Recently I have spent some time trying to understand the differences among various models that deal with strain or displacement discontinuities embedded into finite elements. The table below shows why this has not been an easy task. The number of embedded-crack formulations that emerged in the literature during the past ten years is quite large (not all of them are included in the table), and every author has his (never her in this particular class of problems) prefered notation and style of reasoning.
However, it seems that all the approaches can be classified into three major groups that I tentatively call the
|formulation||authors||parent elements||discontinuity||material law||
|| Belytschko et al. (1988) || quadrilaterals || weak || J2-plasticity || shear bands
|| Larsson and Runesson (1996) || CST || reg. strong || plasticity || mixed-mode fracture
|| Berends (1996) || CST || weak || smeared crack || cracking
|| Sluys (1997) || CST || weak || plasticity || shear bands
|| KOS || Lotfi (1992), Lotfi and Shing (1995) || CST, Q4, QE5, QM6 || strong || cohesive crack || cracking of concrete
|| SKON || Dvorkin et al. (1990) || Q4, QMITC || strong || cohesive crack || cracking of concrete || Klisinski et al. (1991) || Q4 || strong || cohesive crack || cracking
|| Simo and Oliver (1994) || CST || reg. strong || damage, plasticity || shear bands
|| Oloffson et al. (1994) || CST || strong || plastic interface law || cracking
|| Oliver (1996) || CST || reg. strong || damage, plasticity || cracking, shear bands
Table 1: Overview of elements with embedded discontinuities
The symmetric formulations (SOS and KOS) start from a natural static or kinematic condition and then derive the other condition from a variational principle. This is a consistent approach, which however does not lead to a good performance of the element, simply because the most natural static condition is not work-conjugate with the most natural kinematic condition. It seems that the best results can be obtained with the nonsymmetric formulation that simply postulates each of the conditions independently, without regard to variational consistency.
I have played with a constant-strain triangular element of this type, trying to improve its performance on general unstructured meshes. A major problem of the standard formulations is that a displacement discontinuity is introduced into the element right at the onset of cracking. In a practical application, the mesh is usually not very fine and not aligned with the actual direction of crack propagation, and the direction of principal stress/strain often have considerable error. Once the discontinuity is introduced, its direction remains fixed, and if it was mispredicted it can happen that the crack does not separate the nodes that should actually go apart. Spurious stresses are generated in such element, and they can be released only by secondary cracking. This complicates the numerical algorithm and artificially increases energy dissipation.
|The band of cracking elements propagates from left to right. In the rightmost cracking element, the embedded crack is initiated in a completely wrong direction and it separates the nodes incorrectly. This leads to severe stress locking.|
|The band can propagate further only when a second crack forms in the element where the direction was mispredicted. The model must therefore allow for multiple embedded cracks in one element. In more difficult simulations this can lead to problems with convergence.|
A possible remedy is offered by a combination of the embedded crack approach with a smeared crack approach. A smeared rotating crack is also often born in a partially incorrect direction but as the inelastic strain increases the crack usually rotates closer to the ideal direction. All we need for the present purpose is that it finds a position in which it correctly separates the nodes that should end up on the opposite sides of the resulting macroscopic crack. And this almost always happens, because the way the nodes are trying to get apart has a strong influence on the orientation of the inelastic strain. In the final stage of cracking (formation of a stress-free crack) the smeared approach can lead to spurious stress transfer (stress locking) but when we switch to the embedded crack formulation at the right moment, the model should work fine. My simulations, the details of which will be published later, indicate that a model with transition from smeared to embedded crack performs quite well and reduces or completely eliminates secondary cracking.
|Cracks in the left part are already widely open and modeled as discontinuities while cracks in the right part are still modeled by the smeared approach (sorry that they are not plotted in a different color). Each smeared crack can gradually adjust its direction before it turns into an embedded discontinuity with a fixed direction, and so no secondary cracking is necessary.|
The most promising results are obtained when the smeared part of the model is formulated as nonlocal. This dramatically reduces mesh-induced directional bias, as is documented by the following figure. All its parts show in grey the triangular mesh and the cracks that are still in the range modeled by the smeared approach. The dark lines correspond to cracks that have already reached the range modeled by an embedded displacement discontinuity. The smeared part is nonlocal in all examples below.
|The crack path is not enforced to be continuous. The position of each embedded crack is determined by the element center and the direction of maximum principal local strain at the moment of transition from smeared to embedded crack model.|
|A close-up of the previous picture reveals that one of the embedded cracks separates the nodes incorrectly (it is the top one in the center). This leads to severe locking at later stages of the loading process because the macroscopic crack cannot open properly.|
|When continuity of the crack path is enforced, locking disappears and the overall crack direction is correct but locally the crack is tortuous.|
|A perfectly straight crack trajectory in exactly the correct direction is obtained if the crack path continuity is enforced and, in addition, the direction of the segment in each element is taken as orthogonal to the direction of maximum principal nonlocal strain.|