Simulated properties of Kagomé and tetragonal truss core panels
Introduction
Progress in robust, ultra-light metallic materials and systems with topologically configured cores and dense faces has provided several perspectives (Wicks and Hutchinson, 2000; Wallach and Gibson, 2001; Deshpande et al., 2001; Evans et al., 2001; Deshpande and Fleck, 2001; Chiras et al., 2002). Panels having cores with a stochastic cellular topology are not weight efficient, but have utility because of relatively low cost and excellent robustness (Ashby et al., 2000). Periodic truss core panels having tetragonal and pyramidal topology exhibit superior thermo-structural characteristics (Deshpande and Fleck, 2001; Chiras et al., 2002). They are at least as weight efficient as the best competing concepts, especially for curved panels (Evans et al., 1998).
The preferred core topologies are based on the fundamental idea that the trusses should stretch/compress without bending. When realized, the core properties are related to its relative density, , by Wicks and Hutchinson (2000), Wallach and Gibson (2001), Deshpande et al. (2001), Evans et al. (2001), Deshpande and Fleck (2001), Chiras et al. (2002) and Evans et al. (1998):where σY is the yield strength and E the Young’s modulus of the material comprising the trusses, with Gc the shear modulus, τYc the shear strength and σYc the compressive strength. The coefficients A, B and C are functions of truss architecture, loading orientation and node design. Adding the constraint that the core should be nearly isotropic (to minimize compliant orientations), only a small subset of possible truss core topologies appears to satisfy Eqs. , , (Evans et al., 1998; Evans, 2001). Two have been analyzed and characterized experimentally: tetragonal (Deshpande and Fleck, 2001; Chiras et al., 2002) (Fig. 1a) and pyramidal (Deshpande and Fleck, 2001). The shear and compressive response of these cores has been determined, as well as the bending characteristics of panels in near-optimized configurations. The performance of panels with these cores is excellent (Chiras et al., 2002). Nevertheless, improvements appear feasible, based on the following two limitations.
- (i)
The cores have significant anisotropy. The coefficients A and B in , , vary with loading orientation: for the pyramidal design differing by as much as 40% between the maximum and minimum.
- (ii)
The trusses are susceptible to plastic buckling, resulting in a bending asymmetry, particularly for the tetragonal topology.
In order to address these limitations, an objective of this study is to explore the comparative performance of an alternative core topology, known as the 3D Kagomé (Fig. 1b). The genesis of this choice has been the recent finding from topology optimization that 2D Kagomé structures are structurally efficient (Hyun and Torquato, 2002). Namely, their elastic moduli approach the optimal Hashin–Shtrikman upper bounds (Hashin and Shtrikman, 1963; Hashin, 1965) over a wide range of relative densities and they have superior buckling properties. In order to fully capture the failure mechanisms, detailed finite element models, using solid elements, of the two structures (tetragonal and Kagomé) are developed. The models are confirmed by means of experimental measurements described in a companion paper (Wang et al., 2003).
Section snippets
Problem definition
The dimensions of the tetragonal core (core radius, rod length, panel height) are representative of near-optimized sandwich panels (Wicks and Hutchinson, 2000), with relative density, . The same truss radius and panel height are used for the Kagomé core, but to attain the same core density, the truss length is half that for the tetragonal core (Hyun and Torquato, 2002). The displacements imposed on the model are chosen to simulate core compression and shear. For compression, a
Simulation method
The commercial package, ANSYS, was used to generate three-dimensional meshes by utilizing 10-node tetrahedral solid elements, with about 10,000 finite elements and 10,000 nodes needed to obtain reliable convergence. The meshed models were transferred to the commercial finite element solver, ABAQUS, to perform the numerical simulations. The simulations were performed subject to displacement-control, using large displacement theory to capture the softening in the post-buckling state. Small
Compression
The calculated relations between the non-dimensional compressive force, F/πσYR2 (where R is the truss radius) and vertical displacement, Δ/H (Fig. 3) reveal that, for both truss designs, a maximum load capacity is reached, followed by softening. The peak loads are systematically lower for the Al alloy than the Cu/Be alloy, and occur at lower Δ, because of the differences in strain hardening.
The Kagomé geometry sustains appreciably higher loads than the tetrahedron, as well as exhibiting larger
Conclusions
Numerical simulations have been performed of the mechanical responses of two truss structures (tetragonal and Kagomé) subject to compression and shear. Responses of both structures are initially isotropic, but only the Kagomé core maintains the isotropy after yielding; it strain hardens and is resistant to plastic buckling in compression and shear. The tetragonal core is anisotropic in shear with two soft orientations, both governed by the onset of plastic buckling. Accordingly, the Kagomé
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