Continuous and Discontinuous Modelling of Cohesive Frictional Materials,
Lecture Notes in Physics 568, ed. P.A. Vermeer et al., Springer, Berlin, 2001, pp. 17-29.


Milan Jirásek
Swiss Federal Institute of Technology
LSC -DGC, EPFL, 1015 Lausanne, Switzerland


This paper focuses on strain and damage localization due to propagation and coalescence of microcracks in quasibrittle materials such as concrete or rock. The mathematical description and numerical simulation of such phenomena can be based on an indirect representation of the cracking-induced deformation by inelastic strain, or on a direct representation by discontinuities in the displacement field. The first part of the paper gives a general overview and classification of modeling approaches and numerical techniques related to both types of representations and discusses their possible combination. The second part proposes a new technique for computational resolution of localization zones in regularized softening continua, based on special enrichments of the standard finite element approximations.


This paper has briefly sketched the idea of finite elements with embedded localization zones and illustrated it with a simple one-dimensional example. Of course, much additional work is needed to make the proposed concept useful in practical applications. The most difficult step is no doubt the extension to multiple dimensions. In one dimension, the weak compatibility condition (vanishing integral of the enhanced strain mode over the element) is equivalent to the strong compatibility condition (continuity of displacements). In multiple dimensions, this is no longer the case, and the exact way in which weak compatibility is enforced has an important influence on the kinematic properties of the enriched model. For instance, the elements with embedded discontinuities that are derived in a variationally consistent manner from the stress continuity condition are inherently incapable of reproducing a clean separation with no stress transfer across a widely open crack; see Jirasek (2000) for a discussion.

Other issues to be addressed in the future include the optimal choice of enrichment functions and efficient integration schemes. Finally, it is desirable to develop the technique into a truly adaptive   approach, in which the parameters of the enhanced modes are continuously adapted to the evolving shape of the localization profile. These topics are the subject of an ongoing research.

The complete paper can be downloaded in PostScript format (320 kB).

EPFL / 1 February 2001 /