Keywords: fracture, hybrid-Trefftz, interface, softening, crack-insertion, three-dimensional, heterogeneous.

A formulation is presented, using hybrid-Trefftz stress elements for the bulk material and interface elements for the discrete cracks, to model cohesive fracture of heterogeneous materials in three-dimensions using ten-node tetrahedrons. The paper initially outlines the formulation of the hybrid-Trefftz stress elements, following on from the work of Kaczmarczyk and Pearce [1] in two dimensions and extends it into three dimesnions. Hybrid-Trefftz stress elements use two different fields in each element; one to approximate the stresses in the domain of the element and the other to approximate the displacements on the boundary of the element. A stress approximation basis is used to approximate the stresses within the element domain which a priori satisfy the equilibrium condition. The use of these elements allows an accurate representation of the stress state to be found within a body, which is of paramount importance in problems which are investigating fracture.

There are a number of methods for modelling fracture in heterogeneous materials that can broadly be characterised as smeared or discrete. Within this paper discrete cracks are used with all fracture being limited to element boundaries. This approach is adopted because, when carrying out meso-scale analyses, the explicit modelling of heterogeneities in the mesh will strongly influence the direction of cracking and the mesh geometry will be of less importance. The discrete fracture is modelled using continuous interface elements, integrated over the face of the tetrahedron. These interface elements contain a basic linear softening law using fracture energy to capture the softening response of the material. A three-dimensional numerical test is presented where the interface elements have been inserted into the mesh, along a given plane, before the start of the analysis to test the effectiveness of the interface element formulation. Results of this example show that the softening behaviour of the interface element is captured and the elements are behaving in the expected non-linear manner.

Finally, an algorithm is presented for an automatic procedure which allows the interface element to be inserted when the yield stress across an inter element boundary is exceeded. This algorithm is based on the work by Pandolfi and Ortiz [2]. The basic procedure for this algorithm is presented, both to show how faces and nodes were split and to show how the numerical procedure was implemented. A final three-dimensional numerical example is presented to verify the crack insertion algorithm using the continuous interface elements.

1

L. Kaczmarczyk, C.J. Pearce, "A corotational hybrid-Trefftz stress formulation for modelling cohesive cracks", Computer Methods in Applied Mechanics and Engineering, 198(15-16), 1298-1310, 2009. doi:10.1016/j.cma.2008.11.018

2

A. Pandolfi, M. Ortiz, "An Efficient Adaptive Procedure for Three-Dimensional Fragmentation Simulations", Engineering with Computers, 18, 148-159, 2002. doi:10.1007/s003660200013

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