Proc. Computational Modelling of Concrete
Badgastein, Austria, March 31 - April 3, 1998,
ed. R. de Borst, N. Bicanic, H. Mang, and G. Meschke, Balkema, Rotterdam, 291-300
MODELS FOR CONCRETE FRACTURE
Swiss Federal Institute of Technology
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.
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
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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13 January 1998 /