doi:10.1016/S0022-5096(99)00030-7 
Copyright © 1999 Elsevier Science Ltd. All rights reserved.
Effect of imperfections on the yielding of two-dimensional foams
C. Chen, T. J. Lu and N. A. Fleck
, 
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, UK
Received 29 October 1998;
revised 2 April 1999.
Available online 8 September 1999.
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Abstract
The influence of each of the six different types of morphological imperfection—waviness, non-uniform cell wall thickness, cell-size variations, fractured cell walls, cell-wall misalignments, and missing cells—on the yielding of 2D cellular solids has been studied systematically for biaxial loading. Emphasis is placed on quantifying the knock-down effect of these defects on the hydrostatic yield strength and upon understanding the associated deformation mechanisms. The simulations in the present study indicate that the high hydrostatic strength, characteristic of ideal honeycombs, is reduced to a level comparable with the deviatoric strength by several types of defect. The common source of this large knock-down is a switch in deformation mode from cell wall stretching to cell wall bending under hydrostatic loading. Fractured cell edges produce the largest knock-down effect on the yield strength of 2D foams, followed in order by missing cells, wavy cell edges, cell edge misalignments, Γ Voronoi cells, δ Voronoi cells, and non-uniform wall thickness. A simple elliptical yield function with two adjustable material parameters successfully fits the numerically predicted yield surfaces for the imperfect 2D foams, and shows potential as a phenomenological constitutive law to guide the design of structural components made from metallic foams.
Author Keywords: A. Microstructures; B. Elastic-plastic material; Foam material; C. Finite elements
Fig. 1. Six different types of geometric imperfection considered in the present study: (a) waviness; (b) non-uniform wall thickness; (c) fractured cell walls; (d) cell-wall misalignment; (e) Voronoi structure; and (f) missing cells.
Fig. 2. (a) Unit cell model for ideal hexagonal honeycomb, and (b) dependence of the deviatoric yield strength Σ upon the inclination Ω of principal stresses relative to the microstrucure.
Fig. 3. (a) Unit cell of regular honeycomb with cell wall waviness, and (b) its yield surface for the case t/l and selected values of w0/t.
Fig. 4. (a) Unit cell of a perfect honeycomb with non-uniform wall thickness, and (b) its yield surface for the case t/l=0.15 and selected values of w0/t.
Fig. 5. Effects of (a) waviness and (b) non-uniform wall thickness on the yield strengths of perfect honeycombs.
Fig. 6. Typical finite element mesh for the unit cell of a periodic Voronoi structure with (a) Γ-distributed cells and (b) δ-distributed cells.
Fig. 7. Three different types of boundary condition: (a) mixed boundary conditions, (b) prescribed displacement boundary conditions, and (c) periodic boundary conditions.
Fig. 8. Effect of choice of boundary conditions on (a) uniaxial and (b) hydrostatic yield strength of a Γ-distributed Voronoi structure.
Fig. 9. Effect of choice of boundary conditions on (a) uniaxial, (b) hydrostatic stress vs strain behaviour of a Γ-distributed Voronoi structure with .
Fig. 10. Effect of all size variations and cell-wall misalignments on: (a) Young’s modulus; (b) bulk modulus; (c) uniaxial yield strength; and (d) hydrostatic yield strength of 2D cellular foams.
Fig. 11. Ratio of uniaxial to hydrostatic yield strength σU/σH vs relative density for perfect honeycombs, Γ- and δ-distributed Voronoi structures, and honeycombs with cell-wall misalignments (α=0.4).
Fig. 12. Typical finite element mesh for honeycombs with (a) cell-wall misalignment (α=0.2), and (b) fractured cell walls (number fraction=1%).
Fig. 13. Effect of (a) cell-wall misalignments, and (b) fractured cell walls on uniaxial and hydrostatic yield strengths of 2D foams with .
Fig. 14. Finite element mesh for a perfect honeycomb with (a) 1 cell missing, and (b) 7 cells missing.
Fig. 15. Effect of the number of missing cells on uniaxial and hydrostatic yield strengths of (a) perfect honeycomb, and (b) perfect honeycomb with 5% fractured cell edges. The initial relative density of both honeycombs is .
Fig. 16. Elliptical yield surface of a Γ-distributed Voronoi structure with 5% fractured cell edges fitted to the finite element calculated stress paths in the σd−σm space under proportional straining. The initial relative density of the foam is .
Fig. 17. (a) Typical elliptical yield surfaces of Γ-distributed Voronoi structures with and without fractured cell edges (5%) and perfect honeycombs with and without cell-edge misalignments (α=0.4); (b) Taylor yield surface of 2D random strut structures compared with the yield surfaces of perfect honeycombs and Γ-distributed Voronoi structures with 5% fractured cell edges. The relative density of the foams is fixed at .
Table 1. Effect of number of cells on the calculated stiffness and strength of a Γ-distributed Voronoi structure with Not-found=15%a
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