Proc. Computational Modelling of Concrete Structures (EURO-C),
Badgastein, Austria, March 31 - April 3, 1998,
ed. R. de Borst, N. Bicanic, H. Mang, and G. Meschke, Balkema, Rotterdam, 291-300


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


The first part of this paper suggests a systematic approach to the classification of various computational techniques based on the incorporation of a displacement or strain discontinuity into a finite element, with special attention to the type of kinematic enhancement and of the stress continuity condition. Advantages and drawbacks of three principal classes of such techniques are analyzed and discussed. The second part outlines two possible improvements of the standard approach: combination of the smeared and embedded crack approaches, and incorporation of the embedded discontinuities into a nonlocal model.

Concluding Remarks

A number of techniques enriching the standard finite element interpolation by additional terms corresponding to a displacement or strain discontinuity have been presented within a unified framework and critically evaluated. It has been shown that there exist three major classes of these models, called here statically optimal symmetric (SOS), kinematically optimal symmetric (KOS), and statically and kinematically optimal nonsymmetric (SKON). The SOS formulation cannot properly reflect the kinematics of a completely open crack but it gives a natural stress continuity condition, while the KOS formulation describes the kinematic aspects satisfactorily but leads to an awkward relationship between the stress in the continuous part of the element and the tractions across the discontinuity line. These findings motivate the development of the nonsymmetric SKON formulation, which combines the strong points of each of the symmetric formulations. It is not variationally consistent but leads to an improved numerical performance.

The second part of the paper has proposed some improvements of the standard embedded crack technique. It has been shown that a model introducing the discontinuity right at the onset of cracking often leads to a misprediction of the discontinuity direction. As this direction has to remain fixed, there is no chance for its adjustment. Incorrect separation results into stress locking, which must be relaxed by a secondary crack in the same element. This complicates the numerical algorithm and can lead to convergence problems. It has therefore been proposed to use a combined model that represents the early stage of cracking in a smeared manner and introduces a discontinuity only when the crack opens sufficiently wide. If the smeared part is modeled by the rotating crack approach, the crack has a chance to readjust its direction, and there is no need for secondary cracking. Furthermore, the smeared part of the combined model has been reformulated as nonlocal. It has been demonstrated that, in order to avoid locking, it is essential to enforce continuity of the embedded crack trajectory. Optimal performance in terms of insensitivity to mesh-induced directional bias is obtained if the orientation of each embedded crack is determined from the principal directions of nonlocal (rather than local) strain.

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EPFL / 13 January 1998 /