Transition waves in bistable structures. II. Analytical solution: wave speed and energy dissipation

Slepyan, Leonid I.; Cherkaev, Andrej; Cherkaev, Elena

doi:10.1016/j.jmps.2004.08.001

Cited by 81 publications

(70 citation statements)

References 21 publications

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“…As in [21][22][23], we allow the elastic moduli of the two phases to be different (κ I = κ II ) (see Fig. 3).…”

Section: The Discrete Modelmentioning

confidence: 99%

“…When the elastic moduli are different, γ = κ II /κ I = 1, the problem becomes technically more difficult and requires the use of Wiener-Hopf factorization technique. In the absence of long-range interactions it was studied in [21][22][23]. The case γ ≤ 1, which allows for only subsonic phase boundaries, was considered, in [21,23], and nontrivial shock solutions, which occur when γ > 1, were the focus of [22].…”

Section: Introductionmentioning

confidence: 99%

“…In the absence of long-range interactions it was studied in [21][22][23]. The case γ ≤ 1, which allows for only subsonic phase boundaries, was considered, in [21,23], and nontrivial shock solutions, which occur when γ > 1, were the focus of [22]. However, some important details about existence, structure, and stability of the traveling wave solutions and the fundamental difference between shock and kink solutions remained unclear.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Shocks versus kinks in a discrete model of displacive phase transitions

Trofimov

Vainchtein

2010

Continuum Mech. Thermodyn.

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We consider dynamics of phase boundaries in a bistable one-dimensional lattice with harmonic long-range interactions. Using Fourier transform and Wiener-Hopf technique, we construct traveling wave solutions that represent both subsonic phase boundaries (kinks) and intersonic ones (shocks). We derive the kinetic relation for kinks that provides a needed closure for the continuum theory. We show that the different structure of the roots of the dispersion relation in the case of shocks introduces an additional free parameter in these solutions, which thus do not require a kinetic relation on the macroscopic level. The case of ferromagnetic second-neighbor interactions is analyzed in detail. We show that the model parameters have a significant effect on the existence, structure, and stability of the traveling waves, as well as their behavior near the sonic limit.

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“…As in [21][22][23], we allow the elastic moduli of the two phases to be different (κ I = κ II ) (see Fig. 3).…”

Section: The Discrete Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Shocks versus kinks in a discrete model of displacive phase transitions

Trofimov

Vainchtein

2010

Continuum Mech. Thermodyn.

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“…Moreover, the dynamics of atomic-scale configurational changes as well as the mechanisms of damage and fracture have been analyzed by recourse to chains of elements with a nonmonotonic force-displacement (or stress-strain) response; see, e.g., Refs. [64][65][66][67][68][69][70][71].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of periodic mechanical structures containing bistable elastic elements: From elastic to solitary wave propagation

2014

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We investigate the nonlinear dynamics of a periodic chain of bistable elements consisting of masses connected by elastic springs whose constraint arrangement gives rise to a large-deformation snap-through instability. We show that the resulting negative-stiffness effect produces three different regimes of (linear and nonlinear) wave propagation in the periodic medium, depending on the wave amplitude. At small amplitudes, linear elastic waves experience dispersion that is controllable by the geometry and by the level of precompression. At moderate to large amplitudes, solitary waves arise in the weakly and strongly nonlinear regime. For each case, we present closed-form analytical solutions and we confirm our theoretical findings by specific numerical examples. The precompression reveals a class of wave propagation for a partially positive and negative potential. The presented results highlight opportunities in the design of mechanical metamaterials based on negative-stiffness elements, which go beyond current concepts primarily based on linear elastic wave propagation. Our findings shed light on the rich effective dynamics achievable by nonlinear small-scale instabilities in solids and structures.

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“…In these papers, chains made of main links and waiting links are defined as bistable chains. Further research and numerical simulations by Cherkaev, Slepyan and their collaborators showed that bistable chains are able to sustain a higher dynamic loading (in particular, impact and explosion) without breaking, compared to conventional chain structures, since they spread damage across a larger part of the volume [Cherkaev and Zhornitskaya 2004;Slepyan and Ayzenberg 1997;.…”

Section: Introductionmentioning

confidence: 99%

Untitled

2007

JOMMS

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See inside back cover or http://www.jomms.org for submission guidelines.Regular subscription rate: $500 a year. Stiffness based sorting of mixed paper waste at high speed is desirable for sorting board, cardstock, and paper from each other for efficient recycling. We propose a quick, non-contact method for measuring the relative stiffness of paper samples moving on a conveyor at high speeds. The method consists of impinging an air jet on a moving paper sample while it is crossing a gap during transfer from one conveyor belt to another. The deflection caused by the impinging air jet can be measured and used to categorize different grades of paper. In this paper, we report results from a simulation of the proposed method using a finite element model. Influence of conveyor speed, sample orientation and nozzle pressure is studied using the model. Results indicate that the stiffness estimation has sufficient resolution to distinguish between board and paper grades at high conveyor speeds.

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Transition waves in bistable structures. II. Analytical solution: wave speed and energy dissipation

Cited by 81 publications

References 21 publications

Shocks versus kinks in a discrete model of displacive phase transitions

Shocks versus kinks in a discrete model of displacive phase transitions

Dynamics of periodic mechanical structures containing bistable elastic elements: From elastic to solitary wave propagation

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