Research Papers

2013 Koiter Medal Paper: Crack-Tip Fields and Toughness of Two-Dimensional Elastoplastic Lattices

[+] Author and Article Information
H. C. Tankasala, V. S. Deshpande

Cambridge University Engineering Department,
Trumpington Street,
Cambridge CB2 1PZ, UK

N. A. Fleck

Cambridge University Engineering Department,
Trumpington Street,
Cambridge CB2 1PZ, UK
e-mail: naf1@eng.cam.ac.uk

Euler–Bernoulli beam elements are appropriate only for large rotations and small strains as the cross-sectional thickness change is ignored. Timoshenko beam elements use a fully nonlinear formulation so that the strains and rotations can be arbitrarily large.

We note in passing that formula (12) is slightly different from the expression reported in Ref. [13] for the case n = 1 and we ascribe the slight difference to the more refined numerical simulations performed herein.

In agreement with Ref. [11], we find some scatter in predicted toughness from realisation to realisation, but the overall sensitivity of toughness to imperfection is reduced in the nonlinear, ductile case compared to the elastic, brittle case, and the scatter is not shown in Fig. 7.

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 5, 2015; final manuscript received May 15, 2015; published online June 16, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(9), 091004 (Sep 01, 2015) (10 pages) Paper No: JAM-15-1220; doi: 10.1115/1.4030666 History: Received May 05, 2015; Revised May 15, 2015; Online June 16, 2015

The dependence of the fracture toughness of two-dimensional (2D) elastoplastic lattices upon relative density and ductility of cell wall material is obtained for four topologies: the triangular lattice, kagome lattice, diamond lattice, and the hexagonal lattice. Crack-tip fields are explored, including the plastic zone size and crack opening displacement. The cell walls are treated as beams, with a material response given by the Ramberg–Osgood law. There is choice in the criterion for crack advance, and two extremes are considered: (i) the maximum local tensile strain (LTS) anywhere in the lattice attains the failure strain or (ii) the average tensile strain (ATS) across the cell wall attains the failure strain (which can be identified with the necking strain). The dependence of macroscopic fracture toughness upon failure strain, strain hardening exponent, and relative density is obtained for each lattice, and scaling laws are derived. The role of imperfections in degrading the fracture toughness is assessed by random movement of the nodes. The paper provides a strategy for obtaining lattices of high toughness at low density, thereby filling gaps in material property space.

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Grahic Jump Location
Fig. 1

Crack geometry and lattice topologies: (a) coordinate reference frame for the lattice with crack, (b) triangular lattice, (c) kagome lattice, (d) diamond lattice, and (e) hexagonal lattice

Grahic Jump Location
Fig. 2

Mode I plastic zone for (a) triangular lattice, (b) kagome lattice, (c) diamond lattice, and (d) hexagonal lattice

Grahic Jump Location
Fig. 3

Crack-tip opening profile for (a) triangular lattice, (b) kagome lattice, (c) diamond lattice, and (d) hexagonal lattice

Grahic Jump Location
Fig. 6

Imperfect lattice topologies (R/ℓ=0.5) for (a) triangular lattice, (b) kagome lattice, (c) diamond lattice, and (d) hexagonal lattice

Grahic Jump Location
Fig. 7

The normalized fracture toughness versus R/ℓ of imperfect lattices, for the choice ρ¯=0.025 and n=10

Grahic Jump Location
Fig. 8

Dependence of fracture toughness of kagome lattice upon relative density ρ¯ for R/ℓ = 0, 0.3, and 0.5

Grahic Jump Location
Fig. 9

Material property charts (material property ces selector software by Granta Design) for (a) fracture toughness versus density and (b) toughness versus density. Predictions are included for Ti-6Al-4V lattices of cell length ℓ = 10 mm.



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