Library Subscription: Guest
Home Begell Digital Library eBooks Journals References & Proceedings Research Collections
International Journal for Multiscale Computational Engineering

Impact factor: 0.760

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2011002761
pages 599-608


Patrizia Trovalusci
Department of Structural Engineering and Geotechnics, Sapienza University of Rome, 00184 Roma, Italy
Valerio Varano
Department of Structures, Roma Tre University of Rome, 00146 Roma, Italy


This work is based on the formulation of a continuum model with microstructure for the study of the mechanical behavior of microcracked materials. Such a continuum is named multifield continuum because it is characterized by field descriptors accounting for the presence of material internal structure. In particular, the disturbance due to the presence of distributed microcracks in the material is revealed by an additional kinematical field representing the smeared displacement jump over the microcracks. According to the approach of the classical molecular theory of elasticity, the constitutive multifield continuum (macromodel) has been obtained by requiring the energy equivalence with an appropriate discrete micromodel. The stress-strain relations of the continuum have been explicitly identified by selecting the response functions of the interactions of the discrete model and depend on the geometry of the material's internal phases. Attention is here focused on theoretical and numerical investigations on a one-dimensional microcracked bar by varying the microcrack density and size. The effectiveness of the multi-field model, in representing the gross mechanical behavior of such materials with internal structure, is ascertained by comparing the multifield solutions with the numerical solutions obtained by using finite-element simulations for a linear elastic strip having different distributions of voids.


  1. Benvenuti, E., Borino, G., and Tralli, A., A thermodynamically consistent nonlocal formulations for damaging materials. DOI: 10.1016/S0997-7538(02)01220-2

  2. Capecchi, D., Ruta, G., and Trovalusci, P., From classical to Voigt’s molecular models in elasticity. DOI: 10.1007/s00407-010-0065-y

  3. Capriz, G., Continua with Microstructure.

  4. Capriz, G. and Podio-Guidugli, P., Whence the boundary conditions in modern continuumphysics?.

  5. Cauchy, A.-L., Sur l’ èquilibre et le mouvement d’un système de points matèriels sollicitès par des forces d’attraction ou de rèpulsion mutuelle.

  6. Cauchy, A.-L., Sur l’èquilibre et le mouvement d’un système de points matèriels sollicitès par des forces d’attraction ou de rèpulsion mutuelle.

  7. Di Carlo, A., Non-standard format for continuum mechanics.

  8. Erigen, A. C., Microcontinuum Field Theories.

  9. Forest, S., The micromorphic approach for gradient elasticity, viscoplasticity and damage. DOI: 10.1061/(ASCE)0733-9399(2009)135:3(117)

  10. Forest, S. and Trinh, D. K., Generalised continua and the mechanics of heterogeneous materials.

  11. Germain, P., The method of virtual power in continuum mechanics. Part II: Microstructure. DOI: 10.1137/0125053

  12. Goddard, J. D., A general micromorphic theory of kinematics and stress in granular media.

  13. Kouznetsova, V. G., Geers, M. G. D., and Brekelmans, W. A. M., Multi-scale second order computational homogenization of multi-phase materials: A nested finite element solution strategy. DOI: 10.1016/j.cma.2003.12.073

  14. Kunin, I. A., Elastic Media with Microstructure. I: One-dimensional Models.

  15. Mariano, P. M. and Trovalusci, P., Constitutive relations for elastic microcracked bodies: From a lattice model to a multifield continuum description. DOI: 10.1177/105678959900800204

  16. Ortiz, M. and Phillips, R., Nanomechanics of defects in solids.

  17. Palla, P., Ippolito, M., Giordano, S., Mattoni, A., and Colombo, L., Atomistic approach to nanomechanics: Concepts, methods, and (some) applications.

  18. Peerlings, R. H. J. and Fleck, N. A., Computational evaluation of strain gradient elasticity constants.

  19. Pijaudier-Cabot, G. and Bazant, Z. P., Nonlocal damage theory. DOI: 10.1061/(ASCE)0733-9399

  20. Sansalone, V., Trovalusci, P., and Cleri, F., Multiscale modelling of materials by a multifield approach: Microscopic stress and strain distribution in fiber-matrix composites. DOI: 10.1016/j.actamat.2006.03.041

  21. Sluys, L. J., de Borst, R., and Mühlhaus, H.-B., Wave propagation, localization and dispersion in a gradient-dependent medium. DOI: 10.1016/0020-7683(93)90010-5

  22. Trovalusci, P. and Augusti, G., A continuum model with microstructure for materials with flaws and inclusions. DOI: 10.1051/jp4:1998847

  23. Trovalusci, P. and Masiani, R., A multi-field model for blocky materials based on multiscale description. DOI: 10.1016/j.ijsolstr.2005.03.027

  24. Trovalusci, P., Capecchi, D., and Ruta, G., Genesis of multiscale approach for materials with microstructure. DOI: 10.1007/s00419-008-0269-7

  25. Trovalusci, P., Varano, V., and Rega, G., A generalized continuum formulation for composite microcracked materials and wave propagation in a bar. DOI: 10.1115/1.4001639

  26. Voigt, W., Lehrbuch der Kristallphysik.