Modeling of Crack Propagation in Thin-Walled Structures Using a Cohesive Model for Shell Elements

[+] Author and Article Information
Pablo D. Zavattieri

 GM Research and Development Center, 30500 Mound Road, Warren, MI 48090-9055Pablo.zavattieri@gm.com

J. Appl. Mech 73(6), 948-958 (Dec 23, 2005) (11 pages) doi:10.1115/1.2173286 History: Received June 02, 2005; Revised December 23, 2005

A cohesive interface element is presented for the finite element analysis of crack growth in thin specimens. In this work, the traditional cohesive interface model is extended to handle cracks in the context of three-dimensional shell elements. In addition to the traction-displacement law, a bending moment-rotation relation is included to transmit the moment and describe the initiation and propagation of cracks growing through the thickness of the shell elements. Since crack initiation and evolution are a natural outcome of the cohesive zone model without the need of any ad hoc fracture criterion, this model results in automatic prediction of fracture. In particular, this paper will focus on cases involving mode I/III fracture and bending, typical of complex cases existing in industrial applications in which thin-walled structures are subjected to extreme loading conditions (e.g., crashworthiness analysis). Finally, we will discuss how the three-dimensional effects near the crack front may affect the determination of the cohesive parameters to be used with this model.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

(a) Schematics of the separation between two shell elements. The local coordinates are defined in the middle line of the interface elements. The upper-left box shows the cohesive interface elements embedded along quadrilateral shell elements (for illustration purposes, the shell elements have been separated). (b) Traction separation law for pure normal separation. The arrows indicate unloading and loading for λ>λcr. The same triangular law describes the bending moment-rotation relationship under pure rotation (denoted between parentheses).

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Figure 2

(a) Geometry used for the crack propagation analysis of a precracked elastic thin panel under three different loading conditions: tension, torsion, and bending; (b) Hexahedral mesh. (c) Shell mesh.

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Figure 3

(a) Tension test: crack tip position versus displacement for three different loading rates. (b) Torsion test: crack tip position as a function of time for the case where the elastic thin plate is loaded under mode III conditions.

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Figure 4

Schematics of the three-point bending setup. (a) Crack tip position as a function of time for case with thickness t=6mm and vz=1m∕s. (b) Crack tip position as a function of time for the case with thickness t=2mm and vz=10m∕s. (Comment: The dots represent the x-coordinate of the centroid of the interface elements at the time where they fail. Clearly the simulation with solids shows the crack evolution in the different layers of elements.)

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Figure 5

Tensile stress σyy at different times during the propagation of the crack using a shell and solid mesh for the three-point bending configuration for t=2mm and vz=10m∕s.

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Figure 6

(a) Crack tip position as a function of the applied displacement for a mode I ductile crack problem simulated with solid and shell elements. The solution for the solid mesh is given at the midsection of the panel. (b) Tensile stress field for the elasto-plastic and elastic material. (c) Details of the crack tunneling for the case modeled with 3D solid elements. The dark region indicates where Gdis∕GIc=1.0.

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Figure 7

(a) Details of the center-cracked aluminum panel. For simplicity purposes, the solid model considers only one-eighth of the geometry and (b) the shell model one-fourth. Symmetry boundary conditions are applied accordingly. (c) Predicted and measured load-crack growth response using both models (for comparison purposes the shell model uses two cohesive strengths).

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Figure 8

(a) Crack extension and force as a function of the uniformly applied displacement obtained with the fully-three-dimensional model and with the shell model using different cohesive strength values. The two solid lines indicate the crack front position at the middle and outer surface of the specimen for the solid mesh with thickness t=2.3mm. (b) Correction of the cohesive strength for shell elements as a function of the specimen thickness.

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Figure 9

(a) Normal cohesive traction profile developed near the crack front. Each dot represents the value at each integration point and its x-coordinate (initial straight crack front is located at x=0). The solid line represents its average through the thickness T¯n. (b) Effective cohesive law (T¯n vs. u¯n) for three different specimen thicknesses.

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Figure 10

(a) Rod impacting a brittle plate. (b) Bending of an aluminum tube. (c) Fracture of polymer membranes.



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