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.
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