Nonlinear Bloch waves and balance between hardening and softening dispersion

Abstract

The introduction of nonlinearity alters the dispersion of elastic waves in solid media. In this paper, we present an analytical formulation for the treatment of finite-strain Bloch waves in one-dimensional phononic crystals consisting of layers with alternating material properties. Considering longitudinal waves and ignoring lateral effects, the exact nonlinear dispersion relation in each homogeneous layer is first obtained and subsequently used within the transfer matrix method to derive an approximate nonlinear dispersion relation for the overall periodic medium. The result is an amplitude-dependent elastic band structure that upon verification by numerical simulations is accurate for up to an amplitude-to-unit-cell length ratio of one-eighth. The derived dispersion relation allows us to interpret the formation of spatial invariance in the wave profile as a balance between hardening and softening effects in the dispersion that emerge due to the nonlinearity and the periodicity, respectively. For example, for a wave amplitude of the order of one-eighth of the unit-cell size in a demonstrative structure, the two effects are practically in balance for wavelengths as small as roughly three times the unit-cell size.

Oscillating vector solitary waves in soft laminates

“…We recall that for solitary waves to propagate there should be a balance between dispersion and nonlinearity. In our case of a finitely deformed laminate, this balance is achievable by pre-strain tuning: solitary waves can propagate only at specific initial deformations for which the material stiffens along the loading path (Hussein and Khajehtourian, 2018). The quantification of this stiffening is given by the gradient of the characteristic velocity along this path.…”

“…In other words, impacts that give rise to monotonically increasing phase velocities and in turn shocks in a homogeneous Gent material, also give rise to monotonically increasing phase velocities and in turn may form solitary waves in laminates made of the same Gent material whose mass density is modulated. This connection sheds light on the balance between the nonlinearity in the system and its dispersion: the constitutive nonlinearity causes the pulse to steepen, while the scattering at the interfaces disperses it and generates solitary waves (Hussein and Khajehtourian, 2018), otherwise the steepening would have result in shock formation (LeVeque and Yong, 2003). 4 Summary Ziv and Shmuel (2020) have developed a finite-volume method to solve the equations governing finite-amplitude smooth waves with two coupled components in nonlinear compressible laminates.…”

“…To the best of our knowledge, the first numerical observation of one-dimensional solitary waves in elastic periodic continua was by LeVeque (2002b) and Bale et al (2002), using a finite-volume method they developed, which was followed by the studies of LeVeque and Yong (2003), Andrianov et al (2013Andrianov et al ( , 2014, Navarro et al (2015), Hussein and Khajehtourian (2018) and the references therein. As explained in these works, the periodicity of the media causes dispersion, whose balance with geometrical and constitutive nonlinearities forms elastic solitary waves.…”

Soft Adaptive Mechanical Metamaterials

“…Similarly, for substrate-free dissipative bistable structures (with dissipation stemming from, e.g., the intrinsic losses of polymer base materials), transition fronts can act as a topological soliton ( Jin et al, 2020 ), which gradually transforms the structure from unswitched to switched upon propagation. In conventional solitons, an equilibrium between nonlinearity and dispersion leads to a stable wave ( Hussein and Khajehtourian, 2018 ; Khajehtourian and Hussein, 2019 ). By contrast, the topological solitons in dissipative ungrounded multistable metamaterials emerge from the balance between dissipation and kinetic energy, on the one hand, and the energy released when switching unit cells from high-energy (open) stable state to their low-energy (closed) one, on the other hand.…”

Observation of vector solitary waves in soft laminates using a finite volume method

“…This is in contrast with the standard approach that decomposes the conserved vector. These works revealed solitary waves, namely, nonlinear waves that are able to maintain constant speed and profile by virtue of a balance between nonlinear and dispersive features of the system (Dauxois andPeyrard, 2006, Hussein andKhajehtourian, 2018). These one-dimensional elastic solitary waves were later studied in greater detail by LeVeque and Yong (2003a,b); see also the works of Engelbrecht et al (2007), Xu et al (2007).…”