Classification of periodic orbits of two-dimensional homogeneous granular crystals with no pre-compression

In the present study we classify the periodic orbits of a squarely packed, uncompressed and undamped, homogeneous granular crystal, assuming that all elastic granules oscillate with the same frequency (i.e., under condition of 1:1 resonance); this type of Hamiltonian periodic orbits have been labeled as nonlinear normal modes. To this end we formulate an auxiliary system which consists of a two-dimensional, vibro-impact lattice composed of non-uniform “effective particles” oscillating in an anti-phase fashion. The analysis is based on the idea of balancing linear momentum in both horizontal and vertical directions for separate, groups of particles, whereby each such a group is represented by the single effective particle of the auxiliary system. It is important to emphasize that the auxiliary model can be defined for general finite, squarely packed granular crystals composed of n rows and m columns. The auxiliary model is successful in predicting the total number of such periodic orbits, as well as the amplitude ratios for different periodic regimes including strongly localized ones. In fact this methodology enables one to systematically study the generation of mode localization in these strongly nonlinear, highly degenerate dynamical systems. Good correspondence between the results of the theoretical model and direct numerical simulations is observed. The results presented herein can be further extended to study the intrinsic dynamics of the more complex granular materials, such as heterogeneous two-dimensional and three-dimensional granular crystals and multi-layered structures.