Elsevier

Wave Motion

Volume 49, Issue 2, March 2012, Pages 394-410
Wave Motion

Study of wave propagation in strongly nonlinear periodic lattices using a harmonic balance approach

https://doi.org/10.1016/j.wavemoti.2011.12.005Get rights and content

Abstract

This paper presents a general harmonic balance method for studying plane wave propagation in strongly nonlinear periodic media. The proposed approach starts by assuming a multi-wavenumber and frequency solution for the unit cell degrees of freedom. A Galerkin projection then transforms the nonlinear differential equations of motion into a set of nonlinear algebraic equations, which are subsequently solved numerically through a Newton-like iteration scheme. These solutions reveal amplitude-dependent dispersion behavior and group velocities. Specific example systems studied include one-dimensional chains and two-dimensional lattices, both formed by a periodic arrangement of spheres interacting under a Hertzian contact law. Amplitude-dependent dispersion is noted in monatomic and diatomic chains, and in hexagonally close-packed two-dimensional lattices. The validity of the presented technique is assessed through direct numerical simulation of the equations governing finite-extent lattices. Strong agreement is documented for results calculated using the harmonic balance approach and the direct numerical simulations.

Highlights

► Harmonic balance approach presented for predicting plane waves in strongly nonlinear media. ► Dispersion behavior predicted for uniform granular media composed of packed spheres. ► Amplitude-dependent dispersion and group velocities documented. ► Predictions of the harmonic balance approach verified using numerical simulations. ► Predicted behavior may inspire tunable filters and stress-redirecting materials.

Introduction

This paper investigates plane wave propagation in strongly nonlinear, periodic systems. Specific attention is paid to chains and lattices composed of packed spheres undergoing Hertzian contact; however, the presented analysis technique based on the harmonic balance method should be applicable to a wide array of strongly nonlinear, periodic media. These systems are of interest since strongly-nonlinear periodic media exhibit wave phenomena not found in their linear counterparts, including solitary waves, solitons, and response localization.

Uniform granular media composed of packed spheres have received renewed attention as candidate shock wave mitigators, for example in explosive test environments [1], [2]. Daraio et al. [3] demonstrated experimentally that one-dimensional granular media composed of viscoelastic PTFE (polytetrafluoroethylene) as well as elastic (stainless-steel) beads exhibit significant wave speed tunability by varying the induced preload. Nesterenko [4] first revealed the existence of solitary waves that propagate without separation of granules under zero pre-compression. Analytical and experimental results [5], [6], [7], [8], [9], [10], [11], [12], [13] document granular chains under zero pre-compression supporting strongly nonlinear waves. Herbold et al. [14] demonstrated tunability of solitary wave speeds and bandgaps by varying pre-compression and wave amplitude. A more detailed analytical study on an uncompressed chain of beads has been reported by Starosvetsky and Vakakis [15]. With energy arguments, the authors analyze one, two, three and four bead finite chains with applied periodic boundary conditions. A two bead cyclic chain is observed to exhibit only standing waves with out-of-phase mode. As the number of granules increases the authors report the existence of a family of traveling waves with the help of nonlinear modes revealed by Poincare’ maps. As N in an N-bead cyclic chain, the solution tends to a solitary wave as reported by Nesterenko [4]. Also, the authors report localized standing waves leading to the concept of “energy trapping”.

While it is evident from the cited literature that nonlinear periodic structures are capable of supporting multiple solitary and solution wave solutions, very few studies have treated the plane wave case, and those known to the authors have used numerical simulations in their study. Nesterenko and Herbold [16] report the generation of quasi-periodic strain waves in compressed granular chains with two different amplitudes as the harmonic forcing amplitude reaches an initial pre-compression. Numerical simulations impose a single harmonic force excitation at the boundary which leads to the modulation of excitation frequency as the force is transmitted through the chain. Such modulation depends on the ratio of excitation frequency and the characteristic cut-off frequency which depends on the stiffness of contacting spheres and initial static pre-compressive force. Based on the force–time envelope modulation observed in single harmonic excitation, the authors then construct a special composite force function which is essentially a summation of different harmonic force functions. Such a composite force applied at the boundary generates a force–time envelope in the chain that does not appear to change its shape, thus indicating a strongly nonlinear periodic plane wave.

The present paper focuses on analyzing the influence of nonlinearity and wave amplitude on the dispersion properties of plane waves in nonlinear periodic materials, particularly in uniform granular media. Analysis of plane wave propagation in strongly nonlinear media is carried out using a harmonic balance approach and is detailed in Section 2. Section 3 then employs the technique to predict dispersion in one-dimensional Hertzian chains, while Section 4 extends the analysis to two-dimensional granular media. Numerical integration of the equations of motion is also detailed in Sections 3 Monatomic granular chain, 4 Diatomic granular chain, together with a discussion on methods for numerically injecting a plane wave. A desired plane wave can be injected into the structure through point harmonic forcing, or by directly imposing a specific wavelength (wavenumber) into the structure. The latter is analogous to how a plane wave is injected into a piezoelectric substrate by an inter-digital-transducer (IDT) on the surface of an acoustic wave device, and is the preferred method used herein for validating the harmonic balance results.

Section snippets

Geometry description

Consider a two-dimensional (2D) lattice consisting of a periodic arrangement of identical unit cells assembled to cover a portion of the x1,x2 plane — see Fig. 1. Each unit cell is considered to be composed of lumped masses connected to one another by generally nonlinear springs. The periodicity of the lattice is defined by the primitive lattice vectors a1 and a2 which form the basis for the periodic lattice. The position of the ith mass of the unit cell at location n1,n2 can be expressed as

Model description

A monatomic chain consists of a unit cell with a single degree of freedom. A schematic of the monatomic chain formed by spherical beads is shown in Fig. 2. The chain is initially pre-compressed with a static force F0 acting on both ends which compresses the chain by δ0 (compression in each sphere). For the present analysis, the monatomic chain is modeled with the following assumptions:

  • (a)

    The contact force is modeled by a Hertzian contact law, which implies that the strain in each bead does not

Model description and equations of motion

The unit cell for the diatomic chain consists of two different masses, m1 and m2, as shown in Fig. 7. The mass matrix and the nonlinear restoring force array are given as, M=[m100m2],fNL=[Γ(ξj+)3/2Γ(η+)3/2Γ(η+)3/2Γ(ξj+1+)3/2], where ξj=δ0(u1ju2j1), η=δ0(u1ju2j) and the following notation is adopted: ξj+={ξjξj00ξj<0,η+={ηη=00η<0,Γ=4E1E2(1/R2+1/R1)1/23[E2(1ν12)+E1(1ν22)]. In the equations above, E1 and E2, and ν1 and ν2, denote the elastic moduli and Poisson’s ratio for spherical

Two-dimensional hexagonal granular packing

The next example considers a two-dimensional material where spheres are arranged in a hexagonal packing to form a lattice structure. The motivation to study two-dimensional, strongly nonlinear lattices comes in part from the fact that such highly nonlinear systems can exhibit a high degree of variation in directional behavior based on wave amplitude [24]. The present study is restricted to acoustic plane wave propagation with initial pre-compression and aims to explore this amplitude-dependent

Concluding remarks

A generalized harmonic balance method has been presented for predicting plane wave dispersion in strongly nonlinear periodic lattices. Numerical simulations verify the predicted dispersion trends in one- and two-dimensional, uniform granular media. For one-dimensional chains composed of spheres in Hertzian contact, the harmonic balance solutions predict downward shifts in the dispersion relationships as wave amplitude is increased, indicative of a softening system. This effect leads to

References (27)

  • C. Coste

    Sound propagation in a constrained lattice of beads: high-frequency behavior and dispersion relation

    Phys. Rev. E

    (2008)
  • C. Coste et al.

    Solitary waves in a chain of beads under Hertz contact

    Phys. Rev. E

    (1997)
  • C. Daraio et al.

    Strongly nonlinear waves in a chain of Teflon beads

    Phys. Rev. E

    (2005)
  • Cited by (83)

    View all citing articles on Scopus
    View full text