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Research Papers

Lattice Approach in Continuum and Fracture Mechanics

[+] 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: amir_mohammadipour@yahoo.com

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 March 4, 2016; final manuscript received April 1, 2016; published online April 20, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(7), 071003 (Apr 20, 2016) (9 pages) Paper No: JAM-16-1118; doi: 10.1115/1.4033306 History: Received March 04, 2016; Revised April 01, 2016

This study reveals some aspects of lattice formulations to analyze the strain concentration of a famous classic problem in solid mechanics at two different mechanics perspectives: continuum and fracture. A 2D plane stress square panel with a single circular hole is discretized based on Voronoi tessellation. A superelliptic formulation is used to distribute lattice computational points over the panel. From the perspective of linear elasticity under uniaxial and biaxial loading, translational and rotational degrees-of-freedom are considered at each computational node of the lattice to obtain strain concentration factors (SCFs) around the circular perforation. It is observed that the lattice approach is able to approximate the elastic SCFs of three, four, and two in uniaxial, biaxial shear, and equibiaxial tension loadings, respectively. To study the linear elasticity and fracture mechanic (LEFM) and Griffith energy balance in uniaxial loading, a brittle lattice erosion technique is used to compute the energy release rate determined by change in the global stiffness matrix of the mesh with respect to crack extension. This fracture energy is then used to determine the mode I stress intensity factor of the crack emanating from the hole which is validated by the analytical formulation for the same problem. The comparison shows that both methods give very close results for the mode I stress intensity factor. Being simple in terms of constitutive formulation and failure criterion for erosion of brittle material, and also using a propagating crack extension approach, the lattice formulation is used to determine the fracture properties of cohesive/frictional material.

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References

Figures

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Fig. 1

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

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Fig. 2

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 [6]

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Fig. 3

Lattice mesh generation of the rectangular panel using the superellipse formulation: (a) Voronoi particles and their computational points or centroids and (b) lattice mesh struts with smooth transition from polar to Cartesian coordinate system

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Fig. 4

Determining the mesh parameters Smin and Smax according to the circular hole radius, R, and the panel width, 2b, for a selected value of Δθ

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Fig. 5

(a) Part of the lattice mesh and struts with computational points for strain calculations and (b) the equivalent continuum Q4 finite element whose nodes are exactly the computational points of the lattice frame elements

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Fig. 6

The distribution of SCF for εy in a finite thin panel with a single circular hole: (a) the lattice mesh and corresponding boundary conditions, (b) the SCF distribution for the panel with uniaxial tension in y direction, (c) the SCF distribution forthe panel with biaxial tension in x and y directions, and (d) the SCF distribution for the panel with biaxial tension–compression in y and x directions, respectively

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Fig. 7

The distribution of SCF for εx in a finite thin panel with a single circular hole: (a) the lattice mesh and corresponding boundary conditions, (b) the SCF distribution for the panel with uniaxial tension in y direction, (c) the SCF distribution forthe panel with biaxial tension in x and y directions, and (d) the SCF distribution for the panel with biaxial tension–compression in y and x directions, respectively

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Fig. 8

A small region near crack tip along the crack surfaces in a homogeneous domain

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Fig. 9

Two symmetric cracks emanating from a circular hole in a rectangular panel subjected to uniaxial tensile stress [29]

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Fig. 10

Lattice mesh with h/b=2 and R/b=0.5 used for Newman problem in Fig. 9: (a) mesh without crack and (b) mesh with crack emanating from the circular hole emulating the Newman problem

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Fig. 11

Lattice simulation results for Newman problem: (a) load–displacement curve of the panel with circular hole in direct tension and (b) comparing lattice simulations and Newman analytical solution for the function F defined in Eq. (13)

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