International Journal for Numerical Methods in Engineering, 50 (2001), 1269-1290 and 1291-1305.



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


The recently emerged idea of incorporating strain or displacement discontinuities into standard finite element interpolations has triggered the development of powerful techniques that allow efficient modeling of regions with highly localized strains, e.g. of fracture zones in concrete or shear bands in metals or soils. The present paper develops a triangular element with an embedded displacement discontinuity that represents a crack. The constitutive description is formulated within the framework of damage theory, with crack closure effects and friction on the crack faces taken into account. Numerical aspects of the implementation are discussed.

The second part of the paper investigates the behavior of finite elements with embedded discontinuities that represent cracks. Examples of fracture simulations show that an incorrect separation of nodes due to a locally mispredicted crack direction lead to a severe stress locking, which can be alleviated by secondary cracking. As an alternative, the paper advocates a new concept of a model with transition from a smeared to an embedded (discrete) crack. As an additional improvement, the smeared part is reformulated as nonlocal. Various criteria for placing the discontinuity are compared, and the optimal technique is identified.  Remarkable insensitivity of the resulting model to mesh-induced directional bias is demonstrated.

Concluding Remarks

Constitutive description of a damaging interface inspired by continuum damage mechanics has been proposed, and possible extensions that take into account the stiffness recovery upon crack closure and the effects of friction have been outlined. The displacement discontinuity has been incorporated into a constant strain triangle, based on the statically and kinematically optimal nonsymmetric formulation. Algorithms for the evaluation of internal forces and of the element stiffness matrix have been developed and extended to special cases such as the onset of cracking, a completely stress-free crack, or a closed crack. Numerical examples are presented in the companion paper.

Numerical testing has revealed that the conventional embedded crack approach, which introduces a displacement discontinuity right at the onset of cracking, often leads to a misprediction of the discontinuity direction. As the direction has to remain fixed, there is no chance for its adjustment. This inevitably leads to stress locking that must be relaxed by a secondary crack in the same element. Multiple cracking 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.

The combination of the smeared and embedded descriptions of cracking is appealing from the physical point of view. It is intuitively clear that diffuse damage at early stages of material degradation is adequately described by a model dealing with inelastic strain while highly localized fracture is better represented by a displacement discontinuity. Examples of simulations of fracture specimens demonstrate the potential of the developed technique.

As an additional improvement, the smeared part of the combined model has been reformulated as nonlocal. It turns out that for the alleviation of 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 the embedded crack is in each element determined from the principal directions of nonlocal (rather than local) strain.

To summarize, the embedded crack approach seems to be a very appealing technique that certainly has a number of important advantages compared to more traditional approaches. Of course, a large amount of work still remains to be done. For example, a challenging goal would be an extension of the model to three dimensions.

The complete paper can be downloaded in an electronic form:
Part I - PostScript, Part I - PDF, Part II - PostScript, Part II - PDF

EPFL / 6 February 2001 /