uniformly position of natural aggregates and ones with randomly positions of natural aggregates. Volume fraction are set as 10%, 20%, 30%, 40% and 50%. The results are shown in Fig. 13. From Fig. 13 we can see that for the COV of both the equivalent elastic moduli and Poisson’s ratio, the unevenness control group is larger than the uniform group, and the difference is getting even larger when the volume fraction grows up. When the volume fraction is 30%, the difference between the two groups is around 10%. It is demonstrated that the effect of the unevenness of the old mortar might be considerable.

The effect of proportion of old mortar

In the following Monte Carlo simulations, proportions of old mortar in recycled aggregates are set as 0%, 19%, 36%, 51% and 64%, the diameter of recycled aggregate 25 mm, the volume fraction 40% and the replacement rate of RA 100%. The effect of proportion of old mortar on the macro variability is shown in Fig. 14. It is found that elastic modulus of recycled aggregate decreases as proportion of old mortar increases and Poisson’s ratio increases as proportion of old mortar increases. In addition, COV of both parameters decreases as the proportion increases. The conclusion is nearly the same as what we have got in section 5.1.

The effect of aggregate size

Aggregate gradation is related to aggregate size. The study shows that multi-grade concrete is more compact and easier to configure high-strength concrete. So it is necessary to investigate the effect of the aggregate size on macro variability. In this section, aggregates’ diameters are set as 10 mm, 15 mm, 20 mm, 25 mm, 30 mm in the following Monte Carlo simulations. The proportion of old mortar is set as 51%, and the volume fraction of aggregate 50%. The results are given in Fig. 15. It can be found that the mean value of both the equivalent elastic moduli and Poisson’s ratio seems to be irregular. But the mean values of different aggregate size group are nearly the same. That’s to say if the volume fraction of each phase is the same, the mean value of 500-sample Monte Carlo simulation will also be the same. However, the COV of both parameters increases as the aggregate size increases. It is because as the aggregate size increases, the total number of aggregates decreases, and the effect of a single aggregate’s position randomness increases. So if we take the reliability into consideration, the equivalent elastic moduli and Poisson’s ration will decrease as the aggregate size increases.

The effect of elastic moduli of natural aggregate

Different sources of waste concrete will lead to the difference between the properties of recycled aggregates. In this section, the difference of the elastic moduli is considered. Elastic moduli of natural aggregates are set as 7.5 GPa, 15 GPa, 22.5 GPa, 30 GPa, 60 GPa, 90 GPa, 120 GPa and 150 GPa in the following Monte Carlo simulations. The volume fraction is set as 40%, and the diameter of recycled aggregate 25 mm, the size of natural aggregate inside recycled aggregate 20 mm. Results are shown in Fig. 16. It can be seen form Fig. 16(a) that the mean value of equivalent elastic moduli increases as the elastic moduli of natural aggregate increases. Besides, form Fig. 16(b), we can see that the change of elastic moduli of natural aggregate will also cause the change of equivalent Poisson’s ratio. The mean value decreases when the elastic moduli of natural aggregates change from 7.5 GPa to 22.5 GPa. When the elastic modulus of natural aggregates is larger than 22.5 GPa, the mean value increases. The COV of both parameters have the same change regulation. In total, it can be described as the COV becomes large when the difference of the elastic moduli between the aggregates and mortar becomes large. From Fig. 16, it is also found that when elastic modulus of natural aggregate is getting less than one of new mortar, COV increases dramatically. In recycled concrete, there are lots of stoma and microcracks which can be seen as the elastic moduli is very small, nearly 0. And they will make the COV very large. In other words, the stoma and microcracks have significant impact on macro variability. This could be a research content in the future.

Discussion of the future work

In section 5.2.2, a conclusion has been drawn that the equivalent elastic moduli and Poisson’s ratio is nearly follow the normal distribution. So if a large size specimen, such as 1 m², needs to be calculated for its elastic properties, it could be mesh into 100 mm² coarse grid. And the elastic parameters could be given by normal distribution obtained by 500-sample Monte Carlo simulation on small specimen, as shown in Fig. 17. This can preserve the inhomogeneous nature of the concrete.

The research on reliability can use the COV and mean value calculated in section 6. As the conclusion drawn previously, the value with reliability can be obtained from normal distribution table by standard normalization.

Though the random aggregate model and the pixel-element method has been verified in sections 4 and 5, it is worth performing sensitivity analysis on this model and method to quantify the quality of mechanical models soon. Related work can refer to literature [³⁷–³⁹].

Conclusions

In this study, a novel finite pixel-element method is proposed to make Monte Carlo analysis for effect of meso structure on macro elastic properties of recycled aggregate concrete. Representative RAC models are randomly generated with various distribution of aggregates. Based on homogenization theory, effects of recycled aggregate replacement rate, aggregate volume fraction, the unevenness of old mortar, proportion of old mortar, aggregate size and elastic modulus of aggregates on overall material elastic properties are investigated. Results are in good agreement with experimental data in literature. The conclusions made in this study are summarized as follows.

(1) In finite pixel-element method, a 70 × 70 mesh for a 100 mm × 100 mm specimen is accurate enough to obtain the equivalent elastic parameters.

(2) The elastic properties of RAC with different meso structure follow the normal distribution, it can also be fitted by lognormal distribution or Weibull distribution.

(3) The change of meso structure has signification effect on variability of macro elastic properties.

(4) In civil engineering, the effect of the volume fraction of RA on variability of macro elastic properties can be described by a constant.

(5) The effect of the aggregate size on mean value of the equivalent elastic properties is very weak, but the COV of both parameters increases as the aggregate size increases.

(6) Material properties of recycled aggregate concrete might depend on ratio between elastic moduli ratio among various constituents.

(7) The existence of stoma and micro cracks in RAC might cause a large variability on the macro elastic properties.

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