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Modeling crack discontinuities without element‐partitioning in the extended finite element method
Author(s) -
Chin E. B.,
Lasserre J. B.,
Sukumar N.
Publication year - 2016
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.5436
Subject(s) - finite element method , extended finite element method , mathematics , polygon mesh , stiffness matrix , quadrilateral , classification of discontinuities , mixed finite element method , polygon (computer graphics) , mathematical analysis , finite element limit analysis , discontinuity (linguistics) , geometry , structural engineering , computer science , engineering , telecommunications , frame (networking)
Summary In this paper, we model crack discontinuities in two‐dimensional linear elastic continua using the extended finite element method without the need to partition an enriched element into a collection of triangles or quadrilaterals. For crack modeling in the extended finite element, the standard finite element approximation is enriched with a discontinuous function and the near‐tip crack functions. Each element that is fully cut by the crack is decomposed into two simple (convex or nonconvex) polygons, whereas the element that contains the crack tip is treated as a nonconvex polygon. On using Euler's homogeneous function theorem and Stokes's theorem to numerically integrate homogeneous functions on convex and nonconvex polygons, the exact contributions to the stiffness matrix from discontinuous enriched basis functions are computed. For contributions to the stiffness matrix from weakly singular integrals (because of enrichment with asymptotic crack‐tip functions), we only require a one‐dimensional quadrature rule along the edges of a polygon. Hence, neither element‐partitioning on either side of the crack discontinuity nor use of any cubature rule within an enriched element are needed. Structured finite element meshes consisting of rectangular elements, as well as unstructured triangular meshes, are used. We demonstrate the flexibility of the approach and its excellent accuracy in stress intensity factor computations for two‐dimensional crack problems. Copyright © 2016 John Wiley & Sons, Ltd.