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Research Papers

On Energy Release Rates and Configurational Forces for Interfacial Propagating Cracks: A Lattice Approach With a Brittle Erosion Technique

[+] Author and Article Information
Amir Mohammadipour

Department of Civil and
Environmental Engineering,
University of Houston,
N110 Engineering Building 1,
Houston, TX 77204-4003
e-mail: amohammadipour@uh.edu

Kaspar Willam

Department of Civil and
Environmental Engineering,
University of Houston,
N110 Engineering Building 1,
Houston, TX 77204-4003
e-mail: kwillam@uh.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 24, 2016; final manuscript received November 5, 2016; published online November 22, 2016. Editor: Yonggang Huang.

J. Appl. Mech 84(2), 021011 (Nov 22, 2016) (14 pages) Paper No: JAM-16-1521; doi: 10.1115/1.4035181 History: Received October 24, 2016; Revised November 05, 2016

A numerical 2D lattice approach with an erosion algorithm is employed to analyze bimaterial interface fracture quantities in brittle heterogeneous materials in the context of linear elastic fracture mechanics (LEFM). The concept of configurational force is elucidated and the importance of nodal configurational changes in a mesh where no stress–strain analyses are needed is investigated. Three fracture problems, i.e., an infinite panel with a bi-material interface crack, a double-lap shear test, and a prenotched four-point bending masonry beam are then considered. Validated by analytical solutions, the lattice model uses two distinct postprocessing approaches to derive the energy release rates and configurational forces directly at bimaterial interface crack tips. While the first method takes advantage of the change of the lattice mesh's global stiffness matrix before and after crack growth without any stress–strain calculations to obtain crack tip driving forces, the second approach analyzes the configurational forces opposing the crack tip motion using the Eshelby stress tensor and local force balance law in cracked and heterogeneous domains. It is demonstrated that the discrete material forces at crack tips are closely equal to the tip driving forces for the three fracture problems, confirming that the lattice is an appropriate numerical tool to analyze fracture properties of evolving interface cracks. Satisfying C1 continuity by including rotational displacements for frame struts, there is also no need for the lattice to update interior computational points in the mesh to eliminate spurious material forces away from the tip.

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Figures

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Fig. 1

Lattice meshes and boundary conditions for the three fracture problems, (a) center-cracked bimaterial simulation for a square domain, (b) triplet test, and (c) the half model of four-point bending masonry beam. Black, blue and pink struts are related to brick, mortar, and their interface materials, respectively.

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Fig. 2

(a) Degrees-of-freedom and external forces acting on a 2D frame element of the lattice strut in local coordinates, and (b) constitutive relation for a single frame element for linear elastic perfectly brittle behavior

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Fig. 3

Mechanical relationship between two Voronoi particles, (a) embedding translational and rotational stiffness between two particles on the interface, and (b) facet local displacement in t–n coordinates [16]

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Fig. 4

Failure surface for the brick–mortar interface used in this study

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Fig. 5

The comparison of lattice's fracture results for the energy release rate with those of the analytical solutions, (a) the center-cracked bimaterial panel, and (b) the four-point bending masonry beam

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Fig. 6

Eshelby stress distribution for the center-cracked panel problem in Fig. 1(a) at an arbitrarily chosen increment, (a) Pxx, (b) Pxy, (c) Pyx, and (d) Pyy. The values are in lbf/in2.

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Fig. 7

Eshelby stress distribution for the triplet problem in Fig. 1(b) at an arbitrarily chosen increment, (a) Pxx, (b) Pxy, (c) Pyx, and (d) Pyy. The values are in lbf/in2.

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Fig. 8

Eshelby stress distribution for the four-point bend problem in Fig. 1(c) at an arbitrarily chosen increment, (a) Pxx, (b) Pxy, (c) Pyx, and (d) Pyy. The values are in lbf/in2.

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Fig. 9

Four-point bending problem. (a) Material force vectors, F̂ at each node of the mesh including the crack tips due to the same Eshelby stress distributions shown in Fig. (8), and (b) the comparison of two different methods for obtaining the crack tip driving force, G, using Eq. (3) and the material forces tangential to the crack surface at the tip, t·F̂tip, using Eqs. (18) and (19)

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Fig. 10

Center-cracked problem. (a) Material force vectors, F̂ at each node of the mesh including the crack tips due to the same Eshelby stress distributions shown in Fig. 6, and (b) the comparison of two different methods for obtaining the crack tip driving force, G, using Eq. (3) and the material forces tangential to the crack surface at the tip, t·F̂tip, using Eqs. (18) and (19)

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Fig. 11

Triplet problem. (a) Material force vectors, F̂ at each node of the mesh including the crack tips due to the same Eshelby stress distributions shown in Fig. 7, and (b) the comparison of two different methods for obtaining the crack tip driving force, G, using Eq. (3) and the material forces tangential to the crack surface at the tip, t·F̂tip, using Eqs. (18) and (19)

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