From Size-Effect Evaluation to Continuum Models with Strain Softening

  • Gianni Royer-Carfagni
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 61)


Standard uniaxial tensile tests allow to measure the specimen “average” response, but assuming this as a constitutive relationship for 1-D continuum models leads to well-know inconsistencies, such as strain localization or null fracture work, especially when strain softening is involved. Using recent results of variational convergence of discrete functionals (Γ–convegence), a method is here proposed to conceive consistent continuum models starting from the experimental observation of the size-dependent response (size-effect). In general, the resulting model is à la Hillerborg, where softening and fracture are interpreted by the dichotomy between bulk and interface energy.


Fracture Energy Strain Softening Constitutive Relationship Continuous Displacement Variational Convergence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bažant, Z.P.: Size effect in blunt fracture. J. Eng. Mech. ASCE 110, 518–535 (1984)CrossRefGoogle Scholar
  2. 2.
    Carpinteri, A., Chiaia, B., Ferro, G.: Size-effects on nominal tensile strength of concrete structures: multifractality of material ligaments and dimensional transition from order to disorder. Mat. Struct. 28, 311–317 (1995)CrossRefGoogle Scholar
  3. 3.
    Carpinteri, A., Ferro, G.: Size effects on tensile fracture roperties: A unified explanation based on disorder and fractality of concrete microstructure. Mat. Struct. 27, 563–571 (1994)CrossRefGoogle Scholar
  4. 4.
    Braides, A., Gelli, M.S.: Continuum limit of discrete systems without convexity hypotheses. Math. Mech. Solid 7, 41–66 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Hillerborg, A., Modèer, M., Petersson, P.E.: Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement Concrete Res. 6, 773–782 (1976)CrossRefGoogle Scholar
  6. 6.
    Del Piero, G., Truskinovsky, L.: Macro- and micro- cracking in one dimensional elasticity. Int. J. Solid. Struct. 38, 1135–1148 (2001)zbMATHCrossRefGoogle Scholar
  7. 7.
    Choksi, R., Del Piero, G., Fonseca, I., Owen, D.: Structured deformations as energy minimizers in models of fracture and hysteresis. Math. Mech. Solid 4, 321–356 (1999)zbMATHCrossRefGoogle Scholar
  8. 8.
    Gelli, M.S., Royer-Carfagni, G.: Separation of scales in fracture mechanics. From molecular to Continuum theory via Γ–convergence. J. Eng. Mech. ASCE 130, 204–215 (2004)Google Scholar
  9. 9.
    Royer-Carfagni, G.: How the experimental observation of size-effect can help to assess constitutive equations for strain softening bars. In: Proc. XVI National Congress AIMETA, Ferrara (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Gianni Royer-Carfagni
    • 1
    • 2
  1. 1.Dipartimento di Ingegneria Civile, Ambiente, Territorio e ArchitetturaUniversità di ParmaParmaItaly
  2. 2.Istituto per le Applicazioni del Calcolo “M. Picone”Consiglio Nazionale RicercheRomeItaly

Personalised recommendations