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TECHNICAL PAPERS

Stress–Strain Relationship for Metal Hollow Sphere Materials as a Function of Their Relative Density

[+] Author and Article Information
D. Karagiozova

Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China; Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Block 4, Sofia 1113, Bulgaria

T. X. Yu1

Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, Chinametxyu@ust.hk

Z. Y. Gao

Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

1

Corresponding author.

J. Appl. Mech 74(5), 898-907 (Nov 03, 2006) (10 pages) doi:10.1115/1.2712235 History: Received April 24, 2006; Revised November 03, 2006

The stress–strain relationship for uniaxial compression of a metal hollow sphere material in large strains is obtained using a simplified model for the spheres’ deformation within a 3D block assuming a hexagonal packing pattern. The yield strength and material strain hardening are obtained as functions of the relative density in two characteristic loading directions. The expression for the stress–strain relationship consisting of quadratic and linear terms with respect to the relative density is linked to the partitioning of the deformation energy during compression. The theoretical predictions are compared with limited test results on mild steel hollow sphere material and finite element simulation results obtained by our group.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) MHS material; and (b) Hexagonally packed spheres

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Figure 2

Structural representative blocks: (a) compression in the Z direction; (b) compression in the z direction; and (c), (d) characteristic lines of the deformation mechanism

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Figure 3

Description of the deformed shapes of the characteristic lines: (a) S line; (b) C line and M line (Φ0 and Φ stand for the initial and current angles associated with either C line or M line); and (c), (d) variation of the radii of the contact circles and contact angles for the FCC and HCP pattern, respectively; the sectors on a sphere associated with the C, M, and S line are marked

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Figure 4

Predicted shapes of the deformed characteristic lines at 25% strain: (a) compression in the Z direction (FCC); (b) compression in the z direction (HCP); and (c) comparison between the analytical model and finite element simulation (compression in the Z direction)

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Figure 5

Stress–strain curves: (a) compression in the z direction; and (b) compression in the Z direction; the numerical results (as described in Sec. 2) are shown for ρ*∕ρc=0.052

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Figure 6

Ratios between the (bending and membrane) stresses associated with the energy partitioning, σb*∕σm*, in the two characteristic directions of loading

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Figure 7

Yield strength for various cellular solids; σY*=σ*(ϵ=0.05) for the MHS material

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Figure 8

Comparison between quasi-static experimental stress–strain curves and the theoretical predictions (Eq. 25): (a) MHS material with ρ*∕ρs=0.052; and (b) MHS material with ρ*∕ρs=0.045

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Figure 9

MHS material with voids: (a) actual MHS material; (b) assumed regular packing of the spheres with gaps (FCC pattern); and (c) variation of the gap’s size and contacting radius RT=RM for reduced relative densities (t∕R=0.033)

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Figure 10

Stress–strain curves for MHS materials with reduced densities (original density ρ*∕ρs=0.071); (a) reduction by increasing the gaps; and (b) stress–strain curves for MHS materials using two methods for density reduction

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Figure 11

Comparisons of the analytical model predictions for the yield strength, σY,MHS*(ϵ=0.05), and average stress, (σ*)MHSav, with the available test results (7)

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Figure 12

The definitions of the characteristic angles for the C and M lines

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