Research Papers

An Accurate and Efficient Augmented Finite Element Method for Arbitrary Crack Interactions

[+] Author and Article Information
W. Liu

Department of Mechanical and Aerospace Engineering,
University of Miami,
Coral Gables, FL 33124;
Department of Mechanics and Aerospace Engineering,
Peking University,
Beijing 100871, China

Q. D. Yang

Department of Mechanical and Aerospace Engineering,
University of Miami,
Coral Gables, FL 33124
e-mail: qdyang@miami.edu

S. Mohammadizadeh

Department of Mechanical and Aerospace Engineering,
University of Miami,
Coral Gables, FL 33124

X. Y. Su

Department of Mechanics and Aerospace Engineering,
Peking University,
Beijing 100871, China

D. S. Ling

Department of Civil Engineering,
Zhejiang University,
Hangzhou 310058, China

For a weak discontinuity Γc+=Γc- while for a strong discontinuity Γc+andΓc- are separated.

1Corresponding author.

Manuscript received September 9, 2012; final manuscript received October 2, 2012; accepted manuscript posted October 30, 2012; published online May 31, 2013. Editor: Yonggang Huang.

J. Appl. Mech 80(4), 041033 (May 31, 2013) (12 pages) Paper No: JAM-12-1441; doi: 10.1115/1.4007970 History: Received September 09, 2012; Revised October 02, 2012; Accepted October 03, 2012

This paper presents a new augmented finite element method (A-FEM) that can account for path-arbitrary, multiple intraelemental discontinuities with a demonstrated improvement in numerical efficiency by two orders of magnitude when compared to the extended finite element method (X-FEM). We show that the new formulation enables the derivation of explicit, fully condensed elemental equilibrium equations that are mathematically exact within the finite element context. More importantly, it allows for repeated elemental augmentation to include multiple interactive cracks within a single element without additional external nodes or degrees of freedom (DoFs). A novel algorithm that can rapidly and accurately solve the nonlinear equilibrium equations at the elemental level has also been developed for cohesive cracks with piecewise linear traction-separation laws. This efficient new solving algorithm, coupled with the mathematically exact elemental equilibrium equation, leads to dramatic improvement in numerical accuracy, efficiency, and stability when dealing with arbitrary cracking problems. The A-FEM's excellent capability in high-fidelity simulation of interactive cohesive cracks in homogeneous and heterogeneous solids has been demonstrated through several numerical examples.

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Grahic Jump Location
Fig. 1

Notation for an elastic body with a discontinuity

Grahic Jump Location
Fig. 2

The piecewise linear, mixed-mode cohesive zone model used in this study. The circled numbers indicate the respective segment numbers.

Grahic Jump Location
Fig. 3

Illustration of the element augmentation from (a) a regular element with a discontinuity, to (b) an A-FE with two quadrilateral subdomains, or to (c) an A-FE with one triangular subdomain and one pentagonal subdomain

Grahic Jump Location
Fig. 4

Illustration of a 2D A-FE with two intraelemental cracks

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Fig. 5

Single A-FE responses under (a) model I, (b) mode II, (c) mixed-mode, and (d) mode I wedge-opening loading conditions

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Fig. 6

(a) Simulated load-displacement curves and (b) crack trajectories for the shear test with four vastly different meshes. The test configuration is illustrated by the inset of (a).

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Fig. 8

A-FEM simulated nominal stress versus displacement for a single fiber/matrix domain under uniaxial tension. Crack development, including the initial fiber-matrix interface (dashed line), debonding crack, and final kinking cracks in matrix, are indicated in the top and bottom contour plots for the two meshes.

Grahic Jump Location
Fig. 7

(a) X-FEM predicted load-displacement curves compared with the A-FEM predictions. (b) Comparison of the CPU time (right vertical axis) and the numerical error (left axis) as function of mesh sizes.



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