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Non-planar crack growth (X-FEM and fast marching)

N. Sukumar's picture

In the attached manuscript, we have coupled the extended finite element method (X-FEM) to the fast marching method (FMM) for non-planar crack growth simuations. Unlike the level set method, the FMM is ideally-suited to advance a monotonically growing front. The FMM is a single-pass algorithm (no iterations) without any time-step restrictions. The perturbation crack solutions due to Gao and Rice (IJF, 1987) and Lai, Movchan and Rodin (IJF, 2002) are used for the purpose of comparisons. A few of the pertinent cited references can be found off my X-FEM web page. The final version of the manuscript is now attached.

PDF icon xfemfmm.pdf2 MB
PDF icon xfemfmm-final.pdf2 MB


L. Roy Xu's picture

Dear  Suku, 

This is a very challenging topic, and I believe you’ve made significant contributions.   Since you’re using the fast marching method, my impression is that you’re dealing with a dynamic crack.  However, equation (31) is applicable for a static 3-D crack, not a dynamic crack.  Can you use the same approaches to simulate a dynamic crack (for example, the crack tip speed is at least 10% of the shear wave speed)?  Thank you for presenting nice work. 



  The FMM can be applied to quasi-static as well as dynamic processes.  I don't see any reason why what Suku has done could not be extended to dynamic fracture, but will allow him to comment further.  


I believe the sine-cosine expansions in the stress field near the crack tip are based on the work of Westergaard (JAM, 6, p. A49, 1939) and Sneddon (PRSL-A, 187, p. 229, 1946) and can also be derived from the work of Williams (later) and Michell (earlier).  I don't think any of these derivations considered the effect of body forces or inertia.   The sine-cosine expansions might only work for certain forms of the acceleration function (and for negligible accerlerations) - which is probably what Roy is referring to.  You can try plugging in the expressions for displacement in the full Navier equations (including the inertia term) and you will see that it is hard to get the forces to sum up to zero except for special cases.



N. Sukumar's picture

Dear Roy et al.,
Thanks for your comments. Sorry for the time-delay. Probably some further clarifications are needed in the text so as to not mislead the reader. Use of `front velocity' is standard in the FMM/level-set literature, and hence we have used the same (even though dynamic fracture is not considered). As Biswajit pointed out, the expressions for the stresses are for a static/quasi-static (classical) Westergaard's crack solution. No inertial contribution is assumed in the paper. As John alluded to, the FMM is independent from the crack mechanics/physics. Given a crack front and the `velocity' prescribed on it, the FMM allows one to just update the position of the new crack location.  For dynamic fracture, there might be stability issues when too large a `time step for FMM' is taken, but in principle, the algorithm can be applied.

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