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Special Issue Honoring Professor Fazil Erdogan’s Contributions to Mixed Boundary Value Problems of Inhomogeneous and Functionally Graded Materials

Analysis of Interacting Cracks Using the Generalized Finite Element Method With Global-Local Enrichment Functions

[+] Author and Article Information
Dae-Jin Kim, Jeronymo Peixoto Pereira

Department of Civil and Environmental Engineering, Newmark Laboratory, University of Illinois at Urbana-Champaign, 205 North Mathews Avenue, Urbana, IL 61801

Carlos Armando Duarte1

Department of Civil and Environmental Engineering, Newmark Laboratory, University of Illinois at Urbana-Champaign, 205 North Mathews Avenue, Urbana, IL 61801caduarte@uiuc.edu

1

Corresponding author.

J. Appl. Mech 75(5), 051107 (Jul 11, 2008) (12 pages) doi:10.1115/1.2936240 History: Received June 06, 2007; Revised February 14, 2008; Published July 11, 2008

This paper presents an analysis of interacting cracks using a generalized finite element method (GFEM) enriched with so-called global-local functions. In this approach, solutions of local boundary value problems computed in a global-local analysis are used to enrich the global approximation space through the partition of unity framework used in the GFEM. This approach is related to the global-local procedure in the FEM, which is broadly used in industry to analyze fracture mechanics problems in complex three-dimensional geometries. In this paper, we compare the effectiveness of the global-local FEM with the GFEM with global-local enrichment functions. Numerical experiments demonstrate that the latter is much more robust than the former. In particular, the GFEM is less sensitive to the quality of boundary conditions applied to local problems than the global-local FEM. Stress intensity factors computed with the conventional global-local approach showed errors of up to one order of magnitude larger than in the case of the GFEM. The numerical experiments also demonstrate that the GFEM can account for interactions among cracks with different scale sizes, even when not all cracks are modeled in the global domain.

Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 8

Analysis with interacting cracks discretized in the global domain. Global problems are solved with cubic shape functions.

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Figure 9

Analysis with interacting cracks discretized in the global domain. Global problems are solved with cubic shape functions and Westergaard function enrichments. Global problems are solved with cubic shape functions

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Figure 10

Discretization of a problem with two interacting cracks. Front view for the case B∕H=2. The cracks are not discretized in the global domain. (a) The shaded areas represent the local domains extracted from the coarse global mesh. (b) Graded meshes used in the discretization of local problems. (c) Enrichment of global discretization with local solutions. Global nodes enriched with local solutions are represented with squares.

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Figure 11

Analysis with interacting cracks not discretized in the global domain. Global problems are solved with linear shape functions.

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Figure 12

Analysis with interacting cracks not discretized in the global domain. Global problems are solved with cubic shape functions.

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Figure 13

Rectangular panel with a through-the-thickness inclined crack

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Figure 1

Global-local analysis for a structural component with a planar crack surface. (a) Global analysis with a coarse mesh to provide boundary conditions for the extracted local domain. (b) Refined local problem and its solution.

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Figure 2

Construction of a GFEM shape function using a polynomial (a) and a nonpolynomial enrichment (b). Here, φα are the functions at the top, the enrichment functions, Lαi, are the functions in the middle, and the generalized FE shape functions, ϕαi, are the resulting bottom functions.

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Figure 3

Notations for the GFEM with global-local enrichment functions. (a) A global domain containing one macrocrack and several microcracks. (b) A local domain extracted from the global domain in the neighborhood of the macrocrack front.

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Figure 4

Enrichment of the coarse global mesh with a local solution

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Figure 5

Description of a problem with two interacting cracks in an infinite strip

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Figure 6

Discretization of a problem with two interacting cracks using tetrahedral elements. Front view of the strip shown in Fig. 5 for the case B∕H=2. Note that the cracks are discretized in the global domain and a three-dimensional discretization is used. (a) Discretization of cracks in the initial global problem. The shaded areas represent the local domains extracted from the coarse global mesh. (b) Graded meshes used in the discretization of local problems. (c) Enrichment of global discretization with local solutions. Global nodes enriched with local solutions are represented with squares.

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Figure 7

Analysis with interacting cracks discretized in the global domain. Global problems are solved with linear shape functions. Ref. represents the reference SIF values obtained from Ref. 10.

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Figure 14

Discretization of the problem with an inclined crack. The crack is not discretized in the global domain. (a) The shaded area represents the local domain extracted from the coarse global mesh. (b) Enrichment of global discretization with local solutions. Global nodes enriched with local solutions are represented with squares.

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Figure 15

Description of a MSD problem

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Figure 16

Discretization of the MSD problem (front view). Only the main crack is discretized in the global domain while both the main and MSD cracks are discretized in the local domains. (a) Discretization of cracks in the initial global problem. Solid, dashed, and long dash-double dotted lines represent the boundaries of local domains with three different sizes used in this analysis. (b) Graded mesh used in the discretization of the local problem represented by a dashed line in (a). (c) Enrichment of global discretization with the local solution in (b). Global nodes enriched with the local solution are represented by squares.

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Figure 17

Deformed shape of the global domain in the MSD problem before and after enrichment with a local solution. The elements of the local problem nested in the global mesh are visualized in (b). (a) Deformed shape of the global domain before enrichment with a local solution. (b) Deformed shape of the global domain after enrichment with a local solution.

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