Zero thermal expansion

Viscoelasticity

Negative stiffness inclusions

Thermal expansion of solids is associated with the anharmonicity of the interatomic potential energy in the crystalline lattice and has not been considered to be subject to modification. Expansion of composites has been considered to be bounded by the thermal expansion of the individual constituents. Analytical bounds on the physical properties, including thermal expansion, of multiphase media provide limits on properties attainable with variation of phase geometry. One may synthesize, both conceptually and experimentally, material microstructures which permit bounds to be approached or attained. In the following we show how one can substantially exceed traditional two phase bounds via inclusion of void space or slip interfaces. These bounds tacitly assume a perfect bond between the constituents. One can in fact prepare a lattice with two solid phases and a slip interface or a gap between them. Such lattices can have arbitrarily large or negative expansion. Designed lattices can also attain a zero thermal expansion. Negative thermal expansion need not involve negative Poisson's ratio.

These are the first known lattice metamaterials with tunable thermal expansion and extreme thermal expansion. We did not, however, call these materials metamaterials or architected materials or architectured materials. Similarly we did not use the phrase materials by design. Some materials with negative thermal expansion have long been known; our lattice materials allow tuning of negative thermal expansion to arbitrarily large magnitudes.

**Thermal expansion **

Lakes, R. S., **"Cellular solid structures with unbounded thermal expansion"**, *Journal of Materials Science Letters,* 15, 475-477 1996.

Material microstructures are presented which can exhibit coefficients of thermal expansion with a magnitude larger than that of either constituent. The thermal expansion can be either positive or negative depending on the lattice geometry. We conceptualize cellular solids as square (as shown in the image) or hexagonal lattices with two-layer rib elements and determine the thermal expansion coefficient. Thermal expansion increases without bound as the rib elements are made more slender. These cellular solids contain considerable void space. Colossal thermal expansion is possible. Get pdf .

Lakes, R. S., **"Dense solid microstructures with unbounded thermal expansion"**, *J. Mechanical Behav. Mts.,* 7, 85-92, 1996.

We present dense extremal structures which substantially exceed the bounds for thermal expansion of a two-phase composite, by allowing slip at interfaces between phases. New classes of extremal materials with extreme properties are envisaged, based on slip interfaces and void space tending to zero. Extremely high thermal expansion or negative thermal expansion is possible in these laminates.

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Wang, Y. C. and Lakes, R. S., **"Extreme thermal expansion, piezoelectricity, and other coupled field properties in composites with a negative stiffness phase"**, *Journal of Applied Physics*, 90, 6458-6465, Dec. (2001).

Particulate composites with negative stiffness inclusions in a viscoelastic matrix are shown to have higher thermal expansion than that of either constituent and exceeding conventional bounds. It is also shown theoretically that other extreme linear coupled field properties including piezoelectricity and pyroelectricity occur in layer- and fiber-type piezoelectric composites, due to negative inclusion stiffness effects. The causal mechanism is a greater deformation in and near the inclusions than the composite as a whole. A block of negative stiffness material is unstable, but negative stiffness inclusions in a composite can be stabilized by the surrounding matrix and can give rise to extreme viscoelastic effects in lumped and distributed composites. In contrast to prior proposed composites with unbounded thermal expansion, neither the assumptions of void spaces nor slip interfaces are required in the present analysis.

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Lakes, R. S., **"Solids with tunable positive or negative thermal expansion of unbounded magnitude"**, *Applied Phys. Lett.* 90, 221905 (2007).

Material microstructures are presented with a coefficient of thermal expansion larger in magnitude than that of either constituent. Thermal expansion can be large positive, zero, or large negative. Three-dimensional lattices with void space exceed two-phase bounds but obey three-phase bounds. Lattices and normal materials have a trend of expansion decreasing with modulus. Two-phase composites with a negative stiffness phase exceed bounds that assume positive strain energy density. Young's modulus and its relation to thermal expansion are plotted; behavior of these composites is compared with that of homogeneous solids in expansion-modulus maps.

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Lehman, J. and Lakes, R. S., **"Stiff lattices with zero thermal expansion"**, *Journal of Intelligent Material Systems and Structures*, 23 (11) 1263-1268 July (2012).

Lattice microstructures are presented with zero coefficient of thermal expansion. These are made of positive expansion materials. The behavior is primarily stretch dominated, resulting in favorable stiffness. Behavior of these lattices is compared with that of triangular and hexagonal honeycombs in a modulus-density map.

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J.J. Lehman and R.S. Lakes, **"Stiff lattices with zero thermal expansion and enhanced stiffness via rib cross section optimization"** *Int. J. Mech. Mater. Des.* 9, (3), 213-225 September (2013).

For engineering applications that are subject to large fluctuations in temperature, yet dimensional stability is essential, low or even zero thermal expansion materials are desirable. In addition to providing minimal thermal expansion care must be taken to ensure reductions in mechanical stiffness are mitigated. This can be achieved be designing structurally hierarchical materials composed of carefully chosen lattice structures. Within this manuscript honeycombs with thermal expansion coefficients equal to zero are developed analytically. The two dimensional lattice microstructure designs described are made of positive expansion materials. Zero expansion is attained with the use of curved, bi-material rib elements that by the use of thermally induced bending achieves zero overall thermal expansion. This work builds upon previous results, and provides further analysis into creating an optimal rib cross section to increase mechanical stiffness. The design of ribs with Tee shaped and I shaped cross sections is developed. Analytical equations are derived for the overall mechanical stiffness and overall thermal expansion coefficients of the lattices. The behavior of these lattices is compared with that of triangular and regular hexagonal honeycombs having non-zero thermal expansion as well as prior zero expansion lattices with rectangular rib cross sections in a modulus-density map. Lattice relative stiffness is improved by as much as a factor of 2.4 when compared with a curved, triangular, zero thermal expansion lattice with ribs of rectangular section. Thermal shear stress at the material interface is calculated and found to be small.

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Lehman, J. J. and Lakes, R. S. **"Stiff, Strong Zero Thermal Expansion Lattices via the Poisson Effect"**, *Journal of Materials Research*, 29, 2499-2508, September (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.

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Lehman, J. J. and Lakes, R. S., **"Stiff, strong, zero thermal expansion lattices via material hierarchy"**, *Composite Structures* 107, 654-663 (2014).

For engineering applications where tight dimensional tolerances are required, or for applications where materials are subjected to a wide range of temperatures it becomes desirable to reduce a material's coefficient of thermal expansion. By carefully designing lattice microstructures, zero thermal expansion can be achieved. This work describes lattice microstructures that achieve zero expansion by utilizing either the Poisson effect to negate thermal expansion, or a curved, bi-material rib morphology. Previously described microstructures were composed of solid material constituents. The lattices presented here have structural hierarchy in which lattice ribs contain oriented porosity. This gives rise to improved strength and modulus, and provides additional design freedom associated with anisotropy.

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Ha, C. S., Hestekin, E. , Li, J., Plesha, M. E., Lakes, R. S., **"Controllable thermal expansion of large magnitude in chiral negative Poisson's ratio lattices"**, * Physica Status Solidi B*, 252(7), 1431-1434 (2015).

Lattices of controlled thermal expansion are presented based on planar chiral lattice structure with Poisson's ratio approaching -1. Thermal expansion values can be arbitrarily large positive or negative. A lattice was fabricated from bimetallic strips and the properties analyzed and studied experimentally. The effective thermal expansion coefficient of the lattice is about alpha = - 3.5 x 10^{-4} K. This is much larger in magnitude than that of constituent metals. Nodes were observed to rotate as temperature was changed corresponding to a Cosserat thermoelastic solid. This lattice could be called double negative but we did not use such parlance.

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Ha, C. S., Plesha, M. E., Lakes, R. S., **"Simulations of thermoelastic triangular cell lattices with bonded joints by finite element analysis"**, *Extreme Mechanics Letters*, 12, 101-107 (2017).

Thermoelastic triangular cell lattices composed of bi-material curved ribs were designed and analyzed by finite element simulation. Positive, negative, or zero thermal expansion was possible by varying rib curvature if joints can pivot freely, as expected. Welded or bonded joints result in nonzero expansion but smaller in magnitude than that of a constituent material having higher thermal expansion coefficient. The effects of rib curvature variation for bonded joints were found to be negligible. We present a square lattice with bonded joints that has zero net thermal expansion; each curved bi-material rib has zero expansion.

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