Stiff lattices with zero thermal expansion and enhanced stiffness via rib cross section optimization

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.