Generalized beam model for the analysis of wave propagation with a symmetric pattern of deformation in planar pantographic sheets

Dynamics of pantographic sheets presents exotic aspects that deserve investigation. In this paper, we focus the attention on some possible modalities of non-linear wave propagation in planar pantographic sheets. We use a smaller length-scale lattice model, in which the beams and pivots constituting the sheet are described by constrained Euler–Bernoulli beams together with a meso-reduced-order model which belongs to the class of second-gradient elastic materials and with a macro multi-field 1D continuum model, whose displacement is augmented by a specific class of cross-section deformations. Such a three-step reduction process is developed to allow for fast computational analysis of symmetric wave propagation patterns with respect to the longitudinal axis of the sheet. It is conceived by using suitable kinematical hypotheses for the 1D continuum descriptors referring to pantographic sheet sections which are inspired by the numerical evidence obtained performing simulations based on the smaller scale lattice model. The deformation energy of the pantographic sheet, successfully postulated in dell’Isola et al. (2016) for a meso-reduced-order second gradient model, is pivotal in the whole model reduction process: It allows for the determination of generalized 1D deformation energy in terms of the mechanical properties of the micro-lattice model. Performed numerical simulations prove that several waveforms propagate in planar pantographic sheets with low dispersion and motivate further investigations in the subject.