MS18.3: Wave propagation in Mechanical Systems and Nonlinear Metamaterials

Nonlinear meta-materials/structures can enable exotic wave propagation phenomena which ultimately lead to extraordinary functionalities that are not accessible in nature materials. Despite the increasing attention received, the properties and underlying physical mechanisms of band structures and bandgaps in high-dimensional nonlinear metamaterials, such as plates, remain unclear. Here we propose and examine a meta-plate with embedded strongly nonlinear oscillating units to show the evolution process of typical nonlinear band structures using analytical and numerical methods. Based on the band properties of nonlinear meta-plate, we integrate linear and nonlinear meta-plate to break reciprocal wave propagation in the 2D plate. The non-reciprocity properties of the modulated elastic waves may bring about exciting engineering applications. This study provides modelling and computational methods for high-dimensional strongly nonlinear meta-materials/structures, and fills the gaps in the understanding of the mechanism and properties of nonlinear wave propagation in 2D plates.

Dipolar Bose-Einstein condensates emerged as a unique platform permitting the exploration of nonlinear dynamics at the classical-quantum interface, phase-transitions and the realization of supersolidity. In this talk we present theoretical results discussing self-organized pattern-formation in this system, which can proceed via a first- or second-order phase-transition. We show how competing nonlinearities that contain crucial beyond mean-field contributions can give rise to new stable and metastable phases of matter and discuss finite temperature effects.

The paper explores the effect of parametric amplification on interface states in a mass-spring periodic chain system. Particularly, the existence of interface states in the presence of nonlinearity and the impact of parametric amplification is investigated for the case when a Mathieu-Duffing type oscillator at the interface is introduced. Change in dynamic behavior with increasing number of degrees of freedom and damping is studied. The incremental harmonic balance method (IHBM), given in general form and coupled with numerical continuation, is used as a tool for studying and solving a set of second-order differential equations.

We consider the normal form for the celebrated Fermi-Pasta-Ulam-Tsingou (FPUT) lattice for an arbitrary number N of particles, with any number of nonlinear terms in a polynomial expansion ('alpha', 'beta', etc.), and with either periodic or fixed boundary conditions. The normal-form equations of motion truncated up to 5-wave interactions are considered. When 3 divides N and N > 8, this normal form contains non-trivial resonances (Bustamante et al. 2019), and is thus generally non-integrable. This is in contrast to what happens with the normal form truncated up to 4-wave interactions, which is known to be integrable for all values of N. Based on (Harper et al. 2013) we construct new independent quadratic invariants for the normal form truncated up to 5-wave interactions. We show that there is always a significant number of these invariants, no matter how large N is. We provide explicit examples for relatively small N cases. We then move on to compute numerically the number of positive Lyapunov exponents of our FPUT normal form truncated up to 5-wave interactions, for relatively small values of N, and show that it is consistent with the above result on new independent invariants. We then construct a new thermalization theory (with constraints) for the normal-form energies, and validate it numerically, finding very good agreement. Finally, we show numerically that at low enough nonlinearity the normal form truncated up to 5-wave resonances provides a good approximation to the original system.

The free undamped propagation of nonlinear multi-harmonic waves in mechanical 2D metamaterials with inertial amplification is investigated. A Lagrangian model is formulated to describe the weakly nonlinear dynamics of a periodic array of elastically coupled oscillators, pantographically connected with inertia-amplifying auxiliary point masses. The space-time periodic solutions of the differential difference equations of motion, characterized by inertial and elastic nonlinearities, are described asymptotically by means of a perturbation approach. The linear dispersion relations for harmonic waves are determined at the first order. The higher order solutions provide the amplitudes of superharmonic components generated by quadratic and cubic nonlinearities, as explicit functions of the linear harmonic amplitudes. An interesting dynamic phenomenon of superharmonic depolarization is disclosed.