[Paper Review] Wave propagation in pantographic 2D lattices with internal discontinuities

Angela Madeo, Alessandro Della Corte+2 more|arXiv (Cornell University)|Dec 12, 2014

Adhesion, Friction, and Surface Interactions42 references5 citations

TL;DR

This study investigates wave propagation in 2D pantographic lattices composed of orthogonal Euler beams with internal discontinuities, using a nonlinear discrete beam model. Numerical simulations reveal soliton-like wave behavior under large rotations and geometric nonlinearity, particularly when waves travel in opposite directions, suggesting potential for solitary wave emergence in such metamaterials.

ABSTRACT

In the present paper we consider a 2D pantographic structure composed by two orthogonal families of Euler beams. Pantographic rectangular 'long' waveguides are considered in which imposed boundary displacements can induce the onset of traveling (possibly non-linear) waves. We performed numerical simulations concerning a set of dynamically interesting cases. The system undergoes large rotations which may involve geometrical non-linearities, possibly opening the path to appealing phenomena such as propagation of solitary waves. Boundary conditions dramatically influence the transmission of the considered waves at discontinuity surfaces. The theoretical study of this kind of objects looks critical, as the concept of pantographic 2D sheets seems to have promising possible applications in a number of fields, e.g. acoustic filters, vascular prostheses and aeronautic/aerospace panels.

Motivation & Objective

• To analyze wave propagation in 2D pantographic lattices with internal discontinuities using a nonlinear discrete beam model.
• To investigate the influence of boundary conditions and internal hinges on wave transmission and energy localization.
• To explore the emergence of soliton-like waves due to geometric nonlinearity and dispersion in the system.
• To assess the potential of pantographic lattices as tunable acoustic filters or energy-dissipating structures.
• To lay the groundwork for a homogenized theory of pantographic metamaterials with complex internal kinematic constraints.

Proposed method

• A discrete mechanical model based on Euler beam theory with axial and bending strain energy is employed, defined by the energy functional $\mathcal{E} = \int_{\Lambda} \frac{k_M (u'')^2 + k_N (w')^2}{2} \, d\Lambda $.
• The system is modeled as a rectangular lattice of orthogonal beams with spacing $d$, connected via internal hinges allowing free rotation but maintaining beam continuity.
• Boundary conditions include a vertical impulse $\mathfrak{I}(t) = u_0 \cdot \text{sech}[\tau(t - t_0)]$ applied to the upper beam row, with $u_0 = 0.05$ m and $t_0 = 0.005$ s.
• Numerical simulations use material parameters: $\rho = 1450$ kg/m³, $Y = 100$ GPa, $\nu = 0.2$, $k_M = 1.96 \times 10^{-2}$ Nm², $k_N = 7.85 \times 10^4$ N.
• Wave dynamics are analyzed via color maps of cross-sectional rotations and bending moments, particularly at discontinuity surfaces.
• The model incorporates large rotations and geometric nonlinearity, enabling the study of nonlinear wave phenomena such as solitons.

Experimental results

Research questions

• RQ1 How do internal discontinuities, such as hinges or vertical beam connectors, affect wave transmission in 2D pantographic lattices?
• RQ2 Can soliton-like wave behavior emerge in pantographic lattices under large rotations and geometric nonlinearity?
• RQ3 What is the role of coupling stiffness and impulse duration in wave dispersion and localization?
• RQ4 How do boundary conditions influence the shape and velocity of propagating wave fronts in the lattice?
• RQ5 To what extent can the system support localized, shape-preserving wave packets indicative of soliton formation?

Key findings

• Wave propagation in pantographic lattices with internal hinges maintains overall wave character despite discontinuities, indicating robustness in energy transmission.
• When connected via vertical beams, wave energy remains largely confined in the upper half, demonstrating effective damping and potential for filter-like behavior.
• In the case of counter-propagating waves, both wave fronts preserve their shape and speed after interaction, a hallmark of soliton-like behavior.
• The presence of a well-localized, persistent region of maximum perturbation in the wave profile supports the emergence of solitary waves.
• Dispersion effects are observable in simulations, especially when pulse duration is reduced and coupling stiffness is increased.
• The system's ability to sustain large rotations and nonlinear deformations enables the potential for true soliton formation under appropriate nonlinearity.