Microplane Models for Elasticity and Inelasticity of Engineering Materials

  • Ferhun C. Caner
  • Valentín de Carlos Blasco
  • Mercè Ginjaume Egido
Reference work entry


In the traditional approach to the modeling of mechanical behavior of engineering materials, the stress tensor is calculated directly from the prescribed strain tensor either by a closed form tensorial relation as in elasticity or by incremental analysis as in classical plasticity formulations in which the formulation is developed in terms of tensor invariants and their combinations. However, to model the general three-dimensional constitutive behavior of the so-called geomaterials at arbitrary nonproportional load paths that frequently arise in dynamic loadings, such direct approaches do not yield models with desired accuracy. Instead, microplane approach prescribes the constitutive behavior on planes of various orientations of the material microstructure independently, and the second-order stress tensor is obtained by imposing the equilibrium of second-order stress tensor with the microplane stress vectors. In this work, particular attention is devoted to the milestone microplane models for plain concrete, namely, the model M4 and the model M7. Furthermore, a novel autocalibrating version of the model M7 called the model M7Auto is presented as an alternative to both differential and integral type nonlocal formulations since the model M7Auto does not suffer from the shortcomings of these classical nonlocal approaches. Examples of the performance of the models M7 and M7Auto are shown by simulating well-known benchmark test data like three-point bending size effect test data of plain concrete beams using finite element meshes of the same element size and Nooru-Mohamed test data obtained at different load paths using finite element meshes having different element sizes, respectively.


Constitutive model Microplane model Crack band model Concrete Auto-calibrating Microplane model Three dimensional finite element analysis 


  1. Z. Bažant, M. Adley, I. Carol, M. Jirasek, S. Akers, B. Rohani, J. Cargile, F. Caner, Large-strain generalization of microplane model for concrete and application. J. Eng. Mech. ASCE 126(9), 971–980 (2000)CrossRefGoogle Scholar
  2. Z. Bažant, F. Caner, Microplane model M5 with kinematic and static constraints for concrete fracture and anelasticity. II: computation. J. Eng. Mech. ASCE 131(1), 41–47 (2005)Google Scholar
  3. Z. Bažant, F. Caner, M. Adley, S. Akers, Fracturing rate effect and creep in microplane model for dynamics. J. Eng. Mech. ASCE 126(9), 962–970 (2000)CrossRefGoogle Scholar
  4. Z. Bažant, F. Caner, I. Carol, M. Adley, S. Akers, Microplane model M4 for concrete. I: formulation with work-conjugate deviatoric stress. J. Eng. Mech. ASCE 126(9), 944–953 (2000)Google Scholar
  5. Z. Bažant, G. Di Luzio, Nonlocal microplane model with strain-softening yield limits. Int. J. Solids Struct. 41(24–25), 7209–7240 (2004)CrossRefGoogle Scholar
  6. Z. Bažant, P. Gambarova, Crack shear in concrete – crack band microplane model. J. Eng. Mech. ASCE 110(9), 2015–2035 (1984)Google Scholar
  7. Z. Bažant, J.-L. Le, C. Hoover, Nonlocal boundary layer (NBL) model: overcoming boundary condition problems in strength statistics and fracture analysis of quasibrittle materials, in Proceedings of FraMCoS-7, 2010, pp. 135–143Google Scholar
  8. Z. Bažant, B. Oh, Microplane model for progressive fracture of concrete and rock. J. Eng. Mech. ASCE 111(4), 559–582 (1985)CrossRefGoogle Scholar
  9. Z. Bažant, J. Ožbolt, Nonlocal microplane model for fracture, damage, and size effect in structures. J. Eng. Mech. ASCE 116(11), 2485–2505 (1990)CrossRefGoogle Scholar
  10. Z. Bažant, J. Ožbolt, Compression failure of quasibrittle material – nonlocal microplane model. J. Eng. Mech. ASCE 118(3), 540–556 (1992)CrossRefGoogle Scholar
  11. Z. Bažant, G. Pijaudiercabot, Nonlocal continuum damage, localization instability and convergence. J. Appl. Mech. Trans. ASME 55(2), 287–293 (1988)CrossRefGoogle Scholar
  12. Z. Bažant, P. Prat, Microplane model for brittle-plastic material. 1. Theory. J. Eng. Mech. ASCE 114(10), 1672–1687 (1988a)CrossRefGoogle Scholar
  13. Z. Bažant, P. Prat, Microplane model for brittle-plastic material. 2. Verification. J. Eng. Mech. ASCE 114(10), 1689–1702 (1988b)Google Scholar
  14. Z. Bažant, Y. Xiang, M. Adley, P. Prat, S. Akers, Microplane model for concrete. 2. Data delocalization and verification. J. Eng. Mech. ASCE 122(3), 255–262 (1996a)Google Scholar
  15. Z. Bažant, Y. Xiang, P. Prat, Microplane model for concrete. 1. Stress-strain boundaries and finite strain. J. Eng. Mech. ASCE 122(3), 245–254 (1996b)CrossRefGoogle Scholar
  16. Z. Bažant, G. Zi, Microplane constitutive model for porous isotropic rocks. Int. J Numer Anal. methods Geomech. 27(1), 25–47 (2003)CrossRefGoogle Scholar
  17. M. Brocca, Z. Bažant, Microplane constitutive model and metal plasticity. Appl. Mech. Rev. ASME 53(10), 265–281 (2000)CrossRefGoogle Scholar
  18. M. Brocca, Z. Bažant, I. Daniel, Microplane model for stiff foams and finite element analysis of sandwich failure by core indentation. Int. J. Solids Struct. 38(44-45), 8111–8132 (2001)CrossRefGoogle Scholar
  19. M. Brocca, L. Brinson, Z. Bažant, Three-dimensional constitutive model for shape memory alloys based on microplane model. J. Mech. Phys. solids 50(5), 1051–1077 (2002)CrossRefGoogle Scholar
  20. F. Caner, Z. Bažant, Microplane model M4 for concrete. II: algorithm and calibration. J. Eng. Mech. ASCE 126(9), 954–961 (2000)Google Scholar
  21. F. Caner, Z. Bažant, J. Cervenka, Vertex effect in strain-softening concrete at rotating principal axes. J. Eng. Mech. ASCE 128(1), 24–33 (2002)CrossRefGoogle Scholar
  22. F.C. Caner, Z.P. Bažant, Microplane model M7 for plain concrete. I: formulation. J. Eng. Mech. 139(12), 1714–1723 (2013a)Google Scholar
  23. F.C. Caner, Z.P. Bažant, Microplane model M7 for plain concrete. II: calibration and verification. J. Eng. Mech. 139(12), 1724–1735 (2013b)Google Scholar
  24. F.C. Caner, Z.P. Bažant, C.G. Hoover, A.M. Waas, K.W. Shahwan, Microplane model for fracturing damage of triaxially braided fiber-polymer composites. J. Eng. Mater. Tech. Trans. ASME 133(2) 021024-1–021024-12 (2011)CrossRefGoogle Scholar
  25. F.C. Caner, Z. Guo, B. Moran, Z.P. Bažant, I. Carol, Hyperelastic anisotropic microplane constitutive model for annulus fibrosus. J. Biomech. Eng. Trans. ASME 129(5), 632–641 (2007)CrossRefGoogle Scholar
  26. J. Červenka, Z. Bažant, M. Wierer, Equivalent localization element for crack band approach to mesh-sensitivity in microplane model. Int. J. Numer. Methods Eng. 62(5), 700–726 (2005)CrossRefGoogle Scholar
  27. K.-T. Chang, S. Sture, Microplane modeling of sand behavior under non-proportional loading. Comput. Geotech. 33(3), 177–187 (2006)CrossRefGoogle Scholar
  28. X. Chen, Z.P. Bažant, Microplane damage model for jointed rock masses. Int. J. Numer. Anal. Methods Geomech. 38(14), 1431–1452 (2014)CrossRefGoogle Scholar
  29. G. Cusatis, A. Beghini, Z.P. Bažant, Spectral stiffness microplane model for quasibrittle composite larninates – part I: theory. J. Appl. Mech. Trans. ASME 75(2) 021009-1–021009-9 (2008)CrossRefGoogle Scholar
  30. G.Cusatis, A. Mencarelli, D. Pelessone, J. Baylot, Lattice discrete particle model (LDPM) for failure behavior of concrete. II: calibration and validation. Cem. Concr. Compos. 33(9), 891–905 (2011)Google Scholar
  31. G. di Luzio, A symmetric over-nonlocal microplane model {M4} for fracture in concrete. Int. J. Solids Struct. 44(13), 4418–4441 (2007)CrossRefGoogle Scholar
  32. G. Di Luzio, Z. Bažant, Spectral analysis of localization in nonlocal and over-nonlocal materials with softening plasticity or damage. Int. J. Solids Struct. 42(23), 6071–6100 (2005)CrossRefGoogle Scholar
  33. C.G. Hoover, Z.P. Bazant, J. Vorel, R. Wendner, M.H. Hubler, Comprehensive concrete fracture tests: description and results. Eng. Fract. Mech. 114, 92–103 (2013)CrossRefGoogle Scholar
  34. K. Kirane, Z. Bažant, Microplane damage model for fatigue of quasibrittle materials: sub-critical crack growth, lifetime and residual strength. Int. J. Fatigue 70, 93–105 (2015)CrossRefGoogle Scholar
  35. K. Kirane, M. Salviato, Z.P. Bažant, Microplane-triad model for elastic and fracturing behavior of woven composites. J. Appl. Mech. Trans. ASME83(4) 041006-1–041006-14 (2016)CrossRefGoogle Scholar
  36. M.B. Nooru-Mohamed, Mixed-mode fracture of concrete: an experimental approach. Ph.D. thesis, Civil Engineering and Geosciences, Universiteitsdrukkerij, Delft University of Technology, Delft, The Netherlands. (1992, 5)Google Scholar
  37. J. Ožbolt, Z. Bažant, Microplane model for cyclic triaxial behavior of concrete. J. Eng. Mech. ASCE 118(7), 1365–1386 (1992)CrossRefGoogle Scholar
  38. P. Prat, Z. Bažant, Microplane model for triaxial deformation of saturated cohesive soils. J. Geotech. Eng. ASCE 117(6), 891–912 (1991)CrossRefGoogle Scholar
  39. Simulia Corporation, Abaqus User Subroutines Reference Guide. Dassault Systèmes (2014)Google Scholar
  40. R. Wendner, J. Vorel, J. Smith, C.G. Hoover, Z.P. Bazant, G. Cusatis, Characterization of concrete failure behavior: a comprehensive experimental database for the calibration and validation of concrete models. Mater. Struct. 48(11), 3603–3626 (2015)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ferhun C. Caner
    • 1
    • 2
  • Valentín de Carlos Blasco
    • 2
  • Mercè Ginjaume Egido
    • 1
  1. 1.School of Industrial Engineering, Institute of Energy TechnologiesUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Department of Materials Science and Metallurgical EngineeringUniversitat Politècnica de CatalunyaBarcelonaSpain

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