Investigation of the permeability of porous concrete reconstructed using

Investigation of the permeability of porous concrete reconstructed using probabilistic description

methods

Sang-Yeop Chung a ,Tong-Seok Han a ,?,Se-Yun Kim a ,Tae-Hyung Lee b

a Department of Civil and Environmental Engineering,Yonsei University,50Yonsei-ro,Seodaemun-gu,Seoul 120-749,Republic of Korea b

Department of Civil Engineering,Konkuk University,120Neungdong-ro,Gwangjin-gu,Seoul 143-701,Republic of Korea

h i g h l i g h t s

The void distribution inside porous concrete specimen is investigated using CT image. Probability functions are adopted to quantify and to reconstruct porous concrete. Reconstructed virtual specimens have the same characteristics and material responses.

a r t i c l e i n f o Article history:

Received 26April 2014

Received in revised form 9June 2014Accepted 12June 2014

Available online 9July 2014Keywords:

Porous concrete Void distribution CT image

Reconstruction

Probability functions Permeability Phase clustering

a b s t r a c t

Porous concrete has a high void ratio with continuously interconnected void clusters.The void distribu-tion in concrete strongly affects its physical properties,such as strength and percolation.To investigate and quantify the spatial distribution of voids inside porous concrete specimens,computed tomography (CT)images and low-order probability functions can be used.In this study,we reconstruct porous concrete specimens with different void distributions using low-order probability functions.The void distributions of the original and reconstructed porous concrete specimens should exhibit almost the same statistical characteristics.We con?rm that reconstructed porous concrete specimens generated using the proposed probabilistic optimization process have statistically identical characteristics,and exhibit similar material behaviors as with the original model.These reconstructed specimens can be uti-lized for numerical experiments so as to reduce the number of time-consuming real experiments.

1.Introduction

Porous concrete contains a large number of voids.These allow water to drain through the material and develop opened voids in the matrix.Cast concrete contains a large number of voids,and those inside porous concrete are usually dispersed as interconnected clusters.The spatial distribution of these voids has a signi?cant effect on the material’s properties,such as strength [1,2]and perco-lation [3,4].Thus,investigating the void distribution of concrete specimens is important in understanding the characteristics of por-ous concrete.To describe the internal void distribution without damaging the specimen,image-based methods such as scanning electron microscopy (SEM),optical microscopy (OM),and computed tomography (CT)can be used.SEM and OM only provide limited information from their cross-sectional images,and the specimen

can be damaged during preparation processes,such as polishing and drying.In this research,a CT imaging method is utilized to identify the spatial distribution of voids inside porous concrete specimens without in?icting any damage.CT imaging has been widely utilized to investigate the characteristics and properties of porous media.Lu et al.[4]used CT images to investigate the correla-tion between pore connectivity and chloride migration,and Gallucci et al.[5]adopted CT imaging to evaluate the degree of the pore network in hydrated cement.Masad et al.[6]also employed CT to examine the permeability and anisotropy of an asphalt https://www.360docs.net/doc/413802763.html,ing X-ray CT imaging technology,which is a non-destructive and non-invasive method,a series of cross-sectional images of a porous concrete specimen can be generated.We perform a CT imag-ing method that converts images from 8-bit to binary to describe the voids inside the porous concrete specimen.Here,three-dimensional (3D)porous concrete images are generated by stacking two-dimensional (2D)binary images.These are used to evaluate the spatial distribution of the voids inside porous concrete specimens and their reconstructed samples.

?Corresponding author.Tel.:+82221235801;fax:+8223645300.

E-mail address:tshan@yonsei.ac.kr (T.-S.Han).

To describe the characteristics of the void distribution in porous concrete specimens,this research adopts probabilistic methods. Low-order probability-based functions,such as the two-point cor-relation[7,8],lineal-path[9,10],and two-point cluster[10,11] functions that use only a small amount of data to obtain statistical information,can be used to evaluate the spatial distribution of voids in concrete https://www.360docs.net/doc/413802763.html,ing the two-point correlation func-tion,the degree of phase clustering can be evaluated,and the phase connectivity along a speci?c direction can be identi?ed from the lineal-path function[3,10,12].The two-point cluster function is used to describe the material characteristics related to phase con-tinuity,such as permeability and diffusivity[13].

Low-order probability functions can also be used to reconstruct materials.Reconstruction is a methodology whereby a series of models are generated with statistically identical phase distribu-tions using the given material characteristics[10,14].Numerous studies have shown that the reconstruction method can generate samples with almost the same characteristics and material responses using limited probabilistic information[12,14–18].The objective of reconstruction using low-order probability functions is not to produce an identical material to the original,but to gen-erate a sample with similar phase distribution characteristics [10,12].

In this study,the characteristics of the void distribution and material response of porous concrete specimens are evaluated. CT imaging is adopted to obtain void images of porous concrete specimens with different void ratios and clustering distributions, and each void image is considered as a representative volume ele-ment(RVE)of the concrete specimen.The spatial distribution of voids in the specimens is quantitatively described using low-order probability functions.A set of porous concrete specimens that have statistically identical void distributions are generated using the reconstruction method.The characteristics of the void distribution in the original and reconstructed samples are compared to validate the reconstruction method,and the effect of the degree of void clustering on the reconstruction and material characteristics is examined.In addition,the hydraulic conductivity of porous con-crete models is computed to demonstrate the similarity between experimental and simulation data as well as the similarity between the material responses of the original and reconstructed samples. Virtual porous concrete specimens and simulation tools used in this study can be adopted to supplement time-consuming real experiments.

2.Characterization of the void distribution in porous concrete using CT imaging and probability functions

2.1.Porous concrete specimens and CT imaging

Porous concrete is designed to allow water to drain through it, reducing the runoff from a site and allowing groundwater levels to recharge[1,2].Therefore,the cast concrete should contain a large amount of voids.Identifying the void distribution in porous con-crete is dif?cult without damaging the specimen.In this study, we use CT imaging to investigate the spatial distribution of voids without destroying the specimen.CT uses X-rays to produce tomo-graphic images of the scanned object without causing any damage. Fig.1shows the process of CT imaging.Fig.1(a)and(b)are the cylindrical porous concrete specimen and its cross-sectional image. Fig.1(b)can be expressed as an8-bit image in which each pixel is represented by a value between0and255(a total of256values), where0is black and255is white.From the8-bit image,the binary image in Fig.1(c)is generated to classify the voids inside the spec-imen by selecting an appropriate threshold value.In the8-bit image,the pixels with values below the threshold are considered to be voids.In this study,we set the threshold to35on the basis

of experimental results and the Otsu method[19],i.e.,a method

for selecting a threshold by minimizing the cluster variance.In

Fig.1(c),the white regions represent inner and outer voids,and

the black regions are solid parts of the specimen.The binary image

is composed of200?200pixels.The3D image of the porous con-

crete specimen is generated by subsequently stacking430bina-

rized images along the z-direction,as shown in Fig.1(d).Fig.1(e)

is the RVE of the cylindrical specimen.The RVE is selected from

the cylindrical specimen and appropriately rescaled using the

image processing toolbox in MATLAB[20]to enable ef?cient sim-

https://www.360docs.net/doc/413802763.html,ing the image in Fig.1(e),the void distribution and

anisotropy of the porous concrete specimen can be identi?ed.

In this study,three porous concrete specimens with different

void ratios are used to evaluate the void distribution and material

characteristics.Fig.2shows the cylindrical porous concrete speci-

mens and their RVEs generated from the procedure illustrated in

Fig.1.The RVE void ratios based on the number of voxels in

Fig.2are29.11%(Specimen1),28.74%(Specimen2),and22.78%

(Specimen3),and the errors between mix design and numerical

analysis are negligible.To quantitatively describe the void distribu-

tion in the specimen,the probabilistic functions presented in the

following section are used.

2.2.Probability functions for material characterization

An appropriate method is needed to investigate the spatial

distribution of voids in porous concrete specimens.In this study,

low-order probability functions,such as the two-point correlation,

lineal-path,and two-point cluster functions,originally used to

examine the phase distribution of random heterogeneous materi-

als[10],are used to characterize the void distribution.Detailed

descriptions of each function can be found in[3,8,11].

2.2.1.Two-point correlation function

In this study,we use the two-point correlation function

P ijer;h;/Tto investigate the void distribution.P ijer;h;/Tis the prob-ability that any two points are located in phases i and j[8,10,12],

where r is the distance between two points,h is the angle between

a test line and the z-axis,and/is the angle between the projection

of a test line on the xy-plane and the x-axis.As an example of the

binarized concrete image shown in Fig.3(a),i or j in P ijer;h;/Tcan be either void(v)or solid(s).The two-point correlation function only considers phase information at the end points of the test line;

therefore,any additional information between two points is not

included in this function.The limits for the two-point correlation

function are:

lim

r!0

P iier;h;/T?f i;lim

r!0

P ijer;h;/T?0ei–jTe1T

lim

r!1

P iier;h;/T??f i 2;lim

r!1

P ijer;h;/T??f i ?f j ei–jTe2Twhere f i is the volume fraction of phase i.In Eq.(1),when the distance between two points approaches zero,the probability of ?nding the same phase at both points converges to the phase vol-ume fraction.Eq.(2)indicates that P ijerTconverges to the product of the phase volume fraction as r increases.In this study,we apply the two-point correlation function suggested by Gokhale et al.[8], which is an analytical form of the two-point correlation function based on statistical information[21].

P ijer;h;/T?f i f j?1àexp fàe?P Leh;/T

=e2f

f jTTr

g ei–jTe3TP iier;h;/TtP ijer;h;/T?f ie4T

In Eq.(3),?P Leh;/T

is the number of intersections between a test line and the i–j phase interface per unit test line length.For a two-phase material,the probability that both phases are located

S.-Y.Chung et al./Construction and Building Materials66(2014)760–770761

in the same(i th)phase,P ii,can be obtained from P ij(Eq.(3))and Eq.(4).In this paper,the two-point correlation function P vver;h;/Tis only employed to characterize the void distribution, where the subscript v denotes the voids.

2.2.2.Lineal-path function

The lineal-path function L ier;h;/Tgives the probability that a randomly placed line segment of length r lies entirely in phase i [22,23].In Fig.3(b),L aerTand L berTare candidates for the lineal-path function,but L cerTis not regarded as the lineal-path function either for phase v or phase s,because a line segment is not located in a single https://www.360docs.net/doc/413802763.html,ing the lineal-path function,the continuous connectivity of the same phases in a speci?c direction can be identi?ed.The limits for the lineal-path function are:

lim

r!0

L ier;h;/T?f ie5T

lim

r!1

L ier;h;/T?0e6T

In this study,the lineal-path function for voids L ver;h;/Tis only considered for the phase distribution analysis.To calculate the lineal-path function for porous concrete,we adopt the method sug-gested by Coker and Torquato[9,10].From the lineal-path function, the connectivity of speci?c constituents can be examined,because the function considers the entire line segment between the two end points of the test line.

specimen:(a)cylindrical porous concrete specimen,(b)8-bit cross-sectional image,(c)binarized concrete specimen.(Note:In the RVE,the light gray voxels are solids,and the black voxels are voids.)

concrete specimens and their RVEs:(a)Specimen1,(b)Specimen2,(c)Specimen3.(Note:In RVEs,the

Two-point cluster function

two-point cluster function is adopted to examine connectivity of both the phase distribution and the phase clusters.Speci?cally,the two-point cluster function C i er ;h ;/Tgives probability of ?nding any two points in the same cluster of [10,13].Using the two-point cluster function,we can identify clusters in a phase,even when the phase connectivity speci?c direction is interrupted by another phase.In Fig.3example,C a er T;C b er T,and C c er Tare considered as lineal-paths because these line segments are located in one phase.However,and C b er Tare not the same candidates for the two-point relation function,because they are in different solid clusters.Schematics of phase distribution functions (white region:void solid (s )):(a)schematic of P ij er T,(b)schematic of L i er T,(c)schematic Overall void distributions:(a)Specimen 1(void ratio =29.11%),(b)ratio =28.74%),(c)Specimen 3(void ratio =22.78%).(Note:The gray ?gures represent voids.)

case of C derT,it is not a lineal path because a whole line segment is not positioned in one phase.However,it is a candidate for the two-point cluster function,as both end points of a test line are located in the connected void cluster.The general limits for the two-point cluster function are

lim

r!0

C ier;h;/T?f ie7T

lim

r!1

C ier;h;/T?0e8T

In this study,Lee and Torquato’s method[11]is used to obtain the two-point cluster function.This can provide information about the connectivity of both phases and clusters;therefore,it can be utilized to investigate materials that require channels through which a?ow can pass,e.g.,porous concrete.In this study,for the analysis,we are only interested in the two-point cluster function for voids C ver;h;/T.

Using these low-order probability functions,we can identify characteristics of the spatial distribution of voids in porous

764S.-Y.Chung et al./Construction and Building Materials66(2014)760–770

concrete specimens,such as the degree of void clustering and connectivity.In addition,these methods can be utilized in the reconstruction method presented in the following section.

3.Reconstruction and numerical simulation of porous concrete specimens

3.1.Reconstruction method

The reconstruction method is a stochastic realization of proba-bilistic descriptions obtained from the original https://www.360docs.net/doc/413802763.html,ing the reconstruction method with low-order probability functions,we obtain limited information about the phase distribution and gener-ate a series of statistically identical materials.The materials obtained using the reconstruction method exhibit almost the same characteristics and properties[10].Yeong and Torquato[14,15], Kumar et al.[18],and Cule and Torquato[24]performed micro-structure reconstruction to generate statistically equivalent mate-error between the probability functions of the original and tempo-rary materials,is then calculated using the low-order probability functions(P vverT;L verT,and C verT).The virtual energy function E is de?ned as follows:

?^f a

erTàf a

erT 2e9T

where^f a

erTand f a

erTare probability functions for the temporary and original materials.In Eq.(9),a denotes the type of probability function and b indicates the directional component.r represents the distance between two points.In this study,we only consider the probability functions along three perpendicular directions(x, y,and z directions)to calculate E.

After calculating E,we choose voxels from each phase in the temporary material(one from a void,and the other from a solid part),and exchange them to modify the material.The virtual energy E0for the modi?ed temporary material is calculated using Eq.(9).Acceptance of the phase interchange is determined by

specimen and reconstructed samples for each porous concrete specimen.(Note:Recon1,2,and3are reconstructed porous concrete specimens

states.In RVEs,the light gray voxels represent solids,and the black voxels are voids.)

S.-Y.Chung et al./Construction and Building Materials66(2014)760–770765

In this research,the tolerance of E is set to0.003as the ground state.The virtual temperature T is1.0,and is gradually decreased by50%after every5000iterations.To improve the ef?ciency of the void clustering process,the phase interchange is only permit-ted near the boundary of a cluster after120,000iterations.

3.2.Numerical simulation for hydraulic conductivity

To verify the similarity of material responses between the origi-nal and reconstructed porous concrete samples,their hydraulic conductivities are compared using?nite element analysis.Flow simulations have been conducted to evaluate the hydraulic characteristics of various materials[25–27].Here,the?nite ele-ment analysis is performed using the ABAQUS package[28].The Navier–Stokes equation for incompressible?uid motion within porous media is as follows:

q w @v

tq wevá5Tvt5pàg52v?àq w g5ze11T

where q w is the?uid density,v is the?uid velocity tensor,and5p is the pressure gradient.g is the?uid dynamic viscosity,g is the gravitational acceleration,and z is the elevation head.Eq.(11)is combined with a continuity equation(5áv?0)and appropriate boundary conditions.To estimate the macro-scale?uid?ow,we can use Darcy’s law.This describes the?uid?ow in porous media by correlating the effective velocity vector( v)with the hydraulic gradient via the hydraulic conductivity k,and is presented as:

v?àká5he12Twhere5h is the effective hydraulic gradient.By upscaling the Navier–Stokes equation(Eq.(11))to Darcy’s law(Eq.(12)),effective hydraulic conductivity values can be obtained.Details of this proce-dure are given in[25].

In this study,the RVEs of the porous concrete specimens in Fig.2are used for the?uid simulation.The edge length of the RVEs is50mm.Each RVE is discretized into150voxels in x,y,and z directions to maintain computational ef?ciency,and each voxel is represented by a8-noded brick element.The input variables are the water density q w?1000kg=m3,dynamic viscosity g?0:001Pa s,and room temperature of293°K.The pressure boundary conditions are imposed on the porous concrete speci-mens along the z direction,and the pressure drop4p between the inlet(top)and outlet(bottom)surfaces is set to10.0Pa.The other?uid boundaries are speci?ed as no slip.

4.Evaluation of characteristics of porous concrete specimens

The spatial distribution of voids and hydraulic conductivity of the porous concrete specimens in Fig.2are now investigated. The RVE of each specimen used in this study is denoted by‘Speci-men.’Fig.4is the overall void distribution of the RVE in Fig.2. Characteristics of the void distribution in the RVE are described using low-order probability functions.The statistical similarities and material responses of the original and reconstructed speci-mens are compared using probabilistic methods and?nite element analysis.

4.1.Probability functions of the porous concrete sample

Each RVE in Fig.2has a different void ratio and spatial distribu-tion,as shown in Fig.4.In this?gure,the black region represents the void distribution inside the porous concrete specimen.The void distribution in the porous concrete strongly affects the hydraulic conductivity and percolation through the material.To characterize the spatial distribution of voids,we use the low-order probability functions described earlier.Fig.5shows the probability functions for Specimen1(Fig.4(a)).For all specimens,probability functions are obtained for the x,y,and z directions.In Fig.5,D is the edge length of the specimen and r is the distance between the two end points of a test line.Fig.5(a)shows that the values of the two-point correlation function for voids P vverTalong the each direction are almost identical;this indicates that Specimen1has an isotropic void distribution.In Fig.5(b),the lineal-path function for voids L verTthat can describe the connectivity of the voids is almost the same in the x,y,and z directions.This allows us to examine the isotropy of continuously connected voids.The plot

766S.-Y.Chung et al./Construction and Building Materials66(2014)760–770

of the two-point cluster function C verTwhich can describe the connectivity of the void clusters in Fig.5(c)demonstrates that Specimen1has void clusters percolating through the specimen in the x,y,and z directions.For porous concrete,the existence of a void cluster that passes through the specimen is important to ensure good channel?ow.C verTcan also be used to identify the isotropic connectivity of void clusters.

The probability functions for other specimens(Fig.4(b)and(c)) also indicate isotropic void distributions in the x,y,and z https://www.360docs.net/doc/413802763.html,ing probability functions,the void distribution in porous concrete specimens can be quantitatively compared,as shown in Fig.6.In Fig.6(a),P vverTand the magnitude of the initial slope increase with the void ratio of the specimen for the same value of r=D.This indicates that porous concrete with a greater void ratio contains more clustered voids inside the specimen.Fig.6(b)shows that the value of L verTat the same r=D increases with the void ratio. It can also be veri?ed from Fig.6(b)that L verTis zero when r=D is over0.4,because there are no void clusters passing through the specimen parallel to the axial direction.However,Fig.6(c)shows that C verTdoes not converge to zero;thus,continuously connected void clusters exist that pass through the whole specimen.As shown in Fig.6(c),void clusters that are continuously connected through the specimen can be identi?ed using the two-point cluster

S.-Y.Chung et al./Construction and Building Materials66(2014)760–770767

function.These void clusters are strongly related to the hydraulic characteristics of the porous concrete specimen,as they ensure the existence of a channel for?uid?ow.The results shown in Figs.5 and6demonstrate that characteristics of the void distribution related to physical properties of porous concrete can be predicted from probabilistic description methods.

4.2.Reconstruction of porous concrete specimens

Reconstruction of the porous concrete specimens is performed using the optimization process in Section3.1.RVEs with different void ratios(Fig.2)are used as the original samples for the recon-struction process.The void ratios of the specimens are29.11% (Specimen1),28.74%(Specimen2),and22.78%(Specimen3), and each specimen is composed of27,000voxels(30per side of the specimen).All the original specimens have an isotropic void distribution with respect to the x,y,and z directions.

We use P vverT;L verT,and C verTin the reconstruction process. Previous studies have shown that reconstruction using two-point specimens are not exactly the same;however,their probabilistic characteristics in the x direction are very similar,as shown in https://www.360docs.net/doc/413802763.html,paring P vverT;L verT,and C verTin Fig.8shows that the degree of void clustering(P vverT),void continuity(L verT),and clus-ter connectivity(C verT)are(statistically)almost identical in the original and reconstructed samples,even though a difference in C verTappears as r=D increases.Likewise,for the case of Specimens 2and3(second and third rows in Fig.7),the original and three reconstructed specimens also exhibit similar probabilistic charac-teristics,as shown in Figs.9and10.In addition,all the specimens in Fig.7are random heterogeneous materials that have an isotropic void distribution;therefore,for each specimen,the char-acteristics of the void distribution in the y and z directions are the same as for the x direction.

In this study,the convergence rate of the virtual energy E increases as the void ratio of the specimen decreases.The recon-struction method used here starts with a randomly distributed void(phase),and the randomness decreases as the reconstruction progresses.Thus,the convergence rate also decreases as the void ratio and clustering size increase.These investigations con?rm that

contour of the original and reconstructed porous concrete samples.(Note:Fluid percolates from top to bottom of each sample.The

velocity(cm/s).Visualized regions including the transparent regions represent the void/?uid phase,and the velocity contour larger than

regions to emphasize effective percolation paths.)(For interpretation of the references to color in this?gure legend,the reader is referred 768S.-Y.Chung et al./Construction and Building Materials66(2014)760–770

perpendicular planar slices.The velocity is normalized according to the maximum local velocity of each specimen.The average velocity of the specimen increases as the void ratio increases,because a lar-ger void region ensures wider channel?ow;?uids cannot?ow smoothly through a narrow void region.In Fig.11,regions of high ?uid velocity are shown in red,and regions of low?uid velocity are blue.

A comparison of experimental effective hydraulic conductivities with those obtained through our simulations is shown in Table1. The effective vertical hydraulic conductivity,k e,is calculated by averaging the local velocity values at the inlet and outlet planes. In the?gure,the differences between the experimental and the simulation results of the hydraulic conductivity for all cases are within11%;it can be con?rmed that the simulation results of the specimens are in reasonable agreement with the experimental results.The hydraulic conductivity in both the experiment and the simulation decreases as the void ratio of the specimen decreases.This is due to the reduction in channel?ow,and it dem-onstrates that the hydraulic conductivity is strongly affected by the void ratio.To verify the similarity of the material responses,the hydraulic conductivity of the original and reconstructed specimens is compared in Fig.12.As demonstrated in Figs.8–10,the recon-structed porous concrete specimens have almost the same statisti-cal characteristics.In Fig.12,the hydraulic conductivity of the reconstructed samples decreases as the void ratio decreases and deviates from that found in the experiment for small void ratios. This is because the void channel that ensures the?ow is not fully reconstructed for the case of small void ratios;the difference in hydraulic conductivity between the three specimens is relatively small.These results imply that the material responses of the spec-imens are almost identical.Although not shown in this paper,the hydraulic conductivity of the original and reconstructed samples in other directions(x and y)exhibits a similar tendency.The hydrau-lic conductivity of insulating concrete is in?uenced by the void dis-tribution as well as the void ratio,and other material properties are also strongly affected by the void distribution[29,30].Fig.12mainly demonstrates the effect of the void ratio on the hydraulic conductivity when the original and reconstructed samples have similar void distribution.The effect of the void distribution on the hydraulic conductivity was not systematically investigated in this study,and is left for further studies.

The similarities of these material responses of porous concrete specimens are consistent with their probabilistic characteristics. From the results,we can con?rm that reconstructed porous concrete specimens with the same void ratio and void distribution characteristics exhibit an almost identical material response.The validation of the reconstruction method shows that reconstructed specimens can be used for numerical experiments so as to reduce the number of time-consuming real experiments.

5.Conclusions

This paper has described a methodology for representing the void distribution inside porous concrete specimens.The spatial distribution of the voids was visualized using CT images.These cross-sectional images were binarized to describe the void distribution,then stacked to create3D void images.Low-order-probability functions were used to characterize the void distribu-tion.It was con?rmed that these methods can effectively represent the void distribution inside porous concrete without damaging the specimen.We then generated a series of statistically identical por-ous concrete specimens using the reconstruction method.Unlike previous approaches,we incorporated the two-point cluster func-tion in the reconstruction method to enhance the process of void clustering.The statistical similarity of the void distribution charac-teristics was con?rmed by examining the low-order probability functions,and the validity of the reconstructed porous concrete specimens was veri?ed by comparing their hydraulic conductivity using?nite element analysis.

From our results,it can be seen that CT imaging and probabilis-tic methods effectively characterize the void distribution in the porous concrete specimen,and the series of porous concrete spec-imens generated from the reconstruction method show almost the same material characteristics and responses.The reconstruction method investigated in this study can be potentially adopted to generate a series of porous concrete RVEs that can be utilized for numerical experiments so as to reduce cost and save time.Further studies comparing the elastic behavior of porous concrete speci-mens should be conducted to validate the similarity of the mechanical responses.

Acknowledgements

This research was supported by a Korea Research Foundation Grant funded by the Korean Government(NRF-2011-0029212 and NRF-2012R1A1A2006629).This work was also supported by the Industrial Strategic Technology Development Program (10041589)funded by the Ministry of Knowledge Economy (MKE,Korea).In addition,the authors wish to extend their grati-tude to Prof.Young Kug Jo of Chungwoon University,Republic of Korea,who provided porous concrete specimens and experimental data,and to Severance Hospital,Yonsei University,Republic of Korea,for their assistance with CT imaging.

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