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International Journal for Multiscale Computational Engineering
Facteur d'impact: 1.016 Facteur d'impact sur 5 ans: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Imprimer: 1543-1649
ISSN En ligne: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2012002827
pages 343-360


Xia Liu
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, New York 10027, USA
Haim Waisman
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, New York 10027, USA
Jacob Fish
Civil Engineering and Engineering Mechanics, Columbia University, New York, New York 10027, USA


Branch functions are commonly used as crack tip enrichments in the extended finite element method (XFEM). Typically, these are four functions derived from linear elasticity and added as additional degrees of freedom. However, for general inelastic material behavior, where the analytical solution and the order of singularity are unknown, Branch functions are typically not used, and only the Heaviside function is employed. This, however, may introduce numerical error, such as inconsistency in the position of the crack tip. In this paper we propose a special construction of a Ramp function as tip enrichment, which may alleviate some of the problems associated with the Heaviside function when applied to general inelastic materials, especially ones with no analytical solutions available. The idea is to linearly ramp down the displacement jump on the opposite sides of the crack to the actual crack tip, which may stop the crack at any point within an element, employing only one enrichment function. Moreover, a material length scale that controls the slope of the ramping is introduced to allow for better flexibility in modeling general materials. Numerical examples for ideal and hardening elastoplastic and elastoviscoplastic materials are given, and the convergence studies show that a better performance is obtained by the proposed method in comparison with the Heaviside function. Nevertheless, when analytical functions, such as the Hutchinson-Rice-Rosengren (HRR) fields, do exist (for very limited material models), they indeed perform better than the proposed Ramp function. However, they also employ more degrees of freedom per node and hence are more expensive.


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