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©2012 Civil-Comp Ltd |
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G. Edwards, C.J. Pearce and L. Kaczmarczyk
Infrastructure & Environment Research Division, School of Engineering, University of Glasgow, United Kingdom
Keywords: fracture, hybrid-Trefftz, interface, softening, crack-insertion, three-dimensional, heterogeneous.
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reference
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.
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- 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.
- 2
- A. Pandolfi, M. Ortiz, "An Efficient Adaptive Procedure for Three-Dimensional Fragmentation Simulations", Engineering with Computers, 18, 148-159, 2002.
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