Nonlinear dispersive waves in 2-D micropolar lattices

Architected materials, also known as mechanical metamaterials, are solid structures designed by leveraging geometry, rather than composition, to achieve desired mechanical properties. Past studies on mechanical metamaterials have demonstrated properties that are rare or absent in naturally occurring materials, such as negative Poisson's ratio, chirality, and extremely high ratios of bulk to shear modulus and strength to weight. Most of the existing literature on architected materials has been focused on linear behavior, though some works have incorporated nonlinearity, often leveraging mechanical instabilities to re-configure a lattice. In the present work, we explore nonlinear dispersive waves in 2-D architected lattices with rotating microstructure, which are man-made analogues of nonlinear micropolar continua. Specifically, we consider a discrete model that consists of a periodic array of rigid bodies connected by nonlinear springs, focusing on theweakly nonlinear dynamic regime. For wavelengths much longer than the characteristic lengths of the microstructure, we demonstrate that the nonlinear behavior of longitudinal and transverse acoustic wave modes may be described by evolution equations of the Kadomtsev-Petviashvili type, which include effects of weak nonlinearity, weak dispersion, and weak transverse variation. Solutions of these evolution equations will be discussed and compared with direct numerical simulations of the discrete lattice.