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Research Papers

A Conforming Augmented Finite Element Method for Modeling Arbitrary Cracking in Solids

[+] Author and Article Information
Zhaoyang Ma

Department of Mechanics and Engineering Sciences,
Peking University,
Beijing 100871, China;
Department of Mechanical and Aerospace Engineering,
University of Miami,
Coral Gables, FL 33124
e-mail: mazhaoyang@pku.edu.cn

Qingda Yang

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

Xianyue Su

Department of Mechanics and Engineering Sciences,
Peking University,
Beijing 100871, China
e-mail: xyswsk@pku.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received February 20, 2019; final manuscript received March 11, 2019; published online April 12, 2019. Assoc. Editor: Yonggang Huang.

J. Appl. Mech 86(7), 071002 (Apr 12, 2019) (13 pages) Paper No: JAM-19-1086; doi: 10.1115/1.4043184 History: Received February 20, 2019; Accepted March 11, 2019

This paper presents a conforming augmented finite element method (C-AFEM) that can account for arbitrary cracking in solids with similar accuracy of other conforming methods, but with a significantly improved numerical efficiency of about ten times. We show that the numerical gains are mainly due to our proposed new solving procedure, which involves solving a local problem for crack propagation and a global problem for structural equilibrium, through a tightly coupled two-step process. Through several numerical benchmarking examples, we further demonstrate that the C-AFEM is more accurate and mesh insensitive when compared with the original A-FEM, and both C-AFEM and A-FEM are much more robust and efficient than other parallel methods including the extended finite element method (XFEM)/generalized finite element (GFEM) and the conforming embedded discontinuity method.

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Figures

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Fig. 1

(a) Illustration of single element augmentation where internal nodes 5,6,6′, and 5′ are introduced initially for crack displacement calculation but later being condensed, (b) illustration of the local deformation due to nonconformity between internal nodes across element edges that are cut by a crack, (c) the desired local deformation if conformity of internal nodes is enforced

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Fig. 2

Notation for an elastic body with an internal discontinuity that can be described by a cohesive traction-separation process

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Fig. 3

Illustration of the basic idea of the conforming-AFEM (C-AFEM): (a) an example simulation domain with uncracked elements and elements traversed by a crack (shaded elements), (b) local problem: assembly of elements cut by a crack and uses trial displacements of global solution for condensation of all cracked elements, and (c) global problem with uncracked elements + cracked elements with condensed stiffness and nodal forces

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Fig. 4

Illustration of an augmented element (a) with two subelements, (N-M-3-4) and (1-2-L-K), which are connected by a cohesive element (K-L-M-N) with mode I and II cohesive laws in (b) and (c)

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Fig. 5

(a) The flow chart for the solving procedure of the global program and (b) the iterative solving procedure for the local problem

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Fig. 6

(a) A two-element A-FEM model traversed by a cohesive crack in the middle, (b) the equivalent conforming-AFEM model with two cohesive elements that share nodes at junction points, and (c) the equivalent nonconforming A-FEM model with two cohesive elements not sharing nodes at junction points, and (d) comparison of the conforming-AFEM and A-FEM results for the two-element model

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Fig. 7

The geometry and loading of a short double cantilever beam specimen (a), and comparison of simulated load–displacement curves using (b) a coarse mesh with 20 mm mesh resolution (220 elements), (c) an unstructured mesh with 13 mm mesh resolution (779 elements), and (d) a refined mesh with 2 mm mesh resolution (5695 elements)

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Fig. 8

The geometric dimensions and boundary conditions of the FPSB specimen (a), and comparison of load–displacement curves obtained by C-AFEM, AFEM, and abaqus XFEM using (b) 20-mm mesh (410 elements), (c) 13-mm mesh (640 elements), (d) 8-mm mesh (1094 elements), (e) 4-mm mesh (2849 elements), and (f) 2-mm mesh (10,760 elements)

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Fig. 9

Comparison of (a) simulated crack trajectories and (b) CPU time to completion, using C-AFEM, AFEM, and the XFEM in abaqus

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Fig. 10

(a) Illustration of the dimensions (all unit in mm), boundary conditions, and loading scheme of the double-notched specimen, (b) computational mesh (435 elements) with simulated and measured crack trajectories superimposed for comparison, and (c) comparison of simulated load–displacement curves

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