Stiff, strong zero thermal expansion lattices via the Poisson effect
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Roderic Lakesa) Department of Engineering Physics, Materials Science Department and Rheology Research Center, University of Wisconsin–Madison, Madison, Wisconsin 53706-1687 (Received 30 January 2013; accepted 2 May 2013)
Designing structures that have minimal or zero coefficients of thermal expansion (CTE) are useful in many engineering applications. Zero thermal expansion is achievable with the design of porous materials. The behavior is primarily stretch-dominated, resulting in favorable stiffness. Two and three-dimensional lattices are designed using ribs consisting of straight tubes containing two nested shells of differing materials. Differential Poisson contraction counteracts thermal elongation. Tubular ribs provide superior buckling strength. Zero expansion is achieved using positive expansion isotropic materials provided axial deformation is decoupled by lubrication or segmentation. Anisotropic materials allow more design freedom. Properties of two-dimensional zero expansion lattices, of several designs, are compared with those of triangular and hexagonal honeycomb nonzero expansion lattices in a modulus-density map. A three-dimensional, zero expansion, octet-truss lattice is also analyzed. Analysis of relative density, mechanical stiffness, and Euler buckling strength reveals high stiffness in stretch-dominated lattices and enhanced strength due to tubular ribs.
I. INTRODUCTION
It is desirable to design materials with low or zero coefficients of thermal expansion (CTEs) for applications that involve large temperature variations, to reduce thermal stresses and maintain geometric stability. Traditionally the thermal expansion of composite materials is considered to be a weighted average of the thermal expansion coefficients of the constituent materials. For two-phase composites with constituents that are assumed to be isotropic, have positive definite strain energy, and are not porous, the overall thermal expansion is a weighted average based on the constituent volume fractions and bulk moduli.1 It is possible by evading these assumptions, combined with tailored design of the material’s microstructure to obtain large or even negative values of thermal expansion, with positive constituent CTEs.2,3 For use in structural applications, material stiffness optimization is often required. In general, three-dimensional foams (in any direction) and hexagonal honeycombs (in-plane) deform mainly by bending of cell walls.4 This gives rise to a modulus that is quadratic in relative density. Relative density is the density of the porous material divided by the density of the solid phase that comprises the ribs. To optimize material stiffness, lattices are chosen so that the material deformation is axial rather than bending. a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/jmr.2013.154 J. Mater. Res., Vol. 28, No. 17, Sep 14, 2013
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In such stretch-dominated lattices, the modulus is linear in relative density. For low density, such
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