Variegated Collective Behaviors in Mechanical and Thermal Metamaterials: Theory and Analysis

Recent studies facilitated investigations of quasistatic responses of architected material systems in a quest for negative elastic moduli and other unusual basic properties expected at uniform loading conditions. Here, it has been shown that materials with discrete internal structure may also demonstrate strong dependences of their effective (continuum equivalent) mechanical properties on the wavenumber (spatial frequency) of static sinusoidal pressure waves. The Fourier mode stiffness, a ratio of the periodic load amplitude and a response amplitude is a quadratic function of the axial load’s wavenumber in continuum materials. However, in discrete periodic lattices, it has been shown that it has to be a squared sine function of the wavenumber, which flattens down when the wavelength becomes comparable with a unit cell size. Also, a class of beam-like lattice structures, or metabeams under transverse sinusoidal loads is discussed, where the neutral line is shown to deflect either in-phase, out of phase or show no deflection, depending on the beam design parameters, and also on the spatial frequency of the load, contrasting the behavior of continuum beams deflecting always in-phase with static loads. A long-range periodic order and nonlocality of the lattice interaction is essential for this unusual behaviors, and those are particularly pronounced at higher wavenumbers, when the load wavelength becomes comparable with the range of the direct interactions in the lattice. In the discussion of thermal metamaterials, a negative thermal expansion behavior has been discussed and experimentally demonstrated, which occurs rarely in natural materials. Using a universal antichiral metamaterial model with bimetal beams or strips, a generic theory has been developed to predict magnitude of the negative thermal expansion effect from model parameters. Thermal expansivity of the metamaterial is written as an explicit function of temperature and of only three design parameters: relative node size, chirality angle, and a bimetal constant.