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Generalized Reproducing Kernel Particle Method for Fragment-Impact and Fracture Modeling


Accurate modeling of fracture and material fragmentation remains a challenging problem in computational mechanics. Extrinsic enrichment of the approximation functions leads to an increase of degrees of freedom, while intrinsic enrichment near crack tips encounters compatibility issues. Furthermore, although material fragmentation can be effectively represented under a semi-Lagrangian meshfree framework in conjunction with the appropriate damage laws, approximation with high order bases remains difficult to construct in this class of problem due to the need of meeting the kernel support coverage conditions in the fragmented areas. A new approach for constructing the meshfree approximation functions is introduced in this dissertation to enhance the flexibility in their construction and to address the above mentioned difficulties in fracture and fragment-impact modeling of materials and solids.

First, the construction of meshfree approximation functions with arbitrary order of completeness is revisited and extended with a more general approach. Approximation properties of the proposed family of approximation functions are examined and their unique features are utilized to achieve oscillation diminishing properties and $p-$adaptivity. Further, a quasi-linear reproducing kernel particle method (QL-RKPM) is also formulated. The method introduces additional sampling points in the moving least-squares function to avoid singularity of the moment matrix, while providing controllable error in the linear field approximation when the kernel coverage conditions in the conventional reproducing kernel approximation of linear field are not met. This technique is employed in modeling a variety of impact and fragment-impact problems, and is shown to provide robust and accurate results.

The generalized and quasi-linear reproducing kernel approximations are then combined to solve linear elastic fracture mechanics problems. The enriched approximation surrounding crack tips is obtained by an intrinsic enrichment technique using the generalized RKPM approach to minimize the discontinuity at the interface between enriched and non-enriched regions. The displacement discontinuity along the crack surface is captured with the visibility criterion, and the quasi-linear approximation ensures invertibility of the moment matrix even with heavily truncated supports along the crack face. The method is applied to different fracture mechanics boundary value problems, and is shown to be accurate and effective without the need of tedious treatments in the existing methods.

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