Shock Waves in Nonlinear Granular Chain with Viscosity under Hertz Contact

Liang, Tao; Zhu, Xiaolu; Meng, Zhaocang; Duan, Wen-Shan; Yang, Lei

doi:10.7566/jpsj.88.034002

Cited by 6 publications

(8 citation statements)

References 45 publications

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“…This feature of the generalized KK model contrasts with the properties of generalized Hertzian chains with linear dashpots (α > 1, β = 1) for which γ c vanishes with δ [19]. For example, in the numerical simulations performed in [36] for α = 3/2, the underdamped regime occurs when v p is large enough.…”

Section: Numerical Computation Of Traveling Frontsmentioning

confidence: 86%

“…The front profile is underdamped or overdamped depending whether the dissipation constant γ 0 lies below or above a critical value proportional to v p (1−α)/(1+α) , where v p denotes the asymptotic particle velocity after the shock. Consequently, for a given granular chain with KK contact law, the underdamped regime occurs when the downstream velocity v p is small, unlike for β = 1 and α > 1 [19,36].…”

Section: Introductionmentioning

confidence: 99%

“…They can be typically generated by compressing the chain with a piston at constant velocity [41]. A correct modeling of this phenomenon requires to take contact damping into account [20,38]. Depending on the amount of dissipation induced by the material, oscillatory (underdamped) or monotonic (overdamped) shock profiles can be observed experimentally [42] (section 1.8.7 p 80).…”

Section: Introductionmentioning

confidence: 99%

“…In the works [20,38], compression fronts are studied in the case β = 1 (Hertzian spring with linear dashpot). With a precompression applied to the chain, classical multiscale expansions are performed to capture the leading order nonlinear, dispersive and dissipative features of small amplitude long waves.…”

Section: Introductionmentioning

confidence: 99%

“…With a precompression applied to the chain, classical multiscale expansions are performed to capture the leading order nonlinear, dispersive and dissipative features of small amplitude long waves. When a small contact damping is considered, the KdV-Burgers equation [35] is used to approximate traveling fronts and explain the transition from overdamped to underdamped profiles [20,38] (see also [4,14,24] for an application of this technique to other interactions between particles). In particular, the underdamped regime is observed when the impact velocity is large enough [38].…”

Section: Introductionmentioning

confidence: 99%

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Traveling fronts in dissipative granular chains and nonlinear lattices

James

2021

Nonlinearity

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We consider an infinite chain of particles with nonlinear elastic and dissipative nearest neighbors interactions. Assuming the existence of a traveling front between uniformly compressed (or stretched) states, we obtain jump conditions relating the wave speed and limiting particle velocities to the relative displacements at infinity. Using this result, we characterize compression fronts in chains of touching beads, for viscoelastic contact laws that include a nonlinear elastic force (generalized Hertz contact) and viscous dissipation. We compute compression fronts numerically for the generalized Kuwabara-Kono model in which the viscous contact force is proportional to the derivative of the elastic force, without precompression of the chain. Steady fronts are obtained both as the end result of the compression of one end of the chain and using a shooting method which provides numerically exact traveling waves. Depending on the magnitudes of contact damping and strain applied downstream, we obtain either overdamped (monotonic) or underdamped (oscillatory) compression fronts. To explain this transition and approximate the front profiles, we consider a continuum limit valid when the exponent of the contact nonlinearity is close to (and above) unity. Using multiscale expansions, we formally derive two different amplitude equations for long waves, a Burgers equation with logarithmic nonlinearity, and a logarithmic Korteweg-de Vries (KdV)-Burgers equation for small contact damping. Both models possess traveling front solutions that are in good agreement with the front profiles computed numerically in the granular chain. The analysis of the logarithmic KdV-Burgers equation allows one to approximate the critical damping corresponding to the transition from underdamped to overdamped fronts.

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Section: Numerical Computation Of Traveling Frontsmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%