Dynamic Homogenization of a Chain with Bistable Springs. Statistical Approach

Blake, Charlotte; Cherkaev, Andrej

doi:10.1007/978-3-030-38708-2_5

Cited by 1 publication

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“…The advantage of the piecewise linear choice for the stressstrain relation is the possibility to construct the corresponding traveling wave solutions of the FPU problem explicitly using the Wiener-Hopf transform technique [80]. While smoothening the constitutive response around the singular point ε c could make the model more realistic, sometimes even without sacrificing much of analytical transparency [89][90][91], stronger nonlinearity is needed to capture such important physical effects as thermalization of the radiated phonons [92][93][94]. However, such a generalization of the model, which will make its analytical treatment almost impossible without contributing much to the classification of the transition fronts, is outside the scope of this paper.…”

Section: Introductionmentioning

confidence: 99%

Transition fronts and their universality classes

2022

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Steadily moving transition (switching) fronts, associated with local transformation, symmetry breaking, or collapse, are among the most important dynamic coherent structures. The nonlinear mechanical waves of this type play a major role in many modern applications involving the transmission of mechanical information in systems ranging from crystal lattices and metamaterials to macroscopic civil engineering structures. While many different classes of such dynamic fronts are known, the interrelation between them remains obscure. Here we consider a minimal prototypical mechanical system, the Fermi-Pasta-Ulam (FPU) chain with piecewise linear nonlinearity, and show that there are exactly three distinct classes of switching fronts, which differ fundamentally in how (and whether) they produce and transport oscillations. The fact that all three types of fronts could be obtained as explicit Wiener-Hopf solutions of the same discrete FPU problem allows one to identify the exact mathematical origin of the particular features of each class. To make the underlying Hamiltonian dynamics analytically transparent, we construct a minimal quasicontinuum approximation of the FPU model that captures all three classes of the fronts and reveals interrelation between them. This approximation is of higher order than conventional ones (KdV, Boussinesq) and involves mixed space-time derivatives. The proposed framework unifies previous attempts to classify the mechanical transition fronts as radiative, dispersive, topological, or compressive and categorizes them instead as irreducible types of dynamic lattice defects.

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