Research Papers

Representative Volume Element Based Modeling of Cementitious Materials

[+] Author and Article Information
M. M. Shahzamanian

Department of Mechanical Engineering,
The University of Mississippi,
University, MS 38677

T. Tadepalli

Department of Mechanical Engineering,
The University of Mississippi,
University, MS 38677
e-mail: tadepali@olemiss.edu

A. M. Rajendran

Department of Mechanical Engineering,
The University of Mississippi,
University, MS 38677

W. D. Hodo

U.S. Army Engineer Research
and Development Center,
Vicksburg, MS 39180

R. Mohan

Joint School of Nano Science
and Nano Engineering,
North Carolina A&T State University,
Greensboro, NC 27411

R. Valisetty, P. W. Chung, J. J. Ramsey

U.S. Army Research Laboratory,
Aberdeen Proving Ground, MD

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received June 26, 2013; final manuscript received October 28, 2013; published online December 9, 2013. Assoc. Editor: Tetsuya Ohashi.

This material is declared a work of the US Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited.

J. Eng. Mater. Technol 136(1), 011007 (Dec 09, 2013) (16 pages) Paper No: MATS-13-1103; doi: 10.1115/1.4025916 History: Received June 26, 2013; Revised October 28, 2013

The current work focuses on evaluation of the effective elastic properties of cementitious materials through a voxel based finite element analysis (FEA) approach. Voxels are generated for a heterogeneous cementitious material (type-I cement) consisting of typical volume fractions of various constituent phases from digital microstructures. The microstructure is modeled as a microscale representative volume element (RVE) in ABAQUS® to generate cubes several tens of microns in dimension and subjected to various prescribed deformation modes to generate the effective elastic tensor of the material. The RVE-calculated elastic properties such as moduli and Poisson's ratio are validated through an asymptotic expansion homogenization (AEH) and compared with rule of mixtures. Both periodic (PBC) and kinematic boundary conditions (KBC) are investigated to determine if the elastic properties are invariant due to boundary conditions. In addition, the method of “Windowing” was used to assess the randomness of the constituents and to validate how the isotropic elastic properties were determined. The average elastic properties obtained from the displacement based FEA of various locally anisotropic microsize cubes extracted from an RVE of size 100 × 100 × 100 μm showed that the overall RVE response was fully isotropic. The effects of domain size, degree of hydration (DOH), kinematic and periodic boundary conditions, domain sampling techniques, local anisotropy, particle size distribution (PSD), and random microstructure on elastic properties are studied.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Garboczi, E., and Bentz, D. P., 1996, “Multi-Scale Picture of Concrete and Its Transport Properties: Introduction for Non-Cement Researchers,” NISTR 5900, BFRL, NIST, Gaithersburg, MD.
Bernard, O., Ulm, F. J., and Lemarchand, E., 2003, “A Multiscale Micromechanics-Hydration Model for the Early-Age Elastic Properties of Cement-Based Materials,” Cem. Concr. Res., 33(9), pp. 1923–1309. [CrossRef]
Lee, J., Xi, Y., William, K., and Jung, C., 2009, “A Multiscale Model for Modulus of Elasticity of Concrete at High Temperatures,” Cem. Concr. Res., 39(9), pp. 754–762. [CrossRef]
Wu, W., Al-OstazA., Cheng, A., and ChungS., 2010, “Concrete as a Hierarchal Structural Composite Material,” Int. J. Multiscale Comput. Eng., 6(5), pp. 585–595. [CrossRef]
Tennis, P. D., and Jennings, H. M., 2000, “A Model for Two Types of Calcium Silicate Hydrate in the Microstructure of Portland Cement Pastes,” Cem. Concr. Res., 30(6), pp. 855–863. [CrossRef]
Mondal, P., Shah, S. P., and Marks, L. D.,2008, “Nanoscale Characterization of Cementitious Materials,” ACI J. Mater., 105(2), pp. 174–179.
Thomas, J. J., Jennings, H. M., and Allen, A. J., 1998, “The Surface Area of Cement Paste as Measured by Neutron Scattering: Evidence for Two C-S-H Morphologies,” Cem. Concr. Res., 28(6), pp. 897–905. [CrossRef]
Constantinides, G., Ulm, F. J., and Van Vliet, K., 2003,“On the Use of Nanoindentation for Cementitious Materials,” Mater. Struct., 36, pp. 191–196. [CrossRef]
Jennings, H. M., Thomas, J. J., Gevrenov, J. S., Constantinides, G., and Ulm, F. J., 2005, “Relating the Nanostructure of Concrete to Engineering Properties,” 2nd International Symposium on Nanotechnology in Construction, Bilbao, Spain, November 13–16.
Haecker, C. J., Garboczi, E. J., Bullard, J. W., Bohn, R. B., Sun, Z., Shah, S. P., and Voigt, T., 2005, “Modeling the Linear Elastic Properties of Portland Cement Paste,” Cem. Concr. Res., 35(10), pp. 1948–1960. [CrossRef]
Richardson, I. G., 2008, “The Calcium Silicate Hydrates,” Cem. Concr. Res., 38(2), pp. 137–158. [CrossRef]
Pellenq, R. J.-M.Lequeux, N., and Vandamme, H., 2008, “Engineering the Bonding Scheme in C-S-H: The Iono-Covalent Framework,” Cem. Concr. Res., 38(2), pp. 159–174. [CrossRef]
Richardson, I. G., 2000, “The Nature of the Hydration Products in Hardened Cement Pastes,” Cem. Concr. Compos., 22, pp. 97–113. [CrossRef]
Murray, S. J., 2009, “Determination of Strength and Stiffness of Calcium Silicate Hydrate Using Molecular Dynamics,” M.S. thesis, University of Arkansas, Fayetteville, AR.
Lin, F., 2006, “Modeling of Hydration Kinetics and Shrinkage of Portland Cement Paste,” Ph.D. thesis, Columbia University, New York.
SmilauerV., 2005, “Elastic Properties of Hydrating Cement Paste Determined From Hydration Models,” Ph.D thesis, Czech Technical University, Prague.
Allen, A. J., Thomas, J. J., and Jennings, H. M., 2007, “Composition and Density of Nanoscale Calcium–Silicate–Hydrate in Cement,” Nature Mater., 6(4), pp. 311–316. [CrossRef]
Bentz, D. P., Jensenb, O. M., Coats, A. M., and Glasser, F. P., 2000, “Influence of Silica Fume on Diffusivity in Cement-Based Materials I. Experimental and Computer Modeling Studies on Cement Pastes,” Cem. Concr. Res., 30(6), pp. 953–962. [CrossRef]
Chandler, M. Q., Peters, J. F., and Pelessone, D., 2012, “Modeling Nanoindentation of Calcium Silicate Hydrate,” Journal of the Transportation Research Board, No. 2142, Transportation Research Board of the National Academies, Washington, DC, pp. 67–74.
Beaudoin, J. J., Gu, P., and Myers, R. E., 1998, “The Fracture Of C-S-H And C-S-H/Ch Mixtures,” Cem. Concr. Res., 28(3), pp. 341–347. [CrossRef]
Jennings, H. M., Bullard, J. W., Thomas, J. J., Andrade, J. E., Chen, J. J., and Scherer, G. W., 2008, “Characterization and Modeling of Pores and Surfaces in Cement Paste: Correlations to Processing and Properties,” J. Adv. Concr. Technol., 6(1), pp. 5–29. [CrossRef]
Chandler, M. Q., Peters, J. F., and Pelessone, D., “Modeling Nanomechanical Behavior of Calcium-Silicate-Hydrate,” U.S. Army ERDC, Vicksburg, MS, Final Report No. ERDC/GSL TR-12-30.
Moser, R. D., Allison, P. G., Chandler, M. Q., “Investigation of High-Strain Rate Damage in Reactive Powder Concretes Using Instrumented Indentation Techniques,” 4th International Symposium on Nanotechnology in Construction (NICOM 4), Agios Nikolaos, Greece, May 20-22.
Sorelli, L., Constantinides, G., Ulm, F. J., and Toutlemonde, F., 2008, “The Nano-Mechanical Signature of Ultra High Performance Concrete by Statistical Nanoindentation Techniques,” Cem. Concr. Res., 38(12), pp. 1447–1456. [CrossRef]
Dolado, J. S., and Breugel, K. V., 2011, “Recent Advances in Modeling for Cementitious Materials,” Cem. Concr. Res., 41(7), pp. 711–726. [CrossRef]
Kamali-Bernard, S., Bernard, F., and Prince, W., 2009, “Computer Modelling of Tritiated Water Diffusion Test for Cement Based Materials,” J. Com. Mat. Sci., 45(2), pp. 528–535. [CrossRef]
Meier, H. A., Kuhl, E., and Steinmann, P., 2008, “A Note on the Generation of Periodic Granular Microstructures Based on Grain Size Distributions,” Int. J. Numer. Anal. Meth. Geomech., 32(5), pp. 509–522. [CrossRef]
Šmilauer, V., Hlaváček, P., Škvára, F., Šulc, R., Kopecký, L., and Němeček, J., 2011, “Micromechanical Multiscale Model for Alkali Activation of Fly Ash and Metakaolin,” J. Mater. Sci., 46 (20), pp. 6545–6555. [CrossRef]
Andrade, J. E., Fonseca, P. C., and Jennings, H. M., 2011, “A Nanoscale Numerical Model of Calcium Silicate Hydrate,” Mech. Mater., 43(8), pp. 408–419. [CrossRef]
Shahsavari, R., Pellenq, R. J.-M., and Ulm, F. J., 2011, “Empirical Force Fields for Complex Hydrated Calcio Silicate Layered Materials,” Phys. Chem. Chem. Phys., 13(3), pp. 1002–1011. [CrossRef] [PubMed]
Ulm, F. J., Pellenq, R. J.-M., and Vandamme, M., 2010, “Concrete: From Atoms to Concrete Structures,” Computational Modelling of Concrete Structures, N. Bicanic, R. de Borst, H. Mang, and G. Meschke, eds., CRC Press, Boca Raton, FL, pp. 69.
Pellenq, R. J. M., Kushima, A., Shahsavari, R., Van Vlietd, K. J., Buehlerb, M. J., Yipc, S., and Ulm, F.-J., 2009, “A Realistic Molecular Model of Cement Hydrates,” Proc. Natl. Acad. Sci. U.S.A., 106(38), pp. 16102–16107. [CrossRef] [PubMed]
Smilauer, V., and Krejci, T., 2009,“Multiscale Model for Temperature Distribution in Hydrating Concrete Multiscale Model for Temperature Distribution in Hydrating Concrete,” Int. J. Multiscale Comp. Eng., 7(2), pp. 135–151. [CrossRef]
Maekawa, K., Ishida, T., and Kishi, T., 2003, “Multi-Scale Modeling of Concrete Performance: Integrated Material and Structural Mechanics,” J. Adv. Concr. Technol., 1(2), pp. 91–126. [CrossRef]
Ye, G., van Breugel, K., and Fraaij, A. L. A., 2002, “Three-Dimensional Microstructure Analysis of Numerically Simulated Cementitious Materials,” Cem. Concr. Res., 33(3), pp. 215–222. [CrossRef]
Princigallo, A., Lura, P., Levita, G., and van Breugel, K., 2003, “Early Development of Properties in a Cement Paste: A Numerical and Experimental Study,” Cem. Concr. Res., 33(7), pp. 1013–1020. [CrossRef]
Grondin, F., Dumontet, H., Ben Hamida, A., Mounajed, G., and Boussa, H., 2007, “Multi-Scales Modelling for the Behaviour of Damaged Concrete,” Cem. Concr. Res., 37(10), pp. 1453–1462. [CrossRef]
Yeong, C. L. Y., and Torquato, S., 1998, “Reconstructing Random Media. II. Three-Dimensional Media From Two-Dimensional Cuts,” Phys. Rev. E, 58(1), pp. 224–233. [CrossRef]
Bishnoi, S., 2008, “Vector Modelling of Hydrating Cement Microstructure and Kinetics,” Ph. D. thesis No. 4606, École Polytechnique Fédérale De Lausanne, Lausanne, Switzerland.
Wu, W., Al-Ostaz, A. M., Cheng, A. H.-D., and Song, C. R., 2011, “Computation of Elastic Properties of Portland Cement Using Molecular Dynamics,” J. Nanomech. Micromech., 1(2), pp. 84–90. [CrossRef]
Hill, R., 1963, “Elastic Properties of Reinforced Solids: Some Theoretical Principles,” J. Mech. Phys. Solids., 11, pp. 357–372. [CrossRef]
Hashin, Z., and Shtrikman, S., 1963, “A Variational Approach to the Theory of Elastic Behavior of Multiphase Materials,” J. Mech. Phys. Solids., 11, pp. 127–140. [CrossRef]
Willis, J., 1981, “Variational and Related Methods for the Overall Properties of Composites,” Adv. Appl. Mech., 21, pp. 1–78. [CrossRef]
Nemat-Nasser, S., and Hori, M., 1999, Micromechanics: Overall Properties of Heterogeneous Materials, 2nd ed., Elsevier, Amsterdam.
Sab, K., 1992, “On the Homogenization and the Simulation of Random Materials,” Eur. J. Mech. Solids., 11, pp. 585–607.
Huet, C., 1999, “Coupled Size and Boundary-Condition Effects in Viscoelastic Heterogeneous and Composite Bodies,” Mech. Mater., 31(12), pp. 787–829. [CrossRef]
Kanit, T., Forest, S., Galliet, I., Mounoury, V., and Jeulin, D., 2003, “Determination of the Size of the Representative Volume Element for Random Composites: Statistical and Numerical Approach,” Int. J. Solids Struct., 40(13–14), pp. 3647–3679. [CrossRef]
Gitman, I. M., Askes, H., and Sluys, L. J., 2007, “Representative Volume: Existence and Size Determination,” Eng. Fract. Mech., 74(16), pp. 2518–2534. [CrossRef]
Bentz, D. P., 2005, “CEMHYD3D: A Three Dimensional Cement Hydration and Microstructure Development Package,” Version 3.0, NISTIR7232, U.S. Department of Commerce.
Bonen, D., and Diamond, S., 1991, “Application of Image Analysis to a Comparison of Ball Mill and High Pressure Roller Mill Ground Cement,” Proceedings of 13th International Conference Cement Microscopy, Tampa, FL, April 8–11, p. 101.
Bernard, F., Kamali-Bernard, S., and Prince,W., 2008, “3D Multi-Scale Modeling of Mechanical Behaviour of Sound and Leached Mortar,” Cem. Concr. Res., 38(4) pp. 449–458. [CrossRef]
Charmrova, R., 2010, “Modelling and Measurement of Elastic Properties of Hydrating Cement Paste,” Ph.D. thesis No. 4606, École Polytechnique Fédérale De Lausanne, Laussanne, Switzerland.
Kurkuri, S., 2005, “Homogenization of Damaged Concrete Meso-Structures Using Representative Volume Elements—Implementation and Application to Slang,” Master thesis, Institute of Structure Mechanics, Bauhaus–University Weimar, Weimar, Germany.
Ren, Z. Y., and Zheng, Q. S., 2002, “A Quantitative Study of Minimum Sizes of Representative Volume Elements of Cubic Polycrystals—Numerical Experiments,” J. Mech. Phys. Solids., 50(4), pp. 881–893. [CrossRef]
Šmilauer, V., and Bittnar, Z., 2004, “Effects of Representative Cube Size on the Simulation of Portland Cement Hydration in CEMHYD3D Model,” 5th International Doctoral Symposium in Civil Engineering, Delft, Netherlands, June 16–19, J. Blaauwendraad, T. Scarpas, B. Snijder, and J. Walraven, eds., A. A. Balkema, Leiden, Netherlands, pp. 581–587.
Hain,M., and Wrigers, P., 2005, “Simulating the Microstructure of Cement Based Construction Materials” Proc. App. Math. Mech., 5, pp. 401–402. [CrossRef]
Barbero, E. J., 2008, Finite Element Analysis of Composite Materials, 1st ed., CRC Press, Boca Raton, FL.
Chung, P. W., Tamma, K. K., and Namburu, R. R., 2001, “Asymptotic Expansion Homogenization for Heterogeneous Media: Computational Issues and Applications, Compos. Part A, 32(9), pp. 1291–1301. [CrossRef]
Ramsey, J. J., and Chung, P. W., “Massively Parallel Implementation of Asymptotic Expansion Homogenization for Complex Microstructures,” Army Research Lab, Aberdeen Proving Ground, MD (unpublished).
Kanit, T., N'Guyen, F., Forest, S., Jeulin, D., Reed, M., and Singleton, S., 2006, “Apparent and Effective Physical Properties of Heterogeneous Materials: Representativity of Samples of Two Materials From Food Industry,” Comput. Methods Appl. Mech. Eng., 195, pp. 3960–3982. [CrossRef]
Šmilauer,V., and Bittnar, Z., 2006, “Microstructure-Based Micromechanical Prediction of Elastic Properties in Hydrating Cement Paste,” Cem. Concr. Res., 36, pp. 1708–1718. [CrossRef]
Voigt, W., 1966, Lehrbuch der Kristallphysik: mit Ausschluss der Kristalloptik. New York, Johnson Reprint.
Reuss, A., and Angew, Z., 1929, “Berchung der fiessgrenze von mischkristallen auf grund der plastizit atsbedingung f¨ur einkristalle,” Math. Mech., 9, pp. 49–58.
Hashin, Z., 1960, “Elastic Moduli of Heterogeneous Materials,” Technical Report No. 9, Submitted to ONR, Contract No. 1866(02), September.
Wu, W., Al-Ostaz, A., Gladden, J., Cheng, A. H.-D., and Li, G., 2010, “Measurement of Mechanical Properties of Hydrated Cement Paste Using Resonant Ultrasound Spectroscopy,” J. ASTM Int., 7(5), p. JAI102657. [CrossRef]
Kamali, S., Moranville, M., Garboczi, E. G., Pren, S., and Grard, B., 2004, “Hydrate Dissolution Influence on the Young's Modulus of Cement Paste,” 5th International Conference of Fracture Mechanics of Concrete Structures, Vail, CO, April 12–16.
Haeckerd, C.-J., Garboczia, E. J., Bullarda, J. W., Bohnb, R. B., Sunc, Z., Shahc, S. P., and Voigt, T., 2005, “Modeling the Linear Elastic Properties of Portland Cement Paste,” Cem. Concr. Res., 35, pp. 1948–1960. [CrossRef]
Lam, L., Wong, Y. L., and Poon, C. S., 2000, “Degree of Hydration and Gel/Space Ratio of High-Volume Fly Ash/Cement Systems,” Cem. Concr. Res., 30, pp. 747–756. [CrossRef]
Bentz, D. P., 1997, “Three-Dimensional Computer Simulation of Portland Cement Hydration and Microstructure Development,” J. Am. Ceram. Soc., 80(1), pp. 3–21. [CrossRef]


Grahic Jump Location
Fig. 1

Multilevel microstructure of cement-based materials [2]

Grahic Jump Location
Fig. 2

Scaled PSD for initial cement powder in domains of various sizes

Grahic Jump Location
Fig. 3

Work flow of the CEMHYD3D program for generation of cement microstructure

Grahic Jump Location
Fig. 4

Schematic diagram of the nanoscale C–S–H particles [17]

Grahic Jump Location
Fig. 5

Typical volume fractions of major constituents at various stages of curing

Grahic Jump Location
Fig. 6

(a) 1 K, (b) 8 K, (c) 125 K, and (d) 1M FE models of hydrated cement microstructure (PMDs) (not to scale)

Grahic Jump Location
Fig. 7

200 × 200 × 100 μm (4M) FE model of hydrated cement microstructure (PMD)

Grahic Jump Location
Fig. 8

Prescribed kinematic (KBC) (a) tensile deformation (E1) and (b) pure shear (G12) boundary conditions

Grahic Jump Location
Fig. 10

Schematic showing location of windows extracted from the 1M-PMD

Grahic Jump Location
Fig. 11

Deformation corresponding to pure shear (G12)

Grahic Jump Location
Fig. 12

Volume fractions of major phases for 1 K, 8 K, 125 K, and 1M PMDs

Grahic Jump Location
Fig. 13

Variation of (a) principal and (b) shear moduli (KBC) in 1M-PMD for various instances normalized to their respective average

Grahic Jump Location
Fig. 14

Effect of DOH on material bulk properties for 1M-RVE for KBC, PBC, and AEH

Grahic Jump Location
Fig. 15

Effect of domain size on material bulk properties for DOH = 0.8 for KBC, PBC, and AEH

Grahic Jump Location
Fig. 16

The effect of domain size on degree of hydration (α) (CEMHYD3D)

Grahic Jump Location
Fig. 17

Volume fractions of major phases for (a) 1 K element, (b) 8 K element windows, and (c) 125 K element windows

Grahic Jump Location
Fig. 18

Elastic moduli (uniaxial) with increasing window size

Grahic Jump Location
Fig. 19

Elastic shear moduli Gij with increasing window size

Grahic Jump Location
Fig. 20

Development of the compressive strength (f'c) (CEMHYD3D)

Grahic Jump Location
Fig. 21

Development of the Young's modulus (E) (ABAQUS®)



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In