Archive of Applied Mechanics

, Volume 88, Issue 7, pp 1105–1119 | Cite as

A multiscale model for post-peak softening response of concrete and the role of microcracks in the interfacial transition zone

  • Keerthy M. Simon
  • J. M. Chandra Kishen


The effect of microcracks ahead of a macrocrack on the post-peak behavior of concrete-like quasi-brittle material is studied. The critical length of a microcrack is estimated by considering a small element near the macrocrack tip and defining the critical crack opening displacement of the microcrack that exist in the interface region between the aggregate and cement paste. A fracture model is proposed to predict the post-peak response of plain concrete. This model is validated using the experimental results for normal-strength, high-strength and self-consolidating concretes available in the literature. Through a sensitivity analysis, it is observed that the elastic modulus of concrete and the fracture toughness of the interface have a substantial influence on the critical microcrack length.


Microstructure Interfacial transition zone Concrete Fracture process zone 

List of symbols


Interfacial transition zone


Fracture process zone

\(\phi \)

Airy stress function

\(F_n(\theta )\)


\(\lambda _n\)


\(\mu _{\mathrm{micro}}\)

Shear modulus corresponding to the microscale

\(\nu _{\mathrm{micro}}\)

Poisson’s ratio corresponding to the microscale


Mode I fracture toughness of interface


Elastic modulus of interface


Critical length of microcrack

\(\sigma _y\)

Tensile strength of interface

\(\delta _{\mathrm{c}}\)

Critical crack opening displacement of microcrack

\(\delta _{\mathrm{p}}\)

Crack mouth opening displacement corresponding to the peak load


Volume fraction of coarse aggregate


Volume fraction of fine aggregate


Volume fraction of mortar


Volume fraction of cement paste

\(\nu _{\mathrm{eff}}\)

Poissons ratio of concrete

\(\nu _{\mathrm{m}}\)

Poissons ratio of mortar


Elastic modulus of coarse aggregate


Elastic modulus of fine aggregate


Elastic modulus of mortar


Elastic modulus of cement paste


Total volume fraction


Mass of each constituent

\(\rho _i\)

Density of each constituent


Depth of specimen


Span of the specimen


Thickness of the specimen


Crack length


Crack length corresponding to peak load


Critical microcrack length

\(\delta ^{\mathrm{M}}\)

Crack opening displacement corresponding to peak load

\(\delta ^{\mathrm{m}}\)

Microcrack opening displacement


  1. 1.
    Maso, J.: Interfacial Transition Zone in Concrete, RILEM Report, vol. 11. CRC Press, London (2004)Google Scholar
  2. 2.
    Scrivener, K.L., Crumbie, A.K., Laugesen, P.: The interfacial transition zone (ITZ) between cement paste and aggregate in concrete. Interface Sci. 12(4), 411–421 (2004)CrossRefGoogle Scholar
  3. 3.
    Ollivier, J., Maso, J., Bourdette, B.: Interfacial transition zone in concrete. Adv. Cem. Based Mater. 2(1), 30–38 (1995)CrossRefGoogle Scholar
  4. 4.
    Mihai, I.C., Jefferson, A.D.: A material model for cementitious composite materials with an exterior point eshelby microcrack initiation criterion. Int. J. Solids Struct. 48(24), 3312–3325 (2011)CrossRefGoogle Scholar
  5. 5.
    Buyukozturk, O., Nilson, A.H., Slate, F.O.: Deformation and fracture of particulate composite. J. Eng. Mech. Div. 98(3), 581–593 (1972)Google Scholar
  6. 6.
    Liu, T.C., Nilson, A.H., Floyd, F.O.S.: Stress–strain response and fracture of concrete in uniaxial and biaxial compression. In: ACI Journal Proceedings, vol. 69, pp. 291–295. ACI (1972)Google Scholar
  7. 7.
    Struble, L., Skalny, J., Mindess, S.: A review of the cement-aggregate bond. Cem. Concr. Res. 10(2), 277–286 (1980)CrossRefGoogle Scholar
  8. 8.
    Bentur, A., Mindess, S.: The effect of concrete strength on crack patterns. Cem. Concr. Res. 16(1), 59–66 (1986)CrossRefGoogle Scholar
  9. 9.
    Horii, H., Shin, H.C., Pallewatta, T.M.: Mechanism of fatigue crack growth in concrete. Cem. Concr. Compos. 14(2), 83–89 (1992)CrossRefGoogle Scholar
  10. 10.
    Nirmalendran, S., Horii, H.: Analytical modeling of microcracking and bridging in the fracture of quasi-brittle materials. J. Mech. Phys. Solids 40(4), 863–886 (1992)CrossRefGoogle Scholar
  11. 11.
    Mindess, S.: Mechanical properties of the interfacial transition zone: a review. ACI Spec. Publ. 156, 1–10 (1995)Google Scholar
  12. 12.
    Van Mier, J., Vervuurt, A.: Numerical analysis of interface fracture in concrete using a lattice-type fracture model. Int. J. Damage Mech. 6(4), 408–432 (1997)CrossRefGoogle Scholar
  13. 13.
    Prokopski, G., Halbiniak, J.: Interfacial transition zone in cementitious materials. Cem. Concr. Res. 30(4), 579–583 (2000)CrossRefGoogle Scholar
  14. 14.
    Shah, S.P., Swartz, S.E., Ouyang, C.: Fracture Mechanics of Concrete: Applications of Fracture Mechanics to Concrete, Rock and Other Quasi-Brittle Materials. Wiley, New York (1995)Google Scholar
  15. 15.
    Prado, E., Van Mier, J.: Effect of particle structure on mode I fracture process in concrete. Eng. Fract. Mech. 70(14), 1793–1807 (2003)CrossRefGoogle Scholar
  16. 16.
    Sun, C.T., Jin, Z.H.: Fracture Mechanics. Elsevier, Amsterdam (2012)Google Scholar
  17. 17.
    Hillemeier, B., Hilsdorf, H.: Fracture mechanics studies on concrete compounds. Cem. Concr. Res. 7(5), 523–535 (1977)CrossRefGoogle Scholar
  18. 18.
    Huang, J., Li, V.: A meso-mechanical model of the tensile behavior of concrete. Part II: modelling of post-peak tension softening behavior. Composites 20(4), 370–378 (1989)CrossRefGoogle Scholar
  19. 19.
    Yang, C.: Effect of the transition zone on the elastic moduli of mortar. Cem. Concr. Res. 28(5), 727–736 (1998)CrossRefGoogle Scholar
  20. 20.
    Hashin, Z.: The elastic moduli of heterogeneous materials. J. Appl. Mech. 29(1), 143–150 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Bazant, Z.P., Xu, K.: Size effect in fatigue fracture of concrete. ACI Mater. J. 88(4), 390–399 (1991)Google Scholar
  22. 22.
    Shah, S.G., Chandra Kishen, J.M.: Fracture properties of concrete–concrete interfaces using digital image correlation. Exp. Mech. 51(3), 303–313 (2011)CrossRefGoogle Scholar
  23. 23.
    Raghu Prasad, B.K., Vidya Sagar, R.: Relationship between AE energy and fracture energy of plain concrete beams: experimental study. J. Mater. Civ. Eng. 20(3), 212–220 (2008)CrossRefGoogle Scholar
  24. 24.
    Gettu, R., Bazant, Z.P., Karr, M.E.: Fracture properties and brittleness of high-strength concrete. ACI Mater. J. 87(6), 608–618 (1990)Google Scholar
  25. 25.
    Hemalatha, T., Ramaswamy, A., Chandra Kishen, J.M.: Simplified mixture design for production of self-consolidating concrete. ACI Mater. J. 112(2), 277–286 (2015)Google Scholar
  26. 26.
    Mobasher, B., Stang, H., Shah, S.: Microcracking in fiber reinforced concrete. Cem. Concr. Res. 20(5), 665–676 (1990)CrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil EngineeringIndian Institute of ScienceBangaloreIndia

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