Investigating wave transmission and reflection phenomena in pantographic lattices using a second-gradient continuum model

In this study, linear plane wave propagation in pantographic lattices is investigated using a two-dimensional (2D) second-gradient continuum model. Primarily, we aim to investigate wave transmission and reflection phenomena occurring when a pantographic lattice is attached to a material modeled by classical first-gradient theory. From a practical point of view, this would be a possible application for pantographic lattices due to their promising mechanical behavior. From a theoretical point of view, this investigation let us to elucidate wave propagation between second- and first-gradient continua. The adapted second-gradient model is first linearized, and the dispersion relations are obtained. Boundary and jump conditions at the discontinuity surface are identified accordingly. It is shown how elastic constants may affect the wave propagation between the two continua and can be tuned to control such wave propagation phenomena.