Prediction of concrete stiffness characteristics using analytical homogenization

Introduction. Prediction of concrete stiffness characteristics within a scientific framework remains a relevant task of modern concrete science. Considering concrete as a three-phase composite material consisting of a cement matrix, aggregates, and the interfacial transition zone between them, analytical homogenization methods are among the most effective approaches for addressing this problem. However, existing computational models for predicting the stiffness characteristics of concrete based on analytical homogenization methods do not fully account for the specific features of its structure, which necessitates their further development.

Materials and methods. The proposed approach for predicting the stiffness characteristics of concrete involves estimating its effective fourth-order stiffness (elasticity) tensor based on analytical homogenization methods from the micromechanics of composites.

Results. The assumptions of the proposed approach are formulated. Solutions are introduced that consider the grains of coarse aggregate as composite particles, both in the form of spheres and as oblate spheroids for flaky grains, and prolate spheroids for elongated grains surrounded by the interfacial transition zone. An iterative method is also proposed, whereby the stiffness characteristics of concrete are determined sequentially, calculating the stiffness characteristics of the composite system “cement matrix – composite particle of the i-th fraction” at each iteration. A numerical method for calculating the effective elasticity modulus of concrete according to the proposed approach is provided.

Conclusions. The proposed approach based on analytical homogenization is designed to evaluate the stiffness characteristics of concrete, taking into account the features of its structure, such as the spheroidal shape of the aggregates, the presence of an interfacial transition zone around them, and the high volumetric concentration of aggregates.

1. Nilsen A.U., Monteiro P.J.M. Concrete: A three phase material. Cement and Concrete Research. 1993; 23(1):147-151. DOI: 10.1016/0008-8846(93)90145-y

2. Scrivener K.L., Crumbie A.K., Laugesen P. The Interfacial Transition Zone (ITZ) Between Cement Paste and Aggregate in Concrete. Interface Science. 2004; 12(4):411-421. DOI: 10.1023/b:ints.0000042339.92990.4c

3. Bernard O., Ulm F.-J., Lemarchand E. A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials. Cement and Concrete Research. 2003; 33(9):1293-1309. DOI: 10.1016/S0008-8846(03)00039-5

4. Pichler C., Lackner R., Mang H.A. A multiscale micromechanics model for the autogenous-shrinkage deformation of early-age cement-based materials. Engineering Fracture Mechanics. 2007; 74(1-2):34-58. DOI: 10.1016/j.engfracmech.2006.01.034

5. Shi C., Tu Q., Fan H., Li S. An interphase model for effective elastic properties of concrete composites. Journal of Micromechanics and Molecular Physics. 2016; 1(1):1650005. DOI: 10.1142/S2424913016500053

6. Lavergne F., Ben Fraj A., Bayane I., Barthélémy J.F. Estimating the mechanical properties of hydrating blended cementitious materials: An investigation based on micromechanics. Cement and Concrete Research. 2018; 104:37-60. DOI: 10.1016/j.cemconres.2017.10.018

7. Lezgy-Nazargah M., Emamian S.A., Aghasizadeh E., Khani M. Predicting the mechanical properties of ordinary concrete and nano-silica concrete using micromechanical methods. Sādhanā. 2018; 43(12). DOI: 10.1007/s12046-018-0965-0. EDN PIIWMR.

8. Haile B.F., Jin D.W., Yang B., Park S., Lee H.K. Multi-level homogenization for the prediction of the mechanical properties of ultra-high-performance concrete. Construction and Building Materials. 2019; 229:116797. DOI: 10.1016/j.conbuildmat.2019.116797

9. Stefaniuk D., Niewiadomski P., Musiał M., Łydżba D. Elastic properties of self-compacting concrete modified with nanoparticles: Multiscale approach. Archives of Civil and Mechanical Engineering. 2019; 19(4):1150-1162. DOI: 10.1016/j.acme.2019.06.006. EDN EQQVLN.

10. Sharma M., Bishnoi S. Influence of properties of interfacial transition zone on elastic modulus of concrete: Evidence from micromechanical modelling. Construction and Building Materials. 2020; 246:118381. DOI: 10.1016/j.conbuildmat.2020.118381. EDN BCCLEV.

11. Onifade I., Birgisson B. Microstructural integrity characterization of cement-based construction materials. Construction and Building Materials. 2021; 307:125012. DOI: 10.1016/j.conbuildmat.2021.125012. EDN CQWZUK.

12. Li Y., Liu Y., Wang R. Evaluation of the elastic modulus of concrete based on indentation test and multi-scale homogenization method. Journal of Building Engineering. 2021; 43:102758. DOI: 10.1016/j.jobe.2021.102758. EDN WGUTZB.

13. Silva R.M.S.Da., Barboza A.Da.S.R. Concrete modeling using micromechanical multiphase models and multiscale analysis. Revista IBRACON de Estruturas e Materiais. 2023; 16(5). DOI: 10.1590/S1983-41952023000500001. EDN DBYSRR.

14. Zaoui A. Continuum micromechanics: Survey. Journal of Engineering Mechanics. 2002; 128(8):808-816. DOI: 10.1061/(ASCE)0733-9399(2002)128:8(808)

15. Dvorak G. Micromechanics of Composite Materials. Solid Mechanics and Its Applications. 2013. DOI: 10.1007/978-94-007-4101-0

16. Bennoura M., Aboutajeddine A. Predictive capabilities of micromechanical models for composite materials. Journal of Reinforced Plastics and Composites. 2016; 35(14):1115-1125. DOI: 10.1177/0731684416640365. EDN WPQLCD.

17. Ogierman W. Data-driven mean-field homogenization: Enhancing the accuracy of the Mori-Tanaka method. Composite Structures. 2025; 358:118985. DOI: 10.1016/j.compstruct.2025.118985. EDN PGSIWB.

18. Bayat H., Rastgo M., Mansouri Zadeh M., Vereecken H. Particle size distribution models, their characteristics and fitting capability. Journal of Hydrology. 2015; 529(3):872-889. DOI: 10.1016/j.jhydrol.2015.08.067

19. Garboczi E., Berryman J. New Effective Medium Theory for the Diffusivity or Conductivity of a Multi-Scale Concrete Microstructure Model. Concrete Science and Engineering. 2000; 2:88-96.

20. Belamesava K., Paulava I. Analytical model for calculating the amount of basalt fiber to produce dispersion-reinforced concrete with an optimal structure. Vestnik of Polotsk State University. Part F. Constructions. Applied Sciences. 2023; 1:27-31. DOI: 10.52928/2070-1683-2023-33-1-27-31. EDN PRGKBP. (rus.).

21. Christensen R.M. A critical evaluation for a class of micro-mechanics models. Journal of the Mechanics and Physics of Solids. 1990; 38(3):379-404. DOI: 10.1016/0022-5096(90)90005-o

22. Klusemann B., Svendsen B. Homogenization methods for multi-phase elastic composites: Comparisons and benchmarks. Technische Mechanik. 2019; 30(4):374-386.

23. Chatzigeorgiou G., Meraghni F., Charalambakis N. Multiscale Modeling Approaches for Composites. Amsterdam, Elsevier, 2022; 345. DOI: 10.1016/C2019-0-05214-4

24. Boumiz A., Vernet C., Cohen Tenoudji F. Mechanical properties of cement pastes and mortars at early ages. Advanced Cement Based Materials. 1996; 3(3-4):94-106. DOI: 10.1016/S1065-7355(96)90042-5

25. Ramesh G., Sotelino E.D., Chen W.F. Effect of transition zone on elastic moduli of concrete materials. Cement and Concrete Research. 1996; 26(4):611-622. DOI: 10.1016/0008-8846(96)00016-6

26. Mura T. Micromechanics of Defects in Solids. Mechanics of Elastic and Inelastic Solids. 1987. DOI: 10.1007/978-94-009-3489-4

27. Dormand J.R., Prince P.J. A family of embedded Runge-Kutta formulae. Journal of Computational and Applied Mathematics. 1980; 6(1):19-26. DOI: 10.1016/0771-050X(80)90013-3

28. Lebedev V.I., Laikov D.N. A quadrature formula for the sphere of the 131st algebraic order of accuracy. Doklady Mathematics. 1999; 59(3):477-481. EDN LPSYFV.

29. Shalimo M.A. Guide for the concrete and reinforced concrete technology. Minsk, Higher School, 1987; 196. (rus.).