Keywords: fracture, configurational forces, crack path, mesh adaptivity, mesh quality.

This paper presents a finite element based numerical framework for the predictive modelling of three-dimensional crack propagation in brittle solids. The paper briefly sets out the theoretical basis for determining the initiation and direction of propagating cracks, based on the concept of configurational forces. This work broadly follows the work of Gürses and Miehe [1] with a focus here on accurate resolution of cracks by the finite element mesh. Cracks are restricted to the element faces and the mesh is adapted in order to align element faces with the predicted crack path.

We are primarily concerned in this paper with solving crack propagation in large three-dimensional problems. The efficient solution of such problems, with large numbers of degrees of freedom, requires the use of an iterative solver for solving the system of algebraic equations. In such cases, we must control element quality in order to optimise matrix conditioning, thereby increasing the computational efficiency of the solver. A local mesh improvement procedure is developed in conjunction with a mesh quality measure. This approach improves accuracy and solution robustness and reduces the influence of the initial mesh on the direction of propagating cracks. In order to trace the dissipative load-displacement path, an arc-length scheme is adopted.

Unstructured meshes, with four-noded tetrahedral displacement-based finite elements, are used. Furthermore, hierarchic bases of arbitrary polynomial order for approximation of the displacement field are introduced (see Ainsworth and Coyle [2]). The influence of p-adaptivity is studied and its effect on structural response is studied.

The performance of this modelling approach is demonstrated on three numerical examples that qualitatively illustrate its ability to predict complex crack paths, showing good agreement with experimental results. All problems are three-dimensional, including a torsion problem that results in the accurate prediction of a doubly-curved crack. The examples demonstrate that crack propagation can be captured without undue influence from the initial mesh. Furthermore, the influence of mesh quality improvement at the crack front and the influence of global increases in the order of approximation on the solution are also clearly demonstrated.

[1]

E. Gürses, C. Miehe, "A computational framework of three-dimensional configurational-force-driven brittle crack propagation", Computer Methods in Applied Mechanics and Engineering, 198, 1413–1428, 2009. doi:10.1016/j.cma.2008.12.028

[2]

M. Ainsworth, J. Coyle, "Hierarchic finite element bases on unstructured tetrahedral meshes", International Journal for Numerical Methods in Engineering, 58(14), 2103–2130, 2003. doi:10.1002/nme.847

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