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©2012 Civil-Comp Ltd |
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S.Sh. Ghorashi1, N. Valizadeh2, S. Mohammadi3 and T. Rabczuk2
1Research Training Group 1462, 2Institute of Structural Mechanics,
Bauhaus-Universität Weimar, Germany
3School of Civil Engineering, University of Tehran, Iran
Keywords: extended isogeometric analysis, curved crack, enrichment functions, linear elastic fracture mechanics, mapping, stress intensity factor.
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The fracture analysis of an isotropic two-dimensional body with a curved
crack using the extended isogeometric analysis (XIGA) is discussed
in this paper.
The extended finite element method (XFEM) belongs to the class of
partition of unity methods (PUM). The principal feature of this method
is the ability to add, locally, a priori knowledge of the solution
to the approximation space. This enrichment allows for capturing particular
features such as discontinuities and singularities which are present
in the solution exactly. The XFEM, in particular, has been used successfully
to solve crack initiation and propagation problems with no or minimal
remeshing of the moving boundaries. On the other hand, isogeometric
analysis (IGA) is another new robust computational approach, which
offers the possibility of integrating the finite element analysis
(FEA) into conventional NURBS-based CAD design tools. This allows
models to be designed, tested and adjusted in one go, using a common
data set.
The isogeometric formulation using NURBS basis functions has been
recently enriched by the general XFEM methodology to solve linear
fracture mechanics problems with incompatible meshes while obtaining
solutions with higher order convergence rates and high levels of accuracy.
This approach, entitled the extended isogeometric analysis (XIGA),
benefits from the advantages of its origins: XFEM and IGA; while it
is capable of analysing crack growth problems without any remeshing
requirement, complex geometries can be modelled with few elements and
higher order inter-element continuities are satisfied.
XIGA has been successfully employed for analysis of bodies with straight
cracks in the parametric space and parent element. For improving the
accuracy of integration using the Gauss quadrature rule, the "sub-triangles
approach" and the "almost polar technique"
have been utilized for split and crack tip elements, respectively.
The principal difficulty with curved cracks is the special treatment
required for the integration of sub-triangles which is that one of its sides
is not straight in the parametric space.
In this paper, specific mappings based on the blending function
method are used for integration over the elements cut by a non-straight
part of the crack, and is extended to the analysis of structures having
curved cracks in the parametric space. Consequently, no limitation
remains for considering arbitrary non-smooth and smooth strong discontinuities
in the XIGA.
In order to impose Dirichlet boundary conditions, the Lagrange multiplier
method is used. Mixed-mode stress intensity factors (SIFs) are evaluated
by means of the interaction integral to determine the fracture properties
of the domain.
Finally, a plate with an arc-shaped central crack with different
arc angles is analysed using the proposed method. Comparison of the numerical
results with analytical solutions available in the literature shows
the efficiency and validity of the approach.
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